HANKEL DETERMINANTS OF FACTORIAL FRACTIONS

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].

Infinite-Volume Limit of Stochastic s-d System with Single-Component Impurity and s-Electron Spins

A Hamiltonian [Formula: see text], with locally smeared Ising-type s-d exchange between s-electrons and magnetic impurities, in a dilute magnetic alloy, is investigated. The Feynman-Kac theorem, Laplace expansion and Bogolyubov inequality are applied to obtain a lower and upper bound (lb and ub) on the system’s free energy per conducting electron [Formula: see text]. The two bounds differ, in the infinite-volume limit by a term [Formula: see text], linear in impurity concentration: lb[Formula: see text], ub[Formula: see text], [Formula: see text] denoting the Hamiltonian of the approximating mean-field s-d system. [Formula: see text] represents randomly positioned impurities interacting with a mean field implemented by the gas of conduction s-electrons, the latter interacting with the field of barriers and wells (according to the s-electron’s spin orientation) localized at the impurity sites. The inequality [Formula: see text] demonstrates increasing accuracy of the mean-field [Formula: see text]-theory, with decreasing impurity concentration.

"Efficiency Comparison of Algorithms for Finding Determinants Via Permutations "

"Determiners have been used extensively in a selection of applications throughout history. It also biased many areas of mathematics such as linear algebra. There are algorithms commonly used for computing a matrix determinant such as: Laplace expansion, LDU decomposition, Cofactor algorithm, and permutation algorithms. The determinants of a quadratic matrix can be found using a diversity of these methods, including the well-known methods of the Leibniz formula and the Laplace expansion and permutation algorithms that computes the determinant of any n×n matrix in O(n!). In this paper, we first discuss three algorithms for finding determinants using permutations. Then we make out the algorithms in pseudo code and finally, we analyze the complexity and nature of the algorithms and compare them with each other. We present permutations algorithms and then analyze and compare them in terms of runtime, acceleration and competence, as the presented algorithms presented different results.

MULTINOMIAL VANDERMONDE CONVOLUTION VIA PERMANENT

We provide a generalised Laplace expansion for the permanent function and, as a consequence, we re-prove a multinomial Vandermonde convolution. Some combinatorial identities are derived by applying special matrices to the expansion.

Laplace, Pierre-Simon (1749–1827)

In company with Lagrange and Legendre, Laplace was one of the three foremost mathematicians during the half-century of French pre-eminence in science, from the 1770s into the 1820s. Although he invented or developed many mathematical techniques for solving various classes of problems, most notably generating functions, the Laplace transform, the Laplace expansion, the variation of constants, and the generalized gravitational function, he was more interested in what he could do with mathematics than with mathematics itself. His main treatises are Mécanique céleste (Celestial Mechanics) and Théorie analytique des probabilités (Analytical Theory of Probability). In addition he published Exposition du système du monde (Exposition of the System of the World) and Essai philosophique sur les probabilités (Philosophical Essay on Probability). Not exactly popularizations, the latter pair were verbal explanations of the two bodies of subject matter intended for a well-educated public. Both have gone through many editions. Though unrelated in content, the two main branches of Laplace’s work were related philosophically, celestial mechanics being concerned with the structure of the world, and probability with what we can know about it. In both respects Laplace’s motivation was to vindicate the Newtonian system of the world. In the later half of his life he developed an interest in physics and may be considered the impresario if not quite the founder of mathematical physics. He more than any other was responsible for the design of the metric system of weights and measures, a worldwide legacy of the French Revolution. A favourite of Napoleon, Laplace was a principal statesman of science during the Napoleonic period and afterwards.

The (Not So?) Basics

This chapter discusses matrices. Matrices appear in many instances across physics, and it is in this chapter that the background necessary for understanding how to use them in calculations is provided. Although matrices can be a little daunting upon first exposure, they are very handy for a lot of classical physics. This chapter reviews the basics of matrices and their operations. It discusses square matrices, adjoint matrices, cofactor matrices and skew-symmetric matrices. The concepts of matrix multiplication, transpose, inverse, diagonal, identity, Pfaffian and determinant are examined. The chapter also discusses the terms Hermitian, symmetric and antisymmetric, as well as the Levi-Civita symbol and Laplace expansion.

On some properties of generalized Fibonacci and Lucas polynomials

In this paper we investigate some properties of generalized Fibonacci and Lucas polynomials. We give some new identities using matrices and Laplace expansion for the generalized Fibonacci and Lucas polynomials. Also, we introduce new families of tridiagonal matrices whose successive determinants generate any subsequence of these polynomials.

On characteristic and permanent polynomials of a matrix

There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.