All Questions

4
votes
1answer
161 views

Generalizations of the “Curious Tiger” Polygon

I actually don't know, whether the polygon I describe here already has name, but let me explain the problem, that is solved by the polygon, with a little story: Imagine a flat terrain with bushes ...
9
votes
0answers
111 views
+200

Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...
1
vote
1answer
63 views

Spatial dimension of a finite graph

If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ given by $V(G(X,d)) = X$ and $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d(...
3
votes
0answers
90 views

A question about the products of power set sigma algebras

Let $\kappa$ be the least cardinal for which the sigma algebra generated by $\{A \times B: A,B \subseteq \kappa\}$ does not contain every subset of $\kappa \times \kappa$. It is known that $\kappa$ is ...
2
votes
0answers
85 views

Inequalities about tripling and doubling sumsets

Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following: Question 1:...
1
vote
0answers
42 views

How to fix multi-valued function on contour?

I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function. I want to calculate the Fourier transformation of a muti-valued ...
4
votes
1answer
107 views

complex polynomial

Let $N$ be a big integer number and consider the equation : $$ x^{N} + a_{N-1} x^{N-1} + ...+a_{1} x + o(\frac{1}{N})=0,$$ where $o(h)$ is by definition a term such that $\lim_{h \to 0} o(h)/h =0$. ...
2
votes
0answers
45 views

How does the $\lambda$ invariant propagate with extra ramification?

Let $\mathbb{Q}^{cyc}$ denote the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and let $\Lambda$ denote the corresponding Iwasawa algebra. Let $p$ be a prime. Let $S$ denote a finite set of ...
0
votes
0answers
21 views

Find the analytical form of the eigenvalue of a special sparse matrix

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
4
votes
0answers
106 views

Sections of sheaves on limit spaces

Let $\{U_{\nu}\}_{\nu\in I}$ be an inverse system of topological spaces over a filtered index set $I$ with continuous transition maps. Let $A_0$ be a sheaf of abelian groups on $U_{\nu_0}$, for some $...
0
votes
0answers
118 views

Sudden appearance of an eigenvalue of a self-adjoint operator $H = H_0 + \lambda H_1$

In doing some numerical calculation in quantum mechanics, we found something surprising to us. Let the Hamiltonian be $$ H = H_0 + \lambda H_1 , $$ where both $H_0$ and $H_1$ are self-adjoint, and $...
13
votes
1answer
451 views

Holes in double-tileable polynominoes

This question was communicated to me by Evgeniy Romanov. Consider a connected polyomino $P$ that can be completely tiled in two different ways: with disjoint $2 \times 2$ square tetraminoes, and with ...
9
votes
1answer
220 views

Is $\mathbb{C}^n\setminus V(f)$ homotopy equivalent with a “large ball complement”?

Let $f\in\mathbb{C}[x_1,\dots,x_n]$, and let $V(f)$ denote the vanishing locus. Is it true that for large enough $N$, there is a homotopy equivalence $$\mathbb{C}^n\setminus V(f)\simeq B(0,N)\setminus ...
6
votes
0answers
263 views

Adding a dedication to a paper

A paper of mine (relatively junior mathematician) was just accepted in a good journal, and I was considering adding a dedication to the memory of a mathematician in my area that passed away, and whose ...
1
vote
0answers
36 views

When does a collection of sets forming a geometric lattice give the flats of a matroid?

Say we have a matroid on a finite set $X$. The collection of its flats forms a geometric lattice under $\subseteq$, where the join is given by intersection. This question is about the converse to ...
4
votes
1answer
84 views

Conformally flat homogeneous spaces

Let's say we have a homogeneous space $H\backslash G$. Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure? I am particularly ...
-1
votes
1answer
224 views

Write the algebra closure of $F_p$ as union of finite fields [closed]

This question follows Field theory by Steven Roman, Chapter 9, Exercise 20. Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...
3
votes
0answers
49 views

Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
20
votes
5answers
1k views

are quotients by equivalence relations “better” than surjections?

This might be a load of old nonsense. I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an ...
1
vote
0answers
36 views

Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as $$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$ where $X_0,X_1,...,X_k$ are ...
1
vote
0answers
37 views

Optimizing a determinant

Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem: $$ \max_g ~\...
1
vote
1answer
61 views

Integrability of an almost complex structure vs holomorphicity of the section $M\rightarrow \mathcal{J}(M)$

Let's say we have an almost complex manifold $(M, J)$. Consider the complex vector bundle $V\rightarrow M$ whose fiber over $x$ is the space of almost complex structures on $T_x M$. Is there any ...
1
vote
0answers
46 views

When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
1
vote
0answers
48 views

Finite-dimensional graded Lie algebras with $2$ generators

Does anyone know of a classification of those (complex) Lie algebras which are: generated by two elements $\mathbb{Z}$-graded Lie algebras finite dimensional
1
vote
0answers
107 views

A family of crystalline representations

Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...
3
votes
0answers
51 views

Name of a binary matroid coming from the cycle space of a graph

In some of my recent work, I have 'discovered' a binary matroid which I will describe below. Given a graph $G$, let $H_1(G, \mathbb{Z}/2\mathbb{Z})$ denote the cycle space. This is a vector space ...
0
votes
1answer
93 views

Poles of equivariant meromorphic functions on Riemann surfaces

Let $p:\Sigma\to \mathbb{P}^1$ be the cyclic cover of $\mathbb{P}^1$ with Galois group $\Gamma$. Let $\Gamma\cdot p$ be a free $\Gamma$-orbit on $\Sigma$. Given any character $\chi$ of $\Gamma$, does ...
1
vote
1answer
50 views

Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements

Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal: $\...
3
votes
0answers
244 views

Topological approach to create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
2
votes
1answer
133 views

What is the topological/smooth analogue of Nagata compactification

A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...
3
votes
1answer
97 views

Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
0
votes
0answers
31 views

Calculating a side in hyperbolic polygon [on hold]

if I have a hyperbolic quadrilateral ABCD such that $\angle A=\angle B=\frac{2\pi}{3}$, $AD=BC$ and we know the hyperbolic lengths of sides $AB$ and $CD$, how can I calculate the hyperbolic length of $...
0
votes
1answer
45 views

Reference request: Moving source to initial condition and vice versa in PDE problem

I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws. Inhomogeneous Cauchy problem: $$(1) \hspace{1cm} ...
7
votes
1answer
336 views

Simple groups of the same order

I heard that there are no 3 nonisomorphic simple groups of the same order. Question: Is there an elementary proof of this? In case this is not the case, here a modified question: Question: Is ...
0
votes
2answers
415 views

Defining algebraic manifold without referring to schemes

Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions. Is it true that there exists a smooth integral ...
0
votes
0answers
63 views

Coxeter group action on the product of root systems

Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \...
5
votes
0answers
116 views

Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
16
votes
1answer
329 views

Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "...
0
votes
0answers
34 views

Testing relation between two integers in a linear equation [closed]

This is what I read: One can test if there exist integers x and y such that $c\cdot x + j = d \cdot y + k$, if $(k-j)\mod(\gcd(c,d)) = 0$ How can one see that? -- PS: I understand ...
0
votes
3answers
93 views

Almost complex structure corresponds to unique complex structure up to biholomorphism [on hold]

Assume I have a smooth manifold which has two different holomorphic atlases that induce the same almost complex structure on it. How to show that two complex manifolds (corresponding to the two ...
-3
votes
0answers
57 views

Rounding to decimal places by using simple formula and basic math operations [closed]

Is there way to round number to decimal places using just simple math operations in one formula? No converting to text allowed. No considerations (like "if point is..then") allowed. No ...
1
vote
1answer
91 views

Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?

TL;DR. Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $\mathbf{Q}(i)$ seem to have value in $\mu_4$. But apparently our computation via Rohrlich's ...
5
votes
1answer
100 views

Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
0
votes
1answer
43 views

Meaning of “Herbrand-Goedel recursive” in Kleene's “On Notation for Ordinal Numbers”

In Kleene's "On Notation for Ordinal Numbers", Journal of Symbolic Logic, Volume 2, Number 4, December 1938, he says that a function of natural numbers is taken to be effective if it is Herbrand-...
2
votes
1answer
88 views

Pushing forward a complex structure by submersion

I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\...
3
votes
0answers
81 views

How to show mapping cones are homotopy cofibers

In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor $$ {\rm Hom}(-,C)[n]. $$ The cone of a closed morphism $f\colon C \to D$ of degree ...
-1
votes
0answers
63 views

Prove or disprove the non colinearity of three points

Prove or disprove: Let $ABCD$ be a quadrilateral in Pasch Geometry for which the following holds $AB\cap CD=\{E\}$, $AC\cap BD=\{F\}$ and $AD\cap BC=\{G\}$. Then the points $E,F,G$ are not collinear. ...
4
votes
0answers
150 views

How to construct the espace étalé (space of sections) for an arbitrary category?

I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space. In all references I am reading (...
17
votes
2answers
2k views

A Putnam problem with a twist

This question is motivated by one of the problem set from this year's Putnam Examination. That is, Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
2
votes
1answer
80 views

Reference for Minkowski functional when 0 is not in the interior

The Minkowski functional on a normed linear space $E$ is usually defined for convex (or sometimes even non convex) subsets $C$ of $E$ such that $0 \in \operatorname{int}(C)$. Is there any standard ...

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