On q-generalized hyperharmonic numbers with two parameters

Bu çalışma, Fibonacci, Pascal, Stirling ve Bell sayıları gibi özel sayı dizilerini tanıtmak, bu sayı dizilerinin elemanları kullanılarak oluşturulan matrisleri tanımlamak ve bu matrisler arasındaki bazı kombinasyonel özdeşlikleri araştırmak için yapılmıştır.

1 - Matrices

This monograph presents innovative methods for constructing two-dimensional non-equidistant antenna arrays (AAs), particularly designed for radio telescopes operating in the 8–80 MHz range. The work introduces direct and straightforward techniques, emphasizing configurations based on Latin squares and their triangular matrices. These methods ensure full spatial frequency coverage with high rarefaction and minimal lateral radiation. Key contributions include the first use of a Latin square matrix in AA construction, the pioneering combination of circular difference sets and Latin square matrices for planar sparse arrays, and the integration of Latin square and lower triangular matrices in planar sparse array design. The proposed matrix method is more efficient and direct compared to traditional iterative and deletion-based optimization approaches. This book is intended for scientists and specialists involved in developing remote sensing methods and technologies using radiation from existing ground and space radio engineering systems. It is also valuable for students and educators in radiophysics, offering a comprehensive approach to AA design that could advance both academic study and practical applications in radio astronomy.

This paper describes the implementation of the Pan-Reif iterative algorithm for matrix inversion on the re-usable systolic processor array for matrix multiplication. The calculation of initial approximations, matrix norms and matrix transposition are implemented on the same processor array. Further the inversion of several special matrices is also discussed. All phases of the iterative algorithm are integrated so that they can be executed without any communication with a host system. The parallel algorithms and the systolic designs were implemented using Occam.

In this paper we use the Lanczos process for preconditioning discrete ill-posed problems. We show that by few steps of this process one can obtain a well qualified and efficient preconditioner. This is a general method in the sense that it is not limited only to special structured matrices and the matrix–vector multiplications can be carried out in O(n) operations. Also even in problems with structured matrices this preconditioner performs more efficiently than the circulant and Kronecker product approximate preconditioners.

The fastest long-period random number generators currently available are based on linear recurrences modulo 2. So far, software that provides multiple disjoint streams and substreams has not been available for these generators because of the lack of efficient jump-ahead facilities. In principle, it suffices to multiply the state (a k-bit vector) by an appropriate k × k binary matrix to find the new state far ahead in the sequence. However, when k is large (e.g., for a generator such as the popular Mersenne twister, for which k = 19,937), this matrix-vector multiplication is slow, and a large amount of memory is required to store the k × k matrix. In this paper, we provide a faster algorithm to jump ahead by a large number of steps in a linear recurrence modulo 2. The method uses much less than the k2 bits of memory required by the matrix method. It is based on polynomial calculus modulo the characteristic polynomial of the recurrence, and uses a sliding window algorithm for the multiplication.

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

Global stability of a multipatch disease epidemics model

Matrix and vector multiplications are widely used in signal processing in operations such as FIR and IIR filtering, feature extraction and classification. Frequently, signal processing must be done in real time requiring the use of special purpose VLSI hardware. Regular structures such as systolic arrays are well suited for matrix and vector operations and are also amenable to VLSI implementation. This paper describes efficient systolic arrays for matrix and vector multiplication, at both word and bit levels, and outlines three applications relevant for signal processing: a convolver, an IIR filter and a linear classifier.

In this paper, we give some congruences related to $q-$generalized Catalan numbers, $q-$harmonic numbers and alternating $q-$harmonic numbers, using combinatorial identities and some known congruences.

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established.Â

In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which can be inferred straightforwardly from our results.

Fast boundary integration methods are implemented to increase the efficiency of the calculation of the magnetostatic interaction of magnetic particles. We test the treecode algorithm and a hierarchical matrices method to improve the original used matrix-vector multiplication with a fully populated matrix. We compare the CPU-time used for the setup phase and for the matrix-vector multiplication, as well as the required storage.

An accelerated inversion algorithm using the resolvent matrix method

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell