# All Questions

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### Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
2answers
820 views

### Does the Gamma function preserve integers?

Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the ...
1answer
59 views

### Conditions conformal mapping to be expansive

Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that ...
0answers
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### Coding for channels with concentrated error

A note: I accidentally over-simplified my first attempt at this question to the point of triviality - unfortunately, I didn't recognize this until after I had already placed a bounty to draw ...
1answer
82 views

### Points with pairwise integer distances in the plane

Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a ...
1answer
308 views

### Is there any relationship between the Euler class and the Vandermonde determinant?

Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...
1answer
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2answers
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### $(LLP(Epi), Epi)$ is a WFS on any variety of algebras

This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this ...
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0answers
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### Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks. Currently I'm trying to understand the above-mentioned article which can ...
0answers
104 views

### Question about non trivial zeros of Riemann zeta function [on hold]

I would like to know if is it true that $$-\frac{1}{2\pi i}\underset{n\geq1}{\sum}\frac{1}{n\rho^{n}}\in\mathbb{R}-\mathbb{Z}$$where $\rho$ is a non trivial zero of Riemann zeta function. How can I ...
0answers
43 views

### Solution of parabolic PDE system [on hold]

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$. \begin{cases} \frac{\partial}{\partial ...
1answer
159 views

### Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...
0answers
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### Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories ...
0answers
58 views

I try find integral representation to following function ($d_1<d_2$ and $d_2-d_1=1(\mod 2)$ $f(d_1,d_2)=\frac{(-1)^{\frac{d_2-d_1-1}{2}}}{2^{d_1-1} d_1!}\sum_{k=[\frac{d_1}{2}]+1}^{ d_1} { d_1 ... 0answers 44 views ### Binomial theorem [on hold] I have a problem with binomial theorom. What is the result of solving of inequality: (n 1) + (n 2) + (n 3) + ... (n n) > 32 Sorry for this notation. Thanks for answer. 2answers 362 views ### Is every order type of a PA model the \omega of some ZFC model? Let$N$be a model of first-order Peano arithmetic, and let$\sigma$be its order-type. Does it follow that there is a (non-transitive, expect when$M$is the standard model)$ZFC$-model$M$such that ... 1answer 62 views ### CAT spaces and Metric Measure Spaces [on hold] This is very vague question but here it goes: are there any results of analysis on metric spaces in CAT spaces?, for instance as in the paper of Sturm (On the geometry of metric measure spaces) or as ... 1answer 301 views ### what would be the consequences on the distribution of primes of$\Lambda=\infty$? It is widely believed that the quantity$\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where$t_{n}$is the imaginary part of the$n$-th non-trivial zero on the critical line of the ... 0answers 17 views ### Some Galois theory [migrated] I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose$K$and$k$are fields with$k \leq K$and$[K : k] = m$... 0answers 24 views ### Question about Skorokhod embedding problem Let$B=(B_t)_{t\ge 0}$be a standard Brownian motion on some probability space. Now for every centered probability distribution$\mu$on$R$, i.e.$\int_{R}|x|d\mu(x)<+\infty$and ... 0answers 118 views ### Moduli interpretation of Eisenstein series Let$N \geq 11$be an integer and consider the basis of Eisenstein series for$M_2(\Gamma_0(N))$described in Theorem$4.6.2$of Diamond--Shurman's book. Pick and Eisenstein series$F$in this basis. ... 1answer 198 views ### Is$L(\ell_2,\ell_2)$dense in$L(\ell_2,c_0)$? Let$\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$and$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$denote the ... 1answer 37 views ### A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads ... 1answer 69 views ### Weight polytopes of the fundamental representations of simple Lie groups Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups? Thanks in advance 0answers 135 views ### Establishing an upper bound for a dyadic average of a function in$ {L^{p}}([0,1)) $Suppose that$ f $is$ 1 $-periodic and that$ f \in {L^{p}}([0,1)) $, where$ p > 1 $. Let$$(D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j} \stackrel{\text{df}}{=} \left[ ... 2answers 245 views ### Are Banach space norms (up to equivalence) unique? Here is a naive question: is a "completing" norm of a vector space unique (up to equivalence) or can one find a vector space and two non-equivalent norms$\|.\|$and$|||.|||$that both induce a ... 0answers 59 views ### Integral group rings on which stably free modules are free Let$G$be a torsion-free group and$ZG$the integral group rings. Recall that a projective module$P$over$ZG$is stably free if there is an isomorphism$P \oplus ZG^n \cong ZG^m$. Are there known ... 0answers 11 views ### How to restructure adjacency matrix$A$from shortest distance matrix$B$in Network topology inference An undirected graph with$n$nodes could be referred to as an adjacency matrix$A$.$A=[a_{ij}]_{n×n}$with$a_{ij}=a_{ji}=1$standing for there being an edge between node$i$and node$j$, and no ... 0answers 43 views ### Statistics, probability [closed] A statistician-gone-mad has concocted the following multi-part experiment. For the first part of the experiment, a fair, seven sided die is rolled and the upper-most facing number is noted. If the ... 0answers 109 views ### Total variation and Hellinger distance inequality between truncated Gaussians We know that the total variation distance,$d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions$P$and$Q$is lower bounded by their squared Hellinger distance, ... 0answers 70 views ### motivic integration and jacobian ideal When we consider the change of variables in motivic integration, we have a birational map$f:Y\rightarrow X$with Y smooth and we have to consider two invariants the order of the Jacobian ideal of$X$... 1answer 132 views ### What is the ring$A_{\Gamma}$in the Cohen-Lenstra Heuristics? I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of$A$-modules, where ... 0answers 113 views ### “extended TQFT” versus “TQFT with defects” There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related? According to the Atiyah-Segal axioms, a ... 0answers 58 views ### Geodesics on a perturbed submanifold of$\mathbb{R}^m$[closed] Let us consider$M$, a Riemannian manifold of dimension$n$, isometrically embedded in$R^m$. Let us consider a geodesic$\gamma$on$M$. Now, let us "perturb" (in other words, change slightly the ... 0answers 10 views ### 1 dimensional flows and phase portraits [migrated] I have a flow defined by$\dot{x} = x-x^4+1 :=f(x)$. I need to sketch its phase portrait. Firstly, I have to find its fixed points, these occur at$f(x)=x$. So,$x^4=1 \Rightarrow x= \pm1$. Next, I ... 0answers 48 views ### Theorem on the algebraic manipulation of divergent series [closed] There has been much debate over the values of divergente series. Applying the normal rules of algebraic manipulation to series such as 1+2+4+8+... can produce seemingly legit results such as -1, in ... 1answer 145 views ### A generalisation of Narayana-like numbers (walks on the 2D lattice) I apologize in advance if this question has a trivial answer. I am pretty sure this kind of problem was already studied and I am mostly asking for good references. Given integers$0 < k \le n+1,$... 1answer 92 views ### Oddify an even function and vice versa: need a Fourier transform-based formula [closed] I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ... 2answers 126 views ### Is every Montel locally convex vector space compactly generated? Let$X$be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that$X\$ is a semi-Montel space if every ...

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