0
votes
0answers
7 views

List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
0
votes
0answers
13 views

an example for fundamental group of graph of groups

suppose we have a graph $X$ with the vertex set $\left\lbrace v_1,v_2,v_3 \right\rbrace $ and the edge set $\left\lbrace e_1,e_2,e_3 \right\rbrace $ like a triangle. let $(\Gamma,X)$ be a graph of ...
1
vote
0answers
8 views

Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous?

There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve. To construct one example of such a function ...
1
vote
0answers
16 views

Solvable Lie algebra whose nilradical is not characteristic

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p need not be characteristic, but is there an example of a solvable Lie algebra with ...
0
votes
0answers
11 views

HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...
-2
votes
0answers
23 views

Algorithm to rate board of tic-tac-toe [on hold]

At start I want to say that im programmer and I don't want anyone to write me code, just to help me what I can use. Is there any algorith which I can use to rate a board of tic-tac-toe ? What I want ...
4
votes
1answer
65 views

Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of ...
3
votes
0answers
23 views

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 ...
4
votes
0answers
45 views

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 ...
1
vote
0answers
8 views

Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately) $$ g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + ...
3
votes
1answer
53 views

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) ...
0
votes
0answers
22 views

What would be simple way of calculating the area of a 3D PieChart's slices? [on hold]

I have created a 3D Pie Chart able to be rotated. -> http://plnkr.co/edit/QIYu8sJUWPmxcby1ky9l?p=preview I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted ...
4
votes
0answers
23 views

$V(A)$ 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 ...
4
votes
0answers
31 views

When is a crossed-product algebra a division algebra?

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 ...
-3
votes
0answers
16 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.
2
votes
0answers
71 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?
3
votes
1answer
117 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 ...
2
votes
0answers
14 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 $P$. ...
1
vote
0answers
16 views

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 $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...
0
votes
0answers
10 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 ...
2
votes
0answers
57 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, ...
1
vote
0answers
19 views

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 ...
-1
votes
0answers
23 views

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, ...
1
vote
0answers
21 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 $L^1$?
2
votes
1answer
60 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 ...
2
votes
1answer
46 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 ...
-1
votes
1answer
39 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 ...
-10
votes
0answers
81 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 ...
-2
votes
0answers
70 views

Techniques to solve logarithmic functional equations [on hold]

I would like to solve this logarithmic functional equation, but cannot find a standard technique: $$f(f(x)) = log(x)$$
1
vote
1answer
61 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 ...
1
vote
2answers
93 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 ...
-4
votes
0answers
150 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 ...
3
votes
1answer
41 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 ...
9
votes
2answers
227 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.
1
vote
0answers
50 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 ...
-2
votes
0answers
40 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 ...
7
votes
2answers
216 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 ...
-1
votes
0answers
16 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 ...
-2
votes
0answers
21 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 ...
-1
votes
0answers
110 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 ...
5
votes
2answers
118 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 = ...
2
votes
1answer
74 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, ...
-2
votes
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 ...
3
votes
0answers
62 views

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 ...
1
vote
2answers
102 views

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?
-3
votes
0answers
49 views

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 ...
2
votes
1answer
89 views

First proof of the integral representation of the hypergeometric function $F(a,b,c;\cdot)$

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 ...
2
votes
0answers
67 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 ...
5
votes
0answers
73 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 ...
0
votes
0answers
136 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. ...

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