## Conference submissions

#### Resolving non-homogeneous linear differential equations using undetermined coefficients and variation of parameters by means of GeoGebra

Jorge Olivares Funes 1

1Universidad de Antofagasta , Departamento de Matemáticas , Chile

###### Abstract

In this paper, we show how non-homogeneous linear differential equations, especially those of the second order, are solved by means of GeoGebra applets by indeterminate coefficient methods and variation of parameters, for the course of differential equations of engineering students. and pedagogy in mathematics from the University of Antofagasta in Chile. The free software GeoGebra has caused that it is increasingly used in the teaching of mathematics, especially in non-homogeneous linear differential equations, because it facilitates the teaching and learning process.

#### Quantum Field Theory in fractal space-time with negative dimension.

Jaykov Foukzon1

1Israel Institute of Technology, Department of mathematics, Israel

###### Abstract

We introduce Hausdorff-Colombeau measure in respect with negative fractal dimensions. Axiomatic quantum field theory in spacetime with negative fractal dimensions is proposed.Spacetime is modelled as a multifractal subset of $R^{4}$ with positive and negative fractal dimensions.The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum field theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The classical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff $E$ and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff- Colombeau dimensions gives high energy cutoff on natural way. In order to obtain disered physical result we apply the canonical Pauli-Villars regularization up to $E$. It means that there exist the ghost-driven acceleration of the univers hidden in cosmological constant. http://arxiv.org/abs/1004.0451

#### The numerical analysis placement method with GeoGebra in linear second order differential equations

Jorge Olivares Funes 1 , Luis Cortés Vega2 , Pablo Martin3 , Elvis Valero4

1Universidad de Antofagasta , Departamento de Matemáticas , Chile
2University of Antofagasta, matemáticas, Chile
3University of Antofagasta, Physics department, Chile

###### Abstract

In this article we will show how to find approximate solutions, using the numerical analysis placement method with the GeoGebra software to the second order linear differential equations of the form d^2y/dx^2+A(x)dy/dx+B (x)y=Q (x), y (0) = y (a) = 0, where "a" is a positive number. The use of GeoGebra in the numerical analysis allows us to simultaneously, interactively and dynamically view the solutions and approximations of the differential equations.

#### The error function in fractional differential equations

Jorge Olivares Funes 1 , Pablo Martin2 , Fernando Maass3 , Elvis Valero4

1Universidad de Antofagasta , Departamento de Matemáticas , Chile
2University of Antofagasta, Physics department, Chile
3University of Antofagasta, Physics department, Chile

###### Abstract

Fractional differential equations have a great importance and application. That is why the relationship between the fractional derivative and the erf (x) function will be shown below. The objective of this work is to solve the fractional differential equation D ^α y (x) =erf (x), and y(0) = 0, where 0<α<1. We will show the type of generalized hypergeometric solutions obtained by defining the fractional derivative of Caputo and the Laplace transform.

#### New quasi-rational aproximation for the modified Bessel functions I1(x).

Pablo Martin1 , Jorge Olivares2 , Adrian Sotomayor3

###### Abstract

New and more accurate approximation to the modified Bessel function I1 has been found by improving the multipoint quasi -rational approximation method, MPQA. The approximation obtained in previous work (1) , has been improved by using the hyperbolic function sinh, instead of cosh. This change also the structure of the approximation , but there is not change in the structure of the approximation , and the number of parameters is also equal.Three terms of the power series and one term of the asymptotic expansion are also used to obtain the parameters of the approximation. In this way there is a decreasing of the relative error from 0.011 to 0.007 . A detail explication of the new procedure is carry on in this presentation. Ref. P. Martin, J. Olivares and A. Sotomayor, “ Precise Analytic Approximation for the Modified Bessel Function I1”, Rev. Mex. Física 63 (2017) 130- 133 .

#### The solution classical feedback optimal control problem without the Bellman Equation

Jaykov Foukzon1

1Israel Institute of Technology, Department of mathematics, Israel

###### Abstract

A new approach, which is proposed in this paper allows one to construct the Bellman function V(t,x) and optimal control u(t) directly,i.e.,without any reference to the Bellman equation, by way of using strong large deviations principle for the solutions Colombeau-Ito's SDE.

#### Solution to the Troesch Problem for Boundary Equations.

Franco Lindstron1

1Universidad Nacional de La Plata, Matemática, Argentina

###### Abstract

This paper shows, for the first time, that the explicit and exact solution to the Troesch nonlinear twopoint boundary value problem may be computed in a direct and straightforward fashion from the general solution obtained by a generalized Sundman transformation for the related differential equation, which appeared to be a special case of a more general equation. As a result, various initial and boundary value problems may be solved explicitly and exactly.

#### Inverse operator of a chaotic dynamics

yehuda roth1

1Oranim college, science, Israel

###### Abstract

It is known that a dissipative environment is well described by the chaotic process while regular dynamics is associated with animate systems. In this paper, we explore the inverse map of some chaotic maps to find that they are always regular. The result that by reversing a chaotic map we obtain a regular process is associated with the birth of animate systems.

#### Modelling meson clouds using coherent states

Manuel Fiolhais1

1University of Coimbra, Department of Physics, Portugal

###### Abstract

The use of coherent states to describe boson systems goes back to the 1960's in the context of the radiation field. Since the 1970's, they have also been applied to meson clouds, mainly pions, in the context of the description of baryons by means of effective models involving a quark core surrounded by scalar and pseudo-scalar mesons. The use of coherent states allows for an {\em ab-initio} quantum mechanical description of the mesons, therefore going beyond semi-classical approximations. The coherent state e.g. for p-wave pions (with angular momentum and isospin quantum numbers both equal to 1) is given by $|\psi > = {\cal N}(\xi) \exp (\sum_{tm} \xi_{tm} a^\dagger _{tm} ) |B>$ where ${\cal N}$ is a normalization factor, $| B >$ is a bare baryon state and $\xi_{tm}$ are amplitudes to be determined variationally. The $a^\dagger _{tm}$ is the creation operator for a pion state with angular momentum third component, $m$, and isospin third component, $t$. The radial profile of the pion amplitude results from a variational calculation and it is frozen. Hence, only angular momentum and isospin matters to construct the coherent state above. As already mentioned, the idea of mathematically modelling the meson clouds by means of coherent states, having in mind a full quantum mechanical description of baryon systems in the framework of chiral effective models, is not new. Actually, the author, among others, published several papers on the topic, in order to obtain various properties of the nucleon, the delta resonance and other excited states. However, the goal here is to bring together many aspects that are scattered in the literature, focusing on the versatility of the coherent states and stressing their capabilities. In this study, instead of the more realistic chiral effective models of quarks and mesons, we use a toy model whose Hamiltonian is written as $H= \sum_{tm} a^\dagger_{tm} a_{tm} + G \sum _{tm} B_{tm} \left[ a_{tm} + (-1)^{t+m} a_{-t-m}^\dagger \right]\!,$ where $B_{tm}$ is a baryon spin-isospin operator. The model describes a system of non self-interacting pions linearly coupled to a bare baryon core, $G$ being the coupling constant. This model is simple enough for its exact solutions to be worked out in the strong and weak regimes. These accurate solutions are then compared with the variational approximate solutions. Because the multi-particle coherent state, $|\psi>$, cannot directly describe a nucleon, with definite angular momentum and isospin quantum numbers $\left( J={1\over 2}, I={1\over 2} \right)$, the Peiers-Yoccoz angular momentum (and isospin) projection method is used to construct a state, $|\psi_N>$, with the proper nucleon quantum numbers. The variational method consists in minimizing the energy with respect to the amplitudes, i.e. $d< H > / d \xi_{tm}=0$, with the normalization condition $<\psi_N|\psi_N>=1$ dully implemented in the process. We show that the so-called hedgehog configuration for the quark core and for the pion amplitudes minimizes the mean-field energy. On the other hand, we show that the (Peierls-Yoccoz) projected coherent state is an extremely powerful ansatz since it reproduces the accurate solutions of the model both in the strong coupling regime (which is not surprising) but also in the weak coupling regime. We emphasise the use of the variation-after-projection method, for which the variational Hilbert space is larger, therefore with the trial function spanning a larger space than in the simpler variation-before-projection method. The toy model turns out to be a valuable tool to test different approaches which might be used in more realistic models with, for instance, self-interacting mesons.

#### Free vibrations of isotropic FG porous annular and elastically restrained plate using DQM

Yajuvindra Kumar 1

1Government Girls Degree College, Behat, Mathematics, India

###### Abstract

In this paper, author studied free vibrations of a functionally graded (FG) annular plate having porosity. The plate is elastically restrained along the boundary. The material properties of the plate are Porosity dependent. An even porosity distribution is taken in the analysis. The mathematical model of the problem is developed using the concept of physical neutral surface of the plate. The physical neutral surface is taken as the reference plane. Out of many, only first three natural frequencies of the plate are reported using differential quadrature method (DQM). A parametric study is conducted to show the effects of porosity and material distribution parameters on the vibration behavior of the plate.

#### Integration of electromagnetic methods of intuba-tion of stratified mediums on the basis of direct and alternating currents

Yuriy Dimitrienko1 , Igor Krasnov2 , Kirill Zubarev3

1Bauman Moscow State Techical University, Fundamental sciences, Russian Federation
2Bauman Moscow State Techical University, Fundamental sciences, Russian Federation
3Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

###### Abstract

In this work the integration of two methods of electroinves-tigation is considered. One method represents intubation by a direct current, the second intubation by alternating cur-rent. The integration is carried out for the purpose of in-crease in accuracy of results of the solution of the inverse task, the problem is solved in a twodimensional approxima-tion. The direct task for the first and second method is solved numerically. The received values were compared with the experimental datas. The inverse task is formulated as a problem of minimization with the functional considering the experimental values received by both the first and second method of electroinvestigation. The problem of optimization is solved on a compact (for each parameter are set top and bottom border).

#### The soft stadium’s classical dynamics

Julio S Espinoza-Ortiz1 , Roberto E Lagos2

1Federal University of Goias, Physics, Brazil
2UNESP, Rio Claro, SP, Departamento de Fı́sica, IGCE, Brazil

###### Abstract

Billiards are physical models employed to probe experiments that measure the conductivity of quantum dots. In this context, the stadium billiard have been adopted as an standard model for realizations. We study the effect of softening this system in the classical mechanics, pursuing for a more realistic model. This classical approach is a first step towards the truly quantum or semiclassical case. We define the soft stadium as a monomial potential with an exponent {$\alpha\in\Re$} as a parameter, such that for {$\alpha=1$} the system is integrable and the {$\alpha\rightarrow\infty$} limit it converges to the hard billiard. Then, and for computational simplicity, we set up the construction of the classical Poincare map in such a way that it only depends on the partial separability of the system which holds for all {$\alpha$}'s. We present numerical results describing the classical transition from the integrable regime towards the chaotic regime.

###### Acknowledgements:

The authors would like to thank the support of the Goi\'as Research Foundation - FAPEG.

#### Dynamics of a particle periodically driven in the deformable potentiels: stochastic resonance

1University of Yaounde 1, Department of Physics, Cameroon

###### Abstract

In connection with stochastic resonance (SR), we study the dynamics of a particle in the deformable travelling-wave potentials in the presence of the external excitation force and the thermal fluctuations force. We model the deformation of the systems by the i) asymmetric deformable on-site potential (ASDP) and ii) double well deformable on-site potential (DWDP). The phenomenon of SR is known to take place in sinusoidal and nonsinusoidal systems. However, the question of the appeareance of SR in the ASDP as well as DWDP systems has not been resolved. The cooperative effect of noise and external force does show up in these systems. This numerical work presents the characterization of SR through an investigation of the input energy lost by the system to the environment per period of the external force which is also equivalent to the hysteresis loop area or average input energy. SR is characterized by the presence of a peak when the temperature increases. A double SR is observed in the ASDP case, first peak occurring at weak temperature has nothing to do with usual mechanical resonance. But it just associated to intra-well dynamics. However, second peak arising at higher temperature, relates to a classical SR phenomenon. In the DWDP case, only one resonance peak is observed. In both the systems the average input energy of occurrence of SR nonmonotonically depends of the shape parameter. We show that at low temperature the input energy depends very strongly on the initial positions of the particle. For each of the two models, this input energy is confined to two narrow bands in some range of the shape parameter. The input energy distribution of these is also explored. As function of the shape parameter, it can be unimodal or bimodal. Using the DWDP system, we investigated the presence of Chaos in the system in the goal to show that the disappearance of SR in the system can be due to Chaos.

#### FILTERING AND PARALLEL DIFFUSIVE FRACTAL CHARACTERIZATION OF 2-DIMENSIONAL IMAGES

Hafedh Zghidi1

1Silesian University of Technology , Institute of Informatics, Poland

###### Abstract

The article presents a complete solution for filtering and diffusive fractal characterization of 2-dimensional images. This includes preparing the sample by subtracting background, application of random walk procedure and its parallelization using two different approaches. For each technique the processing time is measured to compare speedups with regard to a sequential implementation. To prove the correctness of the results, a black square is used as the reference sample, for which diffusive fractal dimension is known and equls 2. Finally the results for a complex image are elaborated.

#### Modeling the growth of a neural network consisted of diferent types of neural cells,

Pantea Davoudifar1 , Keihanak Rowshan Tabari2

1Research Institute for Astronomy and Astrophysics of Maragha, Astroparticle Physics, Iran (Islamic Republic of)
2Research Institute for Astronomy and Astrophysics of Maragha, , Iran, Islamic Republic Of

###### Abstract

In space physics the use of living organisms is not always possible. To study the environment condition, here a method were developed to create a network of given neurons. Different geometrical structures of the neurons were built using biological constraint. The fluence dose due to cosmic radiation were studied for the resulted structures. A factor of survival were defined and the structures were studied under short and long term radiation dosimetry. The effect of solar cycles and solar events were studied on radiation environment.

#### Using the fuzzy sets for estimating the angular velocity of a small spacecraft rotation motion

Andry Sedelnikov1 , Ekaterina Khnyryova2

1Samara National Research University, , Russian Federation
2Samara national research university , Further Mathematics , Russian Federation

###### Abstract

To know and understand the conditions of carrying out technological processes it is necessary to estimate the rotational motion parameters of the spacecraft. The parameters of the AIST small spacecraft rotational motion around its center of mass were estimated using measurement data of current from solar panels. At the same time, there is a problem in interpretation the telemetry data from small spacecraft: sometimes the significant current was recorded on two opposite solar panels. The paper shows a way to solve this problem using the fuzzy sets. As a membership function it is offered to use the normality condition of the direction cosines. The processing of telemetry data is given for AIST small spacecraft prototype. The offered approach can significantly increase the accuracy of angular velocity estimating using measurements of current from solar battery.

#### Theory and Model of Technological Hype Cycles

Avi Messica1 , Asnat Greenstein-Messica 2

1COMAS, Finance and Quantitative Methods, Israel
2Ben-Gurion University of the Negev, Data Science and Information Systems, Israel

###### Abstract

A new emerging technology, viewed as disruptive, occasionally generates a surge of public expectations over its potential application. This collective excitation (and decay) is generated and diffuses in a complex array of large random networks (e.g. social, media) that are difficult to model via small world models. Former studies of this phenomenon – termed as hype cycle - have focused mainly on descriptive, few case-studies, analysis using a corpus of newspaper articles and explained specific dynamics in a specific context. Motivated by the lack of a mathematical model, we studied a simple two-phase mean field model that is able to explain the dynamics, as well as various patterns, of correlated expectations. Our contribution is as follows, we used an online query data (via Google Trends) as a proxy for public expectations to study more than one hundred technologies. We extended the classification of the diffusion pattern with three new categories to better reflect different observed dynamics. Lastly, we present a data-driven mathematical model that enables to draw useful insights on the rich dynamics of hype cycles.

#### Coherent upper conditional expectations defined by Hausdorff outer measures to make prevision in complex systems

Serena Doria1

1University G.d'Annunzio Chieti-Pescara, Department of Engineering and Geology, Italy

###### Acknowledgements:

This work is supported by RFBR grant no. 18-29-10008 mk.

#### DARK MATTER: REALITY OR A RELATIVISTIC ILLUSION?

Asher Yahalom1

1Ariel University, Electrical & Electronic Engineering, Israel

###### Abstract

Galaxies are huge physical systems having dimensions of many tens of thousands of light years. Thus any change at the galactic center will be noticed at the rim only tens of thousands of years later. Those retardation effects seems to be neglected in present day galactic modelling used to calculate rotational velocities of matter in the rims of the galaxy and surrounding gas. The significant differences between the predictions of Newtonian instantaneous action at a distance and observed velocities are usually explained by either assuming dark matter or by modifying the laws of gravity (MOND). In this presentation we will show that taking general relativity seriously without neglecting retardation effects one can explain the radial velocities of galactic matter without postulating dark matter.

#### EPRB Paradox Resolution. Bell inequalities revisited

Jaykov Foukzon1

1Israel Institute of Technology, Department of mathematics, Israel

###### Abstract

One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell's inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell's inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate.We show that the correlations of the observables involved in the Bohm--Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality reads |++-|≤6.

#### Study of quantum tunneling through two sequential barriers

Zuleima Carrasco1

1Escuela Superior Politecnica de Chimborazo, Physics Research Group - ESPOCH, Ecuador

###### Abstract

Quantum tunneling theory is widely used for many applications in different areas from the fundamental research to technological applications. In particular, the quantum tunneling appears in scanning tunneling microscope, nuclear physics, quantum computing, tunnel diode, among other. Here, we show a simple mathematical derivation of the one-dimensional quantum transmission considering two particular cases: (i) for comparison purposes, the conventional quantum tunneling with one potential barrier and (ii) the quantum tunneling with two potential barriers considering a small distance between barriers, in which a pseudo-potential is created and the particle is non-free. As known, the quantum transmission in the first case, depends only on the incident particle energy (kinetic energy), the mass of particle, the energy barrier (potential energy) and the barrier length that the particle passes through. However, on the second case, an extra dependency is observed, i.e., the separation distance between barriers (the Harmant effect is cancel). We have corrobarated that the first case evidence the major quantum transmition probability respect to second one. Aditionally, we evidence that transmition probability can be modulate and controlled by changing the separation distance between barriers, which produces a region where the particles can be free or trapped. If this discovery is confirmed by future experiments, such as waveguide experiments, it could enhance the particle manipulation at nanometer scale improving the image resolution.

###### Acknowledgements:

I would thanks Physics Research Group - ESPOCH and Dr. Cristian Vacacela Gomez, Ms. Noemi Moreta, Dennys Colcha and Dr. Richard Pachacama for the opportunity to present this work at the event, which help me to increase my expertise and aptitudes taken from my undergraduate career.