All Questions

9
votes
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
2answers
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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 ...

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