Additive Convolution Research Articles

Sums of random polynomials with independent roots

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures μ and ν in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p+q as n tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of μ and ν. More generally, we consider the sum of m independent degree n random polynomials when m is fixed and n tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.

The Development of Scientific Activity in Russian Universities

The aim of this article was to develop scientific activity in Russian universities. Using the survey method, a sample of quantitative indicators of research output for the years 2011–2019 was formed. The respondents were comprised of 934 lecturers from 13 universities in Russia with different work experience, academic degrees, and indicators of scientific activity. The main components method was used to study the structure of scientific activity productivity indicators. Using the method of additive convolution, a five-factor integral model of the structure of scientific activity of a teacher was formed to evaluate the individual and collective productivity of scientific activity. The factor values and the integral indicator of scientific activity output were calculated, showing that the individual growth rates of the young lecturers’ output exceeded the growth rates of the older lecturers. Destructive factors relating to the scientific activity output of the lecturers and researchers in Russian universities (divided into two groups of young scientists and senior scientists) were determined and systematized based on the level of dominant influence. The features of the influence of the factor of emotional burnout on the scientific productivity of university teachers were revealed.

SETTING UP THE GENETIC ALGORITHM FOR THE INDIVIDUALIZED TREATMENT STRATEGY SEARCHING

The genetic algorithm is a verified mechanism for optimization task solutions. Being a heuristic algorithm, it allows speeding up the task solving by the biological evolution principle use. Recently, this algorithm was offered as a method of an individualized treatment strategy search, where it was necessary to optimize the patient’sstatein the distant period, going through various treatment combinations.In that research as an optimization function, the additive convolution function of the patient’sstateindicatorsin the distant period was used, obtained with the help of Saaty analytic hierarchy process, which is one of the multi-criteria decision making methods. Despite showing good results, the genetic algorithm was set with standard parameters. Taking into consideration a big quantity of the parameters, the present study has the aim to find the optimum parameters for the algorithm. First of all, it is necessary for those, who incorporate this algorithm in their work, namely doctors, when they need to prescribe a treatment for a patient. The study describes the analysis of various genetic algorithm parameters and their use in algorithm test launches for individualized treatment strategy search. Also, the optimal patient’s input parameter subsets wereselected,using the correlation feature selection criterion. The selected parameters were necessary for modeling indicators of the patient’sstateafter treatment.Modeling wasperformed via random forest classifier with preliminarydivided total sample intotraining(eighty percent) and testing(twenty percent) ones. Two different databases of patients with congenital heart diseases were usedfor the study, allowing the optimal parameters being more reliable for their future use. In the end, it all allows finding the parameters, which are first of allexclusively recommendatoryto the doctors before using the algorithm.

Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolution

We investigate relations between additive convolution semigroups and max-convolution semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi–Nica semigroup and the max-Belinschi–Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.

Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices

Abstract We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$ and $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.

Polynomial Ensembles and Pólya Frequency Functions

We study several kinds of polynomial ensembles of derivative type which we propose to call P\'olya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of P\'olya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate P\'olya ensembles to P\'olya frequency functions. For illustration we give a number of explicit examples for our results.

ВИБІР НАПРЯМКУ РЕФОРМУВАННЯ ОРГАНІЗАЦІЇ В ПРОЕКТНОМУ СИСТЕМНОМУ УПРАВЛІННІ РОЗВИТКОМ

The subject of research in the publication is the process of managing the development of a high-tech organization. The object of research is project system development management. The work aims to choose the current direction of reforming a high-tech organization in conditions of limited opportunities. The article solves the following tasks: analyzes the problems of managing the development of the organization, identifies the main areas of reform associated with ensuring the competitiveness of the organization in the market of high-tech products. To substantiate the choice of the current direction of reform external experts are involved to assess several possible areas of development, considering the state and level of maturity of the organization, readiness for reforms, existing business process model, financial opportunities for development processes. Then based on expert assessments and the method of full factorial experiment, the analysis, and comparison of reform options is carried out to select the most relevant direction in the task of project development management. The limited resources of the enterprise and the requirements for the timeliness of reforms lead to the task of optimizing resources for the implementation of the reform project. The multiplicity of options for allocating resources at different stages of the reform project leads to the use of targeted search in the task of resource optimization. The optimization problem is set based on integer (Boolean) programming in two views: local and multi-criteria optimization. Costs, project deadlines, and the risk of successful project implementation were used as criteria for assessing the allocated resources and the success of the reform project. For multicriteria optimization, the significance of individual criteria is assessed. As a comprehensive criterion, a simple additive convolution of local criteria was used. To assess the correctness of the chosen reform option, the method of agent modeling is added. A set of agents has been formed, with the help of which all stages of the development project are simulated. This takes into account the time factor, allocated funds (costs of the organization), the risks of individual stages. The results of agent modeling will allow concluding the expediency and relevance of the choice of development direction for the subsequent project management of the reform. The scientific novelty of the publications to create a method by which derivative data are formed for the enterprise reform project. The publication uses the following methods: system analysis to justify and choose the direction of development; expert assessments for the formation of a factor experiment plan; the method of full factorial experiment, which allows to evaluate several options for reform and choose the current direction of development of the organization; method of integer (Boolean) programming for the purposeful choice of means during project implementation for development; the method of agent simulation modeling to simulate the implementation of individual stages of the development project. Conclusions: the proposed approach allows at the stage of a preliminary study of the development of the organization to choose the current direction of reform, considering the state of the high-tech organization, readiness to reform, as well as limited opportunities in the face of challenges and threats in the economy.

FEATURESELECTIONFOR PREDICTING THE PATIENT STATE IN DISTANTPOSTOPERATIVE PERIOD

The optimization problem of patients with congenital heart defects state in the distant postoperative period consists of finding a specific treatment strategy that gives the best rest, taking into account the initial state of a patient. However, there may be too many input variables for this type of task, which significantly increases the risk of worsening the result. The work describes the process of analyzing feature selection algorithms, where features act as indicators of patients with congenital heart defects, applying the mechanism of these algorithms to reduce the dimension (quantity) of input features, and using selected features to predict patients’ state indicators in the distant period. The purpose of the study was to find the optimal composition of indicators that would retain as much information as possible for predicting indicators of the state of patients. Among the analyzed feature selection algorithms, the correlation feature selection method was chosen. The function of the additive convolution of state indicators was used as an output variable. This function was obtained by the Best-Worst method (one of multi-criteria decision making methods). To predict patient state indicators, five classification algorithms were proposed: logistic regression, linear discriminate analysis, random forest, mixed step-by-step algorithm of group method of data handling, and group method of data handling with neurons. Before using them, the total samples weredivided into train (eightypercent) and test (twenty percent) samples. The first three algorithms were programmed in Python, while the others were calculated in GMDH Shell DS software. Algorithms give seventy-eight and four tenths of accuracy on average on the test sample. The results will be used to improve the unified algorithm for optimizing the state of patients, which was obtained in previous studies, and includes a genetic algorithm and analytic hierarchy process.

A methodological approach and criteria for assessing the effectiveness of company's digital transformation

Subject. The article addresses the totality of economic relations and problems emerging in the process of digital transformation of companies. Objectives. The focus is on the development of key indicators to measure the effectiveness of digital transformation implemented by companies. Methods. We employ methods of economic analysis, synthesis, and comparison. The study rests on scholarly works on corporate finance, financial management and digital technology. Results. We define the concept of digital transformation and substantiate its possible directions for companies. We also suggest using the digital transformation index. It includes additive convolution and the use of the distance method. The paper underpins the criteria enabling a comparative analysis, comprehensive and reliable assessment of the effectiveness of digital transformation of company divisions. The digital transformation index can be useful as part of monitoring and managing the implementation of digital transformation measures. Conclusions and Relevance. Practical use of the presented approach to assessing the effectiveness of digital transformation will help create a basis for making managerial decisions aimed at improving the efficiency of the company.

A methodological approach and criteria for assessing the effectiveness of company's digital transformation

Subject. The article addresses the totality of economic relations and problems emerging in the process of digital transformation of companies. Objectives. The focus is on the development of key indicators to measure the effectiveness of digital transformation implemented by companies. Methods. We employ methods of economic analysis, synthesis, and comparison. The study rests on scholarly works on corporate finance, financial management and digital technology. Results. We define the concept of digital transformation and substantiate its possible directions for companies. We also suggest using the digital transformation index. It includes additive convolution and the use of the distance method. The paper underpins the criteria enabling a comparative analysis, comprehensive and reliable assessment of the effectiveness of digital transformation of company divisions. The digital transformation index can be useful as part of monitoring and managing the implementation of digital transformation measures. Conclusions and Relevance. Practical use of the presented approach to assessing the effectiveness of digital transformation will help create a basis for making managerial decisions aimed at improving the efficiency of the company.

On the Hodge Theory of the Additive Middle Convolution

We compute the behavior of Hodge data under additive middle convolution for irreducible variations of polarized complex Hodge structures on punctured complex affine lines.

Personalized nutrition and “digital twins” of food

Mathematization of research is one of the most effective methods of virtual substantiation of foodstuff recipe and technology. This approach allows creating a product that meets consumer's individual needs, i.e. personalized foodstuff (ethnicity, cultural preferences, regional and environmental characteristics, lifestyle), and at the same time reducing the time and cost of decision-making. The article discusses the hypothesis that the “digital twin” of a food product is a virtual model of the product, namely its mathematical model (simulation model). A simulation model is a logical and mathematical description of a food product that is used to conduct a computerized experiment in order to design desired characteristics and properties. The “digital twin” combines all variety of factors from chemical composition, functional and technological properties to organoleptic indicators. The application of the “digital twin” model of the foodstuff will allow: (1) reacting quickly to changes in the composition, properties and types of raw ingredients, (2) adjusting the product recipe in response to changes in consumer preferences, (3) designing products with a given chemical composition, nutritional value and functional orientation, (4) creating functional, specialized products taking into account the metabolism of nutrients (ethnicity, cultural preferences, health status and clinical factors). Products adapted to the needs of small categories of people will help reducing the risks for those who already have diseases, and will meet the needs of those who would like to make their diet more appropriate to individual needs. The proposed approach to creating a model of the “digital twin” of the foodstuff includes several stages. The first stage involves optimization of the nutritional and biological value of the designed product. The second stage is related to designing the food product’s structural forms. But even if the recipe of a food product is optimally selected in the first stage, it does not guarantee its transformation during processing into a stable system with the required structural, mechanical, functional and technological parameters. Evaluation of the developed food product’s efficiency is possible only by analysing numerous and various parameters and indicators. It is convenient to generalize (convolute) many parameters and indicators into a single quantitative dimensionless indicator. To assess the quality and adequacy of the food product, it is suggested to use an integral indicator in the form of additive convolution – the ‘functional’ of the food product quality.

Spectral rigidity for addition of random matrices at the regular edge

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

Variance function of boolean additive convolution

Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formula is used to identify the relation between variance functions under Boolean Bercovici–Pata Bijection between probability measures. We also give the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers.

Development of information technology for planning order fulfillment at a food enterprise

An information technology has been proposed that aims to resolve the task of planning the fulfillment of orders for manufacturing products at food enterprises under conditions of uncertainty and risk. The information technology is based on combining the ant colony, gray wolves, and genetic algorithms, as well as the constructed mathematical model of the operative execution of orders. The advantages of algorithm combination include the formation of alternative variants of plans and the avoidance of local optima. The proposed mathematical model includes the partial criteria, constraints, as well as an evaluation function, for determining the effectiveness of the compiled plan of order execution. The application of a petal diagram and an additive convolution of partial criteria has been suggested to illustrate the clarity of a variant of order fulfillment. The mathematical model makes it possible for a DM to define any set of partial criteria to take into consideration the patterns of order execution. The information technology ensures rapid reconfiguration of the current plan of order execution in the event of emergencies or the need to urgently fulfill a certain order.

СОВРЕМЕННЫЕ АСПЕКТЫ СТРАТЕГИЧЕСКОГО УПРАВЛЕНЧЕСКОГО УЧЕТА И АНАЛИЗА

In the article, the authors considered the parameters of the strategic management process using the mathematical apparatus of fuzzy sets, which allow us to express the internal state of the enterprise through an interval-linguistic assessment of multidirectional changes in economic indicators. The study conducted in the article is based on an effective approach based on the idea of additive con-volution of selected indicators into a single interval with a linguistic assessment of the result at a time to by V. V. Pankov's method (N & P), which allows us to abandon the subjective determination of the weight of indi-vidual coefficients, since in comparison with the forecast balance sheet developed according to the scenario of strategic sustainable development of an economic entity, we obtain primary reliable data.

Поняття економічної стійкості підприємства та її кількісне оцінювання

The article is concerned with the issues of determining and evaluating the economic sustainability of enterprise. The publication analyzes the views of scholars on the definition of «economic sustainability of enterprise». A closer definition of this concept is proposed, according to which under the economic sustainability of enterprise it is advisable to understand its complex ability to respond to and withstand the influence of negative relevant factors of the external and internal environment, while maintaining its internal structure, dynamics of operation, which contributes to the achievement of the main goals of functioning and development of enterprises. The main components of economic sustainability of enterprise are allocated, including: financial, industrial, personnel, organizational, marketing, social, environmental, investment, innovation. The main stages of the process of ensuring economic sustainability of enterprise are proposed, the basis of implementing which is the principles of unity – interconnectedness and subordination of all components of sustainability; complexity, which involves taking into account relevant factors of influence; continuity – implementation of the regulatory process on the basis of permanence and sustainability; adequacy, i.e. compliance of the system with patterns of enterprise development. The proposed components and stages of ensuring and determining the economic sustainability of enterprise are used as the basis of the developed methodical approach concerning its quantitative assessment, which is built up using methods of multidimensional factor, cluster analysis, additive convolution method for calculating the integral indicator of economic sustainability of enterprises. The developed methodical approach was tested on the statistical data of nine machine-building enterprises of Kharkiv region, which proved its applied value and possibility of further use.

Specifics of managerial decisions made by the heat-supplying organizations management during the selection of the partner organizations

The issue of the selection of business partners in the conditions of the uniform heat-supplying organizations functioning is considered in the article. The need of quality and timely evaluation of the potential partners for the purpose of the increase in the level of quality of generation, transportation and distribution of heat energy is proved. The author’s approach of holding the procedure of selection of the most reliable partner was used with the help of the method of the hierarchical analysis. The method of pair comparisons was also applied within the research, the blocks and indicators of the assessment were defined. The model of additive convolution was chosen as the method of the aggregated criterion creation.

Assessment model of regions’ economy in the context of their sustainable development

Currently realizing the new active role of the region as a sustainable development entity is a strategic direction for regional policy’s development. Assessing the sustainable development of regions is an important part of such a policy and facilitates the timely identification of internal and external threats, the development of the necessary stabilization measures to prevent their negative impact, and the formation of strategies aimed at the sustainable functioning of regional systems. The economic system is an important subsystem of the region. The article proposes an approach to assess the level of regions economic development in the context of ensuring its sustainable development. It is based on comprehensive assessment technology. The sustainable economic development composite index is calculated by a weighted additive convolution of partial indicators. A feature of the proposed approach is the simultaneous use of both metric and nonmetric indicators. The metric component is used to calculate the composite index values. Weight coefficients are calculated by the principal component method using the factor loadings of the first principal component. The non-metric part of the initial data is used to refine these weights. The article describes the algorithm for calculating a composite index. The practical testing of the proposed approach is presented for the case Ukraine’s regions. The results lead to the conclusion about significant problems in ensuring sustainable development of the regional economy. Outcomes obtained are very helpful for the public administration bodies to develop and revise the appropriate policy for solving the sustainable development problems in each region.

Conditionally monotone independence and the associated products of graphs

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence on suitably constructed larger algebras. For that purpose, we use the approach developed for a reduction of similar type for boolean, free and monotone independences. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.