# All Questions

**0**

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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**

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**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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. ...