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The number of elements in {1,2,…,a}.{1,2,…,b}, where $ab=n^2$

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 $ab=n^2$?
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0answers
4 views

Can the differentials in a minimal free resolution ever have a “long” row of $0$'s?

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},{...
3
votes
1answer
27 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 ...
7
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4answers
866 views

Interesting results for open Riemann surfaces

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 ...
1
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0answers
73 views

Smooth interpolation of values possible?

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 $f$ ...
1
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0answers
28 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 $p$-...
2
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0answers
27 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
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0answers
25 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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0answers
25 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 $...
4
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1answer
79 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,...
4
votes
1answer
74 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 ...
3
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1answer
81 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
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2answers
113 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}...
3
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0answers
34 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
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0answers
41 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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0answers
38 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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1answer
83 views
+50

Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

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 ...
5
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2answers
378 views

question about commutative diagram in category theory

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), ...
166
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16answers
9k views

Great graduate courses that went online recently

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 ...
2
votes
1answer
49 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 ...
131
votes
22answers
20k views

Good “casual” advanced math books

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 ...
3
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0answers
278 views

Is this result on the set of differentiability of the distance function to the fat cantor set new?

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 ...
2
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1answer
86 views

Weak sequential continuity of certain bilinear forms on Banach algebras

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) &:=& \...
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2answers
516 views

Modified probability distribution

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 ...
0
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0answers
40 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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0answers
24 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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0answers
61 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 ...
0
votes
0answers
95 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(...
13
votes
1answer
3k views

Curvature as infinitesimal holonomy

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 ...
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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 $\...
7
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0answers
84 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) ...
1
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0answers
33 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 ...
2
votes
0answers
39 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 $...
8
votes
2answers
774 views

When do flat holomorphic connections exist?

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, $A(\mathcal{E}) = 0$....
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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)...
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votes
0answers
37 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
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 &...
5
votes
2answers
268 views

Group cohomology of $\mathbf{R}^\ast$ acting on $\mathbf{R}$

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 ...
7
votes
1answer
971 views

Cohomology of Grothendieck topology

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 ...
20
votes
2answers
4k views

Grothendieck's Tohoku Paper and Combinatorial Topology

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 ...
0
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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. $ ...
1
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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 ...
1
vote
0answers
62 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 ...
9
votes
0answers
437 views

Why is the symbol map in Atiyah–Singer paper continuous?

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:...
0
votes
0answers
69 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 ($cd_3(S_5)$).
1
vote
0answers
77 views

Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?

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 $f$ ...
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=...
1
vote
0answers
56 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) = (\...
0
votes
1answer
40 views

Bessel process conditioned to stay positive

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&...
2
votes
0answers
79 views

Measurable selection proof of conditional expectation as Borel function

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}...

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