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Numerical modeling is an important tool when studying various natural processes and phenomena. Fractional diffusion can be used for modeling many processes in biology, for example in silico experiments in molecular biology and medicine design, protein diffusion within cells, complex media geometry, etc. The problem is usually reduced to a system of linear algebraic equations and in many cases this system has a dense coefficient matrix. Numerically solving such problems with the traditional LU factorization is a computationally expensive endeavour - $O(n^3)$. In this paper we explore the use of a hierarchical compression method based on Hierarchical Semi-Separable compression and ULV-like factorization from the STRUctured Matrices PACKage (STRUMPACK) software library for a flow around airfoils problem discretized with Boundary Element Method and fractional diffusion problem discretized with the Finite Element Method. The HSS based method promises better overall computational complexity of $O(r^2n)$ for problems with suitable structure - low rank off-diagonal blocks. Here $r$ is the maximum rank of the off-diagonal blocks. We present analysis of the performance and accuracy of the HSS based method and compare it with the state of the art direct LU factorization solvers. This paper is based on the PhD thesis Composite Numerical Methods and Scalable Tile Algorithms of the author defended on 17.05.2022 in the Institute of Information and Communication Technologies at the Bulgarian Academy of Sciences.

We consider the enzyme kinetic reaction scheme originally proposed by V. HenriВ of single enzyme-substrate dynamics where two fractions of the enzyme - free and bound - are involved. Henri's scheme involves four concentrations and three rate constants and via the mass action law it is translated into a systemВ of four ODE's. On two case studies we demonstrate how the rate constants can be computed whenever time course experimental data are available. The obtained results are compared with analogous results implied by classical Michaelis-Menten model. Our approach focuses on the uncertainties in the experimental data, as well as on the use of contemporary computational tools such as CAS Mathematica.

Browsing the biographies of the generation of Bulgarian teachers who lived and worked in the years around the liberation-bringing Russo-Turkish war (1877--1878), we see the clear outlines of the impediments blocking the education and development endeavors of the young Bulgarian state, which was way behind the times five centuries after it had been engulfed by the Ottoman empire and severed from European culture. In this article we review the activities of eleven Bulgarian teachers who lived and worked around the time of liberation from the Ottoman rule. Their life and activity along the borderline of two epochs strikes us with their self-sacrificial efforts to overcome the enormous backwardness of the Bulgarian nation. We point out their contribution in the spheres of education and science. The biographies of these Bulgarian teachers present a vivid picture of the times and would be an edifying subject to study and remember.

In this paper we study the distance between the sign function and a class of parametric activation functions. The distance is measured in Hausdorff sense, which is natural in a situation when a sign function is involved. Precise upper and lower bounds for the Hausdorff distance have been obtained.

A PhD class of lectures will be jointly developed on the topic of Linear Quadratic Differential Games and Applications. The class is intended to support the learning process of the PhD students as well as to increase their competencies through acquaintance with the latest advances in the field. The total study load is planned to amount to 30 lecture hours.

Initial statistical analysis of genetic data on the Balkan nations showed the extent of their genetic links to each other. A careful review of the data highlighted an interesting feature: relatively large genetic differences between Greek regions. This is clearly expressed in the cases of Northern Greece, whose population is genetically much closer to the population of Bulgaria than to the population of Central and Southern Greece, whose population is genetically closer to the Albanian population than to the population of Northern Greece. These conclusions are based on numerical values from the results of statistical analysis of genetic data from EUPEDIA. The article also presents several historical testimonies that offer an explanation of the established regional genetic features of the population of today's Greece.

Consider a population of two or more different types of cells that at the end of life create two new cells through cell division. We model the population dynamics using a multitype branching stochastic processes. Under consideration are processes of Bieneme-Galton-Watson and of Bellman-Harris for the Markovian case. В drawn Conclusions about the expected number of particles of each type after a random time are drawn. The proposed models could be applicable not only for populations of a unicellular organisms, but also for sets of objects which operate a certain period of time and then split into two new objects or change their type.

This work brings together two recently discussed topics: mathematical modeling of a bioreactor and working with derivatives of non-integer order. Generally, it turns out that it is reasonable to replace the integer order derivatives in some of the already well known mathematical models describing bioprocesses with fractional order ones. However, the specific structure of such type of derivatives makes the study of the properties of the models a real challenge. This work contains primary results for modeling of a bioreactor with appropriately selected numerical approximations. Different scenarios are taken into consideration: starting from the simplest one - without mortality and then complicating by adding nonzero mortality term. In the classical case the solution of the system of differential equations describing the process has a specific behaviour in terms of monotonicity. Therefore, the focus of the further examinations is to find out whether it is possible to generalize the model into a fractional order one such that the key properties considering monotonicity still hold. The results show that the latter requires certain dependencies between the orders of the derivatives in the mathematical model. The hypothesis is based on two types of experiments which are described in detail. Lotka-Volterra and Monod specific growth rate are used in the mathematical model. The paper contains figures which illustrate the results from different numerical computations performed via Wolfram Mathematica software.

The use of graphs is widely applied in modeling and solving problems in the field of computer science and bioinformatics.Therefore, it is essential to develop and improve algorithms reducing their computational complexity and increasing the precision of the solutions generated by them as well as the size of the input data.In this study two well-known algorithms for solving the problem for finding a minimum Hamiltonian cycle in weighted, undirected and complete graph (also known as Travelling Salesman Problem –- TSP) are analyzed.The first algorithm is based on the backtracking method and it always finds the optimal solution, while with the second one, the genetic algorithm (GA), finding the optimal solution is not always guaranteed.The aims of the study are to determine: (1)which of the algorithms can be used so that the resulting solution is optimal or near-optimal and the execution time be reasonable depending on the size of the input data; (2)the influence of GA parameter values on the quality of the resulting solutions for large size of the input data. The parameters determine the number of solutions in each population and the number of all generations.The analysis of the results revealed that:(1) the algorithm that finds all possible solutions can be used for graphs with a small number of vertices (not more than 20), whereas GA can be used for graphs with a large number of vertices; (2) in graphs with a small number of vertices: n<20 (and n*(n-1)/2 edges) GA always finds the optimal solution as long as enough solution space is set. However, the number of all Hamiltonian cycles in a complete graph with n vertices ((n-1)!/2) is bigger than the solution space; (3) all input datasets showed that with the number increase of vertices in the graph it is necessary to increase the number of the current solutions in the population. In this way GA reaches a certain rate of convergence faster, i.e., a generation after which the space of solutions contains only optimal solutions or near optimal ones.Acknowledgments: This work is partially supported by the project of the Bulgarian National Science Fund, entitled: “Bioinformatics research: protein folding, docking and prediction of biological activity”, NSF I02/16, 12.12.14.