# Svetoslav MarkovBulgarian Academy of Sciences | BAS · Institute of Mathematics and Informatics

Svetoslav Markov

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## About

163

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Introduction

I am interested in the relation between mathematical models in biology and reaction network theory.

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## Publications

Publications (163)

Following the ideas given in [13]-[15], in this article we study a hypothetical piecewise smooth modified Schnute growth function. Some numerical examples, using CAS MATHEMATICA are also given.

We mathematically analyze the solutions to the dynamical system induced by the two-step exponential (growth-)decay (2SED) reaction network involving three species and two rate parameters. We study the influence of the rate parameters on the shape of the solutions. We compare the latter to those of the classic Kermack–McKendrick epidemiological SIR...

In this article we will consider the possibility of approximating the input function $s(t)$ (the nutrient supply for cell growth) of the form $s(t)=\frac{1}{1+mt}e^{-mt}$ where $m>0$ is parameter.
We prove upper and lower estimates for the one--sided Hausdorff approximation of the shifted Heaviside function $h_{t^{\ast}}(t)$ by means of the general...

In this paper we study the one-sided Hausdorff approximation of the Heaviside step function by a families of Unit-Logistic (UL), Unit-Weibull (UW) and Topp-Leone (TL) cumulative sigmoids. The estimates of the value of the best Hausdorff approximation obtained in this article can be used in practice as one possible additional criterion in "saturatio...

Sunaga considered all computational procedures, which had been traditionally defined on real numbers, as being too ideal and proposed to replace them by the procedures on real intervals in order to make everything "more realistic". Sunaga studied many different kinds of numerical procedures including the Taylor-series interval solution of the initi...

In this paper we study the one-sided Hausdorff approximation of the shifted Heaviside step function by a family of Exponential-Generalized Extended Gompertz (EGEG) cumulative sigmoid. The model has a certain right of existence insofar as the theory of sigmoidal functions is well developed. The estimates of the value of the best Hausdorff approximat...

In this paper we study the one--sided Hausdorff approximation of the generalized cut function by sigmoidal general n-stage growth model. We show that under some conditions the model is useful insofar as the theory of sigmoidal functions is well developed. The estimates of the value of the best Hausdorff approximation obtained in this article can be...

In this paper we study the one-sided Hausdorff approximation of the generalized cut function by sigmoidal modified three-stage growth model. The model has a certain right of existence insofar as the theory of sigmoidal functions is well developed. The estimates of the value of the best Hausdorff approximation obtained in this article can be used in...

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p+1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the hyper-log-logistic and the shift...

The paper considers the sigmoid function definedthrough the hyper-log-logistic model introduced by Blumberg. We study the Hausdorff distance of this sigmoid to the Heaviside function, which characterises the shape of switching from 0 to 1. Estimates of the Hausdorff distance in terms of the intrinsic growth rate are derived. We construct a family o...

We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the Hyper-log–logistic function. The limiting case of the interval-valued step function is discussed using Hausdorff metric. Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical...

In this note we construct a family of recurrence generated sigmoidal functions based on the Log-logistic function. The study of some biochemical reactions is linked to a precise Log-logistic function analysis. We prove estimates for the Hausdorff approximation of the Heaviside step function by means of this family. Numerical examples, illustrating...

The Hausdorff approximation of the shifted Heaviside function by Log-logistic and quadratic transmuted Log-logistic sigmoid functions is investigated and an expression for the error of the best approximation is obtained. The results of numerical examples performed in the programming environment Mathematica confirm our theoretical conclusions. Some...

In this note we construct a family of parametric Gom-pertz activation function (PGAF) based on hyperbolic tangent function. We prove upper and lower estimates for the Hausdorff approximation of the sign function by means of this family. Some comparisons between the hyperbolic tangent activation function and the new parametric Gompertz activation fu...

Growth models are often used when modelling various processes in life sciences, ecology, demography, social sciences, etc. Dynamical growth models are usually formulated in terms of an ODE (system of ODS's) or by an explicit solution to an ODE, such as the logistic, Gompertz, and Richardson growth models. To choose a suitable growth model it is use...

The subject of this book is multidisciplinary. Sigmoidal functions are of interest for fundamental as well as applied research. We give the readers both directions. From the fundamental science point of view sigmoid functions are of special interest in abstract research in approximation theory, functional analysis and probability theory. From the a...

Biochemical mechanisms with mass action kinetics are usually modeled as systems of ordinary differential equations (ODE) or bipartite graphs. We present a software module for the numerical analysis of ODE models of biochemical mechanisms of chemical species and elementary reactions (BMCSER) within the programming environment of CAS Mathemat-ica. Th...

In this note we construct families of recurrence generated three and four–parametric activation functions. We prove precise upper and lower estimates for the Hausdorff approximation of the sign function by means of these families. Numerical examples, illustrating our results are given. In Appendix 3 we provide an analysis of recurrence generated fa...

In this note the Hausdorff approximation of the Heaviside step function by several sigmoid functions (log-logistic, transmuted log-logistic and generalized logistic functions) is considered and precise upper and lower bounds for the Hausdorff distance are obtained. Numerical examples, that illustrate our results are given, too. © 2015 International...

In this note we construct a family of recurrence generated parametric activation functions. We prove precise upper and lower estimates for the Hausdorff approximation of the sign function by means of this family. Numerical examples, illustrating our results are given.

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.

The education of Bulgarians in the Ottoman Empire in Bulgarian language during the 19 th century is a substantial component of the Bulgarian revival together with the struggle for political autonomy. This education starts almost from zero and reaches considerable growth around the Liberation war without having any support from the side of the state...

Biochemical mechanisms with mass action kinetics are usually modeled as systems of ordinary differential equations (ODE) or bipartite graphs. We present a software module for the numerical analysis of ODE models of biochemical mechanisms of chemical species and elementary reactions (BMCSER) within the programming environment of CAS Mathemat-ica. Th...

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p + 1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the logistic and the shifted logis...

In this paper we study the one-sided Hausdorff distance between the shifted Heaviside step--function and the transmuted Stannard growth function. Precise upper and lower bounds for the Hausdorff distance have been obtained. We present a software module (intellectual property) within the programming environment CAS Mathematica for the analysis of th...

In this paper we study the uniform approximation of the cut function by smooth sigmoid functions such as Nelder and Turner–Blumenstein–Sebaugh growth functions. To illustrate the use of one of the models we have fitted the model to the " classical Verhulst data ". Several numerical examples are presented throughout the paper using the contemporary...

In this work we discuss some methodological aspects of the creation and formulation of mathematical models describing the growth of species from the point of view of reaction kinetics. Our discussion is based on familiar examples of growth models such as logistic growth and enzyme kinetics. We propose several reaction network models for the amiloid...

Algebraic systems, abstracting properties of intervals, are discussed. Certain algebraic structures close to vector spaces, ordered rings, fields and algebras are axiomatically introduced and studied. The properties of errors are studied, as the latter can be considered as centred (zero-symmetric) intervals. It is shown that errors form a specific...

Mathematical models of growth have been developed a long period of time. Estimating the lag time in the growth process is a practically important problem. Any sigmoidal function can be good illustration for the concept of lag time. The Schnute growth model is described by free parameters, each contributing to the characteristics of the curve: an in...

The set of interval Hausdorff continuous functions constitutes the largest space preserving basic algebraic and topological structural properties of continuous functions, such as linearity, ring structure, Dedekind order completeness, etc. Spaces of interval functions have important applications not only in the construction of numerical methods and...

In a recent paper by Tobias and Tasi in this journal the authors discuss the “K-angle” kinetic problem and focus on its analytic solution for \(K\le 5\). The authors correctly mention that the case \(K>5\) requires numerical treatment in the part of the procedure where the eigenvalues of the characteristic polynomial are determined by solving an al...

In this paper we study the uniform approximation of the cut function by smooth sigmoid functions such as Stannard and Richard growth functions. Numerical examples are presented using CAS MATHEMATICA.

Stochastic arithmetic has been developed as a model for computing with imprecise numbers. In this model, numbers are representedby independent Gaussian variables with known mean value and standarddeviation and are called stochastic numbers. The algebraic properties of stochastic numbers have already been studied byseveral authors. Anyhow, in most l...

Proposed are some modifications of bio-economic models with stage structure [3]-[8]. The proposed modifications aim at a better description and understanding of the underlying bio-economic mechanisms. This is achieved by formulating the model in terms of chemical-type reaction steps. Several new modules in the Computer Algebra System MATHEMATICA ar...

In this work we pose the question how reliable the Michaelis constant is as an enzyme kinetic parameter in situations when the Michaelis-Menten equation is not a good approximation of the true substrate dynamics as it may be in the case of metabolic processes in living cells. We compare the Michaelis-Menten substrate-product kinetics with the compl...

We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued step function is discussed using Hausdorff metric. Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Nu...

In this note we prove more precise estimates for the approximation of the step function by sigmoidal logistic functions. Numerical examples, illustrating our results are given, too.

This is a brief report on some international meetings in the field of biomathematics and scientific computations organized within the frames of the Bulgarian Academy of Sciences. Most meetings involved both a scientific conference and a School for Young Scientists aiming to attract doctoral students and young scientists and to promote the integrati...

We focus on some computational, modelling and approximation issues related to the logistic sigmoidal function and to Heaviside step function. The Hausdorff approximation of the Heaviside interval step function by sigmoidal functions is discussed from various computational and modelling aspects. Some relations between Verhulst model and certain bioc...

We give a short description of the annual international conference series in the field of biomathematics "Biomath". A main goal of this conference series is to attract young scientists and integrate learning and research activities during the meetings and the subsequent publication process. We share our experience on the use of specific organizatio...

We consider the enzyme kinetic reaction scheme originally proposed by V. Henri 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 system of four ODEs. In two case studies we demonst...

The subject of this book is cross-disciplinary. Sigmoid functions present a field of interest both for fundamental as well as application-driven research We have tried to give the readers the flavour of both perspectives. From the perspective of fundamental science sigmoid functions are of special interest in abstract areas such as approximation th...

Proposed are some modifications of bio-economic models with stage structure [3]-[8]. The proposed modifications aim at a better description and understanding of the underlying bio-economic mechanisms. This is achieved by formulating the model in terms of chemical-type reaction steps. Several new modules in the Computer Algebra System MATHEMATICA ar...

Simple structured mathematical models of bacterial cell growth are proposed. The models involve fractions of bacterial cells related to their physiological phases. Reaction schemes involving the biomass of the cell fractions, the substrate and the product are proposed in analogy to reaction schemes in enzyme kinetics. Applying the mass action law t...

This paper presents experimental data coming from a batch fermentation process and theoretical models aiming to explain various aspects of these data. The studied process is the production of exopolysaccharides (EPS) by a thermophilic bacterium , Aeribacillus pallidus 418, isolated from the Rupi basin in SouthWest Bulgaria. The modelling approach c...

The BIOMATH 2012 International Conference on Mathematical Methods and Models in Biosciences was held at the Bulgarian Academy of Sciences in Sofia, in June 17-22, 2012, http://www.biomath.bg/2012/. We were happy to meet more than 70 participants from twenty different countries. More than 40 contributions were submitted for publication in the presen...

In February 2012 Prof. Blagovest Sendov turned 80. I first met him in 1963 as a student in mathematics at the Department of Physics and Mathematics of Sofia University. I was fascinated by his lectures in Numerical Methods. His first lecture was devoted to Mathematical modeling. On several realistic case studies Prof. Sendov revealed what was to me...

The 2011 International Conference on Mathematical Methods and Models in Biosciences and School for Young Scientists (BIOMATH 2011) held in Sofia, 15-18 June 2011, was an international meeting that gathered researchers from four different continents and 16 different countries. BIOMATH 2011 continues a tradition of scientific meetings on biomathemati...

We study certain classical basic models for bioreactor simulation in case of batch mode with decay. It is shown that in many cases the two-dimensional differential system describing the dynamics of the substrate and biomass concentrations can be reduced to an algebraic equation for the biomass together with a single differential equation for the su...

Cell movement is a complex process. Cells can move in response to a foreign stimulus in search of nutrients, to escape predation, and for other reasons. Mathematical modeling of cell movement is needed to aid in achieving a deeper understanding of vital ...

We propose a new approach to the mathematical modelling of microbial growth. Our approach differs from familiar Monod type models by considering two phases in the physiological states of the microorganisms and makes use of basic relations from enzyme kinetics. Such an approach may be useful in the modelling and control of biotechnological processes...

This chapter considers interpolation and curve fitting using generalized polynomials under bounded measurement uncertainties from the point of view of the solution set (not the parameter set). Itcharacterizesandpresentstheboundingfunctionsforthe solutionsetusingintervalarithmetic. Numerical algorithms with result verification and corresponding prog...

We propose a new approach to mathematical modelling of microbial growth different to the approaches used in Jacob-Monod type models. Such an approach may be useful in the modelling of biotechnological processes, where microorganisms are used for various biodegradation purposes and are often put under extreme unfavourable conditions, such as prolong...

In this survey paper we focus our attention on dynamical bio‐systems involving uncertainties and the use of interval methods for the modelling study of such systems. The kind of envisioned uncertain systems are those described by a dynamical model with parameters bounded in intervals. We point out to a fruitful symbiosis between dynamical modelling...

A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the Hamilton-Jacobi equation
is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions
can be extended to the wider space of Hausdorff continuous functions while also generalizing some of the...

Stochastic arithmetic has been developed as a model for exact computing with imprecise data. Stochastic arithmetic provides confidence intervals for the numerical results and can be implemented in any existing numerical software by redefining types of the variables and overloading the operators on them. Here some properties of stochastic arithmetic...

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relation...

The paper is devoted to the mathematical modeling of metabolism and nutrition based on enzyme-kinetics reactions described by Michaelis-Menten equations. Proposed are simple models of metabolic processes in a living system which explain certain effects related to various regimes of nutrition and fasting.

The CESTAC method and its implementation known as CADNA software have been created to estimate the accuracy of the solution
of real life problems when these solutions are obtained from numerical methods implemented on a computer. The method takes
into account uncertainties on data and round-off errors. On another hand a theoretical model for this m...

A widely used method to estimate the accuracy of the numerical solution of real life problems is the CESTAC Monte Carlo type
method. In this method, a real number is considered as an N-tuple of Gaussian random numbers constructed as Gaussian approximations of the original real number. This N-tuple is called a “discrete stochastic number” and all it...

In this paper intervals are viewed as approximate real numbers. A revised formula for interval multiplication of generalized intervals is given. This formula will be useful for further axiomatization of interval arithmetic and relevant implementations within computer algebra systems. Relations between multiplication of numbers and multiplication of...

We investigate some algebraic properties of the system of stochastic numbers with the arithmetic operations addition and multiplication by scalars and the relation inclusion and point out certain practically important consequences from these properties. Our idea is to start from a minimal set of empirically known properties and to study these prope...

Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. O...

Hausdorff continuous (H-continuous) functions appear naturally in many areas of mathematics such as Approximation Theory [11],
Real Analysis [1], [8], Interval Analysis, [2], etc. From numerical point of view it is significant that the solutions of
large classes of nonlinear partial differential equations can be assimilated through H-continuous fun...

We show that the operations addition and multiplication on the set $C(\Omega)$ of all real continuous functions on $\Omega\subseteq\mathbb{R}^n$ can be extended to the set $\mathbb{H}(\Omega)$ of all Hausdorff continuous interval functions on $\Omega$ in such a way that the algebraic structure of $C(\Omega)$ is preserved, namely, $\mathbb{H}(\Omega...

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. This result allows us to solve certain practical problems with stochastic numbers and to compare algebraically obtained results with practical applications of stochastic num...

In this work we consider centred zonogons represented implicitly as Minkowski sums of centred segments. Using this representation the order relation inclusion (containment) of zonogons has been studied. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This survey paper aims to promote certain novel mathematical tools, such as computer algebra systems, enclosure methods and interval analysis, to the mathematical modelling and optimization of biotechnological processes.

In the present work we show that the linear operations in the space of Hausdorff continuous functions are generated by an
extension property of these functions. We show that the supremum norm can be defined for Hausdorff continuous functions in
a similar manner as for real functions, and that the space of all bounded Hausdorff continuous functions...

We formulate certain numerical problems with stochastic numbers and compare algebraically obtained results with experimental results provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. The good coincidence between theoretical and...

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. In this work we show how this can be used for the algebraic solution of linear systems of equations with stochastic right-hand sides. On several examples we compare the alge...

In this work the theory of quasivector spaces has been briefly outlined and applied for computation with zonotopes. An approximation problem for zonotopes in the plane has been formulated and an algorithm for its solution has been proposed.

Stochastic arithmetic involving addition and multiplication by scalars is studied with an emphasis on the abstract structure of the set of stochastic numbers. New properties of stochastic numbers are obtained such as a special distributivity relation corresponding to the second distributivity law in a vector space. This allows us to introduce algeb...

Algebraic systems abstracting properties of convex bodies and intervals, with respect to addition and mul-tiplication by scalars, known as quasilinear spaces, are studied axiomatically. We discuss special quasilinear spaces with group structure called quasivector spaces. We show that every quasivector space is a direct sum of a vector space and a s...

Certain algebraic properties of familiar (set-theoretic) interval multiplication are studied diagrammatically. The centred interval multiplication operations (the long known outward multiplication and the newly proposed inward one) are defined and studied diagrammatically in some detail, especially with respect to inclusion isotonicity.

The present work is devoted to computation with zonotopes in the plane. Using ideas from the theory of quasivector spaces we formu- late an approximation problem for zonotopes and propose an algorithm for its solution.

In numerical analysis (absolute) errors can be identified with one-dimensional intervals symmetric with respect to zero. Addition, multiplication and inclusion of errors are well-defined (set-theoretically) in interval analysis. We study axiomatically the algebraic properties of such a system of errors. To this end we introduce and investigate a ne...

The algebraic properties of interval vectors (boxes) are studied. Quasilinear spaces with group structure are studied. Some fundamental algebraic properties are developed, especially in relation to the quasidistributive law, leading to a generalization of the familiar theory of linear spaces. In particular, linear dependence and basis are defined....