Anton Iliev

Plovdiv University "Paisii Hilendarski" · Department of Computer Technologies

The focus of this work is to create a distributed software environment for scientific computations using iterative algorithms and microservice architecture with containers in a cloud infrastructure. This article has two main objectives: first, to analyze and define the necessary requirements for building this kind of software system, including theo...

Following the ideas given in (Kyurkchiev in Int J Differ Equ Appl 21(1):1–17, 2022 [5]), in this article, we study a hypothetical piecewise smooth Gompertz growth function \(G(g_1(t),g_2(t))\). Precise bounds for the Hausdorff distance d between the Heaviside step function \(h_0\) and the sigmoid G are given. The applicability of the new model is p...

A great number of mathematical models of physical systems give rise to differential equation of the Lienard’s–type. Various modifications of this model have been proposed and studied by a number of researchers. In this Book (Part II) we consider extended Lienard system with some ”corrections of polynomial–type”. Number and type of limit cycles, sim...

In this article we consider a new extended Lienard-type system with "corrections" of the first kind Chebyshev's polynomial T_n. The number and type of limit cycles in the light of Melnikov's consideration are also studied. We will explicitly note that the y(t)-components of the differential systems can be used successfully in modeling and approxima...

In this note we will make further computational improvements of Harris algorithm [2, 12]. We improve speed using the technique of least absolute remainder [1]. Numerical experiment give us confidence that we receive new enhanced algorithm.

In this paper we organize new refined enhanced hybrid extended algorithm for searching finding greatest common divisor. We expand the algorithm introduced by us in [24]. For regular numbers extended Eu-clidean algorithm is frequently used [10], [12], [14], [18], [21], [26]. For long numbers binary algorithms are more adequate [22]. Our new algorith...

In this article a model with Chebyshev polynomials of the fifth kind C n (x) as corrections in the Lienard differential system is presented. The type of limit cycles in the light of Melnikov's consideration and level curves are studied. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

In this article we extent the results from [23]. Namely now we will present extended algorithms which work for every integer numbers a != 0 and b != 0.

In this research we present new extended algorithm which is generalization of Tembhurne-Sathe algorithm [3] as well as our previous enhanced versions of Tembhurne-Sathe algorithm [11] and [29]. Also we give new faster organizations of classical Tembhurne-Sathe algorithm [3].

In this article a model with Dickson polynomials of the (m+1)-th kind D n,m (x, a) as corrections in the Lienard differential system is presented. The level curves are studied. Numerical examples, illustrating our results using CAS MATHEMATICA are given. Key Words: Lienard differential system, Dickson polynomials of the (m + 1)-th kind as "correcti...

In this article, we explore a new extended Lienard-type planar system with “corrections” of the second kind Chebyshev’s polynomial Un. The number and type of limit cycles are also studied. The discussion on the y(t)—component of the solution of the Lienard system is connected to searching for the solution of the synthesis of filters and electrical...

In this article a model with Dickson polynomials of the third kind as corrections in the Lienard differential system is presented. The level curves are studied. The model is also considered in the light of Melnikov's approach. Numerical examples, illustrating our results using CAS MATHEMATICA are given. Key Words: Lienard differential system, Dicks...

In this article a model with Dickson polynomials (of first and second kind) as corrections in the Lienard differential system is presented. The model is considered in the light of Melnikov's approach. The level curves are also studied. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

In this article a hypothetical oscillator model with Gegenbauer poly-nomials as corrections in the Lienard differential system is presented. The model is considered in the light of Melnikov's approach. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

In the present article we consider a new modified three-parameter Kies c.d.f. F * (t) = 1 − e −k t 1−k 1 t a where 0 < t < 1 k 1 , k > 0, k 1 > 0 and a is a positive integer. We consider the following hypothetical reaction network: Y ρa(t) −→ X a where ρ a (t) = kat a−1 (1−k 1 t) a+1 is the "rate function". The new model can be written for the grow...

There are a lot of results about the maximum number of limit cycles of Lienard system: x ′ = y; y ′ = g(x) + ǫf (x)y. In this article we offer a natural summary of this dynamic model with a polynomial type correction factors f (x) = fn(x) = [ n 2 ] i=1 (−1) i+1 x n−2i − x n n for n = 3, 7, 11, 15, 19,. .. , (see [8]) and ǫ = n i=1 ǫit i. Some modif...

In this paper we consider a new extended Lienard-type planar system with the polynomial P 2n+1 of the best Hausdorff approximation of the function sgn (x). The number of limit cycles is also studied. The discussions are related to solving some technical problems such as the synthesis of antennas and electrical circuits.

In this note we obtain new hybrid algorithm for finding greatest common divisor (gcd) of two natural numbers a and b. For regular numbers Euclidean algorithm possess good speed [10], [17], [18]. For long numbers binary algorithms are appropriate [19], [20]. The reason of this research is receiving of so called hybrid algorithms which are useful and...

In this note we extend the algorithm in [21] to obtain other new hybrid algorithm for finding greatest common divisor (gcd) of natural numbers a and b. For regular numbers Euclidean algorithm is quite appropriate [10], [17], [18]. For long numbers binary algorithms are extremely useful [19], [20]. Hybrid algorithms are appropriate for the numbers w...

Based on detailed research in [6] new extended oscillator model, in this article we offer a natural summary of this dynamic model with a polynomial type correction factor c(t). We propose a software module within the programming environment CAS Mathematica for the analysis of the considered model. The considered methodological aspects can be succes...

In this note we develop new extended algorithm which generalize Harris algorithm [1], [11]. Our new computational process demonstrates benefits as superior speed and correct results obtained through so-called "hybrid" extended algorithm. We will explicitly note that algorithm in this article is first hybrid extended algorithm known in literature.

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.

A great number of mathematical models of physical systems give rise to differential equation of the Lienard’s-type. Under certain conditions it can be shown that Lienard’s equation has a limit cycle. This result is known as Lienard’s Theorem. It is known that checking the conditions of Lienard’s theorem we find that the Van der Pol’s equation has a...

In this note we obtain new extended algorithm, which is based on idea of least absolute remainder [1], [15]. Numerical experiments demonstrate its superior speed in comparison to Knuth classical algorithm for the same task for regular numbers. By this research we enrich, diversify and extend the theory and practice of so-called Euclidean algorithms...

In this article we study a new class of the growth model by Baranyi and Roberts with ”polynomial variable transfer”. We consider also some applications to the Population Dynamics, Debugging and Test Theory. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

Following the ideas given in [8]-[10] in this article, we analyze a understudied model, such as the Oluyede, Chipepa and Wanduku [6] model. It is shown that the model can be modified in view of their possible application for approximation of data from a real test of software modules. Some numerical examples, using CAS MATHEMATICA are also given.

Following the ideas given in [18], in this article we study a hypothetical piecewise smooth extended Gompertz growth function B(b 1 (t), b 2 (t)). More precise we study the saturation with the new class to the horizontal asymptote with respect to the Hausdorff distance. Some numerical examples, using CAS MATHEMATICA are also given. Studies in this...

Following the ideas given in [1]-[3], in this article we study more sophisticated growth models of the type: h_1 (t) = A − e ^ {1−kt−e^{kt}} and their "hypothetical piecewise smooth functions". Some numerical examples, using CAS MATHEMATICA are also given.

Following the ideas given in [4]-[6] in this article, we analyze some understudied models, such as the Almalki and Bakouch models. It is shown how these models can be modified in view of their possible application for approximation of data from a real test of software modules and platforms. Some numerical examples, using CAS MATHEMATICA are also gi...

This monograph contains three parts. Part I presents some new classes of growth functions generated by reaction networks and based on ''correcting amendments of fractional linear function--type''. The hypothetical smooth growth function gives very good results in approximating data sets in the fields of debugging theory and the propagation of compu...

In this article a hypothetical piecewise smooth Log-Logistic sigmoidal model F (f1(t), f2(t)) is defined. A new class of growth function generated by reaction networks and based on "correcting amendments of fractional linear function-type" is proposed. Some numerical examples, using CAS MATHEMATICA illustrating our results are given.

Following the ideas given in [2] in this paper we define a hypothetical piecewise smooth generalized sigmoidal model Q(q 1 (t), q 2 (t)) using the generalized logistic model. We will discuss precise bounds for the Hausdorff distance d between the Heaviside function h 0 and the sigmoid Q. Studies in this paper can also be applied to a random shifted...

We present new binary extended algorithms that work for every integer numbers a and b for which a != 0 and b != 0. The approach given here generalizes and optimizes the algorithm given in the monograph of A. Menezes, P. Oorschot and S. Vanstone [39] as well our results from [28] and [20]. These computation ways demonstrate high computational effect...

In this note we study properties of some inverted cumulative distribution functions (CDFs). More precisely, we prove estimates for the "saturation"-d about Hausdorff metric using two-parameters generalized inverted exponential c.d.f. The technique used can be successfully applied to other commonly used CDFs in practice. We consider also modified fa...

In the last few years, there have been serious studies in the literature related to the proposed general classes of trigonometric distributions. Various modifications of this ''powerful'' class of functions have been proposed and studied by a number of researchers. We will note that the ''sine potential correction'' can be used to construct other f...

In this article we consider new modified logarithmic transformed adaptive G-families (LTAGM). Some properties for special classes of the families (with "fractional linear correction" and "Sin-G correction) are studied. We study also the "saturation" in the Hausdorff sense for some special cases of the families. Numerical examples, illustrating our...

In this article we study a new class of COS-G family with baseline cumulative function of Volmer-type. We consider also modified transmuted family of "adaptive functions". A justification of the idea for generating the powerful classes of trigonometric-G families is given. Numerical examples, illustrating our results using CAS MATHEMATICA are given...

In this article we study some general classes of trigonometric cumulative distribution functions with baseline inverted exponential (cdf). We consider also modified families of "adaptive functions" with "polynomial variable transfer" with applications to the Antenna-feeder Analysis. We study the "saturation"-d in the Hausdorff sense for some specia...

In this note we study some properties of a new family of adaptive functions. More precisely, we prove estimates for the "saturation"-d about Hausdorff metric. A large proportion of the observed computer viruses are characterized by rapid growth over a relatively short period of time, after which gradual cumulative saturation usually occurs. This wa...

The paper traces the development and scientific growth of the members of the Department of Computer Technology at the FMI of the University of Plovdiv “Paisii Hilendarski” for 25 years since its establishment in January 1996. The scientific directions in which the lecturers work, the leading positions that they held in FMI and PU, the trained...

In this article we will consider some methodological aspects related to the possibility of generating new classes of adaptive functions based on the generalized transmuted family proposed by Shaw and Buckley [1]. Some applications are also given. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

In this article we study some general classes of trigonometric cumulative distribution functions. We consider also modified families of "adaptive functions" with "polynomial variable transfer" with applications to the Antenna-feeder Analysis. We study the "saturation"-d in the Hausdorff sense for some special cases of the families. Numerical exampl...

In this article we study some general classes of trigonometric cumulative distribution functions. We consider also modified families of "adaptive functions" with "fractional linear correction". Here we will also focus on a hypothetical adaptive function, which we will call the "difference adaptive function" (DAF). We study the "saturation"-d in the...

In this article we study properties of a generalized G family of cumulative distribution functions (CDFs) proposed by Altun et all. [1]. More precisely, we prove estimates for the "saturation"- d about Hausdorff metric. The technique used can be successfully applied to other commonly used CDFs in practice. Also we construct and study families of re...

In this note we study some properties of an new TAN–G class of trigonometric cumulative distribution functions proposed by Souza, O. Junior, de Brito, Chesneau, Fernandes and Ferreira [1]. We consider also modified families of ”adaptive functions” with ”polynomial variable transfer” with applications to the Antenna–feeder Analysis. We study the ”sa...

Our research is a natural continuation of previous results in approximating specific data of a strictly exponential nature (e.g., COVID-19 Bulgaria, Cuba, China, South Korea, etc.) using modified logistics and other models in which the typical reaction constants were replaced by "polynomial variable transfer" and showed good results in performing t...

Our research is a natural continuation of previous results in [1]. The ideas given in [1] can be extended for other models of SIRD-type. From a methodological point of view, we recommend the specialists working in this scientific field to study the possibility of using input functions λ(t) and k d (t) of polynomial type (for the MSIRD model [2]). A...

We present new extended algorithms, which work for every integer numbers a and b. These algorithms show high computational effectiveness for regular and long numbers.

In this note we redesign and decrease number of operations in the algorithm [59]. Our benefit from this reduction is the faster calculation of modular multiplicative inverse for the approach from [59].

We develop a novel modification of the classic Kermack–McKendrick SIR (Susceptible–Infectious–Recovered) model. The new SIR model with ”intervention polynomial factor” (SIR-IPF) can be used successfully to model and play different scenarios for the infectious disease spread. A generalized ”reproduction number” is introduced. A similar modifications...

We present new extended algorithms, which are based mainly on ”remainder” and ”sum” operations. Additionally we use the variable for counting the number of the sign changing. These algorithms are quite appropriate for regular as well as long numbers.

In many papers [10]–[44] we present effective iterative and recursive schemes for the all classes of algorithms that concern the classical task of finding greatest common divisor. Here we present a new algorithm for finding modular multiplicative inverse, which is based on combination of ”remainder” and ”difference” operations. The approach present...

We develop new efficient realization of Kronecker symbol calculation. Numerical experiments show that our approach leads to more than 10 times faster algorithm in comparison even to commercial CAS. We will point out that presented here algorithm extents the results by Bach and Shallit [48] for computation the Jacobi symbol in terms of the simple co...

In this paper we present new iterative and recursive versions of Kronecker symbol binary algorithm. Our approach extents the results of Shallit and Sorenson [48] binary algorithm for computation of Jacobi symbol in terms of the simple continued fraction of a rational number a/b because now we give the more general solution in such terms for binary...

In this article we study the Hausdorff approximation of the Haar scaling function by sigmoidal scaling functions. We prove upper and lower estimates for the value of the Hausdorff distance d. A simple dynamic software module using CAS Mathematica and Wolfram Cloud Open Access is developed. Numerical examples are given to illustrate our results.

In [1] H. Bakouch consider a G–family of extended cumulative distribution function (cdf): Fa(t)=aG(t)a+1−G(t);a>0, where G(t) is the baseline cdf. In particular case G(t) = 1 − e−kt, k > 0 we find the following Extended–Bakouch Half–Logistic cdf (EBHL-cdf): Fa(t)=a(1−e−kt)a+e−kt. Similar to our previous studies [2]–[4], in this article we will de...

In this research we present new iterative and recursive versions of Jacobi symbol binary algorithm. Our result speeding up the algorithm BinaryJacobi given in [46].

In this paper we present both iterative and recursive versions of Jacobi symbol algorithm. Our algorithm enhances the algorithm Jacobi2 given in [46].

On the base of the Half Logistic-G family of distributions proposed by Cordeiro, Alizadeh and Marinho [2] some mathematical properties are investigated by Almarashi et al. [1]. We study the "saturation" to the horizontal asymptote: t=1 by the new growth function M(t) in the Hausdorff sense. Similar to our previous studies [3-6], in this article we...

In this note we construct a family of recurrence generated sigmoidal functions based on the Verhulst logistic function with polynomial variable transfer. We prove estimates for the Hausdorff approximation of the Heaviside step function by means of this family. Numerical examples, illustrating our results are given.

In this paper we consider a new mean value function with "polynomial variable transfer". The use of the new model with many free parameters makes it attractive for analysis and approximation of specific data from Software Reliability Growth Analysis and Debugging and Test Theory. Some approximation problems related to the "saturation" in Hausdorff...