A generalization of the Maillet determinant and the Demyanenko matrix

A generalization of the Maillet determinant and the Demyanenko matrix is given.     

Maillet determinants for real abelian number fields and its applications

In this paper, we define an analogue of the Maillet determinant in terms of Clausen function in [6], and we give a class number formula for a real abelian number field. We also give two applications of this class number formula.     

On a problem related to the Vandermonde determinant

This short note describes new properties of the elementary symmetric polynomials, and reveals that the properties give an answer to the conjecture raised by El-Mikkawy in [M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759-769].     

On the determinant of certain strictly dissipative matrices

This paper studies bounds for the modulus and the argument of the determinant of certain strictly dissipative matrices.     

On the determinant of an asymmetric hyperbolic region

Summary A main problem in the geometry of numbers is the evaluation of the so-called determinant of various regions. The author derives a new estimate for the determinant of a certain two-dimensional region bounded by two hyperbolas and applies his result to a problem in the theory of automorphic star bodies. dedicated to Prof.B. Segre.     

On a Toeplitz determinant identity of Borodin and Okounkov

In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants.Supported by National Science Foundation grant DMS-9970879.Supported by National Science Foundation grant DMS-9732687.     

On a superadditivity property of Gram’s determinant

Summary Some superadditivity properties are given for Gram determinants involving inner products on a linear space.     

关于Smarandache素数列及其它的行列式

对任意正整数n≥2,Smarandache下素数列{pp(n))定义为小于或等于n的最大素数;而Smarandache上素数列{Pp(n))表示大于或等于n的最小素数.本文的主要目的是利用初等方法研究Smarandache素数列的性质,并得到由Smarandache素数列组成的行列式的一些性质.     

On a generalization of the Demǰanenko determinant

The Dem?anenko matrices are generalized with the averaged Bernoulli polynomials, and their determinants are computed by the Dedekind-Frobenius formula. This enables us to interpret geometrically the special values at negative integers of Dedekind zeta function of maximal real subfields as well as of imaginary subfields of cyclotomic fields. We utilize the structural properties of the group of reduced residue classes modm and those of (averaged) Bernoulli polynomials, thus appealing to the predominance of Galois theory over the fields themselves, which makes it possible to present very lucid reasoning of the whole picture.     

On the determinant of the adjacency matrix for a planar sublattice

It is proved that the determinant of the adjacency matrix for a finite subgraph G of Z × Z is 1, 0, or ?1, provided that G has no “holes.”     

On maximum entropy interpolants and maximum determinant completions of associated Pick matrices

In this paper we shall show that the value of the maximum -entropy solution of an interpolation problem of the Nevanlinna-Pick type at the point maximizes the determinant of the solution of an associated matrix completion problem. This serves to show that the solutions of two distinct extremal problems coincide.Harry Dym wishes to express his thanks to Renee' and Jay Weiss for endowing the chair which supports his research.     

Permanent and determinant

The n×n permanent is not a projection of the m×m determinant if m ⩽ √2n− 6√n.