Abelian gradings on upper-triangular matrices

Let G be an arbitrary finite abelian group. We describe all possible G-gradings on an upper-triangular matrix algebra over an algebraically closed field of characteristic zero.     

Browder spectra of upper-triangular operator matrices

Let be a 2×2 upper triangular operator matrix acting on the Hilbert space H⊕K. In this paper, for given operators A and B, we prove that

     

Zeta functions of Lie rings of upper-triangular matrices

We prove a two-part theorem on local ideal zeta functions ofLie rings of upper-triangular matrices. First, we prove thatthese local zeta functions display a strong uniformity. Secondly,we prove that these zeta functions satisfy a local functionalequation. Some explicit examples of these zeta functions arealso presented. Finally, we consider certain quotients of theseLie rings, showing that the strong uniformity continues to hold,and that under certain circumstances the functional equationdoes too.     

The growth of varieties generated by upper-triangular matrices algebras

It is shown that if the characteristic of the basic field does not equal two, then there exists no variety of associative algebras whose growth is intermediate between polynomial and exponential. Let UT _s be the algebra of upper triangular matrices of order s over an arbitrary field. V. M. Petrogradsky proved that the exponent of any subvariety of var(UT _s ) exists and is an integer number. In his paper the growth estimates for such varieties are reinforced.     

Mixtures of order matrices and generalized order matrices

This paper deals with matrix representations of linear orders, mixtures of order matrices and the non-integral solutions of the linear systems defining them.     

Neural network closures for nonlinear model order reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.     

On a series of Hadamard matrices of order 2 t and the maximal excess of Hadamard matrices of order 22t

In this paper we give a new series of Hadamard matrices of order 2 ^t . When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 2^2t is 2^3t , which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n ² with the maximal excess 8n ³, then there exist more than one inequivalent Hadamard matrices of order 2^2t n ² with the maximal excess 2^3t n ³ for anyt 2.     

Reciprocal matrices of order 4

In this paper we show that given an arbitrary reciprocal matrix, the vector whose components are the geometric mean of the rows and the vector whose components are the geometric mean of the columns may not be eigenvectors of the matrix. However, for matrices of order 4, if one of the vectors is an eigenvector, then so is the other.     

Classification of categories with matrices of coefficient 2 and order n

In this paper, we present the categories associated to the square matrix with coefficients 2 of order 3. The family of categories associated to these matrices has five isomorphism classes. In the case of a matrix of order higher than 3, we only demonstrate upper and lower bounds for the number of associated categories.     

Closedness of ranges of upper-triangular operators

In this paper, the closedness of ranges and Kato non-singularity of upper triangular operator matrices are investigated.     

Hadamard matrices of order 764 exist

Two Hadamard matrices of order 764 of Goethals-Seidel type are constructed. The author was supported by an NSERC Discovery Grant.     

Maximal difference matrices of order q

Recently, A. B. Evans proved the following Theorem: There is a maximal set of (p ? 3)/2 [(resp. (p ? 1)/2] mutually orthogonal Latin squares of order p if p is a prime p ≡ 3 mod 4 (resp. p ≡ 1 mod 4). In this article I will give a slightly different proof using more geometric arguments and results of Rédei. Further, I discuss possible generalizations of Evans' Theorem to the case of Latin squares whose order is a prime power. © 1993 John Wiley & Sons, Inc.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

Skew-Hadamard matrices of order 2(q + 1)

Szekeres has established the existence of a skew-Hadamard matrix of order 2(q + 1) in the case q ≡ 5 (mod 8), a prime power. His method utilized complementary difference sets in the elementary abelian group of order q. The main result of this paper is to show that, for the same q, there exist skew-Hadamard matrices of order 2(q + 1) that are of the Goethals-Seidel type. This is achieved by using a cyclic relative difference set with parameters $\text{(q + 1, 4, q,}\text{1}\text{4}\text{(q ? 1))}$.