A Strong Szegö-Widom Limit Theorem for operators with almost periodic diagonal

The classical Strong Szegö-Widom Limit Theorem describes the asymptotic behavior of the determinants of the finite sections PnT(a)Pn of Toeplitz operators, i.e., of operators which have constant entries along each diagonal. We generalize these results to operators which have almost periodic sequences as their diagonals.     

The use of jacobi's lemma in unimodularity theory

The paper applies Jacobi's fundamental result on minors of the adjoint matrix to obtain properties on determinants of unimodular matrices.     

Determinant Forms for Extended Heat Functions and Expansions of Solutions of Associated Cauchy Problems

We briefly review series solutions of differential equations problems of the second order that lead to coefficients expressed in terms of determinants. Derivative type formulas involving a generating function with several parameters are developed for these determinant coefficients in first order problems. These permit constructing determinant forms for the heat polynomials and their Appell transforms. Hadamard's theorem for bounding determinants and conical regions are used to deduce simplified versions of expansion theorems involving these polynomials and associated Appell transforms. Extended versions of the heat equation are also considered.     

An explicit formula for the determinant of a skew-symmetric pentadiagonal Toeplitz matrix

We give the explicit determinant of the general skew-symmetric pentadiagonal Toeplitz matrix in terms of the Tchebechev polynomials.     

Compound matrices: properties,numerical issues and analytical computations

This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.      

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

Determinants of Regular Singular Sturm - Liouville Operators

We consider a regular singular Sturm-Liouville operator on the line segment (0,1]. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known (cf. Theorem 1.1 below) that the ζ-function of this operator has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to [RS] the ζ - regularized determinant of L is defined by In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation L_y = 0 generalizing earlier work of S. Levit and U. Smilansky [LS], T. Dreyfus and H. Dym [DD], and D. Burghelea, L. Friedlander and T. Kappeler [BFK1, BFK2). More precisely we prove the formula Here ? ψ is a certain fundamental system of solutions for the homogeneous equation L_y = 0, W(? ψ), denotes their Wronski determinant, and v₀, v₁ are numbers related to the characteristic roots of the regular singular points 0, 1.