query: Explicit components of $y^x=x^y$
positive_passages.docid: 18761
positive_passages.text: Given that $x^y=y^x$, what could $x$ and $y$ be?
negative_passages.docid: 148570
negative_passages.text: Find the all possible real solutions of $x^y=y^x$
negative_passages.docid: 20626
negative_passages.text: Prove that $x^y < y^x$
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 1932
negative_passages.text: How to show this equality of $\rho(X,Y)$
negative_passages.docid: 2205
negative_passages.text: Solving $x^{x^x}=c$ with Lambert's W function
negative_passages.docid: 140988
negative_passages.text: Prove $\exp(x)\exp(y)=\exp(y)\exp(x)$ if $[x,y]=0$.
negative_passages.docid: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
negative_passages.docid: 22004
negative_passages.text: Inverse of $y = x^3 + x $?
negative_passages.docid: 93952
negative_passages.text: Show that $X$ can be represented as a union of disjoint equivalence classes
negative_passages.docid: 139288
negative_passages.text: Formal proof that $e^x$ is not algebraic
negative_passages.docid: 17265
negative_passages.text: Intuition for why equations of the form $k^x=x^c$ are not solvable trivially?
negative_passages.docid: 45555
negative_passages.text: $(x+y)^c\le x^c+y^c$ for $0<c\le1$
negative_passages.docid: 58622
negative_passages.text: Intersection points of $2^x$ and $x^2$
negative_passages.docid: 115799
negative_passages.text: Proving $x^{\log n} = n^{\log x}$
negative_passages.docid: 153355
negative_passages.text: Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?
negative_passages.docid: 47313
negative_passages.text: What is $i^i$ and $(i^i)^i$