Related papers: Note on some Gromoll filtration groups and fundame…

We continue the investigation, that began in [3] and [4], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm…

Motivated by the experimental measurement of the decay rate, $\Gamma$, and the longitudinal polarization, $P_L$, in the Cabibbo favored decay $D_s^+\to \phi {\rho}^{+}$, we have studied theoretical prediction within the context of…

Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex…

Let f(m,n) denote the number of relatively prime subsets of {m+1,m+2,...,n}, and let Phi(m,n) denote the number of subsets A of {m+1,m+2,...,n} such that gcd(A) is relatively prime to n. Let f_k(m,n) and Phi_k(m,n) be the analogous counting…

Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $\Gamma_{p,a}\subset SL(2,\mathbb{R})$ and…

For all integers $p>q>0$ and $k >0$, and all non-elementary torsion-free hyperbolic groups $H$, we construct a hyperbolic group $G$ in which $H$ is a subgroup, such that the distortion function of $H$ in $G$ grows like $\exp^k(n^{p/q})$.…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$, and consider the partition function $M_{G}(\lambda)=\sum_{k=0}^nm_k(G)\lambda^k$. In this paper we show that if $G$ is a $d$--regular graph and…

For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \sigma(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and }…

A composite positive integer $n$ is said to be a {\it weak Carmichael number} if $$ \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv \varphi(n) \pmod{n}. \leqno(1) $$ It is proved that a composite positive integer $n$ is a weak Carmichael…

Let $\Gamma$ be a crystallographic group of dimension $n,$ i.e. a discrete, cocompact subgroup of $\operatorname{Isom}(\mathbb{R}^n)$ = $O(n)\ltimes\mathbb{R}^n.$ For any $n\geq 2,$ we construct a crystallographic group with a trivial…

Using an elementary method, we show that an odd number of Majorana fermions in $8k+1$ dimensions suffer from a gauge anomaly that is analogous to the Witten global gauge anomaly. This anomaly cannot be removed without sacrificing the…

We improve the upper bounds (in terms of $n$) in [9] and [13] on the minimal number of elements required to generate a minimally transitive permutation group of degree $n$.

In this paper, we construct and classify a new family of flips, called generalized Grassmannian flips, by generalizing the construction of standard flips for $\mathbb{P}^m\times \mathbb{P}^n$ to any generalized Grassmannian $G/P$, where $P$…

The aim of this paper is to give a new result of the differential Galois theory of linear ordinary differential equations. In particular, we compute differential Galois group for special type non-resonant Fuchsian system.

We propose a method for determining $\gamma$ using $B^\pm\to D K^\pm$ decays followed by a multibody $D$ decay, such as $D \to K_S \pi^-\pi^+$, $D \to K_S K^-K^+$ and $D \to K_S \pi^-\pi^+\pi^0$. The main advantages of the method is that it…

A set grading on the split simple Lie algebra of type $D_{13}$, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern…

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$…

Let $G$ be the multiplicative group generated by the gamma functions $\Gamma(ax+1)$ $(a=1,2,\dots)$, and $H$ be the subgroup of all elements of $G$ that converge to nonzero constants as $x\rightarrow\infty$. The quotient group $G/H$ is the…

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group,…