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38 views

the choosing of an independent set in a specific $k$-partite graph

Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class ...
273 views

contractible configuration spaces

Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$. My question: is $F(S^\infty,k)$ ...
28 views

Skein theory: How axiomatizing a 2-box space?

Let $(A,+,\times, *)$ with an adjoint operation compatible with $+$, $\times$ and $*$, such that $(A,+,\times)$ and $(A,+,*)$ are finite dimensional ${\rm C}^{*}$-algebras. What are the axioms on ...
86 views

How do you categorify the cycle index series?

Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be $$\sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!}$$ It was observed by Baez and Dolan in their paper ...
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If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ... 0answers 50 views Limit of (n^2+1)^(1/n) [on hold] I am struggling to figure out$\lim\limits_{n \to inf} \sqrt[n]{n^2+1} $. I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find the ... 0answers 47 views Is a vector space with two identical vectors a vector space with one or two vectors? [on hold] I'm new this, and cannot find any answers by searching. If a vector space has 2 identical vectors, in particular the zero vector, is it a vector space with 2 vectors or since they are linearly ... 1answer 307 views Regularized sums of Mobius sequence Do$\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$and$\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$both equal$-2$? Experimentally this seems plausible (up through ... 0answers 102 views Does SL(3,q) have a subgroup of order$q^3.(q^3-1)$[on hold] Let$q=p^n$for$p>3$.I want to know whether the group$G_2(q)$has a subgroup of order$q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ... 0answers 58 views Help in finding the distribution and pdf Considering a set of$n$points that are$d$dimensional and are independently and uniformly distributed on a surface. The points are homogeneous poisson point process. Considering nearest neighbor ... 0answers 19 views exit time of a non degenerate diffusion Let$n, d \geq 1$,$b : \mathbb{R}^n \to \mathbb{R}^n$and$\sigma: \mathbb{R}^n \to \mathbb{R}^{n \times d}$two Lipschitz functions. We assume that \exists \mu >0, \xi^T ... 0answers 37 views Non Normal operator [on hold] Standard example for non normal operator is the shift operator. It is continous but the image of the left shift is not dense. Can we have an example of a non normal operator$A$which is continuous ... 1answer 277 views Group laws in class field theory In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field. In the case of a ... 1answer 108 views Questions about a possible way of representing construcive ordinal numbers Let$K$be the set of all total recursive functions of non-negative integers having only non-negative integers as values. Let$L$be any well-ordered subset of$K$in which the ordering$<$is ... 0answers 45 views Can the GUE be thought of as a uniform point in a high-dimensional polytope I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ... 1answer 139 views What is the growth of the rank of a power of a finite simple group? Which asymptotic bounds (upper and lower) are known for$s_n$- the minimal number of generators of$S^n$where$S$is a nonabelian finite simple group? 1answer 380 views Adding sets not containing arithmetic progressions of length three by forcing Consider the following forcing notion: conditions in$\mathbb{P}$are pairs$(s, N),$where: 1)$s\in 2^{<\omega}$, 2)$N\in \mathbb{N}$, 3) (by identifying$s$with a subset of$lh(s)$)$s$... 0answers 49 views separating parameters in generalized quadratic Gauss sum The normalized generalized quadratic Gauss sum is defined by $$G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right)$$ where$e(x)=\exp(2\pi ix)$. Under what conditions on$c$can we ... 0answers 122 views Tree property and singular strong limit cardinals I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question: Theorem. It is consistent, relative to the existence of large cardinals, that ... 0answers 123 views When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space? Let$\Gamma$be a group admitting a discrete and cocompact action on a CAT(-1) space. Let$d$a word metric on$\Gamma$coming from some finite set of generators. My question is: Does there exist a ... 0answers 166 views Solving a set of equations in a finite symmetric group A standard way to find solutions to a finite set of equations in a finite symmetric group${\rm S}_n$is to take the equations as relators of a finitely presented group, to use the low index subgroups ... 2answers 220 views Tensor calculus on the frame bundle Let$M$be a manifold and let$g$be a tensor on it, say for example a metric$g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on$g$. For instance, taking its ... 0answers 106 views Application of Stickelberger's Theorem to Quadratic field I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ... 1answer 138 views About direct limit of groups Let$G_i$be sequence of groups for$i\in \mathbb N$and Let$\phi_i$be a monomorphism from$G_i$to$G_{i+1}$. Let$\Sigma$be the direcet limits of$G_i$under the embeddings of$\phi_i$. Let ... 0answers 56 views Minimum rank non-negative matrix summations Given matrix$M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$of rank$r$. What is minimum$k$such that $$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains$R,S$... 1answer 58 views Integral Transform with associated Legendre Function of second kind as kernel In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where$\rho,s>1$, ... 0answers 57 views I don't get how -6cos3xsin3x becomes -3sin6x in the later part [on hold] y = cos²3x dy/dx = 2cosx(-sin3x)(3) = -6cos3xsin3x = -3sin6x I found this answer key in my guidebook but I can't find any trigonometric function's or differentiation formula ... 0answers 27 views Unimodular triangulation and affine toric variety Let$\mathcal{K}$be a$pointed$rational cone in$\mathbb{R}^d$with extremal rays generated by$r_1,r_2,\dots, r_m\in \mathbb{Z}^d$. Here, pointed means that all$r_i$lie strictly on one side of ... 0answers 117 views Sum-epimorphisms and prod-monomorphisms Sum-epimorphisms A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition: ... 1answer 94 views Convex hull of the union of two parameterized curves in$\mathbb{R}^3$My goal is to find a way to calculate the convex hull of the union of some parameterized curves. For instance, I had to calculate the convex hull of$A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup ...
It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...