A canonical form for a system of linear difference equations

Summary A step-by-step procedure is presented for findiding non-singular transformations which will reduce an ordinary linear vector difference equation of type(1) to a canonical form, which is particularly convenient if one wishes to write out the asymptotic solutions. To Enrico Bompiani on his scientific Jubilee Part of the work on this paper was done while the author was supported by a grant No. G 14879 from the National Science Foundation.     

Factorization of involutions in characteristic two orthogonal groups: An application of the Jordan form to group theory

This paper uses the theory of the Jordan canonical form for a matrix and the theory of orthogonal sums of isometries in metric vector spaces (quadratic spaces) in order to prove a theorem on the factorization of involutions in the orthogonal groups of metric vector spaces over fields of characteristic two. Using this theorem, a classification scheme for such involutions is devised. This scheme is similar to the scheme for involutions when the field is of characteristic not equal of two.     

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

线性方程组理论的一个应用

利用线性方程组的理论，推导出范德蒙矩阵的求逆公式．     

Krylov-Bogolyubov substitution in the perturbation theory of linear operators

Ukrainian Mathematical Journal -     

An estimate for a linear form in algebraic numbers

M. V. Lomonosov Moscow State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 1, pp. 82–89, January–March, 1991.     

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).     

An inequality for a linear form in the logarithms of algebraic numbers

Let ln ₁, ..., ln _m–1 be the logarithms of fixed algebraic numbers which are linearly independent over the field of rational numbers, b₁, ..., b_m–1 rational integers, > 0. A bound from below is deduced for the height of the algebraic number _m under the condition that ¦b₁ ln ₁+...+_bm–1ln _m– ¦ < exp {–H},H=max ¦ b _k ¦ >0.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 681–689, June, 1969.