0
votes
0answers
5 views

How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
4
votes
0answers
45 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
-4
votes
0answers
34 views

Nauty software package [on hold]

In the latest user guide ''nauty and Traces User's Guide (Version 2.5)'' by using Traces we have the ability to compare and combine different graphs of the same number of vertices! Is it possible to ...
0
votes
1answer
10 views

Diagonalization of a Hermitian matrix under restricted condition

Let us suppose a Hermitian (possibly traceless) matrix $X$ only allows the following transformation using special unitary matrix $U \in SU(N)$, namely \begin{equation} X \rightarrow U X U^* ...
2
votes
0answers
23 views

Complexity :: Integer Programming :: Non-Poly Example

When learning about computational complexity I find that when discussing the NP-Complete problems authors always give examples of such problems that can in fact be proven in poly time. I understand ...
0
votes
0answers
7 views

Non-discrete modularity measure in graph analysis [on hold]

I work in neuroimaging, and right now graph theory is all the rage. Most graph analyses that parcel brain regions into modules do so in a discrete fashion. This might ignore the idea that one brain ...
-5
votes
0answers
32 views

limit exercise which requires ingenuity1 [on hold]

Hello im an eleventh grader in the best mathematics high school in my country. I wanted to see if any of you guys can help me solve a limit without integrals and L'Hospital, as i havent learnt them ...
1
vote
1answer
67 views

Is there a nonzero sheaf with all cohomologies vanish?

Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?
1
vote
0answers
105 views

Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
1
vote
0answers
28 views

Twisting stable maps to C* equivariant space by a line bundle

Let $X$ be a $\mathbb{C}^*$-equivariant algebraic variety. Then there is a notion of a map to $X$ twisted by a line bundle. Namely, let $B$ be a variety and $L/B$ a line bundle. Let $P_L=L\setminus ...
1
vote
0answers
45 views

Runs of consecutive numbers all of which are rebel numbers [migrated]

A positive integer is said to be a rebel number if it is the product of numbers none of which share any of the original number´s digits. Thus 10 = 2 x 5 is a rebel number, while none of the primes is. ...
5
votes
1answer
60 views

Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by $$ I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ By the classical Hardy-Littlewood-Sobolev theorem ...
2
votes
0answers
60 views

A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ ...
0
votes
0answers
29 views

Generalization of a class of sets [on hold]

In topological space, we start with open set, which serves as fundamental set. We know that union of finite disjoint open sets is the smallest set amongst any kind of unions of open sets, so we have a ...
-2
votes
0answers
30 views

Decomposition of ball in Banach Tarski paradox and covering a soccer ball [on hold]

These are 2 separate questions but both related to ball. In both parts, let's use the unit ball (radius = 1) for simplification. Banach Tarski paradox says that it's possible to decompose a ball in ...
0
votes
0answers
23 views

An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
-2
votes
0answers
17 views

MatLab loop which stops after X iterations [on hold]

sorry this is a bit of a simple question but I can't find the answer. I'm trying to halt a loop after 25 iterations like this: ...
-1
votes
1answer
26 views

Can Singular Value Decomposition Optimal Newman Modularity? [on hold]

I face some problem on the way home. I want to optimize Newman Modularity Q , Can I use SVD to do that? Thanks!
1
vote
3answers
103 views

When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?

Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure. Is it always possible to construct a smooth structure on $M$ ...
2
votes
0answers
38 views

When are principal bundles preserved by colimits?

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms ...
-1
votes
0answers
20 views

Estimation VS detection [on hold]

I would like to know what is the difference exactly between estimation (parametric or not) and detection in the statistic signal process. Thanks in advance and have a good day
32
votes
3answers
780 views

Is the set AA+A always at least as large as A+A?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
2
votes
1answer
133 views

The topology of Fano schemes of lines

Is there any references concerning the computation of the fundamental groups and Hodge numbers of Fano schemes of lines in a smooth hypersurface in $\mathbb{P}^n$?
0
votes
0answers
60 views

Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$ maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point $y \in \mathbb{P^1} $ ...
-2
votes
0answers
27 views

Green`s function [on hold]

Find the Green's function $G(x,y, x',y')$ for Laplace's equation in $0<x'<a$, $0<y'<b$; with $G=0$, $x'=0$, $G_{x'}=0$, $x'=a$, $G_{y'}=0$, $y'=0$, $G=0$, $y'=b$, and $0<y'<b$, ...
0
votes
0answers
23 views

On covering by smooth numbers

Denote $P(y)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }y$. Denote $S(x,y)=\{n<x: P(n)<y\}$. Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...
-1
votes
0answers
9 views

Compute Faber polynomials in Matlab [on hold]

I want to compute some faber polynomials associated to an ellipse centered at a point (u,v) (in the complex plane: u+iv) in Matlab. Say the ellipse has minor axis a along the x coordinates (real part) ...
1
vote
1answer
43 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
8
votes
1answer
82 views

Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
5
votes
3answers
238 views

Introductory texts to mathematics [on hold]

I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...
11
votes
2answers
416 views

Mysterious identity between numbers of odd/even meander systems

Definitions: An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting closed semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ...
0
votes
0answers
34 views

Proof for existence of isoperimetric minimizer by compactness theorem

Let $X$ be an $n$-dim closed Riemannian manifold, then given a number $0 <v < vol(X)$, there exists a Borel subset $A\subset X$ attaining $I(v)$, $I$ is the isoperimetric profile. The existence ...
2
votes
1answer
27 views

Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces. Let $B_1, \ldots, B_k \in B(\mathcal{H}).$ Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via ...
1
vote
3answers
120 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
2
votes
0answers
50 views

first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
-4
votes
0answers
28 views

Can anybody help me for this counting question? [on hold]

Peter has 12 pairs of socks and 6 pairs of gloves in different colors. His socks are in green, yellow, black, and grey (3 pairs each). Peter's gloves are either blue, black, or red (2 pairs each). ...
1
vote
0answers
51 views

Lower bound on class number of binary quadratic forms of discriminant of the form n^2+4

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
7
votes
4answers
386 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
1
vote
1answer
72 views

Inner product spaces without symmetry/hermitian axiom

Consider a vector space $X$ over $\mathbb R$ and a bilinear form $ \langle \cdot, \cdot \rangle : X \times X \rightarrow \mathbb R$. We assume furthermore that for any $x \in X$ there exists $y \in ...
1
vote
1answer
98 views

Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...
4
votes
0answers
100 views

Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
4
votes
0answers
213 views

Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ...
12
votes
2answers
512 views

A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ...
1
vote
1answer
115 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
-2
votes
0answers
10 views

Partition on a Closed Set A= [2,3] [migrated]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
2
votes
1answer
54 views

Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
1
vote
0answers
30 views

Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...
8
votes
0answers
84 views

Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ...
1
vote
0answers
104 views

All non-split Cartan subroups of $GL_2(\mathbb{Z}/n\mathbb{Z})$ are conjugate

Let $n>1$ be a positive integer and let $R$ be an order in an imaginary quadratic field with discriminant prime to $n$. Let $A=R/nR$ and let $\lbrace 1, \alpha \rbrace$ be a ...
3
votes
0answers
89 views

SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...

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