# All Questions

**4**

votes

**0**answers

99 views

### Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...

**5**

votes

**3**answers

86 views

### Minimize distance between centroids of subsets of points

In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets.
I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets.
...

**2**

votes

**0**answers

71 views

### representations of dihedral group/quaternion group of order 8

Is there a classification of such representations via unitriangular matrices over characteristic two fields?

**0**

votes

**0**answers

54 views

### Question regarding Grid Graph [on hold]

Let $T_{n^{2}}$ denote the grid $T_{n^{2}}=\{(j,k): 1\leq j \leq n,1 \leq k \leq n\}$.
Given a grid graph $G=(V,E)$ with vertices $V \subseteq T_{n^{2}}$ and ...

**0**

votes

**0**answers

50 views

### Solve quadratic with matrices? [on hold]

The question, pure curiosity, is whether you can solve a quadratic with the use of matrices?
And if yes, does that method also work for higher polynomials?
Say for example I have a quadratic such ...

**0**

votes

**0**answers

23 views

### curvature function of lenght [on hold]

I am doing research on a new method of representation of curves. The curves are expressed in the form k=f(s) where "k" is the curvature (or k=1/R where R is the radius of the osculating circle), "s" ...

**2**

votes

**4**answers

134 views

### Simple bound for generalized geometric series

Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.
Is there a simple way of proving this ...

**-1**

votes

**1**answer

81 views

### What is the definition of maximal ε-separated set

Nowadays, I am just studying the book wrote by Joram Lindenstrauss and Yoav Benyamini,i.e. Geometric Nonlinear Functional Analysis. The putfroward "maximal ε-separated set".I really can not understand ...

**1**

vote

**0**answers

112 views

### Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as
...

**0**

votes

**1**answer

44 views

### perfect Lie algebra with a nonabelian solvable radical

Suppose you want to construct a perfect Lie algebra with a nonabelian solvable radical $\mathfrak{r}$, say with a commutator series of length 2. What are the conditions that guarantee the Lie algebra ...

**-3**

votes

**0**answers

27 views

### Poisson Distribution [on hold]

Connections arrive at a switch at a rate of 12 per ms. The number of arrivals is Poisson distributed:
What is the probability that the number of calls arriving in 2ms is greater that 7 and less ...

**6**

votes

**0**answers

186 views

### Global sections for a locally free sheaf over curves

Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg ...

**0**

votes

**0**answers

28 views

### Explaining of lost probalbity over random loss channel [on hold]

I am reading a paper about packet loss probability over random loss channel. In this paper, the author give a equation about loss probability as $(1)$. However, I cannot understand the meaning of it. ...

**0**

votes

**0**answers

54 views

### Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$ [on hold]

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{p}(T)$ (point spectrum of $T$), and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a ...

**5**

votes

**1**answer

471 views

### Is the set of certain polynomials finite or infinite?

Let us consider the set of all polynomials with the following properties:
i) all coefficients are integer;
ii) the leading coefficient equals one;
iii) all zeros are real and simple and belonging ...

**-1**

votes

**0**answers

33 views

### about the boundary of convex sets with not $C^1$ regularity [on hold]

I am reading a paper, and the author uses the following property:
Let $\Omega \subset R^n$ a open, bounded and convex domain. Let $x_0 \in \partial \Omega$ and suppose that $\partial \Omega$ is not ...

**0**

votes

**1**answer

34 views

### Determinant of block covariance matrix [on hold]

I wonder how to express the determinant of a block covariance matrix. For example, I have a covariance matrix
$\Sigma=\left[
\begin{array}{cc}
\Sigma_1 & \Sigma_{12} \\
\Sigma_{21} ...

**3**

votes

**2**answers

97 views

### Number of monomials of deg D where each variables has low degree

Let $D,n,d$ be three positive integers.
I am looking for the number of monomials of degree $D$ in $n$ variables where each variable appears with exponent at most $d$.
As a result of an application ...

**3**

votes

**2**answers

166 views

### $ \text{Lan}_KN(-/\mathcal C)\cong N(-/K) $

Let $N(-/\mathcal C)\colon \mathcal C\to \mathbf{sSet}$ be the functor sending $c\in\mathcal C$ to the nerve of the coslice category $c/\mathcal C$.
Given a functor $K\colon\mathcal{C}\to ...

**3**

votes

**1**answer

66 views

### Geodesic comparison in Hadamard space

Let $X$ be a hadamard space and $\gamma_1, \gamma_2 \colon \mathbb{R}\rightarrow X$ be two geodesics. Part 2 of Coroallary 2.5 in
...

**3**

votes

**1**answer

119 views

### Polygamous stable marriage/ assignment problem

I'm not sure under which 'algorithm' it falls under, but here is the problem:
I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...

**6**

votes

**1**answer

140 views

### Which ordinals can be proof-theoretic ordinals of a reasonable theory?

When talking to a friend recently he asked a question - are there any reasonable first-order theories which have proof theoretic ordinal equal to small or large Veblen ordinal? I have then extended ...

**0**

votes

**0**answers

19 views

### Adding an edge to a MST generated from a distance matrix [on hold]

Given an N×N distance matrix, but not an adjacency matrix for a connected, weighted, undirected graph G, I've managed to find a minimum spanning tree (with N−1 edges) using Prim's algorithm. Now I ...

**2**

votes

**0**answers

79 views

### Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...

**11**

votes

**1**answer

565 views

### Proving the Irrationality of this Number

I found this problem on Math.SE:
Prove that $\log_35+\log_25$ is irrational.
http://math.stackexchange.com/q/986227/173397.
I labored on it for a few days, and couldn't find an algebraic ...

**5**

votes

**2**answers

535 views

### Decidability of decidability

The questions I'm going to ask are non formal because they concern decidability of decidability, and I couldn't find any references on that after some quick searches. I hope that this thread is still ...

**3**

votes

**0**answers

40 views

### Boundedness Spectral Triple Axioms for de Rham Complex

In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...

**3**

votes

**0**answers

74 views

### Is exponential function in a C*-algebra injective on self-adjoint elements?

I asked this question in stackexchange, but it flashed and disappeared:
Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true ...

**2**

votes

**0**answers

91 views

### generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation

The Atiyah-Hirzebruch spectral sequence
\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}
computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...

**5**

votes

**1**answer

154 views

### Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows
$$
B_6=\begin{bmatrix} & & & & & 6\\
& & & ...

**8**

votes

**1**answer

196 views

### Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(||x||^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.
...

**4**

votes

**0**answers

135 views

### Proving that the kernel of this matrix is of dimension 2

Using a computational software program, I found that the kernel of the following matrix is of dimension 2 but I haven't managed to prove it:
\begin{equation}
\text{for almost all } t_1>0,\quad ...

**9**

votes

**3**answers

222 views

### Locus of complete curves on $\mathcal M_g$

Is the union of the complete curves on $\mathcal M_g$ Zariski dense? ($g \gg 0$)
I know it is hard to find higher-dimensional complete subvarieties of $\mathcal M_g$, but a quasiprojective variety ...

**-4**

votes

**0**answers

29 views

### Probability of coin tosses [on hold]

I have a probability problem.
Given 14 coins what is the likelyhood that on one given toss of all 14 at once 7 land heads and 7 land tails?

**-4**

votes

**0**answers

27 views

### Limits sum explanation required [on hold]

$\lim _{x\to 0}\left(\left(\left(x+bx^2\right)^{0.5}\:-\:x^{0.5}\right)/\left(bx^{1.5}\right)\:\right)$
How do I solve this, i tried online solvers but the ended up giving me the answer 1/2 but not ...

**4**

votes

**0**answers

130 views

### How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_{2},\cdots,k_{d}}^2$ [duplicate]

Assmue that $d$ is give postive integer numbers,and
$$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots ...

**-2**

votes

**1**answer

128 views

### The number of Sylow subgroups of a group [on hold]

Let $ G $ be a finite group of order $ 2^4\times 3\times 7\times 13$. If $13 $-Sylow subgroup of $ G $ is not normal then $ G $ has 14 Sylow $13$-subgroups. Then $ G$ is $2 $-transitive on the set ...

**2**

votes

**0**answers

69 views

### Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...

**4**

votes

**2**answers

159 views

### Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...

**1**

vote

**1**answer

48 views

### Expected value of the inverse of a random, truncated Haar matrix

Let $Q$ be a (say 4x4) unitary matrix, distributed according to the Haar distribution. Denote the upper left 2x2 submatrix of $Q$ as $Q_{1:2,1:2}$. I am interested in the following expectation:
$E(I ...

**2**

votes

**0**answers

104 views

### Uniqueness of scalar curvature

I'm reading Gromov's notes
http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf
and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian ...

**-1**

votes

**0**answers

17 views

### Probability of rolling the same number n or more times in m rolls of a k-sided dice [migrated]

So the only approach I can find to solve this problem is making computer simulations, anyone can explain a mathematical way to solve it? or recommend a book that can explain this topic.
thanks.

**-1**

votes

**1**answer

40 views

### How to construct a semi-positive definite matrix in this form: (L=D-A')

As known, the graph Laplacian $L = D - A$ is semi-positive definite.
What if there is a matrix $A'$ where
$$ A'_{ij} = \begin{cases} A_{ij}, \quad if A_{ij} >0 \\ -\varepsilon, \quad if A_{ij} = ...

**2**

votes

**2**answers

213 views

### System of boolean equations, Satisfiability

Are there any methods to "solve" large systems of boolean equations?
$$x_{i1}\vee x_{i2}\vee x_{i3} = b_i, \quad\text{for}\quad i=1,\dots,N,$$
where $x_i, b_i \in\{0, 1\}$
For example
$$x_{1}\vee ...

**7**

votes

**1**answer

112 views

### Reconstructing a string from random samples

What is known about the following problem?
Reconstruct a string $\sigma$ of known length $n$ over a known
alphabet $\Sigma$ from a collection of uniformly and independently
chosen $k$-long ...

**0**

votes

**0**answers

35 views

### connected Polish groups

We know that a connected locally compact Hausdorff topological group is a pro-Lie group, by the Gleason-Yamabe theorem. Is there a known characterisation of the connected Polish groups?

**0**

votes

**1**answer

101 views

### Complementation in tensor products

This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. Reasonable means that ...

**0**

votes

**1**answer

200 views

### Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...

**0**

votes

**1**answer

114 views

### Divisibility of divisors in some tori and lattices

Let $E$ and $E'$ be two general elliptic curves. We consider the $2$-dimensional torus $A:=\frac{E\times E'}{(u\times u')\left((\mathbb{Z}/2\mathbb{Z})^2\right)}$, where ...

**6**

votes

**1**answer

127 views

### Lie group actions with only one orbit type, but not defining a principal bundle

Searched-for situation: A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the ...