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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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### 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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### 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 ...
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., $A(x)\overline{... 1answer 83 views +50 ### Approximating a subclass of$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 ... 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), ... 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 ... 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 ... 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 ... 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 ... 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) &:=& \... 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 ... 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.$Consider this system ... 0answers 24 views ### What is the difference between Gegenbauer and Legendre polynomials I'm trying to find the Spherical Harmonics decomposition on a function defined on the hypersphere. I wanted to ask what are the main differences between the Gegenbauer and Legendre polynomials? 0answers 61 views ### A question about a truncated object I was hoping someone could help me with the understanding of a particular truncated object. Here are some background: For any object$A$in an abelian category$\mathcal{A}$, we can view$A$as an ... 0answers 95 views ### A characterization of the integral Let$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(... 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 ... 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 \... 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) ... 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 ... 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 ... 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.... 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)... 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 ... 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 &... 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 ... 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 ... 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 ... 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. ... 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 ... 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 ... 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:... 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)). 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])$$for any Schwartz function \phi \in S(\mathbb R^n). Does f ... 1answer 48 views ### A martingale extension/interpolation problem Let (\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P}) be a stochastic basis and let N\in\mathbb{Z}^+, T>0, \{t_n\}_{n=1}^{N} be a partition of [0,T] with t_0=0,t_n<t_{n+1},t_N=... 0answers 56 views ### Birth of chaos due to nonautonomous perturbation Let \sigma, b>0. I want to study the dynamics of the map$$ 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) = (\... 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&...
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}...