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

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}Universidad de Tarapacá, matemáticas, Chile

In this paper, we will show the solutions of certain exact differential equations that are obtained through the interactive GeoGebra software. GeoGebra software has been of great motivational support in the processes of teaching and modeling in mathematics in various universities and colleges, especially in the engineering careers of the University of Antofagasta in the courses of differential equations and calculation of several variables.

Jaykov Foukzon^{1}

^{1} Israel Institute of Technology , Center for Mathematical Sciences, Technion - Israel Institute of Technology. , Israel

Novel explanation of the Active Galactic Nuclei.The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon. J.Foukzon¹, E.Menkova² A.Potapov³ ¹Department of mathematics, Israel Institute of Technology, Haifa, Israel ²All-Russian Research Institute for Optical and Physical Measurements, Moscow,Russia ³Kotel'nikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia In this paper we argue that the current paradigm for AGN and quasars essentially incomplete and rivision is needed. Remind that the current paradigm for AGN and quasars is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed 10¹¹ to 10¹² K. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10.The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than 5.3Gλ, must indeed exceed 2×10¹³K and can reach as high as ~ 3×10¹⁴K. As well known, these visibilities correspond to the structural scales of 30-40 μas oriented along position angles of 25°-30°.These values are indeed close to the width of the inner jet and the normal to its direction.The observed, T_{b,obs}, and intrinsic, $$T_{b,int}$$, brightness temperatures are related by Eq.1: $$T_{b,obs}=δ(1+z)⁻¹T_{b,int}$$ with δ=7.2.The estimeted by Eq.1 a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of $$T_{b,int}>2.910¹² K$$. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about 10¹²K [1].In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below 10¹²K [1]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of $$T_{b,int}>2.910¹²K$$ is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [1] and at least several times larger than the limit established by inverse Compton cooling. Remark 1.Note that if the estimate of the maximum brightness temperature given in [1], is closer to actual values, it would imply $$T_{b;int}=5×10¹³K$$. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons. Remark 2. However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerated it to sublight speed. We argue that these alternative models such as emission from relativistic protons can be suported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities [2]. [1] J. L. Gómez, A. P. Lobanov, G. Bruni, Y. Y. Kovalev,Probing the innermost regions of AGN jets and their magnetic fields with Radio Astron.I.Imaging BL Lacertae at 21 microarcsecond resolution. Astrophysical journal 817 (2016) 96, DOI:10.3847/0004-637X/817/2/96 arXiv:1512.04690 [astro-ph.HE] [2] J.Foukzon,E.Menkova,A.Potapov, Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance.110 pp. Published November 14, 2019 ISBN-13 (15) 978-93-89562-22-4 https://doi.org/10.9734/bpi/mono/978-93-89562-22-4

SANTANU CHATTERJEE^{1}
, Sanjoy Mukherjee^{2}

^{1}RGM International Pvt. Ltd., Civil Department, India

^{2}Vikram Solar Limited, Innovation, India

String theory provides an encouraging way to unify all force fields in our universe into a single framework. Different vibrational patterns of a single string resemble different particles. Bosonic strings require 26 spatial dimensions in order to produce particles in a similar fashion as 10 spatial dimensions are required to produce Fermions states. These extra dimensions (beyond 3 spatial & 1-time dimension) are compactified into very small scale and thus in today’s scale of probing energy it is not possible to detect them experimentally. This idea of extra dimensions, hidden from our perception is tempting & worth in-depth theoretical work. Certain topological features and its application on String theory will also be discussed in this paper. We will try to fathom these small plank scale compactified dimensions & will try to throw some light on the various topological aspects of quantum geometry offered by this remarkable theory.

SANTANU CHATTERJEE^{1}
, Sanjoy Mukherjee^{2}

^{1}RGM International Pvt. Ltd., Civil Department, India

^{2}Vikram Solar Limited, Innovation, India

In today’s era of modern medicines, mathematical modeling plays an important role in analyzing difficult aspects of cell divisions and proliferation to speculate the result of the actual experiment. In these concepts of modeling, differential equation as a technique for determining relation between a function and its derivatives played a very important role. If we know a function and few of its derivatives at a particular point then that information along with differential equation can plot the function over its entire domain [1]. By choosing suitable parameters & boundary conditions different types of differential equations can be formed to represent actual biological process and up-to the complexity levels as we may desire. Thus, mathematical modeling of metastatic cancer & growth of cancerous cell has become an active area of research. In normal cells, hundreds of genes intricately control the process of cell division. Cells become cancerous after mutations accumulate in the various genes that control cell proliferation. In other words, they no longer respond to most of the signals that control cellular growth and death. Mutations in genes can cause cancer by accelerating cell division rates or inhibiting normal controls on the system, such as cell cycle arrest or programmed cell death [2]. In this paper will investigate spherical tumor considering three types of cell formation or layer of cell inside it, namely, proliferating cell layer, quiescent cell layer & necrotic cell layer and we will study their properties & dynamics of expansion or contraction with respect to time. Instead of oxygen (as in; Greenspan, H. (1972)), we will model our case with flow of nutrients that are required for a living cell to be alive and proliferate. Through a simple linear 1st order differential equation, different stages of tumor growth and invasion of tumor cells to surrounding tissue in the form of micro metastatic cells will also be investigated. Our approach will be to modify the 1st order differential equation for standard population dynamics (growth & death type) with considerable changes in the parameters to catch up all three earlier mentioned phases or layers of a growing tumor.

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

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}University of Antofagasta, physics department, Chile

^{3}Universidad de Tarapacá, matemáticas, Chile

The fractional derivative of Caputo, has huge and important applications in various areas of science and engineering. In this case, through the definition of the Caputo derivative and the Laplace transform and its inverse, we propose to show the solutions that can be obtained for each specific value of alpha of the following fractional differential equations $$\frac{d^\alpha y}{{dx}^\alpha}= I_3/4(x)$$ , $$\frac{d^\alpha y}{{dx}^\alpha}= I_-3/4(x)$$ , with $$m-1<α\le m$$ , $$m\ \in N $$. Where the non-homogeneous parts I_3 / 4 (x) and I_-3/4 (x) are the modified Bessel functions of the first species.

MARÍA ROJAS^{1}
, Jorge Olivares Funes^{2}
, Daniza Rojas^{3}

^{1}UNIVERSIDAD DE ANTOFAGASTA, MATEMÁTICAS, Chile

^{2}University of Antofagasta, Department of Mathematics, Chile

^{3}University of Antofagasta, Department of Mathematics, Chile

GeoGebra and Wolfram Αlpha are educational software used in many areas of science, engineering and mathematics. In this work we will use these programs to see the homogeneous reducible differential equations along with their solutions. The present material that will be presented corresponds to the one designed for the course of differential equations for engineering and pedagogy careers in mathematics at the University of Antofagasta-Chile.

Fedor Grigoriev^{1}

^{1}M.V. Lomonosov MSU, Russian Federation, RCC, Russian Federation

Glancing angle deposition (GLAD) is one of the technique for the fabrication of the anisotropic thin films with high porosity and low refractive index. In this technique the incoming flux of the deposited atoms is directed almost parallel to the substrate surface. Due to density fluctuation in the deposited atoms flow and shadow effect, it results in formation of the different separate nanostructures - tree-like column, slanted column, chevron-like structures and so on – on the substrate. GLAD films are widely used in the optical coating due to low reflectance and anisotropy properties. The structural and optical properties of GLAD-films essentially depend on their fabrication conditions. Experimental investigation of these dependencies is still challenge for the existing experimental techniques. On the other hand due to the progress in high performance computing, now it is possible to study thin films deposition process using the atomistic simulation. In the present work the anisotropy of the GLAD SiO2 films is investigated using the classical atomistic simulation and anisotropic Bruggeman effective medium theory. The depolarizing factors defining the difference of the refractive index components are calculated based on the geometry parameters of the pores between the slanted columns forming the large-scale GLAD structure. Averaged shape parameters of these ellipsoids are defined using the density gradient tensor. It is revealed that the values of difference of main components of refractive index tensor are equal 0,03 and 0,05 for deposition angles 60 grad and 80 grad, if the free volume fraction is calculated using the dependence of film density on the deposition angle. The obtained values coincide with experimental results for silicon dioxide films. Simulation is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.

The work is supported by the Russian Science Foundation (grant number 19-11-00053).

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

^{1}University of Antofagasta, Department of Mathematics, Chile

^{2}University of Antofagasta, physics department, Chile

^{3}Universidad de Tarapacá, matemáticas, Chile

Consider $$A∂z/∂y+B∂z/∂x=0$$, Where A, B 𝜖 R. We will solve these types of linear partial differential equations using the GeoGebra Graphical Calculator and their solutions will be seen by GeoGebra 3D. The use of GeoGebra in the study of PDE is a subject that is still being studied and developed with great potential.

Dimitrios Vlachos^{1}

^{1}University of Peloponnese, Department of Informatics and Telecommunications, Greece

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