For every n divisible by 4, we construct a square matrix H of size n, with coeecients 1, such that H H t nI mod 32. This solves the 32-modular version of the classical Hadamard conjecture. We also determine the set of lengths of 16-modular Golay sequences. 0. Introduction Hadamard matrices can be constructed from various binary sequences, either from Golay complementary sequences, or Williamson...

Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1 1− ax − x2 ∗ 1 1− bx − x2 = 1− x2 1− abx − (2 + a2 + b2)x2 − abx3 + x4 , where ∗ denotes the Hadamard product. In a similar way, by considering tilings of a 2× n rectangle with 1× 1 and 1× 2 bricks in the top row, and 1× 1 and 1× n bricks in the bottom row, we find...

This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets leads to several results...

The problems of systems identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, Haar and many other functions. In the present paper, several operational matrices for integration and differentiation are studied. we introduce the Kronecker convol...

By means of Ruscheweyh derivative operator , we introduced and investigated two new subclasses of p-valent analytic functions.The various results obtained here for each of these function class include coefficient bounds and distortion inequalities, associated inclusion relations for the (n, θ)-neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which ar...

This is the first in a series of two papers in which we construct a convolution product for associated homogeneous distributions (AHDs) with support in R. In this paper, we show that if f and g are AHDs with degrees of homogeneity a−1 and b−1, the convolution fa∗gb exists as an AHD, provided the resulting degree of homogeneity a + b − 1 is not a natural number. Under this restriction, it is fou...

We investigate the problem of learning continuous vector representations knowledge graphs for predicting missing links. Recent results suggest that using a Hermitian inner product on complex-valued embeddings or convolutions real-valued can be effective means bring these insights together and propose ConEx—a multiplicative composition 2D convolution with embeddings. ConEx utilizes Hadamard to c...

The Hadamard and SJT product of matrices are two types of special matrix product. The latter was ®rst de®ned by Chen. In this study, they are applied to the di€erential quadrature (DQ) solution of geometrically nonlinear bending of isotropic and orthotropic rectangular plates. By using the Hadamard product, the nonlinear formulations are greatly simpli®ed, while the SJT product approach minimiz...

We present a preliminary study of Schur norms ?M?S=max{?M?C?:?C?=1}, where M is matrix whose entries are ±1, and ? denotes the entrywise (i.e. or Hadamard) product matrices. recover result Johnsen that says that, if such n×n, then its norm bounded by n, equality holds only it Hadamard matrix. develop numerically efficient method computing norms, as an application our results we several almost m...