Search results for: elliptic integral

Authors: Maja Andrić, Josip Pečarić

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Over the last five decades, an enormous amount of work has been done on Opial’s integral inequality, dealing with new proofs, various generalizations, extensions and discrete analogs. The Opial inequality is recognized as a fundamental result in the analysis of qualitative properties of solution of differential equations. We use submultiplicative convex functions, appropriate representations of functions and inequalities involving means to obtain generalizations and extensions of certain known multidimensional integral and discrete Opial-type inequalities.

Keywords: Opial's inequality, Jensen's inequality, integral inequality, discrete inequality

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Authors: Dawei Shen, Ming Xu, Pengfei Liu

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The joints between simply supported bridge decks and abutments need to be regularly repaired, which would greatly increase the cost during the service life of the bridge. Simply supported girder bridges suffered the most severe damage during earthquakes. Another type of bridge, the integral bridge, of which the superstructure and abutment are rigidly connected, was also used in some European countries. Because no bearings or joints exit in the integral bridge, this type of bridge could significantly reduce maintenance requirements and costs. However, conventional integral bridge usually result in high earth pressure on the abutment and surface settlement in the backfill. To solve these problems, a new type of integral bridge, geosynthetic-reinforced soil (GRS) integral bridge, was come up in recent years. This newly invented bridge has not been used in engineering practices. There was a lack of research on the seismic behavior of the conventional and new type of integral abutments. In addition, no common design code could be found for the calculation of seismic pressure of soil behind the abutment. This paper developed a dynamic constitutive model, which can consider the soil behaviors under cyclic loading. Numerical analyses of the seismic response of a full height integral bridge and GRS integral bridge were carried out using the two-dimensional numerical code, FLAC. A parametric study was also performed to investigate the soil-structure interaction. The results are presented below. The seismic responses of GRS integral bridge together with conventional simply supported bridge, GRS conventional bridge and conventional integral bridge were investigated. The results show that the GRS integral bridge holds the highest seismic stability, followed by conventional integral bridge, GRS simply supported bridge and conventional simply supported bridge. Compared with the integral bridge with 1 m thick abutments, the GRS integral bridge with 0.4 m thick abutments is subjected to a smaller bending moment, and the natural frequency and horizontal displacement remains almost the same. Geosynthetic-reinforcement will be more effective when the abutment becomes thinner or the abutment is higher.

Keywords: geosynthetic-reinforced soil integral bridge, nonlinear hysteretic model, numerical analysis, seismic response

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Authors: Farid Saeidi, Serkan Dag

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In this study, fracture analysis of a fibrous composite laminate with variable fiber spacing is carried out using Jk-integral method. The laminate is assumed to be under thermal loading. Jk-integral is formulated by using the constitutive relations of plane orthotropic thermoelasticity. Developed domain independent form of the Jk-integral is then integrated into the general purpose finite element analysis software ANSYS. Numerical results are generated so as to assess the influence of variable fiber spacing on mode I and II stress intensity factors, energy release rate, and T-stress. For verification, some of the results are compared to those obtained using displacement correlation technique (DCT).

Keywords: Jk-integral, Variable Fiber Spacing, Thermoelasticity, T-stress, Finite Element Method, Fibrous Composite.

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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: V. Srikanth, K. Balasubramanian, Rajath R. Bhat, A. S. Arjun, Nandhu Venugopal, Ananthu Unnikrishnan

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Indirect vector control is known to produce high performance in Permanent Magnet Synchronous Motor (PMSM) drives by decoupling flux and torque producing current components of stator current. The most commonly used controller or the vector control of AC motor is Proportional-Integral (P-I) controller. However, the P-I controller has some disadvantages such as high starting overshoot, sensitivity to controller gains and slower response to sudden disturbance. Therefore, the Integral-Proportional controller for PMSM drives to overcome the disadvantages of the P-I controller. Simulations results are presented and analyzed for both controllers and it is observed that Integral-Proportional (I-P) controllers give better responses than the traditional P-I controllers.

Keywords: PMSM, FOC, PI controller, IP controller

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Authors: András Szekrényes

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Delamination is one of the major failure modes in laminated composite structures. Delamination tips are mostly captured by spatial numerical models in order to predict crack growth. This paper presents some mechanical models of delaminated composite shells based on shallow shell theories. The mechanical fields are based on a third-order displacement field in terms of the through-thickness coordinate of the laminated shell. The undelaminated and delaminated parts are captured by separate models and the continuity and boundary conditions are also formulated in a general way providing a large size boundary value problem. The system of differential equations is solved by the state space method for an elliptic delaminated shell having simply supported edges. The comparison of the proposed and a numerical model indicates that the primary indicator of the model is the deflection, the secondary is the widthwise distribution of the energy release rate. The model is promising and suitable to determine accurately the J-integral distribution along the delamination front. Based on the proposed model it is also possible to develop finite elements which are able to replace the computationally expensive spatial models of delaminated structures.

Keywords: J-integral, levy method, third-order shell theory, state space solution

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Authors: Bingjie Wu, C. Q. Ru

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The present work studies several reasonably expected candidate integral forms for self-attractive potential energy of a free monolayer graphene sheet. The admissibility of a specific integral form for ripple formation is verified, while all others most of the candidate integral forms are rejected based on the non-existence of stable periodic ripples. Based on the selected integral form of self-attractive potential energy, some mechanical behavior, including ripple formation and buckling, of a free monolayer grapheme sheet are discussed in details

Keywords: graphene, monolayer, ripples, van der Waals energy

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Authors: Shiji Hu, Lei Li, Wanting Zhou, DaoHong Yang

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The data encryption, is the foundation of today’s communication. On this basis, how to improve the speed of data encryption and decryption is always a problem that scholars work for. In this paper, we proposed an elliptic curve crypto processor architecture based on SM2 prime field. In terms of hardware implementation, we optimized the algorithms in different stages of the structure. In finite field modulo operation, we proposed an optimized improvement of Karatsuba-Ofman multiplication algorithm, and shorten the critical path through pipeline structure in the algorithm implementation. Based on SM2 recommended prime field, a fast modular reduction algorithm is used to reduce 512-bit wide data obtained from the multiplication unit. The radix-4 extended Euclidean algorithm was used to realize the conversion between affine coordinate system and Jacobi projective coordinate system. In the parallel scheduling of point operations on elliptic curves, we proposed a three-level parallel structure of point addition and point double based on the Jacobian projective coordinate system. Combined with the scalar multiplication algorithm, we added mutual pre-operation to the point addition and double point operation to improve the efficiency of the scalar point multiplication. The proposed ECC hardware architecture was verified and implemented on Xilinx Virtex-7 and ZYNQ-7 platforms, and each 256-bit scalar multiplication operation took 0.275ms. The performance for handling scalar multiplication is 32 times that of CPU(dual-core ARM Cortex-A9).

Keywords: Elliptic curve cryptosystems, SM2, modular multiplication, point multiplication.

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Authors: Archit Yajnik, Rustam Ali

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In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method.

Keywords: GHM multiwavelet, fredholm integral equations, quadrature method, function approximation

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Authors: E. V. Krishnan

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In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.

Keywords: travelling wave solutions, Jacobi elliptic functions, solitary wave solutions, Korteweg-de Vries equation

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Authors: Ahmet Kaya

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Quantum mechanism is one of the most important approaches to calculating non-ruin probability. We apply standard Dirac notation to model given Hamiltonians. By using the traditional method and eigenvector basis, non-ruin probability is found for several examples. Also, non-ruin probability is calculated for two different Hamiltonian by using the tensor product. Finally, the path integral method is applied to the examples and comparison is made for stochastic simulations and path integral calculation.

Keywords: quantum physics, Hamiltonian system, path integral, tensor product, ruin probability

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Authors: Serhat Tüzün, Tufan Demirel

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The objective of the present study is to select the most suitable location for a wind power plant station through Choquet integral method. The problem of selecting the location for a wind power station was considered as a multi-criteria decision-making problem. The essential and sub-criteria were specified and location selection was expressed in a hierarchic structure. Among the main criteria taken into account in this paper are wind potential, technical factors, social factors, transportation, and costs. The problem was solved by using different approaches of Choquet integral and the best location for a wind power station was determined. Then, the priority weights obtained from different Choquet integral approaches are compared and commented on.

Keywords: multi-criteria decision making, choquet integral, fuzzy sets, location of a wind power plant

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Authors: Artion Kashuri, Rozana Liko

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In this paper, some generalization integral inequalities of Hermite–Hadamard type for functions whose derivatives are s–convex in modulus are given by using k–fractional integrals. Some applications to special means are obtained as well. Some known versions are recovered as special cases from our results. We note that our inequalities can be viewed as new refinements of the previous results. Finally, our results have a deep connection with various fractional integral operators and interested readers can find new interesting results using our idea and technique as well.

Keywords: Hermite-Hadamard's inequalities, Hölder's inequality, k-Riemann-Liouville fractional integral, special means

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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: Tohru Kawabe

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In this paper, a new SMC (Sliding Mode Control) method with MP (Model Predictive Control) integral action for the slip suppression of EV (Electric Vehicle) under braking is proposed. The proposed method introduce the integral term with standard SMC gain , where the integral gain is optimized for each control period by the MPC algorithms. The aim of this method is to improve the safety and the stability of EVs under braking by controlling the wheel slip ratio. There also include numerical simulation results to demonstrate the effectiveness of the method.

Keywords: sliding mode control, model predictive control, integral action, electric vehicle, slip suppression

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Authors: Shai Sarussi

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Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. A characterization of the property of immediate neighbors in an Alexandroff topological space is given, in terms of closed and open subsets of appropriate subspaces. Moreover, two special subspaces of W are introduced, and a way in which their closed and open subsets induce W is presented.

Keywords: integral domains, Alexandroff topology, immediate neighbors, valuation domains

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Authors: Lila D. Sigdel, Chin J. Leo, Samanthika Liyanapathirana, Pan Hu, Minghao Lu

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Integral bridges are a class of bridges with integral or semi-integral abutments, designed without expansion joints in the bridge deck of the superstructure. Integral bridges are economical alternatives to conventional jointed bridges with lower maintenance costs and greater durability, thereby improving social and economic stability for the community. Integral bridges have also been proven to be effective in lowering the overall construction cost compared to the conventional type of bridges. However, there is significant uncertainty related to the design and analysis of integral bridges in response to cyclic thermal movements induced due to deck expansion and contraction. The cyclic thermal movements of the abutments increase the lateral earth pressures on the abutment and its foundation, leading to soil settlement and heaving of the backfill soil. Thus, the primary objective of this paper is to investigate the soil-abutment interaction under the cyclic translational movement of the abutment. Results from five experiments conducted to simulate different magnitudes of cyclic translational movements of abutments induced by thermal changes are presented, focusing on lateral earth pressure development at the abutment-soil interface. Test results show that the cycle number and magnitude of cyclic translational movements have significant effects on the escalation of lateral earth pressures. Experimentally observed earth pressure distributions behind the integral abutment were compared with the current design approaches, which shows that the most of the practices has under predicted the lateral earth pressure.

Keywords: integral bridge, cyclic thermal movement, lateral earth pressure, soil-structure interaction

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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: Muna Alghabshi, Edmana Krishnan

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A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.

Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method

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Authors: M. Naji, A. R. Khalim, M. Naji

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Integral Abutment Bridges (IAB) are defined as simple or multiple span bridges in which the bridge deck is cast monolithically with the abutment walls. This kind of bridges are becoming very popular due to different aspects such as good response under seismic loading, low initial costs, elimination of bearings and less maintenance. However, the main issue related to the analysis of this type of structures is dealing with soil-structure interaction of the abutment walls and the supporting piles. A two-dimensional, non-linear finite element (FE) model of an integral abutment bridge has been developed to study the effect of lateral time history displacement loading on the soil system.

Keywords: integral abutment bridge, soil structure interaction, finite element modeling, soil-pile interaction

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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: Naser Safdarian, Nader Jafarnia Dabanloo

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In this paper we used four features i.e. Q-wave integral, QRS complex integral, T-wave integral and total integral as extracted feature from normal and patient ECG signals to detection and localization of myocardial infarction (MI) in left ventricle of heart. In our research we focused on detection and localization of MI in standard ECG. We use the Q-wave integral and T-wave integral because this feature is important impression in detection of MI. We used some pattern recognition method such as Artificial Neural Network (ANN) to detect and localize the MI. Because these methods have good accuracy for classification of normal and abnormal signals. We used one type of Radial Basis Function (RBF) that called Probabilistic Neural Network (PNN) because of its nonlinearity property, and used other classifier such as k-Nearest Neighbors (KNN), Multilayer Perceptron (MLP) and Naive Bayes Classification. We used PhysioNet database as our training and test data. We reached over 80% for accuracy in test data for localization and over 95% for detection of MI. Main advantages of our method are simplicity and its good accuracy. Also we can improve accuracy of classification by adding more features in this method. A simple method based on using only four features which extracted from standard ECG is presented which has good accuracy in MI localization.

Keywords: ECG signal processing, myocardial infarction, features extraction, pattern recognition

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Authors: Damir Latypov

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A closed-form solution for 4-D double surface integrals arising in boundary integrals equations of a potential theory is obtained for arbitrary coplanar polygonal surfaces. The solution method is based on the construction of exact differential forms followed by the application of Stokes' theorem for each surface integral. As a result, the 4-D double surface integral is reduced to a 2-D double line integral. By an appropriate change of variables, the integrand is transformed into a separable function of integration variables. The closed-form solutions to the corresponding 1-D integrals are readily available in the integration tables. Previously closed-form solutions were known only for the case of coincident triangle surfaces and coplanar rectangles. Solutions for these cases were obtained by surface-specific ad-hoc methods, while the present method is general. The method also works for non-polygonal surfaces. As an example, we compute in closed form the 4-D integral for the case of coincident surfaces in the shape of a circular disk. For an arbitrarily shaped surface, the proposed method provides an efficient quadrature rule. Extensions of the method for non-coplanar surfaces and other than 1/R integral kernels are also discussed.

Keywords: boundary integral equations, differential forms, integration, stokes' theorem

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Authors: El Hassene Osmani, Mounir Haddou, Naceurdine Bensalem

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In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely, the obstacle problem. We present a relaxed formulation for the problem using smoothing functions. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods, we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm (Interior Point Optimizer) are presented to verify the efficiency of our approach.

Keywords: complementarity problem, IPOPT, Lagrange multipliers, mathematical programming, optimal control, smoothing methods, variationally inequalities

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Authors: Babitha Elizabeth Philip

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This paper aims to study in general about bridges without expansion joints. Integral Abutment Bridges (IAB) fall into this category of bridges. They are having a continuous deck and also the girders are integrated into the abutments. They are most cost effective system in terms of construction, maintenance, and longevity. The main advantage of IAB is that it is corrosion resistant since water is not allowed to pass through the structure. The other attractions of integral bridges are its simple and rapid construction, smooth and uninterrupted deck which provides a safe ride. Also damages to the abutments can be avoided to a great extent due to better load distribution at the bridge ends. Damages due to improper drainage are not seen in IAB because of its properly drained approach slabs thus eliminating the possibility of erosion of the abutment backfill and freeze and thaw damage resulting from saturated backfill.

Keywords: continuous bridge, integral abutment bridge, joint bridge, life cycle cost, soil interaction

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Authors: Samson Davoyan, Ashot Davoyan, Ani Khachatryan

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The indexes (Global Competitiveness Index, Economic Freedom Index, Human Development Index, etc.) developed by different international and non-government organizations in time and space express the quantitative and qualitative features of different fields of various reforms implemented in different countries. The main objective of our research is to develop new methodology that we will use to create integral index based on many indexes and that will include many areas of reforms. To achieve our aim we have used econometric methods (regression model for panel data method). The basis of our methodology is the development of the new integral index based on quantitative assessment of the change of two main parameters: the score of the countries by different indexes and the change of the ranks of countries for following two periods of time. As a result of the usage of methods for analyzes we have defined the indexes that are used to create the new integral index and the scales for each of them. Analyzing quantitatively and qualitatively analysis through the integral index for more than 100 countries for 2009-2014, we have defined comparative efficiency that helps to conclude in which directions countries have implemented reforms more effectively compared to others and in which direction reforms have implemented less efficiently.

Keywords: development, rank, reforms, comparative, index, economic, corruption, social, program

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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: 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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