Jorge Olivares Funes ^{1}

^{1}Universidad de Antofagasta , Departamento de Matemáticas , Chile

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.

Jaykov Foukzon^{1}

^{1}Israel Institute of Technology, Department of mathematics, Israel

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

Jorge Olivares Funes ^{1}
, Luis Cortés Vega^{2}
, Pablo Martin^{3}
, Elvis Valero^{4}

^{1}Universidad de Antofagasta , Departamento de Matemáticas , Chile

^{2}University of Antofagasta, matemáticas, Chile

^{3}University of Antofagasta, Physics department, Chile

^{4}Universidad de Tarapacá, matemáticas, Chile

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.

Jorge Olivares Funes ^{1}
, Pablo Martin^{2}
, Fernando Maass^{3}
, Elvis Valero^{4}

^{1}Universidad de Antofagasta , Departamento de Matemáticas , Chile

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

^{3}University of Antofagasta, Physics department, Chile

^{4}Universidad de Tarapacá, matemáticas, Chile

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.

Pablo Martin^{1}
, Jorge Olivares^{2}
, Adrian Sotomayor^{3}

^{1}Universidad de Antofagasta, Física, Chile

^{2}Universidad de Antofagasta, Matemáticas, Chile

^{3}Universidad de Antofagasta, Matemáticas, Chile

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 .

Jaykov Foukzon^{1}

^{1}Israel Institute of Technology, Department of mathematics, Israel

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.

Franco Lindstron^{1}

^{1}Universidad Nacional de La Plata, Matemática, Argentina

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.

yehuda roth^{1}

^{1}Oranim college, science, Israel

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.

Manuel Fiolhais^{1}

^{1}University of Coimbra, Department of Physics, Portugal

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.

Yajuvindra Kumar ^{1}

^{1}Government Girls Degree College, Behat, Mathematics, India

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.

Yuriy Dimitrienko^{1}
, Igor Krasnov^{2}
, Kirill Zubarev^{3}

^{1}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

^{2}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

^{3}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

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).

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

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

^{2}UNESP, Rio Claro, SP, Departamento de Fı́sica, IGCE, Brazil

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.

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

Jorge Olivares Funes ^{1}
, Pablo Martin^{2}
, Fernando Maass^{3}
, Elvis Valero^{4}

^{1}Universidad de Antofagasta , Departamento de Matemáticas , Chile

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

^{3}University of Antofagasta, Physics department, Chile

^{4}Universidad de Tarapacá, matemáticas, Chile

The fractional differential equations together with the modified Bessel functions have different applications in both Engineering and Electromagnetism. In this paper we will show how through the derivative of Caputo we can find explicit solutions for D ^ α y (x) = I_3 / 4 (x), y (0) = Dy(0)=0 and D ^ α g (x) = K_3 / 4 (x), g (0) = Dg(0)=0 , where 1 <α <2.

Yannick Joel Wadop Ngouongo^{1}

^{1}University of Yaounde 1, Department of Physics, Cameroon

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.

Hafedh Zghidi^{1}

^{1}Silesian University of Technology , Institute of Informatics, Poland

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.

Pantea Davoudifar^{1}
, Keihanak Rowshan Tabari^{2}

^{1}Research Institute for Astronomy and Astrophysics of Maragha, Astroparticle Physics, Iran (Islamic Republic of)

^{2}Research Institute for Astronomy and Astrophysics of Maragha, , Iran, Islamic Republic Of

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.

Andry Sedelnikov^{1}
, Ekaterina Khnyryova^{2}

^{1}Samara National Research University, , Russian Federation

^{2}Samara national research university , Further Mathematics , Russian Federation

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.

Avi Messica^{1}
, Asnat Greenstein-Messica ^{2}

^{1}COMAS, Finance and Quantitative Methods, Israel

^{2}Ben-Gurion University of the Negev, Data Science and Information Systems, Israel

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.

Serena Doria^{1}

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

A new mathematical model of coherent upper conditional expectations based on Hausdorff outer measures is proposed in a metric space $(\Omega,d)$ to make prevision when the conditioning events are fractal sets, i.e. sets with non-integer Hausdorff dimension. The necessity to propose a new tool to define coherent upper conditional expectations arises because they cannot be obtained as extensions of linear conditional expectations defined, by the Radon-Nikodym derivative, in the axiomatic approach (Billingsley, 1986); it occurs because one of the defining properties of the Radon-Nikodym derivative, that is to be measurable with respect to the $\sigma$-field of the conditioning events, contradicts a necessary condition for the coherence. Given a partition $\textbf{B}$ of $\Omega$ for every $B \in \textbf{B}$ denote by $s$ the Hausdorff dimension of $B$ and let $h^{s}$ be the Hausdorff $s$-dimensional Hausdorff outer measure associated to the coherent upper conditional expectation $. For every bounded random variable $X$ a coherent upper conditional expectation $\overline{P}(X|B)$ is defined by the Choquet integral with respect to its associated Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise if the conditioning event has Hausdorff outer measure in its Hausdorff dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability.

Zhong-Cheng Liang^{1}

^{1}Nanjing University of Posts and telecommunications, College of Electronic and Optical Engineering , China

Point-like and wave-like particles are foundations of classical and modern physics. They are so idealized models that lose the physical authenticity. Real particles are three-dimensional body with mass and volume. The spatial intersection of real particles is empty set, thus ensuring limited density of particles. In the centre-of-mass frame of reference, a real particle has three independent modes of motion: vibration, rotation and translation. Three mode energies $(H,L,K)$ constitute a Cartesian space. The energy space is quantized by vibrating quantum $(H_s=Y_s V_s=hv)$, rotating quantum $(L_s=I_s ω_s^2=lz)$ and translating quantum $(K_s=M_s u_s^2=kT)$. The energy space is divided into six phases and three zones. Three zones represent gas, solid and liquid state of object. There are three equilibrium surfaces, each of which has two stable areas and two excited areas. Two types of phase transition are distinguished by two types of phase interface. Complete energy relations and differential equations can be derived through the statistics of particle ensemble. The results show that the order parameters of liquid, solid and gas are the correlative functions of the particle mass $(M)$, the rotary inertia $(I)$ and the elastic modulus $(Y)$, respectively. Thermodynamic laws are the natural inferences of the theoretical results. A physical theory based on the real particle model has simplicity, consistency and universality.

Mayken Espinoza-Andaluz^{1}

^{1}ESPOL, Centro de Energías Renovables y Alternativas, Ecuador

The energy demand to supply micro devices has been increasing during the last years. Considering the power output of the laminar flow microfluidic fuel (LFFC), it appears as a suitable solution to provide the required electrical energy in small devices. Absence of electrolyte and not requirement of platinum as catalytic material are two of the most important advantages of this type of fuel cells. The current study aims to provide a detailed information about the design and characteristics of a LFFC. A complete analysis of the different shape channels has been considered in this study being selected the best option as the Y-shaped channels. The impact of the inclination degree for the inlet channels has been considered to evaluate the average velocity that the flow can acquire into the channel. In addition, the voltage-current behaviour considering the materials, fuel/oxidant and design characteristics has been obtained from a modelling point of view.

The authors kindly acknowledge the financial support from FIMCP-CERA-05-2017. In addition, Åforsk project No 17-331 is gratefully acknowledged.

Vania Rangelova^{1}

^{1}Technical University Sofia - branch Plovdiv, Electrical Engineering, Bulgaria

The biosensor amperometric transducers can work in the case of three basic types of reversible inhibitor enzyme systems – with competitive inhibition, with non-competitive inhibition and mixed inhibition. Tipicaly they work in static mode. Now they are investigated in dynamic mode. The kinetic in those type biosensors is generally discussed in terms of a simple extension to the Michaelis-Menten reaction scheme. The investigated biosensors are amperometric product sensitive. The parameters for simulations are chosen from some real experiments with biosensors. The models are described in non stationary diffusion conditions. Solving system of non-linear partial differential equations is reseived in three dimensional size and the concentration profiles in active membrane of substrate S(x,t), inhibitor I(x,t) and product P(x,t) are reseived. The systems of non-linear differential partial equations have been solved numerically in MATLAB medium. The influence of starting concentration of substrate, inhibitor and kinetic parameters - reaction constants of biosensors for substrate and for inhibitor over output current have been investigated.

Yuriy Dimitrienko^{1}
, Kirill Zubarev^{2}
, Igor Krasnov^{3}

^{1}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

^{2}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

^{3}Bauman Moscow State Techical University, Fundamental sciences, Russian Federation

This article discusses the results of solving the problem of electrical exploration using direct current. The developed methods and algorithms for solving the direct and inverse exploration tasks are tested on several inhomogeneous models of the environment. The task is considered in three-dimensional approximation. To solve using the finite element method. On the basis of the direct problem algorithm, a method for solving the inverse problem was implemented, which consists in finding the minimum of the deviation functional, which in turn leads to the multiple solution of the direct problem. The obtained results were analyzed, the advantages and shortcomings of the developed methods were revealed, the evaluation of the search time for the optimal solution for the inverse problem was carried out, and the dependence of the solution accuracy on the thickening of the grid when solving the problem by the finite element method was discussed.

Sabine Upnere^{1}

^{1}Riga Technical University, Institute of Mechanics, Latvia

The numerical modelling of cross-flow through the rods bundle with triangular arrangement has been done to analyse flow-induced forces on the rod located in the middle of the bundle. Significant problems of rods in the bundle during the operational time of the system can be caused by the cross-flow. At the same time, it is known that the behaviour of the system is strongly related to many parameters such as bundle geometry, flow, rods support and others. Therefore, there is needed to investigate the characteristics of each type of typical bundles. In this paper is analysed flow-induced hydrodynamic forces in closely-packed rods bundle using Computational Fluid Dynamics. Unsteady Reynolds Averaged Navier-Stokes equations are solved using Finite Volume discretization. The impact of the size of the computational domain and the number of rows in it was investigated to find the optimal case for numerical modelling. Obtained results are compared with references from literature and experimental data.

Ashraf Zaher^{1}

^{1}American University of Kuwait, Electrical and Computer Engineering, Kuwait

This paper proposes an efficient technique for detecting mechanical faults in three-phase induction motors, without using mechanical sensors. Only measurements of the currents of every phase are used to identify the fault. The proposed system can diagnose two types of faults corresponding to shaft misalignment or imbalance, along with normal operation. The power spectrum of the experimental data is generated, followed by applying a soft-computing mathematical algorithm that will extract the peaks of the fundamental frequencies and their harmonics, while filtering out noise. These peaks will be compiled in a vector form such that it can be used as inputs to train an artificial neural network (ANN) to produce a decision regarding the operating condition of the motor, via applying intelligent pattern recognition techniques. Mathematical details regarding the structure of the ANN, its training, tuning of its synaptic weights, and the testing/validation phase will be investigated. Detailed analysis of the obtained results is provided to highlight the advantages and limitations of the proposed algorithm. In addition, a comparison is made with similar techniques that use mechanical sensors to contrast their differences and highlight the superiority of the proposed system. The obtained results prove the intelligence and robustness of the proposed system and allows for versatile extensions that promote its application in real-time scenarios for many industrial applications.

This work was supported by a grant from the American University of Kuwait, during the academic year 2018-19.

Santanu Chatterjee^{1}
, Sanjoy Mukherjee^{2}

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

^{2}KEC International Limited, Engineering Services, India

For decades, scientists have worked relentlessly to move forward to see what lies beyond the third dimension and to find out if there is any existence of a unified theory to explain all the workings of the universe from sub-atomic to gigantic inhabitants of cosmos. This insurmountable task has been taken on by many over the last century or so until the emergence of “super string theory” or “string theory” happened showcasing the fact that an answer seemed possible. From the concepts of a “string”, our ideation on this theory started. In this paper we strived to put forward that the concept of a vibrating one-dimensional microscopic object named “string” can be taken as a fundamental ingredient (in place of point particle) for developing non-relativistic quantum mechanics. Unlike point particle, we take a vibrating string as the quantum object & build a perfectly reasonable quantum mechanical description of the microscopic world. Our main objective in this paper to show that complete development of quantum mechanics is possible based on one dimensional open string.

Santanu Chatterjee^{1}
, Sanjoy Mukherjee^{2}

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

^{2}KEC International Limited, Engineering Services, 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.

Nadir MAGHLAOUI^{1}

^{1}Higher School of Applied Sciences, Physics, Algeria

Phased-array transducer are becoming of common use in ultrasound imaging either in medical applications or in the field of non destructive technique. The modelling of the acoustic field emitted by phased-arrays transducers can be realized by different methods developed in the frequency domain or in the time domain. The method proposed here consists in using the Rayleigh integral method where the reflection at the plane interface is taken into account by using the reflection coefficients for harmonic plane waves. The transient field is obtained by an inverse Fourier transform of the harmonic results. The results obtained put in evidence the fact that the ultrasonic waveforms reflected by a liquid-solid interface and detected by the phased-arrays transducer depend strongly on the geometrical and physical parameters of these kind of transducers. The transient representation of these waves have been analysed and discussed by the rays model. . A potential application of this work would be the study of the acoustic signature of materials by using phased-arrays transducer working in pulsed mode. The obtained results have been compared to those obtained by using a finite element method package.

Esha Kashyap^{1}
, Kannan Sr^{2}

^{1}Pondicherry University(A Central University of India), Mathematics, India

^{2}Pondicherry University(A Central University of India), Mathematics, India

This paper reviews the influence of kernel function in unsupervised way of clustering in data analysis. Unsupervised clustering analysis is considered as an explorative data analysis tool that assists in discovering hidden patterns or natural grouping in data, and has been effectively applied in various applications. The implementation of kernel function with the objective functions of unsupervised clustering techniques contribute an effective works for recognizing nonlinear structures of high dimensional databases and the objective functions are robust in clustering the high dimensional databases which contains outliers with heavy noise. This paper mainly discusses the kernels with the objective functions of fuzzy c-means, possiblistic c-means, and intutionistic fuzzy c-means for clustering the non-liner structured databases. Experimental section of the paper supports to understand the effective of kernel in various clustering methods.

This work was financially supported by DST India and MOST Israel