Column Invariants Over Cohen–Macaulay Local Rings

Abstract

We define some numerical invariants over Cohen–Macaulay local rings. These invariants are related to columns of the presenting matrices of maximal Cohen–Macaulay modules and syzygy modules without free summands. We study the relationship between these invariants, and the invariant col(A).

     

Three-dimensional zeropotent algebras over an algebraically closed field of characteristic two

Abstract

We previously classified three-dimensional zeropotent algebras over an algebraically closed field of any characteristic except for two. The exceptional case of characteristic two is special because some of the previous transformation matrices to verify isomorphism are unavailable. In this paper, we give new transformation matrices peculiar to characteristic two and then achieve classification in the exceptional case. We thus accomplish a classification of three-dimensional zeropotent algebras over an algebraically closed field of any characteristic.     

A Combinatorial Property for Semigroups of Matrices

Abstract. In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property . We consider an application of this result to the Burnside problem for groups.     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.     

The distribution of the poles of the best approximating rational functions and the analytical properties of the approximated function

The following theorem was proved in our paper in Math. USSR-Sb.9 (1969): Iff(z)∈H ₂ and the poles of the rational functions of best approximation tend to infinity sufficiently quickly, thenf(z) is an entire function. In the present article we weaken the restrictions on the distribution of poles by assuming only that these poles have no finite limit point (Theorem 1). Some generalizations of this result are also given.     

Elliptic grids, rational functions, and the Padé interpolation

We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of the Padé interpolation with prescribed poles and zeros on the elliptic grids. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21.     

Rank formulae from the perspective of orthogonal projectors

For matrices F and G having the same number of rows and the orthogonal projectors P?=?FF ^? and Q?=?GG ^?, with F ^? and G ^? denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F?:?G) (which coincide with the ranks of PQ and P?+?Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.     

ON POLE ASSIGNABILITY AND FEEDEBACK CYCLISATION FOR SYSTEMS OVER RINGS OF FINITE DIMENSION

ABSTRACT

It is shown that the rank of the projective added to the state space of a system in dynamic feedback can be decreased by assigning some poles to the original system first and by using these to obtain a new system with state space of lower rank on which dynamic feedback can be applied.     

Eigenvalues and switching algorithms for Quasi-Newton updates

Two switching algorithms QNSWl and QNSW2 are proposed in this paper. These algorithms are developed based on the eigenvalues of matrices which are inertial to the symmetric rank-one (SR1) updates and the BFGS updates. First, theoretical results on the eigenvalues and condition numbers of these matrices are presented. Second, switch-ing mechanisms are then developed based on theoretical results obtained so that each proposed algorithm has the capability of applying appropriate updating formulae at each iterative point during the whole minimization process. Third, the performance of

each of the proposed algorithms is evaluated over a wide range of test problems with variable dimensions. These results are then compared to the results obtained by some well-known minimization packages. Comparative results show that among the tested methods, the QNSW2 algorithm has the best overall performance for the problems examined. In some cases, the number of iterations and the number function/gradient calls required by certain existing methods are more than a four-fold increase over that required by the proposed switching algorithms     

Invertible and nilpotent matrices over antirings

In this paper, we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GL_n(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log₂n⌉ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.     

SOME PROPERTIES OF A CLASS OF INTERCHANGE GRAPHS

§１ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＧｂｅａｃｏｎｎｅｃｔｅｄｇｒａｐｈｗｉｔｈｖｅｒｔｅｘ－ｓｅｔＶ（Ｇ）ａｎｄｅｄｇｅ－ｓｅｔＥ（Ｇ）．Ｄｅｎｏｔｅｂｙｅ＝（ｘ，ｙ）ｔｈｅｅｄｇｅｊｏｉｎｉｎｇｔｈｅｖｅｒｔｉｃｅｓｘａｎｄｙｏｆＧ．Ａｍ－ｃｌｉｑ...     

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R ₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

Perimeter preserver of matrices over semifields

For a rank-1 matrix A = ab ^t, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over semifields. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices over semifields if and only if it has the form T(A) = U AV, or T(A) = U A ^t V with some invertible matrices U and V. This work was supported by the research grant of the Cheju National University in 2006.     

On the Norm Equation Over Function Fields

If K is an algebraic function field of one variable over analgebraically closed field k and F is a finite extension ofK, then any element a of K can be written as a norm of someb in F by Tsen's theorem. All zeros and poles of a lead to zerosand poles of b, but in general additional zeros and poles occur.The paper shows how this number of additional zeros and polesof b can be restricted in terms of the genus of K, respectivelyF. If k is the field of all complex numbers, then we use Abel'stheorem concerning the existence of meromorphic functions ona compact Riemann surface. From this, the general case of characteristic0 can be derived by means of principles from model theory, sincethe theory of algebraically closed fields is model-complete.Some of these results also carry over to the case of characteristicp>0 using standard arguments from valuation theory.     

A Class of Bidimensional FMRA Wavelet Frames

This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

Wavelet Bases on the Sphere Obtained by Radial Projection

We present a method for constructing wavelet bases on the unit sphere S² of R³, using the radial projection and an inner product associated to a convex polyhedron having the origin inside. The main advantage of this method is the avoidance of singularities and distortions around poles, which occur in other approaches. Also, we can obtain spherical wavelets with small support, a fact which is crucial in working with large amounts of data, since the algorithms deal with sparse matrices.