Search results for: Jacobi elliptic functions

Authors: Jihad H. Asad

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An expression for the Green’s function (GF) of anisotropic face cantered cubic (IFCC) lattice is evaluated analytically and numerically for a single impurity problem. The density of states (DOS), phase shift and scattering cross section are expressed in terms of complete elliptic integrals of the first kind.

Keywords: lattice Green's function, elliptic integral, physics, cubic lattice

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Authors: Fuad Al Sarari

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The purpose of the present paper is to study subclasses of analytic functions which generalize the classes of Janowski functions introduced by Polatoglu. We study certain convolution conditions. This leads to a study of the sufficient condition and the neighborhood results related to the functions in the class S (T; H; F ): and a study of new subclasses of analytic functions that are defined using notions of the generalized Janowski classes and -symmetrical functions. In the quotient of analytical representations of starlikeness and convexity with respect to symmetric points, certain inherent properties are pointed out.

Keywords: convolution conditions, subordination, Janowski functions, starlike functions, convex functions

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Authors: Kanshio Richard Tyokyaa, Jagadish Singh

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In this paper, we examined the location and stability of Out-Of-Plane Equilibrium points in the elliptic restricted three-body problem of an infinitesimal body when both primaries are taken as oblate spheroids with oblateness up to zonal harmonic J₄. The positions of the Equilibrium points L₆,₇ and their stability depend on the oblateness of the primaries and the eccentricity of their orbits. We explored the problem numerically to show the effects of parameters involved in the position and stability of the Out-Of-Plane Equilibrium points for the systems: HD188753 and Gliese 667. It is found that their positions are affected by the oblateness of the primaries, eccentricity and the semi-major axis of the orbits, but its stability behavior remains unchanged and is unstable.

Keywords: out-of-plane, equilibrium points, stability, elliptic restricted three-body problem, oblateness, zonal harmonic

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Authors: D. M. S. Bandara, Yunqi Lei, Ye Luo

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Fingerprints are suitable as long-term markers of human identity since they provide detailed and unique individual features which are difficult to alter and durable over life time. In this paper, we propose an algorithm to encrypt and decrypt fingerprint images by using a specially designed Elliptic Curve Cryptography (ECC) procedure based on block ciphers. In addition, to increase the confusing effect of fingerprint encryption, we also utilize a chaotic-behaved method called Arnold Cat Map (ACM) for a 2D scrambling of pixel locations in our method. Experimental results are carried out with various types of efficiency and security analyses. As a result, we demonstrate that the proposed fingerprint encryption/decryption algorithm is advantageous in several different aspects including efficiency, security and flexibility. In particular, using this algorithm, we achieve a margin of about 0.1% in the test of Number of Pixel Changing Rate (NPCR) values comparing to the-state-of-the-art performances.

Keywords: arnold cat map, biometric encryption, block cipher, elliptic curve cryptography, fingerprint encryption, Koblitz’s encoding

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Authors: Khaled I. Nawafleh

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In this paper, it provide the canonical approach for studying dissipated oscillators based on the doubling of degrees of freedom. Clearly, expressions for Lagrangians of the elementary modes of the system are given, which ends with the familiar classical equations of motion for the dissipative oscillator. The equation for one variable is the time reversed of the motion of the second variable. it discuss in detail the extended Bateman Lagrangian specifically for a dual extended damped oscillator time-dependent. A Hamilton-Jacobi analysis showing the equivalence with the Lagrangian approach is also obtained. For that purpose, the techniques of separation of variables were applied, and the quantization process was achieved.

Keywords: doubling of degrees of freedom, dissipated harmonic oscillator, Hamilton-Jacobi, time-dependent lagrangians, quantization

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Authors: Yanpei Zhen

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The nonlinear Schrödinger (NLS) equation models wave dynamics in many physical problems related to fluids, plasmas, and optics. The standing periodic waves are known to be modulationally unstable, and rogue waves (localized perturbations in space and time) have been observed on their backgrounds in numerical experiments. The exact solutions for rogue waves arising on the periodic standing waves have been obtained analytically. It is natural to ask if the rogue waves persist on the standing periodic waves in the integrable discretizations of the integrable NLS equation. We study the standing periodic waves in the semidiscrete integrable system modeled by the high-order Ablowitz-Ladik (AL) equation. The standing periodic wave of the high-order AL equation is expressed by the Jacobi cnoidal elliptic function. The exact solutions are obtained by using the separation of variables and one-fold Darboux transformation. Since the cnoidal wave is modulationally unstable, the rogue waves are generated on the periodic background.

Keywords: Darboux transformation, periodic wave, Rogue wave, separating the variables

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Authors: Elyas Shivanian

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In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problems of two-dimensional elliptic PDEs in doubly connected domains. It is obtained the unknown data on the inner boundary of the domain while overspecified boundary data are imposed on the outer boundary of the domain by using the SMRPI. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high order convergence rate. In this way, localization in SMRPI can reduce the ill-conditioning for Cauchy problem. Furthermore, we improve previous results and it is revealed the SMRPI is more accurate and stable by adding strong perturbations.

Keywords: cauchy problem, doubly connected domain, radial basis function, shape function

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Authors: H. Parveen Begam, M. A. Maluk Mohamed

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Grid computing is an environment that allows sharing and coordinated use of diverse resources in dynamic, heterogeneous and distributed environment using Virtual Organization (VO). Security is a critical issue due to the open nature of the wireless channels in the grid computing which requires three fundamental services: authentication, authorization, and encryption. The privacy and anonymity are considered as an important factor while communicating over publicly spanned network like web. To ensure a high level of security we explored an extension of onion routing, which has been used with dynamic token exchange along with protection of privacy and anonymity of individual identity. To improve the performance of encrypting the layers, the elliptic curve cryptography is used. Compared to traditional cryptosystems like RSA (Rivest-Shamir-Adelman), ECC (Elliptic Curve Cryptosystem) offers equivalent security with smaller key sizes which result in faster computations, lower power consumption, as well as memory and bandwidth savings. This paper presents the estimation of the performance improvements of onion routing using ECC as well as the comparison graph between performance level of RSA and ECC.

Keywords: grid computing, privacy, anonymity, onion routing, ECC, RSA

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Authors: Muna Tabuni

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The paper contains an investigation of the behavior of the Zeros of Bargmann functions for one and two-mode systems. A brief introduction to Harmonic oscillator formalism for one and two-mode is given. The Bargmann analytic representation for one and two-mode has been studied. The zeros of Bargmann analytic function for one-mode are considered. The Q Husimi functions are introduced. The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros are discussed. The zeros of Bargmann analytic functions for two-mode are introduced. Various examples have been given.

Keywords: Bargmann functions, two-mode, zeros, harmonic oscillator

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Authors: Song Ni, Junxiang Xu

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This paper concerns quasi-periodic perturbations with parameters of 2-dimensional degenerate systems. If the equilibrium point of the unperturbed system is elliptic-type degenerate. Assume that the perturbation is real analytic quasi-periodic with diophantine frequency. Without imposing any assumption on the perturbation, we can use a path of equilibrium points to tackle with the Melnikov non-resonance condition, then by the Leray-Schauder Continuation Theorem and the Kolmogorov-Arnold-Moser technique, it is proved that the equation has a small response solution for many sufficiently small parameters.

Keywords: quasi-periodic systems, KAM-iteration, degenerate equilibrium point, response solution

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Authors: Hassan Manouzi, Taous-Meriem Laleg-Kirati

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In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Ito chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: eigenvalue problem, Wick product, SPDEs, finite element, Wiener-Ito chaos expansion

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Authors: Alicia C. Sanchez

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This paper investigates the implementation of RAFU functions (radical functions) in robotics and automation. Specifically, the main goal is to show how these functions may be useful in lane-keeping control and the lateral control of autonomous machines, vehicles, robots or the like. From the knowledge of several points of a certain route, the RAFU functions are used to achieve the lateral control purpose and maintain the lane-keeping errors within the fixed limits. The stability that these functions provide, their ease of approaching any continuous trajectory and the control of the possible error made on the approximation may be useful in practice.

Keywords: automatic navigation control, lateral control, lane-keeping control, RAFU approximation

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Authors: Rashidah Omar, Suzeini Abdul Halim, Aini Janteng

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The coefficients estimate problem for Taylor-Maclaurin series is still an open problem especially for a function in the subclass of bi-univalent functions. A function f ϵ A is said to be bi-univalent in the open unit disk D if both f and f^-1 are univalent in D. The symbol A denotes the class of all analytic functions f in D and it is normalized by the conditions f(0) = f’(0) – 1=0. The class of bi-univalent is denoted by  The subordination concept is used in determining second and third Taylor-Maclaurin coefficients. The upper bound for second and third coefficients is estimated for functions in the subclasses of bi-univalent functions which are subordinated to the function φ. An analytic function f is subordinate to an analytic function g if there is an analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f(z) = g[w(z)]. In this paper, two subclasses of bi-univalent functions associated with Hohlov operator are introduced. The bound for second and third coefficients of functions in these subclasses is determined using subordination. The findings would generalize the previous related works of several earlier authors.

Keywords: analytic functions, bi-univalent functions, Hohlov operator, subordination

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Authors: İbrahim Aktaş, Árpád Baricz

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In this paper, the radii of star likeness of the Jackson and Hahn-Exton q-Bessel functions are considered, and for each of them three different normalizations is applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower, and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Keywords: bessel function, lommel function, radius of starlikeness and convexity, Struve function

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Authors: Safa Saoudi, Souheib Yousfi, Riadh Robbana

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E-passport is a relatively new electronic document which maintains the passport features and provides better security. It deploys new technologies such as biometrics and Radio Frequency identification (RFID). The international civil aviation organization (ICAO) and the European union define mechanisms and protocols to provide security but their solutions present many threats. In this paper, a new mechanism is presented to strengthen e-passport security and authentication process. We propose a new protocol based on Elliptic curve, identity based encryption and shared secret between entities. Authentication in our contribution is formally proved with BAN Logic verification language. This proposal aims to provide a secure data storage and authentication.

Keywords: e-passport, elliptic curve cryptography, identity based encryption, shared secret, BAN Logic

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Authors: Debajyoti Choudhuri, Ratan Kumar Giri, Shesadev Pradhan

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The perturbed nonlinear Schrodinger equation involving the p-biharmonic and the p-Laplacian operators involving a real valued parameter and a continuous real valued potential function defined over the N- dimensional Euclidean space has been considered. By the variational technique, an existence result pertaining to a nontrivial solution to this non-linear partial differential equation has been proposed. Further, by the Concentration lemma, the concentration of solutions to the same problem defined on the set consisting of those elements where the potential function vanishes as the real parameter approaches to infinity has been addressed.

Keywords: p-Laplacian, p-biharmonic, elliptic PDEs, Concentration lemma, Sobolev space

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Authors: Tulin Coskun

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We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: Chahid K. Ghaddar

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The spreadsheet engine is exploited via a non-conventional mechanism to enable novel worksheet solver functions for computational calculus. The solver functions bypass inherent restrictions on built-in math and user defined functions by taking variable formulas as a new type of argument while retaining purity and recursion properties. The enabling mechanism permits integration of numerical algorithms into worksheet functions for solving virtually any computational problem that can be modelled by formulas and variables. Several examples are presented for computing integrals, derivatives, and systems of deferential-algebraic equations. Incorporation of the worksheet solver functions with the ubiquitous spreadsheet extend the utility of the latter as a powerful tool for computational mathematics.

Keywords: calculus, differential algebraic equations, solvers, spreadsheet

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Authors: Sherwan Kher Alden Yakub Alsofy

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In this paper, we consider the relativistic Hamilton-Jacobi (HJ) equation and study the Hawking radiation (HR) of scalar particles from uncharged Grumiller black hole (GBH) which is affordable for testing in astrophysics. GBH is also known as Rindler modified Schwarzschild BH. Our aim is not only to investigate the effect of the Rindler parameter A on the Hawking temperature (TH ), but to examine whether there is any discrepancy between the computed horizon temperature and the standard TH as well. For this purpose, in addition to its naive coordinate system, we study on the three regular coordinate systems which are Painlev´-Gullstrand (PG), ingoing Eddington- Finkelstein (IEF) and Kruskal-Szekeres (KS) coordinates. In all coordinate systems, we calculate the tunneling probabilities of incoming and outgoing scalar particles from the event horizon by using the HJ equation. It has been shown in detail that the considered HJ method is concluded with the conventional TH in all these coordinate systems without giving rise to the famous factor- 2 problem. Furthermore, in the PG coordinates Parikh-Wilczek’s tunneling (PWT) method is employed in order to show how one can integrate the quantum gravity (QG) corrections to the semiclassical tunneling rate by including the effects of self-gravitation and back reaction. We then show how these corrections yield a modification in the TH.

Keywords: ingoing Eddington, Finkelstein, coordinates Parikh-Wilczek’s, Hamilton-Jacobi equation

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Authors: Falek Kamel

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Various approaches are usually used in the dynamic analysis of beams vibrating transversally. For this, numerical methods allowing the solving of the general eigenvalue problem are utilized. The equilibrium equations describe the movement resulting from the solution of a fourth-order differential equation. Our investigation is based on the finite element method. The findings of these investigations are the vibration frequencies obtained by the Jacobi method. Two types of the elementary mass matrix are considered, representing a uniform distribution of the mass along with the element and concentrated ones located at fixed points whose number is increased progressively separated by equal distances at each evaluation stage. The studied beams have different boundary constraints representing several classical situations. Comparisons are made for beams where the distributed mass is replaced by n concentrated masses. As expected, the first calculus stage is to obtain the lowest number of beam parts that gives a frequency comparable to that issued from the Rayleigh formula. The obtained values are then compared to theoretical results based on the assumptions of the Bernoulli-Euler theory. These steps are used for the second type of mass representation in the same manner.

Keywords: structural elements, beams vibrating, dynamic analysis, finite element method, Jacobi method

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Authors: Luiz F. J. Maia

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This paper presents an approach to fuzzy control, with the use of new pertinence functions, applied in the case of an inverted pendulum. Appropriate definitions of pertinence functions to fuzzy sets make possible the implementation of the controller with only one control rule, resulting in a smooth control surface. The fuzzy control system can be implemented with analog devices, affording a true real-time performance.

Keywords: control surface, fuzzy control, Inverted pendulum, pertinence functions

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Authors: Mohammad Mehdi Mazarei, Ali Asghar Behroozpoor

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In this paper, we define an optimization problem corresponding to smooth and nonsmooth functions which its optimal solution is the first derivative of these functions in a domain. For this purpose, a linear programming problem corresponding to optimization problem is obtained. The optimal solution of this linear programming problem is the approximate generalized first derivative. In fact, we approximate generalized first derivative of nonsmooth functions as tailor series. We show the efficiency of our approach by some smooth and nonsmooth functions in some examples.

Keywords: general derivative, linear programming, optimization problem, smooth and nonsmooth functions

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Authors: J. Juan Peña, J. Morales, J. García-Ravelo, J. García-Martínes

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It is known that by introducing alternative forms of exponential and logarithmic functions, the Tsallis entropy Sq is itself a generalization of Shannon entropy S. In this work, from a deformation through a scaling function applied to the differential operator, it is possible to generate a q-deformed calculus as well as a q-deformed arithmetic, which not only allows generalizing the exponential and logarithmic functions but also any other standard function. The updated q-deformed differential operator leads to an updated integral operator under which the functions are integrated together with a weight function. For each differentiable function, it is possible to identify its q-deformed partner, which is useful to generalize other algebraic relations proper of the original functions. As an application of this proposal, in this work, a generalization of exponential and logarithmic functions is studied in such a way that their relationship with the thermodynamic functions, particularly the entropy, allows us to have a q-deformed expression of these. As a result, from a particular scaling function applied to the differential operator, a q-deformed arithmetic is obtained, leading to the generalization of the Tsallis entropy.

Keywords: q-calculus, q-deformed arithmetic, entropy, exponential functions, thermodynamic functions

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Authors: Mustapha Benssalah, Mustapha Djeddou, Karim Drouiche

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The random number generation (RNG) presents a significant importance for the security and the privacy of numerous applications, such as RFID technology and smart cards. Since, the quality of the generated bit sequences is paramount that a weak internal generator for example, can directly cause the entire application to be insecure, and thus it makes no sense to employ strong algorithms for the application. In this paper, we propose a new pseudo random number generator (PRNG), suitable for cryptosystems ECC-based, constructed by randomly selecting points from several elliptic curves randomly selected. The main contribution of this work is the increasing of the generator internal states by extending the set of its output realizations to several curves auto-selected. The quality and the statistical characteristics of the proposed PRNG are validated using the Chi-square goodness of fit test and the empirical Special Publication 800-22 statistical test suite issued by NIST.

Keywords: PRNG, security, cryptosystem, ECC

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Authors: S.I.Mukhin, S. Seidov, A. Mukherjee

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The Dicke model is a key tool for the description of correlated states of quantum atomic systems, excited by resonant photon absorption and subsequently emitting spontaneous coherent radiation in the superradiant state. The Dicke Hamiltonian (DH) is successfully used for the description of the dynamics of the Josephson Junction (JJ) array in a resonant cavity under applied current. In this work, we have investigated a generalized model, which is described by DH with a frustrating interaction term. This frustrating interaction term is explicitly the infinite coordinated interaction between all the spin half in the system. In this work, we consider an array of N superconducting islands, each divided into two sub-islands by a Josephson Junction, taken in a charged qubit / Cooper Pair Box (CPB) condition. The array is placed inside the resonant cavity. One important aspect of the problem lies in the dynamical nature of the physical observables involved in the system, such as condensed electric field and dipole moment. It is important to understand how these quantities behave with time to define the quantum phase of the system. The Dicke model without frustrating term is solved to find the dynamical solutions of the physical observables in analytic form. We have used Heisenberg’s dynamical equations for the operators and on applying newly developed Rotating Holstein Primakoff (HP) transformation and DH we have arrived at the four coupled nonlinear dynamical differential equations for the momentum and spin component operators. It is possible to solve the system analytically using two-time scales. The analytical solutions are expressed in terms of Jacobi's elliptic functions for the metastable ‘bound luminosity’ dynamic state with the periodic coherent beating of the dipoles that connect the two double degenerate dipolar ordered phases discovered previously. In this work, we have proceeded the analysis with the extended DH with a frustrating interaction term. Inclusion of the frustrating term involves complexity in the system of differential equations and it gets difficult to solve analytically. We have solved semi-classical dynamic equations using the perturbation technique for small values of Josephson energy EJ. Because the Hamiltonian contains parity symmetry, thus phase transition can be found if this symmetry is broken. Introducing spontaneous symmetry breaking term in the DH, we have derived the solutions which show the occurrence of finite condensate, showing quantum phase transition. Our obtained result matches with the existing results in this scientific field.

Keywords: Dicke Model, nonlinear dynamics, perturbation theory, superconductivity

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Authors: Hussein Shaman, Faris Almansour

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This paper proposes a modified optimum bandstop filter with ultra-wideband stopband. The filter consists of three shunt open-circuited stubs and two non-redundant unit elements. The proposed bandstop filter is designed with unequal electrical lengths of the open-circuited stubs at the mid-stopband. Therefore, the filter can exhibit a quasi-elliptic function response that improves the selectivity and enhances the rejection bandwidth. The filter is designed to exhibit a fractional bandwidth of about 114% at a mid-stopband frequency of 3.0 GHz. The filter is successfully realized in theory, simulated, fabricated and measured. An excellent agreement is obtained between calculated, simulated and measured. The fabricated filter has a compact size with a low insertion loss in the passbands, high selectivity and good attenuation level inside the desired stopband

Keywords: microstrip filter, bandstop filter, UWB filter, transmission line filter

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Authors: Despoina Chochtoula, Aristidis Ilias, Yannis Stamatiou

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Modbus is a protocol that enables the communication among devices which are connected to the same network. This protocol is, often, deployed in connecting sensor and monitoring units to central supervisory servers in Supervisory Control and Data Acquisition, or SCADA, systems. These systems monitor critical infrastructures, such as factories, power generation stations, nuclear power reactors etc. in order to detect malfunctions and ignite alerts and corrective actions. However, due to their criticality, SCADA systems are vulnerable to attacks that range from simple eavesdropping on operation parameters, exchanged messages, and valuable infrastructure information to malicious modification of vital infrastructure data towards infliction of damage. Thus, the SCADA research community has been active over strengthening SCADA systems with suitable data protection mechanisms based, to a large extend, on cryptographic methods for data encryption, device authentication, and message integrity protection. However, due to the limited computation power of many SCADA sensor and embedded devices, the usual public key cryptographic methods are not appropriate due to their high computational requirements. As an alternative, Elliptic Curve Cryptography has been proposed, which requires smaller key sizes and, thus, less demanding cryptographic operations. Until now, however, no such implementation has been proposed in the SCADA literature, to the best of our knowledge. In order to fill this gap, our methodology was focused on integrating Modbus, a frequently used SCADA communication protocol, with Elliptic Curve based cryptography and develop a server/client application to demonstrate the proof of concept. For the implementation we deployed two C language libraries, which were suitably modify in order to be successfully integrated: libmodbus (https://github.com/stephane/libmodbus) and ecc-lib https://www.ceid.upatras.gr/webpages/faculty/zaro/software/ecc-lib/). The first library provides a C implementation of the Modbus/TCP protocol while the second one offers the functionality to develop cryptographic protocols based on Elliptic Curve Cryptography. These two libraries were combined, after suitable modifications and enhancements, in order to give a modified version of the Modbus/TCP protocol focusing on the security of the data exchanged among the devices and the supervisory servers. The mechanisms we implemented include key generation, key exchange/sharing, message authentication, data integrity check, and encryption/decryption of data. The key generation and key exchange protocols were implemented with the use of Elliptic Curve Cryptography primitives. The keys established by each device are saved in their local memory and are retained during the whole communication session and are used in encrypting and decrypting exchanged messages as well as certifying entities and the integrity of the messages. Finally, the modified library was compiled for the Android environment in order to run the server application as an Android app. The client program runs on a regular computer. The communication between these two entities is an example of the successful establishment of an Elliptic Curve Cryptography based, secure Modbus wireless communication session between a portable device acting as a supervisor station and a monitoring computer. Our first performance measurements are, also, very promising and demonstrate the feasibility of embedding Elliptic Curve Cryptography into SCADA systems, filling in a gap in the relevant scientific literature.

Keywords: elliptic curve cryptography, ICT security, modbus protocol, SCADA, TCP/IP protocol

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Authors: Khosrow Maleknejad, Yaser Rostami

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In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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The Bessel function J₀(x) is very important in Electrodynamics and Physics, as well as its zeros. In this work, a method to obtain high accuracy approximation is presented through an application to that function. In most of the applications of this function, the values of the zeros are very important. In this work, analytic approximations for this function have been obtained valid for all positive values of the variable x, which have high accuracy for the function as well as for the zeros. The approximation is determined by the simultaneous used of the power series and asymptotic expansion. The structure of the approximation is a combination of two rational functions with elementary functions as trigonometric and fractional powers. Here us in Pade method, rational functions are used, but now there combined with elementary functions us fractional powers hyperbolic or trigonometric functions, and others. The reason of this is that now power series of the exact function are used, but together with the asymptotic expansion, which usually includes fractional powers trigonometric functions and other type of elementary functions. The approximation must be a bridge between both expansions, and this can not be accomplished using only with rational functions. In the simplest approximation using 4 parameters the maximum absolute error is less than 0.006 at x ∼ 4.9. In this case also the maximum relative error for the zeros is less than 0.003 which is for the second zero, but that value decreases rapidly for the other zeros. The same kind of behaviour happens for the relative error of the maximum and minimum of the functions. Approximations with higher accuracy and more parameters will be also shown. All the approximations are valid for any positive value of x, and they can be calculated easily.

Keywords: analytic approximations, asymptotic approximations, Bessel functions, quasirational approximations

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Authors: Priestly Malambo, Sonja Van Putten, Hanlie Botha, Gerrit Stols

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The relevance of what is content is taught in tertiary teacher training has long been in question. This study attempts to understand how advanced mathematics courses equip student teachers to teach functions at secondary school level. This paper reports on an investigation that was conducted in an African university, where preservice teachers were purposefully selected for participation in individual semi-structured interviews after completing a test on functions as taught at secondary school. They were asked to justify their reasoning in the test and to explain functions in a way that might bring about understanding of the topic in someone who did not know how functions work. These were final year preservice mathematics teachers who had studied advanced mathematics courses for three years. More than 50% of the students were not able to explain concepts or to justify their reasoning about secondary school functions in a coherent way. The results of this study suggest that the study of advanced mathematics does not automatically enable students to teach secondary school functions, and that, although these students were able to do advanced mathematics, they were unable to explain the working of functions in a way that would allow them to teach this topic successfully.

Keywords: secondary school, mathematical reasoning, student-teachers, functions

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