Related papers: Commutators of elementary subgroups: curiouser and…

We show that the standard set of elementary generators of an elementary isotropic reductive group over a connected finitely generated ring is a Kazhdan subset. This generalizes the corresponding result of M. Ershov, A. Jaikin-Zapirain, and…

Let k be a field of characteristic zero. Given an ordered 3-tuple of positive integers a=(a,b,c) and for j in N, a family of sequences a_j = (j,a+j,a+b+j, a+b+c+j), we consider the collection of monomial curves in A^{4} associated with a_j.…

Let $E$ be a elementary abelian $p$-group of order $q=p^n$. Let $W$ be a faithful indecomposable representation of $E$ with dimension 2 over a field $k$ of characteristic $p$, and let $V= S^m(W)$ with $m<q$. We prove that the rings of…

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the…

Let $(E,V)$ be a general generated coherent system of type $(n,d,n+m)$ on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of $E$ to the semistability of the kernel of the…

There are several results in the literature concerning $p$-groups $G$ with a maximal elementary abelian normal subgroup of rank $k$ due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators…

In the conventional formulation of N=1 supersymmetry, a vector multiplet is supposed to be in the adjoint representation of a given gauge group. We present a new formulation with a vector multiplet in the non-adjoint representation of SO(N)…

This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup,…

The {\em abeliant} is a polynomial rule for producing an $n$ by $n$ matrix with entries in a given ring from an $n$ by $n$ by $n+2$ array of elements of that ring. The theory of abeliants, first introduced in an earlier paper of the author,…

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to…

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the…

We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2:…

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections \mu; z_1,...,z_n, where \mu \in…

Artin's Conjecture on Primitive Roots states that a non-square nonunit integer $a$ is a primitive root modulo $p$ for the positive proportion of $p$. This conjecture remains open, but on average, there are many results due to P. J.…

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the…

The paper is primarily concerned with the asymptotic behavior as $N\to\infty$ of averages of nonconventional arrays having the form $N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is…

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite ring $R$…

A commutative diagram that connects the basic objects of commutative algebra with the main objects of commutative analysis is constructed. Namely, with the help of five types of canonical embeddings we constructed a diagram between two sets…

Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the "geometric subring" $A$ of a polynomial ring $R[T]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that…

Let $R$ be a commutative Noetherian ring and $D$ be a discrete Hodge algebra over $R$ of dimension $d>\text{dim}(R)$. Then we show that (i) the top Euler class group $E^d(D)$ of $D$ is trivial. (ii) if $d>\text{dim}(R)+1$, then $(d-1)$-st…