# All Questions

**9**

votes

**1**answer

102 views

### Representations of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimensions are $p^k$

I should preface this by saying that I am not a representation theorist, so I apologize if this can easily be found in standard sources (but sadly I cannot seem to extract it from any of the books I ...

**-2**

votes

**0**answers

32 views

### Odds of 7 straight spins on a roulette wheel falling within the same group of 12 numbers [on hold]

What are the odds of 7 straight spins on a roulette wheel rendering a number within the same group of 12 numbers? (i.e. 7 numbers within 1-12, 7 within 13-24 or 7 within 25-36). This would be a ...

**2**

votes

**2**answers

62 views

### Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?

**0**

votes

**1**answer

97 views

### A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...

**-3**

votes

**0**answers

49 views

### Eigenfunction and fourier transform [on hold]

Assume that $f_1$ is the first eigenfunction of the Dirichlet Laplacian $-\Delta f = \lambda f$, $U=\{z\in \mathbf{C}:|z|<1\}$ and $f|_{|z|=1}=0$. Assume also that $\lambda_1 $ is the first ...

**0**

votes

**0**answers

19 views

### topologically finitely generated residually finite group

Suppose that $G$ is a topologically finitely generated profinite group and $H$ is a subgroup of countably infinite index. Can I say that $H$ must be topologically finitely generated with the subspace ...

**3**

votes

**2**answers

215 views

### How can one determine if a singularity is simple?

Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin.
If we have the following types of ...

**0**

votes

**0**answers

23 views

### Does a singularly perturbed cadlag process has sample paths in a Polish space?

In the theory of stochastic processes it is often said in the broader literature that Polish state spaces are the only important ones appearing in practice. Are there also examples of stochastic ...

**1**

vote

**0**answers

100 views

### Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...

**1**

vote

**1**answer

256 views

### Kodaira dimension of co-adjoint orbit

Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...

**-2**

votes

**0**answers

38 views

### Why is the constraint “ Rank (W) = 1” nonconvex? [on hold]

The SDR (semidefinite relaxation) is introduced to handle the SDP (semidefinite programming) problem with Rank (W) = 1, where W is a positive semidefinite matrix.
I wonder why is the constraint " ...

**8**

votes

**0**answers

144 views

### Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$:
$$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$
thus, for instance, ...

**6**

votes

**2**answers

285 views

### Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable ...

**1**

vote

**1**answer

156 views

### Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...

**1**

vote

**1**answer

42 views

### countably-infinite-index subgroup of a finitely generated profinite group

Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with ...

**0**

votes

**0**answers

46 views

### Integrating new vectors of GL(n,F)

I'd be interested in the following: Let $\pi$ be an irreducible admissible generic representation of $GL(2n,F)$, $F$ a p-adic field. Assume that $\pi$ is ramified and let $W$ be a (non-trivial) new ...

**0**

votes

**0**answers

125 views

### Problem regarding orthogonal vectors

Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...

**1**

vote

**1**answer

65 views

### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

**2**

votes

**2**answers

182 views

### Find all faces in a graph from list of edges

I have the information from a undirected graph stored in a 2D array. The array stores all of the edges between nodes, e.g. graph[3] might be equal to [1,8,30] and represents the fact that node 3 ...

**1**

vote

**0**answers

55 views

### Differential equation related to the Schwarzschild metric

How can one find solutions of the following second-order diﬀerential equation
$$\frac{d^2W}{dr^2}-\frac{1}{r}\frac{dW}{dr}=\frac{C}{W^2}\frac{dW}{dr}$$
with the boundary condition $W(r)\to r^2$ at ...

**0**

votes

**0**answers

63 views

### A question on banach or $C^{*}$ algebra [on hold]

Let $A$ be a $C^{*}$ algebra ($B$ be a semi simple commutative Banach algebra) and $T$ be a bounded derivation on it. Assume that an element $x$ of $A$ ($B$) satisfies the following property:
...

**-3**

votes

**0**answers

52 views

### Math Model help [on hold]

At time t, the distribution for a dynamical model is: a1(t),a2(t),a3(t),…,an(t) as the system evolves it may be expected that if the number of samples in a species is less than the critical number ...

**-1**

votes

**0**answers

20 views

### Inequality of gamma distribution [on hold]

Let $X_{\alpha} \sim \text{Gammma}(\alpha,1),\alpha >0$ with distribution function $G_{\alpha}$ and $X_{\beta} <_{(c)} X_{\alpha}\,\forall ~0<\alpha<\beta<\infty$.
Then show ...

**0**

votes

**0**answers

29 views

### Real centrally-symmetric plane algebraic curves

I am looking for a reference regarding the topology of real centrally-symmetric plane algebraic curves. By this I mean the curves defined by
$$
P(x,y)=0,
$$
where $P$ is a degree $m$ polynomial, ...

**-3**

votes

**0**answers

50 views

### Rank- nullity theorem of linear transformations. Help! [on hold]

I have tried my best to prove the following statement but still fail to do so. Hope that you could help me with this.
Let U, V and W be the vector spaces (over real numbers) and let S: U --> V and
...

**-3**

votes

**0**answers

47 views

### Asymptotic solutions to transcendental equations? [on hold]

x exp(x) = t^(-1) as t goes to infinity
e^(-x) = x^(t) as t goes to infinity
x^2 - ln (x) = t as t goes to infinity
How to find asymtotic solutions to these problems?
The answers could be based on ...

**-1**

votes

**0**answers

26 views

### RANSAC Multivariate Regression [on hold]

I am using RANSAC as my robust regression method. I saw many examples for a line and a plane but what if there are many independent variables as in multivariate regression. Is there anyway handle ...

**2**

votes

**0**answers

60 views

### Is a parametric subvariety singular?

Suppose I have a parametric subvariety $V^k$ of $A^{2n}$ (so, given by $2n$ polynomials in in $k$ variables). The question is: how does one tell whether $V^n$ intersects itself? In the $k=1$ case, ...

**22**

votes

**2**answers

738 views

### What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...

**0**

votes

**1**answer

95 views

### On the inverse problem of Dobrushin

Dobrushin, in this paper, looked into the following problem. Suppose We are given a Markov kernel (conditional distribution) $P_{Y|X}$. Information theorist usually call $W$ a channel. It is known ...

**8**

votes

**1**answer

225 views

### Existence of certain “nondegenerate” function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...

**3**

votes

**2**answers

140 views

### Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ >> log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds ...

**2**

votes

**1**answer

123 views

### Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...

**0**

votes

**1**answer

138 views

### Coherent sheaves on Proj

Roughly speaking , the question is : when a f.g. graded module induces a trivial coherent sheave on Proj ? More precisely, let S be a (complex) graded polynomial algebra, where the variables have ...

**25**

votes

**1**answer

696 views

### Why is there no Brauer scheme?

Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed).
Then the ...

**5**

votes

**0**answers

128 views

### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...

**8**

votes

**3**answers

267 views

### Unknotting number of knot diagrams

Define the "diagram unknotting number" of a knot diagram $D$ as the minimal number of crossings that need to be changed in $D$ in order to get a diagram of the trivial knot (the usual unknotting ...

**6**

votes

**2**answers

308 views

### Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard.
$S$ is a smooth complex projective surface with a fibration $f$ over ...

**5**

votes

**2**answers

155 views

### Can this way of comparing numbers of the form a+b sqrt(K) be generalized?

So I want to make a system for computing with various classes of numbers. One of those is a class of number closed under the standard arithmetic operators ($+$, $-$, $*$ and $/$) along with square ...

**2**

votes

**0**answers

91 views

### Hodge numbers of l-adic sheaves?

Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. ...

**1**

vote

**0**answers

71 views

### Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?

Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...

**4**

votes

**1**answer

178 views

### If $S\subset\mathbb R$ is a $G_\delta$ there is a function $\mathbb R\to\mathbb R$ continuous exactly on $S$. Reference?

Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function ...

**0**

votes

**1**answer

45 views

### Slope decomposition of a product of operators

I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product.
First I'll give some background, for ...

**8**

votes

**1**answer

127 views

### Freely adding degeneracies does not change the homotopy type

Background: Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already ...

**1**

vote

**0**answers

135 views

### Dropping the closed requirement from the symplectic manifold definition?

A symplectic manifold is a pair $(M,\omega)$, where $\omega$ is a non-degenerate closed two-form. When $M$ is compact, Hodge decomposition implies that such manifolds have non-zero second ...

**1**

vote

**1**answer

60 views

### Maximal minimum of Bessel functions

This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...

**0**

votes

**0**answers

18 views

### Distribution of average and median of a random variable [on hold]

sorry if this is a trivial question, but I am a practical engineer, which now needs to have some statistical problems solved, but I just can’t extract the answer from my memories of my long ago taken ...

**1**

vote

**0**answers

123 views

### Campbell-Baker-Hausdorff formula for $\log(\exp(X+Y)\exp(X-Y))$ [migrated]

Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\log(\exp X\exp Y)$ (wiki), is there any convenient way to collect the terms of $\log(\exp(X+Y)\exp(X-Y))$ that involve ...

**-2**

votes

**0**answers

71 views

### A basic doubt on the quantity $\ln E[e^X]$ [on hold]

I heard that the quantity $\ln E[e^X]$ expresses variance of $X$ other than $E[X]$. But, I can't prove it formally ? any help will be appreciated.
i.e. I want to see how $\ln E[e^X] \geq E[X]$ (other ...

**6**

votes

**1**answer

228 views

### Cusps forms for $\Gamma (N)$

I know how to build a basis of the vector space of cusp forms for the congruence subgroups $\Gamma_1 (N)$ and $\Gamma_0 (N)$, but I couldn't find in the literature how to build a basis for ...