The early historical roots of Lee-Yang theorem

A deep and detailed historiographical analysis of a particular case study concerning the so-called Lee-Yang theorem of theoretical statistical mechanics of phase transitions, has emphasized what real historical roots underlie such a case study. To be precise, it turned out that some well-determined aspects of entire function theory have been at the primeval origins of this important formal result of statistical physics.Comment: History of Physics case study. arXiv admin note: substantial text overlap with arXiv:1106.4348, arXiv:math/0601653, arXiv:0809.3087, arXiv:1311.0596 by other author

The Tur\'an and Laguerre inequalities for quasi-polynomial-like functions

This paper deals with both the higher order Tur\'an inequalities and the Laguerre inequalities for quasi-polynomial-like functions -- that are expressions of the form $f(n)=c_l(n)n^l+\cdots+c_d(n)n^d+o(n^d)$, where $d,l\in\mathbb{N}$ and $d\leqslant l$. A natural example of such a function is the $A$-partition function $p_{A}(n)$, which enumerates the number of partitions of $n$ with parts in the fixed finite multiset $A=\{a_1,a_2,\ldots,a_k\}$ of positive integers. For an arbitrary positive integer $d$, we present efficient criteria for both the order $d$ Tur\'an inequality and the $d$th Laguarre inequality for quasi-polynomial-like functions. In particular, we apply these results to deduce non-trivial analogues for $p_A(n)$.Comment: 18 pages, 6 figure

Some Numerical Significance of the Riemann Zeta Function

In this paper, the Riemann analytic continuation formula (RACF) is derived from Euler’s quadratic equation. A nonlinear function and a polynomial function that were required in the derivation were also obtained. The connections between the roots of Euler’s quadratic equation and the Riemann Zeta function (RZF) are also presented in this paper. The method of partial summation was applied to the series that was obtained from the transformation of Euler’s quadratic equation (EQE). This led to the derivation of the RACF. A general equation for the generation of the zeros of the analytic continuation formula of the Riemann Zeta equation via a polynomial approach was also derived and thus presented in this work. An expression, which was based on a polynomial function and the products of prime numbers, was also obtained. The obtained function thus afforded us an alternative approach to defining the analytic continuation formula of the Riemann Zeta equation (ACFR). With the new representation, the Riemann Zeta function was shown to be a type of function. We were able to show that the solutions of the RACF are connected to some algebraic functions, and these algebraic functions were shown to be connected to the polynomial and the nonlinear functions. The tables and graphs of the numerical values of the polynomial and the nonlinear function were computed for a generating parameter, k, and shown to be some types of the solutions of some algebraic functions. In conclusion, the RZF was redefined as the product of a derived function, R(tn,s), and it was shown to be dependent on the obtained polynomial function

Fractional Calculus Operators and the Mittag-Leffler Function

This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others