On the conjugacy classes in the orthogonal and symplectic groups over algebraically closed fields

Let F$\mathbb{F}$ be an algebraically closed field. Let V$\mathbb{V}$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form B   over F$\mathbb{F}$. Suppose the characteristic of F$\mathbb{F}$ is sufficiently large  , i.e. either zero or greater than the dimension of V$\mathbb{V}$. Let I(V,B)$I(\mathbb{V},\mathbb{B})$ denote the group of isometries. Using the Jacobson–Morozov lemma we give a new and simple proof of the fact that two elements in I(V,B)$I(\mathbb{V},\mathbb{B})$ are conjugate if and only if they have the same elementary divisors.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Finite traces and representations of the group of infinite matrices over a finite field

The article is devoted to the representation theory of locally compact infinite-dimensional group GLB$\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with q   elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate n=∞$n=\infty$ analogue of general linear groups GL(n,q)$\mathbb{GL}(n,q)$. It serves as an alternative to GL(∞,q)$\mathbb{GL}(\infty ,q)$, whose representation theory is poor.     

Characterization of full rank ACI-matrices over fields

An ACI-matrix over a field F$\mathbb{F}$ is a matrix whose entries are polynomials with coefficients on F$\mathbb{F}$, the degree of these polynomials is at most one in a number of indeterminates, and where no indeterminate appears in two different columns. In 2011 Huang and Zhan characterized the m×n$m\times n$ ACI-matrices such that all its completions have rank equal to min{m,n}$\min \{m,n\}$ whenever |F|?max{m,n+1}$|\mathbb{F}|?\max \{m,n+1\}$. We will give a characterization for arbitrary fields by introducing two classes of ACI-matrices: the maximal and the minimal full rank ACI-matrices.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

A 3-manifold group which is not four dimensional linear

We give examples of closed orientable graph 3-manifolds having a fundamental group which is not a subgroup of GL(4,F)$\mathit{GL}(4,\mathbb{F})$ for any field F$\mathbb{F}$. This answers a question in the Kirby problem list from 1977 which is credited to the late William Thurston.     

A new class of superregular matrices and MDP convolutional codes

This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F$\mathbb{F}$. Such construction works for any given choice of characteristic of the field F$\mathbb{F}$ and code parameters (n,k,δ)$(n,k,\delta )$ such that (n−k)|δ$(n-k)|\delta$. We also discuss the size of F$\mathbb{F}$ needed so that the proposed matrices are superregular.