# All Questions

126,249
questions

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### 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 $p$-...

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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 ...

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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. ...

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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 $...

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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 ...

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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 ...

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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{...

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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 ...

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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?

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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 ...

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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 ...

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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) ...

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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,...

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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 ...

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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 ...

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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 &...

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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 $...

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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 ...

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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 $...

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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 ...

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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 $f(1/n_1,1/n_2) = x_n$ ...

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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)...

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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 ...

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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(...

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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 ...

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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 ($cd_3(S_5)$).

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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=...

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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 $\...

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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 ...

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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\...

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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 ...

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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 ...

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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"...

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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(...

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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}...

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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) = (\...

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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(\...

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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 ...

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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 $p$...

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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 ...

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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 ...

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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: ...

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**1**answer

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....

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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. ...

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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 $P$ ...

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**1**answer

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^...

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**1**answer

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}) ...

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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 ...

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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. $ ...

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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 ...