# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 X \rightarrow U X U^* ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $i=0,1,2...$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f,g$. ...
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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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 $y \in ... 1answer 98 views ### Could we extend the exact sequence$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)$is a ... 0answers 100 views ### Koopman representation, weakly compact action, Ozawa Popa Given a weakly compact action (Ozawa-Popa) of a discrete group$\Gamma$on p.m space$X$, consider the Koopman representation$\pi$on$L^2(X)$. Compose this representation with the Calkin projection. ... 0answers 213 views ### Order theory as a foundation of mathematics? I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ... 2answers 512 views ### A combinatorial question about orthonormal bases Suppose that$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)$over an orthonormal basis is independent of the basis. ... 1answer 115 views ### Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ... 0answers 10 views ### Partition on a Closed Set A= [2,3] [migrated] Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty? 1answer 54 views ### Computionally efficient vertex enumeration for (convex) polytopes Let$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$can be ... 0answers 30 views ### Optimization with random matrix Consider$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)$denotes the vector where ... 0answers 84 views ### Coloring a Ferrers diagram I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ... 0answers 104 views ### All non-split Cartan subroups of$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$be a ... 0answers 89 views ### SubGROUPs of Banach spaces, when are they dense in a vector subspace? It’s relatively easy to show that if$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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