All Questions

Let $A_{a,b}$=$\{mn:m\leq a,n\leq b\}$, where $ab=n^2$ are two fixed integers. How large is $A_{a,b}$? Must $A_{a,b}$ no less then $[1,2,...,n].[1,2,...,n]$ for all $a,b$ such that $ab=n^2$?

Assume just for sake of simplicity that $R = k[x_1 , \dots , x_n]$ is a standard graded polynomial ring over a field. If one considers the ideal $$I = \left({x}_{1}{x}_{3},{x}_{2}^{2},{x}_{2}{x}_{3},{...

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 ...

As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...

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 the unit square $[0,1]^2$ such that $f$ ...

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 $p$-...

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 ...

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. ...

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 $...

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,...

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 ...

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 $...

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}...

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 ...

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 ...

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{...

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...

I am reading the article Maurice Auslander, Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay 21-22 mai 1987), ...

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...

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 ...

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...

Definitions: Let $C \subset [0, 1]$ be a fat Cantor set, for parameter $0 < r < 1/3$. Thus intervals of width $r^n$ are removed from the middle of the previous intervals at each step. For the ...

Let $A$ be a Banach algebra and $Bil(A)$ denote the space of bounded bilinear forms on $A$. $Bil(A)$ is a Banach $A$-bimodule with the module operations \begin{eqnarray*} \beta a(x,y) &:=& \...

My question entails finding a continuous function equation that is the continuous function equivalent of a modified discrete probability calculation. This is in support of research that I have been ...

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 ...

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?

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 ...

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(...

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...

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 $\...

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) ...

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 ...

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 $...

Let $X$ be a smooth projective variety over $\mathbb{C}$. I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$....

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)...

$\{A_k\}$ 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 ...

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 &...

I am interested in computing the first group cohomology $H^1(\mathbf{R}^\ast, \mathbf{R})$, where $\mathbf{R}^\ast$ is acting on $\mathbf{R}$ by multiplication (here $\mathbf{R}$ denotes the real ...

My naïve cartoon picture of the construction of étale cohomology is this: start with a scheme, associate to it a Grothendieck topology (making a site). A functor from the Grothendieck topology to ...

I've read some discussions of Grothendieck's famous Tohoku Paper, and I understand that one reason it was a landmark paper was that it introduced abelian categories and gave us sheaf cohomology as a ...

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. $ ...

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 ...

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 ...

I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...

I need to know the set of all irreducible Brauer character degrees of $S_5$ for p=3 ($cd_3(S_5)$).

Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying $$\langle f(t), \phi \rangle \in L^1([0,T])$$ for any Schwartz function $\phi \in S(\mathbb R^n)$. Does $f$ ...

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=...

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) = (\...

This question has also been asked on https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x&...

It is well know that if $X,Y\in L^2(\Omega,\mathcal{F},\mathbb{P})$, for some probability space $(\Omega,\mathcal{F},\mathbb{P})$, then there exists a Borel function $h:\mathbb{R}\rightarrow \mathbb{R}...