All Questions

Filter by
Sorted by
Tagged with
0
votes
0answers
8 views

Isogenies of complex multiplication elliptic curves

This is a slight continuation of a previous question of mine. Given an elliptic curve $E$ over $\mathbb{Q}$ which has complex multiplication. How would one find each $p$ such that $E$ admits a $p$-...
1
vote
0answers
13 views

Is anything written about winning the “Dollar Game” in the minimal number of moves?

I run some Master's projects on Chip-Firing games, using the Holly Krieger's Numberphile video on the topic as an initial motivation, and going on to prove the main theorem stated there (that you can ...
1
vote
0answers
11 views

Proof that any hyperbolic group has Rapid Decay property

A classic result that states that any hyperbolic group in the sense of Gromov has Rapid Decay property in the sense of Jolisaint. But the original proof of that fact is contained in an old Ph.D. ...
0
votes
0answers
21 views

Robustness of ergodic dynamical systems

Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space. For an ergodic measure preserving transformation $T$, we define the ergodic robustness $\mathcal R(T)$ of $T$ as follows: For $...
1
vote
0answers
18 views

Analytic sets and Turing determinacy

I wonder whether the following question have a positive answer within $ZFC$. Question If $\{A_n\}_{n\in \omega}$ is a sequence of analytic sets so that $\bigcup_n A_n=2^{\omega}$, then there must be ...
-1
votes
0answers
28 views

First hitting time with infinite recursion

Suppose $(i,j)$ is a process that evolves by: $(i,j) \to (i-1,j)$ with rate $\mu$, $(i,j) \to (i,j+1)$ with rate $\theta$. Given that $T(0,j)=0$ for all $j$. I want to calculate the first hitting time ...
1
vote
0answers
35 views

On hypergeometric functions over finite fields

Let $\mathbb{F}_q$ be a finite field of $q$ elements. Let $A,B,C,\cdots$ denote the multiplicative character over $\mathbb{F}_q$, and let $\overline{A}$ denote the inverse of $A$, i.e., $A(x)\overline{...
2
votes
0answers
37 views

Folding Polygons into 'Vessels'

This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
0
votes
0answers
20 views

What is the difference between Gegenbauer and Legendre polynomials

I'm trying to find the Spherical Harmonics decomposition on a function defined on the hypersphere. I wanted to ask what are the main differences between the Gegenbauer and Legendre polynomials?
0
votes
0answers
56 views

A question about a truncated object

I was hoping someone could help me with the understanding of a particular truncated object. Here are some background: For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
2
votes
1answer
47 views

Asymptotic behavior of infinite product of cosines

Consider the function $$F(z) := \cos(z) \cos(z/3) \cos(z/5) \cos(z/7) \cdots$$ Note that $\cos(z/n) = 1 - \frac{z^2}{n^2} + \cdots$ so the product is absolutely convergent to an entire function. I ...
6
votes
0answers
77 views

What is the precise definition of “Hypergeometric motives over $\mathbb{Q}$”?

The question is as in the title, but here is some background: Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
3
votes
1answer
55 views

Is the fixed subring a symmetric algebra?

Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A,...
0
votes
0answers
36 views

How to estimate the spectral radius of a matrix series?

$\{A_k\}$ is a uniformly bounded sequence of matrices whose eigenvalues are in $(0,\rho]$, $\rho<1$. Let $\Phi(k+1,j)= A_kA_{k-1}\dotsm A_j$ and $Q_k=\sum_{j=k}^{\infty}\Phi^T(j,k)\Phi(j,k)$, how ...
1
vote
0answers
31 views

Monoidal functors and their projection functors

I first posted this post as a math I as I am instructions proceed. Suppose $M$ and $N$ are monoidal categories and let $M\times N$ denote the associated product category. $M\times N$ comes equipped ...
1
vote
0answers
12 views

Choice of splitting in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
2
votes
1answer
63 views

Can we always choose 2 nonisomorphic simple modules to satisfy the following nonvanishing extension conditions?

Let A be a finite dimensional algebra over some field k and S a nonprojective simple left A-module. Suppose the projective dimension $pd_A(S)$ of S is finite. Let n be a nonnegative integer such that $...
3
votes
0answers
32 views

Stronger form of countable dense homogeneity

I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove ...
2
votes
0answers
37 views

A polar open set in a topological subspace?

Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar? A set $...
0
votes
0answers
39 views

The solutions of a system of differential equations

Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y $, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$ Consider this system ...
1
vote
0answers
66 views

Smooth interpolation of values

Consider a sequence of points $(x_n)_{n \in \{0,\ldots,N\}^2}.$ The finite element method tells us how to find for example a piecewise linear function $f$ on $[0,1]^2$ such that $f(1/n_1,1/n_2) = x_n$ ...
0
votes
0answers
78 views

Finding which members of a family of (possibly infinite-dimensional) matrices have trivial null space

Background I have a set $S$ (that is possibly infinite) and a correspondence between functions $c:S^3\to\mathbb{R}$ (I will write $c(i,j,k)$ as $c_{ijk}$) and matrices $M$ with rows indexed by $(i,j,k)...
1
vote
0answers
61 views

'Spherically complete' normed fields

A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is ...
0
votes
0answers
94 views

A characterization of the integral

Let $I(f)$ be an endomorphism of the smooth functions with zero value in zero such that: $$\ln[1+I(f)]=I\left(\frac{f}{1+I(f)}\right). $$ Then, does it exist $g$ smooth such that: $$I(f)(x)=\int_0^x f(...
1
vote
0answers
35 views

A semimartingale interpolation problem

This question is a direct extension of this one. Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...
0
votes
0answers
68 views

Brauer character degrees of $S_5$

I need to know the set of all irreducible Brauer character degrees of $S_5$ for p=3 ($cd_3(S_5)$).
0
votes
1answer
48 views

A martingale extension/interpolation problem

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a partition of $[0,T]$ with $t_0=0,t_n<t_{n+1},t_N=...
0
votes
1answer
118 views

Graph $G=(V,E)$ with $\chi(G)$ finite and $\text{Col}(G)$ infinite

Let $G = (V,E)$ be a simple, undirected graph. For $v\in V$ we let $N(v) = \{w \in V: \{v,w\} \in E\}$. We define the coloring number $\text{Col}(G)$ of the graph $G$ to be the smallest cardinal $\...
4
votes
1answer
67 views

Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$

Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
0
votes
0answers
115 views

Proof of “Hochschild-Serre” spectral sequence

I'm looking for a detailed proof of the Hochschild-Serre spectral sequence for Galois Cohomology: If we have $H$ a normal subgroup of $G$ (profinite group) and $T$ is a $G-$module, we get: $$ 0\...
1
vote
0answers
80 views

Help to use Statistics and algebra books for community

My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me I just need to know if we can donate these ...
3
votes
0answers
50 views

Jacobi fields on non-symmetric spaces

I am searching for examples of manifolds which are not symmetric spaces but where Jacobi fields can be computed in closed form. For now, I am aware of Gaussian distribution with the Wasserstein ...
9
votes
1answer
496 views

Is there any non-commutative ring such that every element other than the identity is a zero divisor?

A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
3
votes
0answers
87 views

Does there exists a $C^*$-algebra corresponding to every Banach ternary algebra?

Let $V$ be a TRO (closed subspace of $B(H,K)$, closed under the product $xyz\to xy^*z$). Let $C(V)$, $D(V)$ denote the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^*V$ respectively. We define $A(...
3
votes
2answers
108 views

Tangent cone of null sets

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that $$ \liminf_{r \to 0+} r^{-1}...
1
vote
0answers
55 views

Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map $$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that $$T_{\sigma,b}(n,\theta,y) = (\...
5
votes
1answer
250 views

Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$?

As the title says, I am interested to know Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$. There is some confusion in the literature. Let recall that the compactness theorem in $L^p(\...
2
votes
0answers
66 views

On induced maps in group homology

I have some problem calculating the value of some specific (but quite common) induced maps I stumbled on while reading some papers on group (co)homology and I would like to know if there are general ...
4
votes
0answers
44 views

Is Broué's abelian defect conjecture true for finite groups with abelian TI Sylow p-subgroups?

I am now interested in Broué's abelian defect conjecture and I have read many papers concerning it. For a prime $p$, I informally define a finite group to be a $p$-ATI-group if it has abelian Sylow $p$...
1
vote
0answers
106 views

Genus of a curve given by self intersection of a very ample line bundle

Let $X$ be a smooth, integral and projective $d$-dimensional variety over a field $k$ of characteristic 0. Let $H$ be a very ample line bundle over $X$. Assume that there exists a smooth and ...
1
vote
0answers
52 views

Tight upper bounds on trigonometric polynomials

According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
5
votes
0answers
104 views

A model structure on CDGAs

Assume our base ring $k=\mathbb{Z}_{(p)}$ or $\mathbb{F}_p$ and for simplicity let us consider non-negative-graded connected algebras $A$ with an augmentation, i.e. $A^0=k,A^{<0}=0$. The question: ...
0
votes
1answer
53 views

Large chromatic number in hypergraphs with large edges

Let $H=(V,E)$ be a hypergraph. If $\kappa \neq \emptyset$ is a cardinal, we call a map $c:V\to \kappa$ a coloring if for each $e\in E$ with $|e|>1$ the restriction $c\restriction_e$ is non-constant....
3
votes
1answer
157 views

Finiteness of Hausdorff measure of balls

Let $(X,d)$ be an arbitrary metric space and let $\Bbb B(x,r)$ denote the closed ball with center $x \in X$ and radius $r>0$. For $p\geq 0$, let $H^p$ denote the $p$- dimensional Hausdorff measure. ...
3
votes
1answer
55 views

The product of the lengths of two line segments that belong to Newton line

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
1
vote
1answer
55 views

Some basic inequalities in the theory of symmetric normed space

I am learning about symmetric normed space and have trouble figuring out the following inequality. Let $c_c(\mathbb{N})$ be the space of compactly supported sequence. Let $||a||_{p,\omega}:=\sup_{n}(n^...
2
votes
1answer
79 views

Decoding the Reed–Muller $RM(m, m)$ code, and a matrix related to Pascal's triangle

The Reed–Muller $RM(m, m)$ code sends the message $p(X_0, \ldots , X_{m - 1}) = \sum_{S \subset \{0, \ldots , m - 1\}} \alpha_S \cdot X_S$ to its set of evaluations $\left\{ p(x_0, \ldots , x_{m - 1}) ...
1
vote
0answers
43 views

PDE for the area-preserving non-parametric curve shortening flow?

In dimension $1$, it is well-known that the motion of a (unparametrized) plane curve $\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$ under the curve shortening flow can be encoded by the PDE $$\partial_t ...
0
votes
0answers
72 views

Show that $ \sum_{i=0}^{2k} {n \choose i+1} u(2k,i) +\sum_{i=0}^{2k} {-n-1 \choose i+1} u(2k,i)=0. $

Let $u(k,j) = 1$ if $j=0$, $0$ if $j > k$, or else it is $j*u(k-1,j-1) +(j+1)*u(k-1,j) $. Prove that $ \sum_{i=0}^{2k} {n \choose i+1} u(2k,i) +\sum_{i=0}^{2k} {-n-1 \choose i+1} u(2k,i)=0. $ ...
5
votes
1answer
290 views

Largeness of the set of zeroes of a Brownian motion

Definitions: A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...

15 30 50 per page
1
2 3 4 5
2525