diff --git "a/bench_data/arxiv_math.jsonl" "b/bench_data/arxiv_math.jsonl" --- "a/bench_data/arxiv_math.jsonl" +++ "b/bench_data/arxiv_math.jsonl" @@ -1,3015 +1,3015 @@ -{"pdf": "arxiv_math/2503.04048_pg46.pdf", "page": 1, "id": "2503.04048_pg46_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathcal{V}}(\\psi_m)\\rightarrow +\\infty"} -{"pdf": "arxiv_math/2503.07228_pg12.pdf", "page": 1, "id": "2503.07228_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq k \\leq 2^{N}-1"} -{"pdf": "arxiv_math/2503.04993_pg18.pdf", "page": 1, "id": "2503.04993_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{d}{d\\epsilon} J_1(u_1 + \\epsilon v_1, u_2) \\Big|_{\\epsilon=0}."} -{"pdf": "arxiv_math/2503.04993_pg18.pdf", "page": 1, "id": "2503.04993_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{d}{d\\epsilon} J_1(u_1 + \\epsilon v_1, u_2) \\Big|_{\\epsilon=0}=I_1 + I_2,"} -{"pdf": 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"page": 1, "id": "2503.09588_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{ab}\\colon X\\to S_{ab}"} -{"pdf": "arxiv_math/2503.09588_pg25.pdf", "page": 1, "id": "2503.09588_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "c'\\circ\\alpha = \\overline{\\alpha} \\circ c"} -{"pdf": "arxiv_math/2503.09588_pg25.pdf", "page": 1, "id": "2503.09588_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{cd}\\colon X\\to S_{cd}"} -{"pdf": "arxiv_math/2503.04448_pg10.pdf", "page": 1, "id": "2503.04448_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "d(X_1,X_2) + d(X_2,X_1) = 1"} -{"pdf": "arxiv_math/2503.04448_pg10.pdf", "page": 1, "id": "2503.04448_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_i/(\\sum_j \\lambda_j)"} -{"pdf": "arxiv_math/2503.05358_pg8.pdf", "page": 1, "id": "2503.05358_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_{1,{\\max}} = \\sqrt{C3} + v^E = \\sqrt{C3} + \\sqrt {\\frac{\\mu^S}{r^E}},"} -{"pdf": "arxiv_math/2503.05358_pg8.pdf", "page": 1, "id": "2503.05358_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cos(\\theta_{12}) = \\frac{1}{e} \\left( \\frac{p}{r^F} - 1 \\right),"} -{"pdf": "arxiv_math/2503.05358_pg8.pdf", "page": 1, "id": "2503.05358_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "l^F_2 = \\Omega^F + \\omega^F + n^F (t_2 - t^F_p),"} -{"pdf": "arxiv_math/2503.05358_pg8.pdf", "page": 1, "id": "2503.05358_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\max} = \\frac{v^2_{1,{\\max}}}{2} - \\frac{\\mu^S}{r^E} \\to a_{\\max} = \\frac{-\\mu^S}{2E_{\\max}} \\to e_{max} = 1 - \\frac{r^E}{a_{\\max}},"} -{"pdf": "arxiv_math/2503.05358_pg8.pdf", "page": 1, "id": "2503.05358_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{\\min} = \\frac{r^E + r^F}{2} \\to e_{\\min} = 1 - \\frac{r^E}{a_{\\min}}."} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})\\to L\\in\\left[0,\\sqrt{\\mu_{s,t}}\\right]"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})<\\sqrt{\\mu_{s,t}}=\\|u_k\\|_{\\dot{H}^s}"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "L=0<\\sqrt{\\mu_{s,t}}\\,"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{s,t}^{\\frac{1}{2_s^*(t)-2}}u"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})\\leq d(u_k,0)=\\|u_k\\|_{\\dot{H}^s}=\\sqrt{\\mu_{s,t}}<\\infty"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{s,t}(u_n)\\to \\beta"} -{"pdf": "arxiv_math/2503.06716_pg12.pdf", "page": 1, "id": "2503.06716_pg12_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u_k\\|^2_{\\dot{H}^s}=\\mu_{s,t}"} -{"pdf": "arxiv_math/2503.08597_pg10.pdf", "page": 1, "id": "2503.08597_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\Sigma} \\triangleq \\max_{(d_1,\\cdots, d_K)\\in \\mathcal{D}}(d_1+\\cdots +d_K)"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf T}=\\{e^{t T}, t \\in \\R \\}"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi \\in \\mathfrak{k}"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\R H \\oplus \\R L"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p(\\R_{\\bf T}) = \\overline{p(\\Gamma)}"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Psi \\in \\mathfrak{k}"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lfloor \\frac{n}{2} \\rfloor"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf T}=\\lambda+\\Phi+\\omega"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R_{\\bf H} \\times \\R_{\\bf L}"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R"} -{"pdf": "arxiv_math/2503.08614_pg8.pdf", "page": 1, "id": "2503.08614_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf H} \\times \\R_{\\bf L} \\simeq \\R^2"} -{"pdf": "arxiv_math/2503.05558_pg18.pdf", "page": 1, "id": "2503.05558_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "E = 41.9 log_2(B)^{-1}-1.92"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde \\Omega, \\tilde{\\mathscr{F}}, \\tilde{\\mathbb{P}})"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma: [0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^{d \\times \\tilde{d}}"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "b:[0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^d"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r:=\\{r(t)\\}_{t\\geq 0}"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^r := \\{\\tilde{\\mathscr F}_t^r\\}_{t\\geq 0}"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^{B}:=\\{\\tilde{\\mathscr F}_t^{B}\\}_{t\\geq 0}"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "B:=\\{B(t)\\}_{t\\geq 0}"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{q}_{j_0 k_0}\\geq 0"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_0^B"} -{"pdf": "arxiv_math/2503.06658_pg1.pdf", "page": 1, "id": "2503.06658_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_t:=\\tilde{\\mathscr F}_t^{B} \\vee \\tilde{\\mathscr F}_t^r"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f_\\pm(x_1,x_2)=\\pm \\Delta(x_1,x_2)(x_1,x_2,1),"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "m(u,v)=(u^2+v^2+1)^{-\\frac{1}{2}(n-1)}"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta(x_1,x_2)=(x_1^2+x_2^2+1)^{-\\frac{1}{2}}"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Fix}(Q)\\subset\\mathbb{R}^m"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_i:U_i\\rightarrow\\mathbb{R}^2"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Q(x_1,\\dots,x_m)=(x_1,\\dots,x_r,-x_{r+1},\\dots,-x_m)."} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_+=\\{y\\in\\mathbb{S}^2:y_3>0\\}"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} &v^n\\;m(u,v)\\left(Q\\left(\\frac{1}{v},\\frac{u}{v}\\right)-uP\\left(\\frac{1}{v},\\frac{u}{v}\\right),-vP\\left(\\frac{1}{v},\\frac{u}{v}\\right)\\right) \\text{ in } U_1, \\\\ &v^n\\;m(u,v)\\left(P\\left(\\frac{u}{v},\\frac{1}{v}\\right)-uQ\\left(\\frac{u}{v},\\frac{1}{v}\\right),-vQ\\left(\\frac{u}{v},\\frac{1}{v}\\right)\\right) \\text{ in } U_2, \\\\ &m(u,v)(P(u,v),Q(u,v)) \\text{ in } U_3, \\end{aligned}"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{S}^2=\\{y=(y_1,y_2,y_3)\\in\\mathbb{R}^3:y_1^2+y_2^2+y_3^2=1\\}"} -{"pdf": "arxiv_math/2503.05436_pg25.pdf", "page": 1, "id": "2503.05436_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_-=\\{y\\in\\mathbb{S}^2:y_3<0\\}"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\left (\\bigcup_{x\\in \\partial M} i |P_n\\left [ -\\sqrt{\\max(0,N^2)},\\sqrt{\\max(0,N^2)}\\right ]\\right )^c"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 w_v) = 0."} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 z_v) + \\frac{g_0'}{c^2} \\cdot \\nabla \\times (\\rho_0 w_v) + \\frac{\\rho_0 g_0'}{c^2} \\cdot z_v = 0 , \\quad n \\cdot z_v|_{\\partial M} = 0."} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\times (\\rho_0 z_v) = 0."} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "w \\in \\operatorname{Ker}(T)"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\sigma_{ess}(L)^c"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w = \\nabla \\times (\\rho_0 w_v) + \\nabla \\varphi_v"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_0 z_v = \\nabla \\varphi_v"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\operatorname{Ker}(T)"} -{"pdf": "arxiv_math/2503.05428_pg17.pdf", "page": 1, "id": "2503.05428_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "u = \\nabla \\times (\\rho_0 w_u) + \\rho_0 z_u"} -{"pdf": "arxiv_math/2503.04438_pg16.pdf", "page": 1, "id": "2503.04438_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "O^\\sigma = L(O_{h\\Sigma_n})\\times_{LB\\Sigma_n}\\{\\sigma\\}"} -{"pdf": "arxiv_math/2503.07105_pg18.pdf", "page": 1, "id": "2503.07105_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\set{1,\\ldots,n}"} -{"pdf": "arxiv_math/2503.07105_pg18.pdf", "page": 1, "id": "2503.07105_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "w_{k+1} := w_k + \\tau_k r(w_k)"} -{"pdf": "arxiv_math/2503.07105_pg18.pdf", "page": 1, "id": "2503.07105_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "2 (w_l - w_r) \\frac{q_{1,1}^e q_{2,2}^e - q_{2,1}^e q_{1,2}^e}{2} ."} -{"pdf": "arxiv_math/2503.07105_pg18.pdf", "page": 1, "id": "2503.07105_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(w_{k+1} - w_k)/\\tau_k = r(w_k)"} -{"pdf": "arxiv_math/2503.03754_pg10.pdf", "page": 1, "id": "2503.03754_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{XY}=\\begin{pmatrix} 1-s & 0 \\\\ sd & s(1-d) \\end{pmatrix}"} -{"pdf": "arxiv_math/2503.03754_pg10.pdf", "page": 1, "id": "2503.03754_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_2=\\frac{m-svt}{1-st}"} -{"pdf": "arxiv_math/2503.03754_pg10.pdf", "page": 1, "id": "2503.03754_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\frac{m-x_2}{t(x_1-x_2)}."} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{2}\\delta x_{s}^{1}(\\xi_i)}"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l}\\delta x_{s}^{l-1}(\\xi_i)}"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{Z}^{W\\delta x}"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{det}(C) \\neq 0"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathring{\\chi}_{t, i}"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{L\\top}dh_{s}^{L}(\\xi_i)}"} -{"pdf": "arxiv_math/2503.09565_pg19.pdf", "page": 1, "id": "2503.09565_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l\\top}dh_{s}^{l}(\\xi_i)}"} -{"pdf": 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"|\\sigma(\\mathbf{u})^{-1}(P_N\\mathbf{w})|\\leq C_0|P_N\\mathbf{w}|,\\text{ for all }\\mathbf{u},\\mathbf{w}\\in H."} -{"pdf": "arxiv_math/2503.09285_pg9.pdf", "page": 1, "id": "2503.09285_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V := \\{\\mathbf{u}\\in H^1(D)^2 :\\nabla\\cdot\\mathbf{u}=0,\\; \\mathbf{u}|_{\\partial D}=0\\}."} -{"pdf": "arxiv_math/2503.09285_pg9.pdf", "page": 1, "id": "2503.09285_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "P_NH\\subset \\text{Range}\\;(\\sigma(\\mathbf{u})),\\text{ for all }\\mathbf{u}\\in H."} -{"pdf": "arxiv_math/2503.09285_pg9.pdf", "page": 1, "id": "2503.09285_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_k^{-1}:P_NH\\rightarrow H"} -{"pdf": "arxiv_math/2503.09285_pg9.pdf", "page": 1, "id": "2503.09285_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma=(\\sigma_1,\\dots,\\sigma_m)"} -{"pdf": 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"\\theta\\in[-\\frac{\\pi}{2},\\frac{\\pi}{2}]"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\hat{\\tau},\\hat{v},\\theta)"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q} \\in \\mathbb{K}"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{C}^{n\\times m}"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}_{r}=[x_{r,0},x_{r,1},\\cdots,x_{r,M_r-1}]^T"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{a}(\\mathbf{x}_t,\\alpha)=[1,e^{j\\frac{2\\pi}{\\lambda}x_{t,1}\\sin\\alpha},\\cdots,e^{j\\frac{2\\pi}{\\lambda}x_{t,M_t-1}\\sin\\alpha}]^T"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{r,m}=\\frac{\\lambda}{2}m,m\\in\\mathcal{M}_r"} -{"pdf": "arxiv_math/2503.04407_pg3.pdf", "page": 1, "id": "2503.04407_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q}\\neq c_{m',q}, \\forall q, m\\neq m'"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_0 \\in C^0(\\overline\\Omega)"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_h^0 = \\pi^h \\varphi_0"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta_o = |\\max_o |\\phi_h^n| - 1 | > 0.5"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "h_c = \\frac{L_d}{N_c}"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_f = 8N_c = 2^{3+k} L_d"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\prod_{i=1}^d (0,L_i) \\subset \\mathbb R^d"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "S^h = \\{ \\zeta \\in C^0(\\overline{\\Omega}) \\, : \\, \\zeta \\vert_{o} \\in P_1(o) \\quad \\forall o \\in \\mathcal{T}_h\\},"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\phi_h^{n+1}, \\mu_h^{n+1}) \\in S^h \\times S^h"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "h_f = \\frac{L_d}{N_f}"} -{"pdf": "arxiv_math/2503.09581_pg21.pdf", "page": 1, "id": "2503.09581_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_0 : \\overline\\Omega \\to \\mathbb R"} -{"pdf": "arxiv_math/2503.05742_pg2.pdf", "page": 1, "id": "2503.05742_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."} -{"pdf": "arxiv_math/2503.05742_pg2.pdf", "page": 1, "id": "2503.05742_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ", this means that there is no prior knowledge about the ideal control. Furthermore, the coefficient"} -{"pdf": "arxiv_math/2503.05742_pg2.pdf", "page": 1, "id": "2503.05742_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."} -{"pdf": "arxiv_math/2503.05742_pg2.pdf", "page": 1, "id": "2503.05742_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": ". The desired control function"} -{"pdf": "arxiv_math/2503.07895_pg2.pdf", "page": 1, "id": "2503.07895_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\dfrac{\\partial}{\\partial z}"} -{"pdf": "arxiv_math/2503.07895_pg2.pdf", "page": 1, "id": "2503.07895_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=g(z)\\dfrac{\\partial}{\\partial z}"} -{"pdf": "arxiv_math/2503.07895_pg2.pdf", "page": 1, "id": "2503.07895_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f(z)=1/\\left(1-e^z\\right)"} -{"pdf": "arxiv_math/2503.09424_pg5.pdf", "page": 1, "id": "2503.09424_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{2,\\ldots,n-1\\}"} -{"pdf": "arxiv_math/2503.09432_pg6.pdf", "page": 1, "id": "2503.09432_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k+1}(f)^2\\leq \\lambda _k(f)\\lambda _{k+1}(f)"} -{"pdf": "arxiv_math/2503.09432_pg6.pdf", 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null, "math": "\\begin{aligned} A_{a,b,c} = 2(a+b+c) + 3 \\\\ C_{a,b|c,d} = 3 (2a+1)(2b+1) \\\\ G_{a} = (2a+1)(2a^2 + 2a + 1) \\end{aligned}"} -{"pdf": "arxiv_math/2503.07583_pg22.pdf", "page": 1, "id": "2503.07583_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_{[8,7]} = 8^3 + 7^3 = 855 = 9^3 + 5^3 + 1^3 = \\varepsilon_{[9,5,1]}"} -{"pdf": "arxiv_math/2503.05183_pg15.pdf", "page": 1, "id": "2503.05183_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_4\\|\\Z\\|_{F,1}^\\psi"} -{"pdf": "arxiv_math/2503.05183_pg15.pdf", "page": 1, "id": "2503.05183_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "r^0 = \\min\\{n_1, n_2\\}"} -{"pdf": "arxiv_math/2503.05183_pg15.pdf", "page": 1, "id": "2503.05183_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z)=\\left\\lbrace j \\mid\\|\\Z(:,j,:)\\| \\neq 0, \\, j=1, \\ldots, r\\right\\rbrace"} -{"pdf": 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\\zeta\\in \\partial\\widetilde{W}."} -{"pdf": "arxiv_math/2503.07128_pg28.pdf", "page": 1, "id": "2503.07128_pg28_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R} (t_n) \\widetilde{W}"} -{"pdf": "arxiv_math/2503.07128_pg28.pdf", "page": 1, "id": "2503.07128_pg28_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(t,x)\\in[0,+\\infty)\\times\\R^n"} -{"pdf": "arxiv_math/2503.07128_pg28.pdf", "page": 1, "id": "2503.07128_pg28_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_n := x_n - h_n \\in [0,1)^N"} -{"pdf": "arxiv_math/2503.07128_pg28.pdf", "page": 1, "id": "2503.07128_pg28_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "(t_n+t) \\widetilde{W}"} -{"pdf": "arxiv_math/2503.07128_pg28.pdf", "page": 1, "id": "2503.07128_pg28_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\in\\partial\\big(t_n\\widetilde{W}-\\{x_n\\}\\big)"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", 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"arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J}=\\left\\{\\mathcal{J}_h: \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}\\rightarrow \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}\\right\\}_{h=1}^n"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "F:D\\subset\\mathbb{C}^n\\to \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in D\\iff\\overline{z}^h\\in D"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F:D\\to \\mathbb{R}_m\\otimes\\mathbb{R}^{2^n}"} -{"pdf": "arxiv_math/2503.04329_pg7.pdf", "page": 1, "id": "2503.04329_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_{J_1}\\times...\\times\\phi_{J_n}:\\mathbb{C}^n\\ni(z_1,...,z_n)\\mapsto\\left(\\phi_{J_1}(z_1),...,\\phi_{J_n}(z_n)\\right)\\in(\\mathbb{R}^{m+1})^n,"} -{"pdf": "arxiv_math/2503.08360_pg10.pdf", "page": 1, "id": "2503.08360_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}_\\ell(\\mathcal{T}_h, \\mathbb{R}^{m\\times n})"} -{"pdf": "arxiv_math/2503.08360_pg10.pdf", "page": 1, "id": "2503.08360_pg10_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}_\\ell(\\mathcal{F}_h, \\mathbb{R}^{m\\times n})"} -{"pdf": "arxiv_math/2503.04623_pg54.pdf", "page": 1, "id": "2503.04623_pg54_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(0, \\mu^{\\sharp\\prime}, \\ldots, \\mu^{\\sharp\\prime})"} -{"pdf": "arxiv_math/2503.04623_pg54.pdf", "page": 1, "id": "2503.04623_pg54_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu\\in X_\\bullet(G^*)"} -{"pdf": "arxiv_math/2503.04623_pg54.pdf", "page": 1, "id": "2503.04623_pg54_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu^\\sharp\\in X_\\bullet(G^\\sharp)"} -{"pdf": "arxiv_math/2503.04623_pg54.pdf", "page": 1, "id": "2503.04623_pg54_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_1^\\sharp, \\ldots, \\omega_{n-2}^\\sharp, \\omega_{n-1}^\\sharp+\\omega_{n-2}^\\sharp, 2\\omega_{n-1}^\\sharp, 2\\omega_{n-2}^\\sharp,"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{1}{p}, \\frac{1}{q})"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{2}{3}, \\frac{1}{3})"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{3}{8}, \\frac{1}{8})"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\leq r \\leq \\frac{3}{2}"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{s}_{C, \\epsilon}(p, q, r) = \\begin{cases} -\\frac{2}{q} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{p} \\leq \\frac{1}{4}; \\\\ \\frac{2}{p} - \\frac{2}{q} - \\frac{1}{2} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{4} < \\frac{1}{p} < \\frac{1}{2} - \\frac{1}{q}; \\\\ \\frac{1}{p} - \\frac{3}{q} + \\epsilon, & \\text{for } q \\geq 3p' \\text{ and } \\frac{1}{p} \\geq \\frac{1}{2} - \\frac{1}{q}; \\\\ \\frac{3}{2p} - \\frac{3}{2q} - \\frac{1}{2} + \\epsilon, & \\text{for } p' < q < 3p'; \\\\ \\frac{2}{p} - \\frac{1}{q} - 1 + \\epsilon, & \\text{for } q \\leq p'. \\end{cases}"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\frac{1}{2}, \\frac{1}{6})"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{s}_{C, 0}(p, q, r) < 0"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\leq r < p \\leq q \\leq \\infty"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "1 + (1 + \\omega)\\left(\\frac{1}{q} - \\frac{1}{p}\\right) > 0"} -{"pdf": "arxiv_math/2503.05140_pg22.pdf", "page": 1, "id": "2503.05140_pg22_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} \\frac{1}{q} > \\frac{2}{3p} - \\frac{1}{6}, & \\text{for } q \\geq 3p'; \\\\ \\frac{1}{q} > \\frac{1}{p} - \\frac{1}{3}, & \\text{for } p' < q < 3p'; \\\\ \\frac{1}{q} > \\frac{2}{p} - 1, & \\text{for } q \\leq p'. \\end{cases}"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_0 = 4(y_0 + y_1 - 2y_3), b_1 = 4(y_0 - y_3 + y_4 - y_5), b_2 = - 3y_0 - y_1 + 4y_3"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b_3 = 4(y_0 - y_3 + y_4 - y_5), b_4 = 4(y_0 + y_2 - 2y_5), b_5 = -3y_0 - y_2 + 4y_5"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a_0 = 4(x_0 + x_1 - 2x_3), a_1 = 4(x_0 - x_3 + x_4 - x_5), a_2 = - 3x_0 - x_1 + 4x_3"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_0 = \\arctan(\\sqrt{\\frac{\\phi_0}{\\phi_2}} \\phi_1)"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{0}^{1} (\\xi + \\phi_1)^p \\sqrt{\\phi_0(\\xi + \\phi_1 )^2 + \\phi_2}\\rm{d}\\xi = (\\frac{\\phi_2}{\\phi_0})^{\\frac{p+1}{2}} \\sqrt{\\phi_2} \\int_{\\theta_0}^{\\theta_1} tan^p \\theta sec^3 \\theta \\rm{d} \\theta,"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\frac{\\partial x}{\\partial \\xi} = a_0\\xi + a_1\\eta + a_2, \\\\ \\frac{\\partial x}{\\partial \\eta} = a_3\\xi + a_4\\eta + a_5, \\\\ \\frac{\\partial y}{\\partial \\xi} = b_0\\xi + b_1\\eta + b_2, \\\\ \\frac{\\partial y}{\\partial \\eta} = b_3\\xi + b_4\\eta + b_5, \\\\ \\end{aligned}"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_3 = 4(x_0 - x_3 + x_4 - x_5), a_4 = 4(x_0 + x_2 - 2x_5), a_5 = -3x_0 - x_2 + 4x_5"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_n = \\int \\sec^n \\theta \\rm{d} \\theta = \\frac{1}{n - 1} (tan\\theta \\sec^{n - 2} \\theta + (n - 2)I_{n-2}). \\nonumber"} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\tan^{2k+1} \\theta \\sec^3 \\theta \\rm{d} \\theta = \\int (sec^2 \\theta - 1)^k sec^2 \\theta \\rm{d} \\sec \\theta."} -{"pdf": "arxiv_math/2503.04493_pg31.pdf", "page": 1, "id": "2503.04493_pg31_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\tan^{2k} \\theta \\sec^3 \\theta \\rm{d} \\theta = \\int (sec^2 \\theta - 1)^k sec^2 \\theta \\rm{d} \\theta,"} -{"pdf": "arxiv_math/2503.09254_pg3.pdf", "page": 1, "id": "2503.09254_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "M = \\{ m_1 , ..., m_r \\}"} -{"pdf": "arxiv_math/2503.09254_pg3.pdf", "page": 1, "id": "2503.09254_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "G := \\{ m_1 - \\overline{m_1}^{G_<} , ..., m_r - \\overline{m_r}^{G_<} \\}"} -{"pdf": "arxiv_math/2503.04917_pg1.pdf", "page": 1, "id": "2503.04917_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "D(-\\Delta_{g})=H^{2}(\\Omega)\\cap H_{0}^{1}(\\Omega)."} -{"pdf": "arxiv_math/2503.06838_pg3.pdf", "page": 1, "id": "2503.06838_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{X} \\in \\R^{N \\times n}"} -{"pdf": "arxiv_math/2503.06838_pg3.pdf", "page": 1, "id": "2503.06838_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T \\in \\R^{n \\times d}"} -{"pdf": "arxiv_math/2503.06838_pg3.pdf", "page": 1, "id": "2503.06838_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{T} \\|\\mathbf{X} T - \\mathbf{Y}\\|^2"} -{"pdf": "arxiv_math/2503.06838_pg3.pdf", "page": 1, "id": "2503.06838_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{Y} \\in \\R^{N \\times d}"} -{"pdf": "arxiv_math/2503.04026_pg2.pdf", "page": 1, "id": "2503.04026_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{\\nu }=(\\nu _{1},\\nu _{2},\\nu _{3})"} -{"pdf": "arxiv_math/2503.04026_pg2.pdf", "page": 1, "id": "2503.04026_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\mathfrak{e}=\\left( e_{1},e_{2},e_{3}\\right)"} -{"pdf": "arxiv_math/2503.07449_pg17.pdf", "page": 1, "id": "2503.07449_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = 30 \\ {\\rm \\frac{J}{m^2}}"} -{"pdf": 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"arxiv_math/2503.06804_pg14.pdf", "page": 1, "id": "2503.06804_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{ \\text{x}}^{\\text{Test}}>0"} -{"pdf": "arxiv_math/2503.06804_pg14.pdf", "page": 1, "id": "2503.06804_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C_L \\left(u^L_n X^{ \\text{Work}}_n,0\\right),"} -{"pdf": "arxiv_math/2503.09076_pg12.pdf", "page": 1, "id": "2503.09076_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{1, 2, \\ldots, t\\}"} -{"pdf": "arxiv_math/2503.08411_pg18.pdf", "page": 1, "id": "2503.08411_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\not\\in Z\\cap \\bigcap_{i=1}^nN(J_i)"} -{"pdf": "arxiv_math/2503.08411_pg18.pdf", "page": 1, "id": "2503.08411_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\in Z\\cap \\bigcap_{i=1}^nN(J)"} -{"pdf": "arxiv_math/2503.08411_pg18.pdf", "page": 1, "id": 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"math": "X^\\mathbb{G}:= X^\\odot \\cup \\bigcup\\limits_{x \\in X} \\mathrm{L}_\\mathbb{G}(x)"} -{"pdf": "arxiv_math/2503.08411_pg18.pdf", "page": 1, "id": "2503.08411_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in Y\\cap \\bigcap_{J\\in \\mathcal{J}}N(J)"} -{"pdf": "arxiv_math/2503.08411_pg18.pdf", "page": 1, "id": "2503.08411_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Cont}^\\triangle(X,\\mathbb{G})"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d})"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{2}|\\mathcal{B}_{C}| \\leq (1+o(1))\\frac{n}{3}"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,4}) \\leq 0.8327 n"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "h = 1, 2, \\dots, \\omega"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,3})\\leq (1+o(1))\\frac{n}{3}"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d}) = 4"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha(\\mathcal{G}_{n,4}) \\leq 0.41635"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "s(\\mathcal{G}_{n,3}) = (1+o(1))\\frac{n}{4}"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z = \\sum_{i=0}^{i_{\\max}-1}\\sum_{h=1}^{\\omega}Z_{h}^{(i)},"} -{"pdf": "arxiv_math/2503.07335_pg33.pdf", "page": 1, "id": "2503.07335_pg33_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d}) = (1+o(1))\\frac{n}{\\alpha(\\mathcal{G}_{n,d})} = (1+o(1))\\frac{d}{2\\log d}"} -{"pdf": "arxiv_math/2503.07741_pg8.pdf", "page": 1, "id": "2503.07741_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\rho(x_0,t)\\rangle"} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}(Y^{(\\nu)}\\in A | X^{(\\nu)} = x) = p(A|x), \\quad (x\\in \\mathcal{X}, A\\subset L^1([0, L)^d, \\mathcal{X}))."} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "respectively and satisfy"} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\mathscr{M}^T(\\mathcal{X})} \\varepsilon_{\\nu}\\, d\\lambda(\\nu) = \\mathbb{E}\\left(\\frac{1}{L^d} \\int_{[0, L)^d} \\mathbf{d}\\left(T^t X, Y_t\\right) d\\mathbf{m}(t)\\right) +\\delta < \\varepsilon."} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(X; Y) \\geq \\int_{\\mathscr{M}^T(\\mathcal{X})} I(\\nu, p)\\, d\\lambda(\\nu) = \\int_{\\mathscr{M}^T(\\mathcal{X})} I\\left(X^{(\\nu)}; Y^{(\\nu)}\\right) d\\lambda(\\nu)."} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "we take random variables"} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "be the regular conditional distribution of"} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_{\\nu} = \\mathbb{E}\\left(\\frac{1}{L^d} \\int_{[0, L)^d} \\mathbf{d}\\left(T^t X^{(\\nu)}, Y^{(\\nu)}_t\\right)d\\mathbf{m}(t)\\right) +\\delta."} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "p(A|x) = \\mathbb{P}(Y\\in A|X=x), \\quad (x\\in \\mathcal{X}, A\\subset L^1([0, L)^d, \\mathcal{X}))."} -{"pdf": "arxiv_math/2503.06851_pg17.pdf", "page": 1, "id": "2503.06851_pg17_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": ". Suppose that random variables"} -{"pdf": "arxiv_math/2503.05976_pg3.pdf", "page": 1, "id": "2503.05976_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in \\set{0} \\cup \\N \\cup \\set{\\infty}"} -{"pdf": "arxiv_math/2503.05976_pg3.pdf", "page": 1, "id": "2503.05976_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "z = (z_1, \\dots, z_n)"} -{"pdf": "arxiv_math/2503.05976_pg3.pdf", "page": 1, "id": "2503.05976_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta = (\\zeta_1, \\dots, \\zeta_n)"} -{"pdf": "arxiv_math/2503.07897_pg25.pdf", "page": 1, "id": "2503.07897_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_r(0)=0 \\quad\\forall r\\geq 0"} -{"pdf": "arxiv_math/2503.07897_pg25.pdf", "page": 1, "id": "2503.07897_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{\\Delta}(t) = \\mu_{r + 1}(t) - \\mu_r(t) = \\frac{\\gamma}{\\rho} \\; \\ln{(1 + \\rho \\; t)},"} -{"pdf": "arxiv_math/2503.07897_pg25.pdf", "page": 1, "id": "2503.07897_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_r'(t)= \\lambda_r(t)"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "U\\in C([0,\\infty),X_p)"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_p\\!:\\!D(A_p)\\!\\subset \\!X_p\\to X_p"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "F^M(U(\\cdot))\\in L^1((0,t);X_p)"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f_\\nu^M,\\psi^M,\\phi^M"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to0}e^{tA_p}U=U"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C^1([0,\\infty);X_p)\\cap C([0,\\infty);D_p(A))"} -{"pdf": "arxiv_math/2503.07156_pg35.pdf", "page": 1, "id": "2503.07156_pg35_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "E:=\\{U\\in C([0,\\infty),X_p) : \\|U\\|_E=\\sup_{t\\ge0}e^{-(\\omega_p+\\theta)t}\\|U(t)\\|_{p}<\\infty\\}\\,,"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_h = \\hat v_h \\circ \\bm F_K^{-1}"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{diam} T \\le Ch"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "B(\\bm z; r) = \\{\\bm x \\mid |\\bm x - \\bm z| \\le r\\}"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat v_h \\in \\mathbb P_k(\\hat T)"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\Delta g = \\eta \\quad\\text{in }\\; \\Omega, \\qquad g = 0 \\quad\\text{on }\\; \\Gamma,"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A(\\bm z; r, R) = \\{\\bm x \\mid r \\le |\\bm x - \\bm z|\\le R\\}"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{j=\\ell_1}^{\\ell_2} d_j^\\alpha"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{supp} \\eta \\subset \\Omega"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in W^{m, p}(\\tilde\\Omega)"} -{"pdf": "arxiv_math/2503.09190_pg4.pdf", "page": 1, "id": "2503.09190_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\emptyset \\neq T \\cap \\Gamma_h \\subset S"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1(0,0)\\sharp b_2(0,0)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_2(z_1,z_2)\\sharp b_1(z_1,z_2)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}''_2(z_1,z_2)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}''_1(z_1,z_2), \\tilde{\\gamma}''_2(z_1,z_2))"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}_1''=\\tilde{\\gamma}''_1(0,0)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma_2''=\\tilde{\\gamma}''_2(0,0)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1,z_2)=(0,0) \\in [0,1]_{z_1} \\times [0,1]_{x_2}"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma'_i(z_1,z_2)= a_i(z_1,z_2)\\sharp b_i(z_1,z_2)"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1, 0) \\in [0,1] \\times [0,1]"} -{"pdf": "arxiv_math/2503.07543_pg42.pdf", "page": 1, "id": "2503.07543_pg42_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}'_1, \\tilde{\\gamma}'_2)=(\\tilde{\\gamma}'_1(0,0), \\tilde{\\gamma}'_2(0,0))"} -{"pdf": "arxiv_math/2503.05716_pg6.pdf", "page": 1, "id": "2503.05716_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x_1,x_2,t) = t^4 + \\sin(x_1)\\cdot\\sin(x_2)\\cdot\\sin(t)."} -{"pdf": "arxiv_math/2503.05716_pg6.pdf", "page": 1, "id": "2503.05716_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_1=[0,2\\pi]\\times[0,2\\pi], t\\in(0,2)"} -{"pdf": "arxiv_math/2503.05716_pg6.pdf", "page": 1, "id": "2503.05716_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_2=[0,10\\pi]\\times[0,10\\pi], t\\in(0,10)"} -{"pdf": "arxiv_math/2503.05716_pg6.pdf", "page": 1, "id": "2503.05716_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{GELU}(x) = x \\cdot \\frac{1}{2}[1+erf(\\frac{x}{\\sqrt{2}})]"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi\\in C^\\infty([0,\\infty))"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(r)\\sim cr^{\\frac32}"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "r^{\\frac32}, r^{-\\frac12}"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle L_1 \\phi , \\chi_b(r) r^{\\frac12}\\rho_1\\rangle = \\lambda_0 \\langle \\phi, \\chi_b(r) r^{\\frac12}\\rho_1\\rangle"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1(r^{\\frac12}\\rho_1)=0"} -{"pdf": "arxiv_math/2503.07345_pg12.pdf", "page": 1, "id": "2503.07345_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1\\phi =\\lambda_0\\phi"} -{"pdf": "arxiv_math/2503.08272_pg14.pdf", "page": 1, "id": "2503.08272_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = (0.62,0.18,0.1,0.1)"} -{"pdf": "arxiv_math/2503.08272_pg14.pdf", "page": 1, "id": "2503.08272_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-(1+\\frac{2}{3})^{-1} = 0.4"} -{"pdf": "arxiv_math/2503.08272_pg14.pdf", "page": 1, "id": "2503.08272_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat q = (0.5,0.5,0,0)"} -{"pdf": "arxiv_math/2503.08272_pg14.pdf", "page": 1, "id": "2503.08272_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{2}(\\omega )"} -{"pdf": "arxiv_math/2503.08272_pg14.pdf", "page": 1, "id": "2503.08272_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{1}(\\omega )"} -{"pdf": "arxiv_math/2503.05594_pg4.pdf", "page": 1, "id": "2503.05594_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} \\big(D^X(s-)+\\tfrac12 \\gamma(s) \\Delta X(s)\\big)^\\top \\Delta X(s) & = D^X(s-)\\Delta X(s-) + \\tfrac12 (\\Delta X(s))^\\top \\gamma(s) \\Delta X(s) . \\end{split}"} -{"pdf": "arxiv_math/2503.05594_pg4.pdf", "page": 1, "id": "2503.05594_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C(x,d,X) = \\int_{[0,T]} (D^X(s-))^\\top\\, dX(s) + \\frac12 \\int_{[0,T]} (\\Delta X(s))^\\top \\gamma(s)\\, dX(s) ."} -{"pdf": "arxiv_math/2503.05594_pg4.pdf", "page": 1, "id": "2503.05594_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "D^X(s)=D^X(s-)+\\Delta D^X(s) = D^X(s-)+ \\gamma(s)\\Delta X(s)."} -{"pdf": "arxiv_math/2503.06167_pg12.pdf", "page": 1, "id": "2503.06167_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{u}(u) := \\rho \\left[ \\frac{u}{\\rho}\\right]"} -{"pdf": "arxiv_math/2503.06167_pg12.pdf", "page": 1, "id": "2503.06167_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l: \\mathbb{R} \\mapsto \\mathbb{R}"} -{"pdf": "arxiv_math/2503.06167_pg12.pdf", "page": 1, "id": "2503.06167_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{x_j} f_j(k)"} -{"pdf": "arxiv_math/2503.06167_pg12.pdf", "page": 1, "id": "2503.06167_pg12_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l(\\partial_{x_j} f_j(k))"} -{"pdf": "arxiv_math/2503.09500_pg70.pdf", "page": 1, "id": "2503.09500_pg70_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi=\\mathrm{BC}_E(\\pi)"} -{"pdf": "arxiv_math/2503.09500_pg70.pdf", "page": 1, "id": "2503.09500_pg70_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi=\\bigotimes_v\\pi_v"} -{"pdf": "arxiv_math/2503.09500_pg70.pdf", "page": 1, "id": "2503.09500_pg70_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi_{v_1}\\simeq \\pi_{v_1}\\otimes \\pi_{v_1}"} -{"pdf": "arxiv_math/2503.09500_pg70.pdf", "page": 1, "id": "2503.09500_pg70_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f' = \\sum_\\tau p_{\\tau,!}(f^\\tau)"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = E_{\\Gamma_0}(\\theta) + \\mu_j"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta)=2\\sum_{i=1}^d \\cos 2\\pi\\theta_i"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta) = 2\\cos 2\\pi\\theta_1+2\\cos 2\\pi\\theta_2+2\\cos 2\\pi(\\theta_1+\\theta_2)"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = (1+\\mu_j) E_{\\Gamma_0}(\\theta)+\\mu_j"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_3=\\Gamma_0 \\boxtimes G_F"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_1 = \\Gamma_0 \\mathop\\square G_F"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle a\\rangle_p = \\sum_{q=1}^{\\nu_F} \\langle a(\\cdot+v_q)\\rangle \\sum_{s=1}^{\\nu'_F} |P_{\\mu_s}(v_p,v_q)|^2\\,,"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_2 = \\Gamma_0 \\times G_F"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = \\mu_j E_{\\Gamma_0}(\\theta)"} -{"pdf": "arxiv_math/2503.08595_pg4.pdf", "page": 1, "id": "2503.08595_pg4_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{\\mu_s}(v_p,v_q) = \\sum_{j\\,\\mu_j=\\mu_s} w_j(v_p)\\overline{w_j(v_q)}"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\langle L_I,h\\rangle| \\leq \\alpha \\int_0^T \\int_{\\mathbb{R}^N} |\\xi(t,x)|dx dt + 2\\beta \\|I\\|_{L^\\infty}\\|h\\|_{L^\\infty}T + \\gamma \\int_{\\mathbb{R}^N} |\\xi(T,x)|dx \\leq C_1 \\|h\\|_{L^\\infty(0,T)},"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I \\in \\mathcal{U}_{ad}"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi_I(\\varepsilon)= \\dfrac{J(I^\\varepsilon) - J(I)}{\\varepsilon}"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_1, h_2 \\in L^{\\infty}(0,T)"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "h = h_1 + \\lambda h_2"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\mathbb{R}"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "I^\\varepsilon = I + \\varepsilon h"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{\\varepsilon \\to 0} \\left\\|\\frac{G(I + \\varepsilon h) - G(I)}{\\varepsilon} - (\\xi, \\eta)\\right\\|_X = \\lim_{\\varepsilon \\to 0} \\left\\|(\\tilde{\\xi}, \\tilde{\\eta})\\right\\|_X = 0."} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L_I: L^{\\infty}(0,T) \\to \\mathbb{R}"} -{"pdf": "arxiv_math/2503.09208_pg18.pdf", "page": 1, "id": "2503.09208_pg18_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(p^\\varepsilon, d^\\varepsilon) = G(I^\\varepsilon)"} -{"pdf": "arxiv_math/2503.06055_pg5.pdf", "page": 1, "id": "2503.06055_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_\\tau \\, v \\preceq T_\\sigma \\, v"} -{"pdf": "arxiv_math/2503.04122_pg13.pdf", "page": 1, "id": "2503.04122_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g\\in\\{2,3,\\cdots 10\\}"} -{"pdf": "arxiv_math/2503.04122_pg13.pdf", "page": 1, "id": "2503.04122_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": ". A straightforward computation leads to this initial segment and the corresponding sequence of differences:"} -{"pdf": "arxiv_math/2503.05685_pg4.pdf", "page": 1, "id": "2503.05685_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_m(j(E_1), j(E_2))"} -{"pdf": "arxiv_math/2503.05685_pg4.pdf", "page": 1, "id": "2503.05685_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\gg_\\epsilon( \\# S)^{5+\\epsilon}"} -{"pdf": "arxiv_math/2503.04567_pg38.pdf", "page": 1, "id": "2503.04567_pg38_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{ij} \\ne \\tau_{ji}"} -{"pdf": "arxiv_math/2503.03772_pg1.pdf", "page": 1, "id": "2503.03772_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g \\cdot (h \\cdot x) = (gh) \\cdot x"} -{"pdf": "arxiv_math/2503.03772_pg1.pdf", "page": 1, "id": "2503.03772_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot : G \\times X \\to X"} -{"pdf": "arxiv_math/2503.03772_pg1.pdf", "page": 1, "id": "2503.03772_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(g \\cdot x) = g \\cdot \\tau(x)"} -{"pdf": "arxiv_math/2502.15977_pg21.pdf", "page": 1, "id": "2502.15977_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_1 + ... + \\theta_n"} -{"pdf": "arxiv_math/2503.06731_pg12.pdf", "page": 1, "id": "2503.06731_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x.\\omega=\\omega(S(x).-),\\; \\forall\\omega\\in V^\\ast, x\\in A\\,."} -{"pdf": "arxiv_math/2503.07447_pg1.pdf", "page": 1, "id": "2503.07447_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{-1}\\log^2 n \\ll p \\ll n^{-1/2}\\log^{1/4} n,"} -{"pdf": "arxiv_math/2503.07447_pg1.pdf", "page": 1, "id": "2503.07447_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p \\gg n^{-2/3}\\log^{2/3} n"} -{"pdf": "arxiv_math/2503.05871_pg24.pdf", "page": 1, "id": "2503.05871_pg24_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta_2 = \\partial_y^2-\\alpha^2"} -{"pdf": "arxiv_math/2503.05871_pg24.pdf", "page": 1, "id": "2503.05871_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta\\phi_{\\pm\\alpha}(t,y)"} -{"pdf": "arxiv_math/2503.05871_pg24.pdf", "page": 1, "id": "2503.05871_pg24_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(t, \\pm 1) = \\partial_y \\Phi(t,\\pm 1) = 0"} -{"pdf": "arxiv_math/2503.05871_pg24.pdf", "page": 1, "id": "2503.05871_pg24_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_y\\delta\\phi_0"} -{"pdf": "arxiv_math/2503.05871_pg24.pdf", "page": 1, "id": "2503.05871_pg24_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vec{\\nabla}\\cdot \\vec{u} = 0"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H \\coloneqq (\\delta_{m+i, j})_{i=1, \\dots, d-m; \\: j=1, \\dots, d}"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\gamma^{(n)} - \\gamma\\|_2 \\to 0"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lVert Z_n - Z \\rVert_2 \\to 0"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "X_{m+1}(t),\\dots,X_d(t)"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma \\in L_t(X, d')"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in \\R^{d\\times d}"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "Z_n \\in \\mathcal{E}(Y, t, d')"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_018", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|I_t H - Z\\| = \\lim_{n \\to \\infty} \\|Z_n - Z\\| = 0"} -{"pdf": "arxiv_math/2503.05588_pg16.pdf", "page": 1, "id": "2503.05588_pg16_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\Sigma(0,-1):=\\mathrm{Cov}(X(0))"} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\log\\left( \\frac{P_1^{\\rho_0+\\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_2^{\\rho_0}(X_{(k-1)/n},X_{k/n};1/n)}\\right) = \\log\\left(\\frac{\\beta}{\\alpha}\\right) -\\frac{(X_{k/n}-X_{(k-1)/n})^2}{2/n}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right) + R_{k,\\theta}"} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g_k := \\log\\left(\\frac{\\beta}{\\alpha}\\right) - \\frac{n}{2}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right)(X_{k/n}-X_{(k-1)/n})^2"} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_4(\\theta)}"} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{k,\\theta} = \\log\\left( \\frac{1-\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\alpha^2/n}(X_{k/n}-\\rho_0-\\theta)(X_{(k-1)/n}-\\rho_0-\\theta)\\right)}{1+\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\beta^2/n}(X_{k/n}-\\rho_0)(X_{(k-1)/n}-\\rho_0)\\right)}\\right)."} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_2(\\theta)"} -{"pdf": "arxiv_math/2503.07022_pg75.pdf", "page": 1, "id": "2503.07022_pg75_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_5(\\theta)}"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "V_j= B(z,(1+u)2^{-j})"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r/100}"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "B(z,2^{-j_0})\\subseteq D"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{F}_{W,r/100}"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r} = \\{(\\ell^1_i,\\ell^2_i,y_i)\\}_{i=1,\\ldots,M^*}"} -{"pdf": "arxiv_math/2503.08958_pg35.pdf", "page": 1, "id": "2503.08958_pg35_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W_{z,j},r}"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{C}_{m}(S,H)"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\varphi(i),\\varphi(j)) \\in E(S)"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_0, w_1,\\ldots,w_m,w_{m+1}"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(H)=|E(H)|-|V(H)|"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_A = \\{ (u,v) \\in E: u,v \\in A \\}"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Dist}_{H}(u,v)"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_{\\setminus A} = (V,E_{\\setminus A})"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Diam}(H)=\\max_{u,v \\in V(H)} \\mathsf{Dist}_H(u,v)"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "H,S \\subset \\mathcal{K}_n"} -{"pdf": "arxiv_math/2503.06464_pg8.pdf", "page": 1, "id": "2503.06464_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "V(H)=\\{ u,v,w_1,\\ldots,w_m \\}"} -{"pdf": "arxiv_math/2503.09087_pg6.pdf", "page": 1, "id": "2503.09087_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "L_\\alpha^{\\red(w)}(G)"} -{"pdf": "arxiv_math/2503.06589_pg21.pdf", "page": 1, "id": "2503.06589_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{23},\\;B_{23},\\;R,\\;Y"} -{"pdf": "arxiv_math/2503.06589_pg21.pdf", "page": 1, "id": "2503.06589_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[x_{\\rm min},x_{\\rm max}]"} -{"pdf": "arxiv_math/2503.06589_pg21.pdf", "page": 1, "id": "2503.06589_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mu-m)e^2+(a^2-n^2)\\mu<0,"} -{"pdf": "arxiv_math/2503.06605_pg1.pdf", "page": 1, "id": "2503.06605_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t=(b_{ij}^t)_{n\\times n}"} -{"pdf": "arxiv_math/2503.06605_pg1.pdf", "page": 1, "id": "2503.06605_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf x}_t=(x_{1;t},\\ldots,x_{n;t})"} -{"pdf": "arxiv_math/2503.06605_pg1.pdf", "page": 1, "id": "2503.06605_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat y_{k;t}={\\bf x}_t^{B_t{\\bf e}_k}"} -{"pdf": "arxiv_math/2503.06605_pg1.pdf", "page": 1, "id": "2503.06605_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F_u^t(y_1,\\ldots,y_n)\\in\\mathbb Z[y_1,\\ldots,y_n]"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\det A_{n;3}(x_1,x_2)=0"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.97,\\sqrt{0.97})"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{rcl} p_n(x,\\sqrt{x})&=&-k_n(x)k_n(1/x)-2k_n(\\sqrt{x})k_n(1/\\sqrt{x})\\\\ \\ q_n(x,\\sqrt{x})&=&k_n(x)k_n(1/\\sqrt{x})^2+k_n(1/x)k_n(\\sqrt{x})^2. \\end{array}"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.098,\\sqrt{0.098})"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "P=(0.098,\\sqrt{0.098})"} -{"pdf": "arxiv_math/2503.08374_pg13.pdf", "page": 1, "id": "2503.08374_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\det A_{n;3}(x,\\sqrt{x})=\\delta_n^3+p_n(x,\\sqrt{x})\\delta_n+q_n(x,\\sqrt{x})=0,"} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", "page": 1, "id": "2503.07147_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "|N_{G}(U\\setminus B)|\\geq \\frac{s|U\\setminus B|}{2r}"} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", "page": 1, "id": "2503.07147_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|B|\\leq \\frac{|U|}{10r}+\\frac{|U|}{10}\\le \\frac{|U|}{2},"} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", "page": 1, "id": "2503.07147_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "N_{G,r}(U\\setminus B)\\subseteq X"} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", "page": 1, "id": "2503.07147_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|U'|\\leq \\frac{r|X \\cap L|}{t}\\le \\frac{r}{t}\\cdot |L| \\leq \\frac{r}{t}\\cdot \\frac{|U|\\cdot t}{10r}= \\frac{|U|}{10}."} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", "page": 1, "id": "2503.07147_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "X\\subseteq V(G)\\setminus U"} -{"pdf": "arxiv_math/2503.07147_pg16.pdf", 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x}))\\lambda({\\bf x})"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "J_{R_I}(\\theta) := \\int_{\\Omega} \\left( \\frac{1}{2} |\\nabla \\widetilde{U}({\\bf x};\\theta)|^2- f({\\bf x}) \\widetilde{U}({\\bf x};\\theta)\\right) \\, d{\\bf x}"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{U}({\\bf x};\\theta)"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "U({\\bf x};\\theta_{R})"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{R,M}(\\theta,\\lambda) := J_{R,M}(\\theta)+\\frac{1}{X(\\partial \\Omega)} \\sum_{{\\bf x} \\in X(\\partial \\Omega)} (U({\\bf x};\\theta)-g({\\bf x}))\\lambda({\\bf x}),"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "U({\\bf x};\\theta_{P})"} -{"pdf": "arxiv_math/2503.09086_pg5.pdf", "page": 1, "id": "2503.09086_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda=\\lambda+ \\alpha \\nabla_{\\lambda} L_{P,M} \\quad \\text{or} \\quad \\lambda=\\lambda+\\alpha \\nabla_{\\lambda} L_{R,M}."} -{"pdf": "arxiv_math/2503.07355_pg11.pdf", "page": 1, "id": "2503.07355_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}(2k)\\simeq \\mathbb{C}(2^k) \\qquad \\text{and}\\qquad \\mathcal{C}(2k+1)\\simeq \\mathbb{C}(2^k)\\oplus\\mathbb{C}(2^k),"} -{"pdf": "arxiv_math/2503.07355_pg11.pdf", "page": 1, "id": "2503.07355_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{r,s}=\\begin{cases} 2 \\text{ if } r-s=2,4 \\text{ mod 4},\\\\ 1 \\text{ if } r-s=1,3 \\text{ 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\\delta_{ij}."} -{"pdf": "arxiv_math/2503.06570_pg6.pdf", "page": 1, "id": "2503.06570_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{\\partial_{t_j}} \\left( L(\\tau,z) \\alpha \\right) =0"} -{"pdf": "arxiv_math/2503.06570_pg6.pdf", "page": 1, "id": "2503.06570_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu(\\phi_\\ell)=\\frac{1}{2} (\\deg \\phi_\\ell - \\dim X) \\phi_\\ell,"} -{"pdf": "arxiv_math/2503.06570_pg6.pdf", "page": 1, "id": "2503.06570_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\tau , d \\rangle \\to -\\infty"} -{"pdf": "arxiv_math/2503.06570_pg6.pdf", "page": 1, "id": "2503.06570_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^2(X) \\times \\mathbb{C}^\\times"} -{"pdf": "arxiv_math/2503.06570_pg6.pdf", "page": 1, "id": "2503.06570_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d \\in \\mathsf{Eff}_{\\neq 0}"} 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\\left\\Vert \\nabla u_{n}\\right\\Vert _{L^{m}}:p_{n}^{-}\\geqslant m\\right\\}"} -{"pdf": "arxiv_math/2503.07785_pg20.pdf", "page": 1, "id": "2503.07785_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "N_x\\times N_v=49\\times97"} -{"pdf": "arxiv_math/2503.05880_pg5.pdf", "page": 1, "id": "2503.05880_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{0}\\in \\mathbf{R}^{d}"} -{"pdf": "arxiv_math/2503.05880_pg5.pdf", "page": 1, "id": "2503.05880_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( W\\left( x+x_{0}\\right) -W\\left( x_{0}\\right) \\right) _{x\\in \\mathbf{R}^{d}}"} -{"pdf": "arxiv_math/2503.05880_pg5.pdf", "page": 1, "id": "2503.05880_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\Vert x\\right\\Vert"} -{"pdf": "arxiv_math/2503.05880_pg5.pdf", "page": 1, "id": "2503.05880_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( W\\left( x\\right) \\right) _{x\\in \\mathbf{R}^{d}}"} -{"pdf": "arxiv_math/2503.04498_pg27.pdf", "page": 1, "id": "2503.04498_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bmod p^{\\prime \\prime}"} -{"pdf": "arxiv_math/2503.03952_pg5.pdf", "page": 1, "id": "2503.03952_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} f(t,0;q)=\\frac{t-q}{1-q}. \\end{aligned}"} -{"pdf": "arxiv_math/2503.03952_pg5.pdf", "page": 1, "id": "2503.03952_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} f(t,q/t;q)=\\frac{t+q/t-2q}{1-q}. \\end{aligned}"} -{"pdf": "arxiv_math/2503.03952_pg5.pdf", "page": 1, "id": "2503.03952_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} (x;q)_\\infty=\\prod_{k=0}^\\infty (1-xq^k),\\qquad (x;q)_n=\\prod_{k=0}^{n-1}(1-xq^k),\\qquad (x;q)_0=1, \\end{aligned}"} -{"pdf": "arxiv_math/2503.03952_pg5.pdf", "page": 1, "id": "2503.03952_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "I_N(q^{1/2}, q^{1/2}; q)"} -{"pdf": "arxiv_math/2503.06701_pg5.pdf", "page": 1, "id": "2503.06701_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "1.262 \\cdot \\left(\\lvert e \\rvert\\right)^{\\frac{1}{5}} + 2"} -{"pdf": "arxiv_math/2503.06701_pg5.pdf", "page": 1, "id": "2503.06701_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-\\lvert e \\rvert )/70"} -{"pdf": "arxiv_math/2503.06701_pg5.pdf", "page": 1, "id": "2503.06701_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-2 \\times 10^{-6} \\cdot i - 10^{-6} \\cdot c"} -{"pdf": "arxiv_math/2503.06701_pg5.pdf", "page": 1, "id": "2503.06701_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-\\lvert e \\rvert )/20"} -{"pdf": "arxiv_math/2503.05392_pg19.pdf", "page": 1, "id": "2503.05392_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{1},K_{2}\\in\\mathcal{K}_o^n"} -{"pdf": "arxiv_math/2503.05392_pg19.pdf", "page": 1, "id": "2503.05392_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[S_{e_n}(K^\\circ)]^\\circ"} -{"pdf": "arxiv_math/2503.05392_pg19.pdf", "page": 1, "id": "2503.05392_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K_1|e_n^\\perp=K_2|e_n^\\perp"} -{"pdf": "arxiv_math/2503.05392_pg19.pdf", "page": 1, "id": "2503.05392_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{1},K_{2}\\in\\mathcal{K}^n_o"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1\\coloneqq\\sup\\{t\\ge0\\ |\\ d_X(o,\\gamma(t))\\le R\\}"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(Z_n,\\rho_n)\\xrightarrow{AI\\ conv.} (Z,\\rho_0)"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\xi|\\eta)_o\\ge(\\gamma_1(R)|\\gamma_2(R))_o= \\frac{1}{2}\\bigg(d_X(o,\\gamma_1(R))+d_x(o,\\gamma_2(R))-d_X(\\gamma_1(R),\\gamma_2(R))\\bigg)\\ge R-\\frac{\\hbox{diam}(U)}{2}."} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma(t_0)\\in S_X(o,R)"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\gamma_1(t)|\\gamma_2(t))_o\\uparrow (\\xi|\\eta)_o"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} d_X(x_n,p)&\\le d_X(x_n,p_n)+d_X(p_n,p)\\\\ &=d_X(o,p_n)-d_X(o,x_n)+ d_X(p_n,p)\\\\ &=d_X(o,p_n)-R+d_X(p_n,p)\\xrightarrow{n\\to \\infty} d_X(o,p)-R+0=0. \\end{split}"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\partial_P X_n,\\rho_{x_n})\\xrightarrow{AI conv.}(\\partial_P X,\\rho_x)"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1(R),\\gamma_2(R)\\in U"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(Z,\\rho_0)=(\\partial_P X,\\rho_x)"} -{"pdf": "arxiv_math/2503.09284_pg29.pdf", "page": 1, "id": "2503.09284_pg29_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "2(x|y)_o=d_X(x,o)+d_X(y,o)-d_X(x,y)=2R-d_X(x,y)."} -{"pdf": "arxiv_math/2503.06334_pg4.pdf", "page": 1, "id": "2503.06334_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{C_{v_0},\\cdots,C_{v_{n-1}}\\}"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{\\xi}=\\cup_{\\xi(\\pi(m_i))\\neq1}l_i."} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L_{\\xi}=l_{i_1}\\cup \\cdots l_{i_k}"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m_1,...,m_d \\in \\pi_1(M-L;\\Z)"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_1(M-L) \\to \\pi_1(M-L)^{ab}"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ker \\tilde F_n\\subset \\ker \\tilde F_{n-1}"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|H_1(M_{\\pi};\\Z)|=\\frac{|G|}{\\prod_{\\xi \\in \\hat{G}^{(1)}}|1-\\xi(\\pi(m_{i(\\xi)}))|}\\prod_{\\xi \\in \\hat{G}}|\\Delta_{L_{\\xi}}(\\xi(\\pi(m_{i_1})),...,\\xi(\\pi(m_{i_k})))|."} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varprojlim_nG_n\\cong\\Z_p^{\\,d}"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_{\\Gamma}:\\pi_1(M-L)\\to \\Z_p^d/\\Gamma"} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{G}^{(1)}=\\{\\xi \\in \\hat{G}\\mid L_{\\xi} \\text{ has a single component}\\}."} -{"pdf": "arxiv_math/2503.06194_pg10.pdf", "page": 1, "id": "2503.06194_pg10_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Z^d \\hookrightarrow \\Z_p^d"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[\\|\\hat{x}_\\eta-x^*\\|^2]\\leq \\tfrac{2}{\\mu_f}\\left(\\|\\nabla f(x^*)\\| \\,\\sqrt[\\kappa]{\\tfrac{1}{\\alpha}\\left(\\texttt{Err}_{\\eta}+\\eta M\\right)} + \\tfrac{1}{\\eta}\\texttt{Err}_{\\eta}\\right)"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tfrac{\\mu_f}{2}\\mathbb{E}[\\|\\hat{x}_\\eta-x^*\\|^2]+\\nabla f(x^*)^\\top\\mathbb{E}\\left[\\left(\\hat{x}_\\eta- \\Pi_{X^*_h}[\\hat{x}_\\eta]\\right)\\right]\\leq \\mathbb{E}[f(\\hat{x}_\\eta)] - f^*"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi_{X^*_h}[\\hat{x}_\\eta] \\in X^*_h"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta\\le\\tfrac{\\alpha}{2\\|\\nabla f(x^*)\\|}"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\le \\mathbb{E}[h(\\hat{x}_\\eta)]-h^*\\le \\texttt{Err}_{\\eta}+\\eta M"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\le \\mathbb{E}[ h(\\hat{x}_\\eta)]-h^*\\le \\texttt{Err}_{\\eta}+\\eta M"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta)] -h^*\\geq 0"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta)]-h^*\\leq \\texttt{Err}_{\\eta}+\\eta M"} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta) + \\eta f(\\hat{x}_\\eta)] -f^*_\\eta \\leq \\texttt{Err}_{\\eta}."} -{"pdf": "arxiv_math/2503.08634_pg6.pdf", "page": 1, "id": "2503.08634_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "f^*- \\mathbb{E}[ f(\\hat{x}_\\eta)] \\lambda_2(\\nu)"} -{"pdf": "arxiv_math/2503.04612_pg8.pdf", "page": 1, "id": "2503.04612_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_2 = \\int \\log |a| \\, d\\pi"} -{"pdf": "arxiv_math/2503.04612_pg8.pdf", "page": 1, "id": "2503.04612_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega = (\\omega)_{n \\in \\Z} \\in \\Omega"} -{"pdf": "arxiv_math/2503.04612_pg8.pdf", "page": 1, "id": "2503.04612_pg8_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "X(\\omega) = \\sum_{n=0}^\\infty a(\\omega_{-1}) a(\\omega_{-2}) \\cdots a(\\omega_{-n}) b(\\omega_{-n-1}) \\, ."} -{"pdf": "arxiv_math/2503.04612_pg8.pdf", "page": 1, "id": "2503.04612_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lbrace \\mathbf{E}_1(\\omega), \\mathbf{E}_2(\\omega)\\rbrace = S"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c_1x_1^\\alpha + c_2 x_2^\\beta + g(x_1,x_2)"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v(D) = v(f_t^*D) = pq."} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-K_X\\cdot D = f_t^*\\left(\\sum_i C_i\\right)\\cdot D_t =\\left( (p+q)E_t + \\sum_i C_{i,t}\\right) \\cdot D_t = p+q"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(K_{Y_t} + D_t) \\cdot D_t = -2"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq (K_{Y_t} + D_t) \\cdot D_t = \\left(-\\sum_{i}C_{i,t} - E_t + D_t\\right) \\cdot D_t = D_t^2 - E_t\\cdot D_t < -E_t\\cdot D_t,"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq D_t^2 - E_t\\cdot D_t"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p+q = D\\cdot C \\geq (D\\cdot C)_x = \\alpha + \\beta \\geq p + q,"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*D = D_t + pq E_t"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*(\\sum_{i}C_i)\\cdot D_t = -K_X \\cdot D> 0"} -{"pdf": "arxiv_math/2503.06612_pg19.pdf", "page": 1, "id": "2503.06612_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t = \\sum_{i=1}^k a_iC_{i,t}"} -{"pdf": "arxiv_math/2503.07452_pg8.pdf", "page": 1, "id": "2503.07452_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} & \\underset{\\mathbf{b}}{\\text{minimize}} & & \\epsilon_d(\\mathbf{b}) = \\dfrac{1}{n_{MP}} \\sum_{i=1}^{n_{MP}} (\\mathbf{u}_{r_i} - \\mathbf{u}_{s_i}(\\mathbf{b}))^2\\\\ & \\text{subject to} & & \\mathbf{b}_l \\mathbf{\\leq b} \\leq \\mathbf{b}_u \\end{aligned}"} -{"pdf": "arxiv_math/2503.09042_pg5.pdf", "page": 1, "id": "2503.09042_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_1 = Y_2 = \\cdots = Y_{k-1}"} -{"pdf": "arxiv_math/2503.09042_pg5.pdf", "page": 1, "id": "2503.09042_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{g(X) \\geq k \\text{ or } g(-X) \\geq k} \\leq 2\\epsilon"} -{"pdf": "arxiv_math/2503.09042_pg5.pdf", "page": 1, "id": "2503.09042_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{g(-X) \\geq k} = \\Pr{g(X) \\geq k} \\leq \\epsilon"} -{"pdf": "arxiv_math/2503.09042_pg5.pdf", "page": 1, "id": "2503.09042_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{X_{f(Y)}=X_{f(-Y)}}{Y=y} \\geq 1/2"} -{"pdf": "arxiv_math/2503.09042_pg5.pdf", "page": 1, "id": "2503.09042_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pr{B_Y} = 2^{-(k-2)}"} -{"pdf": "arxiv_math/2503.05138_pg21.pdf", "page": 1, "id": "2503.05138_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle Au,u-v\\rangle\\le\\Phi(v)-\\Phi(u)+I_\\Delta(\\psi^0(\\gamma_\\psi u;\\gamma_\\psi v-\\gamma_\\psi u))-\\langle f,v-u\\rangle."} -{"pdf": "arxiv_math/2503.05138_pg21.pdf", "page": 1, "id": "2503.05138_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle Au^h, u^h-v^h\\rangle \\le \\Phi(v^h)-\\Phi(u^h) + I_\\Delta(\\psi^0(\\gamma_\\psi u^h;\\gamma_\\psi v^h-\\gamma_\\psi u^h))-\\langle f,v^h-u^h\\rangle."} -{"pdf": "arxiv_math/2503.05138_pg21.pdf", "page": 1, "id": "2503.05138_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi^0(z;z_1+z_2)\\le \\psi^0(z;z_1)+\\psi^0(z;z_2)\\quad \\forall\\,z,z_1,z_2\\in \\mathbb{R}^m,"} -{"pdf": "arxiv_math/2503.05348_pg3.pdf", "page": 1, "id": "2503.05348_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{-\\infty}^t F_X(x)dx\\geq \\int_{-\\infty}^t F_Y(x)dx\\ \\text{ for all }t."} -{"pdf": "arxiv_math/2503.05348_pg3.pdf", "page": 1, "id": "2503.05348_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to -\\infty} \\pi_X(t)-\\pi_Y(t)=0"} -{"pdf": "arxiv_math/2503.05348_pg3.pdf", "page": 1, "id": "2503.05348_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_t^{\\infty} \\bar F_X(x)dx\\leq \\int_t^{\\infty} \\bar F_Y(x)dx\\ \\text{ for all }t."} -{"pdf": "arxiv_math/2503.05348_pg3.pdf", "page": 1, "id": "2503.05348_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar F_X\\leq \\bar F_Y"} -{"pdf": "arxiv_math/2503.05348_pg3.pdf", "page": 1, "id": "2503.05348_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(t)=\\int_t^{\\infty} \\bar F(x)dx"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(x)=e^{|x|}(1+|x|)-1."} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+1}=G_e^{2t+1}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+2}=G_e^{2t+2}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t+1} \\cap E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}\\equiv \\partial T_e^{2t}"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "[n]\\setminus V(G_e^{2t})"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\partial G_e^{2t}| \\le (2k\\lambda+2)^t \\log n"} -{"pdf": "arxiv_math/2503.08984_pg35.pdf", "page": 1, "id": "2503.08984_pg35_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(2k\\lambda +2)^t \\log n = n^{o(1)}"} -{"pdf": "arxiv_math/2503.07624_pg4.pdf", "page": 1, "id": "2503.07624_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x = a r\\cos\\theta, y = b r \\sin\\theta"} -{"pdf": "arxiv_math/2503.07624_pg4.pdf", "page": 1, "id": "2503.07624_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_1 = \\frac{\\cos^2\\theta}{a^2} + \\frac{\\sin^2\\theta}{b^2}, \\qquad \\omega_2 = \\frac{\\cos^2\\theta}{b^2} + \\frac{\\sin^2\\theta}{a^2}, \\qquad \\omega_3 = \\sin2\\theta(\\frac{1}{a^2}-\\frac{1}{b^2})."} -{"pdf": "arxiv_math/2503.07624_pg4.pdf", "page": 1, "id": "2503.07624_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\bigl\\{ (x, y): \\tfrac{x^2}{a^2} + \\tfrac{y^2}{b^2} \\leq 1 \\bigr\\}."} -{"pdf": "arxiv_math/2503.04040_pg2.pdf", "page": 1, "id": "2503.04040_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial\\left(\\cdot\\right)"} -{"pdf": "arxiv_math/2503.04040_pg2.pdf", "page": 1, "id": "2503.04040_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A} \\succeq \\mathbf{0}"} -{"pdf": "arxiv_math/2503.04040_pg2.pdf", "page": 1, "id": "2503.04040_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{M \\times N}"} -{"pdf": 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"2503.04040_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\left(\\mathbf{x}\\right)"} -{"pdf": "arxiv_math/2503.04040_pg2.pdf", "page": 1, "id": "2503.04040_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{\\mathbf{x}}^2f\\left(\\mathbf{x}\\right)"} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix} & \\alpha'& \\beta'& \\gamma'&&& \\delta'\\\\ {\\rm angles:}& 67 & 68 & 71 & 74 & 5 & 14 & /78\\\\ \\times ~3: & 15& 16& 19& 22& 5 & 14 & /26 \\end{matrix}"} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat\\beta,~~ \\beta'=\\widehat\\gamma,~~ ~ {\\rm and}~~ \\delta'=\\widehat\\alpha~."} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat{\\beta}"} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat\\beta,\\quad \\beta'=\\widehat\\gamma,\\quad {\\rm and}\\quad \\delta'= \\widehat\\alpha~."} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\gamma=\\beta'"} -{"pdf": "arxiv_math/2503.08868_pg80.pdf", "page": 1, "id": "2503.08868_pg80_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix}& \\gamma & & & \\delta & \\alpha &\\beta\\\\ {\\rm angles:}& 16& 19 & 22& 31 & 40 & 41 & /78 \\\\ \\times ~3: & 16 & 19& 22& 5 & 14 & 15 & /26 \\end{matrix}"} -{"pdf": "arxiv_math/2503.08572_pg3.pdf", "page": 1, "id": "2503.08572_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{x}_\\ell(\\omega)"} -{"pdf": 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\\ddot{\\theta}_\\ell(t) + \\dot{\\theta}_\\ell(t) = \\omega_\\ell + \\text{Im}\\left( e^{-i\\theta_\\ell(t)} \\zeta_\\ell(t) \\right),"} -{"pdf": "arxiv_math/2503.08572_pg3.pdf", "page": 1, "id": "2503.08572_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "x_\\ell = e^{i\\theta_\\ell}"} -{"pdf": "arxiv_math/2503.08572_pg3.pdf", "page": 1, "id": "2503.08572_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_z(\\omega) = \\frac{1}{N} \\sum_{\\ell=1}^{N} |\\tilde{x}_\\ell(\\omega)|^2."} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^1 = 0.001\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^1"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\gamma} = 1/\\alpha_{\\gamma}"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\varepsilon}^i \\sim \\mathcal{N}(0, \\pmb{I}_l)"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^2 = 0.1\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^2"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Nt_{\\text{Filter}}=50"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_{\\text{Darcy}} = 0.0001\\epsilon_D"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": "2503.07617_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_i = 1/d_i, i=1, 2"} -{"pdf": "arxiv_math/2503.07617_pg27.pdf", "page": 1, "id": 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-{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(q^n_m(e^n_i, z_1, \\ldots, z_n), q^n_m(e^n_j, z_1, \\ldots, z_n)) \\in \\theta"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(b_1, \\ldots, b_n, 0, \\ldots, 0) \\in T(m)"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{T} \\simeq \\mathbf{N}"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lnot c=\\bigvee\\{a_k : k \\neq i\\}"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^n_i, e^n_j) \\in \\theta"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "B \\simeq B / \\theta(c,1) \\times B/\\theta(c,0)"} -{"pdf": "arxiv_math/2503.04430_pg9.pdf", "page": 1, "id": "2503.04430_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "a:=(b_1, \\ldots, b_n) \\in T(n)"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{n_j}=0,~~ j=0, 1, \\ldots, n"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_{n_j}, j=0, 1, 2,\\ldots,n"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{n_j}\\ne a_{n_{j+1}}, j=0, 1, 2, \\cdots, k"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p(q)=\\sum_{\\nu=0}^{n}q^{n_j}a_{n_j}"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{n_j}=a_{n_j}, ~~j=0, 1, \\ldots, n"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{n_0}\\le \\alpha_{n_1}\\le \\cdots\\le \\alpha_{n_k}"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{n_l}=\\alpha_{n_l}+\\beta_{n_l}i"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "p(q)=\\sum_{\\nu=0}^{k} q^{n_\\nu} a_{n_\\nu}"} -{"pdf": "arxiv_math/2503.08618_pg9.pdf", "page": 1, "id": "2503.08618_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": 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\\lambda_{-1}(\\B) \\\\ % \\lambda^{\\prime}_{-1}(0) &= -\\vect{v}_{-1}(\\B)^T \\C \\vect{v}_{-1}(\\B) =: - \\xi %"} -{"pdf": "arxiv_math/2503.09469_pg13.pdf", "page": 1, "id": "2503.09469_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{-1}(\\omega) \\le \\lambda_{-1}(0) + \\lambda^{\\prime}_{-1}(0) \\omega"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\circ T = \\sigma \\circ f"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(-1)^m(d_m - x_{k+m}) < 0"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{\\beta}\\setminus P_{\\beta}^{\\Z}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 211 211 \\cdots 211 \\bullet 2(1)^{\\infty}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 210 210 \\cdots 210 \\bullet 2 (1)^{\\infty}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{k+1} \\cdots x_{k+m-1} = d_1\\cdots d_{m-1}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d(l_{\\beta}, \\beta) = \\bullet2 (1)^{\\infty}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdots 210 210 210 \\cdots 210 \\bullet (2)^{\\infty}"} -{"pdf": "arxiv_math/2503.06597_pg18.pdf", "page": 1, "id": "2503.06597_pg18_math_009", "type": "math", "max_diffs": 0, 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null, "math": "\\mathrm{H}_c\\subset \\mathrm{G}_\\mathrm{R}"} -{"pdf": "arxiv_math/2503.05407_pg5.pdf", "page": 1, "id": "2503.05407_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(Q,{\\bf d})=(Q',{\\bf d}|_{Q'})\\#_i(Q'',{\\bf d}|_{Q''})."} -{"pdf": "arxiv_math/2503.05407_pg5.pdf", "page": 1, "id": "2503.05407_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota:\\mathbb{Z}(Q_{\\bf d})_0\\rightarrow\\mathbb{Z}(Q_{\\bf e})_0."} -{"pdf": "arxiv_math/2503.05407_pg5.pdf", "page": 1, "id": "2503.05407_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\iota:Q_{\\bf d}\\rightarrow Q_{\\bf e},\\;\\;\\; (i,k)\\mapsto(i,k+e_i-d_i)"} -{"pdf": "arxiv_math/2503.05407_pg5.pdf", "page": 1, "id": "2503.05407_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{\\widehat{\\bf d}}^{\\rm flag}(Q_{\\bf d})"} -{"pdf": "arxiv_math/2503.05407_pg5.pdf", "page": 1, "id": "2503.05407_pg5_math_004", "type": 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h_2t_2^{2m}h_2^{-1})=\\sum_{k,l\\in\\mathbb{Z}}c_{t_1^k,t_2^l}t_1^kh_1t_1^{2n}h_1^{-1}\\otimes t_2^lh_2t_2^{2m}h_2^{-1}."} -{"pdf": "arxiv_math/2503.09548_pg18.pdf", "page": 1, "id": "2503.09548_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{(t_1^n,t_2^m)}\\ne0"} -{"pdf": "arxiv_math/2503.09548_pg18.pdf", "page": 1, "id": "2503.09548_pg18_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{t_1}\\otimes\\mathbb{E}_{t_2}(a)(h_1t_1^{2n}h_1^{-1}\\otimes h_2t_2^{2m}h_2^{-1})\\in\\mathcal{A}"} -{"pdf": "arxiv_math/2503.09548_pg18.pdf", "page": 1, "id": "2503.09548_pg18_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "t_2^mh_2t_2^{2m}h_2^{-1}\\in\\langle v'\\rangle"} -{"pdf": "arxiv_math/2503.09548_pg18.pdf", "page": 1, "id": "2503.09548_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^nh_1t_1^nh_1^{-1}\\otimes t_2^mh_2t_2^mh_2^{-1}\\in\\mathcal{A}"} -{"pdf": "arxiv_math/2503.09548_pg18.pdf", "page": 1, "id": "2503.09548_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "t_1^nh_1^2t_1^nh_1^{-2}\\otimes t_2^mh_2^2t_2^mh_2^{-2}\\in\\mathcal{A}."} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V}:=\\{(v,q)\\in H^1_0(\\Omega)\\times L^2_0(\\Omega)\\colon -\\Delta v+\\nabla q \\in L^2(\\Omega), \\nabla\\cdot v\\in P\\}"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sin\\lambda\\omega=\\pm\\lambda\\sin\\omega"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Re}\\lambda\\in[\\frac12,4]"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(y,p)\\in L^2(\\Omega)\\times P'"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(u,p)\\in H^1(\\Omega)\\times L^2_0(\\Omega)"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "U(\\omega)=-(1-\\lambda)c_3U^{(1)}(\\omega) -(1+\\lambda)c_4U^{(2)}(\\omega)+ c_3U^{(3)}(\\omega)+c_4U^{(4)}(\\omega)."} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\min(1,\\xi)<\\lambda\\le0"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\in H^s(\\Omega),\\quad p\\in H^{s-1}(\\Omega) \\quad\\forall s<1+\\lambda,"} -{"pdf": "arxiv_math/2503.05437_pg6.pdf", "page": 1, "id": "2503.05437_pg6_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "P=\\{v\\in H^1(\\Omega)\\cap L^2_0(\\Omega): r^{-1} v\\in L^2(\\Omega)\\}"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X}"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\longrightarrow \\omega_{\\mathcal{A}^\\vee/X,\\tau_i}\\longrightarrow\\H(\\mathcal{A}/X)_{\\tau_i}\\longrightarrow \\textnormal{Lie}_{\\mathcal{A}/X,\\tau_i}\\longrightarrow 0,"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p\\mathcal{O}_E=\\mathfrak{q}\\mathfrak{q}^c"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle\\cdot,\\cdot\\rangle:\\H(\\mathcal{A}/X)_{\\tau_i}\\times \\H(\\mathcal{A}/X)_{\\tau_i^c}\\longrightarrow \\mathcal{O}_X"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(m_i,n_i)_{1\\le i\\le N}"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i^c}=\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}^\\perp"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}_Y(\\lambda)"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\tau_1,\\dots,\\tau_N\\}"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{L}_Y(\\lambda)\\cdot C)"} -{"pdf": "arxiv_math/2503.08119_pg4.pdf", "page": 1, "id": "2503.08119_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in\\Omega\\setminus\\tilde{\\Omega}"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(0,\\infty)\\ni s\\mapsto\\dfrac{\\Phi(x,s)}{s^{r-1}}"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_1,w_2\\in W^{1,p(x)}(\\Omega)"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla w_1(x)|,|\\nabla w_2(x)|>0"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\Phi(x,|\\nabla w_1(x)|)}{|w_1(x)|^{r-1}}=\\dfrac{\\Phi(x,\\lambda|\\nabla w_1(x)|)}{\\big (\\lambda|w_1(x)|\\big )^{r-1}}"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{|\\nabla w_2(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{\\lambda(x)|\\nabla w_1(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\lambda(x)=\\dfrac{w_2(x)}{w_1(x)}"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla w_2=\\lambda\\nabla w_1=0"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla w_1(x)|\\cdot |\\nabla w_2(x)|=\\nabla w_1(x)\\cdot\\nabla w_2(x)"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{w_1}{w_2},\\ \\dfrac{w_2}{w_1}\\in L^{\\infty}(\\Omega)"} -{"pdf": "arxiv_math/2503.06630_pg14.pdf", "page": 1, "id": "2503.06630_pg14_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\nabla w_1(x)}{w_1(x)}=\\dfrac{\\nabla w_2(x)}{w_2(x)}=0"} -{"pdf": "arxiv_math/2503.06581_pg8.pdf", "page": 1, "id": "2503.06581_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_f,\\mathcal{I}_p,\\mathcal{I}_s,\\mathcal{I}_E"} -{"pdf": "arxiv_math/2503.06581_pg8.pdf", "page": 1, "id": "2503.06581_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho\\in L^2(\\mathbb R^3)"} -{"pdf": "arxiv_math/2503.06581_pg8.pdf", "page": 1, "id": "2503.06581_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "J\\in\\left(H^1(\\mathbb R^3)\\right)^3"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "H^* = \\operatorname{Span}\\{f_1, f_2, \\cdots, f_n\\}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\beta_i\\}_{i\\leq n}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b_i\\}_{i\\leq n} \\subseteq X_{\\leq 1+\\sigma}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1 = r(y_1) >\\dfrac{1}{2}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in X^{\\ast\\ast}\\backslash Q(X)"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_2 = \\operatorname{Span}\\{b_1, b_2\\}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_n\\|\\sum_{i \\leq n}e_i\\| < \\infty"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "y_2 \\in X^*_{\\leq 1}\\cap E_1^{\\perp}"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda_1, \\lambda_2, \\cdots, \\lambda_n)\\in\\mathbb{C}^n"} -{"pdf": "arxiv_math/2503.06880_pg36.pdf", "page": 1, "id": "2503.06880_pg36_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_n = \\sup\\big\\{ r(y)\\,\\vert\\, y\\in X^{\\ast}_{= 1}\\cap E_n^{\\perp}\\big\\}"} -{"pdf": "arxiv_math/2503.06466_pg22.pdf", "page": 1, "id": "2503.06466_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha(e^{-1}) = (\\alpha(e))^{-1}"} -{"pdf": "arxiv_math/2503.06466_pg22.pdf", "page": 1, "id": "2503.06466_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha: D(\\Gamma) \\rightarrow G"} -{"pdf": "arxiv_math/2503.06466_pg22.pdf", "page": 1, "id": "2503.06466_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\Gamma^{\\alpha}) = V(\\Gamma) \\times G"} -{"pdf": "arxiv_math/2503.06466_pg22.pdf", "page": 1, "id": "2503.06466_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e = (u,v) \\in D(\\Gamma)"} -{"pdf": "arxiv_math/2503.08808_pg5.pdf", "page": 1, "id": "2503.08808_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\frac{1}{2\\pi i} \\right)^2 \\int_{c - i\\infty}^{c + i\\infty} \\int_{c - i\\infty}^{c + i\\infty} \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k} e^{z_1 x_1 + z_2 x_2} \\,dz_1 dz_2 \\notag \\\\ = \\frac{1}{\\Gamma(k) b^{k-1}} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right)."} -{"pdf": "arxiv_math/2503.08808_pg5.pdf", "page": 1, "id": "2503.08808_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b = \\frac{\\sqrt{\\rho}}{\\sigma(1-\\rho)}"} -{"pdf": "arxiv_math/2503.08808_pg5.pdf", "page": 1, "id": "2503.08808_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{\\Gamma(k) b^{k-1}} \\int_0^{\\infty} \\int_0^{\\infty} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right) e^{-z_1 x_1 - z_2 x_2} \\,dx_1 dx_2 \\notag \\\\ = \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k}."} -{"pdf": "arxiv_math/2503.08808_pg5.pdf", "page": 1, "id": "2503.08808_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = \\frac{1}{\\sigma(1-\\rho)}"} -{"pdf": "arxiv_math/2503.07729_pg8.pdf", "page": 1, "id": "2503.07729_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T \\Delta E \\to \\infty"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "id^*\\nabla^\\mathrm{flat}=\\nabla^\\mathrm{flat}"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\nabla}\\in[\\overline{\\nabla}]"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y\\in\\mathfrak{L}(\\mathcal{F}^M)"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "id:\\mathbb{R}^3\\rightarrow\\mathbb{R}^3"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(F_*\\hat{\\nabla}^M)_XY=W"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(F^*\\hat{\\nabla}^N)_XY=Z"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y \\in\\mathfrak{X}(M)"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z(p):=d\\Phi_p^{-1}\\left(\\hat{\\nabla}^N_{\\Phi_*X}\\Phi_*Y\\right)_{\\Phi(p)}"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "W(q):=d\\Phi_p\\left(\\hat{\\nabla}_{\\Phi^*X}\\Phi^*Y\\right)_{p}"} -{"pdf": "arxiv_math/2503.06344_pg15.pdf", "page": 1, "id": "2503.06344_pg15_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "V\\in\\mathfrak{X}(\\mathcal{F}^M)"} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "p(\\tau, \\tau \\sqcup \\{ij\\}) = \\left\\{ \\begin{array}{cl} \\displaystyle \\frac{\\alpha_{ij} w_{i, ij}}{\\sum_{k=1}^d \\alpha_{ik} w_{i, ik}} \\alpha_i , & \\text{if } i \\in \\partial \\tau, \\\\[15pt] \\alpha_{ij}, & \\text{if } i \\notin \\partial \\tau \\text{ and } ij \\notin \\tau, \\end{array} \\right."} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i = \\alpha_{i1} + \\ldots + \\alpha_{id} \\in [0, 1]"} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{6} \\left(\\frac{w_1 + w_2}{2}\\right)^3 = \\frac{1}{12} \\left(\\frac{w_1 + w_2}{2}\\right)^3 + \\frac{1}{48} \\sum_{1 \\leq i, j \\leq 2} w_i w_j \\left(\\frac{w_1 + w_2}{2}\\right)."} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\prod_{i \\in \\overset{\\circ}{\\tau}} \\frac{\\alpha_i}{\\sum_{j=1}^d \\alpha_{ij} w_{i, ij}}}=\\left(\\frac{d}{\\sum_{j=1}^dw_j}\\right)^{|\\tau|- |\\partial \\tau|},"} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "j \\in \\{1, \\ldots, d\\}"} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{n}{k_1,\\cdots,k_d}"} -{"pdf": "arxiv_math/2503.08172_pg25.pdf", "page": 1, "id": "2503.08172_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^{(j)} \\sqcup \\bigsqcup_{l\\neq j} \\sigma^{(l)}"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon} \\rightarrow e(F,\\rho)"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I_i = \\frac 43 \\int \\rho_i^{\\frac 32} dt"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon}"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e \\geq e_{\\gamma_i, \\rho_i}"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L_\\varepsilon"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e(0,0,F(t),t) = 0, \\forall t \\in [0,1]"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(t)= \\begin{cases} \\frac{1}{a^2(3-\\sqrt{8a})^2}, & \\text{if } t \\leq 2a \\\\ \\frac{2}{at(3-\\sqrt{8a})^2}, & \\text{if } 2a \\leq t \\leq 1. \\end{cases}"} -{"pdf": "arxiv_math/2503.09486_pg4.pdf", "page": 1, "id": "2503.09486_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "e_{\\gamma, \\rho}(x_1,t_1,x_2,t_2) = - \\infty"} -{"pdf": "arxiv_math/2503.09231_pg4.pdf", "page": 1, "id": "2503.09231_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "G(U)=\\begin{pmatrix}g_{1}(U) \\\\ g_{2}(U) \\\\ g_{3}(U) \\end{pmatrix}=\\begin{pmatrix} m P(H(M)-a_{1} \\lambda M ) \\\\ -m a_{2} \\lambda M S \\\\m_{s}S(1-M)-\\eta M P \\end{pmatrix},\\quad U_{R}=(0,S_{R},M_{R})\\quad \\text{and}\\quad U_{0}=(P_0,S_{0},M_{0})."} -{"pdf": "arxiv_math/2503.09231_pg4.pdf", "page": 1, "id": "2503.09231_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{L^1([0,T]; X)} = \\int_0^T \\|f(t)\\|_X dt."} -{"pdf": "arxiv_math/2503.09231_pg4.pdf", "page": 1, "id": "2503.09231_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C(\\Omega_R)} = \\sup_{x \\in \\Omega_R} |f(x)| ."} -{"pdf": "arxiv_math/2503.09231_pg4.pdf", "page": 1, "id": "2503.09231_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{2,1}(\\Omega_R) = \\{u \\in L^1(\\Omega_R) : D^\\alpha u \\in L^1(\\Omega_R) \\quad\\text{for all}\\quad |\\alpha| \\leq 2\\}"} -{"pdf": "arxiv_math/2503.09231_pg4.pdf", "page": 1, "id": "2503.09231_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C([0,T]; X)} = \\sup_{t \\in [0,T]} \\|f(t)\\|_X ."} -{"pdf": 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P(t,x)\\right),\\operatorname{div}\\left(\\vec{\\alpha_{s}}(t,x)\\left(\\int_{ \\Omega_{R}}\\gamma_{s}(x-y)S(t,y)d y \\right)S(t,x)\\right),-D\\Delta M\\right]"} -{"pdf": "arxiv_math/2503.03873_pg5.pdf", "page": 1, "id": "2503.03873_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}} \\in L^1(H^{\\perp})"} -{"pdf": "arxiv_math/2503.03873_pg5.pdf", "page": 1, "id": "2503.03873_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{\\perp} \\cong \\widehat{A/H}"} -{"pdf": "arxiv_math/2503.03873_pg5.pdf", "page": 1, "id": "2503.03873_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_{A/H}(\\phi^H) = \\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}}"} -{"pdf": "arxiv_math/2503.03873_pg5.pdf", "page": 1, "id": "2503.03873_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A= (\\Z/N\\Z)^{d_1} \\times \\R^{d_2}"} -{"pdf": "arxiv_math/2503.03873_pg5.pdf", 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"2503.06939_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma = \\left[ \\frac{1}{N} \\sum_{k=1}^N \\, (z_k-\\nu)^2 \\right]^{\\frac{1}{2}} \\; , \\quad \\nu = \\frac{1}{N} \\sum_{k=1}^N z_k \\; ."} -{"pdf": "arxiv_math/2503.06939_pg20.pdf", "page": 1, "id": "2503.06939_pg20_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\wp_\\Upsilon(\\upsilon,t)"} -{"pdf": "arxiv_math/2503.07916_pg19.pdf", "page": 1, "id": "2503.07916_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{\\varphi }=\\pi /100"} -{"pdf": "arxiv_math/2503.07916_pg19.pdf", "page": 1, "id": "2503.07916_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon ,\\lambda \\right)"} -{"pdf": "arxiv_math/2503.07916_pg19.pdf", "page": 1, "id": "2503.07916_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{0}^{\\xi _{0}}\\left( \\mathbf{x},\\mathbf{x}_{0}\\right)"} -{"pdf": 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"type": "math", "max_diffs": 0, "checked": null, "math": "V \\in L^1((0,T); L^\\infty(\\R^n))"} -{"pdf": "arxiv_math/2503.06205_pg1.pdf", "page": 1, "id": "2503.06205_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\in L^2(\\R^n) \\mapsto u \\in C([0, T]; L^2(\\R^n))"} -{"pdf": "arxiv_math/2503.06205_pg1.pdf", "page": 1, "id": "2503.06205_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ u(t, \\centerdot) : t \\in [0, T] \\}"} -{"pdf": "arxiv_math/2503.06205_pg1.pdf", "page": 1, "id": "2503.06205_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{U}_T : f \\in L^2(\\R^n) \\mapsto u(T, \\centerdot) \\in L^2(\\R^n),"} -{"pdf": "arxiv_math/2503.06205_pg1.pdf", "page": 1, "id": "2503.06205_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "V \\in L^1((0, T); L^\\infty (\\R^n))"} -{"pdf": "arxiv_math/2503.04047_pg8.pdf", "page": 1, "id": "2503.04047_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "C_T = \\frac{\\partial \\mathbb{E}[E(T)] }{\\partial T}"} -{"pdf": "arxiv_math/2503.04047_pg8.pdf", "page": 1, "id": "2503.04047_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "t^* = \\underset{t \\geq M}{\\arg\\max} \\, \\hat{C}(t)"} -{"pdf": "arxiv_math/2503.04047_pg8.pdf", "page": 1, "id": "2503.04047_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^2(\\{f(x_{t-M+1}), \\cdots, f(x_{t}) \\})"} -{"pdf": "arxiv_math/2503.05323_pg16.pdf", "page": 1, "id": "2503.05323_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-c_0 n^{-\\tilde{\\epsilon}}"} -{"pdf": "arxiv_math/2503.05323_pg16.pdf", "page": 1, "id": "2503.05323_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i \\neq j \\in [n] : |i-j|\\leq 2}\\frac{1}{(|\\lambda_{j}-\\lambda_{i}| + n^{-1.5-\\tilde{\\epsilon}})^2} \\leq 8c_0 n^{3.5 + 2\\tilde{\\epsilon}}."} -{"pdf": 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\\frac{1}{(\\lambda_j-\\lambda_i)^2} \\leq 32c_0 n^{3.5 + 2\\tilde{\\epsilon}} + \\pi^2 n^{17/5+2\\tilde{\\epsilon}/5} \\leq n^{3.5 + 3\\tilde{\\epsilon}},"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "w\\colon E(H)\\to \\mathbb{R}^+"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "c \\colon V(G)\\to \\{1,\\dots,k\\}"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n w(v_{i-1}v_i)"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "P=(v_0,v_1,\\dots,v_n)"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w\\colon E(H) \\to \\mathbb{N}"} -{"pdf": "arxiv_math/2503.07448_pg3.pdf", "page": 1, "id": "2503.07448_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi\\colon V(G)\\to V(H)"} -{"pdf": "arxiv_math/2503.09299_pg16.pdf", "page": 1, "id": "2503.09299_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "W_1(x, y) = \\sqrt{|x - y|}"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{C}\\otimes \\bm{u})"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\cdot \\nabla)\\bm{C}"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{u}\\otimes \\bm{u})=(\\bm{u}\\cdot \\nabla)\\bm{u}+(\\nabla\\cdot\\bm{u})\\bm{u}=(\\bm{u}\\cdot \\nabla)\\bm{u}"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\otimes\\bm{v})_{ij}=u_i v_j, ~(\\bm{C}\\otimes\\bm{u})_{ijk}= C_{ij}u_k"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\bm{u}\\cdot \\nabla)\\bm{u}"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{u}\\otimes \\bm{u})"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "F:=\\partial K^+\\bigcap \\partial K^-"} -{"pdf": "arxiv_math/2503.08127_pg2.pdf", "page": 1, "id": "2503.08127_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla\\cdot(\\bm{C}\\otimes \\bm{u})=(\\bm{u}\\cdot \\nabla)\\bm{C}"} -{"pdf": "arxiv_math/2503.04465_pg3.pdf", "page": 1, "id": "2503.04465_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\in L^2((0,T)\\times G_0)"} -{"pdf": "arxiv_math/2503.04465_pg3.pdf", "page": 1, "id": "2503.04465_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi\\mapsto y(T,\\cdot;\\xi;y_0;u)"} -{"pdf": "arxiv_math/2503.04465_pg3.pdf", "page": 1, "id": "2503.04465_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}\\left[\\omega\\;:\\; y(T,\\cdot;\\alpha(\\omega);y_0;u)=0\\right]=0."} -{"pdf": "arxiv_math/2503.04465_pg3.pdf", "page": 1, "id": "2503.04465_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "y_0\\in L^2(G)\\setminus\\{0\\}"} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S = \\begin{pmatrix} 1/4 & 0 & 0 \\\\ 0 & 1/2 & 0 \\\\ 0 & 0 & 1/4 \\end{pmatrix}, \\qquad \\mathcal Q = \\begin{pmatrix} -2 & 2 & 0 \\\\ 1 & -2 & 1 \\\\ 0 & 2 & -2 \\end{pmatrix},"} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "a_0=\\frac1{2+\\omega L}"} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left< \\cdot, \\cdot \\right>_S"} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu = \\Omega\\left(\\frac{\\omega}{1 + (\\omega L)^2}\\right)."} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu \\propto \\omega^{-1} L^{-2}"} -{"pdf": "arxiv_math/2503.04238_pg26.pdf", "page": 1, "id": "2503.04238_pg26_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left_S = x^\\top S y"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1} = (I_{i} \\setminus \\{i\\})\\cup\\{j\\}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1}\\setminus I_{i} = \\{j\\}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_f = (I_1, \\dots, I_n)"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{f}^{-1}_{\\mathcal{I}}(i) = j"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{a \\in [n] \\mid a <_{i} \\bar{f}(a)\\}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I = \\{i \\in [n] \\mid f(i) = i+n\\}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1} = I_{i} = (I_{i}\\setminus\\{i\\})\\cup\\{i\\}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{i-1}\\setminus I_{i}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\mapsto \\mathcal{I}_{f}"} -{"pdf": "arxiv_math/2503.04923_pg9.pdf", "page": 1, "id": "2503.04923_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\{i \\in [n] : f(i) > n\\}| = k"} -{"pdf": "arxiv_math/2503.09195_pg13.pdf", "page": 1, "id": "2503.09195_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{v_0,+,j_1} \\frac{1}{n_0+2}. \\end{cases}"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\left( \\sigma(W) \\geq \\frac{1}{k} \\right)>0"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P(C^{\\leq k,w})=P(C^{k,w})"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "P(X_1 = X_2 = \\dots =X_{n_0} = 1) > 0."} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\left( N=n_0 \\text{ for the sequence generated by }\\{X_i\\}_{i> n_0}\\right) > 0,"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{A(n)}{n} \\geq \\begin{cases} 1 & \\text{if } n \\leq n_0 \\\\ \\frac{1}{2} & \\text{if } n_0 < n \\leq 2n_0 \\\\ \\frac{1}{k} & \\text{if } n > 2n_0 \\end{cases},"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(s) = \\inf_{n}{\\frac{A(n)}{n}},"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{n=1}^{\\infty}{P(N=n)} = 1"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "d(\\{W_n\\}_{n \\in \\mathbf{N}}) > \\frac{1}{k}"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "A(n) = |\\{i: s_i \\leq n\\}|"} -{"pdf": "arxiv_math/2503.05177_pg10.pdf", "page": 1, "id": "2503.05177_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "N = \\min\\left\\{n:\\forall m \\geq n, \\quad \\frac{A(m)}{m} > \\frac{1}{k}\\right\\}."} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "n-|\\mu|-\\mu_1-(|\\eta|-1)>i"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{maj}(\\sigma)=2+1+1+1=5"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_n(\\sigma)=\\sigma'"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma:\\nu\\rightarrow \\N"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "n\\geq |\\mu|+\\mu_1+|\\eta|+i"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_n:M^{(n)}\\rightarrow M^{(n+1)}"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{maj}(\\sigma):=\\sum_{u\\in \\text{Des}(\\sigma)}(\\text{leg}(u)+1),"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle [q^it^j]\\tilde{H}_{\\mu[n]},h_{\\eta[n]}\\rangle"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{st}(\\sigma)(u),\\text{st}(\\sigma)(w),\\text{st}(\\sigma)(v)"} -{"pdf": "arxiv_math/2503.04950_pg23.pdf", "page": 1, "id": "2503.04950_pg23_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma:\\nu\\rightarrow \\{1,\\cdots,|\\nu|\\}"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma' \\colon X' \\longrightarrow X"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda'_{j}-\\lambda_{j}=\\nu(T'_{j},V)-\\nu(T_{j},V)=0"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{j} \\in \\{\\theta_{j}\\}"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_{j}=(c \\|\\{\\theta_{j}\\}\\|+\\frac{\\epsilon}{2c_{V}})^{-1}\\frac{\\lambda'_{j}-\\lambda_{j}}{2},"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nu(T'_{j},V) = \\nu(R'_{j},V')=\\nu(R_{j},V') = \\nu(T_{j},V)"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathscr{S} \\cap \\mathscr{B}"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "T'_{1}, \\dots , T'_{m}"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{j} = (\\sigma')^{*}T_{j}"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "[V] \\wedge \\omega^{n-m} \\neq 0"} -{"pdf": "arxiv_math/2503.09233_pg16.pdf", "page": 1, "id": "2503.09233_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{1}', \\dots, R'_{m}"} -{"pdf": "arxiv_math/2503.04555_pg2.pdf", "page": 1, "id": "2503.04555_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot \\ : M\\times R\\rightarrow M"} -{"pdf": "arxiv_math/2503.04555_pg2.pdf", "page": 1, "id": "2503.04555_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} a(b+c) & = ab + ac \\\\ (a+b)c & = ac + bc \\end{split}"} -{"pdf": "arxiv_math/2503.04041_pg20.pdf", "page": 1, "id": "2503.04041_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\epsilon=2.22\\times10^{-16}"} -{"pdf": "arxiv_math/2503.04041_pg20.pdf", "page": 1, "id": "2503.04041_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\kappa(A)=\\frac{\\sigma_{\\max}(A)}{\\sigma_{\\text{min}}(A)}"} -{"pdf": "arxiv_math/2503.04041_pg20.pdf", "page": 1, "id": "2503.04041_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "L=\\left(\\begin{array}{cccc} 3 & 1 & & \\\\ 1 & \\ddots & \\ddots & \\\\ & \\ddots & \\ddots & 1 \\\\ & & 1 & 3 \\end{array}\\right),"} -{"pdf": "arxiv_math/2503.07260_pg6.pdf", "page": 1, "id": "2503.07260_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0=3H^{2}-\\frac{3}{4}\\dot{\\beta}_{+}^{2}-\\frac{1}{4}\\dot{\\beta}_{-}^{2}% -3\\dot{\\phi}\\left( H+\\dot{\\beta}_{+}\\right) \\left( 2H-\\dot{\\beta}_{+}% -\\dot{\\beta}_{-}\\right) \\left( 2H-\\dot{\\beta}_{-}+\\dot{\\beta}_{+}\\right) -2\\Lambda, \\label{ai.08}%"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d^p= d^q \\Rightarrow p=q"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d_p= d_q \\Rightarrow p=q"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I^+(p)=I^+(q)\\Rightarrow p=q"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell(\\gamma\\vert_{[\\delta,1]})> m"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "d^r(p)= d^r(q) \\Rightarrow p=q"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I^-(p)=I^-(q)\\Rightarrow p=q"} -{"pdf": "arxiv_math/2503.04382_pg8.pdf", "page": 1, "id": "2503.04382_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d_p(r)\\ge \\ell(\\gamma\\vert_{[\\delta,1]})> m"} -{"pdf": "arxiv_math/2503.09157_pg22.pdf", "page": 1, "id": "2503.09157_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_1,I_2 \\in \\mathcal{C}^0([0,T], \\R)"} -{"pdf": "arxiv_math/2503.09157_pg22.pdf", "page": 1, "id": "2503.09157_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}^0([0,T],\\R)"} -{"pdf": "arxiv_math/2503.09157_pg22.pdf", "page": 1, "id": "2503.09157_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "2T\\|\\partial_Ir\\|_\\infty r_M<1"} -{"pdf": "arxiv_math/2503.09157_pg22.pdf", "page": 1, "id": "2503.09157_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned}\\|\\mu(n_1)-\\mu(n_2)\\|_X&\\leq 2T\\|\\partial_Ir\\|_\\infty \\|I_1-I_2\\|_\\infty. \\\\ &\\leq 2T\\|\\partial_Ir\\|_\\infty r_M\\|n_1-n_2\\|_X \\end{aligned}"} -{"pdf": "arxiv_math/2503.09157_pg22.pdf", "page": 1, "id": "2503.09157_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r_1(t,x):=r(x,I_1(t))"} -{"pdf": 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"type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} x_1=x_3 \\\\ x_2=x_4+v_{11}x_3x_4+v_{02}x_4^2+v_{21}x_3^2x_4+v_{12}x_3x_4^2+v_{03}x_4^3+O(\\|x\\|^4) \\end{cases}"} -{"pdf": "arxiv_math/2503.06185_pg6.pdf", "page": 1, "id": "2503.06185_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{k}^{RBB}\\in[\\alpha_{k}^{BB1}, \\ \\alpha_{k}^{BB2}]"} -{"pdf": "arxiv_math/2503.06185_pg6.pdf", "page": 1, "id": "2503.06185_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{k}^{RBB}=\\mathop{\\text{argmin}}_{\\alpha\\in\\mathbb{R}} \\Big\\{\\Vert\\alpha\\Delta y^{k-1}-\\Delta\\Psi^{k-1}\\Vert_{2}^{2} + \\tau_{k}\\Vert\\alpha \\sqrt{H_{k}}\\Delta y^{k-1}- \\sqrt{H_{k}}\\Delta\\Psi^{k-1}\\Vert_{2}^{2}\\Big\\},"} -{"pdf": "arxiv_math/2503.06185_pg6.pdf", "page": 1, "id": "2503.06185_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{k}\\in[0,\\infty)"} -{"pdf": 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P_{k}\\oplus P_{k-1}\\oplus \\cdots \\oplus P_{1}\\quad \\text{with}\\quad N_{k}=N_{k+1}\\oplus P_{k+1},"} -{"pdf": "arxiv_math/2503.07623_pg2.pdf", "page": 1, "id": "2503.07623_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F^{-1}|S|+F(U)+F(\\mathcal{T})+\\|divC\\|_{HS(V)} \\leq K_0,"} -{"pdf": "arxiv_math/2503.07623_pg2.pdf", "page": 1, "id": "2503.07623_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "C(X,Y,Z):=C_{ijk}X^iY^jZ^k=\\frac{1}{4}\\frac{\\partial^3F^2(x,y)}{\\partial y^i\\partial y^j\\partial y^k}X^iY^jZ^k,"} -{"pdf": "arxiv_math/2503.07623_pg2.pdf", "page": 1, "id": "2503.07623_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "G^i=\\frac12\\Gamma^i_{jk}y^jy^k"} -{"pdf": "arxiv_math/2503.07623_pg2.pdf", "page": 1, "id": "2503.07623_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X,Y,Z\\in TM\\setminus\\{0\\}"} -{"pdf": "arxiv_math/2503.07623_pg2.pdf", "page": 1, "id": 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"id": "2503.04590_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{V}(u) \\leq -K.\\big(V(u)\\big)^p \\quad \\forall u\\in U\\setminus \\{u^*\\},"} -{"pdf": "arxiv_math/2503.04590_pg7.pdf", "page": 1, "id": "2503.04590_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)=0"} -{"pdf": "arxiv_math/2503.05643_pg19.pdf", "page": 1, "id": "2503.05643_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "contains at most two 1s in each row and column. If a row (or column) contains two 1s, each of them appears in different adjacent blocks. All other entries in the matrix are zeros. In the corresponding graphical invariants, the symbol"} -{"pdf": "arxiv_math/2503.07166_pg4.pdf", "page": 1, "id": "2503.07166_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^+)^2 - (e^-)^2 = e^+ + e^-"} -{"pdf": "arxiv_math/2503.04415_pg2.pdf", "page": 1, "id": "2503.04415_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma\\in[0,\\frac{1-\\gamma}{2})"} -{"pdf": "arxiv_math/2503.04415_pg2.pdf", "page": 1, "id": "2503.04415_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "N = \\left\\lfloor \\frac{1}{\\gamma} \\right\\rfloor"} -{"pdf": "arxiv_math/2503.06000_pg17.pdf", "page": 1, "id": "2503.06000_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[1,173] \\times [1,60] \\times [95,200]"} -{"pdf": "arxiv_math/2503.08475_pg18.pdf", "page": 1, "id": "2503.08475_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[c,d]_\\rho+[c+1,d+1]_\\rho\\mapsto [c,d+1]_\\rho+[c+1,d]_\\rho"} -{"pdf": "arxiv_math/2503.08475_pg18.pdf", "page": 1, "id": "2503.08475_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_1\\times\\ldots \\times \\pi_k"} -{"pdf": "arxiv_math/2503.08475_pg18.pdf", "page": 1, "id": "2503.08475_pg18_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "[a,b]_\\rho+\\ldots+[a+o(\\rho)-1,b+o(\\rho)-1]_\\rho"} -{"pdf": "arxiv_math/2503.08475_pg18.pdf", "page": 1, "id": "2503.08475_pg18_math_027", "type": "math", "max_diffs": 0, "checked": null, "math": "l([1,b']_\\rho)\\ge l([c,d+1]_\\rho)"} -{"pdf": "arxiv_math/2503.08475_pg18.pdf", "page": 1, "id": "2503.08475_pg18_math_028", "type": "math", "max_diffs": 0, "checked": null, "math": "l([1,b']_\\rho)\\le l([c,d+1]_\\rho)"} -{"pdf": "arxiv_math/2503.08924_pg2.pdf", "page": 1, "id": "2503.08924_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overbrace{ \\begin{pmatrix} \\begin{matrix}a_m&\\ldots&a_0\\\\ 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"2503.08924_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hbox{\\bf detpol}(\\Delta)=\\sum_{k=0}^{n-m}\\det(\\Delta_k)x^{n-m-k}"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{0,\\infty}(\\omega)=L^{\\infty}(\\omega)"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "h|_{T}=h_{T} :={\\rm diam}(T)"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{0}(\\omega)=L^{2}(\\omega)"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{m,p,\\omega}"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H_{D}^{1}(\\Omega)=\\{v\\in H^{1}(\\Omega): v|_{\\Gamma_{D}}=0\\}."} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V_{h}\\subset H^{1}(\\Omega)"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\cdot,\\cdot)_{\\omega}"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{m,2}(\\omega)=H^{m}(\\omega)"} -{"pdf": "arxiv_math/2503.06555_pg3.pdf", "page": 1, "id": "2503.06555_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{0,\\infty,\\omega}"} -{"pdf": "arxiv_math/2503.06889_pg12.pdf", "page": 1, "id": "2503.06889_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0.3, 0.5, 0.5, 0.5, 0.5"} -{"pdf": 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null, "math": "\\mathcal{A}_{\\varepsilon}(\\Omega;\\mathbb{R}^{m})=\\{u:\\mathbb{R}^{d} \\rightarrow \\mathbb{R}^{m}: u\\ \\text{constant on}\\ \\alpha +[0,\\varepsilon)^{d}\\ \\text{for any}\\ \\alpha \\in \\Omega_{\\varepsilon}\\}."} -{"pdf": "arxiv_math/2503.06723_pg3.pdf", "page": 1, "id": "2503.06723_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in \\mathcal{A}_{\\varepsilon}(\\Omega;\\mathbb{R}^{m})"} -{"pdf": "arxiv_math/2503.06723_pg3.pdf", "page": 1, "id": "2503.06723_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega \\subset \\mathbb{R}^{d}"} -{"pdf": "arxiv_math/2503.06723_pg3.pdf", "page": 1, "id": "2503.06723_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi \\in \\mathbb{Z}^{d}"} -{"pdf": "arxiv_math/2503.06723_pg3.pdf", "page": 1, "id": "2503.06723_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon\\mathbb{Z}^{d}"} -{"pdf": 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\\overset{\\circ}{g}"} -{"pdf": "arxiv_math/2503.09234_pg7.pdf", "page": 1, "id": "2503.09234_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g= u^{\\frac{4}{n-4}} g_{\\rm euc}"} -{"pdf": "arxiv_math/2503.09234_pg7.pdf", "page": 1, "id": "2503.09234_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "u:\\R^n \\backslash \\{ 0 \\} \\rightarrow (0,\\infty)"} -{"pdf": "arxiv_math/2503.08105_pg5.pdf", "page": 1, "id": "2503.08105_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f(z) = h(z) + \\overline{g(z)},"} -{"pdf": "arxiv_math/2503.08105_pg5.pdf", "page": 1, "id": "2503.08105_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f^2=(h + \\overline{g})^2 = h^2 +\\overline{g}^2+ 2h\\overline{g}"} -{"pdf": "arxiv_math/2503.08105_pg5.pdf", "page": 1, "id": "2503.08105_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial h}{\\partial z}= 0, ~~ ~~ \\frac{\\partial 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"id": "2503.07361_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{v_1},\\dots,S_{v_h}"} -{"pdf": "arxiv_math/2503.07361_pg12.pdf", "page": 1, "id": "2503.07361_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "y_k-y_j> k-1-j \\geq 2"} -{"pdf": "arxiv_math/2503.07361_pg12.pdf", "page": 1, "id": "2503.07361_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "G=(V,E_s \\cup E_\\ell)"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Im}(B_t^p(u,u)) \\leq \\frac{M}{\\nu}\\mathrm{Re}(B_t^p(u,u))"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "U_p : (s,\\infty) \\rightarrow L^2_\\omega(\\mathbb{R}^n)"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(B^p_t)_{{t \\in \\mathbb{R}}}"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{H}^\\star v=0"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "L^2((s,\\mathfrak{T});H^1_\\omega(\\mathbb{R}^n))"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Re}(B_t^p(\\cdot,\\cdot))"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "U_p(t):=\\Gamma_p(t,s)f"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_x U_p(t) \\in L^2_\\omega(\\mathbb{R}^n)"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\in \\mathbb{R}^{2n}"} -{"pdf": "arxiv_math/2503.07569_pg19.pdf", "page": 1, "id": "2503.07569_pg19_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta \\in \\mathcal{D}(\\mathbb{R})"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i=\\alpha_{i+1}"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{i_1},s_{i_2},\\ldots,s_{i_{\\ell(\\sigma)}}"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\sigma \\pi)_i = \\sigma_{\\pi_i}"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "s_1, s_2, \\ldots, s_{n-1}"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "s_{i_1}s_{i_2}\\cdots s_{i_{\\ell(\\sigma)}}"} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\alpha}^{\\sigma}(\\mathbf{x}; q, t) = E_{\\alpha}^{\\sigma s_i}(\\mathbf{x}; q, t)."} -{"pdf": "arxiv_math/2503.06051_pg2.pdf", "page": 1, "id": "2503.06051_pg2_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_{i+1} = \\sigma_i \\pm 1"} -{"pdf": "arxiv_math/2503.07337_pg40.pdf", "page": 1, "id": "2503.07337_pg40_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall x\\in (u^{-1}(0))^{\\alpha_0}, \\qquad |\\nabla u(x)|\\ge \\frac{k}{2}."} -{"pdf": "arxiv_math/2503.07337_pg40.pdf", "page": 1, "id": "2503.07337_pg40_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(K)^t =\\Set{p\\in \\R^n\\;|\\; d(p,K)\\le t}"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dim N(S(\\lambda))=\\dim N(T(\\lambda))<\\infty"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i_{\\lambda}\\bigl(N(S(\\lambda))\\bigr)\\supseteq N(T(\\lambda))"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "L^{\\infty}[-\\tau,\\tau]"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\colon X_-\\oplus X_+\\to \\R^d"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R(T(\\lambda))+ V= L^{\\infty}(\\R)"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "S^\\pm(\\lambda):X_{\\pm}\\to Y_{\\pm}"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "p(u):=u|_{[-\\tau,\\tau]}"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "[-\\tau,\\tau]=[-\\tau,0]\\cup [0,\\tau]"} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "D(S(\\lambda)) := \\set{u\\in W^{1,\\infty}[-\\tau,\\tau]\\mid u(-\\tau)\\in N(P_\\lambda^-(-\\tau)),\\, u(\\tau)\\in R(P_\\lambda^+(\\tau))}."} -{"pdf": "arxiv_math/2503.07221_pg16.pdf", "page": 1, "id": "2503.07221_pg16_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "Ju:=(u_-,u_+) \\text{ and } J_{\\lambda}v:=(v_-,v_+)"} -{"pdf": "arxiv_math/2503.08820_pg1.pdf", "page": 1, "id": "2503.08820_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha\\in\\mathbb{Z}^n_{\\ge 0}"} -{"pdf": "arxiv_math/2503.05688_pg1.pdf", "page": 1, "id": "2503.05688_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\mathcal{H}}"} -{"pdf": "arxiv_math/2503.08848_pg1.pdf", "page": 1, "id": "2503.08848_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}^{\\text {Airy }}"} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix}\\varphi^{\\xi,+}_{n}\\\\\\varphi^{\\xi,-}_{n}\\end{bmatrix}=e^{2\\pi in\\theta}\\begin{bmatrix}\\check{\\phi}^{+}(\\xi+n\\Phi)\\\\\\check{\\phi}^{-}(\\xi+n\\Phi)\\end{bmatrix}=\\frac1{\\sqrt2}e^{2\\pi in\\theta}\\begin{bmatrix}\\check\\psi^+(\\xi+n\\Phi)+i\\check\\psi^-(\\xi+n\\Phi)\\\\i\\check\\psi^+(\\xi+n\\Phi)+\\check\\psi^-(\\xi+n\\Phi)\\end{bmatrix}."} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_1,\\lambda_2,\\Phi,\\theta}"} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi=\\left[\\psi^{+},\\psi^{-}\\right]^\\top"} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_{1},\\lambda_{2},\\Phi,\\theta}\\psi=z\\psi"} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi=\\varphi^\\xi=\\left[\\varphi^{\\xi,+},\\varphi^{\\xi,-}\\right]^\\top"} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in\\Sigma_{\\lambda_1,\\lambda_2,\\Phi}."} -{"pdf": "arxiv_math/2503.06710_pg5.pdf", "page": 1, "id": "2503.06710_pg5_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sin2\\pi(\\theta+n\\Phi)|<\\exp(-|n|^{\\frac{1}{2\\tau}})"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ", because it matters less if the sequence starts at the beginning of a sequence or not if there are many"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "being a sequence of tokens. However, as discussed in the main text, the"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-tuple of tokens with"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "prior tokens to conditional on. Therefore, we focusing on reducing the bias for small"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": ". Then, the entropy of the distribution can be estimated naively as \\begin{equation} \\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}}, \\end{equation} where the summation runs over all possible combination of tokens"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "starts at the beginning of a sentence, but LLMs model distributions conditioned on BOS token. To mitigate this issue, we use"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "term suffers from an additional bias---we cannot guarantee that"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}},"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is small, we can iterate over the dataset and construct a histogram for the"} -{"pdf": "arxiv_math/2503.04725_pg19.pdf", "page": 1, "id": "2503.04725_pg19_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "and the total number of samples with"} -{"pdf": "arxiv_math/2503.07281_pg12.pdf", "page": 1, "id": "2503.07281_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "% H^1_{\\Theta}=K^1_{\\Theta}\\oplus \\Theta H^1"} -{"pdf": "arxiv_math/2503.07281_pg12.pdf", "page": 1, "id": "2503.07281_pg12_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1_\\Theta/(K^1_{\\Theta}\\oplus \\Theta H^1 )"} -{"pdf": "arxiv_math/2503.07281_pg12.pdf", "page": 1, "id": "2503.07281_pg12_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K^1_{\\Theta}\\oplus \\Theta H^1 \\subsetneq H^1_{\\Theta},"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_\\varepsilon=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*) L_{i,t}=0"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_{\\varepsilon}:=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\varepsilon}:=(\\rho_{1,\\varepsilon},\\cdots,\\rho_{n,\\varepsilon})"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda_{I,N}(\\rho_\\varepsilon)=-\\sum_{i=1}^n\\frac{2}{N}\\Big(D_{i,t}+\\sum_{s\\neq t}\\frac{e^{2H_{i,s}(p^*)}h_i(p_{s}^*)}{e^{2H_{i,t}(p^*)}h_i(p_{t}^*)} D_{i,s}+o(1)\\Big)\\varepsilon_t^{2}"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i\\in\\hat{I}}\\sum_{t=1}^N e^{(\\hat{m}^*-2)H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_1,\\cdots,\\varepsilon_N"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{i,\\varepsilon}\\to\\rho_i^*"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^N \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"} -{"pdf": "arxiv_math/2503.07467_pg6.pdf", "page": 1, "id": "2503.07467_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)L_{i,t}\\neq 0"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}H= e^{-i\\alpha z}\\mathcal{H}(E)"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(D-\\lambda I)\\mathcal{J},~\\lambda\\in \\mathbb{C}"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "R=(D-\\overline{\\lambda}I|\\mathcal{J})^{-1}"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\lambda\\notin \\mathcal{J}"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\lambda} \\notin \\sigma(D|\\mathcal{J})"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(D|\\mathcal{J})=\\emptyset"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} \\in \\mathscr{J}"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_{\\lambda}(x) = \\varphi(V(I-\\overline{\\lambda}V)^{-1}x)=\\varphi(Rx) = 0,"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} = C^{\\infty}(D)"} -{"pdf": "arxiv_math/2503.08443_pg27.pdf", "page": 1, "id": "2503.08443_pg27_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}\\varphi(\\lambda) = 0"} -{"pdf": "arxiv_math/2503.04958_pg4.pdf", "page": 1, "id": "2503.04958_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x', x \\rangle \\ge 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\\quad (st)^{\\ast} = t^{\\ast}."} -{"pdf": "arxiv_math/2503.06329_pg24.pdf", "page": 1, "id": "2503.06329_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{L}_{a_1a_5} \\times \\widetilde{R}_{a_1a_5}"} -{"pdf": "arxiv_math/2503.06329_pg24.pdf", "page": 1, "id": "2503.06329_pg24_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W_1 W_2 \\cdots W_{m_e}"} -{"pdf": "arxiv_math/2503.06329_pg24.pdf", "page": 1, "id": "2503.06329_pg24_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_{LC_n}(X) \\neq 0"} -{"pdf": "arxiv_math/2503.06329_pg24.pdf", "page": 1, "id": "2503.06329_pg24_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{L}_e \\times \\widetilde{R}_e"} -{"pdf": "arxiv_math/2503.06329_pg24.pdf", "page": 1, "id": "2503.06329_pg24_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{c|cccc} & a_1a_5 & a_1a_2a_5 & a_1a_5a_6 & a_1a_2a_5a_6 \\\\ 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"checked": null, "math": "S_{i j}^{\\prime}=\\left[L_{j}, U_{j}\\right]"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A^{-}=\\left[\\begin{array}{llllllllll} 0.70 & 0.70 & 0.32 & 0.44 & 0.00 & 0.16 & 0.20 & 0.50 & 0.40 & 0.39 \\\\ 0.70 & 0.65 & 0.14 & 0.12 & 0.80 & 0.76 & 0.00 & 1.00 & 0.15 & 0.79 \\\\ 0.17 & 0.24 & 0.20 & 0.20 & 0.06 & 0.25 & 0.13 & 0.19 & 0.22 & 0.02 \\\\ 0.14 & 0.10 & 0.04 & 0.00 & 0.10 & 0.00 & 0.14 & 0.02 & 0.15 & 0.08 \\\\ 0.70 & 0.04 & 0.27 & 0.36 & 0.60 & 0.40 & 0.48 & 0.50 & 0.50 & 0.50 \\\\ 0.66 & 0.63 & 0.14 & 0.73 & 0.53 & 0.46 & 0.61 & 0.85 & 0.85 & 0.39 \\\\ 0.00 & 0.15 & 0.15 & 0.05 & 0.02 & 0.03 & 0.10 & 0.12 & 0.18 & 0.09 \\\\ 0.63 & 0.03 & 0.55 & 0.77 & 0.79 & 0.49 & 0.21 & 0.32 & 0.80 & 0.71 \\\\ 0.27 & 0.30 & 0.35 & 0.24 & 0.35 & 0.07 & 0.29 & 0.35 & 0.20 & 0.75 \\\\ 0.59 & 0.34 & 0.26 & 0.38 & 0.02 & 0.60 & 0.52 & 0.43 & 0.27 & 0.44 \\end{array}\\right]"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "A^{+}=\\left[\\begin{array}{llllllllll} 0.25 & 0.32 & 0.41 & 0.19 & 0.70 & 0.13 & 0.44 & 0.37 & 0.28 & 0.50 \\\\ 0.80 & 0.73 & 0.64 & 0.79 & 0.80 & 0.22 & 0.80 & 0.56 & 0.10 & 0.28 \\\\ 0.11 & 0.20 & 0.12 & 0.13 & 0.05 & 0.25 & 0.40 & 0.25 & 0.20 & 0.18 \\\\ 0.10 & 0.23 & 0.25 & 0.15 & 0.12 & 0.05 & 0.02 & 0.01 & 0.15 & 0.15 \\\\ 0.45 & 0.35 & 0.70 & 0.50 & 0.41 & 0.27 & 0.39 & 0.48 & 0.17 & 0.39 \\\\ 0.60 & 0.70 & 0.25 & 0.38 & 0.63 & 0.58 & 0.46 & 0.47 & 0.85 & 0.33 \\\\ 0.01 & 0.02 & 0.15 & 0.09 & 0.12 & 0.10 & 0.15 & 0.15 & 0.04 & 0.25 \\\\ 0.75 & 0.64 & 0.32 & 0.29 & 0.39 & 0.61 & 0.57 & 0.34 & 1.00 & 0.46 \\\\ 0.22 & 0.20 & 0.35 & 0.23 & 0.30 & 0.18 & 0.29 & 0.25 & 0.35 & 0.10 \\\\ 0.41 & 0.25 & 0.50 & 0.20 & 0.56 & 0.60 & 0.59 & 0.60 & 0.47 & 0.31 \\end{array}\\right]"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathscr{I}=\\{1,2, \\ldots, 10\\}"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall i \\in \\mathscr{I}"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall j \\in \\mathscr{J}"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{i j}^{\\prime}=S_{i j} \\cap I_{j}=\\left\\{L_{j}\\right\\}"} -{"pdf": "arxiv_math/2503.05874_pg8.pdf", "page": 1, "id": "2503.05874_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{i j}^{\\prime}=\\left\\{L_{j}\\right\\}"} -{"pdf": "arxiv_math/2503.04056_pg25.pdf", "page": 1, "id": "2503.04056_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v(t) = {v}(x_{i},y_{i}, t)"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial \\Omega\\in C^2"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\Gamma_\\sigma}g(d(x))\\ dx %=\\int_{\\Gamma_\\sigma}g(d(x))\\ |\\nabla d(x)| dx =\\int_0^\\sigma g(t)\\mathcal H^{N-1}(\\Gamma_\\sigma\\cap\\{d=t\\})\\ dt,"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "c_1H^{N-1}(\\partial \\Omega)\\le H^{N-1}(\\Gamma_\\sigma\\cap\\{d=t\\})\\le c_1H^{N-1}(\\partial \\Omega)"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_\\sigma\\cap\\{d=t\\}"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g: (0,\\sigma)\\to \\mathbb R"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{B\\subset\\Omega}\\left(\\frac1{|B|}{\\int_Bw\\ dx }\\right)\\left(\\frac1{|B|}{\\int_Bw^{-1}}dx \\right)<+\\infty"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega \\subset\\mathbb R^N"} -{"pdf": "arxiv_math/2503.08649_pg4.pdf", "page": 1, "id": "2503.08649_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "w(x):=d^\\beta(x), \\quad{\\rm \\ \\ }"} -{"pdf": "arxiv_math/2503.05000_pg9.pdf", "page": 1, "id": "2503.05000_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau^*\\frac{\\partial S_t}{\\partial t}"} -{"pdf": "arxiv_math/2503.05000_pg9.pdf", "page": 1, "id": "2503.05000_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(f, g)\\colon \\begin{array}{ccc} \\Omega^1_0(M) & \\to & \\Omega^1_0(M)\\\\ \\zeta &\\mapsto & \\zeta^*[f,\\tau^*\\zeta^*g] \\end{array}."} -{"pdf": "arxiv_math/2503.05000_pg9.pdf", "page": 1, "id": "2503.05000_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f, g\\in J_\\xi^\\infty,"} -{"pdf": "arxiv_math/2503.05000_pg9.pdf", "page": 1, "id": "2503.05000_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(J_\\xi^\\infty,\\triangleright)"} -{"pdf": "arxiv_math/2503.05000_pg9.pdf", "page": 1, "id": "2503.05000_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\abs{\\xi}=1,\\quad \\abs{\\eta(f, g)}=\\abs{f}+\\abs{g}. %"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{G}(tX,t^\\beta Y) = t^{d-e}\\widetilde{G}(X,Y)"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f_-(t) = \\sum_{k = 0}^m (-1)^{d-rk}a_k t^{sk}"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq m,n\\leq \\lfloor d/r\\rfloor"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = t^d F(X,Y)= t^d X^e \\widetilde{F}(X,Y)."} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_-(t) = \\sum_{k = 0}^n (-1)^{d-rk}b_k t^{sk}"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{F}(tX,t^\\beta Y) = t^{d-e}\\widetilde{F}(X,Y)"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a,b\\in\\R\\setminus\\{0\\}"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g_+(t) = \\sum_{k = 0}^n b_k t^{sk}"} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = (tX)^e \\widetilde{F}(tX, t^\\beta Y) = t^e X^e \\widetilde{F}(tX,t^\\beta Y)."} -{"pdf": "arxiv_math/2503.06022_pg26.pdf", "page": 1, "id": "2503.06022_pg26_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\deg g_+ = \\deg g_- = sn"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "D:=\\{\\omega\\in\\Omega\\colon \\|z(\\omega)\\|>\\sigma\\}"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "D\\setminus N=\\bigcup_{i=1}^\\infty E_i"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_0:=(E\\setminus D)\\cup N\\cup C"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\lambda a(\\omega)+\\nu b(\\omega)\\|=\\|\\lambda a(\\omega)\\|+\\|\\nu b(\\omega)\\|"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\label{eq: ||int_(E_i) z dmu|| > (1-eps) int_(E_i) ||z|| dmu} \\begin{aligned} \\biggl\\|\\int_{E_i} z\\,d\\mu\\biggr\\| &\\geq\\re\\zs_i\\biggl(\\int_{E_i} z\\,d\\mu\\biggr) =\\int_{E_i}\\bigl(\\re\\zs_i(z_i)+\\re\\zs_i(z-z_i)\\bigr)\\,d\\mu\\\\ &\\geq\\int_{E_i}(\\|z_i\\|-\\|z-z_i\\|)\\,d\\mu \\geq\\int_{E_i}(\\|z\\|-2\\|z-z_i\\|)\\,d\\mu\\\\ &>\\int_{E_i}(\\|z\\|-\\eps\\sigma)\\,d\\mu \\geq\\int_{E_i}(\\|z\\|-\\eps\\|z\\|)\\,d\\mu =(1-\\eps)\\int_{E_i}\\|z\\|\\,d\\mu. \\end{aligned}"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "C:=\\bigcup_{i=n+1}^\\infty E_i"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bigcup_{i=0}^n E_i=E"} -{"pdf": "arxiv_math/2503.09182_pg4.pdf", "page": 1, "id": "2503.09182_pg4_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\alpha a(\\omega)+\\beta b(\\omega)\\|=\\alpha\\|a(\\omega)\\|+\\beta\\|b(\\omega)\\|"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\bar{\\textbf{H}}_{1,U}}"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textbf{H}_{1,\\text D}"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\delta} = \\frac{\\rho_1 e^{j\\omega_1}}{\\rho_2 e^{j\\omega_2}}."} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textbf{H}_{2,\\text D}"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}^{'}_{2,D}} = \\frac{\\mathrm{\\textbf{H}_{2,D}}}{\\rho_2 e^{j\\omega_2}},"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}_\\text D}"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}_D}"} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}} = [\\mathrm{\\textbf{H}^{'}_{1,U}},\\mathrm{\\textbf{H}^{'}_{2,D}} ]."} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}^{'}_{1,U}} = \\frac{\\mathrm{\\bar{\\textbf{H}}}_{1,U}}{\\delta}."} -{"pdf": "arxiv_math/2503.06651_pg13.pdf", "page": 1, "id": "2503.06651_pg13_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{\\textbf{H}_{1,U}}"} -{"pdf": "arxiv_math/2503.04535_pg2.pdf", "page": 1, "id": "2503.04535_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X=C_1\\times \\ldots \\times C_n"} -{"pdf": "arxiv_math/2503.04535_pg2.pdf", "page": 1, "id": "2503.04535_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L=L_1\\boxtimes\\ldots\\boxtimes L_n"} -{"pdf": "arxiv_math/2503.04535_pg2.pdf", "page": 1, "id": "2503.04535_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "X=C_1\\times\\ldots\\times C_n"} -{"pdf": "arxiv_math/2503.04535_pg2.pdf", "page": 1, "id": "2503.04535_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d_1\\geq d_2\\geq\\ldots\\geq d_n>0"} -{"pdf": "arxiv_math/2503.08465_pg8.pdf", "page": 1, "id": "2503.08465_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "w_\\perp \\in \\overline{E}_{\\leq\\rho\\Lambda}^\\perp"} -{"pdf": "arxiv_math/2503.08465_pg8.pdf", "page": 1, "id": "2503.08465_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": 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"\\frac{1}{2}|\\!\\operatorname{Tr} gh| = \\left|\\sinh\\left(\\frac{t_g}{2}\\right)\\sinh\\left(\\frac{t_h}{2}\\right) + \\cosh\\left(\\frac{t_g}{2}\\right)\\cosh\\left(\\frac{t_h}{2}\\right)\\cos\\theta_P\\right|."} -{"pdf": "arxiv_math/2503.07247_pg2.pdf", "page": 1, "id": "2503.07247_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{PSL}^{\\pm}_2(\\mathbb{R}) = \\left\\{ \\pm \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\middle| a, b, c, d \\in \\mathbb{R}, ad - bc = \\pm 1 \\right\\}."} -{"pdf": "arxiv_math/2503.07247_pg2.pdf", "page": 1, "id": "2503.07247_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Tr} g = 0"} -{"pdf": "arxiv_math/2503.07247_pg2.pdf", "page": 1, "id": "2503.07247_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\operatorname{Tr} g| = 2"} -{"pdf": "arxiv_math/2503.07247_pg2.pdf", "page": 1, "id": "2503.07247_pg2_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\operatorname{Tr} g| > 2"} -{"pdf": "arxiv_math/2503.08986_pg11.pdf", "page": 1, "id": "2503.08986_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\mathcal{C}}_{\\mathrm{sum},u}=\\overline{\\mathcal{C}}_{\\mathrm{c},u}+\\overline{\\mathcal{C}}_{\\mathrm{p},u}"} -{"pdf": "arxiv_math/2503.06722_pg17.pdf", "page": 1, "id": "2503.06722_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Q \\colon \\mathbf{2} \\to \\mathbf{Fin}"} -{"pdf": "arxiv_math/2503.06722_pg17.pdf", "page": 1, "id": "2503.06722_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{Fun}(\\mathbf{2},\\mathbf{Fin})"} -{"pdf": "arxiv_math/2503.05476_pg14.pdf", "page": 1, "id": "2503.05476_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_{n}(t)\\le\\psi(t)"} -{"pdf": "arxiv_math/2503.05476_pg14.pdf", "page": 1, "id": "2503.05476_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_{n+1}(t) \\leq \\psi(t)"} -{"pdf": "arxiv_math/2503.05476_pg14.pdf", "page": 1, "id": "2503.05476_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi_n(t) \\leq \\psi(t)"} -{"pdf": "arxiv_math/2503.05476_pg14.pdf", "page": 1, "id": "2503.05476_pg14_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(t) = 1 \\geq \\psi_{n+1}(t)"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}_{k, n, w}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{i+1, i+2, \\dots, n\\}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^{i_0-1}(k-1)k^{k-1}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) \\leq \\frac{k^n(k-1)}{4} \\sum_{i=1}^{i_0-1}(k^{n-i}-k^{n-i-(n-i)}) = \\frac{k(k-1)}{4} \\sum_{i=1}^{i_0-1}k^{n-1}(k^{n-i}-1) = \\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{n-i}}{2}."} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) = \\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{(c_w)_{i}} }{2}k^{2n-2i-2(c_w)_i} = \\frac{k-1}{4}\\sum_{i=1}^{i_0-1}(k^{2n-i} - k^{2n-i-(c_w)_i})."} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^{i_0-1} (k-1)k^{i-1}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(k, n, w) = \\frac{k^n(k-1)}{4}\\sum_{i=1}^n (k^{n-i} - k^{n-i-(c_w)_i}) = \\frac{k^n(k-1)}{4}\\sum_{i=1}^{i_0-1}(k^{n-i} - k^{n-i-w_i+1})"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{k}{2}\\sum_{i=1}^{i_0-1} k^{i-1}\\binom{k^{n-i}}{2}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\{i_0, i_0+1, \\dots, n \\}"} -{"pdf": "arxiv_math/2503.09577_pg20.pdf", "page": 1, "id": "2503.09577_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{k}{2} \\sum_{i=1}^{i_0-1}k^{i-1}\\binom{k^{n-i}}{2}"} -{"pdf": "arxiv_math/2503.05642_pg4.pdf", "page": 1, "id": "2503.05642_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "D_s(G):=|\\{(u,v)~|~u,v\\in V,~d_{u,v}=s\\}|"} -{"pdf": "arxiv_math/2503.05642_pg4.pdf", "page": 1, "id": "2503.05642_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} k(e_{u_1,v_1},e_{u_2,v_2})=k_v(l_{u_1},l_{u_2})\\cdot k_e(d_{u_1,v_1},d_{u_2,v_2})\\cdot k_v(l_{v_1},l_{v_2}) \\end{aligned}"} -{"pdf": "arxiv_math/2503.05642_pg4.pdf", "page": 1, "id": "2503.05642_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} k_{\\mathit{SP}}(G^1, G^2)=\\sum\\limits_{u_1,v_1\\in V^1,u_2,v_2\\in V^2}k(e_{u_1,v_1},e_{u_2,v_2}) \\end{aligned}"} -{"pdf": "arxiv_math/2503.04523_pg4.pdf", "page": 1, "id": "2503.04523_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{R}^{n_1\\times n_2\\times\\cdots\\times n_d}"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(D\\setminus \\{u\\})\\cup \\{v\\}"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e\\in E(G) \\, \\colon e"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_G(v), N_G[v], N_G(X),"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{v\\in V(G)\\setminus X \\, \\colon v \\text{ is adjacent to a vertex in } X\\}"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "N(v)\\cap D\\subseteq A_D"} -{"pdf": "arxiv_math/2503.08088_pg5.pdf", "page": 1, "id": "2503.08088_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{v \\in V(G) \\setminus D \\, \\colon N(v) \\cap D = \\{u\\}\\}"} -{"pdf": "arxiv_math/2503.05886_pg3.pdf", "page": 1, "id": "2503.05886_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i,j}\\tilde p_{ij}\\log\\frac{\\tilde p_{ij}}{p_{ij}},"} -{"pdf": "arxiv_math/2503.05886_pg3.pdf", "page": 1, "id": "2503.05886_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L= \\sum_{i,j}\\left(\\tilde p_{ij}\\log\\frac{\\tilde p_{ij}}{p_{ij}}+\\lambda_i(\\tilde p_{ij}-\\tilde \\alpha_i)+\\gamma_j(\\tilde p_{ij}-\\tilde \\beta_j)\\right),"} -{"pdf": "arxiv_math/2503.05886_pg3.pdf", "page": 1, "id": "2503.05886_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\rho_1\\neq \\sum_{i,k,j}L_{ikj}(1,0)\\tilde\\rho_0L_{ikj}(1,0)^\\dagger."} -{"pdf": "arxiv_math/2503.04086_pg3.pdf", "page": 1, "id": "2503.04086_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in (\\Z/n)^{\\times}."} -{"pdf": "arxiv_math/2503.04086_pg3.pdf", "page": 1, "id": "2503.04086_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gcd(a,n) = \\gcd(b,n)"} -{"pdf": "arxiv_math/2503.03909_pg14.pdf", "page": 1, "id": "2503.03909_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u_{xx} + u_{yy} + \\lambda e^{u} = 0, \\ \\ (x,y) \\in [0,1] \\times [0,1],"} -{"pdf": "arxiv_math/2503.03909_pg14.pdf", "page": 1, "id": "2503.03909_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{k+1}(i,j) = G(i,j;X^{k},\\alpha) \\equiv X^k(i,j) + \\alpha M(G_{\\rm B}(i,j;X^{k}))."} -{"pdf": "arxiv_math/2503.03909_pg14.pdf", "page": 1, "id": "2503.03909_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "X(i,j) \\approx u(x_i,y_j)"} -{"pdf": "arxiv_math/2503.03909_pg14.pdf", "page": 1, "id": "2503.03909_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\rm TOL}=10^{-6}, \\hat{m}=5, \\theta=0.9, \\alpha = 0.125h_x^2"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y}=t\\in T\\setminus\\{0\\}"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_i|=2^{2m-t}+2^m-2^{m-t}"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1\\star x + y_2\\star x=z"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-2^{-t})(2^{2m}-2^m)"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y} \\in T"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_1|=|f_1^{-1}(f_1(0,0))| = 2\\cdot 2^m-1+(2^{m-t}-1)(2^m-1)."} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(1-\\frac{1}{2^m}\\right)\\big(2^n-2^{n/2}\\big)"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{y_1}(x)\\neq \\alpha_{y_2}(x)"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma'(y_1)\\neq \\sigma'(y_2)"} -{"pdf": "arxiv_math/2503.03905_pg7.pdf", "page": 1, "id": "2503.03905_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "W_f(a,b)=\\begin{cases} \\pm 2^{n/2} & \\text{if $b\\neq 0$,} \\\\ 0 & \\text{if $a\\neq 0$, $b=0$,} \\\\ 2^n & \\text{if $a=0$, $b=0$.} \\end{cases}"} -{"pdf": "arxiv_math/2503.05503_pg13.pdf", "page": 1, "id": "2503.05503_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\cdot, \\cdot \\rangle"} -{"pdf": "arxiv_math/2503.05503_pg13.pdf", "page": 1, "id": "2503.05503_pg13_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{l,m}= \\sqrt{\\frac{(2 l+1)}{4 \\pi} \\frac{(l-m)!}{(l+m)!}}"} -{"pdf": "arxiv_math/2503.05503_pg13.pdf", "page": 1, "id": "2503.05503_pg13_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ Y_l^m : l\\in \\N_0, \\, -l \\le m \\le l \\}"} -{"pdf": "arxiv_math/2503.07286_pg6.pdf", "page": 1, "id": "2503.07286_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in [\\underline{H},\\overline{H}],"} -{"pdf": "arxiv_math/2503.07286_pg6.pdf", "page": 1, "id": "2503.07286_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathbb E}X^2(t)= \\sum_{j=0}^{+\\infty} \\sum_{k=0}^{2^{j}-1} \\left|\\int_{0}^{1} (t-s)_{+}^{H_{j}(k/{2^j})-{1}/{2}} h_{j,k}(s)ds\\right|^2 < +\\infty."} -{"pdf": "arxiv_math/2503.07286_pg6.pdf", "page": 1, "id": "2503.07286_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda,x) \\in [\\underline{H},\\overline{H}] \\times \\mathbb{R},"} -{"pdf": "arxiv_math/2503.07421_pg26.pdf", "page": 1, "id": "2503.07421_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial H^h/\\partial l_i"} -{"pdf": "arxiv_math/2503.07421_pg26.pdf", "page": 1, "id": "2503.07421_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "l_{\\sigma}=(l_{12}(l_{23}, l_{24}, l_{34}), l_{13}(l_{23}, l_{24}, l_{34}), l_{14}(l_{23}, l_{24}, l_{34}), l_{23}, l_{24}, l_{34}),"} -{"pdf": "arxiv_math/2503.07421_pg26.pdf", "page": 1, "id": "2503.07421_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^h(l^h)=H(l^*,l^h)=H(l)= cov(l) - 2\\pi\\sum_{i\\in E} l_i,"} -{"pdf": "arxiv_math/2503.07421_pg26.pdf", "page": 1, "id": "2503.07421_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial H^h}{\\partial l_i} = - K_i, \\quad e_i\\in E^h."} -{"pdf": "arxiv_math/2503.07421_pg26.pdf", "page": 1, "id": "2503.07421_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(l_{23}, l_{24}, l_{34})=l^h_{\\sigma} \\in \\R^3"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ". We obtain the \\textit{conilpotent filtration}"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "is conilpotent. Similar definitions can be made for comodules over"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "to be the dual of the group algebra"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "be an augmented coalgebra over a field"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "be the reduced coproduct. Then we call"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "in the profinite topology of"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ". For a profinite group, we define"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded algebra and for a graded comodule"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded coalgebra. Additionally, if"} -{"pdf": "arxiv_math/2503.09264_pg5.pdf", "page": 1, "id": "2503.09264_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "we define its group coalgebra"} -{"pdf": "arxiv_math/2503.09478_pg23.pdf", "page": 1, "id": "2503.09478_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_k = \\|\\boldsymbol{x}_k-\\boldsymbol{x}_*\\|,\\quad f(k) = -\\ln \\xi_k."} -{"pdf": "arxiv_math/2503.09478_pg23.pdf", "page": 1, "id": "2503.09478_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty}\\frac{\\xi_{k+1}}{\\xi_k^q}=Q_q, \\quad q>1,\\quad 00"} -{"pdf": "arxiv_math/2503.08011_pg19.pdf", "page": 1, "id": "2503.08011_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b^{(1_n)}_1\\}_{n=1}^\\infty"} -{"pdf": "arxiv_math/2503.08011_pg19.pdf", "page": 1, "id": "2503.08011_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq b^{(1_n)}_2 \\leq a_2"} -{"pdf": "arxiv_math/2503.08011_pg19.pdf", "page": 1, "id": "2503.08011_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n\\to\\infty} \\sum_{i=1}^\\infty |b^{(n_n)}_i - c_i| = 0"} -{"pdf": "arxiv_math/2503.08206_pg4.pdf", "page": 1, "id": "2503.08206_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f(x) = \\begin{dcases} 1/q & \\text{ if } x=p/q\\text{ with } (p,q)=1;\\\\ 0 & \\text{ if } x \\text{ is irrational;} \\end{dcases}"} -{"pdf": "arxiv_math/2503.08206_pg4.pdf", "page": 1, "id": "2503.08206_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta^+(x) := B^+(x)-2B_0^+(x)\\quad \\text{ and }\\quad \\Delta^-(x) := W^-(x)-2B_0^-(x)."} -{"pdf": "arxiv_math/2503.05844_pg8.pdf", "page": 1, "id": "2503.05844_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Q=\\begin{bmatrix} 10& 0\\\\ 0& 50\\\\ \\end{bmatrix}"} -{"pdf": "arxiv_math/2503.05844_pg8.pdf", "page": 1, "id": "2503.05844_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0= [0.2; 0; 0; 0.1; 0]^T"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in {\\Sigma_\\vartheta}"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta: {\\Sigma_\\vartheta} \\times \\mathbb{T} \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto (T_\\omega (x) , T_\\omega(y))."} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta^{(2)} : {\\Sigma_\\vartheta} \\times \\mathbb{T}^2 \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}^2"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "T^n_\\omega (x) \\coloneqq T_{\\omega_{n-1}} \\circ \\cdots \\circ T_{\\omega_0} (x)"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Omega_\\vartheta"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto T^{(2)}_\\omega (x,y)"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Sigma_\\vartheta"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\Sigma_\\vartheta} \\mu^{(2)} \\left( \\left( T^{(2)}_\\omega\\right)^{-1} (A) \\right) \\, d\\mathbb{P} (\\omega) = \\mu^{(2)}(A),"} -{"pdf": "arxiv_math/2503.08244_pg9.pdf", "page": 1, "id": "2503.08244_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma \\omega \\coloneqq (\\omega_{i+1})_{i\\in\\mathbb{N}}"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "and the fact that for every"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{\\P(\\overline{W}_{n} \\in (x,x+1])}{ n \\P(\\overline{X}_{1} \\in (x,x+1])} -1 \\right|=0,"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": ". This will indeed imply that"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\overline{W}_{n}}{\\widehat{b}_{n}} = \\frac{X_{1}+ \\cdots+X_{n}-b_{n}}{a_{n}} \\cdot \\frac{a_{n}}{\\widehat{b}_{n}}+ \\frac{b_{n}+\\gamma n}{\\widehat{b}_{n}},"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "which implies tightness since"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "and that condition (3.3) there holds also with"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\mathrm{TV}}\\left((\\overline{X}_i^{(n)}:1\\leq i\\leq n-1),(\\overline{X}_i:1\\leq i\\leq n-1)\\right) \\to 0,"} -{"pdf": "arxiv_math/2503.07530_pg7.pdf", "page": 1, "id": "2503.07530_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{ \\P({X}_{1} \\in(x-m_{n},x-m_{n}+1])}{ \\P({X}_{1} \\in(x-\\gamma,x-\\gamma+1])} -1 \\right|=0,"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{\\circ}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{*}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},k_{2})\\in \\mathcal{A}_{2}^{*}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},\\ldots,k_{m-1})\\in \\mathcal{A}_{m-1}^{*}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{K}_{m}(3,2^{m-3},1,k) = 2(k-m)+3,\\quad \\mathbb{K}_{m-1}(2^{m-3},1,k) = k-m+2,"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{*}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{\\circ}(3,1,2)"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "3m(k-1)-2k+5\\not\\equiv 2\\pmod{3},\\quad\\text{that is,}\\quad k\\not\\equiv 0\\pmod{3}."} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "3\\leqslant r\\leqslant m-1"} -{"pdf": "arxiv_math/2503.08176_pg20.pdf", "page": 1, "id": "2503.08176_pg20_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "3(k-m)+5\\not\\equiv 2\\pmod{3}"} -{"pdf": "arxiv_math/2503.07696_pg3.pdf", "page": 1, "id": "2503.07696_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1<\\gamma_2<\\ldots"} -{"pdf": "arxiv_math/2503.07696_pg3.pdf", "page": 1, "id": "2503.07696_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\arg(\\eta(s))\\ne\\pm\\pi/2"} -{"pdf": "arxiv_math/2503.07696_pg3.pdf", "page": 1, "id": "2503.07696_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda^\\prime\\ne\\rho^\\prime"} -{"pdf": "arxiv_math/2503.07696_pg3.pdf", "page": 1, "id": "2503.07696_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta^{\\prime\\prime}(s)"} -{"pdf": "arxiv_math/2503.05513_pg15.pdf", "page": 1, "id": "2503.05513_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\colon X \\to \\R \\cup \\{-\\infty\\}"} -{"pdf": "arxiv_math/2503.05513_pg15.pdf", "page": 1, "id": "2503.05513_pg15_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u|_{\\partial U} \\leq h|_{\\partial U}"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge x_k\\not\\leqslant b"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\neq(a_1 \\wedge a_2) \\wedge n_i = (a_1 \\wedge a_2) \\wedge (n_i^1 \\wedge n_i^2) =(a_1 \\wedge n_i^1) \\wedge (a_2 \\wedge n_i^2)."} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge x_1\\wedge x_2\\not\\leqslant b"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "((b\\vee c)\\wedge x_k)>b"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1 \\wedge n_i^1 \\neq 0"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\wedge a\\wedge x_1\\wedge x_2\\neq 0"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "a_1 \\wedge \\left( \\bigwedge_{i=1}^2 n_i^1 \\right) \\neq 0\\quad \\text{and}\\quad a_2 \\wedge \\left( \\bigwedge_{i=1}^2 n_i^2 \\right) \\neq 0."} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{1,3\\}^\\uparrow={\\{1\\}\\vee\\{3\\}}^\\uparrow"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "a\\wedge x_1\\wedge x_2\\neq 0"} -{"pdf": "arxiv_math/2503.06739_pg8.pdf", "page": 1, "id": "2503.06739_pg8_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(a_1 \\wedge a_2) \\wedge \\left( \\bigwedge_{i=1}^2 n_i \\right) = (a_1 \\wedge a_2) \\wedge \\left( \\bigwedge_{i=1}^2 (n_i^1 \\wedge n_i^2) \\right) \\neq 0."} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\left( 0_{X},0_{Y},-1\\right) \\right\\}"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\cap \\mathbb{A}% ^{-1}\\left( D\\right)"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\cap \\mathbb{A}^{-1}\\left( D\\right) ,"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{U}\\cap V=U\\cap V,"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\left\\{ \\left( 0_{X^{\\prime }},0\\right) \\right\\}"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f:X\\longrightarrow \\overline{\\mathbb{R}}"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{A})\\Longleftrightarrow (\\mathcal{B}% )"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ x\\in X:f\\left( x\\right) \\geq 0\\right\\} ,"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\notin \\overline{U},"} -{"pdf": "arxiv_math/2503.04226_pg3.pdf", "page": 1, "id": "2503.04226_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{A}% )\\Longleftrightarrow (\\mathcal{B})"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{(c)}c_{(1)}\\otimes c_{(2)}=\\sum_{(c)}c_{(2)}\\otimes c_{(1)}."} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(m)=\\sum_{i} {m_{(0)}}_i\\otimes {m_{(1)}}_i\\in M\\otimes C"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(H, \\Delta, \\varepsilon)"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta\\colon H\\to H\\otimes H"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{(m)}m_{(0)}\\otimes m_{(1)}"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\colon M\\rightarrow M'"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu\\colon H\\otimes H\\to H"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(C,\\Delta, \\varepsilon)"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\colon (M,\\rho)\\rightarrow (M', \\rho')"} -{"pdf": "arxiv_math/2503.04897_pg4.pdf", "page": 1, "id": "2503.04897_pg4_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(c\\otimes c')=c'\\otimes c"} -{"pdf": "arxiv_math/2503.05275_pg9.pdf", "page": 1, "id": "2503.05275_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X\\subseteq V(H)\\setminus V (P)"} -{"pdf": "arxiv_math/2503.05275_pg9.pdf", "page": 1, "id": "2503.05275_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S'_{A}=\\{w_1,w_2,\\dots,w_{k-\\ell}\\}"} -{"pdf": "arxiv_math/2503.05275_pg9.pdf", "page": 1, 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null, "math": "\\|f\\|_{\\omega_i,T}:=\\max\\limits_{t\\in [t_0,T]}\\omega_i(t)\\|f(t)\\|."} -{"pdf": "arxiv_math/2503.09471_pg2.pdf", "page": 1, "id": "2503.09471_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_i(s)=\\|\\Omega_i(p,s)\\|"} -{"pdf": "arxiv_math/2503.09471_pg2.pdf", "page": 1, "id": "2503.09471_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i, \\beta_i, \\gamma_i, \\delta_i"} -{"pdf": "arxiv_math/2503.09471_pg2.pdf", "page": 1, "id": "2503.09471_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in B_{\\eta}\\cap K\\implies \\lim_{t\\to\\infty}\\|x(t;t_0,x_0)\\|=0"} -{"pdf": "arxiv_math/2503.09471_pg2.pdf", "page": 1, "id": "2503.09471_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\in C([t_0,T];L(\\mathcal H_i))"} -{"pdf": "arxiv_math/2503.09471_pg2.pdf", "page": 1, "id": "2503.09471_pg2_math_011", "type": "math", "max_diffs": 0, "checked": null, 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"math", "max_diffs": 0, "checked": null, "math": "u, v \\in C^{1+\\beta, \\frac{1+\\beta}{2}}(\\Omega_{T_m-\\varepsilon})"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\leq u(x, t) \\leq C_1, \\quad |h'(t)| + |g'(t)| \\leq C_2, \\quad |h(t)|, |g(t)| \\leq C_2 t + h_0."} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u(t, \\cdot)\\|_{C^{2+\\beta}([g(t), h(t)])} \\leq Q^*."} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t,x,u)\\leq \\bar f(u)<0"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "h, g \\in C^{1+\\frac{\\beta}{2}}((0, T_m)), \\quad u \\in C^{1+\\frac{\\beta}{2}, 2+\\beta}(\\Omega_{T_m}),"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u\\|_{C^{2+\\beta,1+\\frac{\\beta}{2} }(\\Omega_{T_m - \\varepsilon} \\setminus \\Omega_{T_0})} \\leq Q^*"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{u}(t, x) := \\max\\{q_0(ct - x - L), q_0(ct + x - L)\\}"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{(t,x): t>0, x\\in [g(t), h(t)]\\}"} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{u}(0, x) \\leq u_0(x) \\quad \\text{for } x \\in [-h_0, h_0]."} -{"pdf": "arxiv_math/2503.06020_pg21.pdf", "page": 1, "id": "2503.06020_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega_{T_m} := \\{(t, x) : t \\in (0, T_m), x \\in [g(t), h(t)]\\}"} -{"pdf": "arxiv_math/2503.07208_pg11.pdf", "page": 1, "id": "2503.07208_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(D_{\\sigma})"} -{"pdf": "arxiv_math/2503.07208_pg11.pdf", "page": 1, "id": "2503.07208_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_B(T_{\\sigma}, P_{\\sigma})"} -{"pdf": "arxiv_math/2503.07208_pg11.pdf", "page": 1, "id": "2503.07208_pg11_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(D_{\\sigma},S)"} -{"pdf": "arxiv_math/2503.07208_pg11.pdf", "page": 1, "id": "2503.07208_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\textsf{Sfas}(T_{\\sigma}) = \\textsf{Sfas}(T_{\\sigma} [A_I(T_{\\sigma}, P_{\\sigma})])+\\textsf{Sfas}(T_{\\sigma} [A_B(T_{\\sigma}, P_{\\sigma})])"} -{"pdf": 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null, "math": "\\widetilde{X}^{n}\\in\\mathrm{co}\\{X^{m}:m\\geq n\\}"} -{"pdf": "arxiv_math/2503.06350_pg3.pdf", "page": 1, "id": "2503.06350_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{co}\\left\\{\\left\\vert{\\int_{0}^{t}\\xi dX^{n}}\\right\\vert:n\\in\\mathbb{N},\\xi\\textrm{ is predictable and }\\vert{\\xi}\\vert\\leq1\\right\\}"} -{"pdf": "arxiv_math/2503.06350_pg3.pdf", "page": 1, "id": "2503.06350_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(M^{n})_{n=1}^{\\infty}"} -{"pdf": "arxiv_math/2503.04467_pg7.pdf", "page": 1, "id": "2503.04467_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{ \\begin{aligned} \\frac{d}{dt}u + vAu &= B(u_{2})-B(u_{1})\\\\ u(0) &= 0 \\end{aligned} \\right. ,\\quad u \\in L^{\\infty}(0,T;V) \\cap L^{2}(0,T,D(A))"} -{"pdf": "arxiv_math/2503.04467_pg7.pdf", "page": 1, "id": "2503.04467_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": 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E[I_C] = \\bigcup_{i \\in I_C} E[\\{i\\}]."} -{"pdf": "arxiv_math/2503.07398_pg15.pdf", "page": 1, "id": "2503.07398_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "|I| = |I| \\times \\aleph_0 = |J| \\times \\aleph_0 = |J|."} -{"pdf": "arxiv_math/2503.07398_pg15.pdf", "page": 1, "id": "2503.07398_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|I_C| \\times \\sup_{i \\in I_C} |E[\\{i\\}]|"} -{"pdf": "arxiv_math/2503.06549_pg8.pdf", "page": 1, "id": "2503.06549_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_i^{(N)},\\lambda_i^{(N-1)}, \\lambda_i^{(N-2)}\\ldots"} -{"pdf": "arxiv_math/2503.06549_pg8.pdf", "page": 1, "id": "2503.06549_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\langle {\\bm w}_i^{(N)},{\\bm w}_j^{(N-k)}\\rangle|\\ll 1"} -{"pdf": "arxiv_math/2503.06549_pg8.pdf", "page": 1, "id": "2503.06549_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, 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"checked": null, "math": "\\hat{F}_1 = 0.0099s_{xx}+0.9955u"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{l}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{k}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{F}_1 = 0.0098s_{xx} + u - 0.0083s + 0.0046"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,t) = (\\pi x)/5+u_1(t)"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma} = -1/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma = -\\Delta^{-1} s"} -{"pdf": "arxiv_math/2503.08059_pg6.pdf", "page": 1, "id": "2503.08059_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,y,t) = u_3(t)\\times \\{0.1\\sin[2\\pi (x + y)] + \\cos[2\\pi (x + y)]\\}"} -{"pdf": "arxiv_math/2503.06944_pg2.pdf", "page": 1, "id": "2503.06944_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_{r}^{\\text{NLoS}} \\in \\mathbb{C}^{M_r \\times N}"} -{"pdf": "arxiv_math/2503.06944_pg2.pdf", "page": 1, "id": "2503.06944_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "e^{j 2 \\pi d_{\\text{R}} \\sin(\\gamma) \\left[ \\lfloor \\frac{n-1}{N_x} \\rfloor \\sin(\\zeta) + ((n-1)-\\lfloor \\frac{n-1}{N_x} \\rfloor N_x) \\cos(\\zeta) \\right]/\\lambda }, n = 1, 2, \\cdots, N"} -{"pdf": 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"math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_d \\in \\mathbb{C}^{M_r \\times M_t}"} -{"pdf": "arxiv_math/2503.06944_pg2.pdf", "page": 1, "id": "2503.06944_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_n = e^{j \\theta_n}"} -{"pdf": "arxiv_math/2503.06944_pg2.pdf", "page": 1, "id": "2503.06944_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_d^{\\text{AoA}}"} -{"pdf": "arxiv_math/2503.06944_pg2.pdf", "page": 1, "id": "2503.06944_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "M_s \\leq \\min\\left\\{M_t, M_r\\right\\}"} -{"pdf": "arxiv_math/2503.08947_pg7.pdf", "page": 1, "id": "2503.08947_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{ab}{}^I = - u^\\mu_a u^\\nu_b \\nabla_{(\\mu} n^I_{\\nu)} =n^I_\\mu D_a u^\\mu_b + \\frac{1}{2} \\hat \\upsilon^I \\tau_{ab},"} -{"pdf": "arxiv_math/2503.08947_pg7.pdf", "page": 1, "id": "2503.08947_pg7_math_001", 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\\sigma + \\frac{1}{2} F_{Ib} \\partial_a \\sigma ."} -{"pdf": "arxiv_math/2503.08947_pg7.pdf", "page": 1, "id": "2503.08947_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{R}^a{}_{b cd}"} -{"pdf": "arxiv_math/2503.05572_pg22.pdf", "page": 1, "id": "2503.05572_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_k = \\Theta(e^{k^\\beta})"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_{i,l,j,l'}=D_{j}(\\frac{(E_{d,j})^{l'}.a_{i,l}}{l'!})"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\sum_{i_0=0}^\\infty\\sum_{i_1=0}^\\infty\\cdots\\sum_{i_{d-1}=0}^\\infty c_{i_0...i_{d-1}}(x_{0}-x_{0,n})^{i_0}\\cdots(x_{d-1}-x_{d-1,n})^{i_{d-1}}"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{d,i}(x_i-x_{i,n})=1"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Z^{-1}E_{d,i}Z=E_{d,i}"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "Z=\\left( \\begin{matrix} I_{d\\times d}&0\\\\ (z_0,z_1,...,z_{d-1})&1 \\end{matrix} \\right)"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{i,l}=D_i(\\frac{(E_{d,i})^l.s}{l!})"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\sum_{l=0}^\\infty a_{i,l}(x_{i}-x_{i,n})^l"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{d,i}c_{i_0...i_{d-1}}=0"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_023", "type": "math", "max_diffs": 0, "checked": null, "math": "a_{i,l}=\\sum_{k=0}^\\infty b_{i,l,j,l'}(x_j-x_{j,n})^{l'}"} -{"pdf": "arxiv_math/2503.09352_pg5.pdf", "page": 1, "id": "2503.09352_pg5_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "z=[z_0:z_1:...:z_{d-1}:1]"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\mathcal{L}_{\\text{SIMax}} = \\mathbb{E}_{z \\sim {p_{\\theta}(z|x)}} \\{ & \\mathbb{E}_{t \\sim T} \\left[-\\log q_{\\phi_t}(y_{t}|z)\\right] \\\\ & - \\alpha \\log \\frac{\\exp(\\text{sim}(z, z^{+}) / \\tau)}{\\sum_{z' \\in \\mathcal{B^{+}}} \\exp(\\text{sim}(z, z') / \\tau)} \\}, \\end{aligned}"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I(X; Z) \\geq I(Z; Z^\\prime)"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_t \\rightarrow X \\rightarrow Z \\rightarrow Z_t \\rightarrow \\hat{Y}_t"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "q_{\\phi_t}(y_{t} | z)"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{t=1}^{|T|}[I(Y_t;Z)]"} -{"pdf": "arxiv_math/2503.04667_pg3.pdf", "page": 1, "id": "2503.04667_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "Z \\leftarrow X \\rightarrow Z^\\prime"} -{"pdf": "arxiv_math/2503.07924_pg3.pdf", "page": 1, "id": "2503.07924_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "J_{ij} = -\\frac{Q_{ij}}{4}"} -{"pdf": 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j} x_{i,D} x_{j,D} \\\\ & + P_3 \\sum_{i \\not\\in \\{S,D\\}} \\left( \\sum_{(i,j) \\in E} \\sum_{\\substack{(i,k) \\in E \\\\ k \\neq j}} x_{i,j} x_{i,k} \\right. \\\\ & \\quad + \\left. \\sum_{(j,i) \\in E} \\sum_{\\substack{(k,i) \\in E \\\\ k \\neq j}} x_{j,i} x_{k,i} - 2 \\sum_{(i,j) \\in E} \\sum_{(k,i) \\in E} x_{i,j} x_{k,i} \\right) \\end{aligned}"} -{"pdf": "arxiv_math/2503.07924_pg3.pdf", "page": 1, "id": "2503.07924_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "H(\\sigma) = -\\sum_{i,j} J_{ij} \\sigma_i \\sigma_j - \\sum_i h_i \\sigma_i"} -{"pdf": "arxiv_math/2503.07924_pg3.pdf", "page": 1, "id": "2503.07924_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "h_i = -\\frac{q_i}{2} - \\frac{\\sum_{i \\neq j} (Q_{ij} + Q_{ji})}{4}"} -{"pdf": "arxiv_math/2503.07924_pg3.pdf", "page": 1, "id": "2503.07924_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_i \\in \\{-1,1\\}"} -{"pdf": 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"math": "ad^r_{x}(x+y)=ad^r_{x}(y)=0"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "[x+y,_{n}x]=[y,_{n}x]=0"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Z_{i+1}(L')\\leq Z_{k+m}(L)"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "C_A(L)=C_A(x_1,..,x_n)"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "m=1+\\sum_{i=1}^{k} (n_i-1)"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "[y,_m L]\\leq Z_i(L')\\leq Z_k(L)"} -{"pdf": "arxiv_math/2503.06230_pg7.pdf", "page": 1, "id": "2503.06230_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_0=H\\lhd H_1 \\lhd...\\lhd H_n=L."} -{"pdf": "arxiv_math/2503.07705_pg1.pdf", "page": 1, "id": "2503.07705_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sqrt{g(r,\\theta)}drd\\theta"} -{"pdf": "arxiv_math/2503.07705_pg1.pdf", "page": 1, "id": "2503.07705_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\tau_\\mathrm{min}}^\\infty p(x,y;t)\\,dt<\\infty"} -{"pdf": "arxiv_math/2503.07705_pg1.pdf", "page": 1, "id": "2503.07705_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "ds^2=dr^2+f(r)^2\\,d\\theta^2"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "E_7: f(x,y,z)=x^2+y^3+yz^3"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v_3\\in (\\mathbb{R}_{\\geq 0})^3"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C_{v_3}(f)=\\langle (1,0,0),(0,1,0),(n+1,0,1),(0,n+1,1) \\rangle"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_6: f(x,y,z)=z^2+y^3+x^4"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "In_{{v_3}}(f)=-z^{n+1}"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "C_{v_1}(f)=\\langle (n+1,0,1),(0,n+1,1) \\rangle"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A_n: f(x,y,z)=xy-z^{n+1}, \\ \\ n\\in \\mathbb{N}"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "D_n: f(x,y,z)=z^2-x(y^2+x^{n-2}), \\ \\ n\\in \\mathbb{N}, n\\geq 4"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "-s\\in \\mathbb{Z}_{\\leq 0}"} -{"pdf": "arxiv_math/2503.08118_pg5.pdf", "page": 1, "id": "2503.08118_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "X=V(f)\\subset \\mathbb{C}^3"} -{"pdf": "arxiv_math/2503.03949_pg1.pdf", "page": 1, "id": "2503.03949_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "s(D) \\in \\{0,\\cdots,N-2\\}"} -{"pdf": "arxiv_math/2503.03949_pg1.pdf", "page": 1, "id": "2503.03949_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta(D) \\in [1,\\frac{1}{2}(N-s(D))]"} -{"pdf": "arxiv_math/2503.08955_pg9.pdf", "page": 1, "id": "2503.08955_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\mapsto {}^{\\natural}x^{-1}"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "a \\leq \\|f\\|_{L^2(\\Omega)} \\leq b."} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{f}_s \\in U_{ad} \\subset L^2(\\Omega)"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "f_s \\rightharpoonup f"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f_s\\|_{H^{-s}(\\Omega)}"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "N,\\Omega \\text{and} s"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "s\\in(0,1),\\Omega \\subset \\R^N"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "J_s(\\bar{f}_s):=\\min_{f_s \\in U_{ad}} J_s(f_s),"} -{"pdf": "arxiv_math/2503.09386_pg6.pdf", "page": 1, "id": "2503.09386_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_s=\\left\\{f_s\\right\\}_{0 0"} -{"pdf": "arxiv_math/2503.05873_pg4.pdf", "page": 1, "id": "2503.05873_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}(\\underline{\\hat{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}}) \\neq \\underline{V}_{\\mathcal{L}}) \\leq \\epsilon_e"} -{"pdf": "arxiv_math/2503.05873_pg4.pdf", "page": 1, "id": "2503.05873_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(\\mathcal{V},p_V\\right)"} -{"pdf": "arxiv_math/2503.05873_pg4.pdf", "page": 1, "id": "2503.05873_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{V}_{\\mathcal{L}}"} -{"pdf": "arxiv_math/2503.05873_pg4.pdf", "page": 1, "id": "2503.05873_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V} \\in \\{0,1\\}"} -{"pdf": "arxiv_math/2503.05873_pg4.pdf", "page": 1, "id": "2503.05873_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\underline{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}})"} -{"pdf": "arxiv_math/2503.07310_pg2.pdf", "page": 1, "id": "2503.07310_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in \\mathcal{X}\\subseteq \\mathbb{R}^{n_x}"} -{"pdf": "arxiv_math/2503.08266_pg4.pdf", "page": 1, "id": "2503.08266_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "is the inverse of a (fictitious) mass matrix. The solutions to the equations of motion resulting from this Hamiltonian can be used as a preliminary step in the HMC algorithm. The time reversal invariance of Hamiltonian systems guarantees that the detailed balance holds. In general, the equations of motions are solved with a numerical integrator, and detailed balance is satisfied only in the limit of time step"} -{"pdf": "arxiv_math/2503.08266_pg4.pdf", "page": 1, "id": "2503.08266_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ". In the condition of detailed balance, the kinetic energy arising from the Metropolis acceptance probability and that coming from the Maxwell--Boltzmann term in the {\\it a priori} probability"} -{"pdf": "arxiv_math/2503.08266_pg4.pdf", "page": 1, "id": "2503.08266_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "as the momenta associated to the weights"} -{"pdf": "arxiv_math/2503.08266_pg4.pdf", "page": 1, "id": "2503.08266_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": ", we make use of a modified HMC algorithm that damps fluctuations in the direction opposite to"} -{"pdf": "arxiv_math/2503.08266_pg4.pdf", "page": 1, "id": "2503.08266_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "intermediate pivot points. Although this method allows for more flexible connections, it is computationally demanding, requiring multiple training runs to optimize the pivot locations. Furthermore, the number of required pivots can grow significantly, particularly in less overparameterized settings, making the approach increasingly impractical in such regimes. As an alternative to generate trajectories from"} -{"pdf": "arxiv_math/2503.04646_pg2.pdf", "page": 1, "id": "2503.04646_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset \\{\\mu \\in P(X \\times Y): \\mu_X^* \\in P(X) \\text{ is the } X\\text{-marginal of } \\mu\\}"} -{"pdf": "arxiv_math/2503.04646_pg2.pdf", "page": 1, "id": "2503.04646_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu^* \\in P(X\\times Y)"} -{"pdf": "arxiv_math/2503.04646_pg2.pdf", "page": 1, "id": "2503.04646_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset P(X\\times Y)"} -{"pdf": "arxiv_math/2503.04604_pg3.pdf", "page": 1, "id": "2503.04604_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "00"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "C^\\infty_c(E\\setminus B_{R_0}(o))"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "[0,L]\\times \\epsilon\\mathbb S^{n-1}"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\hat V_{\\hat j},\\angle_{Tits})"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "n\\omega_n^{1/n}\\theta^{1/n}"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat {\\mathcal D}_{r}(\\Omega)\\cap S_t"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat A_{r}:=\\{y\\in A_r\\ :\\ \\Phi_r(y)\\not\\in B_{R_0}\\}"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(t)\\mapsto [\\sigma]"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "M\\setminus B_{R_0}(o)"} -{"pdf": "arxiv_math/2503.08279_pg19.pdf", "page": 1, "id": "2503.08279_pg19_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal H^{n-1}(\\hat V_{\\hat j})"} -{"pdf": "arxiv_math/2503.09483_pg1.pdf", "page": 1, "id": "2503.09483_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_k \\in \\mathbb{R}^{k_f\\times k_f}"} -{"pdf": "arxiv_math/2503.09528_pg16.pdf", "page": 1, "id": "2503.09528_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "P=\\{b^{l_1}_1/b^{l_2}_2: (l_1,l_2)\\in\\mathbb{Z}^2_{>0}\\}\\subset\\mathbb{C}."} -{"pdf": "arxiv_math/2503.09528_pg16.pdf", "page": 1, "id": "2503.09528_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "b_1,\\dots,b_k\\in\\mathbb{Z}[i]"} -{"pdf": "arxiv_math/2503.04245_pg7.pdf", "page": 1, "id": "2503.04245_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon} \\cdot \\underline{x} = \\sum_{i=1}^{5} \\varepsilon_i x_i \\leq 1, \\quad \\underline{x}\\in \\mathbb{H}_5"} -{"pdf": "arxiv_math/2503.04245_pg7.pdf", "page": 1, "id": "2503.04245_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Isom}(\\mathbb{H}_5)"} -{"pdf": "arxiv_math/2503.04245_pg7.pdf", "page": 1, "id": "2503.04245_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "h_x = % (1-\\norm{x}^2)^{-1} g_x"} -{"pdf": "arxiv_math/2503.04245_pg7.pdf", "page": 1, "id": "2503.04245_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon}"} -{"pdf": "arxiv_math/2503.04245_pg7.pdf", "page": 1, "id": "2503.04245_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod \\varepsilon_i=1"} -{"pdf": "arxiv_math/2503.07801_pg15.pdf", "page": 1, "id": "2503.07801_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "p+1 \\mid \\delta(p+1) \\mid (p-1)\\ell(p)."} -{"pdf": "arxiv_math/2503.07801_pg15.pdf", "page": 1, "id": "2503.07801_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\ell(p) \\ge \\frac{p+1}{2}."} -{"pdf": "arxiv_math/2503.07801_pg15.pdf", "page": 1, "id": "2503.07801_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(p-1)\\ell(p) = (p+1)\\ell(p) - 2\\ell(p)"} -{"pdf": "arxiv_math/2503.07801_pg15.pdf", "page": 1, "id": "2503.07801_pg15_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g_i\\ge\\prod_{\\substack{p~\\text{inert in~}K\\\\(p+1)\\mid M_{x_i}}}2>\\exp\\left((\\log2)\\exp\\left(C \\frac{\\log x_i}{\\log\\log x_i}\\right)\\right)>i"} -{"pdf": "arxiv_math/2503.07801_pg15.pdf", "page": 1, "id": "2503.07801_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L(g_i)\\le M_{x_i}\\le x_i^2=(\\log i)^{(4/C)\\log\\log\\log i}<(\\log g_i)^{c_0\\log\\log\\log g_i}"} -{"pdf": "arxiv_math/2503.07573_pg1.pdf", "page": 1, "id": "2503.07573_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{P_* T} \\alpha = \\int_T P^* \\alpha."} -{"pdf": "arxiv_math/2503.07573_pg1.pdf", "page": 1, "id": "2503.07573_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\alpha(x)| \\lesssim_N \\langle x\\rangle^{-N}"} -{"pdf": "arxiv_math/2503.07573_pg1.pdf", "page": 1, "id": "2503.07573_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha \\in \\mathscr S"} -{"pdf": "arxiv_math/2503.07573_pg1.pdf", "page": 1, "id": "2503.07573_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x\\rangle := \\sqrt{1 + |x|^2}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l} \\in \\mathbf{R}^{m}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\in \\mathbb{R}^{n \\times n}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l}_i= \\mathbf{u}_i"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x} \\in \\mathbb{R}^{n}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "A \\in \\mathbb{R}^{m \\times n}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} \\in \\mathbb{R}^{m}"} -{"pdf": "arxiv_math/2503.05941_pg1.pdf", "page": 1, "id": "2503.05941_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{q} \\in \\mathbb{R}^{n }"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{L}(n)\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}>\\tau_n)\\le \\bar{L}(|x-x_0|),\\qquad x\\in B^c_{r_0}(x_0)."} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "O\\subseteq B_{r_0}(x_0)"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(\\R^d)"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(B_{r_0}(x_0))"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(0,\\infty)\\times \\bar{B}_{r_0}(x_0)\\times\\bar{B}_{r_0}(x_0)"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}=\\infty)\\le \\frac{\\bar{L}(|x-x_0|)}{\\bar L(\\infty)}<1,\\qquad x\\in B^c_{r_0}(x_0),"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=\\mathcal{B}(B_{r_0}(x_0))."} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}<\\infty)>0"} -{"pdf": "arxiv_math/2503.06111_pg6.pdf", "page": 1, "id": "2503.06111_pg6_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in \\mathcal{B}(\\R^d)"} -{"pdf": "arxiv_math/2503.08227_pg2.pdf", "page": 1, "id": "2503.08227_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial f / \\partial \\vec{n}"} -{"pdf": "arxiv_math/2503.04674_pg19.pdf", "page": 1, "id": "2503.04674_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "t\\in [-\\frac{1}{2},0]"} -{"pdf": "arxiv_math/2503.04674_pg19.pdf", "page": 1, "id": "2503.04674_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(t,x,y)=\\mathrm{e}^{-t}x(1-x)y(1-y)"} -{"pdf": "arxiv_math/2503.06958_pg1.pdf", "page": 1, "id": "2503.06958_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta(\\varphi)=\\sup_{\\mu\\in \\mathcal{M}_T(X)}\\int \\varphi\\ d\\mu"} -{"pdf": "arxiv_math/2503.06958_pg1.pdf", "page": 1, "id": "2503.06958_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}^{\\mathbb{Z}^d}"} -{"pdf": "arxiv_math/2503.06958_pg1.pdf", "page": 1, "id": "2503.06958_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{M}_{{\\rm max}}(\\varphi)"} -{"pdf": "arxiv_math/2503.06958_pg1.pdf", "page": 1, "id": "2503.06958_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi:X\\rightarrow \\mathbb{R}"} -{"pdf": "arxiv_math/2503.05104_pg18.pdf", "page": 1, "id": "2503.05104_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = 0.3, 0.6, 0.9"} -{"pdf": "arxiv_math/2503.05885_pg8.pdf", "page": 1, "id": "2503.05885_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{B}^\\nu_{h,\\alpha} := \\big\\{ r \\in [1,\\infty) : m^\\nu([r, r+h]) \\geq \\frac{\\alpha h}{ r}\\big\\}"} -{"pdf": "arxiv_math/2503.05885_pg8.pdf", "page": 1, "id": "2503.05885_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{1,R}(\\overline{B}^\\nu_{h,\\alpha}) \\leq C\\alpha^{-1}."} -{"pdf": 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"math", "max_diffs": 0, "checked": null, "math": "B_{v_{\\infty}} \\coloneq E[p^{\\infty}]^{G_{K_{\\infty, v_{\\infty}}}}"} -{"pdf": "arxiv_math/2503.09034_pg7.pdf", "page": 1, "id": "2503.09034_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n,v_{n}}"} -{"pdf": "arxiv_math/2503.09034_pg7.pdf", "page": 1, "id": "2503.09034_pg7_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in \\Sigma \\setminus \\{ p \\}"} -{"pdf": "arxiv_math/2503.09034_pg7.pdf", "page": 1, "id": "2503.09034_pg7_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n, v_{n}}"} -{"pdf": "arxiv_math/2503.09034_pg7.pdf", "page": 1, "id": "2503.09034_pg7_math_019", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])"} -{"pdf": "arxiv_math/2503.09034_pg7.pdf", "page": 1, "id": "2503.09034_pg7_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": 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"\\tilde{\\mathbb{H}}^k\\mathbb{E}^0 = (\\mathbb{E}^0\\otimes 1)\\tilde{\\mathbb{H}}^k"} -{"pdf": "arxiv_math/2503.08457_pg18.pdf", "page": 1, "id": "2503.08457_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(X, \\underline{F})"} -{"pdf": "arxiv_math/2503.08457_pg18.pdf", "page": 1, "id": "2503.08457_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\mathcal{S}}_X(X)"} -{"pdf": "arxiv_math/2503.08457_pg18.pdf", "page": 1, "id": "2503.08457_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\mathcal{S}}_X"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "X(\\theta_{i}): \\begin{bmatrix}x_{i} \\\\ p_{i} \\end{bmatrix} \\mapsto \\begin{bmatrix} \\cos(\\theta_{i}) & \\sin(\\theta_{i}) \\\\ -\\sin(\\theta_{i}) & \\cos(\\theta_i) \\end{bmatrix}\\begin{bmatrix}x_{i} \\\\ p_{i} \\end{bmatrix}."} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "F_{SL(2n)} \\coloneqq \\frac{E_{SL(2n)}}{E[f_{0}]}"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{Sp(2)} = E_{SL(2)}"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}\\circ \\phi^{-1} = \\frac{1}{|B(R)|} \\Theta(R^2-\\epsilon^{-1/2}(x^2+p_{x}^2+p_{y}^2)-\\epsilon^{3/2}y^2)"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{i}^{H} \\lambda_{n+1-i}^{V} \\equiv \\text{const}"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}(\\mathbf{z},R) = \\frac{6}{R^2 |B(R)|} \\chi_{B(R)} = \\frac{6}{R^2|B(R)|} \\Theta(R^2-|\\mathbf{z}|^2)"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\epsilon) = \\text{diag}(1,\\epsilon^2,1,1)"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{V} = \\text{diag}( 1, \\epsilon^{-1/2}, 1, \\epsilon^{1/2})"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}(\\mathbf{z},\\epsilon) = x^2 + \\epsilon^2 y^2 + p_{x}^2 + p_{y}^2"} -{"pdf": "arxiv_math/2503.07965_pg7.pdf", "page": 1, "id": "2503.07965_pg7_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "E[f_{0}] = (3 + \\epsilon)"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_2= \\frac{D_{\\rm W}}{4}"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta = \\frac{c^2}{16\\pi^2 f_c^2 }"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\psi}_m= (x_m,y_m,0)"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{h}_{mk}=\\alpha_{mk}h_{mk}"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\boldsymbol \\psi}_m^{\\rm Pin}=(x_m,\\beta_m,d)"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1=-\\frac{D_{\\rm W}}{4}"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_m=-\\frac{D_{\\rm W}}{2}+(m-1)\\frac{D_{\\rm W}}{M}+\\frac{D_{\\rm W}}{2M}"} -{"pdf": "arxiv_math/2503.08554_pg1.pdf", "page": 1, "id": "2503.08554_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{mk}=\\frac{\\sqrt{\\eta} e^{-2\\pi j \\left(\\frac{ 1}{\\lambda}\\left| {\\boldsymbol \\psi}_m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| +\\frac{1}{\\lambda_g}\\left| {\\boldsymbol \\psi}_0^{\\rm Pin} - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| \\right)}}{ \\left| {\\boldsymbol \\psi} _m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right|}"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{B}(\\mathcal{H})"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_s\\,\\mathcal{E}_t \\subseteq \\mathcal{E}_{s+t}"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}, \\quad\\text{and for each integer } t \\ge 1,"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\mathcal{E}_t\\}_{t \\ge 0} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H}),"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_t^* = \\mathcal{E}_t"} -{"pdf": "arxiv_math/2503.08736_pg1.pdf", "page": 1, "id": "2503.08736_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H}),"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in\\{\\frac{1}{3},\\frac{2}{3}\\}"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{\\bm{y}}[F(\\bm{y})]"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\bm{y})=G(u^s(\\cdot,\\bm{y}))=u^s(x_0,\\bm{y}),"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1,\\ldots,y_s\\overset{i.i.d.}{\\sim}N(0,1)"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a^s(x,\\bm{y})= \\exp\\left(\\sum_{j=1}^{s}\\frac{1}{j^2}\\sin(2j\\pi x)y_j \\right),"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_j(\\frac{n}{N})"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_1 = \\frac{1}{2}\\left(b_1+\\sqrt{b_1^2+1-\\frac{1}{2\\lambda}}\\right),"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "u^s(0,\\bm{y})=u^s(1,\\bm{y})=0."} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\frac{d}{dx}(a^s(x,\\bm{y})\\frac{du^s(x,\\bm{y})}{dx})=1,"} -{"pdf": "arxiv_math/2503.05334_pg17.pdf", "page": 1, "id": "2503.05334_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{y}\\in\\mathbb{R}^s"} -{"pdf": "arxiv_math/2503.05610_pg5.pdf", "page": 1, "id": "2503.05610_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\inf\\{|\\lambda-\\lambda'|:\\lambda\\neq\\lambda',\\lambda,\\lambda'\\in \\sigma(\\Delta)\\}=0"} -{"pdf": "arxiv_math/2503.05610_pg5.pdf", "page": 1, "id": "2503.05610_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^k(x_1)\\leq\\phi_\\zeta^k(\\gamma_m)\\leq\\phi_\\zeta^k(x_2), \\forall k"} -{"pdf": "arxiv_math/2503.05610_pg5.pdf", "page": 1, "id": "2503.05610_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^m(x_j)\\to\\zeta, j=1,2"} -{"pdf": "arxiv_math/2503.05610_pg5.pdf", "page": 1, "id": "2503.05610_pg5_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "x_1, x_2 \\in \\sigma(\\Delta_n)"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "not being defined on the whole"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "where the left value is the result of the Riemann integral computed in"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\int_{[a, b]} f \\right)^{M} = \\left( \\int_{[a, b]} g \\right)^{N} \\! \\! \\! \\!,"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "we have a bounded real-valued function defined on some rectangle"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "is unique except in a Lebesgue measure zero set: if in"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ", then there exists some measure zero set"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": ", and the right one is the result of the integral computed in"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "is a \\emph{Boolean algebra} if"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bounded function. Then,"} -{"pdf": "arxiv_math/2503.08799_pg3.pdf", "page": 1, "id": "2503.08799_pg3_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "is another Riemann integrable function on"} -{"pdf": "arxiv_math/2503.07030_pg8.pdf", "page": 1, "id": "2503.07030_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\Phi^{T_F},\\mathbf{u},\\mathbf{y},\\mathbf{p})=0"} -{"pdf": "arxiv_math/2503.07030_pg8.pdf", "page": 1, "id": "2503.07030_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\sigma}_\\alpha (u^k,y^k,\\mathbf{p})"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_2"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "R=\\{(\\alpha,\\beta,\\gamma)\\in[0,1]^3:\\alpha\\beta+\\gamma>1,\\alpha\\gamma+\\beta>1,\\beta\\gamma+\\alpha>1\\},"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E(H)\\subset \\binom{[n]}{r}"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_1"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma)=\\alpha+\\beta+\\gamma-2"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha=1-1/n+\\alpha'n^{(\\delta_a-2)},\\beta=1-1/n+\\beta'n^{(\\delta_b-2)},\\gamma=1-1/n+\\gamma'n^{(\\delta_c-2)}"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|V_1|=|V_2|=\\cdots=|V_r|=n"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|B_1|=\\alpha n^2-n(n-1)"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma) = 2\\sqrt{\\alpha\\beta(1-\\gamma)}+2\\gamma-2"} -{"pdf": "arxiv_math/2503.05218_pg5.pdf", "page": 1, "id": "2503.05218_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma)"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "H_i(x, 1) = K_i(x,1) \\in A"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\in \\complement \\overline{V_i} \\cap U_i"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_i(\\overline{W_i}) = 1"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_i(u,t) = \\begin{cases} p_i(H(u,2t)) & \\text{if } 0 \\leq t \\leq \\frac12 \\\\ G(h_i(u), 2t-1) & \\text{if } \\frac12 \\leq t \\leq 1. \\end{cases}"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi\\colon X\\to T^n(\\mu)"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "L_i \\colon U_i \\times I\\to Z_h"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\complement V_i= X\\backslash V_i"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(X) \\subseteq T^n(\\mu)"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(U_i)_{0\\leq i \\leq n}"} -{"pdf": "arxiv_math/2503.06969_pg11.pdf", "page": 1, "id": "2503.06969_pg11_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "p_i(H(u,1)) = p_i ((t_n \\circ \\Phi)(u)) = h_i(u) = G(h_i(u),0)"} -{"pdf": "arxiv_math/2503.07910_pg7.pdf", "page": 1, "id": "2503.07910_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c \\in \\mathbb{F}_p^{\\times}"} -{"pdf": "arxiv_math/2503.07910_pg7.pdf", "page": 1, "id": "2503.07910_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x)"} 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S^2\\setminus B(p_-, h(Lr))\\subset S^2\\setminus B(p_-, Lr)"} -{"pdf": "arxiv_math/2503.08553_pg25.pdf", "page": 1, "id": "2503.08553_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta\\coloneqq \\min\\left\\{\\frac{ l_0^2}{2\\pi}, \\frac{\\varepsilon}{10 C_{\\mathbf{I}}}\\right\\} \\quad\\text{and}\\quad L\\coloneqq 2e^{\\delta^{-1}k_\\mathbf{I}(e_\\mathbf{I}(\\varphi) + 10^{-1}\\varepsilon)},"} -{"pdf": "arxiv_math/2503.08553_pg25.pdf", "page": 1, "id": "2503.08553_pg25_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta\\colon S^2\\to S^2"} -{"pdf": "arxiv_math/2503.07509_pg2.pdf", "page": 1, "id": "2503.07509_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{s} \\in [0, 1]^d"} -{"pdf": "arxiv_math/2503.07509_pg2.pdf", "page": 1, "id": "2503.07509_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x = \\sqrt{\\alpha P}s_1 +\\sqrt{\\bar \\alpha P}s_2"} -{"pdf": 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s \\subset [d]\\}"} -{"pdf": "arxiv_math/2503.09181_pg5.pdf", "page": 1, "id": "2503.09181_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat y^{(n)} = f(x_s^{(n)},z^{(n)})"} -{"pdf": "arxiv_math/2503.09181_pg5.pdf", "page": 1, "id": "2503.09181_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{\\theta,\\phi} \\mathbb{E}_{\\mathbf{y}, \\mathbf{x}} l(f_\\theta(\\mathbf{x}_s,z),\\mathbf{y}),"} -{"pdf": "arxiv_math/2503.09181_pg5.pdf", "page": 1, "id": "2503.09181_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s \\leftarrow s \\cup \\pi(x_s,z)"} -{"pdf": "arxiv_math/2503.09181_pg5.pdf", "page": 1, "id": "2503.09181_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x^{(n)}_s, z^{(n)}) \\in \\Lambda^{(n)}"} -{"pdf": "arxiv_math/2503.06825_pg2.pdf", "page": 1, "id": "2503.06825_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v, w \\in \\mathbb{R}^n"} 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"arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "L=0<\\sqrt{\\mu_{s,t}}\\,"} +{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{s,t}^{\\frac{1}{2_s^*(t)-2}}u"} +{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d(u_k,\\mathcal{M})\\leq d(u_k,0)=\\|u_k\\|_{\\dot{H}^s}=\\sqrt{\\mu_{s,t}}<\\infty"} +{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "I_{s,t}(u_n)\\to \\beta"} +{"pdf": "arxiv_math/2503.06716_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06716", "page": 1, "id": "2503.06716_pg12_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u_k\\|^2_{\\dot{H}^s}=\\mu_{s,t}"} +{"pdf": "arxiv_math/2503.08597_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08597", "page": 1, "id": "2503.08597_pg10_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\Sigma} \\triangleq \\max_{(d_1,\\cdots, d_K)\\in \\mathcal{D}}(d_1+\\cdots +d_K)"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf T}=\\{e^{t T}, t \\in \\R \\}"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi \\in \\mathfrak{k}"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\R H \\oplus \\R L"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p(\\R_{\\bf T}) = \\overline{p(\\Gamma)}"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Psi \\in \\mathfrak{k}"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lfloor \\frac{n}{2} \\rfloor"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf T}=\\lambda+\\Phi+\\omega"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R_{\\bf H} \\times \\R_{\\bf L}"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{p(\\Gamma)} = \\R"} +{"pdf": "arxiv_math/2503.08614_pg8.pdf", "url": "https://arxiv.org/pdf/2503.08614", "page": 1, "id": "2503.08614_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R_{\\bf H} \\times \\R_{\\bf L} \\simeq \\R^2"} +{"pdf": "arxiv_math/2503.05558_pg18.pdf", "url": "https://arxiv.org/pdf/2503.05558", "page": 1, "id": "2503.05558_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "E = 41.9 log_2(B)^{-1}-1.92"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde \\Omega, \\tilde{\\mathscr{F}}, \\tilde{\\mathbb{P}})"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma: [0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^{d \\times \\tilde{d}}"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "b:[0, T]\\times \\mathbb{R}^d \\times S \\mapsto \\mathbb{R}^d"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r:=\\{r(t)\\}_{t\\geq 0}"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^r := \\{\\tilde{\\mathscr F}_t^r\\}_{t\\geq 0}"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathbb{F}}^{B}:=\\{\\tilde{\\mathscr F}_t^{B}\\}_{t\\geq 0}"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "B:=\\{B(t)\\}_{t\\geq 0}"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{q}_{j_0 k_0}\\geq 0"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_0^B"} +{"pdf": "arxiv_math/2503.06658_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06658", "page": 1, "id": "2503.06658_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\mathscr F}_t:=\\tilde{\\mathscr F}_t^{B} \\vee \\tilde{\\mathscr F}_t^r"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f_\\pm(x_1,x_2)=\\pm \\Delta(x_1,x_2)(x_1,x_2,1),"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "m(u,v)=(u^2+v^2+1)^{-\\frac{1}{2}(n-1)}"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta(x_1,x_2)=(x_1^2+x_2^2+1)^{-\\frac{1}{2}}"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\operatorname{Fix}(Q)\\subset\\mathbb{R}^m"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_i:U_i\\rightarrow\\mathbb{R}^2"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "Q(x_1,\\dots,x_m)=(x_1,\\dots,x_r,-x_{r+1},\\dots,-x_m)."} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_+=\\{y\\in\\mathbb{S}^2:y_3>0\\}"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} &v^n\\;m(u,v)\\left(Q\\left(\\frac{1}{v},\\frac{u}{v}\\right)-uP\\left(\\frac{1}{v},\\frac{u}{v}\\right),-vP\\left(\\frac{1}{v},\\frac{u}{v}\\right)\\right) \\text{ in } U_1, \\\\ &v^n\\;m(u,v)\\left(P\\left(\\frac{u}{v},\\frac{1}{v}\\right)-uQ\\left(\\frac{u}{v},\\frac{1}{v}\\right),-vQ\\left(\\frac{u}{v},\\frac{1}{v}\\right)\\right) \\text{ in } U_2, \\\\ &m(u,v)(P(u,v),Q(u,v)) \\text{ in } U_3, \\end{aligned}"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{S}^2=\\{y=(y_1,y_2,y_3)\\in\\mathbb{R}^3:y_1^2+y_2^2+y_3^2=1\\}"} +{"pdf": "arxiv_math/2503.05436_pg25.pdf", "url": "https://arxiv.org/pdf/2503.05436", "page": 1, "id": "2503.05436_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H_-=\\{y\\in\\mathbb{S}^2:y_3<0\\}"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\left (\\bigcup_{x\\in \\partial M} i |P_n\\left [ -\\sqrt{\\max(0,N^2)},\\sqrt{\\max(0,N^2)}\\right ]\\right )^c"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 w_v) = 0."} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\cdot (\\rho_0 z_v) + \\frac{g_0'}{c^2} \\cdot \\nabla \\times (\\rho_0 w_v) + \\frac{\\rho_0 g_0'}{c^2} \\cdot z_v = 0 , \\quad n \\cdot z_v|_{\\partial M} = 0."} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla \\times (\\rho_0 z_v) = 0."} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "w \\in \\operatorname{Ker}(T)"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\sigma_{ess}(L)^c"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "w = \\nabla \\times (\\rho_0 w_v) + \\nabla \\varphi_v"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_0 z_v = \\nabla \\varphi_v"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\operatorname{Ker}(T)"} +{"pdf": "arxiv_math/2503.05428_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05428", "page": 1, "id": "2503.05428_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "u = \\nabla \\times (\\rho_0 w_u) + \\rho_0 z_u"} +{"pdf": "arxiv_math/2503.04438_pg16.pdf", "url": "https://arxiv.org/pdf/2503.04438", "page": 1, "id": "2503.04438_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "O^\\sigma = L(O_{h\\Sigma_n})\\times_{LB\\Sigma_n}\\{\\sigma\\}"} +{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "i \\in \\set{1,\\ldots,n}"} +{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "w_{k+1} := w_k + \\tau_k r(w_k)"} +{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "2 (w_l - w_r) \\frac{q_{1,1}^e q_{2,2}^e - q_{2,1}^e q_{1,2}^e}{2} ."} +{"pdf": "arxiv_math/2503.07105_pg18.pdf", "url": "https://arxiv.org/pdf/2503.07105", "page": 1, "id": "2503.07105_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(w_{k+1} - w_k)/\\tau_k = r(w_k)"} +{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{XY}=\\begin{pmatrix} 1-s & 0 \\\\ sd & s(1-d) \\end{pmatrix}"} +{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_2=\\frac{m-svt}{1-st}"} +{"pdf": "arxiv_math/2503.03754_pg10.pdf", "url": "https://arxiv.org/pdf/2503.03754", "page": 1, "id": "2503.03754_pg10_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s=\\frac{m-x_2}{t(x_1-x_2)}."} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{2}\\delta x_{s}^{1}(\\xi_i)}"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l}\\delta x_{s}^{l-1}(\\xi_i)}"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{Z}^{W\\delta x}"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{det}(C) \\neq 0"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathring{\\chi}_{t, i}"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{L\\top}dh_{s}^{L}(\\xi_i)}"} +{"pdf": "arxiv_math/2503.09565_pg19.pdf", "url": "https://arxiv.org/pdf/2503.09565", "page": 1, "id": "2503.09565_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{Z}^{W_{0}^{l\\top}dh_{s}^{l}(\\xi_i)}"} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u}\\in V\\cap H^2(D)^2."} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H_N = \\text{span\\;}\\{e_k : k = 1,\\dots, N \\}"} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} &|\\sigma(\\mathbf{u})|^2=\\sum_{k=1}^{m}|\\sigma_k(\\mathbf{u})|^2\\leq B_0,\\text{ for all }\\mathbf{u}\\in H;\\\\ &|\\sigma(\\mathbf{u})-\\sigma(\\mathbf{v})|^2=\\sum_{k=1}^{m}|\\sigma_k(\\mathbf{u})-\\sigma_k(\\mathbf{v})|^2\\leq L|\\mathbf{u}-\\mathbf{v}|^2,\\text{ for all }\\mathbf{u},\\mathbf{v}\\in H. \\end{aligned}"} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sigma(\\mathbf{u})^{-1}(P_N\\mathbf{w})|\\leq C_0|P_N\\mathbf{w}|,\\text{ for all }\\mathbf{u},\\mathbf{w}\\in H."} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V := \\{\\mathbf{u}\\in H^1(D)^2 :\\nabla\\cdot\\mathbf{u}=0,\\; \\mathbf{u}|_{\\partial D}=0\\}."} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "P_NH\\subset \\text{Range}\\;(\\sigma(\\mathbf{u})),\\text{ for all }\\mathbf{u}\\in H."} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_k^{-1}:P_NH\\rightarrow H"} +{"pdf": "arxiv_math/2503.09285_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09285", "page": 1, "id": "2503.09285_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma=(\\sigma_1,\\dots,\\sigma_m)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma_t(\\cdot )=\\sigma(\\cdot ,t)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x,T\\circ (\\sigma_t)^{-1}(x)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\sigma(E,t)} \\rho_0(x)\\,dx=\\int_E \\rho_0(x)\\,dx"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{T^{-1}(E)}\\rho_0(x)\\,dx=\\int_E \\rho_1(x)\\,dx"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "T:(\\Omega_0,\\rho_0)\\to (\\Omega_1,\\rho_1)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "T\\circ (\\sigma_t)^{-1}"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "T\\circ (\\sigma_t)^{-1}:(\\Omega_0,\\rho_0)\\to (\\Omega_1,\\rho_1)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(y_1,t+0)=\\sigma(y_2,t+0)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_0(\\sigma(z,t))\\,J_\\sigma(z)=\\rho_0(z)"} +{"pdf": "arxiv_math/2503.04536_pg9.pdf", "url": "https://arxiv.org/pdf/2503.04536", "page": 1, "id": "2503.04536_pg9_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(t-\\delta,t+\\delta)\\subset J"} +{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1(X, \\mathscr O_X) = 0"} +{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1( X;\\mathbb C) = 0"} +{"pdf": "arxiv_math/2503.03861_pg30.pdf", "url": "https://arxiv.org/pdf/2503.03861", "page": 1, "id": "2503.03861_pg30_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "R := \\mathbb Z[1/2|G|]"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{n}(t)=[n_0(t),n_1(t),\\cdots,n_{M_r-1}(t)]^T"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{t,m}=\\sum_{i=0}^{m}d_{t,i},m\\in \\mathcal{M}_t\\triangleq[1,2,\\cdots,M_t-1]"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta\\in[-\\frac{\\pi}{2},\\frac{\\pi}{2}]"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\hat{\\tau},\\hat{v},\\theta)"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q} \\in \\mathbb{K}"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{C}^{n\\times m}"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}_{r}=[x_{r,0},x_{r,1},\\cdots,x_{r,M_r-1}]^T"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{a}(\\mathbf{x}_t,\\alpha)=[1,e^{j\\frac{2\\pi}{\\lambda}x_{t,1}\\sin\\alpha},\\cdots,e^{j\\frac{2\\pi}{\\lambda}x_{t,M_t-1}\\sin\\alpha}]^T"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "x_{r,m}=\\frac{\\lambda}{2}m,m\\in\\mathcal{M}_r"} +{"pdf": "arxiv_math/2503.04407_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04407", "page": 1, "id": "2503.04407_pg3_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{m,q}\\neq c_{m',q}, \\forall q, m\\neq m'"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_0 \\in C^0(\\overline\\Omega)"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_h^0 = \\pi^h \\varphi_0"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta_o = |\\max_o |\\phi_h^n| - 1 | > 0.5"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "h_c = \\frac{L_d}{N_c}"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "N_f = 8N_c = 2^{3+k} L_d"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\prod_{i=1}^d (0,L_i) \\subset \\mathbb R^d"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "S^h = \\{ \\zeta \\in C^0(\\overline{\\Omega}) \\, : \\, \\zeta \\vert_{o} \\in P_1(o) \\quad \\forall o \\in \\mathcal{T}_h\\},"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\phi_h^{n+1}, \\mu_h^{n+1}) \\in S^h \\times S^h"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "h_f = \\frac{L_d}{N_f}"} +{"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_0 : \\overline\\Omega \\to \\mathbb R"} +{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."} +{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ", this means that there is no prior knowledge about the ideal control. Furthermore, the coefficient"} +{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."} +{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": ". The desired control function"} +{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\dfrac{\\partial}{\\partial z}"} +{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=g(z)\\dfrac{\\partial}{\\partial z}"} +{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f(z)=1/\\left(1-e^z\\right)"} +{"pdf": "arxiv_math/2503.09424_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09424", "page": 1, "id": "2503.09424_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{2,\\ldots,n-1\\}"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k+1}(f)^2\\leq \\lambda _k(f)\\lambda _{k+1}(f)"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\times f:X\\times X\\rightarrow X\\times X"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\max _{j=0,\\ldots ,n}\\lambda _j(Fr_q^s\\circ f^t)"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k}(f)=\\lambda _k(f)"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda _k(Fr_q^s\\circ f^t)"} +{"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "b_0,b_1,\\ldots ,b_{2n}"} +{"pdf": "arxiv_math/2503.07583_pg22.pdf", "url": "https://arxiv.org/pdf/2503.07583", "page": 1, "id": "2503.07583_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} A_{a,b,c} = 2(a+b+c) + 3 \\\\ C_{a,b|c,d} = 3 (2a+1)(2b+1) \\\\ G_{a} = (2a+1)(2a^2 + 2a + 1) \\end{aligned}"} +{"pdf": "arxiv_math/2503.07583_pg22.pdf", "url": "https://arxiv.org/pdf/2503.07583", "page": 1, "id": "2503.07583_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_{[8,7]} = 8^3 + 7^3 = 855 = 9^3 + 5^3 + 1^3 = \\varepsilon_{[9,5,1]}"} +{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": 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"page": 1, "id": "2503.05183_pg15_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Z \\in \\mathbb{R}^{n_2 \\times r \\times b}"} +{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z^t)=\\Gamma(\\Z^{t+1})"} +{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma(\\Z^t)\\neq\\Gamma(\\Z^{t+1})"} +{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": "2503.05183_pg15_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\Z^{t+1}-\\Z^t\\| < \\min\\{(2\\hat{\\lambda}_4(1-p))^{1/(2-p)}, \\nu\\}"} +{"pdf": "arxiv_math/2503.05183_pg15.pdf", "url": "https://arxiv.org/pdf/2503.05183", "page": 1, "id": 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\\alpha_h}+\\mathcal{J}_h\\left(\\dfrac{\\partial F}{\\partial\\beta_h}\\right)\\right)."} +{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "F\\in Stem(D)\\cap\\mathcal{C}^1(D)"} +{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "z=(z_1,\\dots,z_n)\\in\\mathbb{C}^n"} +{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{e_K\\}_{K\\in\\mathcal{P}(n)}"} +{"pdf": "arxiv_math/2503.04329_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04329", "page": 1, "id": "2503.04329_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J}=\\left\\{\\mathcal{J}_h: 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"2503.05140_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{s}_{C, 0}(p, q, r) < 0"} +{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "1 \\leq r < p \\leq q \\leq \\infty"} +{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "1 + (1 + \\omega)\\left(\\frac{1}{q} - \\frac{1}{p}\\right) > 0"} +{"pdf": "arxiv_math/2503.05140_pg22.pdf", "url": "https://arxiv.org/pdf/2503.05140", "page": 1, "id": "2503.05140_pg22_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} \\frac{1}{q} > \\frac{2}{3p} - \\frac{1}{6}, & \\text{for } q \\geq 3p'; \\\\ \\frac{1}{q} > \\frac{1}{p} - \\frac{1}{3}, & \\text{for } p' < q < 3p'; \\\\ \\frac{1}{q} > \\frac{2}{p} - 1, & \\text{for } q \\leq p'. \\end{cases}"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "b_0 = 4(y_0 + y_1 - 2y_3), b_1 = 4(y_0 - y_3 + y_4 - y_5), b_2 = - 3y_0 - y_1 + 4y_3"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b_3 = 4(y_0 - y_3 + y_4 - y_5), b_4 = 4(y_0 + y_2 - 2y_5), b_5 = -3y_0 - y_2 + 4y_5"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "a_0 = 4(x_0 + x_1 - 2x_3), a_1 = 4(x_0 - x_3 + x_4 - x_5), a_2 = - 3x_0 - x_1 + 4x_3"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_0 = \\arctan(\\sqrt{\\frac{\\phi_0}{\\phi_2}} \\phi_1)"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{0}^{1} (\\xi + \\phi_1)^p \\sqrt{\\phi_0(\\xi + \\phi_1 )^2 + \\phi_2}\\rm{d}\\xi = (\\frac{\\phi_2}{\\phi_0})^{\\frac{p+1}{2}} \\sqrt{\\phi_2} \\int_{\\theta_0}^{\\theta_1} tan^p \\theta sec^3 \\theta \\rm{d} \\theta,"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\frac{\\partial x}{\\partial \\xi} = a_0\\xi + a_1\\eta + a_2, \\\\ \\frac{\\partial x}{\\partial \\eta} = a_3\\xi + a_4\\eta + a_5, \\\\ \\frac{\\partial y}{\\partial \\xi} = b_0\\xi + b_1\\eta + b_2, \\\\ \\frac{\\partial y}{\\partial \\eta} = b_3\\xi + b_4\\eta + b_5, \\\\ \\end{aligned}"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a_3 = 4(x_0 - x_3 + x_4 - x_5), a_4 = 4(x_0 + x_2 - 2x_5), a_5 = -3x_0 - x_2 + 4x_5"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "I_n = \\int \\sec^n \\theta \\rm{d} \\theta = \\frac{1}{n - 1} (tan\\theta \\sec^{n - 2} \\theta + (n - 2)I_{n-2}). \\nonumber"} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": "https://arxiv.org/pdf/2503.04493", "page": 1, "id": "2503.04493_pg31_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int \\tan^{2k+1} \\theta \\sec^3 \\theta \\rm{d} \\theta = \\int (sec^2 \\theta - 1)^k sec^2 \\theta \\rm{d} \\sec \\theta."} +{"pdf": "arxiv_math/2503.04493_pg31.pdf", "url": 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Suppose that random variables"} +{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in \\set{0} \\cup \\N \\cup \\set{\\infty}"} +{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "z = (z_1, \\dots, z_n)"} +{"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta = (\\zeta_1, \\dots, \\zeta_n)"} +{"pdf": "arxiv_math/2503.07897_pg25.pdf", "url": "https://arxiv.org/pdf/2503.07897", "page": 1, "id": "2503.07897_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_r(0)=0 \\quad\\forall r\\geq 0"} +{"pdf": "arxiv_math/2503.07897_pg25.pdf", "url": 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"https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}''_2(z_1,z_2)"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}''_1(z_1,z_2), \\tilde{\\gamma}''_2(z_1,z_2))"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma}_1''=\\tilde{\\gamma}''_1(0,0)"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma_2''=\\tilde{\\gamma}''_2(0,0)"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1,z_2)=(0,0) \\in [0,1]_{z_1} \\times [0,1]_{x_2}"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde\\gamma'_i(z_1,z_2)= a_i(z_1,z_2)\\sharp b_i(z_1,z_2)"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(z_1, 0) \\in [0,1] \\times [0,1]"} +{"pdf": "arxiv_math/2503.07543_pg42.pdf", "url": "https://arxiv.org/pdf/2503.07543", "page": 1, "id": "2503.07543_pg42_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tilde{\\gamma}'_1, \\tilde{\\gamma}'_2)=(\\tilde{\\gamma}'_1(0,0), \\tilde{\\gamma}'_2(0,0))"} +{"pdf": "arxiv_math/2503.05716_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05716", 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"id": "2503.07345_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi\\in C^\\infty([0,\\infty))"} +{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(r)\\sim cr^{\\frac32}"} +{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "r^{\\frac32}, r^{-\\frac12}"} +{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle L_1 \\phi , \\chi_b(r) r^{\\frac12}\\rho_1\\rangle = \\lambda_0 \\langle \\phi, \\chi_b(r) r^{\\frac12}\\rho_1\\rangle"} +{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1(r^{\\frac12}\\rho_1)=0"} +{"pdf": "arxiv_math/2503.07345_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07345", "page": 1, "id": "2503.07345_pg12_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "L_1\\phi =\\lambda_0\\phi"} +{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "q = (0.62,0.18,0.1,0.1)"} +{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-(1+\\frac{2}{3})^{-1} = 0.4"} +{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat q = (0.5,0.5,0,0)"} +{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{2}(\\omega )"} +{"pdf": "arxiv_math/2503.08272_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08272", "page": 1, "id": "2503.08272_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta R^{1}(\\omega )"} +{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} \\big(D^X(s-)+\\tfrac12 \\gamma(s) \\Delta X(s)\\big)^\\top \\Delta X(s) & = D^X(s-)\\Delta X(s-) + \\tfrac12 (\\Delta X(s))^\\top \\gamma(s) \\Delta X(s) . \\end{split}"} +{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "C(x,d,X) = \\int_{[0,T]} (D^X(s-))^\\top\\, dX(s) + \\frac12 \\int_{[0,T]} (\\Delta X(s))^\\top \\gamma(s)\\, dX(s) ."} +{"pdf": "arxiv_math/2503.05594_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05594", "page": 1, "id": "2503.05594_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "D^X(s)=D^X(s-)+\\Delta D^X(s) = D^X(s-)+ \\gamma(s)\\Delta X(s)."} +{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{u}(u) := \\rho \\left[ \\frac{u}{\\rho}\\right]"} +{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l: \\mathbb{R} \\mapsto \\mathbb{R}"} +{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_{x_j} f_j(k)"} +{"pdf": "arxiv_math/2503.06167_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06167", "page": 1, "id": "2503.06167_pg12_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_l(\\partial_{x_j} f_j(k))"} +{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi=\\mathrm{BC}_E(\\pi)"} +{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi=\\bigotimes_v\\pi_v"} +{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Pi_{v_1}\\simeq \\pi_{v_1}\\otimes \\pi_{v_1}"} +{"pdf": "arxiv_math/2503.09500_pg70.pdf", "url": "https://arxiv.org/pdf/2503.09500", "page": 1, "id": "2503.09500_pg70_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f' = \\sum_\\tau p_{\\tau,!}(f^\\tau)"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = E_{\\Gamma_0}(\\theta) + \\mu_j"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta)=2\\sum_{i=1}^d \\cos 2\\pi\\theta_i"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_{\\Gamma_0}(\\theta) = 2\\cos 2\\pi\\theta_1+2\\cos 2\\pi\\theta_2+2\\cos 2\\pi(\\theta_1+\\theta_2)"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = (1+\\mu_j) E_{\\Gamma_0}(\\theta)+\\mu_j"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_3=\\Gamma_0 \\boxtimes G_F"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_1 = \\Gamma_0 \\mathop\\square G_F"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle a\\rangle_p = \\sum_{q=1}^{\\nu_F} \\langle a(\\cdot+v_q)\\rangle \\sum_{s=1}^{\\nu'_F} |P_{\\mu_s}(v_p,v_q)|^2\\,,"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_2 = \\Gamma_0 \\times G_F"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "E_j(\\theta) = \\mu_j E_{\\Gamma_0}(\\theta)"} +{"pdf": "arxiv_math/2503.08595_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08595", "page": 1, "id": "2503.08595_pg4_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "P_{\\mu_s}(v_p,v_q) = \\sum_{j\\,\\mu_j=\\mu_s} w_j(v_p)\\overline{w_j(v_q)}"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\langle L_I,h\\rangle| \\leq \\alpha \\int_0^T \\int_{\\mathbb{R}^N} |\\xi(t,x)|dx dt + 2\\beta \\|I\\|_{L^\\infty}\\|h\\|_{L^\\infty}T + \\gamma \\int_{\\mathbb{R}^N} |\\xi(T,x)|dx \\leq C_1 \\|h\\|_{L^\\infty(0,T)},"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I \\in \\mathcal{U}_{ad}"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi_I(\\varepsilon)= \\dfrac{J(I^\\varepsilon) - J(I)}{\\varepsilon}"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "h_1, h_2 \\in L^{\\infty}(0,T)"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "h = h_1 + \\lambda h_2"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in \\mathbb{R}"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "I^\\varepsilon = I + \\varepsilon h"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{\\varepsilon \\to 0} \\left\\|\\frac{G(I + \\varepsilon h) - G(I)}{\\varepsilon} - (\\xi, \\eta)\\right\\|_X = \\lim_{\\varepsilon \\to 0} \\left\\|(\\tilde{\\xi}, \\tilde{\\eta})\\right\\|_X = 0."} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L_I: L^{\\infty}(0,T) \\to \\mathbb{R}"} +{"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(p^\\varepsilon, d^\\varepsilon) = G(I^\\varepsilon)"} +{"pdf": "arxiv_math/2503.06055_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06055", "page": 1, "id": "2503.06055_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_\\tau \\, v \\preceq T_\\sigma \\, v"} +{"pdf": "arxiv_math/2503.04122_pg13.pdf", "url": "https://arxiv.org/pdf/2503.04122", "page": 1, "id": "2503.04122_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g\\in\\{2,3,\\cdots 10\\}"} +{"pdf": "arxiv_math/2503.04122_pg13.pdf", "url": "https://arxiv.org/pdf/2503.04122", "page": 1, "id": "2503.04122_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": ". A straightforward computation leads to this initial segment and the corresponding sequence of differences:"} +{"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_m(j(E_1), j(E_2))"} +{"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\gg_\\epsilon( \\# S)^{5+\\epsilon}"} +{"pdf": "arxiv_math/2503.04567_pg38.pdf", "url": "https://arxiv.org/pdf/2503.04567", "page": 1, "id": "2503.04567_pg38_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{ij} \\ne \\tau_{ji}"} +{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g \\cdot (h \\cdot x) = (gh) \\cdot x"} +{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\cdot : G \\times X \\to X"} +{"pdf": "arxiv_math/2503.03772_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03772", "page": 1, "id": "2503.03772_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(g \\cdot x) = g \\cdot \\tau(x)"} +{"pdf": "arxiv_math/2502.15977_pg21.pdf", "url": "https://arxiv.org/pdf/2502.15977", "page": 1, "id": "2502.15977_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_1 + ... + \\theta_n"} +{"pdf": "arxiv_math/2503.06731_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06731", "page": 1, "id": "2503.06731_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "x.\\omega=\\omega(S(x).-),\\; \\forall\\omega\\in V^\\ast, x\\in A\\,."} +{"pdf": "arxiv_math/2503.07447_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07447", "page": 1, "id": "2503.07447_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "n^{-1}\\log^2 n \\ll p \\ll n^{-1/2}\\log^{1/4} n,"} +{"pdf": "arxiv_math/2503.07447_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07447", "page": 1, "id": "2503.07447_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p \\gg n^{-2/3}\\log^{2/3} n"} +{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta_2 = \\partial_y^2-\\alpha^2"} +{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta\\phi_{\\pm\\alpha}(t,y)"} +{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(t, \\pm 1) = \\partial_y \\Phi(t,\\pm 1) = 0"} +{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial_y\\delta\\phi_0"} +{"pdf": "arxiv_math/2503.05871_pg24.pdf", "url": "https://arxiv.org/pdf/2503.05871", "page": 1, "id": "2503.05871_pg24_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\vec{\\nabla}\\cdot \\vec{u} = 0"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H \\coloneqq (\\delta_{m+i, j})_{i=1, \\dots, d-m; \\: j=1, \\dots, d}"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\gamma^{(n)} - \\gamma\\|_2 \\to 0"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lVert Z_n - Z \\rVert_2 \\to 0"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "X_{m+1}(t),\\dots,X_d(t)"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma \\in L_t(X, d')"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in \\R^{d\\times d}"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "Z_n \\in \\mathcal{E}(Y, t, d')"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_018", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|I_t H - Z\\| = \\lim_{n \\to \\infty} \\|Z_n - Z\\| = 0"} +{"pdf": "arxiv_math/2503.05588_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05588", "page": 1, "id": "2503.05588_pg16_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\Sigma(0,-1):=\\mathrm{Cov}(X(0))"} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\log\\left( \\frac{P_1^{\\rho_0+\\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_2^{\\rho_0}(X_{(k-1)/n},X_{k/n};1/n)}\\right) = \\log\\left(\\frac{\\beta}{\\alpha}\\right) -\\frac{(X_{k/n}-X_{(k-1)/n})^2}{2/n}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right) + R_{k,\\theta}"} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g_k := \\log\\left(\\frac{\\beta}{\\alpha}\\right) - \\frac{n}{2}\\left(\\frac{1}{\\alpha^2}-\\frac{1}{\\beta^2}\\right)(X_{k/n}-X_{(k-1)/n})^2"} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_4(\\theta)}"} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "R_{k,\\theta} = \\log\\left( \\frac{1-\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\alpha^2/n}(X_{k/n}-\\rho_0-\\theta)(X_{(k-1)/n}-\\rho_0-\\theta)\\right)}{1+\\frac{\\alpha-\\beta}{\\alpha+\\beta}\\exp\\left(-\\frac{2}{\\beta^2/n}(X_{k/n}-\\rho_0)(X_{(k-1)/n}-\\rho_0)\\right)}\\right)."} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_2(\\theta)"} +{"pdf": "arxiv_math/2503.07022_pg75.pdf", "url": "https://arxiv.org/pdf/2503.07022", "page": 1, "id": "2503.07022_pg75_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\mathcal{I}_5(\\theta)}"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "V_j= B(z,(1+u)2^{-j})"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r/100}"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "B(z,2^{-j_0})\\subseteq D"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{F}_{W,r/100}"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W,r} = \\{(\\ell^1_i,\\ell^2_i,y_i)\\}_{i=1,\\ldots,M^*}"} +{"pdf": "arxiv_math/2503.08958_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08958", "page": 1, "id": "2503.08958_pg35_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak F_{W_{z,j},r}"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathfrak{C}_{m}(S,H)"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\varphi(i),\\varphi(j)) \\in E(S)"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "w_0, w_1,\\ldots,w_m,w_{m+1}"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau(H)=|E(H)|-|V(H)|"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E_A = \\{ (u,v) \\in E: u,v \\in A \\}"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Dist}_{H}(u,v)"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "H_{\\setminus A} = (V,E_{\\setminus A})"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathsf{Diam}(H)=\\max_{u,v \\in V(H)} \\mathsf{Dist}_H(u,v)"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "H,S \\subset \\mathcal{K}_n"} +{"pdf": "arxiv_math/2503.06464_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06464", "page": 1, "id": "2503.06464_pg8_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "V(H)=\\{ u,v,w_1,\\ldots,w_m \\}"} +{"pdf": "arxiv_math/2503.09087_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09087", "page": 1, "id": "2503.09087_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "L_\\alpha^{\\red(w)}(G)"} +{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{23},\\;B_{23},\\;R,\\;Y"} +{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "[x_{\\rm min},x_{\\rm max}]"} +{"pdf": "arxiv_math/2503.06589_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06589", "page": 1, "id": "2503.06589_pg21_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mu-m)e^2+(a^2-n^2)\\mu<0,"} +{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t=(b_{ij}^t)_{n\\times n}"} +{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\bf x}_t=(x_{1;t},\\ldots,x_{n;t})"} +{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat y_{k;t}={\\bf x}_t^{B_t{\\bf e}_k}"} +{"pdf": "arxiv_math/2503.06605_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06605", "page": 1, "id": "2503.06605_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F_u^t(y_1,\\ldots,y_n)\\in\\mathbb Z[y_1,\\ldots,y_n]"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\det A_{n;3}(x_1,x_2)=0"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.97,\\sqrt{0.97})"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{rcl} p_n(x,\\sqrt{x})&=&-k_n(x)k_n(1/x)-2k_n(\\sqrt{x})k_n(1/\\sqrt{x})\\\\ \\ q_n(x,\\sqrt{x})&=&k_n(x)k_n(1/\\sqrt{x})^2+k_n(1/x)k_n(\\sqrt{x})^2. \\end{array}"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A_{n;3}(0.098,\\sqrt{0.098})"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "P=(0.098,\\sqrt{0.098})"} +{"pdf": "arxiv_math/2503.08374_pg13.pdf", "url": "https://arxiv.org/pdf/2503.08374", "page": 1, "id": "2503.08374_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\det A_{n;3}(x,\\sqrt{x})=\\delta_n^3+p_n(x,\\sqrt{x})\\delta_n+q_n(x,\\sqrt{x})=0,"} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "|N_{G}(U\\setminus B)|\\geq \\frac{s|U\\setminus B|}{2r}"} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|B|\\leq \\frac{|U|}{10r}+\\frac{|U|}{10}\\le \\frac{|U|}{2},"} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "N_{G,r}(U\\setminus B)\\subseteq X"} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "|U'|\\leq \\frac{r|X \\cap L|}{t}\\le \\frac{r}{t}\\cdot |L| \\leq \\frac{r}{t}\\cdot \\frac{|U|\\cdot t}{10r}= \\frac{|U|}{10}."} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": "https://arxiv.org/pdf/2503.07147", "page": 1, "id": "2503.07147_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "X\\subseteq V(G)\\setminus U"} +{"pdf": "arxiv_math/2503.07147_pg16.pdf", "url": 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0, "checked": null, "math": "|U\\setminus B|\\geq \\frac{|U|}2"} +{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{GC}_\\text{id}(M)=\\mathcal{GC}(M)"} +{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{id}=(\\text{id},\\ldots,\\text{id})"} +{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(w,x_1),\\ldots,(w,x_s)"} +{"pdf": "arxiv_math/2503.07368_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07368", "page": 1, "id": "2503.07368_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "c\\colon V\\rightarrow 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null, "math": "\\begin{split} d_X(x_n,p)&\\le d_X(x_n,p_n)+d_X(p_n,p)\\\\ &=d_X(o,p_n)-d_X(o,x_n)+ d_X(p_n,p)\\\\ &=d_X(o,p_n)-R+d_X(p_n,p)\\xrightarrow{n\\to \\infty} d_X(o,p)-R+0=0. \\end{split}"} +{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\partial_P X_n,\\rho_{x_n})\\xrightarrow{AI conv.}(\\partial_P X,\\rho_x)"} +{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma_1(R),\\gamma_2(R)\\in U"} +{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": "https://arxiv.org/pdf/2503.09284", "page": 1, "id": "2503.09284_pg29_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(Z,\\rho_0)=(\\partial_P X,\\rho_x)"} +{"pdf": "arxiv_math/2503.09284_pg29.pdf", "url": 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|a| \\, d\\pi"} +{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega = (\\omega)_{n \\in \\Z} \\in \\Omega"} +{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "X(\\omega) = \\sum_{n=0}^\\infty a(\\omega_{-1}) a(\\omega_{-2}) \\cdots a(\\omega_{-n}) b(\\omega_{-n-1}) \\, ."} +{"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lbrace \\mathbf{E}_1(\\omega), \\mathbf{E}_2(\\omega)\\rbrace = S"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c_1x_1^\\alpha + c_2 x_2^\\beta + g(x_1,x_2)"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v(D) = v(f_t^*D) = pq."} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "-K_X\\cdot D = f_t^*\\left(\\sum_i C_i\\right)\\cdot D_t =\\left( (p+q)E_t + \\sum_i C_{i,t}\\right) \\cdot D_t = p+q"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(K_{Y_t} + D_t) \\cdot D_t = -2"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq (K_{Y_t} + D_t) \\cdot D_t = \\left(-\\sum_{i}C_{i,t} - E_t + D_t\\right) \\cdot D_t = D_t^2 - E_t\\cdot D_t < -E_t\\cdot D_t,"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "-2\\leq D_t^2 - E_t\\cdot D_t"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "p+q = D\\cdot C \\geq (D\\cdot C)_x = \\alpha + \\beta \\geq p + q,"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*D = D_t + pq E_t"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f_t^*(\\sum_{i}C_i)\\cdot D_t = -K_X \\cdot D> 0"} +{"pdf": "arxiv_math/2503.06612_pg19.pdf", "url": "https://arxiv.org/pdf/2503.06612", "page": 1, "id": "2503.06612_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "B_t = \\sum_{i=1}^k a_iC_{i,t}"} +{"pdf": "arxiv_math/2503.07452_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07452", "page": 1, "id": "2503.07452_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} & \\underset{\\mathbf{b}}{\\text{minimize}} & & \\epsilon_d(\\mathbf{b}) = \\dfrac{1}{n_{MP}} \\sum_{i=1}^{n_{MP}} (\\mathbf{u}_{r_i} - \\mathbf{u}_{s_i}(\\mathbf{b}))^2\\\\ & \\text{subject to} & & \\mathbf{b}_l \\mathbf{\\leq b} \\leq \\mathbf{b}_u \\end{aligned}"} +{"pdf": "arxiv_math/2503.09042_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09042", "page": 1, "id": "2503.09042_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Y_1 = Y_2 = \\cdots = Y_{k-1}"} +{"pdf": 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"url": "https://arxiv.org/pdf/2503.05348", "page": 1, "id": "2503.05348_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(t)=\\int_t^{\\infty} \\bar F(x)dx"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(x)=e^{|x|}(1+|x|)-1."} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+1}=G_e^{2t+1}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T_e^{2t+2}=G_e^{2t+2}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "E^{2t+1} \\cap E^{2t} \\cap \\{B_u = \\tilde{B}_u, \\forall u \\in \\partial G_e^{2t}\\}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in \\partial G_e^{2t}\\equiv \\partial T_e^{2t}"} +{"pdf": "arxiv_math/2503.08984_pg35.pdf", "url": "https://arxiv.org/pdf/2503.08984", "page": 1, "id": "2503.08984_pg35_math_009", "type": "math", 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\\frac{\\sin^2\\theta}{b^2}, \\qquad \\omega_2 = \\frac{\\cos^2\\theta}{b^2} + \\frac{\\sin^2\\theta}{a^2}, \\qquad \\omega_3 = \\sin2\\theta(\\frac{1}{a^2}-\\frac{1}{b^2})."} +{"pdf": "arxiv_math/2503.07624_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07624", "page": 1, "id": "2503.07624_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega = \\bigl\\{ (x, y): \\tfrac{x^2}{a^2} + \\tfrac{y^2}{b^2} \\leq 1 \\bigr\\}."} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial\\left(\\cdot\\right)"} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A} \\succeq \\mathbf{0}"} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": 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\\mathbf{a} \\rVert_2"} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{CN}(\\mathbf{0}, \\sigma^2 \\mathbf{I})"} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\left(\\mathbf{x}\\right)"} +{"pdf": "arxiv_math/2503.04040_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04040", "page": 1, "id": "2503.04040_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{\\mathbf{x}}^2f\\left(\\mathbf{x}\\right)"} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix} & \\alpha'& \\beta'& \\gamma'&&& \\delta'\\\\ {\\rm angles:}& 67 & 68 & 71 & 74 & 5 & 14 & /78\\\\ \\times ~3: & 15& 16& 19& 22& 5 & 14 & /26 \\end{matrix}"} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat\\beta,~~ \\beta'=\\widehat\\gamma,~~ ~ {\\rm and}~~ \\delta'=\\widehat\\alpha~."} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat{\\beta}"} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha'=\\widehat\\beta,\\quad \\beta'=\\widehat\\gamma,\\quad {\\rm and}\\quad \\delta'= \\widehat\\alpha~."} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat\\gamma=\\beta'"} +{"pdf": "arxiv_math/2503.08868_pg80.pdf", "url": "https://arxiv.org/pdf/2503.08868", "page": 1, "id": "2503.08868_pg80_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{matrix}& \\gamma & & & \\delta & \\alpha &\\beta\\\\ {\\rm angles:}& 16& 19 & 22& 31 & 40 & 41 & /78 \\\\ \\times ~3: & 16 & 19& 22& 5 & 14 & 15 & /26 \\end{matrix}"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{x}_\\ell(\\omega)"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "S_z(\\omega) = \\frac{1}{N} \\sum_{\\ell=1}^N S_{x_\\ell}(\\omega)."} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\langle e^{i(\\theta_m(t') - \\theta_n(t))} \\right\\rangle = \\delta_{mn} \\left\\langle e^{i(\\theta_m(t') - \\theta_m(t))} \\right\\rangle,"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{x_\\ell}(\\omega) = \\lim_{T \\to \\infty} \\frac{1}{T} \\left\\langle |\\tilde{x}_\\ell(\\omega)|^2 \\right\\rangle, \\quad \\tilde{x}_\\ell(\\omega) = \\int_0^T e^{i\\omega t} x_\\ell(t)\\,dt,"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\ddot{\\theta}_\\ell(t) + \\dot{\\theta}_\\ell(t) = \\omega_\\ell + \\text{Im}\\left( e^{-i\\theta_\\ell(t)} \\zeta_\\ell(t) \\right),"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "x_\\ell = e^{i\\theta_\\ell}"} +{"pdf": "arxiv_math/2503.08572_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08572", "page": 1, "id": "2503.08572_pg3_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "S_z(\\omega) = \\frac{1}{N} \\sum_{\\ell=1}^{N} |\\tilde{x}_\\ell(\\omega)|^2."} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^1 = 0.001\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^1"} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\gamma} = 1/\\alpha_{\\gamma}"} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\varepsilon}^i \\sim \\mathcal{N}(0, \\pmb{I}_l)"} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\omega}^2 = 0.1\\sqrt{\\Delta{t}_{\\text{Filter}}}\\tilde{\\varepsilon}^2"} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "Nt_{\\text{Filter}}=50"} +{"pdf": "arxiv_math/2503.07617_pg27.pdf", "url": "https://arxiv.org/pdf/2503.07617", "page": 1, "id": "2503.07617_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_{\\text{Darcy}} = 0.0001\\epsilon_D"} +{"pdf": 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\\in \\widetilde{S}_s^{m-1, k}"} +{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "Q(a e^{i \\frac{\\varphi}{h}}) = Q'_\\varphi({a}) e^{i \\frac{\\varphi}{h}} + R(a), \\tag{3.12}"} +{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(x, y', \\xi') \\to (x, y', \\Sigma(x, y', \\xi')),"} +{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi(x, \\xi') - \\varphi(y, \\xi') = \\Sigma(x, y', \\xi')(x' - y'),"} +{"pdf": "arxiv_math/2503.04189_pg11.pdf", "url": "https://arxiv.org/pdf/2503.04189", "page": 1, "id": "2503.04189_pg11_math_006", "type": "math", "max_diffs": 0, 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"2503.05437_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "u\\in H^s(\\Omega),\\quad p\\in H^{s-1}(\\Omega) \\quad\\forall s<1+\\lambda,"} +{"pdf": "arxiv_math/2503.05437_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05437", "page": 1, "id": "2503.05437_pg6_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "P=\\{v\\in H^1(\\Omega)\\cap L^2_0(\\Omega): r^{-1} v\\in L^2(\\Omega)\\}"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X}"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\longrightarrow \\omega_{\\mathcal{A}^\\vee/X,\\tau_i}\\longrightarrow\\H(\\mathcal{A}/X)_{\\tau_i}\\longrightarrow \\textnormal{Lie}_{\\mathcal{A}/X,\\tau_i}\\longrightarrow 0,"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "p\\mathcal{O}_E=\\mathfrak{q}\\mathfrak{q}^c"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle\\cdot,\\cdot\\rangle:\\H(\\mathcal{A}/X)_{\\tau_i}\\times \\H(\\mathcal{A}/X)_{\\tau_i^c}\\longrightarrow \\mathcal{O}_X"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(m_i,n_i)_{1\\le i\\le N}"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i^c}=\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}^\\perp"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}_Y(\\lambda)"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\tau_1,\\dots,\\tau_N\\}"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\mathcal{L}_Y(\\lambda)\\cdot C)"} +{"pdf": "arxiv_math/2503.08119_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08119", "page": 1, "id": "2503.08119_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega_{\\mathcal{A}^\\vee/X,\\tau_i}"} 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"https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\Phi(x,|\\nabla w_1(x)|)}{|w_1(x)|^{r-1}}=\\dfrac{\\Phi(x,\\lambda|\\nabla w_1(x)|)}{\\big (\\lambda|w_1(x)|\\big )^{r-1}}"} +{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{|\\nabla w_2(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\dfrac{|\\nabla w_1(x)|}{w_1(x)}=\\dfrac{\\lambda(x)|\\nabla w_1(x)|}{w_2(x)}\\ \\Longrightarrow\\ \\lambda(x)=\\dfrac{w_2(x)}{w_1(x)}"} +{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla w_2=\\lambda\\nabla w_1=0"} +{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla w_1(x)|\\cdot |\\nabla w_2(x)|=\\nabla w_1(x)\\cdot\\nabla w_2(x)"} +{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{w_1}{w_2},\\ \\dfrac{w_2}{w_1}\\in L^{\\infty}(\\Omega)"} +{"pdf": "arxiv_math/2503.06630_pg14.pdf", "url": "https://arxiv.org/pdf/2503.06630", "page": 1, "id": "2503.06630_pg14_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dfrac{\\nabla w_1(x)}{w_1(x)}=\\dfrac{\\nabla w_2(x)}{w_2(x)}=0"} +{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{I}_f,\\mathcal{I}_p,\\mathcal{I}_s,\\mathcal{I}_E"} +{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho\\in L^2(\\mathbb R^3)"} +{"pdf": "arxiv_math/2503.06581_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06581", "page": 1, "id": "2503.06581_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "J\\in\\left(H^1(\\mathbb R^3)\\right)^3"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "H^* = \\operatorname{Span}\\{f_1, f_2, \\cdots, f_n\\}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\beta_i\\}_{i\\leq n}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b_i\\}_{i\\leq n} \\subseteq X_{\\leq 1+\\sigma}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1 = r(y_1) >\\dfrac{1}{2}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in X^{\\ast\\ast}\\backslash Q(X)"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "E_2 = \\operatorname{Span}\\{b_1, b_2\\}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_n\\|\\sum_{i \\leq n}e_i\\| < \\infty"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "y_2 \\in X^*_{\\leq 1}\\cap E_1^{\\perp}"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda_1, \\lambda_2, \\cdots, \\lambda_n)\\in\\mathbb{C}^n"} +{"pdf": "arxiv_math/2503.06880_pg36.pdf", "url": "https://arxiv.org/pdf/2503.06880", "page": 1, "id": "2503.06880_pg36_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_n = \\sup\\big\\{ r(y)\\,\\vert\\, y\\in X^{\\ast}_{= 1}\\cap E_n^{\\perp}\\big\\}"} +{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha(e^{-1}) = (\\alpha(e))^{-1}"} +{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha: D(\\Gamma) \\rightarrow G"} +{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V(\\Gamma^{\\alpha}) = V(\\Gamma) \\times G"} +{"pdf": "arxiv_math/2503.06466_pg22.pdf", "url": "https://arxiv.org/pdf/2503.06466", "page": 1, "id": "2503.06466_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e = (u,v) \\in D(\\Gamma)"} +{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\frac{1}{2\\pi i} \\right)^2 \\int_{c - i\\infty}^{c + i\\infty} \\int_{c - i\\infty}^{c + i\\infty} \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k} e^{z_1 x_1 + z_2 x_2} \\,dz_1 dz_2 \\notag \\\\ = \\frac{1}{\\Gamma(k) b^{k-1}} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right)."} +{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b = \\frac{\\sqrt{\\rho}}{\\sigma(1-\\rho)}"} +{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{\\Gamma(k) b^{k-1}} \\int_0^{\\infty} \\int_0^{\\infty} (x_1 x_2)^{\\frac{k-1}{2}} e^{-\\alpha (x_1 + x_2)} I_{k-1} \\left( 2b \\sqrt{x_1 x_2} \\right) e^{-z_1 x_1 - z_2 x_2} \\,dx_1 dx_2 \\notag \\\\ = \\frac{1}{\\left[ (z_1 + a)(z_2 + a) - b^2 \\right]^k}."} +{"pdf": "arxiv_math/2503.08808_pg5.pdf", "url": "https://arxiv.org/pdf/2503.08808", "page": 1, "id": "2503.08808_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = \\frac{1}{\\sigma(1-\\rho)}"} +{"pdf": "arxiv_math/2503.07729_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07729", "page": 1, "id": "2503.07729_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T \\Delta E \\to \\infty"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "id^*\\nabla^\\mathrm{flat}=\\nabla^\\mathrm{flat}"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\nabla}\\in[\\overline{\\nabla}]"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y\\in\\mathfrak{L}(\\mathcal{F}^M)"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "id:\\mathbb{R}^3\\rightarrow\\mathbb{R}^3"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(F_*\\hat{\\nabla}^M)_XY=W"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(F^*\\hat{\\nabla}^N)_XY=Z"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "X, Y \\in\\mathfrak{X}(M)"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z(p):=d\\Phi_p^{-1}\\left(\\hat{\\nabla}^N_{\\Phi_*X}\\Phi_*Y\\right)_{\\Phi(p)}"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "W(q):=d\\Phi_p\\left(\\hat{\\nabla}_{\\Phi^*X}\\Phi^*Y\\right)_{p}"} +{"pdf": "arxiv_math/2503.06344_pg15.pdf", "url": "https://arxiv.org/pdf/2503.06344", "page": 1, "id": "2503.06344_pg15_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "V\\in\\mathfrak{X}(\\mathcal{F}^M)"} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "p(\\tau, \\tau \\sqcup \\{ij\\}) = \\left\\{ \\begin{array}{cl} \\displaystyle \\frac{\\alpha_{ij} w_{i, ij}}{\\sum_{k=1}^d \\alpha_{ik} w_{i, ik}} \\alpha_i , & \\text{if } i \\in \\partial \\tau, \\\\[15pt] \\alpha_{ij}, & \\text{if } i \\notin \\partial \\tau \\text{ and } ij \\notin \\tau, \\end{array} \\right."} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i = \\alpha_{i1} + \\ldots + \\alpha_{id} \\in [0, 1]"} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{1}{6} \\left(\\frac{w_1 + w_2}{2}\\right)^3 = \\frac{1}{12} \\left(\\frac{w_1 + w_2}{2}\\right)^3 + \\frac{1}{48} \\sum_{1 \\leq i, j \\leq 2} w_i w_j \\left(\\frac{w_1 + w_2}{2}\\right)."} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\prod_{i \\in \\overset{\\circ}{\\tau}} \\frac{\\alpha_i}{\\sum_{j=1}^d \\alpha_{ij} w_{i, ij}}}=\\left(\\frac{d}{\\sum_{j=1}^dw_j}\\right)^{|\\tau|- |\\partial \\tau|},"} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "j \\in \\{1, \\ldots, d\\}"} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\binom{n}{k_1,\\cdots,k_d}"} +{"pdf": "arxiv_math/2503.08172_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08172", "page": 1, "id": "2503.08172_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^{(j)} \\sqcup \\bigsqcup_{l\\neq j} \\sigma^{(l)}"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon} \\rightarrow e(F,\\rho)"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I_i = \\frac 43 \\int \\rho_i^{\\frac 32} dt"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal M_{\\varepsilon}"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "e \\geq e_{\\gamma_i, \\rho_i}"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal L_\\varepsilon"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "e(0,0,F(t),t) = 0, \\forall t \\in [0,1]"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho(t)= \\begin{cases} \\frac{1}{a^2(3-\\sqrt{8a})^2}, & \\text{if } t \\leq 2a \\\\ \\frac{2}{at(3-\\sqrt{8a})^2}, & \\text{if } 2a \\leq t \\leq 1. \\end{cases}"} +{"pdf": "arxiv_math/2503.09486_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09486", "page": 1, "id": "2503.09486_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "e_{\\gamma, \\rho}(x_1,t_1,x_2,t_2) = - \\infty"} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "G(U)=\\begin{pmatrix}g_{1}(U) \\\\ g_{2}(U) \\\\ g_{3}(U) \\end{pmatrix}=\\begin{pmatrix} m P(H(M)-a_{1} \\lambda M ) \\\\ -m a_{2} \\lambda M S \\\\m_{s}S(1-M)-\\eta M P \\end{pmatrix},\\quad U_{R}=(0,S_{R},M_{R})\\quad \\text{and}\\quad U_{0}=(P_0,S_{0},M_{0})."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{L^1([0,T]; X)} = \\int_0^T \\|f(t)\\|_X dt."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C(\\Omega_R)} = \\sup_{x \\in \\Omega_R} |f(x)| ."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{2,1}(\\Omega_R) = \\{u \\in L^1(\\Omega_R) : D^\\alpha u \\in L^1(\\Omega_R) \\quad\\text{for all}\\quad |\\alpha| \\leq 2\\}"} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C([0,T]; X)} = \\sup_{t \\in [0,T]} \\|f(t)\\|_X ."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{L^{\\infty}(\\Omega_R)} = \\operatorname{ess\\,sup}_{x \\in \\Omega_R} |f(x)|."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\left([0,T],C^{1}_{b}(\\R^d)\\right)"} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|f\\|_{C^1_b(\\Omega_R)} = \\|f\\|_{C(\\Omega_R)} + \\|\\nabla f\\|_{C(\\Omega_R)}."} +{"pdf": "arxiv_math/2503.09231_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09231", "page": 1, "id": "2503.09231_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\displaystyle \\mathcal{A}\\left(U\\right)(t,x)=\\left[\\operatorname{div}\\left(\\vec{\\alpha_{p}}(t,x)\\left(\\int_{ \\Omega_{R}}\\gamma_{p}(x-y)P(t,y)d y \\right) P(t,x)\\right),\\operatorname{div}\\left(\\vec{\\alpha_{s}}(t,x)\\left(\\int_{ \\Omega_{R}}\\gamma_{s}(x-y)S(t,y)d y \\right)S(t,x)\\right),-D\\Delta M\\right]"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}} \\in L^1(H^{\\perp})"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{\\perp} \\cong \\widehat{A/H}"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_{A/H}(\\phi^H) = \\mathcal{F}_A(\\phi)\\rvert_{H^{\\perp}}"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "A= (\\Z/N\\Z)^{d_1} \\times \\R^{d_2}"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_H \\phi(xh)dh = \\int_{H^{\\perp}} \\mathcal{F}_A(\\phi)(\\chi)\\chi(x)d\\chi,"} +{"pdf": "arxiv_math/2503.03873_pg5.pdf", "url": "https://arxiv.org/pdf/2503.03873", "page": 1, "id": "2503.03873_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi^H(xH) = \\int_H \\phi(xh)dh"} +{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\wp_\\Upsilon(\\upsilon,t)d\\upsilon"} +{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Upsilon(t)\\in[\\upsilon,\\upsilon+d\\upsilon]"} +{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{\\sigma}^{-1}(\\kappa)"} +{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma = \\left[ \\frac{1}{N} \\sum_{k=1}^N \\, (z_k-\\nu)^2 \\right]^{\\frac{1}{2}} \\; , \\quad \\nu = \\frac{1}{N} \\sum_{k=1}^N z_k \\; ."} +{"pdf": "arxiv_math/2503.06939_pg20.pdf", "url": "https://arxiv.org/pdf/2503.06939", "page": 1, "id": "2503.06939_pg20_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\wp_\\Upsilon(\\upsilon,t)"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{\\varphi }=\\pi /100"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon ,\\lambda \\right)"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{0}^{\\xi _{0}}\\left( \\mathbf{x},\\mathbf{x}_{0}\\right)"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x}\\in \\partial \\Omega"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon \\right) ,"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\alpha ,\\varepsilon \\right)"} +{"pdf": "arxiv_math/2503.07916_pg19.pdf", "url": "https://arxiv.org/pdf/2503.07916", "page": 1, "id": "2503.07916_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "% \\lambda \\in \\left[ 1,5\\right] ."} +{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in C([0,T]; L^2(\\R^n))"} +{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "V \\in L^1((0,T); L^\\infty(\\R^n))"} +{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f \\in L^2(\\R^n) \\mapsto u \\in C([0, T]; L^2(\\R^n))"} +{"pdf": "arxiv_math/2503.06205_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06205", "page": 1, "id": "2503.06205_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ u(t, \\centerdot) : t \\in [0, T] \\}"} +{"pdf": 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"arxiv_math/2503.04047_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04047", "page": 1, "id": "2503.04047_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma^2(\\{f(x_{t-M+1}), \\cdots, f(x_{t}) \\})"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "1-c_0 n^{-\\tilde{\\epsilon}}"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i \\neq j \\in [n] : |i-j|\\leq 2}\\frac{1}{(|\\lambda_{j}-\\lambda_{i}| + n^{-1.5-\\tilde{\\epsilon}})^2} \\leq 8c_0 n^{3.5 + 2\\tilde{\\epsilon}}."} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "1-n^{-\\tilde{\\epsilon}}"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\epsilon} = \\epsilon/6"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{i \\in [n-1]} |\\lambda_{i+1}-\\lambda_i| \\geq n^{-1.5-\\tilde{\\epsilon}}"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\min_{i \\in [n-3]} |\\lambda_{i+3}-\\lambda_i| \\leq n^{-6/5-\\tilde{\\epsilon}/5}"} +{"pdf": "arxiv_math/2503.05323_pg16.pdf", "url": "https://arxiv.org/pdf/2503.05323", "page": 1, "id": "2503.05323_pg16_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": 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"https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I^-(p)=I^-(q)\\Rightarrow p=q"} +{"pdf": "arxiv_math/2503.04382_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04382", "page": 1, "id": "2503.04382_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "d_p(r)\\ge \\ell(\\gamma\\vert_{[\\delta,1]})> m"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "I_1,I_2 \\in \\mathcal{C}^0([0,T], \\R)"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{C}^0([0,T],\\R)"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "2T\\|\\partial_Ir\\|_\\infty r_M<1"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned}\\|\\mu(n_1)-\\mu(n_2)\\|_X&\\leq 2T\\|\\partial_Ir\\|_\\infty \\|I_1-I_2\\|_\\infty. \\\\ &\\leq 2T\\|\\partial_Ir\\|_\\infty r_M\\|n_1-n_2\\|_X \\end{aligned}"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "r_1(t,x):=r(x,I_1(t))"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall t \\in [0,T],\\qquad \\|\\mu_1(t)-\\mu_2(t)\\|_1 \\leq \\|\\mu_1(0)-\\mu_2(0)\\|_1+2\\|\\partial_Ir\\|_\\infty\\int_0^t\\|I_1(\\tau)-I_2(\\tau)\\|_\\infty d\\tau"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_1(0)= \\mu_2(0) = n_\\text{ini}"} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall n \\in X, \\qquad \\Phi(n,F(n)) = 0."} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\mu_1-\\mu_2\\|_X\\leq 2T\\|\\partial_Ir\\|_\\infty \\|I_1-I_2\\|_\\infty."} +{"pdf": "arxiv_math/2503.09157_pg22.pdf", "url": "https://arxiv.org/pdf/2503.09157", "page": 1, "id": "2503.09157_pg22_math_009", "type": "math", 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"type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{h}, \\boldsymbol{x} \\in \\mathbb{C}^n"} +{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "|(D_p x)_i|_p = \\frac{1}{p^{\\text{ord}_p((D_p x)_i)}}"} +{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "D_p \\in \\mathbb{M}_{n\\times n}(\\mathbb{Q}_p)"} +{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|x|_p = p^{-k}, \\quad \\text{where } k \\in \\mathbb{N} \\text{ and } k \\geq 1."} +{"pdf": "arxiv_math/2503.04182_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04182", "page": 1, "id": "2503.04182_pg3_math_003", 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\\delta_p(\\delta_p(x))= \\delta_p \\circ\\delta_p, \\dots, \\delta_p^n(x), \\dots"} +{"pdf": "arxiv_math/2503.08711_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08711", "page": 1, "id": "2503.08711_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{BSPA}_{0.1,30,30,90}"} +{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle A,\\wedge,\\vee\\rangle"} +{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{A}=\\langle A,\\wedge,\\vee, \\cdot,\\backslash,/,1\\rangle"} +{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{K}=\\mathbf{S}_3"} +{"pdf": "arxiv_math/2503.06657_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06657", "page": 1, "id": "2503.06657_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle A,\\cdot,1\\rangle"} +{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\frac{\\|\\cdot\\|^2}{2}"} +{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "L_2 = \\mu_1 - \\mu_F \\geq -\\mu_F"} +{"pdf": "arxiv_math/2503.04486_pg21.pdf", "url": "https://arxiv.org/pdf/2503.04486", "page": 1, "id": "2503.04486_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_2 > -\\mu_1 = L_2 + \\mu_F"} +{"pdf": "arxiv_math/2503.05342_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05342", "page": 1, "id": 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"\\partial_{\\lambda_{i}} \\Phi = \\sum_{j=1}^n l_{ij} - 1"} +{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(L,\\lambda):=D(L,M) +\\sum_{i=1}^n \\lambda_i \\left(\\sum_{j=1}^n l_{ij} - 1\\right)"} +{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{t\\to \\infty} \\frac{h(tx)}{t}=\\infty"} +{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": "2503.07145_pg13_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\argmin_{L\\in C_2} D(L,M) = \\left(\\frac{m_{ij}}{\\sum_{k=1}^n m_{kj}}\\right)_{1\\le i, j\\le n}."} +{"pdf": "arxiv_math/2503.07145_pg13.pdf", "url": "https://arxiv.org/pdf/2503.07145", "page": 1, "id": 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f\\|_\\infty"} +{"pdf": "arxiv_math/2503.08578_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08578", "page": 1, "id": "2503.08578_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi f\\in W^{k,p}(\\Omega)"} +{"pdf": "arxiv_math/2503.06459_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06459", "page": 1, "id": "2503.06459_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\sqrt{n-1}}{\\sqrt{n-2}}\\delta"} +{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "A=(\\theta_--\\theta_+)/2"} +{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": "https://arxiv.org/pdf/2503.09574", "page": 1, "id": "2503.09574_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda(du)=\\lambda(\\delta_1(du)+ \\delta_{-1}(du))"} +{"pdf": "arxiv_math/2503.09574_pg27.pdf", "url": 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"\\begin{cases} x_3=x_1 \\\\ x_4=x_2+\\frac{b_{11}}{b_{10}}x_1x_2+\\frac{b_{20}}{b_{10}^2}x_2^2+\\frac{b_{12}}{b_{10}}x_1^2x_2+\\frac{b_{21}}{b_{10}^2}x_1x_2^2+\\frac{b_{30}}{b_{10}^3}x_1^3+O(\\|x\\|^4) \\end{cases}"} +{"pdf": "arxiv_math/2503.09472_pg15.pdf", "url": "https://arxiv.org/pdf/2503.09472", "page": 1, "id": "2503.09472_pg15_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{cases} x_1=x_3 \\\\ x_2=x_4+v_{11}x_3x_4+v_{02}x_4^2+v_{21}x_3^2x_4+v_{12}x_3x_4^2+v_{03}x_4^3+O(\\|x\\|^4) \\end{cases}"} +{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{k}^{RBB}\\in[\\alpha_{k}^{BB1}, \\ \\alpha_{k}^{BB2}]"} +{"pdf": "arxiv_math/2503.06185_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06185", "page": 1, "id": "2503.06185_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": 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I} N_i \\rightarrow \\oplus_{i \\in I} M / N_i"} +{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(P_{k})_{k=1}^{\\infty}"} +{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "(N_{k})_{k=1}^{\\infty}"} +{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{N_1, N_2, \\ldots, N_n\\right\\}"} +{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07271", "page": 1, "id": "2503.07271_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "N_1 \\cap N_2 \\cap \\cdots \\cap N_n"} +{"pdf": "arxiv_math/2503.07271_pg3.pdf", "url": 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y^i\\partial y^j\\partial y^k}X^iY^jZ^k,"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "G^i=\\frac12\\Gamma^i_{jk}y^jy^k"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "X,Y,Z\\in TM\\setminus\\{0\\}"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "V(u)=\\exp(\\frac12F^{*2}(Du))"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "N^i_j=\\frac{\\partial G^i}{\\partial y^j}"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\delta}{\\delta x^i}=\\frac{\\partial}{\\partial x^i}-N^j_i\\frac{\\partial}{\\partial y^j}"} +{"pdf": "arxiv_math/2503.07623_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07623", "page": 1, "id": "2503.07623_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "divC=FC^{i}_{\\,jk\\vert i}dx^j\\otimes dx^k"} +{"pdf": "arxiv_math/2503.08397_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08397", "page": 1, "id": "2503.08397_pg18_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{T}=\\sum_{k\\in \\{0\\}\\cup [K]} T_k"} +{"pdf": "arxiv_math/2503.08397_pg18.pdf", "url": "https://arxiv.org/pdf/2503.08397", "page": 1, "id": "2503.08397_pg18_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon=o\\left({1}/{\\sqrt{N^{1-\\alpha_s}}}\\right)"} +{"pdf": 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jk_1"} +{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "d_2 = \\big[\\frac{t^2}{2!}\\big]d(t) = \\frac{d}{dt}^2d(t)\\bigr|_{t=0}"} +{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "c(t) =\\sum_{i=0}^\\infty c_i\\frac{t^i}{i!}"} +{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{n,k} = [t^n]d(t)h(t)^k,\\nonumber"} +{"pdf": "arxiv_math/2503.05034_pg12.pdf", "url": "https://arxiv.org/pdf/2503.05034", "page": 1, "id": "2503.05034_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{n,k}=\\Big[\\frac{t^n}{n!}\\Big]d(t)\\frac{h(t)^k}{k!}.\\nonumber"} 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\\Phi_{\\overline{\\alpha} h}"} +{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{O}(h^{2p+1})"} +{"pdf": "arxiv_math/2503.08453_pg23.pdf", "url": "https://arxiv.org/pdf/2503.08453", "page": 1, "id": "2503.08453_pg23_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_3 \\in \\mathbb{R}"} +{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\label{time1} T(u(0)) \\leq \\dfrac{V(u(0))^{1-p}}{K(1-p)}"} +{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T(u)\\equiv f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)"} +{"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": 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If a row (or column) contains two 1s, each of them appears in different adjacent blocks. All other entries in the matrix are zeros. In the corresponding graphical invariants, the symbol"} +{"pdf": "arxiv_math/2503.07166_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07166", "page": 1, "id": "2503.07166_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^+)^2 - (e^-)^2 = e^+ + e^-"} +{"pdf": "arxiv_math/2503.04415_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04415", "page": 1, "id": "2503.04415_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma\\in[0,\\frac{1-\\gamma}{2})"} +{"pdf": "arxiv_math/2503.04415_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04415", "page": 1, "id": "2503.04415_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "N = \\left\\lfloor \\frac{1}{\\gamma} \\right\\rfloor"} +{"pdf": "arxiv_math/2503.06000_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06000", "page": 1, "id": "2503.06000_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "[1,173] \\times [1,60] \\times [95,200]"} +{"pdf": 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"https://arxiv.org/pdf/2503.05688", "page": 1, "id": "2503.05688_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\mathcal{H}}"} +{"pdf": "arxiv_math/2503.08848_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08848", "page": 1, "id": "2503.08848_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{L}^{\\text {Airy }}"} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{bmatrix}\\varphi^{\\xi,+}_{n}\\\\\\varphi^{\\xi,-}_{n}\\end{bmatrix}=e^{2\\pi in\\theta}\\begin{bmatrix}\\check{\\phi}^{+}(\\xi+n\\Phi)\\\\\\check{\\phi}^{-}(\\xi+n\\Phi)\\end{bmatrix}=\\frac1{\\sqrt2}e^{2\\pi in\\theta}\\begin{bmatrix}\\check\\psi^+(\\xi+n\\Phi)+i\\check\\psi^-(\\xi+n\\Phi)\\\\i\\check\\psi^+(\\xi+n\\Phi)+\\check\\psi^-(\\xi+n\\Phi)\\end{bmatrix}."} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_1,\\lambda_2,\\Phi,\\theta}"} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi=\\left[\\psi^{+},\\psi^{-}\\right]^\\top"} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W_{\\lambda_{1},\\lambda_{2},\\Phi,\\theta}\\psi=z\\psi"} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi=\\varphi^\\xi=\\left[\\varphi^{\\xi,+},\\varphi^{\\xi,-}\\right]^\\top"} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in\\Sigma_{\\lambda_1,\\lambda_2,\\Phi}."} +{"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sin2\\pi(\\theta+n\\Phi)|<\\exp(-|n|^{\\frac{1}{2\\tau}})"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ", because it matters less if the sequence starts at the beginning of a sequence or not if there are many"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "being a sequence of tokens. However, as discussed in the main text, the"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-tuple of tokens with"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "prior tokens to conditional on. Therefore, we focusing on reducing the bias for small"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": ". Then, the entropy of the distribution can be estimated naively as \\begin{equation} \\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}}, \\end{equation} where the summation runs over all possible combination of tokens"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "starts at the beginning of a sentence, but LLMs model distributions conditioned on BOS token. To mitigate this issue, we use"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "term suffers from an additional bias---we cannot guarantee that"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}},"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is small, we can iterate over the dataset and construct a histogram for the"} +{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "and the total number of samples with"} +{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "% H^1_{\\Theta}=K^1_{\\Theta}\\oplus \\Theta H^1"} +{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1_\\Theta/(K^1_{\\Theta}\\oplus \\Theta H^1 )"} +{"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K^1_{\\Theta}\\oplus \\Theta H^1 \\subsetneq H^1_{\\Theta},"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_\\varepsilon=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*) L_{i,t}=0"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{u_{\\varepsilon}:=(u_{1,\\varepsilon},\\cdots,u_{n,\\varepsilon})\\}_\\varepsilon"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{\\varepsilon}:=(\\rho_{1,\\varepsilon},\\cdots,\\rho_{n,\\varepsilon})"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Lambda_{I,N}(\\rho_\\varepsilon)=-\\sum_{i=1}^n\\frac{2}{N}\\Big(D_{i,t}+\\sum_{s\\neq t}\\frac{e^{2H_{i,s}(p^*)}h_i(p_{s}^*)}{e^{2H_{i,t}(p^*)}h_i(p_{t}^*)} D_{i,s}+o(1)\\Big)\\varepsilon_t^{2}"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i\\in\\hat{I}}\\sum_{t=1}^N e^{(\\hat{m}^*-2)H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_1,\\cdots,\\varepsilon_N"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\rho_{i,\\varepsilon}\\to\\rho_i^*"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^N \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)D_{i,t}\\neq 0"} +{"pdf": "arxiv_math/2503.07467_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07467", "page": 1, "id": "2503.07467_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{i=1}^n \\sum_{t=1}^N e^{2H_{i,t}(p^*)}h_i(p_{t}^*)L_{i,t}\\neq 0"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}H= e^{-i\\alpha z}\\mathcal{H}(E)"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(D-\\lambda I)\\mathcal{J},~\\lambda\\in \\mathbb{C}"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "R=(D-\\overline{\\lambda}I|\\mathcal{J})^{-1}"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\lambda\\notin \\mathcal{J}"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{\\lambda} \\notin \\sigma(D|\\mathcal{J})"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(D|\\mathcal{J})=\\emptyset"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} \\in \\mathscr{J}"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_{\\lambda}(x) = \\varphi(V(I-\\overline{\\lambda}V)^{-1}x)=\\varphi(Rx) = 0,"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{J} = C^{\\infty}(D)"} +{"pdf": "arxiv_math/2503.08443_pg27.pdf", "url": "https://arxiv.org/pdf/2503.08443", "page": 1, "id": "2503.08443_pg27_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{W}\\varphi(\\lambda) = 0"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x', x \\rangle \\ge 0"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x, x' \\rangle\\geq 0"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma=\\overline{\\mathrm{conv}(S')}"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma \\subseteq V'_+"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x'', x' \\rangle \\ge \\gamma > \\langle x'', x'_0 \\rangle"} +{"pdf": "arxiv_math/2503.04958_pg4.pdf", "url": "https://arxiv.org/pdf/2503.04958", "page": 1, "id": "2503.04958_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x, x' \\rangle \\ge \\gamma > \\langle x, x'_0 \\rangle"} +{"pdf": "arxiv_math/2503.04607_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04607", "page": 1, "id": "2503.04607_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "g(X_1 \\# X_2) = g(X_1) + g(X_2)"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "LTR(3645712)=\\{ 3,6,7\\}"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\tau_1 +n)\\cdots (\\tau_m +n)"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi =\\pi_1 \\cdots \\pi_n \\in S_n"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi^{r}=\\pi_n \\cdots \\pi_1"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau =\\tau_1 \\cdots \\tau_m \\in S_m"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi_i =\\max \\{ \\pi_j \\, |\\, j\\leq i\\}"} +{"pdf": "arxiv_math/2503.08285_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08285", "page": 1, "id": "2503.08285_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "S=\\bigcup_{n\\in \\mathbb{N}}S_n"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p = r_p + \\beta n_p"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "g_{ij}=\\langle \\partial x_i, \\partial x_j\\rangle"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_+ =\\lambda_{\\max}\\left(\\frac{M_{+}+M_{+}^T}{2}\\right),\\qquad \\mu_{-} =\\lambda_{\\max}\\left(\\frac{{M_{-}}+{M_{-}}^T}{2}\\right),"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Sigma\\in\\mathbb{R}^{r\\times r}"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla_{r_p}X\\in \\text{Range}(\\nabla X|^*_p)"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\nabla_{v_p} X, v_p\\rangle = \\langle w_p, (\\nabla X|_p)^{-1}w_p\\rangle \\leq \\|(\\nabla X|_p)^{-1}\\|\\,\\|w_p\\|^2 = \\|(\\nabla X|_p)^{-1}\\|\\, \\|\\nabla_{v_p} X|\\|^2"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "v_p^T g \\mathcal{A}_{X,p}v_p \\leq -\\alpha_p v_p^T \\mathcal{A}_{X,p}^T g \\mathcal{A}_{X,p} v_p,\\quad\\forall v_p\\in\\mathbb{R}^d"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{Range}(\\nabla X|_p)"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\nabla_{r_p} X, r_p\\rangle + \\beta\\langle \\nabla_{r_p} X, n_p\\rangle \\leq -\\alpha\\|\\nabla_{r_p}X\\|^2"} +{"pdf": "arxiv_math/2503.09434_pg25.pdf", "url": "https://arxiv.org/pdf/2503.09434", "page": 1, "id": "2503.09434_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\alpha_p = \\max_{1\\leq i \\leq d} \\lambda_i\\left(\\frac{M+M^T}{2}\\right)"} +{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "f_i \\in C^{1,\\frac{1}{2}}(L_i)"} +{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma : [0,\\ell_i] \\rightarrow \\R^2"} +{"pdf": "arxiv_math/2503.05399_pg6.pdf", "url": "https://arxiv.org/pdf/2503.05399", "page": 1, "id": "2503.05399_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_i \\cup \\tilde{\\Gamma}_i"} +{"pdf": "arxiv_math/2503.08283_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08283", "page": 1, "id": "2503.08283_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "t_{p}, 1.5 t_{p}, 3 t_{p}, 4 t_{p}, 6 t_{p}"} +{"pdf": "arxiv_math/2503.05390_pg14.pdf", 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\\{a_1a_5, a_1a_4a_5, a_na_1a_5, a_na_1a_4a_5\\}"} +{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{R}_{a_1a_5} = \\{a_1a_5, a_1a_2a_5, a_1a_5a_6, a_1a_2a_5a_6\\}."} +{"pdf": "arxiv_math/2503.06329_pg24.pdf", "url": "https://arxiv.org/pdf/2503.06329", "page": 1, "id": "2503.06329_pg24_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{array}{c|ccc} & a_1 & a_2 & 1 \\\\ \\hline a_1 & a_1 & \\cdot & a_1 \\\\ a_2 & \\cdot & a_2 & a_2 \\\\ 1 & a_1 & a_2 & 1 \\end{array}"} +{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": "2503.08488_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S_k=0, \\ \\text{if}\\ k \\in \\partial\\mathcal{L};"} +{"pdf": "arxiv_math/2503.08488_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08488", "page": 1, "id": 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n+2"} +{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^{\\mathcal{Q}} = 7"} +{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{r}_i = \\begin{bmatrix} b_{i,1}, & b_{i,2}, & \\cdots, & b_{i,n} \\end{bmatrix}."} +{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "N = 26 \\times (3^2 + 4^2)"} +{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07567", "page": 1, "id": "2503.07567_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\text{min}}^\\mathcal{C}"} +{"pdf": "arxiv_math/2503.07567_pg6.pdf", "url": 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"type": "math", "max_diffs": 0, "checked": null, "math": "S_{i j}^{\\prime}=\\left[L_{j}, U_{j}\\right]"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "A^{-}=\\left[\\begin{array}{llllllllll} 0.70 & 0.70 & 0.32 & 0.44 & 0.00 & 0.16 & 0.20 & 0.50 & 0.40 & 0.39 \\\\ 0.70 & 0.65 & 0.14 & 0.12 & 0.80 & 0.76 & 0.00 & 1.00 & 0.15 & 0.79 \\\\ 0.17 & 0.24 & 0.20 & 0.20 & 0.06 & 0.25 & 0.13 & 0.19 & 0.22 & 0.02 \\\\ 0.14 & 0.10 & 0.04 & 0.00 & 0.10 & 0.00 & 0.14 & 0.02 & 0.15 & 0.08 \\\\ 0.70 & 0.04 & 0.27 & 0.36 & 0.60 & 0.40 & 0.48 & 0.50 & 0.50 & 0.50 \\\\ 0.66 & 0.63 & 0.14 & 0.73 & 0.53 & 0.46 & 0.61 & 0.85 & 0.85 & 0.39 \\\\ 0.00 & 0.15 & 0.15 & 0.05 & 0.02 & 0.03 & 0.10 & 0.12 & 0.18 & 0.09 \\\\ 0.63 & 0.03 & 0.55 & 0.77 & 0.79 & 0.49 & 0.21 & 0.32 & 0.80 & 0.71 \\\\ 0.27 & 0.30 & 0.35 & 0.24 & 0.35 & 0.07 & 0.29 & 0.35 & 0.20 & 0.75 \\\\ 0.59 & 0.34 & 0.26 & 0.38 & 0.02 & 0.60 & 0.52 & 0.43 & 0.27 & 0.44 \\end{array}\\right]"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "A^{+}=\\left[\\begin{array}{llllllllll} 0.25 & 0.32 & 0.41 & 0.19 & 0.70 & 0.13 & 0.44 & 0.37 & 0.28 & 0.50 \\\\ 0.80 & 0.73 & 0.64 & 0.79 & 0.80 & 0.22 & 0.80 & 0.56 & 0.10 & 0.28 \\\\ 0.11 & 0.20 & 0.12 & 0.13 & 0.05 & 0.25 & 0.40 & 0.25 & 0.20 & 0.18 \\\\ 0.10 & 0.23 & 0.25 & 0.15 & 0.12 & 0.05 & 0.02 & 0.01 & 0.15 & 0.15 \\\\ 0.45 & 0.35 & 0.70 & 0.50 & 0.41 & 0.27 & 0.39 & 0.48 & 0.17 & 0.39 \\\\ 0.60 & 0.70 & 0.25 & 0.38 & 0.63 & 0.58 & 0.46 & 0.47 & 0.85 & 0.33 \\\\ 0.01 & 0.02 & 0.15 & 0.09 & 0.12 & 0.10 & 0.15 & 0.15 & 0.04 & 0.25 \\\\ 0.75 & 0.64 & 0.32 & 0.29 & 0.39 & 0.61 & 0.57 & 0.34 & 1.00 & 0.46 \\\\ 0.22 & 0.20 & 0.35 & 0.23 & 0.30 & 0.18 & 0.29 & 0.25 & 0.35 & 0.10 \\\\ 0.41 & 0.25 & 0.50 & 0.20 & 0.56 & 0.60 & 0.59 & 0.60 & 0.47 & 0.31 \\end{array}\\right]"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathscr{I}=\\{1,2, \\ldots, 10\\}"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall i \\in \\mathscr{I}"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall j \\in \\mathscr{J}"} +{"pdf": "arxiv_math/2503.05874_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05874", "page": 1, "id": "2503.05874_pg8_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "S_{i j}^{\\prime}=S_{i j} \\cap I_{j}=\\left\\{L_{j}\\right\\}"} +{"pdf": 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H^{N-1}(\\Gamma_\\sigma\\cap\\{d=t\\})\\ dt,"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "c_1H^{N-1}(\\partial \\Omega)\\le H^{N-1}(\\Gamma_\\sigma\\cap\\{d=t\\})\\le c_1H^{N-1}(\\partial \\Omega)"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Gamma_\\sigma\\cap\\{d=t\\}"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g: (0,\\sigma)\\to \\mathbb R"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{B\\subset\\Omega}\\left(\\frac1{|B|}{\\int_Bw\\ dx }\\right)\\left(\\frac1{|B|}{\\int_Bw^{-1}}dx \\right)<+\\infty"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Omega \\subset\\mathbb R^N"} +{"pdf": "arxiv_math/2503.08649_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08649", "page": 1, "id": "2503.08649_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "w(x):=d^\\beta(x), \\quad{\\rm \\ \\ }"} +{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau^*\\frac{\\partial S_t}{\\partial t}"} +{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(f, g)\\colon \\begin{array}{ccc} \\Omega^1_0(M) & \\to & \\Omega^1_0(M)\\\\ \\zeta &\\mapsto & \\zeta^*[f,\\tau^*\\zeta^*g] \\end{array}."} +{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f, g\\in J_\\xi^\\infty,"} +{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "(J_\\xi^\\infty,\\triangleright)"} +{"pdf": "arxiv_math/2503.05000_pg9.pdf", "url": "https://arxiv.org/pdf/2503.05000", "page": 1, "id": "2503.05000_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "%\\abs{\\xi}=1,\\quad \\abs{\\eta(f, g)}=\\abs{f}+\\abs{g}. %"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{G}(tX,t^\\beta Y) = t^{d-e}\\widetilde{G}(X,Y)"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f_-(t) = \\sum_{k = 0}^m (-1)^{d-rk}a_k t^{sk}"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq m,n\\leq \\lfloor d/r\\rfloor"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = t^d F(X,Y)= t^d X^e \\widetilde{F}(X,Y)."} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "g_-(t) = \\sum_{k = 0}^n (-1)^{d-rk}b_k t^{sk}"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widetilde{F}(tX,t^\\beta Y) = t^{d-e}\\widetilde{F}(X,Y)"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a,b\\in\\R\\setminus\\{0\\}"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "g_+(t) = \\sum_{k = 0}^n b_k t^{sk}"} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F(tX,t^\\beta Y) = (tX)^e \\widetilde{F}(tX, t^\\beta Y) = t^e X^e \\widetilde{F}(tX,t^\\beta Y)."} +{"pdf": "arxiv_math/2503.06022_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06022", "page": 1, "id": "2503.06022_pg26_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\deg g_+ = \\deg g_- = sn"} +{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "D:=\\{\\omega\\in\\Omega\\colon \\|z(\\omega)\\|>\\sigma\\}"} +{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "D\\setminus N=\\bigcup_{i=1}^\\infty E_i"} +{"pdf": "arxiv_math/2503.09182_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09182", "page": 1, "id": "2503.09182_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "E_0:=(E\\setminus D)\\cup N\\cup C"} +{"pdf": "arxiv_math/2503.09182_pg4.pdf", 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"https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "u_{xx} + u_{yy} + \\lambda e^{u} = 0, \\ \\ (x,y) \\in [0,1] \\times [0,1],"} +{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "X^{k+1}(i,j) = G(i,j;X^{k},\\alpha) \\equiv X^k(i,j) + \\alpha M(G_{\\rm B}(i,j;X^{k}))."} +{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "X(i,j) \\approx u(x_i,y_j)"} +{"pdf": "arxiv_math/2503.03909_pg14.pdf", "url": "https://arxiv.org/pdf/2503.03909", "page": 1, "id": "2503.03909_pg14_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\rm TOL}=10^{-6}, \\hat{m}=5, \\theta=0.9, \\alpha = 0.125h_x^2"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y}=t\\in T\\setminus\\{0\\}"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_i|=2^{2m-t}+2^m-2^{m-t}"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1\\star x + y_2\\star x=z"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(1-2^{-t})(2^{2m}-2^m)"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\star\\frac{x}{y} \\in T"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|E_1|=|f_1^{-1}(f_1(0,0))| = 2\\cdot 2^m-1+(2^{m-t}-1)(2^m-1)."} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(1-\\frac{1}{2^m}\\right)\\big(2^n-2^{n/2}\\big)"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{y_1}(x)\\neq \\alpha_{y_2}(x)"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma'(y_1)\\neq \\sigma'(y_2)"} +{"pdf": "arxiv_math/2503.03905_pg7.pdf", "url": "https://arxiv.org/pdf/2503.03905", "page": 1, "id": "2503.03905_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "W_f(a,b)=\\begin{cases} \\pm 2^{n/2} & \\text{if $b\\neq 0$,} \\\\ 0 & \\text{if $a\\neq 0$, $b=0$,} \\\\ 2^n & \\text{if $a=0$, $b=0$.} \\end{cases}"} +{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\cdot, \\cdot \\rangle"} +{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "c_{l,m}= \\sqrt{\\frac{(2 l+1)}{4 \\pi} \\frac{(l-m)!}{(l+m)!}}"} +{"pdf": "arxiv_math/2503.05503_pg13.pdf", "url": "https://arxiv.org/pdf/2503.05503", "page": 1, "id": "2503.05503_pg13_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{ Y_l^m : l\\in \\N_0, \\, -l \\le m \\le l \\}"} +{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda \\in [\\underline{H},\\overline{H}],"} +{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\mathbb E}X^2(t)= \\sum_{j=0}^{+\\infty} \\sum_{k=0}^{2^{j}-1} \\left|\\int_{0}^{1} (t-s)_{+}^{H_{j}(k/{2^j})-{1}/{2}} h_{j,k}(s)ds\\right|^2 < +\\infty."} +{"pdf": "arxiv_math/2503.07286_pg6.pdf", "url": "https://arxiv.org/pdf/2503.07286", "page": 1, "id": "2503.07286_pg6_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\lambda,x) \\in [\\underline{H},\\overline{H}] \\times \\mathbb{R},"} +{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial H^h/\\partial l_i"} +{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "l_{\\sigma}=(l_{12}(l_{23}, l_{24}, l_{34}), l_{13}(l_{23}, l_{24}, l_{34}), l_{14}(l_{23}, l_{24}, l_{34}), l_{23}, l_{24}, l_{34}),"} +{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^h(l^h)=H(l^*,l^h)=H(l)= cov(l) - 2\\pi\\sum_{i\\in E} l_i,"} +{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial H^h}{\\partial l_i} = - K_i, \\quad e_i\\in E^h."} +{"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(l_{23}, l_{24}, l_{34})=l^h_{\\sigma} \\in \\R^3"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ". We obtain the \\textit{conilpotent filtration}"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "is conilpotent. Similar definitions can be made for comodules over"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "to be the dual of the group algebra"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "be an augmented coalgebra over a field"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "be the reduced coproduct. Then we call"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "in the profinite topology of"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ". For a profinite group, we define"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded algebra and for a graded comodule"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded coalgebra. Additionally, if"} +{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "we define its group coalgebra"} +{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_k = \\|\\boldsymbol{x}_k-\\boldsymbol{x}_*\\|,\\quad f(k) = -\\ln \\xi_k."} +{"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty}\\frac{\\xi_{k+1}}{\\xi_k^q}=Q_q, \\quad q>1,\\quad 00"} +{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{b^{(1_n)}_1\\}_{n=1}^\\infty"} +{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "0\\leq b^{(1_n)}_2 \\leq a_2"} +{"pdf": "arxiv_math/2503.08011_pg19.pdf", "url": "https://arxiv.org/pdf/2503.08011", "page": 1, "id": "2503.08011_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n\\to\\infty} \\sum_{i=1}^\\infty |b^{(n_n)}_i - c_i| = 0"} +{"pdf": "arxiv_math/2503.08206_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08206", "page": 1, "id": "2503.08206_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f(x) = \\begin{dcases} 1/q & \\text{ if } x=p/q\\text{ with } (p,q)=1;\\\\ 0 & \\text{ if } x \\text{ is irrational;} \\end{dcases}"} +{"pdf": "arxiv_math/2503.08206_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08206", "page": 1, "id": "2503.08206_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Delta^+(x) := B^+(x)-2B_0^+(x)\\quad \\text{ and }\\quad \\Delta^-(x) := W^-(x)-2B_0^-(x)."} +{"pdf": "arxiv_math/2503.05844_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05844", "page": 1, "id": "2503.05844_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "Q=\\begin{bmatrix} 10& 0\\\\ 0& 50\\\\ \\end{bmatrix}"} +{"pdf": "arxiv_math/2503.05844_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05844", "page": 1, "id": "2503.05844_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0= [0.2; 0; 0; 0.1; 0]^T"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in {\\Sigma_\\vartheta}"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta: {\\Sigma_\\vartheta} \\times \\mathbb{T} \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto (T_\\omega (x) , T_\\omega(y))."} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta^{(2)} : {\\Sigma_\\vartheta} \\times \\mathbb{T}^2 \\to {\\Sigma_\\vartheta} \\times \\mathbb{T}^2"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "T^n_\\omega (x) \\coloneqq T_{\\omega_{n-1}} \\circ \\cdots \\circ T_{\\omega_0} (x)"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Omega_\\vartheta"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(x,y) \\mapsto T^{(2)}_\\omega (x,y)"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Sigma_\\vartheta"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\Sigma_\\vartheta} \\mu^{(2)} \\left( \\left( T^{(2)}_\\omega\\right)^{-1} (A) \\right) \\, d\\mathbb{P} (\\omega) = \\mu^{(2)}(A),"} +{"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma \\omega \\coloneqq (\\omega_{i+1})_{i\\in\\mathbb{N}}"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "and the fact that for every"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{\\P(\\overline{W}_{n} \\in (x,x+1])}{ n \\P(\\overline{X}_{1} \\in (x,x+1])} -1 \\right|=0,"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": ". This will indeed imply that"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\overline{W}_{n}}{\\widehat{b}_{n}} = \\frac{X_{1}+ \\cdots+X_{n}-b_{n}}{a_{n}} \\cdot \\frac{a_{n}}{\\widehat{b}_{n}}+ \\frac{b_{n}+\\gamma n}{\\widehat{b}_{n}},"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "which implies tightness since"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "and that condition (3.3) there holds also with"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\mathrm{TV}}\\left((\\overline{X}_i^{(n)}:1\\leq i\\leq n-1),(\\overline{X}_i:1\\leq i\\leq n-1)\\right) \\to 0,"} +{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{ \\P({X}_{1} \\in(x-m_{n},x-m_{n}+1])}{ \\P({X}_{1} \\in(x-\\gamma,x-\\gamma+1])} -1 \\right|=0,"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{\\circ}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{*}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},k_{2})\\in \\mathcal{A}_{2}^{*}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},\\ldots,k_{m-1})\\in \\mathcal{A}_{m-1}^{*}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{K}_{m}(3,2^{m-3},1,k) = 2(k-m)+3,\\quad \\mathbb{K}_{m-1}(2^{m-3},1,k) = k-m+2,"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{*}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{r}^{\\circ}(3,1,2)"} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "3m(k-1)-2k+5\\not\\equiv 2\\pmod{3},\\quad\\text{that is,}\\quad k\\not\\equiv 0\\pmod{3}."} +{"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "3\\leqslant r\\leqslant m-1"} +{"pdf": 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L^1(\\R^{nm},\\gamma_{n,m})"} +{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\nabla D^m\\psi(x_1,\\dots,x_m)=\\left\\{\\nabla \\psi(x_i) - \\nabla \\psi\\left(-\\sum_{i=1}^m x_i\\right) \\right\\}_{i=1}^m."} +{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "V_\\epsilon^\\star = \\frac{|x|^2}{2} - \\epsilon \\psi(x) + \\frac{\\epsilon^2}{2}|\\nabla \\psi|^2 + o(\\epsilon^3),"} +{"pdf": "arxiv_math/2503.06191_pg26.pdf", "url": "https://arxiv.org/pdf/2503.06191", "page": 1, "id": "2503.06191_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\nabla D^m\\psi(x_1,\\dots,x_m)|^2=\\sum_{i=1}^m\\left|\\nabla \\psi(x_i) - \\nabla \\psi\\left(-\\sum_{i=1}^m x_i\\right)\\right|^2."} +{"pdf": 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"arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_i(s)=\\|\\Omega_i(p,s)\\|"} +{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_i, \\beta_i, \\gamma_i, \\delta_i"} +{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in B_{\\eta}\\cap K\\implies \\lim_{t\\to\\infty}\\|x(t;t_0,x_0)\\|=0"} +{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": "https://arxiv.org/pdf/2503.09471", "page": 1, "id": "2503.09471_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\in C([t_0,T];L(\\mathcal H_i))"} +{"pdf": "arxiv_math/2503.09471_pg2.pdf", "url": 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"id": "2503.05270_pg21_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "b = \\beta_+^{-{2}/{3}}"} +{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left({n^{1/3}}\\middle/{\\log^{{1}/{3}}}\\right)\\exp\\left\\{-g_n\\right\\}\\to\\infty"} +{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{n}{\\log n}\\exp\\left\\{-g_n\\right\\}\\to\\infty"} +{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{n^{1/3}}{\\log^{1/3}n}\\exp\\left\\{- \\frac{\\beta_+^2b^3\\log n }{6}\\right\\} = \\frac{n^{1/3}}{\\log^{1/3}n} \\exp\\left\\{-\\frac{1}{6}\\log n\\right\\} = \\frac{n^{1/6}}{\\log^{1/3}n}\\to \\infty \\text{, for } n\\to \\infty."} +{"pdf": "arxiv_math/2503.05270_pg21.pdf", "url": "https://arxiv.org/pdf/2503.05270", "page": 1, "id": "2503.05270_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\exp\\left\\{-\\frac{\\beta_+^2b^2\\log^{2/3}n}{4n^{2/3}}-\\frac{\\beta_+^2b\\log^{{1}/{3}}n}{12n^{4/3}}\\right\\}\\to 1 \\text{, for } n\\to\\infty,"} +{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "meg(G)=\\vert Man(G)\\vert"} +{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "d(x'_1,y)=d(x'_1,v_{i_l})"} +{"pdf": "arxiv_math/2503.06086_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06086", "page": 1, "id": "2503.06086_pg17_math_002", 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"math": "1 \\le \\ell \\le \\lfloor \\frac{p-k-1}{2k-1} \\rfloor"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R \\subseteq A \\times A"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\R \\subseteq A \\times B"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| > (1+\\delta)p - O(1)"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| = 6 \\lfloor \\frac{p}{11} \\rfloor - 3"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| = p - (k-1) \\lfloor \\frac{p-k-1}{2k-1} \\rfloor - k + 1"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| \\ge p-k+\\lfloor \\frac{p-k-1}{2k-1} \\rfloor + 4 > \\frac{2k}{2k-1} p - k + 2"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "|A| + |B| \\ge (1+\\delta)p"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09121", "page": 1, "id": "2503.09121_pg4_math_016", "type": "math", "max_diffs": 0, "checked": null, "math": "|B| = k \\lfloor \\frac{p-k-1}{2k-1} \\rfloor + 2"} +{"pdf": "arxiv_math/2503.09121_pg4.pdf", "url": 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h_0."} +{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|u(t, \\cdot)\\|_{C^{2+\\beta}([g(t), h(t)])} \\leq Q^*."} +{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t,x,u)\\leq \\bar f(u)<0"} +{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "h, g \\in C^{1+\\frac{\\beta}{2}}((0, T_m)), \\quad u \\in C^{1+\\frac{\\beta}{2}, 2+\\beta}(\\Omega_{T_m}),"} +{"pdf": "arxiv_math/2503.06020_pg21.pdf", "url": "https://arxiv.org/pdf/2503.06020", "page": 1, "id": "2503.06020_pg21_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": 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null, "math": "\\mu_i^{(1)}(t), \\mu_i^{(2)}(t)"} +{"pdf": "arxiv_math/2503.06549_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06549", "page": 1, "id": "2503.06549_pg8_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "k(a)\\sim N^{2(1-a)/3}"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{I}\\left|f_n\\left(t_0\\right)(x)-f\\left(t_0\\right)(x)\\right| d x \\geq \\int_{\\tilde{I}}\\left|f_n\\left(t_0\\right)(x)-f\\left(t_0\\right)(x)\\right| d x>\\epsilon_0 \\cdot m(\\tilde{I})"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{\\mathcal{Z}}"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\forall n \\in \\mathbb{N}"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "C\\left([0, T] ; L^2(I ; \\mathbb{R} \\otimes \\mathbb{R})\\right)"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "(f_n)_{n=1}^{\\infty} \\in \\mathcal{Z}"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t_0)(x) > \\bar{A} + \\epsilon_0"} +{"pdf": "arxiv_math/2503.09107_pg9.pdf", "url": "https://arxiv.org/pdf/2503.09107", "page": 1, "id": "2503.09107_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "f(t_0)(x) < -\\bar{A} -\\epsilon_0"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{F}_1 = 0.0099s_{xx}+0.9955u"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{l}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\bm{k}/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{F}_1 = 0.0098s_{xx} + u - 0.0083s + 0.0046"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,t) = (\\pi x)/5+u_1(t)"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tilde{\\gamma} = -1/(\\bm{k}^2+\\bm{l}^2)\\tilde{s}"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\gamma = -\\Delta^{-1} s"} +{"pdf": "arxiv_math/2503.08059_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08059", "page": 1, "id": "2503.08059_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "u(x,y,t) = u_3(t)\\times \\{0.1\\sin[2\\pi (x + y)] + \\cos[2\\pi (x + y)]\\}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_{r}^{\\text{NLoS}} \\in \\mathbb{C}^{M_r \\times N}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "e^{j 2 \\pi d_{\\text{R}} \\sin(\\gamma) \\left[ \\lfloor \\frac{n-1}{N_x} \\rfloor \\sin(\\zeta) + ((n-1)-\\lfloor \\frac{n-1}{N_x} \\rfloor N_x) \\cos(\\zeta) \\right]/\\lambda }, n = 1, 2, \\cdots, N"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{n}_{\\text{BS},i} \\sim \\mathcal{CN} \\left( \\mathbf{0}_{M_t}, \\sigma_{\\text{BS}}^2 \\mathbf{I}_{M_t} \\right)"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\boldsymbol{\\varphi} = [\\varphi_1, \\varphi_2, \\cdots, \\varphi_N]^T"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_t \\in \\mathbb{C}^{N \\times M_t}, \\mathbf{H}_r \\in \\mathbb{C}^{M_r \\times N}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta \\in \\left[0, \\pi\\right)"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{H}_d \\in \\mathbb{C}^{M_r \\times M_t}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_n = e^{j \\theta_n}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta_d^{\\text{AoA}}"} +{"pdf": "arxiv_math/2503.06944_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06944", "page": 1, "id": "2503.06944_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "M_s \\leq \\min\\left\\{M_t, M_r\\right\\}"} +{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{ab}{}^I = - u^\\mu_a u^\\nu_b \\nabla_{(\\mu} n^I_{\\nu)} =n^I_\\mu D_a u^\\mu_b + \\frac{1}{2} \\hat \\upsilon^I \\tau_{ab},"} +{"pdf": "arxiv_math/2503.08947_pg7.pdf", "url": "https://arxiv.org/pdf/2503.08947", "page": 1, "id": "2503.08947_pg7_math_001", "type": "math", 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"2503.09352_pg5_math_025", "type": "math", "max_diffs": 0, "checked": null, "math": "z=[z_0:z_1:...:z_{d-1}:1]"} +{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{aligned} \\mathcal{L}_{\\text{SIMax}} = \\mathbb{E}_{z \\sim {p_{\\theta}(z|x)}} \\{ & \\mathbb{E}_{t \\sim T} \\left[-\\log q_{\\phi_t}(y_{t}|z)\\right] \\\\ & - \\alpha \\log \\frac{\\exp(\\text{sim}(z, z^{+}) / \\tau)}{\\sum_{z' \\in \\mathcal{B^{+}}} \\exp(\\text{sim}(z, z') / \\tau)} \\}, \\end{aligned}"} +{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I(X; Z) \\geq I(Z; Z^\\prime)"} +{"pdf": "arxiv_math/2503.04667_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04667", "page": 1, "id": "2503.04667_pg3_math_003", "type": "math", "max_diffs": 0, 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"max_diffs": 0, "checked": null, "math": "J_s(\\bar{f}_s):=\\min_{f_s \\in U_{ad}} J_s(f_s),"} +{"pdf": "arxiv_math/2503.09386_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09386", "page": 1, "id": "2503.09386_pg6_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{F}_s=\\left\\{f_s\\right\\}_{0 0"} +{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}(\\underline{\\hat{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}}) \\neq \\underline{V}_{\\mathcal{L}}) \\leq \\epsilon_e"} +{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left(\\mathcal{V},p_V\\right)"} +{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{V}_{\\mathcal{L}}"} +{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V} \\in \\{0,1\\}"} +{"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\underline{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}})"} +{"pdf": "arxiv_math/2503.07310_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07310", "page": 1, "id": "2503.07310_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in \\mathcal{X}\\subseteq \\mathbb{R}^{n_x}"} +{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "is the inverse of a (fictitious) mass matrix. The solutions to the equations of motion resulting from this Hamiltonian can be used as a preliminary step in the HMC algorithm. The time reversal invariance of Hamiltonian systems guarantees that the detailed balance holds. In general, the equations of motions are solved with a numerical integrator, and detailed balance is satisfied only in the limit of time step"} +{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ". In the condition of detailed balance, the kinetic energy arising from the Metropolis acceptance probability and that coming from the Maxwell--Boltzmann term in the {\\it a priori} probability"} +{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "as the momenta associated to the weights"} +{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": ", we make use of a modified HMC algorithm that damps fluctuations in the direction opposite to"} +{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "intermediate pivot points. Although this method allows for more flexible connections, it is computationally demanding, requiring multiple training runs to optimize the pivot locations. Furthermore, the number of required pivots can grow significantly, particularly in less overparameterized settings, making the approach increasingly impractical in such regimes. As an alternative to generate trajectories from"} +{"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset \\{\\mu \\in P(X \\times Y): \\mu_X^* \\in P(X) \\text{ is the } X\\text{-marginal of } \\mu\\}"} +{"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu^* \\in P(X\\times Y)"} +{"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset P(X\\times Y)"} +{"pdf": "arxiv_math/2503.04604_pg3.pdf", "url": "https://arxiv.org/pdf/2503.04604", "page": 1, "id": "2503.04604_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "00"} +{"pdf": 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"math": "\\mathcal H^{n-1}(\\hat V_{\\hat j})"} +{"pdf": "arxiv_math/2503.09483_pg1.pdf", "url": "https://arxiv.org/pdf/2503.09483", "page": 1, "id": "2503.09483_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d_k \\in \\mathbb{R}^{k_f\\times k_f}"} +{"pdf": "arxiv_math/2503.09528_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09528", "page": 1, "id": "2503.09528_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "P=\\{b^{l_1}_1/b^{l_2}_2: (l_1,l_2)\\in\\mathbb{Z}^2_{>0}\\}\\subset\\mathbb{C}."} +{"pdf": "arxiv_math/2503.09528_pg16.pdf", "url": "https://arxiv.org/pdf/2503.09528", "page": 1, "id": "2503.09528_pg16_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "b_1,\\dots,b_k\\in\\mathbb{Z}[i]"} +{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon} \\cdot \\underline{x} = \\sum_{i=1}^{5} \\varepsilon_i x_i \\leq 1, \\quad \\underline{x}\\in \\mathbb{H}_5"} +{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathrm{Isom}(\\mathbb{H}_5)"} +{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "h_x = % (1-\\norm{x}^2)^{-1} g_x"} +{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{\\varepsilon}"} +{"pdf": "arxiv_math/2503.04245_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04245", "page": 1, "id": "2503.04245_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\prod \\varepsilon_i=1"} +{"pdf": 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x_i}{\\log\\log x_i}\\right)\\right)>i"} +{"pdf": "arxiv_math/2503.07801_pg15.pdf", "url": "https://arxiv.org/pdf/2503.07801", "page": 1, "id": "2503.07801_pg15_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L(g_i)\\le M_{x_i}\\le x_i^2=(\\log i)^{(4/C)\\log\\log\\log i}<(\\log g_i)^{c_0\\log\\log\\log g_i}"} +{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{P_* T} \\alpha = \\int_T P^* \\alpha."} +{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\alpha(x)| \\lesssim_N \\langle x\\rangle^{-N}"} +{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha \\in \\mathscr S"} +{"pdf": "arxiv_math/2503.07573_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07573", "page": 1, "id": "2503.07573_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle x\\rangle := \\sqrt{1 + |x|^2}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l} \\in \\mathbf{R}^{m}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "P \\in \\mathbb{R}^{n \\times n}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{l}_i= \\mathbf{u}_i"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{x} \\in \\mathbb{R}^{n}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "A \\in \\mathbb{R}^{m \\times n}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} \\in \\mathbb{R}^{m}"} +{"pdf": "arxiv_math/2503.05941_pg1.pdf", "url": "https://arxiv.org/pdf/2503.05941", "page": 1, "id": "2503.05941_pg1_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{q} \\in \\mathbb{R}^{n }"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar{L}(n)\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}>\\tau_n)\\le \\bar{L}(|x-x_0|),\\qquad x\\in B^c_{r_0}(x_0)."} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "O\\subseteq B_{r_0}(x_0)"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(\\R^d)"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in\\mathcal{B}(B_{r_0}(x_0))"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(0,\\infty)\\times \\bar{B}_{r_0}(x_0)\\times\\bar{B}_{r_0}(x_0)"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}=\\infty)\\le \\frac{\\bar{L}(|x-x_0|)}{\\bar L(\\infty)}<1,\\qquad x\\in B^c_{r_0}(x_0),"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=\\mathcal{B}(B_{r_0}(x_0))."} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}^x(\\tau_{B_{r_0}(x_0)}<\\infty)>0"} +{"pdf": "arxiv_math/2503.06111_pg6.pdf", "url": "https://arxiv.org/pdf/2503.06111", "page": 1, "id": "2503.06111_pg6_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "B\\in \\mathcal{B}(\\R^d)"} +{"pdf": "arxiv_math/2503.08227_pg2.pdf", "url": "https://arxiv.org/pdf/2503.08227", "page": 1, "id": "2503.08227_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\partial f / \\partial \\vec{n}"} +{"pdf": "arxiv_math/2503.04674_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04674", "page": 1, "id": "2503.04674_pg19_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "t\\in [-\\frac{1}{2},0]"} +{"pdf": "arxiv_math/2503.04674_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04674", "page": 1, "id": "2503.04674_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi(t,x,y)=\\mathrm{e}^{-t}x(1-x)y(1-y)"} +{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta(\\varphi)=\\sup_{\\mu\\in \\mathcal{M}_T(X)}\\int \\varphi\\ d\\mu"} +{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}^{\\mathbb{Z}^d}"} +{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{M}_{{\\rm max}}(\\varphi)"} +{"pdf": "arxiv_math/2503.06958_pg1.pdf", "url": "https://arxiv.org/pdf/2503.06958", "page": 1, "id": "2503.06958_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi:X\\rightarrow \\mathbb{R}"} +{"pdf": "arxiv_math/2503.05104_pg18.pdf", "url": "https://arxiv.org/pdf/2503.05104", "page": 1, "id": "2503.05104_pg18_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha = 0.3, 0.6, 0.9"} +{"pdf": "arxiv_math/2503.05885_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05885", "page": 1, "id": "2503.05885_pg8_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\overline{B}^\\nu_{h,\\alpha} := \\big\\{ r \\in [1,\\infty) : m^\\nu([r, r+h]) \\geq \\frac{\\alpha h}{ r}\\big\\}"} +{"pdf": "arxiv_math/2503.05885_pg8.pdf", "url": "https://arxiv.org/pdf/2503.05885", "page": 1, "id": "2503.05885_pg8_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu_{1,R}(\\overline{B}^\\nu_{h,\\alpha}) \\leq C\\alpha^{-1}."} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])^{\\vee} \\simeq \\Lambda_{n}^{2}/\\omega_{n}^{\\pm}\\Lambda_{n}^{2}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{\\infty, w}, E[p^{\\infty}]) \\coloneq \\varinjlim H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "E[p^{\\infty}]^{G_{K_{\\infty, p}}}=0"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{\\infty, w}, E[p^{\\infty}])^{\\vee} \\simeq \\Lambda^{2}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "B_{v_{\\infty}} \\coloneq E[p^{\\infty}]^{G_{K_{\\infty, v_{\\infty}}}}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n,v_{n}}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "v \\in \\Sigma \\setminus \\{ p \\}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K_{\\infty, v_{\\infty}}/K_{n, v_{n}}"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_019", "type": "math", "max_diffs": 0, "checked": null, "math": "H^{1}_{\\pm}(K_{n, p}, E[p^{\\infty}])"} +{"pdf": "arxiv_math/2503.09034_pg7.pdf", "url": "https://arxiv.org/pdf/2503.09034", "page": 1, "id": "2503.09034_pg7_math_021", "type": "math", "max_diffs": 0, "checked": null, "math": "E[p^{\\infty}]^{G_{K_{\\infty}}} = 0"} +{"pdf": "arxiv_math/2503.07088_pg3.pdf", "url": 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\\binom{[10]}{5}"} +{"pdf": "arxiv_math/2503.06377_pg8.pdf", "url": "https://arxiv.org/pdf/2503.06377", "page": 1, "id": "2503.06377_pg8_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "u \\in X_1 \\cup X_{-1}"} +{"pdf": "arxiv_math/2503.09115_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09115", "page": 1, "id": "2503.09115_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v_1 < v_2 < \\ldots < v_k"} +{"pdf": "arxiv_math/2503.09115_pg3.pdf", "url": "https://arxiv.org/pdf/2503.09115", "page": 1, "id": "2503.09115_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "v_1 <_x v_2 <_x \\ldots <_x v_k"} +{"pdf": "arxiv_math/2503.09123_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09123", "page": 1, "id": "2503.09123_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in\\mathbb{I}_{P^{(k)}}"} +{"pdf": "arxiv_math/2503.09123_pg4.pdf", "url": "https://arxiv.org/pdf/2503.09123", "page": 1, "id": 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0, "checked": null, "math": "E_{Sp(2)} = E_{SL(2)}"} +{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}\\circ \\phi^{-1} = \\frac{1}{|B(R)|} \\Theta(R^2-\\epsilon^{-1/2}(x^2+p_{x}^2+p_{y}^2)-\\epsilon^{3/2}y^2)"} +{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lambda_{i}^{H} \\lambda_{n+1-i}^{V} \\equiv \\text{const}"} +{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, "id": "2503.07965_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "f_{0}(\\mathbf{z},R) = \\frac{6}{R^2 |B(R)|} \\chi_{B(R)} = \\frac{6}{R^2|B(R)|} \\Theta(R^2-|\\mathbf{z}|^2)"} +{"pdf": "arxiv_math/2503.07965_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07965", "page": 1, 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\\psi}_m^{\\rm Pin}=(x_m,\\beta_m,d)"} +{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_1=-\\frac{D_{\\rm W}}{4}"} +{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\beta_m=-\\frac{D_{\\rm W}}{2}+(m-1)\\frac{D_{\\rm W}}{M}+\\frac{D_{\\rm W}}{2M}"} +{"pdf": "arxiv_math/2503.08554_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08554", "page": 1, "id": "2503.08554_pg1_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "h_{mk}=\\frac{\\sqrt{\\eta} e^{-2\\pi j \\left(\\frac{ 1}{\\lambda}\\left| {\\boldsymbol \\psi}_m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| +\\frac{1}{\\lambda_g}\\left| {\\boldsymbol \\psi}_0^{\\rm Pin} - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right| \\right)}}{ \\left| {\\boldsymbol \\psi} _m - {\\boldsymbol \\psi}_k^{\\rm Pin}\\right|}"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{B}(\\mathcal{H})"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_s\\,\\mathcal{E}_t \\subseteq \\mathcal{E}_{s+t}"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_0 = \\mathbb{C}\\,I_{\\mathcal{H}}, \\quad\\text{and for each integer } t \\ge 1,"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\{\\mathcal{E}_t\\}_{t \\ge 0} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H}),"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E}_t^* = \\mathcal{E}_t"} +{"pdf": "arxiv_math/2503.08736_pg1.pdf", "url": "https://arxiv.org/pdf/2503.08736", "page": 1, "id": "2503.08736_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{E} \\;\\subseteq\\; \\mathcal{B}(\\mathcal{H}),"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "x_0\\in\\{\\frac{1}{3},\\frac{2}{3}\\}"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}_{\\bm{y}}[F(\\bm{y})]"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\bm{y})=G(u^s(\\cdot,\\bm{y}))=u^s(x_0,\\bm{y}),"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "y_1,\\ldots,y_s\\overset{i.i.d.}{\\sim}N(0,1)"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "a^s(x,\\bm{y})= \\exp\\left(\\sum_{j=1}^{s}\\frac{1}{j^2}\\sin(2j\\pi x)y_j \\right),"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta_j(\\frac{n}{N})"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_1 = \\frac{1}{2}\\left(b_1+\\sqrt{b_1^2+1-\\frac{1}{2\\lambda}}\\right),"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "u^s(0,\\bm{y})=u^s(1,\\bm{y})=0."} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "-\\frac{d}{dx}(a^s(x,\\bm{y})\\frac{du^s(x,\\bm{y})}{dx})=1,"} +{"pdf": "arxiv_math/2503.05334_pg17.pdf", "url": "https://arxiv.org/pdf/2503.05334", "page": 1, "id": "2503.05334_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bm{y}\\in\\mathbb{R}^s"} +{"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\inf\\{|\\lambda-\\lambda'|:\\lambda\\neq\\lambda',\\lambda,\\lambda'\\in \\sigma(\\Delta)\\}=0"} +{"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^k(x_1)\\leq\\phi_\\zeta^k(\\gamma_m)\\leq\\phi_\\zeta^k(x_2), \\forall k"} +{"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^m(x_j)\\to\\zeta, j=1,2"} +{"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "x_1, x_2 \\in \\sigma(\\Delta_n)"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "not being defined on the whole"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "where the left value is the result of the Riemann integral computed in"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\int_{[a, b]} f \\right)^{M} = \\left( \\int_{[a, b]} g \\right)^{N} \\! \\! \\! \\!,"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "we have a bounded real-valued function defined on some rectangle"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "is unique except in a Lebesgue measure zero set: if in"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ", then there exists some measure zero set"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": ", and the right one is the result of the integral computed in"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "is a \\emph{Boolean algebra} if"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bounded function. Then,"} +{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "is another Riemann integrable function on"} +{"pdf": "arxiv_math/2503.07030_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07030", "page": 1, "id": "2503.07030_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\Phi^{T_F},\\mathbf{u},\\mathbf{y},\\mathbf{p})=0"} +{"pdf": "arxiv_math/2503.07030_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07030", "page": 1, "id": "2503.07030_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\sigma}_\\alpha (u^k,y^k,\\mathbf{p})"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_2"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "R=\\{(\\alpha,\\beta,\\gamma)\\in[0,1]^3:\\alpha\\beta+\\gamma>1,\\alpha\\gamma+\\beta>1,\\beta\\gamma+\\alpha>1\\},"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "E(H)\\subset \\binom{[n]}{r}"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_1"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma)=\\alpha+\\beta+\\gamma-2"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha=1-1/n+\\alpha'n^{(\\delta_a-2)},\\beta=1-1/n+\\beta'n^{(\\delta_b-2)},\\gamma=1-1/n+\\gamma'n^{(\\delta_c-2)}"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "|V_1|=|V_2|=\\cdots=|V_r|=n"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "|B_1|=\\alpha n^2-n(n-1)"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma) = 2\\sqrt{\\alpha\\beta(1-\\gamma)}+2\\gamma-2"} +{"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "T_{min}(\\alpha,\\beta,\\gamma)"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "H_i(x, 1) = K_i(x,1) \\in A"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x \\in \\complement \\overline{V_i} \\cap U_i"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_i(\\overline{W_i}) = 1"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "L_i(u,t) = \\begin{cases} p_i(H(u,2t)) & \\text{if } 0 \\leq t \\leq \\frac12 \\\\ G(h_i(u), 2t-1) & \\text{if } \\frac12 \\leq t \\leq 1. \\end{cases}"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi\\colon X\\to T^n(\\mu)"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "L_i \\colon U_i \\times I\\to Z_h"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\complement V_i= X\\backslash V_i"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi(X) \\subseteq T^n(\\mu)"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "(U_i)_{0\\leq i \\leq n}"} +{"pdf": "arxiv_math/2503.06969_pg11.pdf", "url": "https://arxiv.org/pdf/2503.06969", "page": 1, "id": "2503.06969_pg11_math_015", "type": "math", "max_diffs": 0, "checked": null, "math": "p_i(H(u,1)) = p_i ((t_n \\circ \\Phi)(u)) = h_i(u) = G(h_i(u),0)"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "c \\in \\mathbb{F}_p^{\\times}"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x)"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv 1 + F(x) \\equiv (x+1)^m \\pmod{p}"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv c \\cdot (x+1)^m \\pmod{p}"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\text{Frob}_p: \\, a \\mapsto a^p"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv 1 \\pmod{n}"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(C_n^{\\boxtimes d}, x) \\equiv (x+1)^m \\pmod{p}"} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "(x+1)^{p^d} \\equiv x^{p^d} + 1 \\pmod{p}."} +{"pdf": "arxiv_math/2503.07910_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07910", "page": 1, "id": "2503.07910_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "F(x) = \\sum_{S \\in \\mathcal{F}_p} x^{|S|}"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "S^2\\setminus B(p_+, r_1)"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "E_\\mathbf{I}(u\\circ\\eta|_{B(p, r_0)})\\leq E_\\mathbf{I}(u|_{S^2\\setminus B(\\bar{p}, L\\bar{r})}) \\leq \\frac{\\varepsilon_\\mathbf{I}}{5}."} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varrho\\circ\\delta_r\\circ\\psi"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(B(p, r_0)) \\subset B(\\eta(p), r\\kappa^2r_0) \\subset B(\\eta(p), r)"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi\\colon S^2\\to X"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(B(p, r_0))\\subset S^2\\setminus B(p_-, Lr)"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "0<\\varepsilon<\\varepsilon_{\\mathbf{I}}"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta(B(p_+, 2r_1)) = S^2\\setminus B(p_-, h(Lr))\\subset S^2\\setminus B(p_-, Lr)"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\delta\\coloneqq \\min\\left\\{\\frac{ l_0^2}{2\\pi}, \\frac{\\varepsilon}{10 C_{\\mathbf{I}}}\\right\\} \\quad\\text{and}\\quad L\\coloneqq 2e^{\\delta^{-1}k_\\mathbf{I}(e_\\mathbf{I}(\\varphi) + 10^{-1}\\varepsilon)},"} +{"pdf": "arxiv_math/2503.08553_pg25.pdf", "url": "https://arxiv.org/pdf/2503.08553", "page": 1, "id": "2503.08553_pg25_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\eta\\colon S^2\\to S^2"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{s} \\in [0, 1]^d"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "x = \\sqrt{\\alpha P}s_1 +\\sqrt{\\bar \\alpha P}s_2"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat {\\mathbf{s}}^{(i)}"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\mathbf{s}} = g_{\\theta'}(\\mathbf{u}) = \\sigma(W'\\mathbf{u} + \\mathbf{b}')"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\bar \\alpha \\triangleq 1 - \\alpha"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta = \\{W, \\mathbf{b}\\}"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma(\\mathbf{x}) = [\\sigma(\\mathbf{x}_1), \\dots, \\sigma(\\mathbf{x}_d)]^T"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\theta' = \\{W', \\mathbf{b}'\\}"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} = f_\\theta(\\mathbf{s}) = \\sigma(W\\mathbf{s} + \\mathbf{b})"} +{"pdf": "arxiv_math/2503.07509_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07509", "page": 1, "id": "2503.07509_pg2_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbf{u} \\in [0, 1]^{d'}"} +{"pdf": "arxiv_math/2503.04397_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04397", "page": 1, "id": "2503.04397_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "{\\left\\| {\\cdot} \\right\\|_2}"} +{"pdf": 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\\mathbb{E}_{\\mathbf{y}, \\mathbf{x}} l(f_\\theta(\\mathbf{x}_s,z),\\mathbf{y}),"} +{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "s \\leftarrow s \\cup \\pi(x_s,z)"} +{"pdf": "arxiv_math/2503.09181_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09181", "page": 1, "id": "2503.09181_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\pi(x^{(n)}_s, z^{(n)}) \\in \\Lambda^{(n)}"} +{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "v, w \\in \\mathbb{R}^n"} +{"pdf": "arxiv_math/2503.06825_pg2.pdf", "url": "https://arxiv.org/pdf/2503.06825", "page": 1, "id": "2503.06825_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "B \\in \\mathbb{R}^{n \\times l}"} +{"pdf": 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"page": 1, "id": "2503.06725_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "v_{\\rm cpt}(x) = \\begin{cases} v^+_{\\rm cpt}(x) = (x - x_{\\rm ref})^{\\alpha_{\\rm cpt}}, ~x \\geq x_{\\rm ref};\\\\ v^-_{\\rm cpt}(x) = -\\lambda_{\\rm cpt} (x_{\\rm ref} - x)^{\\beta_{\\rm cpt}}, ~x < x_{\\rm ref}, \\end{cases}"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\mathcal{P}}"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_m, \\beta_m>0, \\forall m"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle s(t), a(t), r(t), s(t+1) \\rangle_{t=0}^{T_{\\rm e}}"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "w_{\\rm cpt}(x)=x, \\forall x"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "g_m(t; y_m(t)) = \\operatorname{min}\\left\\{ 1, \\frac{y_m^{\\alpha_m-1}(t) (1 - y_m(t))^{\\beta_m-1}}{\\operatorname{B}(\\alpha_m, \\beta_m)} \\right\\}"} +{"pdf": "arxiv_math/2503.06725_pg7.pdf", "url": "https://arxiv.org/pdf/2503.06725", "page": 1, "id": "2503.06725_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\alpha_{\\rm cpt}=\\beta_{\\rm cpt}=0.5"} +{"pdf": "arxiv_math/2503.07083_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07083", "page": 1, "id": "2503.07083_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{P}(T\\to \\infty)"} +{"pdf": "arxiv_math/2503.07083_pg1.pdf", "url": "https://arxiv.org/pdf/2503.07083", "page": 1, "id": "2503.07083_pg1_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Theta(2)\\equiv \\langle T\\rangle =2"} +{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "h^{\\mathrm{BdG}}_8(\\mathbf{k})=\\frac{1}{2}\\left[\\sin k_x\\sigma^x+\\sin k_y \\sigma^z +\\sin k_z\\tau^z\\sigma^y\\right]."} +{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat S_0(\\textbf{k}) = \\tau^z \\qquad \\hat{Q}_0=\\sum_\\mathbf{k} d_{\\mathbf{k}}^\\dag \\hat S_0(\\mathbf{k})d_{\\mathbf{k}} \\\\ \\hat S_1(\\textbf{k}) = \\cos k_z \\tau^z + \\sin k_z \\tau^x \\qquad \\hat{Q}_1=\\sum_\\mathbf{k} d_{\\mathbf{k}}^\\dag \\hat S_1(\\mathbf{k})d_{\\mathbf{k}}."} +{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "d^\\dag_\\mathbf{k}\\equiv(c^\\dag_{\\mathbf{k}\\uparrow},c^\\dag_{\\mathbf{k}\\downarrow},c_{-\\mathbf{k}\\uparrow},c_{-\\mathbf{k}\\downarrow})"} +{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "m_j(\\mathbf{k})\\eta^z\\sigma^j"} +{"pdf": "arxiv_math/2503.07708_pg3.pdf", "url": "https://arxiv.org/pdf/2503.07708", "page": 1, "id": "2503.07708_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "e_{\\mathbf{k}}\\equiv(c_{\\mathbf{k}-\\mathbf{K}},c^\\dag_{-\\mathbf{k}+\\mathbf{K}},c_{\\mathbf{k}+\\mathbf{K}},c^\\dag_{-\\mathbf{k}-\\mathbf{K}})^T"} +{"pdf": 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