Search results for: Asymmetric Traveling Salesman Problem

Authors: Ashwani Kumar

Abstract:

Use of NIR light for imaging the biological tissue and to quantify its optical properties is a good choice over other invasive methods. Optical tomography involves two steps. One is the forward problem and the other is the reconstruction problem. The forward problem consists of finding the measurements of transmitted light through the tissue from source to detector, given the spatial distribution of absorption and scattering properties. The second step is the reconstruction problem. In X-ray tomography, there is standard method for reconstruction called filtered back projection method or the algebraic reconstruction methods. But this method cannot be applied as such, in optical tomography due to highly scattering nature of biological tissue. A hybrid algorithm for reconstruction has been implemented in this work which takes into account the highly scattered path taken by photons while back projecting the forward data obtained during Monte Carlo simulation. The reconstructed image suffers from blurring due to point spread function.

Keywords: NIR light, tissue, blurring, Monte Carlo simulation

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Authors: Srushti Bhatt, M. B. Kiran

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Two-sided assembly line balancing problem is yet to be addressed simply to compete for the global market for manufacturers. The task assigned in an ordered sequence to get optimum performance of the system is known as assembly line balancing problem mainly classified as single and two sided. It is very challenging in manufacturing industries to balance two-sided assembly line, wherein the set of sequential workstations the task operations are performed in two sides of the line. The conflicting major objective in two-sided assembly line balancing problem is either to maximize /minimize the performance parameters. The present study emphases on combining different evolutionary algorithm; ant colony, Tabu search and petri net method; and compares their results of an algorithm for solving two-sided assembly line balancing problem. The concept of multi objective optimization of performance parameters is now a day adopted to make a decision involving more than one objective function to be simultaneously optimized. The optimum result can be expected among the selected methods using multi-objective optimization. The performance parameters considered in the present study are a number of workstation, slickness and smoothness index. The simulation of the assembly line balancing problem provides optimal results of classical and practical problems.

Keywords: Ant colony, petri net, tabu search, two sided ALBP

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Authors: Gopi Ram, Durbadal Mandal, Rajib Kar, Sakti Prasad Ghoshal

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A design problem of non-uniform circular antenna arrays for maximum reduction of both the side lobe level (SLL) and first null beam width (FNBW) is dealt with. This problem is modeled as a simple optimization problem. The method of Firefly algorithm (FFA) is used to determine an optimal set of current excitation weights and antenna inter-element separations that provide radiation pattern with maximum SLL reduction and much improvement on FNBW as well. Circular array antenna laid on x-y plane is assumed. FFA is applied on circular arrays of 8-, 10-, and 12- elements. Various simulation results are presented and hence performances of side lobe and FNBW are analyzed. Experimental results show considerable reductions of both the SLL and FNBW with respect to those of the uniform case and some standard algorithms GA, PSO, and SA applied to the same problem.

Keywords: circular arrays, first null beam width, side lobe level, FFA

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Authors: Peng Li, Er-xiang Song

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Numerical analysis of longitudinal tunnel seismic response due to spatial variation of earthquake ground motion is an important issue that cannot be ignored in the design and safety evaluation of tunnel structures. In this paper, numerical methods for analysis of tunnel longitudinal response under asynchronous seismic wave is extensively studied, including the improvement of the 1D time-domain finite element method, three dimensional numerical simulation technique for the site asynchronous earthquake response as well as the 3-D soil-tunnel structure interaction analysis. The study outcome will be beneficial to aid further research on the nonlinear meticulous numerical analysis and seismic response mechanism of tunnel structures under asynchronous earthquake motion.

Keywords: asynchronous input, longitudinal seismic response, tunnel structure, numerical simulation, traveling wave effect

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Authors: Halil Ibrahim Demir, Caner Erden, Abdullah Hulusi Kokcam, Mumtaz Ipek

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Process planning, scheduling, and due date assignment are three important manufacturing functions which are studied independently in literature. There are hundreds of works on IPPS and SWDDA problems but a few works on IPPSDDA problem. Integrating these three functions is very crucial due to the high relationship between them. Since the scheduling problem is in the NP-Hard problem class without any integration, an integrated problem is even harder to solve. This study focuses on the integration of these functions. Sum of weighted tardiness, earliness, and due date related costs are used as a penalty function. Random search and hybrid metaheuristics are used to solve the integrated problem. Marginal improvement in random search is very high in the early iterations and reduces enormously in later iterations. At that point directed search contribute to marginal improvement more than random search. In this study, random and genetic search methods are combined to find better solutions. Results show that overall performance becomes better as the integration level increases.

Keywords: process planning, genetic algorithm, hybrid search, random search, weighted due-date assignment, weighted scheduling

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Authors: Maatoug Hassine, Mourad Hrizi

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In this paper, we consider a geometric inverse source problem for the heat equation with Dirichlet and Neumann boundary data. We will reconstruct the exact form of the unknown source term from additional boundary conditions. Our motivation is to detect the location, the size and the shape of source support. We present a one-shot algorithm based on the Kohn-Vogelius formulation and the topological gradient method. The geometric inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from a source function. Then, we present a non-iterative numerical method for the geometric reconstruction of the source term with unknown support using a level curve of the topological gradient. Finally, we give several examples to show the viability of our presented method.

Keywords: geometric inverse source problem, heat equation, topological optimization, topological sensitivity, Kohn-Vogelius formulation

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Authors: Muhammad Umair, Syed Qasim Gilani

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A technique using Independent Component Analysis (ICA) for blind receiver signal processing is investigated. The problem of the receiver signal processing is viewed as of signal equalization and implementation imperfections compensation. Based on this, a model similar to a general ICA problem is developed for the received signal. Then, the use of ICA technique for blind signal equalization in the time domain is presented. The equalization is regarded as a signal separation problem, since the desired signal is separated from interference terms. This problem is addressed in the paper by over-sampling of the received signal. By using ICA for equalization, besides channel equalization, other transmission imperfections such as Direct current (DC) bias offset, carrier phase and In phase Quadrature phase imbalance will also be corrected. Simulation results for a system using 16-Quadraure Amplitude Modulation(QAM) are presented to show the performance of the proposed scheme.

Keywords: blind equalization, blind signal separation, equalization, independent component analysis, transmission impairments, QAM receiver

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Authors: Xiaofei Hu, Zhiyu Cai, Weian Yao

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V-notch problem under dynamic loading condition is considered in this paper. In the time domain, the precise time domain expanding algorithm is employed, in which a self-adaptive technique is carried out to improve computing accuracy. By expanding variables in each time interval, the recursive finite element formulas are derived. In the space domain, a Symplectic Analytical Singular Element (SASE) for V-notch problem is constructed addressing the stress singularity of the notch tip. Combining with the conventional finite elements, the proposed SASE can be used to solve the dynamic stress intensity factors (DSIFs) in a simple way. Numerical results show that the proposed SASE for V-notch problem subjected to dynamic loading condition is effective and efficient.

Keywords: V-notch, dynamic stress intensity factor, finite element method, precise time domain expanding algorithm

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Authors: Veronica Diaz Quezada

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In this article, the performance and errors are featured and analysed in the limit problems solving of a real-valued function, in correspondence to competency-based education in engineering careers, in the south of Chile. The methodological component is contextualised in a qualitative research, with a descriptive and explorative design, with elaboration, content validation and application of quantitative instruments, consisting of two parallel forms of open answer tests, based on limit application problems. The mathematical competences and errors made by students from five engineering careers from a public University are identified and characterized. Results show better performance only to solve routine-context problem-solving competence, thus they are oriented towards a rational solution or they use a suitable problem-solving method, achieving the correct solution. Regarding errors, most of them are related to techniques and the incorrect use of theorems and definitions of real-valued function limits of real variable.

Keywords: engineering education, errors, limits, mathematics competences, problem solving

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Authors: Babak H. Tabrizi, Seyed Farid Ghaderi

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Concurrent planning of project scheduling and material ordering has been increasingly addressed within last decades as an approach to improve the project execution costs. Therefore, we have taken the problem into consideration in this paper, aiming to maximize schedules quality robustness, in addition to minimize the relevant costs. In this regard, a bi-objective mathematical model is developed to formulate the problem. Moreover, it is possible to utilize the all-unit discount for materials purchasing. The problem is then solved by the constraint method, and the Pareto front is obtained for a variety of robustness values. The applicability and efficiency of the proposed model is tested by different numerical instances, finally.

Keywords: e-constraint method, material ordering, project management, project scheduling

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Authors: M. Bouinoun, M. Bouhadef

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The present study is concerned with the problem of determining the shape of the free surface flow in a hydraulic channel which has an uneven bottom. For the mathematical formulation of the problem, the fluid of the two-dimensional irrotational steady flow in water is assumed inviscid and incompressible. The solutions of the nonlinear problem are obtained by using the usual conformal mapping theory and Hilbert’s technique. An experimental study, for comparing the obtained results, has been conducted in a hydraulic channel (subcritical regime and supercritical regime).

Keywords: free-surface flow, experiments, numerical method, uneven bottom, supercritical regime, subcritical regime

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Authors: Vesna Kirandziska, Nevena Ackovska, Ana Madevska Bogdanova

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The problem of emotion recognition is a challenging problem. It is still an open problem from the aspect of both intelligent systems and psychology. In this paper, both voice features and facial features are used for building an emotion recognition system. A Support Vector Machine classifiers are built by using raw data from video recordings. In this paper, the results obtained for the emotion recognition are given, and a discussion about the validity and the expressiveness of different emotions is presented. A comparison between the classifiers build from facial data only, voice data only and from the combination of both data is made here. The need for a better combination of the information from facial expression and voice data is argued.

Keywords: emotion recognition, facial recognition, signal processing, machine learning

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Authors: H. Mashhady, H. A. Manzoori, M. Doosti, M. Fatollahi

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This study reports a descriptive research in the form of a genre analysis of postgraduates' theses and dissertations at three Iranian universities, including Ferdowsi, Tehran, and Tarbiat Moddares universities. The researchers sought to depict the generic structure of “statement of the problem” section of PhD dissertations and MA theses. Moreover, researchers desired to find any probable variety based on the year the dissertations belonged, to see weather genre-consciousness developed among Iranian postgraduates. To obtain data, “statement of the problem” section of 90 Ph.D. dissertations and MA theses from 2001 to 2013 in Teaching English as a Foreign Language (TEFL) at above-mentioned universities was selected. Frequency counts was employed for the quantitative method of data analysis, while genre analysis was used as the qualitative method. Inter-rater reliability was found to be about 0.93. Results revealed that students in different degrees at each of these universities used various generic structures for writing “statement of the problem”. Moreover, comparison of different time periods (2001-2006, and 2007-2013) revealed that postgraduates in the second time period, regardless of their degree and university, employed more similar generic structures which can be optimistically attributed to a general raise in genre awareness.

Keywords: genre, genre analysis, Ph.D. and MA dissertations, statement of the problem, generic structure

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Authors: Eylem Koç, Hasan Arda Burhan

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Location selection presents a crucial decision problem in today’s business world where strategic decision making processes have critical importance. Thus, location selection has strategic importance for companies in boosting their strength regarding competition, increasing corporate performances and efficiency in addition to lowering production and transportation costs. A right choice in location selection has a direct impact on companies’ commercial success. In this study, a store location selection problem of Carglass Turkey which operates in vehicle glass branch is handled. As this problem includes both tangible and intangible criteria, Analytic Network Process (ANP) was accepted as the main methodology. The model consists of control hierarchy and BOCR subnetworks which include clusters of actors, alternatives and criteria. In accordance with the management’s choices, five different locations were selected. In addition to the literature review, a strict cooperation with the actor group was ensured and maintained while determining the criteria and during whole process. Obtained results were presented to the management as a report and its feasibility was confirmed accordingly.

Keywords: analytic network process (ANP), BOCR, multi-actor decision making, multi-criteria decision making, real-life problem, location selection

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Authors: Mintu Philip, Renumol V. G.

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An introductory programming course is a major challenge faced in Computing Education. Many of the introductory programming courses fail because student concentrate mainly on writing programs using a programming language rather than involving in problem solving. Computational thinking is a general approach to solve problems. This paper proposes a new preliminary course that aims to develop computational thinking skills in students, which may help them to become good programmers. The proposed course is designed based on the four basic components of computational thinking - abstract thinking, logical thinking, modeling thinking and constructive thinking. In this course, students are engaged in hands-on problem solving activities using a new problem solving model proposed in this paper.

Keywords: computational thinking, computing education, abstraction, constructive thinking, modelling thinking

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Authors: Se-Mi Lim, Won-Jae Jung, Jin-Sup Kim, Jun-Seok Park, Hyung-Il Chae

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This paper presents a current-mode vertical hall device (VHD) structure using metal layers in a CMOS process. The proposed metal layer based vertical hall device (MLVHD) utilizes vertical connection among metal layers (from M1 to the top metal) to facilitate hall effect. The vertical metal structure unit flows a bias current Ibias from top to bottom, and an external magnetic field changes the current distribution by Lorentz force. The asymmetric current distribution can be detected by two differential-mode current outputs on each side at the bottom (M1), and each output sinks Ibias/2 ± Ihall. A single vertical metal structure generates only a small amount of hall effect of Ihall due to the short length from M1 to the top metal as well as the low conductivity of the metal, and a series connection between thousands of vertical structure units can solve the problem by providing NxIhall. The series connection between two units is another vertical metal structure flowing current in the opposite direction, and generates negative hall effect. To mitigate the negative hall effect from the series connection, the differential current outputs at the bottom (M1) from one unit merges on the top metal level of the other unit. The proposed MLVHD is simulated in a 3-dimensional model simulator in COMSOL Multiphysics, with 0.35 μm CMOS process parameters. The simulated MLVHD unit size is (W) 10 μm × (L) 6 μm × (D) 10 μm. In this paper, we use an MLVHD with 10 units; the overall hall device size is (W) 10 μm × (L)78 μm × (D) 10 μm. The COMSOL simulation result is as following: the maximum hall current is approximately 2 μA with a 12 μA bias current and 100mT magnetic field; This work was supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIP) (No.R7117-16-0165, Development of Hall Effect Semiconductor for Smart Car and Device).

Keywords: CMOS, vertical hall device, current mode, COMSOL

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Authors: A. Guezane-Lakoud, S. Bensebaa

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In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained.

Keywords: positive solution, fractional caputo derivative, Banach contraction principle, Avery and Peterson fixed point theorem

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Authors: Aquil Ahmed, Srikant Gupta, Irfan Ali

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In this paper, we study a multi-objective capacitated transportation problem (MOCTP) with mixed constraints. This paper is comprised of the modelling and optimisation of an MOCTP in a fuzzy environment in which some goals are fractional and some are linear. In real life application of the fuzzy goal programming (FGP) problem with multiple objectives, it is difficult for the decision maker(s) to determine the goal value of each objective precisely as the goal values are imprecise or uncertain. Also, we developed the concept of linearization of fractional goal for solving the MOCTP. In this paper, imprecision of the parameter is handled by the concept of fuzzy set theory by considering these parameters as a trapezoidal fuzzy number. α-cut approach is used to get the crisp value of the parameters. Numerical examples are used to illustrate the method for solving MOCTP.

Keywords: capacitated transportation problem, multi objective linear programming, multi-objective fractional programming, fuzzy goal programming, fuzzy sets, trapezoidal fuzzy number

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Authors: Mushfiqul Alam, Kashyapa Narenathreyas

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The demand for efficient transonic transport has been growing every day and may turn out to be the most pressed innovation in coming years. Oblique wing configuration was proposed as an alternative to conventional wing configuration for supersonic and transonic passenger aircraft due to its aerodynamic advantages. This paper re-demonstrates the aerodynamic advantages of oblique wing configuration using open source CFD code. The aerodynamic data were generated using Panel Method. Results show that Oblique Wing concept with elliptical wing planform offers a significant reduction in drag at transonic and supersonic speeds and approximately twice the lift distribution compared to conventional operating aircrafts. The paper also presents a preliminary conceptual aircraft sizing which can be used for further experimental analysis.

Keywords: aerodynamics, asymmetric sweep, oblique wing, swing wing

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Authors: S. Feli, S. Tavousi Tafreshi, A. Ghasemi

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The present paper aims at evaluating the response of three-dimensional buildings with in-plan stiffness irregularities that have been subjected to two-way excitation ground motion records simultaneously. This study is broadly-based fragility assessment with greater emphasis on structural response at in-plan flexible and stiff sides. To this end, three type of three-dimensional 5-story steel building structures with stiffness eccentricities, were subjected to extensive nonlinear incremental dynamic analyses (IDA) utilizing Ibarra-Krawinkler deterioration models. Fragility assessment was implemented for different configurations of braces to investigate the losses in buildings with center of resisting (CR) eccentricities.

Keywords: Ibarra-Krawinkler, fragility assessment, flexible and stiff side, center of resisting

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Authors: Shao-Yuan Huang

Abstract:

We study the shape of the bifurcation curve of positive solutions for the semipositone problem with the Minkowski-curvature operator. The Minkowski-curvature problem plays an important role in certain fundamental issues in differential geometry and in the special theory of relativity. In addition, it is well known that studying the multiplicity of positive solutions is equivalent to studying the shape of the bifurcation curve. By the shape of the bifurcation curve, we can understand the change in the multiplicity of positive solutions with varying parameters. In this paper, our main technique is a time-map method used in Corsato's PhD Thesis. By this method, studying the shape of the bifurcation curve is equivalent to studying the shape of a certain function T with improper integral. Generally speaking, it is difficult to study the shape of T. So, in this paper, we consider two cases that the nonlinearity is convex or concave. Thus we obtain the following results: (i) If f''(u) < 0 for u > 0, then the bifurcation curve is C-shaped. (ii) If f''(u) > 0 for u > 0, then there exists η>β such that the bifurcation curve does not exist for 0 η. Furthermore, we prove that the bifurcation is C-shaped for L > η under a certain condition.

Keywords: bifurcation curve, Minkowski-curvature problem, positive solution, time-map method

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Authors: Kemal Polat

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In this study, it has been estimated whether there is skin by using RBG values obtained from the camera and k-nearest neighbor (k-NN) classifier. The dataset used in this study has an unbalanced distribution and a linearly non-separable structure. This problem can also be called a big data problem. The Skin dataset was taken from UCI machine learning repository. As the classifier, we have used the k-NN method to handle this big data problem. For k value of k-NN classifier, we have used as 1. To train and test the k-NN classifier, 50-50% training-testing partition has been used. As the performance metrics, TP rate, FP Rate, Precision, recall, f-measure and AUC values have been used to evaluate the performance of k-NN classifier. These obtained results are as follows: 0.999, 0.001, 0.999, 0.999, 0.999, and 1,00. As can be seen from the obtained results, this proposed method could be used to predict whether the image is skin or not.

Keywords: k-NN classifier, skin or non-skin classification, RGB values, classification

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Authors: Boumesbah Asma, Chergui Mohamed El-amine

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Since it has been shown that the multi-objective minimum spanning tree problem (MOST) is NP-hard even with two criteria, we propose in this study a hybrid NSGA-II algorithm with an exact mutation operator, which is only used with low probability, to find an approximation to the Pareto front of the problem. In a connected graph G, a spanning tree T of G being a connected and cycle-free graph, if k edges of G\T are added to T, we obtain a partial graph H of G inducing a reduced size multi-objective spanning tree problem compared to the initial one. With a weak probability for the mutation operator, an exact method for solving the reduced MOST problem considering the graph H is then used to give birth to several mutated solutions from a spanning tree T. Then, the selection operator of NSGA-II is activated to obtain the Pareto front approximation. Finally, an adaptation of the VNS metaheuristic is called for further improvements on this front. It allows finding good individuals to counterbalance the diversification and the intensification during the optimization search process. Experimental comparison studies with an exact method show promising results and indicate that the proposed algorithm is efficient.

Keywords: minimum spanning tree, multiple objective linear optimization, combinatorial optimization, non-sorting genetic algorithm, variable neighborhood search

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Authors: Elias Udeaja, Elijah Udoh

Abstract:

Exploring an extended SVAR model (SVAR-X), we use the case of Nigeria to hypothesize for the role of uncertainty as the underlying source of asymmetries in the response of monetary policy to output and inflation. Deciphered the empirical finding is the potential of monetary policy exhibiting greater sensitive to shocks due to output growth than they do to shocks due to inflation in recession periods, while the reverse appears to be the case for a contractionary monetary policy. We also find the asymmetric preference in the response of monetary policy to changes in output and inflation as relatively more pronounced when we control for uncertainty as the underlying source of asymmetries.

Keywords: asymmetry response, developing economies, monetary policy shocks, uncertainty

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Authors: Xiaoman Dong

Abstract:

The refugee problem has a long history, but the scale and severity of modern refugee crises demand us to consider if the progress of political history exacerbates the refugee problem. This paper argues that although sovereignty owes its legitimacy to the protection of human rights, the modern state system complicates the refugee problem by first introducing then blurring the line between human rights and civil rights, and making national identity indispensable to basic livelihood and dignity. This paper first explains the source of the modern state system’s legitimacy by putting it in the context of social contract theories and the politics of nation-building. It then discusses how states create the concept of statelessness, which leads to more violations on human rights. Using historical records of the League of Nations High Commission for Refugees and the United Nations High Commissioner for Refugees, this paper reveals that neither the refugee problem of the Cold-War period nor the current refugee crisis is collateral damage of war, but rather the consequence of intentional exclusionary policies produced out of political interests. Finally, it contends that if the modern state system is to sustain, it cannot prioritize the protection of civil rights of a particular group over the protection of basic human rights of all.

Keywords: burden sharing, human rights, legitimacy of state, positive externality, sovereignty

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Authors: Neetesh Saxena, Narendra S. Chaudhari

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Nowadays, SMS is a very popular mobile service and even the poor, illiterate people and those living in rural areas use SMS service very efficiently. Although many mobile operators have already started 3G and 4G services, 2G services are still being used by the people in many countries. In 2G (GSM), only encryption provided is between the MS and the BTS, there is no end-to-end encryption available. Sometimes we all need to send some confidential message to other person containing bank account number, some password, financial details, etc. Normally, a message is sent in plain text only to the recipient and it is not an acceptable standard for transmitting such important and confidential information. Authors propose an end-to-end encryption approach by proposing a terminal for sending/receiving a secure message. An asymmetric key exchange algorithm is used in order to transmit secret shared key securely to the recipient. The proposed approach with terminal device provides authentication, confidentiality, integrity and non-repudiation.

Keywords: AES, DES, Diffie-Hellman, ECDH, A5, SMS

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Authors: Hari Mohan Kushwaha, Santosh Kumar Sahu

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In the present study, analysis of heat transfer is carried out in the slip flow region for the fluid flowing between two parallel plates by employing the asymmetric heat fluxes at surface of the plates. The flow is assumed to be hydrodynamically and thermally fully developed for the analysis. The second order velocity slip and viscous dissipation effects are considered for the analysis. Closed form expressions are obtained for the Nusselt number as a function of Knudsen number and modified Brinkman number. The limiting condition of the present prediction for Kn = 0, Kn2 = 0, and Brq1 = 0 is considered and found to agree well with other analytical results.

Keywords: Knudsen number, modified Brinkman number, slip flow, velocity slip

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Authors: Arun Kumar, Neelesh Haulder

Abstract:

This paper identifies and resolves some key issues relating to a specific aspect within the supply chain logistics of the public health care industry in the Republic of Mauritius. The analysis and the proposed solution are performed using soft systems methodology (SSM). Through the application of this relevant systematic approach at problem solving, the aim is to obtain an in-depth analysis of the problem, incorporating every possible world view of the problem and consequently to obtain a well explored solution aimed at implementing relevant changes within the current supply chain logistics of the health care industry, with the purpose of tackling the key identified issues.

Keywords: soft systems methodology, CATWOE, healthcare, logistics

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Authors: Panupat Chaiworn, Soraya lama

Abstract:

We study the motion of particles in cylinders which are subjected to a sinusoidal vertical vibration. We measure the rising time of a large intruder from the bottom of the container to free surface of the bed particles and find that the rising time as a function of intruder density increases to a maximum and then decreases monotonically. The result is qualitatively accord to the previous findings in experiments using relative humidity of the bed particles and found speed convection of the bed particles containers it moving slowly, and the rising time of the intruder where a minimal instead of maximal rising time in the small density region was found. Our experimental results suggest that the topology of the container plays an important role in the Brazil nut effect.

Keywords: granular particles, Brazil nut effect, cylinder container, vertical vibration, convection

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Authors: Hassan Samadi, Mustapha El Jarroudi

Abstract:

In this work, we consider the homogenization of a non-linear problem in periodic medium with two periodic connected media exchanging a heat flux throughout their common interface. The interfacial exchange coefficient λ is assumed to tend to zero or to infinity following a rate λ=λ(ε) when the size ε of the basic cell tends to zero. Three homogenized problems are determined according to some critical value depending of λ and ε. Our method is based on Γ-Convergence techniques.

Keywords: variational methods, epiconvergence, homogenization, convergence technique

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