On the intersection problem for linear sets in the projective line

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

Linear recurring sequence subgroups in finite fields

Given a finite field and a linear recurrence relation over it is possible to find, in a fairly “obvious” way, a finite extension of and a subgroup M of the multiplicative group of such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence. Here we investigate this phenomenon for second-order recurrences; the situation in which has prime order and the characteristic polynomial of the relation is irreducible over is described.     

On the Toeplitz pencil conjecture

The validity of a conjecture on a singular Toeplitz pencil over ?[x] stated by Schmale and Sharma in connection with an old problem in control theory is presented in this article. The aim of this work is to give a proof of this conjecture.     

Group Extensions and Hall Polynomials

In this paper, we give two proofs of a formula containing the numbers of automorphisms of an Abelian group, of its subgroups, and of its quotient groups. The first proof is based on the use of the theory of Hall polynomials, while the second one uses extension theory for Abelian groups.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 180–185.Original Russian Text Copyright © 2005 by G. V. Voskresenskaya.     

Unimodular polynomial matrices over finite fields

Journal of Algebraic Combinatorics - We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory, we give a proof of a result of Lieb, Jordan...     

A new proof of the i' 4 - terms in the middle '- theorem

In this note we give a short proof of a theorem of Bautista and Brenner [1,2] which once again demonstrates the usefulness of quadratic forms in representation theory. Let λ denote a finite dimensional algebra over an algebraically closed field k , and mod λ the category of finite dimensional λ-module.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

Finite Models of Some Substructural Logics

We give a proof of the finite model property (fmp) of some fragments of commutative and noncommutative linear logic: the Lambek calculus, BCI, BCK and their enrichments, MALL and Cyclic MALL. We essentially simplify the method used in [4] for proving fmp of BCI and the Lambek ca culus and in [5] for proving fmp of MALL. Our construction of finite models also differs from that used in Lafont [8] in his proof of fmp of MALL (we do not use cut elimination).     

On common linear copositive Lyapunov functions for pairs of stable positive linear systems

We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.     

B(■)的生成元

设H是复数域C上的可分Hilbert空间,B(■)是■上有界线性算子的全体.本文证明B(■)可以由三个投影生成,改进了C.Davis[2]的方法,使得第三个投影E3的取法更为灵活,证明更为简洁,获得了这样{E1,E2,E3}更多的族,并且给出当■是有限维情形时的证明。     

Doubly transitive permutation groups with abelian stabilizers

Summary We prove that any doubly transitive permutation group with abelian stabilizers is the group of linear functions over a suitable field. The result is not new: for finite groups it is well known, for infinite groups it follows from a more general theorem of W. Kerby and H. Wefelscheid on sharply doubly transitive groups in which the stabilizers have finite commutator subgroups. We give a direct and elementary proof.     

Artin Algebras of Finite Type and Finite Categories of Δ-Good Modules

We give an alternative proof to the fact that, if the square of the infinite radical of the module category of an Artin algebra is equal to zero, then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of Δ-good modules of a quasi-hereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type.     

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems, II

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. We give the proof of some result stated in [AT5], and in addition we prove uniqueness of the Riccati operator, provided its final datum is suitably regular. Accepted 14 October 1998     

A constructive proof of the Peter‐Weyl theorem

We present a new and constructive proof of the Peter‐Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*‐algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

多项式带余除法定理的一种新证明

利用线性空间的理论,在高观点下给出了多项式带余除法定理的一个新证明.它避免了使用构造法这种技巧性较高的数学方法,使学生比较容易轻松掌握.     

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.     

Estimates from above of certain double trigonometric sums

In this paper we consider double trigonometric sums. Expressions of this type appear in some problems of quantum chaos and number theory. We are interested in rotation numbers of bounded type. We prove a uniform linear bound on double trigonometric sums along the subsequence of denominators of the continued fraction. The proof uses elementary techniques and the analysis of cancellations in sums of certain oscillatory functions over rotations. We also include a proof of a result on discrepancy for rotations of bounded type and in the Appendix we give an elementary proof of a result by Hardy and Littlewood.     

McKay箭图

Using the connection between McKay quiver and Loewy matrix,and the properties of characteristic polynomial of Loewy matrix,we give a generalized way to determine the McKay quiver for a finite subgroup of a generalized linear group.