nixiesearch/nixie-querygen-v2
Text Generation • 7B • Updated • 87 • 3
Datasets:
Tasks:
Question Answering
Modalities:
Text
Sub-tasks:
closed-domain-qa
Languages:
English
Size:
1M - 10M
License:
title stringlengths 20 150 | upvoted_answer stringlengths 0 54.3k |
|---|---|
Embedding a manifold in the disk | The proof of this has two steps.
Step 1: Denote the map $M \to S^{n+k}$ by $f$. Denote the inclusion $S^{n+k} \to D^{n+k+1}$ by $i$. Then the map $i \circ f : M \to D^{n+k+1}$ extends to a smooth function $g : W \to D^{n+k+1}$. You can define this extension in a variety of ways. A natural extension would be to ... |
Polar equation of an ellipse in polar axis with pole not in origin | The parametric equations are
$$x=4\cos (t ) $$
$$y=2+3\sin(t) $$ |
$f(x) = 1 / \lvert x \rvert^2$, $x\in \mathbb{R}^3$ , for the Fourier transform F, prove by scaling: $ F(f) (y) = C \frac{1}{\lvert y\rvert}. $ | So first $|x|^{-a}$ is in $L^1_{\mathrm{loc}}(\mathbb{R}^d)$ as soon as $a < d$ since by a radial change of variable
$$
\int_{|x|<1} \frac{\mathrm{d}x}{|x|^a} = \omega_d\int_0^1 r^{d-1-a} \mathrm{d}r = \frac{\omega_d}{d-a} < \infty
$$
where $\omega_d = \frac{2\pi^{d/2}}{\Gamma(d/2)}$ is the size of the unit sp... |
Hundred-Digit Challenge - problem 2 - math's idea of solution | A complete, purely mathematical approach would imply a high number of tedious passages and calculations (the solution to this problem is known to require $14$ reflections before reaching $t=10$). However, if I interpret correctly the question, it asks for a simple general idea of solution.
If this is the case, we cou... |
If $z_1$ and $z_2$ are complex numbers, find minimum value of $|z_1-z_2|$ | Following Fabian's hint, show that
$|z_1|=2$ implies $z_1$ is on the circle of radius $2$ centered at the origin, and
if you rewrite $(1-i)z_2 + (1+i) \bar{z}-2 = 8\sqrt{2}$ using $z_2=x+iy$, the equation becomes $x+y=4\sqrt{2}$, which is a line in the complex plane with intercepts $4\sqrt{2}$ and $4\sqrt{2}i$.
If ... |
Non unique factorization of integer valued polynomials | This can even be done with one variable:
$$
2\cdot \left(\frac{x(x+1)}{2}\right)=\big(x\big)\cdot\big(x+1\big).
$$
If you prefer to avoid irreducibles that become units in $\mathbb{Q}$:
$$
\left(\frac{x(x+1)}{2}\right)\cdot\left(\frac{(x+2)(x+3)}{2}\right) = \left(\frac{x(x+3)}{2}\right)\cdot\left(\frac{(x+1)(x+2)}{2}\... |
Angle between 2 faces of tetrahedron | Put $C$ at the origin, $B$ on the $x$-axis, and $A$ on the $y$-axis; $D$ is in the $yz$-plane, $\overline{DA}$ is perpendicular to the $xy$-plane, $\overline{DB}$ makes a $30$º degree angle with the $xy$-plane (so that $\angle DBA = 30$º), $\angle CBD = 45$º, and what’s wanted is $\angle DCA$.
$\triangle DCB$ is an is... |
Dimension of the span | Take then null $n\times n$ matrix. It has $n$ columns, but the dimension of the span of the column vectors is $0$. |
Finding a Topology from a Subbase | You seem to have forgotten the 1-fold intersections: The sets of $\cal S$ itself.
With this base, the number $5$ would not be covered, and $\{3,4,5\}$ would not be open.
The base is $\mathcal B=\{ \emptyset, \{1\}, \{3\}, \{4\}, \{2,3\}, \{3,4,5\} \}$. With this you can compute the interior of $B=\{1,5\}$. Just find ... |
Symmetric, commuting matrices in $\mathrm{SL}(3,\mathbb{Z})$ | No. We can take
$$
M_1=1, \quad M_2=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} , \quad M_3=\begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix} ,
$$
with $A\in SL(2,\mathbb Z)$ symmetric, but not a multiple of the identity. |
On if $X \sim N(0,1)$ Then $ \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^X e^{-x^2/2} \, dx \sim U[0,1] $ | There is a somewhat more general proposition regarding this problem.
If $X$ is a random variable with a continuous and strictly increasing distribution function $F$, then $F(X) \sim U(0, 1)$.
Proof: For any real $0 < x < 1$,$$
P(F(X) \leqslant x) = P(X \leqslant F^{-1}(x)) = F(F^{-1}(x)) = x,
$$
therefore $... |
Orthogonality and projection of a vector | I ran the calculation as well...yes your numbers check out OK...
... and the scalar projection you need to do, that is simply the dot product...take the dot product of AC with the unit vector of AB... |
Inscribed angle is always the same and twice the central angle -- is this absolute? | The measure of an inscribed angle in the hyperbolic plane is always less than half the measure of the central angle. Here is a picture using the Poincaré disk model:
As you can see, the angle $\alpha$ is always less than half of the angle $\beta$. |
Calculating the volume of a surfboard | If you use any modeling software, it probably has a volume function in it. Otherwise, one approach which I used before is to approximate the surface with a triangulation, then used calculated the volume from the tetrahedra formed by each face and the origin (some triangles have to be summed in, while others have to sub... |
Laplace Transform of $f(x)=\frac{\sqrt{x}}{1+x}$ | We want $\displaystyle F(t)=\int^{\infty}_0 \frac{\sqrt{x}}{1+x} e^{-t x} dx$
$F(t) e^{-t}=\int^{\infty}_0 \frac{\sqrt{x}}{1+x} e^{-t(x+1)} dx$
$\frac{d}{dt}(F(t) e^{-t})=-\int^{\infty}_0 \sqrt{x}e^{-t(x+1)} dx$
$e^t\frac{d}{dt}(F(t) e^{-t})=-\int^{\infty}_0 \sqrt{x}e^{-tx} dx=-t^{-1/2-1}\Gamma(1/2+1)=-\sqrt{\pi}/(2t... |
If $f$ is a cut edge in $G-e$, is $e$ a cut edge in $G-f$? | For a 2-edge connected graph, $G$, if $f$ is a cut-edge of $G-e$ that would mean that ($G-e-f$) is disconnected. Removing of edges is commutative, so $G-f-e$ is also disconnected. Finally, $G-f$ would have $e$ as a cut-edge since $G-f-e$ is disconnected.
As for the general statement if $G$ were 1 edge-connected, th... |
Inverse function theorem for manifolds | Assume $\text{dim}\ M=\text{dim}\ N=n.$ In what follows, we use the Einstein convention for all sums.
The point is that if $(U_p, \phi_p)$ is a chart about $p$ in $M$ and $(V_{f(p)}, \psi_{f(p)})$ is a chart about $f(p)$ in $N$ then $(f_*)_p: T_pM \to T_{f(p)}N$ is a linear transformation, so it has a matrix represen... |
A Binomial Coefficient Sum: $\sum_{m = 0}^{n} (-1)^{n-m} \binom{n}{m} \binom{m-1}{l}$ | This is a special case of the identity $$\sum_k \binom{l}{m+k} \binom{s+k}{n} (-1)^k = (-1)^{l+m} \binom{s-m}{n-l},$$ which is identity 5.24 on p. 169 of Concrete Mathematics, 2nd edition. With $l = n$, $m = 0$, $s = -1$, $k = m$, and $n = l$, we see that the OP's sum is $$(-1)^{2n} \binom{-1}{l-n} = \binom{-1}{l-n}.$... |
Linearity in first argument of $\langle X,Y\rangle =X^*MY $ | Generally one requires that an inner product on a complex vector space be linear in one argument and conjugate linear in the other. What you have proven is precisely that this inner product is conjugate linear in the first argument.
Note that linearity in the second argument and the conjugate symmetry together imply c... |
Linear function between normed spaces is continous. | Since $X$ is finite dimensional space, all norms are equivalent in X, so let $(e_i)_i$ be a finite algebriac basis of $X$ and we equipe $X$ by the norme:
$$
\|x:=x_1e_1+\dots x_ne_n\|=\sum_i |x_i|
$$
so
$$
\|Ax\|=\|\sum_i x_i Ae_i\|\leq\sum_i \|x_i Ae_i\| \leq\max_i \|Ae_i\|\sum_i |x_i|=\alpha \|x\|
$$
Where $\alpha=\... |
For what values of $k$ is $p(x) = k(1-r^2)^x$ a valid probability mass function | Summing over $x$, $$1=\sum_{x=0}^\infty P(x) = k\sum_{x=0}^\infty (1-r^2)^x = \frac{k}{1-(1-r^2)}$$
by the geometric series formula. So $k=r^2$. |
Help with limit $\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt$.. | Note first that
$$
\frac{1}{h}\int_x^{x+h}P(t,h)dt-P(x,y)=\frac{1}{h}\int_x^{x+h}(P(t,y)-P(x,y))dt
$$
for all $|h|$ small enough such that $[x,x+h]\times\{y\}$ or $[x+h,x]\times\{y\}$ is contained in $D$.
Fix $\epsilon>0$.
Now by continuity of $t\longmapsto P(t,y)$ at $x$, there exists $\delta>0$ such that $|P(... |
Basis for nullspace - Free variables and basis for $N(A)$ | Columns 1 and 3 have the pivots. So the other two columns (2 and 4) correspond to the free variables.
Then call $x_4=t$. Then the last equation says $$x_3 - x_4 =0 \leftrightarrow x_3 = x_4= t$$
Call the other free variable $x_2=s$.
Then the first equation becomes, after substiting what we know so far:
$$x_1 - s + ... |
Length of chord on ellipse | If your chord makes an angle $\phi$ with the $x$-axis, then its length is
$$
\frac{2ab}{\sqrt{ {b^2}\cos^2{\phi} + {a^2}\sin^2{\phi} }}
$$
See this answer for a bit more detail.
The angle $\theta$ used in apurv's answer is not the angle between the chord and the $x$-axis. That's probably the reason for his question,... |
How can I compute the 'average rank' of an infinite set? | There are several things here.
There's no reasonable way to take an infinite sum here. For one, cardinals do not admit inverses, and since ordinals measure order and cardinals measure cardinality, it is unclear if you mean that the sum itself is an ordinal or a cardinal summation, and if it is an ordinal sum, what is... |
Counterexample to Cauchy product theorem | No, a counterexample satisfying the property you seek cannot exist, which we can see by Cesàro's theorem.
For a series $\sum d_n$, the Cesàro sum is the limit of the average of the partial sums:
$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\sum_{i=1}^nd_i$$
If a sum is classically summable, then it is also Cesàro summa... |
Why does the double integral for the area of a circle of radius$1$ equal $\pi/8 $ instead of $\pi/4$? | It's just not true that this integral is "the representation of a quarter circle of radius 1 in polar coordinates." The domain over which you are integrating is a quarter disk, but you're integrating a function $f(r,\theta) = 1-r^2$ over that domain.
There is a danger in calculus (especially in the transition from si... |
Conceptual question on differentiation in calculus? | We have $y'=\frac{dy}{dx}=y$ where $y$ is a function of $x$. Therefore, by rearranging the equation,
\begin{align}
\frac{dy}{y}=dx\,.
\end{align}
Integrating with respect to $x$ on each side,
\begin{align}
\int\frac{dy}{y}=\int dx\implies \ln(y)=x+C\,,
\end{align}
where $C$ is an integration constant. Exponentiatin... |
How Can you factor $1/2$ out of integral of $\cos2x$? | When you do a $u$-substitution, you have to substitute for every $x$ in the integral, including the $dx$. Starting from $u=2x$, you can differentiate both sides and get $\frac{du}{dx}=2$. We rewrite this as $\frac12 du=dx$.
Thus, when you substitute in the integral, you replace the $2x$ with a simple $u$, and you repl... |
Objects whose morphisms are all injective | One way to define fields is as precisely the commutative rings which have no nontrivial ideals; in other words, as precisely the commutative rings which have no nontrivial quotients. You can play this game in other categories too. For example, the analogous subcategory for groups is simple groups (the trivial group is ... |
Any simple function which behaves like this? | Looks somewhat like a rescaled version of $y = x e^{-x}$. Try $y=cx e^{1-cx}$ with various choices of $c>1$. For example, try entering (5x)e^(1-5x) from 0 to 1 into Wolfram Alpha. |
Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic | If you know the structure theorem for finite Abelian groups, you will
be able to prove that either a finite Abelian group $G$ is cyclic,
or that its exponent is $<|G|$, that is there is $m<|G|$
with $g^m=e$ for for all $g\in G$.
An alternative attack: show that for $d\mid (p-1)$ the group
$G=(\Bbb Z/\Bbb Z)^*$ h... |
Does the congruence $x^2 - 3x - 1 \equiv 0$ (mod 31957) have any solutions? | It's true, you can use the quadratic formula in this context. You obtain:
$x \equiv 2^{-1}(3\pm\sqrt{9+4}) \equiv 15979(3\pm\sqrt{13}) \pmod{31957}$
Note that the multiplicative inverse of $2$ is the number whose product with $2$ is congruent to $1$, modulo $31957$. This is going to work if and only if $13$ is a perf... |
Finding coefficient of $x^8$ | The coefficient of $x^8$ of $\prod_{k=1}^{10}(x-k)$ is given by
$$\sum_{1\leq j<k\leq 10}kj=\frac{1}{2}\left(\left(\sum_{k=1}^{10}k\right)^2-\sum_{k=1}^{10}k^2\right)=
\frac{1}{2}\left(55^2-385\right)=1320.$$
See also Vieta's formulas.
In other words, here the coefficient of $x^8$ is given by the sum of all product... |
Statistics and the addition rule. | $$
\frac 2 3 \times \frac 2 3 = \frac {2\times2}{3\times3} = \frac 4 9.
$$
The probability of $3$ or $5$ on the first trial is $1/3.$
The probability that that happens on the first toss or the second is not $\frac 1 3 + \frac 1 3$ because those two events are not mutually exclusive: they could both happen.
One way t... |
Proving the equivalence of the properties of onto functions | Take $y\in Y$. Our goal is to prove that $f^{-1}(\{y\})$ is non-empty. If it is empty, then, we see that
$$f^{-1}(\{y\}) = \varnothing = f^{-1}(\varnothing)$$
so, $(3)$ would implies that $\{y\} = \varnothing$, a contradiction. Therefore, $f^{-1}(\{y\}) \neq \varnothing$ and hence, there exists some $x\in f^{-1}(\{y\})... |
An inequality involving the AM-GM inequality: $| x + \frac1x | \ge 2 $ (for $x<0$). | You can prove the result at once by writing
$$\left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \ge 2\sqrt{x^2\cdot \frac{1}{x^2}} + 2 = 4,$$
then taking square roots. |
Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix. | You have $A = PJP^{-1}$ where $J$ is in Jordan form. Write $J = D + N$ where $D$ is the diagonal and $N$ is the rest, which is strictly upper triangular and thus nilpotent. Then $A = PDP^{-1} + PNP^{-1}$. The former is clearly diagonalizable, while the latter is nilpotent; just note that $(PNP^{-1})(PNP^{-1}) = PN(P^{-... |
In a ring where $(a-b)^2 = a - b$ for fixed $a,b$, then $(a-b)(a+b) = 1 \iff a^2 - b^2 = 1$. | Idempotents are always a joy to work with. For the converse implication denote $a-b=e$ and notice that $e$ is an idempotent by hypothesis; substituting $a=b+e$ in the relation $a^2=b^2+1_A$ leads (after a few simple calculations and cancellations) to $e+be+eb=1_A$ and hence to
$$eb+be=1_A-e \tag{1}$$
Multiplying rela... |
Rudin Theorem 1:11: understanding why $L \subset S$ | Simply the sentence 'Let $L$ be the set of all lower bounds of $B$.' should be implicitly understood as
Let $L$ be the set of all lower bounds $s\in S$ of $B$. |
Solving a first order linear system matrix | Method 1: Notice that the third equation is decoupled from the other two.
We have $$z' = -z \implies z(t) = c e^{-t}$$
Substituting $z(t)$ into the second equation, we have
$$y' = -y + 2 z = -y + 2 c e^{-t} \implies y(t) = (b + 2ct)e^{-t}$$
Substituting $y(t)$ and $z(t)$ into the first equation, we have
$$x' = -x... |
Launguages in Discrete Mathematical Structures II | Note that in order to get rid of the initial symbol $v_0$, you must at some point apply the production $v_0\to yv_1$, at which point you have the non-terminal symbol $v_1$ in your string. In order to get rid of $v_1$, you must at some point apply $v_1\to z$. Thus, every derivable terminal string contains the symbol $z$... |
Prove $\limsup\limits_{n \rightarrow \infty} b_n \leq \limsup\limits_{n \rightarrow \infty} a_n$, given $b_n = \frac{a_1+ \cdots +a_n}{n}$. | Fix $m$ and pick $n\geqslant m$ large. Then $$\begin{align}
\frac{{{a_1} + {a_2} + \cdots + {a_m} + {a_{m + 1}} + \cdots + {a_n}}}{n} &\leqslant \frac{{{a_1} + {a_2} + \cdots + {a_m} + \left( {n - m} \right)\mathop {\sup }\limits_{k > m} {a_k}}}{n} \cr
&\leqslant \frac{{{a_1} + {a_2} + \cdots +... |
Convergence in a normed space | What you did is fine. This proves that$$\|x-y\|_{\mathbb{X}}\leqslant\varepsilon+\sup_{n\geqslant N}\|x_n-y\|_{\mathbb{X}}.$$But note that $\sup_{n\geqslant N}\|x_n-y\|_{\mathbb{X}}\leqslant\sup_{n\in\mathbb N}\|x_n-y\|_{\mathbb{X}}$. Therefore, you proved that$$\|x-y\|_{\mathbb{X}}\leqslant\varepsilon+\sup_{n\in\mathb... |
Algebraic independence and $\overline{\mathbb{Q}}-$ linear independence | If the set $\{\alpha_1,\dots,\alpha_n\}$ is algebraically independent over $\mathbb{Q}$ then it is also algebraically independent over $\bar{\mathbb{Q}}$.
Otherwise we can assume, without loss of generality, that $\alpha_n$ is algebraic over $\bar{\mathbb{Q}}(\alpha_1,\dots,\alpha_{n-1})$.
Let $f$ be the minimal poly... |
Generating recursive equation for urn question | First, compute $P(W_{n+1}=w)$. Notice that total number of balls in the urn always stay unchanged, $a+b$.
We have
$$
P(W_{n+1}=w)=P(W_n=w-1)\left(1-\frac{w-1}{a+b}\right)+P(W_n=w)\frac{w}{a+b}
$$
(you could either have $w$ balls and draw a white ball, or $w-1$ white balls and draw a black one so $1$ gets added)
mul... |
How to Show that $1+\left(\left\lceil\dfrac{x}{n}\right\rceil -1\right)n\leq x$? | By the property of the ceiling function we have:
$$
\left\lceil\dfrac{x}{n}\right\rceil <\dfrac{x}{n}+1,
$$
which gives me:
$$
1+\left(\left\lceil\dfrac{x}{n}\right\rceil-1\right)n <x+1.
$$
What I missed which seems some kind of obvious is that $\left(1+\left(\left\lceil\dfrac{x}{n}\right\rceil-1\right)n\right)... |
Trouble computing an index | $NC/N$ is a nontrivial subgroup of the infinite cyclic group $G/N$, so $|G:NC| = |\frac{G}{N}:\frac{NC}{N}|$ is finite.
Now $|NC:N_2C| \le |N:N_2| \le 2$ is finite, so $|G:N_2C| = |G:NC| \times |NC:N_2C|$ is finite. |
Find the coordinates of the inflexion points of $A(\beta)=8\pi-16\sin(2\beta)$ in $\mathbb{R}$ | $A''(\beta) = 64\sin(2\beta) = 0$
Let $\theta = 2\beta$, then where is $\sin\theta = 0$?
$\theta = k\pi$, where $k \in \mathbb{Z}.$ Thus $\beta = k\frac{\pi}{2}$ |
Graded modules over $k[t,t^{-1}]$ | A low-tech way:
Let $M$ be a graded $k[t^{\pm1}]$-module, so that in particular $M=\bigoplus_{n\in\mathbb Z}M_i$ as a vector space. For each $n\in\mathbb Z$ the map $M_n\to M_{n+1}$ given by multiplication by $t$ is a linear bijection, with inverse given by multiplication by $t^{-1}$, of course. It follows that for al... |
Proof of Banach-Alaoglu theorem by Douglas | If two linear functionals $f$ and $g$ are equal on the closed unit ball they are equal everywhere: For any $x$ with $\|x\| >1$ we have $f(x)=\|x\|f(\frac x {\|x\|})=\|x\|g(\frac x {\|x\|})=g(x)$ by linearity of $f$ and $g$.
The product topology is the topology of convergence of each coordinate and weak* convergence... |
Prove an equation with summation and binomial coefficients | We write $n$ instead of $\gamma$, focus on the essentials and skip the constant $(-1)^{a-b}2^{-2(a-b)}$.
We obtain for integers $a\geq b>0$
\begin{align*}
\color{blue}{\sum_{n=0}^{a-b}}&\color{blue}{(-1)^n\frac{b+n}{b}\binom{2a}{a-b-n}\binom{2b-1+n}{n}}\tag{1}\\
&=\sum_{n=0}^{a-b}\frac{b+n}{b}\bin... |
In how many orders can perfumes and colognes be sprayed? | Visualize placing the perfumes and colognes in a sequence, and spraying them in that sequence.
Part (i): Out of 7 possible places, we must choose 4 for the perfumes. Thus, the answer is:
$\binom{7}{4} = \boxed{35}$
Part (ii): Since both end places must be taken up by perfumes, we eliminate them from the count. Now,... |
Metrizability of quotient spaces of metric spaces | A very simple way in which the quotient can fail to be metrizable is if there are equivalence classes that are not closed. Take your original space to be $\mathbb{R}$ and let $\sim$ have as its equivalence classes $(0,1)$ and all singletons $\{x\}$ with $x\notin(0,1)$. In the quotient topology, you can not separate $(0... |
Solutions to $\frac{1}{n} = \frac{1}{a} + \frac{1}{b}$ | Okay so, let's solve $a+b\mid ab$. First, set $d=\gcd(a,b)$ and set $a=da'$ and $b=db'$. Also see that $a+b\mid ab-a(a+b)$ so that $a+b\mid a^2$, and similarly $a+b\mid b^2$.
Now, set $k$ and $l$ so that $d(a'+b')k=(a+b)k=a^2=d^2a'$ and $d(a'+b')l=d^2b'$. This means $(a'+b')k=da'$ and $(a'+b')l=db'$. Since $a'$ doesn'... |
Choose a basis of $\mathbb{F}_q/\mathbb{Z}_p$ to do inverse quickly. | Well, I'm sure you know this, and probably it is not what you're looking for, but if you represent $\mathbf F_q$ as a quotient $\mathbf F_p[x]/(f)$ for some irreducible polynomial $f$ of degree $n$, then you can simply take
$$
\varepsilon_k=\varepsilon_k'=x^k
$$
for $k=0,\ldots,n-1$,
and determine $i_0',\ldots,i_{n-1}'... |
Reasoning that $ \sin2x=2 \sin x \cos x$ | $$\color{red}{\sin 2x}=\color{blue}{2\sin x}\cos x$$ |
Computing the residues of a function with a single pole. | Let $$f(z) = \sum_{m=-\infty}^{+\infty} a_m z^m$$ on $\mathbb{C}\backslash \{0\}.$ You want to compute $a_{-1}$. Since your function is even, you get $$\sum_{m=-\infty}^{+\infty} a_m z^m = \sum_{m=-\infty}^{+\infty} (-1)^ma_m z^m$$ and hence $$a_{-1} = (-1)^{-1} a_{-1}.$$ You can conclude easily that $a_{-1}=0.$ |
The alternating Fourier series associated with the fourth Bernoulli polynomial | Let $$f_4(t):=\sum_{n>0} 2\frac{\cos(2\pi n t)}{n^4}$$
and
$$g_4(t):=\sum_{n>0} (-1)^n 2\frac{\cos(2\pi n t)}{n^4}$$
You are right: the expression of $g_4(t)$ on $[0,1]$ is the piecewise polynomial
$$\begin{cases}-\frac{2^4}{4!}\pi^4B_4(t+\tfrac12)&\text{for}&0 \le t \le \tfrac12\\
-\frac{2^4}{4!}\pi^4B_4... |
How does one show $\tan(nz)$ converges uniformly to $-i$ in the upper half plane? | Hmm... I'm not convinced the statement is true. Perhaps you mean
$$\lim_{n\rightarrow\infty} \tan(nz) = i?$$
Note that the problem is ripe for experimentation, so use your favorite numerical tool to check the value of $\tan(nz)$ for some fixed $z$ and some increasing sequence of $n$s, then you'll see why I'm guessing t... |
How to find a limit in implicit function | I will show that
$A
> a
> A-\frac1{N}\int_0^1 p(x)dx
$,
so
$\lim_{N \to \infty} a
=A
$.
First of all,
$\int_0^1 \frac{1}{1-(p(x)/(p(x)+a))^N}dx
> 1$,
so
$A > a$.
Then,
using Bernoulli's inequality,
$\begin{array}\\
(p(x)+a)/p(x)
&=1+(a/p(x))\\
\text{so}\\
((p(x)+a)/p(x))^N
&=(1+(a/p(x)))^N\\
&... |
Proof that Effros Borel space is standard | The citations below refer, as in the OP, to Kechris' book.
This is what Theorem $4.14$ does:
Every separable metrizable space is homeomorphic to a subset of the Hilbert cube.
This gives the desired embedding result. Note that Kechris uses a nonstandard (in my experience) notion of "compactification," in Def... |
uniform boundedness principle for $L^{1}$ | I wouldn't know about the proof in the book, but here's a proof. It could probably be streamlined some - you should see what it looked like a few days ago. Going to change some of the notation; this is going to be enough typing as it is.
Going to assume we're talking about real-valued functions, so that for every $f$ ... |
difference between normed linear space and inner product space | If you have an inner product space $\left(E, \varphi\right)$, it has a natural structure as a normed vector space: $\left(E,x\mapsto \sqrt{\varphi(x,x)}\right)$ but the other way around isn't true. There are norms that do not come from inner products.
And example with $E=\Bbb R^2$
If you take $\varphi:\left(\left(x_1... |
How to write $A D A^T x$ as $\sum_{j=1}^p A_j D_{jj} A_j^T x$? | Check the fact that $ADA^Tx=(AD)A^Tx$. Then,
$y=\sum_{j=1}^p (A_j D_{j,j})A^T_jx$
This is a consequence of matrix multiplication being associative, thus $AD$ is the linear combination of the row vectors of $A$ by the column vectors of $D$.
Hope this hint helps. |
Is it acceptable to say that if $a^b ≡ a^c\;(mod\;p)$, then $b≡c\;(mod\;p)\;?$ | No, it's not. For example,
$$1^2\equiv 1^3\mod 5$$
but obviously $2\not\equiv3\mod 5$.
If you'd tried this out with any number $a\not\equiv0\mod p$, you'd have found that $a^b\equiv a^c\mod p$ pretty much never means $b\equiv c\mod p$.
I cannot stress this enough, but try numerical examples to your theorem before yo... |
Determining if the set is a basis for the vector space | Let V be a vector space, in your case $\mathbb{R}^4$. A linearly independent spanning set for V is called a basis. You have to show that $S$ is linearly independent and that it spans $\mathbb{R}^4$. |
Show that a specific $w$ cannot be the root of an quadratic with integer coefficients. | Note that, by Euclidien division
$$x^3-x-1=Q(x)(ax^2+bx+c)+x \left(\frac{b^2}{a^2}-\frac{c}{a}-1\right)+\frac{b c}{a^2}-1$$
So, if $w$ is a root of both $x^3-x-1=0$ and $ax^2+bx+c=0~$, then we would have
$$
w \left(\frac{b^2}{a^2}-\frac{c}{a}-1\right)+\frac{b c}{a^2}-1=0
$$
So if $a^2+a c-b^2\ne0~$ then
$$
w=\frac{b c-... |
Prove that ray CE intersects $\triangle ABC$ at a point $D$ on $AB$ and that $D$ must lie strictly between $A$ and $B$. | In most situations, the given statement would be accepted as obvious. I'm not sure whether this is what Allison is looking for, but here is a proof based on the late 19th-early 20th century axiomatization of Euclidean geometry, especially the fact (equivalent to Pasch's Axiom) that every line divides the plane into two... |
Do $A$ and $A^{2}$ share eigenvectors if both are real and symmetric? | No, not necessarily. For instance, suppose $A=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$. Then $A^2=I$, so every vector is an eigenvector of $A^2$. But, for instance, $(1,1)$ is not an eigenvector of $A$.
More generally, if there is a number $c\neq0$ such that both $c$ and $-c$ are eigenvalues of $A$ (wi... |
How to find this recurrence relation, | Suppose $a_{n+1}=xa_n+ya_{n-1}$
Then $4=2x+y$ and $9=3x+y$ $\implies$ $x=5,y=-6$
Hence $a_{n+1}=5a_n-6a_{n-1}$, whose general term is $a_n=p2^n+q3^n$
When $a_1=a_2=1$, $2p+3q=1$ and $4p+9q=1$, Hence $p=1 ,q=-{1\over 3}$ |
Examine function differentiability | Your function is not differentiable at $(0,0)$. If it was, the function $x\mapsto0+\cos\left(\sqrt[3]{x^2}\right)$ would be differentiable at $0$. But it is not. |
axioms of real numbers without multiplication | Yes, this has been done by Tarski.
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals |
What is the probability that the 3 remaining cards of the suit are in one player's hand? | When you condition, you get that you have 26 cards left and 3 of them are of the particular suit. There are $26 \choose 13$ ways of assigning these remaining 26 cards among E and W (because once you assign 13 cards to E, the remaining 13 cards automatically go to W. You get that one player has all 3 cards if either E h... |
Complex analysis proof about $|f(z)|$ | Michael M gave a wondeful hint above.
Define the function $g:U\to \mathbb{C}$, where $U$ consists of $z$ where $|z| < \frac{1}{R}$, and $g(z) = f(z^{-1})$, with $g(0) = 0$. Check that this function is holomorphic. Here is a hint for this step. Prove that $g$ on the open set $U\setminus \{0\}$ is holomorphic and th... |
Should I be worried that I am doing well in analysis and not well in algebra? | I believe that I may be of some consolation.
I had a very similar experience to you. I started doing "serious" math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis--baby Rudin. My ego took a hit when I reached college and realized that while I... |
How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers? | My answer would be $\frac{1}{3}\left(\binom{6m+2}{2}-1\right)+1$.
$\binom{6m+2}{2}$ is the number of ways of writing $6m$ as the sum of three non-negative integers.
We count the one case where all the values are the same seperately. That yields one garland.
The other cases, the equations:
$$6m=a+b+c=b+c+a=c+a+b$$ ... |
Every rational function of $f \in k(x_1,x_2,\dots,x_n)$ is transcendental over $k$. | Since I've been a little rough in my comment, here is my trivial answer to make it up.
You can consider $f$ as a rational function in $\bar k(x_1,\dotsc,x_n)$, where $\bar k$ is an algebraic closure, right ? And $f$ is a constant if and only if it is a constant in this new field. But if $f$ is algebraic over $k$, it i... |
Integrals of functions with compact support. | The standard mollifier $f\ast\varphi_{\epsilon}$ of $f$ is such that $f\ast\varphi_{\epsilon}\rightarrow f$ in $L^{1}$, now passing to an a.e. convergence subsequence and you are done. |
The principal form, up to equivalanc, is the only integral form of Discriminant $\mathbf{D}$, which represent one. | Yes, you can.
Since a common factor of $u$ and $v$ would end up as a squared factor of any number of the form $g(u,v),$ we know that $u$ and $v$ are relatively prime. Thus, we may find integers $\alpha$ and $\beta$ with $\beta u - \alpha v = 1$. The coefficient of $x^2$ in the form
$$ f(x,y) = g(ux + \alpha y, vx... |
How to inter change of norm and limit in the Banach algebra? | In every Banach space absolutely convergent series converge and you have
$$\|\sum_{n=0}^\infty x_n\|\le \sum_{n=0}^\infty \|x_n\|.$$ The proof is exactly as in the case of real or complex numbers (the partial sums form a Cauchy sequence and the bound for the norms of the partial sums carries over to the limit). Therefo... |
Every continuous open mapping $\mathbb{R} \to \mathbb{R}$ is monotonic | Hint: Supose that $f$ is not monotonic, then exist a interval $[x,y]$ and a point $t \in (x,y)$, such that $f(t)= \max_{s \in [x,y]}{f(s)}$ or $f(t)= \min_{s \in [x,y]}{f(s)}$. Once you have that, study the set $f((x,y))$. |
Number of Points on the Jacobian of a Hyperelliptic Curve | In fact, you do not have to assume anything about the genus, nor is it relevant that the curve is hyperelliptic, nor does the cardinality of the finite field matter:
Theorem. Let $C$ be a smooth projective curve of genus $g$ over a finite field $\mathbb F_q$ and let
$$ Z(C;t)=\frac{L(t)}{(1-t)(1-qt)} $$
denote the zet... |
Determine the value of $ \frac{1}{\log_m (mn)}+\frac {1}{\log_n (mn)}$ | Let $x = \log_m n$. Then $\log_m mn = \log_m m + \log_m n = 1 + x$ and $\log_n mn = \log_n n + \log_n m = 1 + \frac1x$. Therefore
$$\frac{1}{\log_m mn} + \frac{1}{\log_n mn} = \frac{1}{1 + x} + \frac{1}{1 + \frac1x} = \frac{1}{1 + x} + \frac{x}{x + 1} = 1.$$ |
Need help with simple system of differential equations | OK, I'll pitch two solution methods at y'all, one based on linear algebra and one, surprisingly enough, somewhat akin to our OP newuser's exploratory attempt centered around the derived equation
$\dfrac{\dot x_1}{\dot x_2} = -\dfrac{x_2}{x_1}. \tag{1}$
Note that I prefer the use of the "$\dot y$" notation over the "$... |
The union of two closure | Your second part is ok, but the first is incomplete. You actually demonstrated that $Cl(A_1\cup A_2)\subset Cl(A_1)\cup Cl(A_2)$, but this doesn't mean they are equal. You should show also that $Cl(A_1)\cup Cl(A_2)\subset Cl(A_1\cup A_2)$. |
Dirichlet's Test Remark in Apostol | Note that the partial sums of $\{a_n\}$ are bounded means that $\lvert A_k \rvert \leq M$ for all $k$ and some $M > 0$. Hence, we have that
\begin{align}
\left \lvert \sum_{k \leq n} A_k(b_k - b_{k+1}) \right \rvert & \leq \sum_{k \leq n} \left(\left \lvert A_k(b_k - b_{k+1}) \right \rvert \right) & (\becaus... |
Linear algebra elementary row operation | If you interchange the rows and then perform the operation you mentioned - the matrix will be in upper triangular form. I guess that's what the book is referring to. |
Is the source (and/or target) a group, or just its underlying set? | $G$ and $H$ certainly can be groups, but the function will not necessarily preserve any structure, so it will just be a map between sets.
If $f$ is a homomorphism, this preserves structure by definition, so it will be a map between groups by necessity. |
How do I evaluate this limit :$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$.? | I see you're a high school teacher so you're familiar with the following concepts :
$\bullet$ $\sin(\frac{1}{x}) \simeq \frac{1}{x} - \frac{1}{6x^3} \text{ } [\text{as x $\rightarrow$ $\infty$}]$
$\bullet $ $ \lim_{x \to \infty} (1-\frac{k}{x})^x = e^{-k} $
Compile these facts to get :
$$\underset{x \to... |
Geometrical Interpretation of the Cauchy-Goursat Theorem? | It sounds like you want a kind of "visual" proof, or at least intuition. The go-to source for that is Needham's Visual Complex Analysis. Check out page 435 (of the pdf) of the linked book, which offers a few different explanations. Personally, I find the geometric intuition to be the following: if you can shrink the co... |
Continuous real-valued functions on the first uncountable ordinal | Yes, every continuous $f \colon X \to \mathbb{R}$ is eventually constant.
There is a sequence $(\alpha_n)$ in $X$ such that for all $n$
$$\sup_{\beta > \alpha_n} \lvert f(\beta) - f(\alpha_n)\rvert \leqslant 2^{-n}.$$
For otherwise, there would be a $k\in \mathbb{N}$ such that for every $\alpha \in X$ there is a ... |
every projective module has a free complement. | If $P$ is projective and $Q$ is any complement of $P$ in a free module,then $P\oplus Q\oplus P\oplus Q\oplus P\oplus Q\oplus\cdots$ is a free complement.
In your example, no finitely generated free module is a complement to $P$: indeed, every f.g. free module has $6^n$ elements for some $n$ and $P$ has $2$, so that th... |
Orthogonality and cross product | The statement is trivial if $x$ and $y$ are linearly dependent, because then $x\times y=0$ and so $v=0.(v\times y)$.
Otherwise, $\dim\operatorname{span}(\{x,y\})=2$, and therefore $\dim\operatorname{span}(\{x,y\})^\perp=1$. So, since $v,x\times y\in\operatorname{span}(\{x,y\})^\perp$, and since $x\times y\ne0$, $v$ is ... |
Question on proving quotient space is homeomorphic to circle | Label each point on the circle by the angle $\theta\in [0,2\pi)$. Let $f:S/{\sim}\rightarrow S$ be defined by $f(\theta)=2\theta$. You should be able to prove this is well-defined on $S/{\sim}$, and is in fact a bijection between the two spaces. Then to prove it's a homeomorphism, you just need to show that the image/p... |
Rate of change in length of hypotenuse | Check your question. Are you sure it's a rate of change you're asked to find for $x$ rather than simply a change for $x$?
Assuming what they're asking for is the estimate of a small change of $x$ for a small change of $\theta$ of $-0.05$ radian, you would work it out like so:
The initial conditions help you verify th... |
if $ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$ then $\sum_{n=1}^\infty a_n = L$ | The statement in the title is false. But it's true under the additional assumption that $a_n\ge0$. And it's actually quite easy.
Since $a_n\ge0$ there exists $S\in[0,\infty]$ such that $$\sum_{n=0}^\infty a_n=S.$$(Note we allowed the possibility $S=\infty$.) Now for a given $N$,
$$\sum_{n=0}^Na_n=\lim_{x\to1}\sum_{n=0... |
$\Bbb Z[\sqrt{-5}]$ is not a PID | Hint: use the fact that PID are unique factorization domains.
Let $N$ be the norm, $N(1+\sqrt{-5})=6$, suppose that $1 + \sqrt{-5}=ab, N(ab)=N(a)N(b)=6$, set $a=u+v\sqrt{-5}, N(a)=1$ implies $u^2+5v^2=1$ this implies $v=0, u^2=1$, $N(a)=2$ implies also $v=0, u^2=2$ impossible, you cannot have $N(a)=3$ with a similar a... |
Integral calculus, find actual volume of cone | The diameters of the frustrums (frustra) are decreasing linearly, hence the volumes quadratically.
$$v_n=\frac Vn\left(\frac{n-k}n\right)^2,$$ where $\dfrac{V}{n}$ denotes the volume of the corresponding cylindrical slices.
Then the total volume
$$V'=\frac Vn\sum_{k=0}^{n-1}\left(\frac {n-k}n\right)^2=\frac Vn\sum_{... |
Weak topology and subspaces | I don’t have at hand the definition of the limit map between direct the limits, but I guess the question can have a negative answer. Let $X=\Bbb R^\omega$ be a subspace of a Tychonoff product $\Bbb R^\omega$ consisting of all sequences $x=(x_i)$ such that all but finitely many $x_n$ are zeroes. Then the space $X$ is a ... |
End of preview. Expand in Data Studio
Dataset Card Creation Guide
Dataset Summary
We automatically extracted question and answer (Q&A) pairs from Stack Exchange network. Stack Exchange gather many Q&A communities across 50 online plateform, including the well known Stack Overflow and other technical sites. 100 millon developpers consult Stack Exchange every month. The dataset is a parallel corpus with each question mapped to the top rated answer. The dataset is split given communities which cover a variety of domains from 3d printing, economics, raspberry pi or emacs. An exhaustive list of all communities is available here.
Languages
Stack Exchange mainly consist of english language (en).
Dataset Structure
Data Instances
Each data samples is presented as follow:
{'title_body': 'How to determine if 3 points on a 3-D graph are collinear? Let the points $A, B$ and $C$ be $(x_1, y_1, z_1), (x_2, y_2, z_2)$ and $(x_3, y_3, z_3)$ respectively. How do I prove that the 3 points are collinear? What is the formula?',
'upvoted_answer': 'From $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$ we can get their position vectors.\n\n$\\vec{AB}=(x_2-x_1,y_2-y_1,z_2-z_1)$ and $\\vec{AC}=(x_3-x_1,y_3-y_1,z_3-z_1)$.\n\nThen $||\\vec{AB}\\times\\vec{AC}||=0\\implies A,B,C$ collinear.',
'downvoted_answer': 'If the distance between |AB|+|BC|=|AC| then A,B,C are collinear.'}
This particular exampe corresponds to the following page
Data Fields
The fields present in the dataset contain the following informations:
title_body: This is the concatenation of the title and body from the questionupvoted_answer: This is the body from the most upvoted answerdownvoted_answer: This is the body from most downvoted answertitle: This is the title from the question
Data Splits
We provide three splits for this dataset, which only differs by the structure of the fieds which are retrieved:
titlebody_upvoted_downvoted_answer: Includes title and body from the question as well as most upvoted and downvoted answer.title_answer: Includes title from the question as well as most upvoted answer.titlebody_answer: Includes title and body from the question as well as most upvoted answer.
| Number of pairs | |
|---|---|
titlebody_upvoted_downvoted_answer |
17,083 |
title_answer |
1,100,953 |
titlebody_answer |
1,100,953 |
Dataset Creation
Curation Rationale
We primary designed this dataset for sentence embeddings training. Indeed sentence embeddings may be trained using a contrastive learning setup for which the model is trained to associate each sentence with its corresponding pair out of multiple proposition. Such models require many examples to be efficient and thus the dataset creation may be tedious. Community networks such as Stack Exchange allow us to build many examples semi-automatically.
Source Data
The source data are dumps from Stack Exchange
Initial Data Collection and Normalization
We collected the data from the math community.
We filtered out questions which title or body length is bellow 20 characters and questions for which body length is above 4096 characters. When extracting most upvoted answer, we filtered to pairs for which their is at least 100 votes gap between most upvoted and downvoted answers.
Who are the source language producers?
Questions and answers are written by the community developpers of Stack Exchange.
Additional Information
Licensing Information
Please see the license information at: https://archive.org/details/stackexchange
Citation Information
@misc{StackExchangeDataset,
author = {Flax Sentence Embeddings Team},
title = {Stack Exchange question pairs},
year = {2021},
howpublished = {https://huggingface.co/datasets/flax-sentence-embeddings/},
}
Contributions
Thanks to the Flax Sentence Embeddings team for adding this dataset.
- Downloads last month
- 109
Homepage:
stackexchange
Repository:
flax-sentence-embeddings
Number of rows:
2,218,989
Total file size:
934 MB