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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Can someone explain this integration trick for log-sine integrals?

I was working on this rather challenging log-sine integral: $$\int_{0}^{2\pi}x^{2}\ln^{2}\left(2\sin\left(x \over 2\right)\right)\,{\rm d}x = {13\pi^{5} \over 45}$$ The upper limit is a waiver ...
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### Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
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### A question about the divisibility of sum of 2 consecutive primes.

Well as I was curious about the sum of $2$ consecutive primes, after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least ...
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### Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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### How to find a total order with constrained comparisons

There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task? ...
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### Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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### complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
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### Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime. In the book, ...
Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt$$ We can use this to evaluate integrals. For example, consider $f(x)=... 1answer 692 views ### Hahn-Banach From Systems of Linear Equations In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables$c_i = A_{ij} x_j$: \begin{... 1answer 2k views ###$C^{k}$-manifolds: how and why? First of all, I have a specific question. Suppose$M$is an$m$-dimensional$C^k$-manifold, for$1 \leq k < \infty$. Is the tangent space to a point defined as the space of$C^k$derivations on the ... 0answers 2k views ### Is OEIS A248049 an integer sequence? The OEIS sequence A248049 defined by $$a_n \!=\! (a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})/a_{n-4} \;\text{ with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$ is apparently an integer sequence but I ... 0answers 1k views ### Does the average primeness of natural numbers tend to zero? Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ... 3answers 605 views ### Can$ f\colon \mathbb{R}^k \to \mathbb{R}^n$such that$ \forall y \in \operatorname{im}(f)$,$f^{-1}(y) = \{a_y,b_y\} $be continuous? This is the problem we want to solve: Can$f\colon \mathbb{R}^k \to \mathbb{R}^n$such that$ \forall y \in \operatorname{im}(f)$,$ f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y $be continuous? ... 1answer 503 views ### Prove that$\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $I'm sure that's a coincidence, but the Laplace transform of$1/\Gamma(x)$at$s=1$turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0.... 0answers 1k views ### What Rubik's Twist configuration has the lowest visible surface area? The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ... 0answers 630 views ### \forall n\in\mathbb N:n^x\in\mathbb Q implies x\in\mathbb Z - elementary proof? Consider the following two problems: Show that if for some x\in\mathbb R and for each n\in\mathbb N we have n^x\in\mathbb N, then x\in\mathbb N. Show that if for some x\in\mathbb R ... 0answers 1k views ### Why are functions with vanishing normal derivative dense in smooth functions? Question Let M be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in C^\infty(M) in the H^1 norm? Here I define ... 0answers 598 views ### Is there a proof of Bézout's theorem via residue theory? Let's define intersection numbers as follows. Consider a collection f_1,\dots, f_n of holomorphic functions on some neighborhood of zero in \mathbb C^N cutting out divisors D_1, all of which ... 0answers 939 views ### Why do universal \delta-functors annihilate injectives? Let \mathcal{A} and \mathcal{B} be abelian categories. Suppose \mathcal{A} has enough injectives, and consider a universal (cohomological) \delta-functor T^\bullet from \mathcal{A} to \... 2answers 5k views ### Showing that \sqrt{9+9\sqrt{9+9\sqrt{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1?$$\sqrt{9+9\sqrt{9+9\sqrt{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$In the second nested radical, the repeating pattern is (-,-,+). I approached this ... 0answers 679 views ### Can some proof that \det(A) \ne 0 be checked faster than matrix multiplication? We can compute a determinant of an n \times n matrix in O(n^3) operations in several ways, for example by LU decomposition. It's also known (see, e.g., Wikipedia) that if we can multiply two n \... 2answers 501 views ### If I stretch a convex polygon, does the original fit into the streched version? Suppose you have a convex polygon P=\mathrm{conv}(\{(x_1,y_1),\dots, (x_k,y_k)\}) and you stretch it in one dimension, that is, we choose \alpha>1 and get a new polygon P^\alpha=\mathrm{conv}(\... 1answer 1k views ### The Modulus of all the roots of a Polynomial are equal to 1 Suppose the real number \lambda \in (0,1), and let n be a positive integer. Prove that all roots of the polynomial$$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{... 0answers 834 views ### Constructing an infinite chain of subfields of 'hyper' algebraic numbers? This has now been cross posted to MO. Let$F$be a subset of$\mathbb{R}$and let$S_F$denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ... 0answers 484 views ### How are topological invariants obtained from TQFTs used in practice? Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ... 0answers 840 views ### Stronger versions of Wilson's Theorem Problem Let$c \in \mathbb{N};\exists$a prime$p$for which: $$p^c \mid (p-1)!+1$$ Does$\existsM\in\mathbb{N};\forallc \geqslant M;\nexistsp\$ ...
Updated question This "reverse" pattern can be plotted as a function of a triangle, read by rows: $$T(n,k) = (\delta)^k F\binom{n}{k} \left\lfloor f(t(k)) \right\rfloor ,\delta\in\{1,-1\}.$$...