# All Questions

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### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
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### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
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### Get angle of Trajectory of a projectile [on hold]

Formula1 Since a view hours I'm desperately trying to solve this equation after alpha. I can't use Formula2 because my launch starts at the height h. Thanks for your guys guidance and help.
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### When is $(1^2+1)(2^2+1)\dots (n^2+1)$ a perfect square? [duplicate]

Find all such $n$. Natural guess is that $n=3$ is the only solution. It is natural to try something like Bertrand's postulate for Gaussian integers with imaginary part 1, but what is known?
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### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ...
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### Carving a rectilinear polygon

In this question, carving a polygon $P$ means removing an axis-parallel rectangle adjacent to the boundary of $P$. Carving $P$ might break it into two or more polygons. You are given a square $P$. ...
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### Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbart space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
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### Points in Convex Configuration with Trivial Optimal Tour

Which property guarantees, that for set of $n$ points of the Euclidean plane, that are convex configuration, the optimal tour visiting all points consists of the $n$ shortest edges of the induced ...
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### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation: $$g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).$$ Here, ...
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### A quantitative version of Pelczynski's property ($V^{*}$)

Let me first fix some notations. Let $A$ be a bounded subset of a Banach space $X$. Set $$wk_{X}(A)=\widehat{d}(\overline{A}^{w^{*}},X)=\sup\{d(x^{**},X):x^{**}\in \overline{A}^{w^{*}}\},$$ where ...
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### finite Projective plane [on hold]

Let Z=PG(q,2) be a finite Projective plane over a finite field Fq, q a prime power. Show that there exists a commutative binary operation * in Z such that (i) x*y is neither x nor y for any x and y, ...
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### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$?
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### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...
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### Area of an irregular, n-sided, non-intersecting (edges) polygon algorithm

I need to generate an irregular, n-sided polygon of non-intersecting edges (n= 200, for example) with the smallest area possible. The position of the vertex is random and I've tried designing a couple ...
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How to get to do paid mathematics reserach in graph theory pure mathematics in private by myself by getting some fund to help me support my family any advise? At present I am doing my post doctoral ...
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### Computing the inverse of a Cholesky decomposition [on hold]

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...
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Find the definition of a locale and its dual (i.e. a frame) and consider the definition of a Grothendieck topology. Discuss the differences between this concept and an ordinary topology on a set ...
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### look for a right technique to solve logarithmic functional equations [on hold]

I would like to solve this equation but can not find a standard technique f(f(x)) = log(x)
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### 3-dimensional vectors satisfying certain equalities

Question: Are there 5 distinct vectors $u,v,w,x,y \in \mathbb{R}^3$, all on the unit sphere (i.e. $||u||=||v||=||w||=||x||=||y||=1$), such that: $||u+v+x||=||u+v+y||=||u+w+x||=||u+w+y||=1$ ? Also, ...
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### Best algorithm to compute the first eigenvector of symmetric matrix [migrated]

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
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### When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is a smooth algebraic variety with free divisor $D$ so that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a ...
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### Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
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### Does this numerical series have any special name? [on hold]

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers. Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...
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Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true $$F(a,b,c;x)=\sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n} x^n=\frac{\Gamma ( c ... 0answers 61 views ### 2nd order partial differential equation with non-constant coefficients During my research I came across the following differential equation:$$f(x,y) = \left(y^2 \partial_{x}^{2} + x^2 \partial_{y}^{2} \right)f(x,y)$$Any ideas how to solve it without using series ... 0answers 72 views ### Sets of matrices which are irreducible but not strongly irreducible A set of d \times d real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and d. A stronger ... 0answers 133 views ### One-dimension Algebraic groups I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let K be an algebraically closed field. ... 0answers 21 views ### 2D convolution property [on hold] If I have three square matrices a,b, and c of equal size. say each of them are 3x3 matrices. then practically it is possible that d = (a.b) * c .....(1) = a * (b.c) .....(2) that is 2D convolution ... 0answers 53 views ### Extension of a valuation on a function field Let K be a field, and K(x) the field of rational functions over K. Consider the degree valuation v on K(x), That is v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f). So for every f(x)\in ... 1answer 185 views ### Up to 2000, A145722(n-1) \equiv \sigma(4n-3) \pmod{5} A145722 is Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions. Using the pari program and offset 0, up to ... 1answer 130 views ### Does every ultrafilter contain sets of sup-measure 0? Let \mathbb{N} be the set of positive integers and for A\subseteq {\mathbb{N}} set$$m(A) = \text{lim sup}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}. Does every ultrafilter ${\cal U}$ on ...
Let $A$, $B$ be well ordered sets. If there exist order-preserving functions $f : A \to B$ and $g : B \to A$ (do not need to preserve initial segments), are $A$ and $B$ isomorphic? I know this is not ...