CAS Mathematica Research Articles

On Topp-Leone-G Power Series: Saturation in the Hausdorff Sense and Applications

This paper discusses the Topp-Leone-G power series class of distributions. The greatest attention is paid to the investigation of intrinsic characteristic “saturation” to the horizontal asymptote in the Hausdorff sense. Some estimates for the value of the Hausdorff distance are obtained. We present a new family of recurrence generated adaptive functions with corresponding applications. The usefulness of the obtained results is demonstrated in a simulation study of some real data sets from the medical sector and insurance. Some suitable software modules within the programming environment CAS MATHEMATICA are proposed.

On a Class of Orthogonal Polynomials as Corrections in Lienard Differential System: Applications

In this paper we demonstrate some specialized modules for investigating the dynamics of differential models, an integral part of a planned much more general Web-based application for scientific computing. As “corrections” in the Lienard differential system is presented a class of orthogonal polynomials (also known as “shell polynomials”). We will note that some specifics of the amplitudes of these polynomials open up the possibility of modeling signals from the field of antenna-feeder techniques. Algorithms and modules have been consistently used for: automatic generation of a theorem on the number and type of limit cycles (in the light of Melnikov’s considerations); study of the Hamiltonian of the system and “level curves”; for the study of catastrophic surfaces (in the light of Zeeman’s considerations), etc. Similar studies have been carried out for associated polynomials. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

On a Hypothetical Model with Second Kind Chebyshev’s Polynomial–Correction: Type of Limit Cycles, Simulations, and Possible Applications

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 circuits. Numerical experiments, depicting our outcomes using CAS MATHEMATICA, are presented.

A note on the piecewise smooth modified Schnute growth model. Applications

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.

COMMENTS ON SOME INVERTED CUMULATIVE DISTRIBUTIONS: “SATURATION IN THE HAUSDORFF SENSE”, APPLICATIONS

In this note we study properties of some inverted cumulative distri-bution 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 prac-tice. We consider also modified families of adaptive functions with “polynomial variabletransfer” with applications to the Antenna–feeder Analysis. Numerical examples, illus-trating our results using CAS MATHEMATICA are given.

DYNAMICS OF MODIFIED LOTKA–VOLTERRA MODEL WITH POLYNOMIAL INTERVENTION FACTORS. METHODOLOGICAL ASPECTS. III

In the present article is considered a modification of the classicalLotka–Volterra model. The ’input functions alpha(t) and gamma(t) are algebraic polynomi-als of degree n, alpha(t) is monotonically increasing (with ”saturation”) and gamma(t) > 0 ismonotonically decreasing in the considered interval. This model is very sensitive withrespect to the coefficients of the polynomials ai and bi. Therefore it is attractive forconducting computer simulations, including other modified predator-prey models andactivator-inhibitor mechanisms. Numerical examples, illustrating our results are givenusing CAS Mathematica.

A LOOK AT SOME TRIGONOMETRIC-G FAMILIES WITH BASELINE INVERTED EXPONENTIAL (CDF). APPLICATIONS

In this article we study some general classes of trigonometric cumu-lative distribution functions with baseline inverted exponential (cdf). We consideralso modified families of ”adaptive functions” with ”polynomial variable transfer” withapplications to the Antenna–feeder Analysis. We study the ”saturation” - d in theHausdorff sense for some special cases of the families. Numerical examples, illustratingour results using CAS MATHEMATICA are given.Numerical examples using CAS Mathematica, illustrating our results are given.

Properties and Applications of a Tan–G Family of ”Adaptive Functions”

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 ”saturation” - d in the Hausdorff sense for some special cases of this family. The article discusses only model aspects of the possible application that this new family finds in the above area. The models are very sensitive to the coefficients of the polynomial (respectively the number and location of its zeros). This makes these models attractive for simulations. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

A LOOK AT THE MODIFIED SIRD MODELS WITH ”INTERVENTION POLYNOMIAL FACTOR”. METHODOLOGICAL ASPECTS. II

Our research is a natural continuation of previous results in [1]. Theideas given in [1] can be extended for other models of SIRD-type. From a method-ological point of view, we recommend the specialists working in this scientific field tostudy the possibility of using input functions lambda(t) and k_d(t) of polynomial type (forthe MSIRD model [2]). A similar modifications are proposed for the modified modelproposed in [3]. The specialists working in the field of ”reaction-kinetic mechanisms”have the word. Numerical examples, illustrating our results are given using CAS Math-ematica.

Statistical approach for analyzing the transient reliability of boiler System using CAS Mathematica

The focal point of this manuscript is to address the reliability analysis of Boiler unit in the Thermal power plant driven from coal. It is made up of six subunits laid out in series and parallel manner, alongwith some standby units. The system under consideration is modelled in transient state aided by Markov birth-death process and CAS Mathematica is deployed to solve the resulting complicated system of probabilistic equations arising out of 8 working states.

INVESTIGATIONS ON SOME NEW CUMULATIVE DISTRIBUTIONS VIA COSINE AND SINE FUNCTIONS. APPLICATIONS

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 examples, illustrating our results using {\it CAS MATHEMATICA} are given. Numerical examples using {\it CAS Mathematica}, illustrating our results are given.

ON SOME ADAPTIVE G--FAMILIES. APPLICATIONS

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 Hausdorff sense for some special cases of the families.Numerical examples, illustrating our results using CAS MATHEMATICA are given.

PROPERTIES OF A GENERALIZED FAMILY OF DISTRIBUTIONS BY ALTUN ET ALL [1

In this article we study properties of a generalized G family of cu-mulative distribution functions (CDFs) proposed by Altun et all. [1]. More precisely,we prove estimates for the ”saturation” - d about Hausdorff metric. The techniqueused can be successfully applied to other commonly used CDFs in practice. Also weconstruct and study families of recurrence generated adaptive functions based on theG–family. We consider modified families of adaptive functions with ”polynomial vari-able transfer” with applications to the Antenna–feeder Analysis. Numerical examples,illustrating our results using CAS MATHEMATICA are given.

Some Notes on the Omega Distribution and the Pliant Probability Distribution Family

In 2020 Dombi and Jónás (Acta Polytechnica Hungarica 17:1, 2020) introduced a new four parameter probability distribution which they named the pliant probability distribution family. One of the special members of this family is the so-called omega probability distribution. This paper deals with one of the important characteristic “saturation” of these new cumulative functions to the horizontal asymptote with respect to Hausdorff metric. We obtain upper and lower estimates for the value of the Hausdorff distance. A simple dynamic software module using CAS Mathematica and Wolfram Cloud Open Access is developed. Numerical examples are given to illustrate the applicability of obtained results.

Comments on the Half Logistic Inverse Rayleigh (HLIR) cdf with “Polynomial Variable Transfer”. Some Applications

Abstract 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 will define and analyze in detail the new family. We will call this family the “Half-Logistic-Inverse-Rayleigh cdf with Polynomial Variable Transfer” (HLIRPVT) cdf. Section 3 shows the potentiality of proposed new model under four real data sets. Some numerical examples using CAS MATHEMATICA are given.

A New Class of “Growth Functions” with Polynomial Variable Transfer Generated by Real Reaction Networks

Abstract In [4, 5], two classes of growth models with “exponentially variable transfer” and “correcting amendments of Bateman-Gompertz-Makeham-type” based on a specific extended reaction network have been studied [1]. In this article we will look at the new scheme with “polynomial variable transfer”. The consideration of such a dynamic model in the present article is dictated by our passionate desire to offer an adequate model with which to well approximate specific data in the field of computer viruses propagation, characterized by rapid growth in the initial time interval. Some numerical examples, using CAS Mathematica illustrating our results are given.

Some Intrinsic Properties of Tadmor–Tanner Functions: Related Problems and Possible Applications

In this paper, we study some properties of an exponentially optimal filter proposed by Tadmor and Tanner. More precisely, we consider the problem for approximating the function of rectangular type F(t) by the class of exponential functions σadapt(t) about the Hausdorff metric. We prove upper and lower estimates for “saturation”—d (in the case q=2). New activation and “semi-activation” functions based on σadapt(t) are defined. Some related problems are discussed. We also consider modified families of functions with “polynomial variable transfer”. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

A LOOK AT THE INVERSE WEIBULL ”ADAPTIVE FUNCTION”: PROPERTIES AND APPLICATIONS

In this note we study some properties of an new inverse Weibull cu-mulative function proposed by Afify, Shawky and Nassar [1]. More precisely, we proveestimates for the ”saturation” - d about Hausdorff metric. A new activation functionsare defined. We consider also modified families of functions with ”polynomial vari-able transfer” with applications to the Antenna–feeder Analysis. Numerical examples,illustrating our results using CAS MATHEMATICA are given.

Analysis of the Reliability of the Butter-Oil Processing Plant using CAS Mathematica and Maxima

The aim of the present paper is to analysis the reliability of the system by using CAS Mathematica and also, it’s comparative study with CAS Maxima. The butter oil manufacturing plant consists of seven units i.e. separator, pasteurizer, continuous butter making, melting vats, butter oil clarifier, packaging and standby state. Model is developed by using Markov birth-death process. The first order differential equations are derived by using model and then solved for comparative study of reliability to give accuracy in result. By using Mathematica, we calculate probability of each state, which is beneficial to plant owners because better accuracy in result enhance reliability of the plant Graphs are plotted and tables are developed with the help of CAS Mathematica and Maxima, graphs show the variation and tables shows the fluctuations in reliability in comparative manner.

A NOTE ON THE EXTENDED SONG–CHANG–PHAM’S SOFTWARE RELIABILITY MODEL. II.

The determination of compulsory in area of the Software ReliabilityTheory components, such as confidence intervals and confidence bounds, should alsobe accompanied by a serious analysis of the value of the best Hausdorff approximation- the subject of study in the present paper.For example we study the Hausdorff approximation of the shifted Heaviside functionht0(t) by cumulative function based on the extended Song–Chang–Pham’s model [1](see, also [2]).We propose a software module within the programming environment CAS Mathe-matica for the analysis of the considered family of functions.We give real example with dataset using the new extended Song–Chang–Pham’ssoftware reliability model.