Additive Convolutions Research Articles

Some Limiting Measures Related to Free and Boolean Convolutions

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

Some Limiting Laws in Non-Commutative Probability

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory.

A unified approach to infinitesimal freeness with amalgamation

In this paper, we consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of [Formula: see text] upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness.

MULTI-CRITERIA EVALUATION OF THE MULTIFACTOR STOCHASTIC SYSTEMS EFFECTIVENESS

Subject of the research in the article is the evaluation of complex multifactor stochastic systems functioning effectiveness according to a variety of correlated criteria. The problem actuality is determined by the fact that an independent evaluation of system effectiveness for each of the mutually correlated criteria for the system under consideration is not informative. In well-known works in the direction of multiple correlation, a relatively simple problem of estimating the correlation between one resulting parameter and a set of influencing factors is considered, which is not enough for the analysis and management of multicriteria systems. In addition, the known results do not take into account possible significant differences in influencing factors mutual correlation values. Purpose of Work is to develop a methodology for a comprehensive assessment of system effectiveness according to a variety of interrelated criteria. Tasks to be Solved: splitting the set of system parameters into two subsets (parameters determining the effectiveness of system and parameters affecting it), forming additive convolutions of parameters included in subsets, developing a methodology for calculating the multiple correlation coefficient between the components of the selected subsets, developing a method for differentiating a scalar function from a vector argument by this argument. Applied Methods: nonlinear programming, multidimensional correlation analysis, method of differentiation of scalar functions by vector argument. These methods are used for forming and calculating a multiple correlation coefficient between the set of system effectiveness complex criterion components values and its control parameters set values. Results Obtained: proposed methodology provides the possibility of solving the problems of system management, taking into account the revealed relationship between the multi-criteria evaluation of system effectiveness and values of its controlled parameters. At the same time, an important advantage of the obtained result lies in the possibility that arises when using it to take into account the joint (group) influence of control variables on the complex criterion of system efficiency. The developed technology of scalar functions differentiation by vector argument has great practical importance which expands the arsenal of computational mathematics.

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

Superconvergence and regularity of densities in free probability

The phenomenon of superconvergence, first observed in the central limit theorem of free probability, was subsequently extended to arbitrary limit laws for free additive convolution. We show that the same phenomenon occurs for the multiplicative versions of free convolution on the positive line and on the unit circle. We also show that a certain Hölder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free (additive and multiplicative) convolutions with arbitrary freely infinitely divisible distributions.

Asymptotic identities for additive convolutions of sums of divisors

AbstractIn a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$ , and proved an asymptotic formula for it when a and b are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real a and b. Ramanujan’s conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term.In this paper, we give a new proof of Ramanujan’s conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author’s thesis and which served as our initial motivation in studying this sum.

Additive representation functions and discrete convolutions

For a set A of non-negative integers, let RA(n) denote the number of solutions to the equation \(n = a + a'\) with a, \(a' \in A\). Denote by \(\chi_{A}(n)\) the characteristic function of A. Let \(b_{n} > 0\) be a sequence satisfying \({{\rm limsup}_{n \to \infty}b_{n} < 1} \). In this paper, we prove some Erdős–Fuchs-type theorems about the error terms appearing in the approximation formulæ for \(R_{A}(n)=\sum_{k = 0}^{n}\chi_{A}(k)\chi_{A}(n - k)\) and \(\sum_{n=0}^{N} R_{A}(n)\) having principal terms \(\sum_{k=0}^{n}b_{k}b_{n-k}\) and \(\sum_{n=0}^{N}\sum_{k=0}^{n}b_{k}b_{n-k}\), respectively.

Finite free convolutions of polynomials

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

Relations between convolutions and transforms in operator-valued free probability

We introduce a class of independence relations, which include free, boolean and monotone independence, in operator-valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, boolean and monotone additive convolutions, we will focus on two other important additive convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescu's fully matricial function theory. In the end, we study relations between certain convolutions and transforms in C⁎-operator-valued probability.

Homomorphisms relative to additive convolutions and max-convolutions: Free, boolean and classical cases

We introduce new homomorphisms relative to additive convolutions and max-convolutions in free, boolean and classical cases. Crucial roles are played by the limit distributions for free multiplicative law of large numbers.

Middle multiplicative convolution and hypergeometric equations

Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations.

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.

Connecting the q-multiplicative convolution and the finite difference convolution

In [1], Brändén, Krasikov, and Shapiro study root location preservation properties of finite difference operators. They then conjecture a characterization of linear finite difference operators on polynomials of fixed degree which preserve root mesh properties, via a natural polynomial convolution. We prove this conjecture using two methods. The first develops a novel connection between the additive (Walsh) and multiplicative (Grace-Szegö) convolutions, which can be generically used to transfer results from multiplicative to additive. We then use this to transfer an analogous result, due to Lamprecht [4], which demonstrates logarithmic root mesh preservation properties of a certain q-multiplicative convolution operator. The second method proves the result directly using a modification of Lamprecht's proof of the logarithmic root mesh result. We present his original argument in a streamlined fashion and then make the appropriate alterations to apply it to the additive case.

Optimizing Cost and Quality of Innovative Solutions at Design Stages of Life Cycle of Engineering High-Tech Items

The article develops a method for optimizing the cost and quality of innovative solutions at design stages of a life cycle of engineering high-tech items. The method allows evaluating and comparing a whole set of permissible options of innovative solutions on the basis of all partial and generalized criteria at the stage of production design engineering.The method is based upon a bipartite graph model of component interconnections in engineering high-tech items and the criteria of optimal option selection. To evaluate the options of innovative solutions, the authors form a set of partial criteria united in the categories: production criteria; investment criteria; criteria of evaluating the quality of manufactured products while using a high-tech item; economic criteria; financial criteria; criteria of evaluation on risk types; social criteria; environmental criteria; budget criteria; integral criteria evaluating the quality of an engineering high-tech item on the basis of all indices. To evaluate the cost and quality of innovative solutions, the authors form integrated efficiency functions presented as additive convolutions of partial criteria.The developed method is automated with the use of software tools MS Excel, MS Access, C#. The method industrial approbation demonstrates its operation efficiency and universal nature for solving various problems of the comparative evaluation of innovative solutions at the design stages of a life cycle of engineering high-tech items.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

On Double Sums with Multiplicative Characters

An estimate for double sums of multiplicative characters with additive convolutions of the characteristic functions of sets having small additive doubling is obtained.

Analytic subordination for bi-free convolution

In this paper we study some analytic properties of bi-free additive convolution, both scalar- and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove several results about atoms of bi-free additive convolutions.

Boolean convolutions and regular variation

In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the notion of classical and free subexponentiality. We show that the distributions with regularly varying tails belong to the class of Boolean subexponential distributions. As an application we also study the behaviour of the Belinschi-Nica map. Breiman's theorem study the classical product convolution between regularly varying measures. We derive an analogous result to Breiman's theorem in case of multiplicative Boolean convolution. In proving these results we exploit the relationship of regular variation with different transforms and their Taylor series expansion.

METHODS OF FORMING VIRTUAL PRODUCTION FOR REPAIRING LOCOMOTIVES

In the paper the need to transform of repair production of locomotives of railways in modern conditions is formulated. To expand the technological capabilities of repair productions it is proposed to use new organizational and production forms - virtual enterprises. It was determined that the main criterion for the formation of virtual enterprises for the repair of locomotives should be obtainment a synergistic effect. For this purpose, it is proposed the formation of virtual enterprises by the principle of supplementing the competences of the components of repair production with the account of the previously determined coefficients of conformity of production. Coefficients of conformity of repair production are complex indicators which are calculated by four components of repair production: the equipment and the tool, the personnel, the working environment, the documentation. The presence of weight coefficients of particular criteria the conformity of productions allows to solve the problem of multicriteria optimization by methods that use procedures of ranking criteria and comparing alternatives. The objective function is represented as the minimization of criteria. A comparison of the results of the solution of the optimization problem of multicriteria optimization showed that the methods of additive and multiplicative convolutions are characterized by the greatest adequacy in such conditions.