# All Questions

126,250
questions

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18 views

### 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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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

**1**answer

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**

votes

**4**answers

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 ...

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vote

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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 $f$ ...

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vote

**0**answers

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 $p$-...

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votes

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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 ...

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vote

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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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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 $...

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votes

**1**answer

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,...

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votes

**1**answer

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 ...

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votes

**1**answer

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 $...

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votes

**2**answers

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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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 ...

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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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**0**answers

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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vote

**1**answer

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 ...

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votes

**2**answers

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**

votes

**16**answers

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 ...

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votes

**1**answer

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

**22**answers

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 ...

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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 ...

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votes

**1**answer

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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votes

**2**answers

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 ...

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**0**answers

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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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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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 ...

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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(...

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votes

**1**answer

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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**1**answer

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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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) ...

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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 ...

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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 $...

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votes

**2**answers

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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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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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 ...

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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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votes

**2**answers

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 ...

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votes

**1**answer

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 ...

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**2**answers

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 ...

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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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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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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 ...

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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:...

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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 ($cd_3(S_5)$).

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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 $f$ ...

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votes

**1**answer

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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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) = (\...

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**1**answer

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&...

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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}...