# All Questions

98,519 questions

**4**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**5**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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 ...