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1

On the existence of complex Hadamard submatrices of the Fourier matrices

Bond Bailey Madison, Glickfield R. Alexander, Herr John E.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

1--9

EN

We use a theorem of Lam and Leung to prove that a submatrix of a Fourier matrix cannot be Hadamard for particular cases when the dimension of the submatrix does not divide the dimension of the Fourier matrix. We also make some observations on the trace-spectrum relationship of dephased Hadamard matrices of low dimension.

2

A generalized Walsh system for arbitrary matrices

Harding Steven N., Picioroaga Gabriel

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

40--55

EN

In this paper we study in detail a variation of the orthonormal bases (ONB) of L2 [0,1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2),1128-1139] by means of representations of the Cuntz algebra ON on L2 [0,1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.

3

On evaluation of influence coefficients for edge and intermediate boundary elements in 3D problems involving strong field concentrations

Rybarska-Rusinek L.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 1

69--76

EN

The paper presents a tool for accurate evaluation of high field concentrations near singular lines, such as contours of cracks, notches and grains intersections, in 3D problems solved the BEM. Two types of boundary elements, accounting for singularities, are considered: (i) edge elements, which adjoin a singular line, and (ii) intermediate elements, which while not adjoining the line, are still under strong influence of the singularity. An efficient method to evaluate the influence coefficients and the field intensity factors is suggested for the both types of the elements. The method avoids time expensive numerical evaluation of singular and hypersingular integrals over the element surface by reduction to 1D integrals. The method being general, its details are explained by considering a representative examples for elasticity problems for a piecewise homogeneous medium with cracks, inclusions and pores. Numerical examples for plane elements illustrate the exposition. The method can be extended for curvilinear elements.

4

Applications of Hadamard matrices

Evangelaras H., Koukouvinos C., Seberry J.

Journal of Telecommunications and Information Technology

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2003

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nr 2

3-10

EN

We present a number of applications of Hadamard matrices to signal processing, optical multiplexing, error correction coding, and design and analysis of statistics.