Bell Polynomials and Nonlinear Inverse Relations

By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).

Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

Minimization of Binary Decision Diagrams for Systems of Completely Defined Boolean Functions using Algebraic Representations of Cofactors

In design systems for digital VLSI (very large integrated circuits), the BDD is used for VLSI verification, as well as for technologically independent optimization, performed as the first stage in the synthesis of logic circuits in various technological bases. BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. Having constructed BDD representation for systems of Boolean functions, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction or conjunction of other cofactors of this BDD level. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from library logic gates.

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].

The Parseval identity for q-Sturm–Liouville problems with transmission conditions

In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions.

ON THE EXPANSION FORMULA FOR A SINGULAR STURM-LIOUVILLE OPERATOR

In this study, on the semi-axis, Sturm - Liouville problem under boundary condition depending on spectral parameter is considered. In what follows scattering data is defined and its properties are given for the problem. The kernel of resolvent operator which is Green function is constructed. Using Titchmarsh method, expansion is obtained according to eigenfunctions and expansion formula is expressed with the scattering data.