# All Questions

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### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable, locally connected and has finite topological dimension, yet fails to be locally compact?
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### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
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### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von Neumann ...
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Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra $$S(L,G,\gamma) = \bigoplus_{g\in ... 0answers 19 views ### Get angle of Trajectory of a projectile [on hold] Formula1 Since a view hours I'm desperately trying to solve this equation after alpha. I can't use Formula2 because my launch starts at the height h. Thanks for your guys guidance and help. 0answers 80 views ### When is (1^2+1)(2^2+1)\dots (n^2+1) a perfect square? [duplicate] Find all such n. Natural guess is that n=3 is the only solution. It is natural to try something like Bertrand's postulate for Gaussian integers with imaginary part 1, but what is known? 1answer 186 views ### Can we always attain another prime via inserting digits between the digits of a fixed prime? The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ... 0answers 27 views ### Carving a rectilinear polygon In this question, carving a polygon P means removing an axis-parallel rectangle adjacent to the boundary of P. Carving P might break it into two or more polygons. You are given a square P. ... 0answers 17 views ### Difference between Schmidt decomposition and singular value decomposition [migrated] Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let O be an operator acting on the Hilbart space \mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}. ... 0answers 17 views ### Points in Convex Configuration with Trivial Optimal Tour Which property guarantees, that for set of n points of the Euclidean plane, that are convex configuration, the optimal tour visiting all points consists of the n shortest edges of the induced ... 0answers 84 views ### Exponential analogue of formal connections Let F=\mathbb{C}((t)). Let G=GL_n. Then G(F) acts on \mathfrak{g}(F) by gauge transformation:$$ g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F). $$Here, ... 0answers 34 views ### A quantitative version of Pełczyński's property (V^{*}) Let me first fix some notations. Let A be a bounded subset of a Banach space X. Set$$wk_{X}(A)=\widehat{d}(\overline{A}^{w^{*}},X)=\sup\{d(x^{**},X):x^{**}\in \overline{A}^{w^{*}}\},$$where ... 0answers 45 views ### Stochastic calculus in L^1 Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on L^1 space, or some Orlicz space between L^2 and L^1? For examples: are there: Ito ... 1answer 114 views ### Opposite of an E2-algebra Suppose C is the monoidal \infty-category of modules over an \mathcal{E}_2-ring spectrum A. Let C' = C as a category, but with opposite monoidal structure to C. Is C' the category of ... 1answer 53 views ### Area of an irregular, n-sided, non-intersecting (edges) polygon algorithm I need to generate an irregular, n-sided polygon of non-intersecting edges (n= 200, for example) with the smallest area possible. The position of the vertex is random and I've tried designing a couple ... 1answer 47 views ### Computing the inverse of a Cholesky decomposition [on hold] I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ... 0answers 92 views ### Please help me for answers to question. best regards [on hold] Find the definition of a locale and its dual (i.e. a frame) and consider the definition of a Grothendieck topology. Discuss the differences between this concept and an ordinary topology on a set ... 0answers 86 views ### Techniques to solve logarithmic functional equations [on hold] I would like to solve this logarithmic functional equation, but cannot find a standard technique:$$f(f(x)) = log(x)$$1answer 92 views ### Discrete spectrum and almost periodicity According to Vershik, an ergodic invertible measure-preserving transformation T on a Lebesgue space X has discrete spectrum if and only if for every bounded measurable function f\colon X \to ... 2answers 103 views ### Probability of at most K consecutive zeroes in a sequence of 0s and 1s [on hold] I want to prove that in a sequence W of length n, consisting of 1s and 0s, P( in W there is at most \frac{\log_2n}2 consecutive zeroes ) \leq \frac{K}{n}  for some constant K. Can anyone ... 0answers 158 views ### What area of maths have I reinvented? [on hold] I was trying to solve the issue of calculating (total) utilities with infinite numbers of people. This is problematic, because many infinite series are divergent, so there does not appear to be a way ... 1answer 50 views ### Converging to moments obeying Carleman's condition I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that \mu_N are a sequence of measures on \mathbb{R}. Let m_{N,k} be the k-th ... 2answers 262 views ### Stable homotopy groups of RP^{\infty} Are the stable homotopy groups \pi^s_i(\mathbb R P^{\infty}) known for small i? In particular, I would be interested in the values for i = 5,6. A quick Internet search did not lead to anything. 0answers 56 views ### Counting growing tree trajectories I am looking for help: Beginning with a single node (\circ), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is ... 0answers 44 views ### concentric spheres with common radius [on hold] I am trying to think of a way to solve this problem but I am not sure if it is doable or not. Here goes: Assume we have n spheres that share a common radius (x0,y0,z0). For each sphere we have one ... 2answers 227 views ### Circle Action on Quaternionic Projective Space Quoting from Wikipedia article on quaternionic projective space: Therefore the quotient manifold$$ \mathbb{HP}^{2}/\mathrm{U}(1) $$may be taken, writing U(1) for the circle group. It has ... 0answers 21 views ### Probability of an event based on percentage in fixed lapse of time [on hold] I am a software engineer. I am also a former triathlete that rides with a large group of friends every time we have a chance. i am trying to come up with a little software to distribute among us ... 0answers 22 views ### Find the number of connected components in pseudospectra [on hold] Suppose: B_i \in \mathbb{C}^{n \times n}, 0<w_i\in \mathbb{R} (i = 0,1,2,\ldots,m) {\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0 is a matrix polynomial, and x  is a complex ... 0answers 119 views ### The eigenvalue of operator -\Delta Let \Omega\subset \mathbb R^N be open bounded, smooth boundary. We know the eigenfunction of Laplacian operator -\Delta is an orthonormal basis of L^2. Let \{\omega_n\} denote the ... 2answers 157 views ### Splitting subspaces and finite fields Hellow. I'm sure that the following is truth, but I can't prove it. Let R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn}) be a chain of finite fields and A = ... 1answer 90 views ### 3-dimensional vectors satisfying certain equalities Question: Are there 5 distinct vectors u,v,w,x,y \in \mathbb{R}^3, all on the unit sphere (i.e. ||u||=||v||=||w||=||x||=||y||=1), such that: ||u+v+x||=||u+v+y||=||u+w+x||=||u+w+y||=1 ? Also, ... 0answers 10 views ### Best algorithm to compute the first eigenvector of symmetric matrix [migrated] Assume that we have a real symmetric matrix \mathbf{A}\in\mathbb{R}^{n\times n} obtained as following :$$\mathbf{A}=\mathbf{N}-\mathbf{P}, with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
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 ...