Self-similar cauchy problems and generalized Mittag-Leffler functions

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

A remark on the distribution of products of independent normal random variables

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

Convergence trends in the “economy – education – digitalization – national security” chain

Purpose. To identify the current level and trends of convergence to justify the directions of adjustment of approaches to the management of the national economy. Methodology. The methodological basis of the study is economic and mathematical modelling using Barro-regression and variational analysis. Integral indicators for the characteristics of the components of the studied chain are defined as the arithmetic mean of partial indicators of economic development (24 indicators), educational development (28 indicators), digitalization (12indicators) and national security (53 indicators), normalized by the method of natural normalization. To assess the pairwise, triple and complex convergent relationships in the studied chain, a multiplicative convolution of the corresponding integral indicators characterizing a pair, triple or four of the studied concepts, was performed. The sample consisted of 11 countries from Central and Eastern Europe (Croatia, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovakia, Slovenia, Ukraine). The research period includes 19992020. Findings. The existence of dynamic convergent links in the national security digitalization, education national security digitalization chains is confirmed, which indicates the need for further interstate integration of regulatory practices in the field of digitalization impact on the national security (including digital education effects). The links in the economy education, economy national security, education national security, economy education national security chains have a fairly high static level of convergence, which indicates the need to level the differences in national practices of regulation of these directions. At the same time, current trends in the digitalization of education and the digitalization of the economy remain quite diversified, which determines the need to apply specific national government practices in this area. Originality. Methodological principles of integrated assessment of convergent relationships in the economy education national security digitalization chain differ from the existing ones by using integrated indicators of characteristics of single, pair, triple and complex relationships within the studied chain to determine the levels of their - and -convergence. This allowed identifying the presence of the achieved level of convergence and dynamic convergent trends that arise in the process of economic and educational transformations in the context of overcoming security challenges in the national economy in the context of digitalization. Practical value. The achieved significant level of convergence of the economy, education and digitalization of the studied countries has been revealed, as well as stable convergent links of integrated development of their economy, education and national security have been formed. The results obtained can be used as a scientific substantiation of adjustment of directions of state regulation of economy and education in the conditions of digitalization and in the context of overcoming security challenges.

Boolean cumulants and subordination in free probability

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

Regularity Properties of Free Multiplicative Convolution on the Positive Line

Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}_{+}=[0,\infty )$, we prove that their free multiplicative convolution $\mu \boxtimes \nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{-1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty )$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu \boxtimes \nu$ is another Jacobi measure whose density has square root decay at the edges of its support.