Jorge Olivares Funes^{1}
, Elvis Valero^{2}

^{1}Universidad de Antofagasta, Mathematics , Chile

^{2}University Tarapaca, mathematics, Chile

In this work, we will show the algebraic and graphic expressions, which were obtained through the Euler method and Lagrange interpolation with GeoGebra software for some Second-order linear differential equations. These applets were designed for the course of differential equations and numerical calculation for Engineering careers at the University of Antofagasta, Chile.

Jorge Olivares Funes^{1}
, Elvis Valero^{2}

^{1}Universidad de Antofagasta, Mathematics , Chile

^{2}University Tarapaca, mathematics, Chile

In this paper, we will show the approximations, which can be obtained, by means of the spline method of order 1, for Hermite's differential equations With very interactive examples of GeoGebra applets. The GeoGebra program today has many applications and uses, and among them, we will dedicate ourselves to making known, the great benefit that can be obtained, in the process of generating new mathematical knowledge for learning and teaching the numerical solutions of differential equations.

Agustin Vazquez-Sanchez^{1}
, Carlos Alberto Cruz Villar^{2}
, Francisco Delgado^{3}
, Ezequiel Chavez-Alcaraz^{4}
, Jesus Chong-Quero^{5}

^{1}Tecnologico de Monterrey, Mechatronics, Mexico

^{2}Tecnologico de Monterrey, Mechatronics, Mexico

^{3}Tecnologico de Monterrey , Physics and Mathematics, Mexico

^{4}Tecnologico de Monterrey, Mechatronics, Mexico

^{5}Tecnologico de Monterrey, Mechatronics, Mexico

Around the world, Covid-19 outbreak caused a sudden and forced migration from face-to-face education to online education generating an unprecedented phenomenon in the history of education. In Mexico, the most affected Education level was Basic Public Education, the least unprepared while Private Higher Education has experienced by years alternative models using technology. Despite, around the world, new findings arose evidencing that students could require emotional support under the confinement due to the extended lockdown and an intense effort to follow their new educative plans revealing behavioral issues as success factors of that extended online education in the emergent strategy. Based on a statistical model of exploratory factor analysis of data applied to Freshman and Sophomore engineering students, this work presents a roadmap of statistical modelling and testing for the analysis of several dimensions of more effective causal in the success of the forced online education paradigm implementation. Obtaining a Cronbach-alpha value of 0.817, it points to meaningful internal reliability to discriminate and rearrange those causal and dimensions into a more comprehensive education ecosystem.

Authors would like to acknowledge the financial support of Novus Grant with PEP no.PHHT023-1920ZZ00023 and Writinglab, Institute of Future Education, both initiatives ofTecnologico de Monterrey, Mexico, in the development and production of this work.

Carlos Cardoso-Isidoro^{1}
, Francisco Delgado^{2}

^{1}Tecnologico de Monterrey, Physics and Mathematics, Mexico

^{2}Tecnologico de Monterrey , Physics and Mathematics, Mexico

Quantum teleportation is a notable basement of quantum processing. It has been experimentally tested with an outstanding growing success by introducing improvements and applied advances in the last two decades. Its quantum non-local properties have let to discover and to introduce novel implementations based on it in quantum processing, cryptography, quantum resources generation among others. In the current work, we develop a scheme performing double teleportation on two different virtual receivers, while the sender is still able to post-select the final target of teleportation. In addition, we present the theoretical modelling of several interesting effects thus generated, where virtual teleported states are processed and combined to reach certain non-local effects useful in the terrains of cryptography, quantum communication, and quantum resources generation on demand. Last effects require that sender and receivers play certain cooperation roles in the process. Notable effects surpassing their classical analogues are then presented, thus showing the outstanding value of teleportation in the quantum mechanics applications. They are analysed in terms of their parametric behavior, limitations, and scalability.

Authors would like to acknowledge the financial support of Novus Grant with PEP no. PHHT023-1920ZZ00018 and Writinglab, Institute of Future Education, both initiatives of Tecnologico de Monterrey, Mexico, in the development and production of this work.

Fazal Rehman^{1}

^{1}GHSS GUMBAT KOHAT PAKISTAN, Maths, Pakistan

This paper propose geometrical technique to obtain the value of π from splitting a circle in 4 Equilateral triangles 1 square and 4 sectors.It is first ever study to calculate the value of π by splitting a circle in these shapes and then by combining areas of these shapes value of π can be calculated.

Fazal Rehman^{1}

^{1}GHSS GUMBAT KOHAT PAKISTAN, Maths, Pakistan

I am thankful to Mr.Qadeem Khan,Professor Ayub Khan ,Dr.Irshad and Mr.Zaheen Gul for helpful comments.

Jorge Olivares Funes^{1}
, Elvis Valero^{2}
, Pablo Martin^{3}

^{1}Universidad de Antofagasta, Mathematics , Chile

^{2}University Tarapaca, mathematics, Chile

^{3}University Antofagasta, Physical, Chile

Consider $$\frac{d^2y}{dx^2} + P(x)\ y=Q(x,a) $$ , y(0)=y(1)=0 , $$x ,a\in(0,1)$$ In the next paper, we will solve these kinds of differential equations, using Galerkin's numerical method with GeoGebra support. The differential equations that will be shown to me will be those used in various applications of science and engineering.

Abdeen Omer^{1}

^{1}UON, Energy Research Institute (ERI), United Kingdom

Geothermal heat pumps (GSHPs), or direct expansion (DX) ground source heat pumps, are a highly efficient renewable energy technology, which uses the earth, groundwater or surface water as a heat source when operating in heating mode or as a heat sink when operating in a cooling mode. It is receiving increasing interest because of its potential to reduce primary energy consumption and thus reduce emissions of the greenhouse gases (GHGs). The main concept of this technology is that it utilises the lower temperature of the ground (approximately <32°C), which remains relatively stable throughout the year, to provide space heating, cooling and domestic hot water inside the building area. The main goal of this study is to stimulate the uptake of the GSHPs. Recent attempts to stimulate alternative energy sources for heating and cooling of buildings has emphasised the utilisation of the ambient energy from ground source and other renewable energy sources. The purpose of this study, however, is to examine the means of reduction of energy consumption in buildings, identify GSHPs as an environmental friendly technology able to provide efficient utilisation of energy in the buildings sector, promote using GSHPs applications as an optimum means of heating and cooling, and to present typical applications and recent advances of the DX GSHPs. The study highlighted the potential energy saving that could be achieved through the use of ground energy sources. It also focuses on the optimisation and improvement of the operation conditions of the heat cycle and performance of the DX GSHP. It is concluded that the direct expansion of the GSHP, combined with the ground heat exchanger in foundation piles and the seasonal thermal energy storage from solar thermal collectors, is extendable to more comprehensive applications.

Volker Thürey^{1}

^{1}None, Bremen, Germany

In the first part we deal with the question which points we have to connect to generate a nonself-intersectioning polygon. Afterwards we introduce polyholes, which is a generalization of polygons. Roughly spoken a polyhole is a big polygon, where we cut out a finite number of small polygons. In the second part we present two `centers', which we call center and barycenter. In the case that both centers coincide, we call these polygons as nice. We show that if a polygon has two symmetry axes, it is nice. We yield examples of polygons with a single symmetry axis which are nice and which are not nice. In a third part we introduce the Spieker center and the Point center for polygons. We define beautiful polygons and perfect polygons. We show that all symmetry axes intersect in a single point.

Alokmay Datta^{1}

^{1}CSIR-Central Glass and Ceramic Research Institute, Materials Characterization and Instrumentation, India

Spontaneous ordering in a monolayer of three-atom models of lipid molecules at the surface of water has been observed through Monte Carlo simulation. The monolayer is the simplest planar bio-mimic of the cell wall in contact with water. Ordering has been studied in the absence and presence of physiologically relevant cations at different cation ratios to understand the role of the Na/K ratio on cell membrane dynamics. Comparison with experimental results obtained from compression studies of the monolayers is also presented.

Moa'ath Oqielat^{1}

^{1}Al balqa Applied University, Mathematics, Jordan

The goals of the research presented in this paper are first, construction a leaf surface from large real 3D scanned data points using a new hybrid interpolation finite element method so-called a hybrid clough-Tocher cubic polynomial interpolation method (CT-CPI). Secondly, a comparison between the hybrid CT-CPI method and hybrid CT-Taylor series method (CT-TS) for the leaf surface reconstruction is presented. Realistic leaf surfaces models are essential for many applications in the sciences of plant, such as modelling spray and spreading droplet movement on the surface, photosynthesis and a canopy light environment or it can be implemented for visual purposes only. For these goals, a precise mathematical depiction of the boundary and surface is mandatory. Although an operative method is to apply either CT-TS or CT-CPI algorithm to recreate the surface of the leaf from 3D scanned data, difficulties occur when dealing with Anthurium leaves, which tend to have many branches. To solve this issue, we implemented interpolating in combined with the CT method. Our algorithm uses finite element methods to represent the surface as a mesh of triangles. Numerical results confirm that the CT-CPI techniques produces more realistic virtual representations of Anthurium leaves than using CT-TS method.

yousef yousefi^{1}

^{1}Payam-e-Noor, physics, Iran, Islamic Republic Of

The temperature dependence of the magnetic susceptibility of the Mn12 magnetic molecular cluster is investigated in different fields. Until now, only dipole excitations were considered in calculations to calculate magnetic susceptibility, but due to the spin number of this molecular cluster and to obtain higher computational accuracy, more multipolar excitations had to be included in the calculations. For this purpose, in this paper, in the calculations related to the magnetic susceptibility of the molecule, in addition to dipolar excitations, quadrupole excitations are also considered, and the relevant diagrams are drawn. Calculations show that the results obtained in a situation where quadrupole excitations are considered, are more consistent with the experimental results.

Jorge Olivares Funes^{1}
, Pablo Martin^{2}
, Elvis Valero^{3}

^{1}Universidad de Antofagasta, Mathematics , Chile

^{2}University Antofagasta, Physical, Chile

^{3}University Tarapaca, mathematics, Chile

Bessel's spherical functions have had many important applications in engineering and optics and science. In this work that is a continuation of The error function in fractional differential equations, showed how to solve the fractional differential equation $$\frac{d^\alpha y}{{dx}^\alpha}= j_0(x)$$, $$y^{\left(k\right)}(0)= 0$$ ,$$ k=0..m-1$$, with m=1,2,3... . Where the nonhomogenous part is the function Bessel Spherical j0 (x)

Paola Lecca^{1}

^{1}Free University of Bozen-Bolzano, Faculty of Computer Science, Italy

In this study we aim to answer through a mathematical model and its numerical simulation the question whether the kinetic rate constants of chemical reactions are influenced by the strength of gravitational field. In order to calculate the effects of gravity on the kinetic rate constants, we correct the mathematical model of the expression of these constants known from the collision theory of chemical kinetics, recasting it in the context of general relativity. The reformulation of the classical chemical kinetic model is based on the remodelling of (i) the Maxwell-Bolzann distribution and (ii) the collision frequency between molecules/atoms of chemical species due to time dilation. The model predicts that the relative velocity between the colliding molecules/atoms is higher at higher gravitational potential. Using numerical simulations, we quantitatively estimate the orders of magnitude by which the kinetic constants vary as a function of varying the strength of the gravitational field. In addition, we present a numerical estimation of the two effects affecting the rate constant, namely the spatial reorganisation of the particle distribution due to gravitational force and the effect of time slowing down as the intensity of the gravitational field increases.

Mohammad Hatami^{1}

^{1} K.N. Toosi University of Technology, Physics, Iran, Islamic Republic Of

In this work, we numerically investigate the sheath dynamics of an electropositive plasma around a defect on the cathode by using the hydrodynamics equations. It is assumed that the plasma consists of electrons and singly charged positive ions. The finding of this work has a great importance in plasma ion implantation technique.

Andrei Melekhin^{1}

^{1}Moscow State University of Civil Engineering, heat, gas supply and ventilation, Russian Federation

The author has developed an engineering calculator for calculating the heat flow for heating buildings according to the enlarged parameters of the object. The algorithm of the calculator is based on the method of determining the amount of heat energy and heat carrier in water supply systems of urban heat supply. The author carried out a systematic analysis of the heat loads on the heating of buildings in Russia according to the data of implemented construction projects. With this in mind, new coefficients a, n were calculated to determine the specific thermal characteristics of the building for newly constructed buildings. Adjusted the algorithm for calculating the heat supply of buildings according to the enlarged parameters of the object. The calculation algorithm is implemented in the software product using DHTML programming.

Tomasz Kuczerski^{1}

^{1}Military Institute of Armament Technology, Zielonka, Poland

The paper describes a practical application of parallel computing using state superposition. Simulations are conducted in Python language using computational libraries. The author presents the possibilities of using ready-made computational libraries to implement modern algorithms based on superposition of states. As a result, the results of the algorithm implemented using parallel computing are presented. Furthermore, the possibilities of using the above algorithms to accelerate classical computations are described.

Paola Lecca^{1}
, Angela Re^{2}

^{1}Free University of Bozen-Bolzano, Faculty of Computer Science, Italy

^{2}Fondazione Istituto Italiano di Tecnologia,, Centre for Sustainable Future Technologies, Environment Park - Parco Scientifico Tecnologico per l’Ambiente, Italy

This study presents an asymptotic analysis of a model of a bioreactor converting carbon monoxide (CO) gas into ethanol through a C. autoethanogenum biocatalyst [1]. The configuration is a bubble column reactor with co-current gas-liquid flows where gas feed is introduced a gas distributor placed at the bottom of the column. A pure culture of C. autoethanogenum is subsequently injected at the bottom of the column; therein cells are dispersed in the liquids and consume the dissolved gas and release by-products such as ethanol and acetic acid. A spatial gradient establishes in the column since gas concentration decreases as gas flows up due to cellular consumption. Consequently, cellular growth and byproduct secretion are affected by spatially varying dissolved gas concentrations. The model accounts for four species representing the biomass, the CO substrate in the liquid phase, and two by-products - ethanol and acetic acid. While the equations of the ethanol and acetate dynamics are taken from Chen’s work [2], in our model the dynamics of the substrate is described by an advection-diffusion equation with no sink/source terms depending on the biomass. This allows the calculation of closed-form solution for the substrate dynamics, that is used in the calculation of the bacterial growth rate [3]. We investigate the asymptotic stability of this solution, i.e. the stability of the solution at the top of the bioreactor. The asymptotic stability property is particularly important to ensure the usability of measurements made in regions of the bioreactor away from the microorganism and nutrient input (output) point to establish controllability and/or optimal controllability of the bacterial fermentation process. The controllability of the system in turn allows the possibility of modulating the ethanol production efficiency. In the absence of asymptotic stability, the exploitation of physical mass transport processes other than free advection and diffusion is not recommended. Possible scenarios will be proposed and analysed. References [1] Norman, R.O.J, Millat T., Winzer, K., Minton,, N.P., Hodgman, C. Progress towards platform chemical production using Clostridium autoethanogenum. Biochem Soc Trans. 2018; 46(3):523-535. doi: https://doi.org/10.1042/BST20170259 [2] Chen J., Gomez,J.A., Hoeffner, K., Barton, P.I., Henson, M.A.: Metabolic modeling of synthesis gas fermentation in bubble column reactors. Biotechnol. Biofuels 8(1) (2015). https://doi.org/10.1186/s13068-015-0272-5 [3] Lecca P., Re A.: Observability of Bacterial Growth Models in Bubble Column Bioreactors. In: Cazzaniga P., Besozzi D., Merelli I., Manzoni L. (eds) Computational Intelligence Methods for Bioinformatics and Biostatistics. CIBB 2019. Lecture Notes in Computer Science, vol 12313 (2020). Springer, Cham. https://doi.org/10.1007/978-3-030-63061-4_27

Aubain Nzokem^{1}

^{1}York University, Mathematics & Statistics, Canada

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the probability density function and its derivatives; and broadly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed and the distributional proprieties are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modeling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY historical data, the SPDR S\&P 500 ETF (SPY). The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the cumulative distribution of the sample data than the CLM. The best VG model comes from the FRFT estimation.

Irina Andreeva^{1}

^{1}Peter the Great St.Petersburg Polytechnic University, Higher Mathematics, Russian Federation

A family of differential dynamic systems is considered on a real plane of their phase variables x, y. The main common feature of systems under consideration is: every particular system includes equations with polynomial right parts of the third order in one equation and of the second order in another one. These polynomials are mutually reciprocal, i.e. their decompositions into forms of lower orders do not contain common multipliers. The whole family of dynamic systems has been split into subfamilies according to the numbers of different reciprocal multipliers in the decompositions and depending on an order of sequence of different roots of polynomials. Every subfamily has been studied in a Poincare circle using Poincare mappings. A plan of the investigation for each selected subfamily of dynamic systems includes the following steps. We determine a list of singular points of systems of the fixed subfamily in a Poincare circle. For every singular point in the list we use the notions of a saddle (S) and node (N) bundles of adjacent to this point semi trajectories, of a separatrix of the singular point, and of a topo-dynamical type of the singular point (its TD – type). Further we split the family under consideration to subfamilies of different levels with proper numbers. For every chosen subfamily we reveal topo-dynamical types of singular points and separatrices of them. We investigate the separatrices’ behavior for all singular points of systems belong to the chosen subfamily. Very important are: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question of a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all subfamilies of studied systems. The presented work is devoted to the original study. The main task of the work is to depict and describe all different in the topological meaning phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family under consideration, and to its numerical subfamilies of different hierarchical levels. This is a theoretical work, but due to special research methods it may be useful for applied studies of dynamic systems with polynomial right parts. Author hopes that this work may be interesting and useful for researchers as well as for students and postgraduates. As a result we depict phase portraits of dynamic systems of a given family and outline the criteria of every portrait appearance.

Yuriy Gubarev^{1}
, Yang Liu^{2}

^{1}Lavrentyev Institute for Hydrodynamics, , Russian Federation

^{2}Novosibirsk State University, Department for Differential Equations, Russian Federation

The Vlasov-Poisson model of boundless collisionless electron gas in self-consistent electric field continues to be one of the basic models for a number of modern physics areas, such as particle physics, electrodynamics, plasma physics, etc. This is due to simplicity, clarity, and obvious effectiveness of the model in describing complicated processes of the micro world. For example, the Vlasov-Poisson model is very successfully used for development and subsequent operation of accelerators with colliding beams, which make it possible to accelerate elementary particles additionally by means of hot electron gas. Despite the fact that this model has been intensively studied for a long time, from the point of view of the mathematical stability theory, it was possible to establish, by and large, only sufficient conditions for the theoretical stability (at semi-infinite time intervals) of a number of dynamic equilibrium states with respect to both small and finite perturbations, but from some incomplete unclosed subclasses. In this report, we consider spatial motions of boundless collisionless electron Vlasov-Poisson gas in three-dimensional Cartesian coordinate system: $$ \frac{\partial f}{\partial t} + v_i\frac{\partial f}{\partial x_i} + \frac{\partial \varphi }{\partial x_i}\frac{\partial f}{\partial v_i} = 0, $$ $$ \frac{\partial ^2\varphi }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f({\bf x}, {\bf v}, t)d{\bf v}; (1) $$ $$ i = 1, 2, 3; f = f({\bf x}, {\bf v}, t) \geq 0; f({\bf x}, {\bf v}, 0) = f_0({\bf x}, {\bf v}). $$ Here $f$ denotes the distribution function of electrons (for reasons of convenience, their charges and masses are assumed to be equal to unity); $t$ is time; ${\bf x} = (x_1, x_2, x_3)$ and ${\bf v} = (v_1, v_2, v_3)$ denote coordinates and velocities of electrons; $\varphi ({\bf x}, t)$ is the potential of self-consistent electric field; $f_0({\bf x}, {\bf v})$ denotes the initial data for function $f$. We suppose that the distribution function $f$ asymptotically approaches zero as $|{\bf v}| \rightarrow \infty $, and this function along with the potential $\varphi $ are periodic in argument ${\bf x}$ or asymptotically approach zero as $|{\bf x}| \rightarrow \infty $ too. Summation is performed on repeating lower index $i$ throughout the report. It is assumed that the mixed problem (1) has the following exact stationary solutions: $$ f = f^0({\bf x}, {\bf v}) \geq 0, \varphi = \varphi ^0({\bf x}); $$ $$ v_i\frac{\partial f^0}{\partial x_i} = \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^0}{\partial v_i}, (2) $$ $$ \frac{\partial ^2\varphi ^0}{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^0({\bf x}, {\bf v})d{\bf v}. $$ The aim of this report is to prove the absolute linear instability for the spatial states of dynamic equilibrium (2) of boundless collisionless electron Vlasov-Poisson gas with respect to small three-dimensional perturbations $f^\prime ({\bf x}, {\bf v}, t)$ and $\varphi ^\prime ({\bf x}, t)$: $$ \frac{\partial f^\prime }{\partial t} + v_i\frac{\partial f^\prime }{\partial x_i} + \frac{\partial \varphi ^\prime }{\partial x_i}\frac{\partial f^0}{\partial v_i} + \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^\prime }{\partial v_i} = 0, $$ $$ \frac{\partial ^2\varphi ^\prime }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^\prime ({\bf x}, {\bf v}, t)d{\bf v}; (3) $$ $$ f^\prime ({\bf x}, {\bf v}, 0) = f_0^\prime ({\bf x}, {\bf v}), $$ where $f_0^\prime ({\bf x}, {\bf v})$ denotes the initial data for function $f^\prime $. In the report, a transition from kinetic equations (1) which describe the spatial motions of electron gas under study to an infinite system of relations similar to the equations of isentropic flow of a compressible fluid medium in the vortex shallow water and Boussinesq approximations was carried out. In the course of instability proof, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of dynamic equilibrium states (2) with respect to one incomplete unclosed subclass of small spatial perturbations was conversed. Also, some linear ordinary differential second-order inequality with constant coefficients was obtained for the Lyapunov functional. An a priori exponential lower estimate for growth of small three-dimensional perturbations (3) follows from this inequality when the sufficient conditions for linear practical instability of the considered dynamic equilibrium states found in this report are satisfied. Since the obtained estimate was deduced without any additional restrictions on the dynamic equilibrium states under study, then the absolute linear instability of the spatial states (2) of dynamic equilibrium of boundless collisionless electron Vlasov-Poisson gas with respect to small three-dimensional perturbations (3) was thereby proved. The report results are fully consistent with the classical Earnshaw instability theorem from electrostatics. This theorem states that any equilibrium configuration of point electric charges is unstable if, besides its own Coulomb forces of attraction and repulsion, no other forces act on them. At present, the area of applicability for the Earnshaw theorem is expanded from electrostatics to kinetics, namely, to the boundless collisionless electron Vlasov-Poisson gas. Finally, constructiveness is inherent in the sufficient conditions for linear practical instability established here, which allows them to be used as a testing and control mechanism for conducting physical experiments and performing numerical calculations.

Authors would like to acknowledge the financial support of China Scholarship Council.

Aubain Nzokem^{1}

^{1}York University, Mathematics & Statistics, Canada

We are interested in describing the infected size of the SIS Epidemic model using Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined in a constant size ($M$) population. The life span of each individual in the population is modelled by an exponential distribution with parameter $\alpha$, and the disease spreads within the population is modelled by a Poisson process with a rate $\lambda_{I}$. $\lambda_{I}=\beta I(1-\frac{I}{M}) $ is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number $R=\frac{\beta} {\alpha} $ is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows the infected size dynamics through sample-path simulations and the relationship $R$. As $M$ becomes large, some stable statistical characteristics of the infected size distribution appear. And the infected size is shown analytically to follow a normal distribution with mean $(1-\frac{1}{R}) M$ and Variance $\frac{M}{R} $

I would like to express my special thanks to Prof.\ Neal Madras for providing advice and feedback on this article

Julio Espinoza-Ortiz^{1}
, Roberto Lagos-Monaco^{2}

^{1} Federal University of Catalão, Physics, Brazil

^{2}Universidade Estadual Paulista, Departamento de Física-IGCE, Brazil

We consider a Bunimovich like quarter-stadium by softening the non zero y-boundary. The smoothing is performed via an exponent monomial potential, the system becomes not completely reflective but preserves the particle's translation and rotational motion. Increasing the exponent value, the stadium's boundaries become rigid and thus the system's chaoticity increases. We consider a leaking soft stadium family by considering an opening limited region, located at some place of its basis's boundary, throughout which the particles can leak out. We chase the particle's trajectory and focus on the stadium transient behavior by mean of the statistical analysis of the survival probability, belonging to the marginal orbits that never leave the system, the so called bouncing ball orbits. A comparison of these family orbits is done with the billiard's transient chaos orbits.