Search results for: Manoj Kumar Dutta

Authors: Manoj Kumar Dutta

Abstract:

Wavelength Division Multiplexing (WDM) technology is the most promising technology for the proper utilization of huge raw bandwidth provided by an optical fiber. One of the key problems in implementing the all-optical WDM network is the packet contention. This problem can be solved by several different techniques. In time domain approach the packet contention can be reduced by incorporating Fiber Delay Lines (FDLs) as optical buffer in the switch architecture. Different types of buffering architectures are reported in literatures. In the present paper a comparative performance analysis of three most popular FDL architectures are presented in order to obtain the best contention resolution performance. The analysis is further extended to consider the effect of different fiber non-linearities on the network performance.

Keywords: WDM network, contention resolution, optical buffering, non-linearity, throughput.

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Authors: Kumar Manoj, Ramesh Kumar, Kumari Arti

Abstract:

A wireless Ad-hoc network consists of wireless nodes communicating without the need for a centralized administration, in which all nodes potentially contribute to the routing process.In this paper, we report the simulation results of four different scenarios for wireless ad hoc networks having thirty nodes. The performances of proposed networks are evaluated in terms of number of hops per route, delay and throughput with the help of OPNET simulator. Channel speed 1 Mbps and simulation time 600 sim-seconds were taken for all scenarios. For the above analysis DSR routing protocols has been used. The throughput obtained from the above analysis (four scenario) are compared as shown in Figure 3. The average media access delay at node_20 for two routes and at node_20 for four different scenario are compared as shown in Figures 4 and 5. It is observed that the throughput will degrade when it will follow different hops for same source to destination (i.e. it has dropped from 1.55 Mbps to 1.43 Mbps which is around 9.7%, and then dropped to 0.48Mbps which is around 35%).

Keywords: Throughput, Delay, DSR, OPNET, MANET, DSSS

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Authors: Manoj Kumar, Rajesh Kumar

Abstract:

With the favourable geographical conditions at Indian-subcontinent, it is suitable for flourishing renewable energy. Increasing amount of dependence on coal and other conventional sources is driving the world into pollution and depletion of resources. This paper presents the statistics of energy consumption and energy generation in Indian Sub-continent, which notifies us with the increasing energy demands surpassing energy generation. With the aggrandizement in demand for energy, usage of coal has increased, since the major portion of energy production in India is from thermal power plants. The increase in usage of thermal power plants causes pollution and depletion of reserves; hence, a paradigm shift to renewable sources is inevitable. In this work, the capacity and potential of renewable sources in India are analyzed. Based on the analysis of this work, future potential of these sources is estimated.

Keywords: Energy consumption and generation, depletion of reserves, pollution, estimation, renewable sources.

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Authors: Arvind Kumar, Manoj Kumar, Dattatraya H. Nagaraj, Amanpreet Singh, Jayanthi Prattapati

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In recent years, the use of renewable energy resources instead of pollutant fossil fuels and other forms has increased. Photovoltaic generation is becoming increasingly important as a renewable resource since it does not cause in fuel costs, pollution, maintenance, and emitting noise compared with other alternatives used in power applications. In this paper, Perturb and Observe and Incremental Conductance methods are used to improve energy conversion efficiency under different environmental conditions. PI controllers are used to control easily DC-link voltage, active and reactive currents. The whole system is simulated under standard climatic conditions (1000 W/m2, 250C) in MATLAB and the irradiance is varied from 1000 W/m2 to 300 W/m2. The use of PI controller makes it easy to directly control the power of the grid connected PV system. Finally the validity of the system will be verified through the simulations in MATLAB/Simulink environment.

Keywords: Incremental conductance algorithm, modeling of PV panel, perturb and observe algorithm, photovoltaic system and simulation results.

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Authors: Arvind Kumar, Manoj Kumar, Dattatraya H. Nagaraj, Amanpreet Singh, Jayanthi Prattapati

Abstract:

The purpose of this paper is to study and compare two maximum power point tracking (MPPT) algorithms in a photovoltaic simulation system and also show a simulation study of maximum power point tracking (MPPT) for photovoltaic systems using perturb and observe algorithm and Incremental conductance algorithm. Maximum power point tracking (MPPT) plays an important role in photovoltaic systems because it maximize the power output from a PV system for a given set of conditions, and therefore maximize the array efficiency and minimize the overall system cost. Since the maximum power point (MPP) varies, based on the irradiation and cell temperature, appropriate algorithms must be utilized to track the (MPP) and maintain the operation of the system in it. MATLAB/Simulink is used to establish a model of photovoltaic system with (MPPT) function. This system is developed by combining the models established of solar PV module and DC-DC Boost converter. The system is simulated under different climate conditions. Simulation results show that the photovoltaic simulation system can track the maximum power point accurately.

Keywords: Incremental conductance Algorithm, Perturb and Observe Algorithm, Photovoltaic System and Simulation Results.

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Authors: Mousmita Sarma, Krishna Dutta, Kandarpa Kumar Sarma

Abstract:

Speech corpus is one of the major components in a Speech Processing System where one of the primary requirements is to recognize an input sample. The quality and details captured in speech corpus directly affects the precision of recognition. The current work proposes a platform for speech corpus generation using an adaptive LMS filter and LPC cepstrum, as a part of an ANN based Speech Recognition System which is exclusively designed to recognize isolated numerals of Assamese language- a major language in the North Eastern part of India. The work focuses on designing an optimal feature extraction block and a few ANN based cooperative architectures so that the performance of the Speech Recognition System can be improved.

Keywords: Filter, Feature, LMS, LPC, Cepstrum, ANN.

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Authors: J. Dutta, P. Pranith Kumar Reddy

Abstract:

This research paper portrays a comparative analysis of thermomechanical behaviour of Shielded Metal Arc Welding (SMAW) and Gas Tungsten Arc Welding (GTAW) of low carbon steel of AISI 1020 grade butt joints. The thermal history has been obtained by experimental work. We have focused on temperature dependent cooling rate as depicted by Adam’s two-dimensional model. The effect of moving point heat source of SMAW and GTAW on mechanical properties has been judged by optical and scanning electron micrographs of different regions in weld joints. The microhardness study has been carried to visualize the joint strength due to formation of different phases.

Keywords: Shielded metal arc welding, gas tungsten arc welding, low carbon steel, microhardness study, thermal history, microscopic morphology.

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Authors: Raksha Sharma, Vishnu Kant Soni, Manoj Kumar Mishra, Prachet Bhuyan

Abstract:

Grid computing is a form of distributed computing that involves coordinating and sharing computational power, data storage and network resources across dynamic and geographically dispersed organizations. Scheduling onto the Grid is NP-complete, so there is no best scheduling algorithm for all grid computing systems. An alternative is to select an appropriate scheduling algorithm to use in a given grid environment because of the characteristics of the tasks, machines and network connectivity. Job and resource scheduling is one of the key research area in grid computing. The goal of scheduling is to achieve highest possible system throughput and to match the application need with the available computing resources. Motivation of the survey is to encourage the amateur researcher in the field of grid computing, so that they can understand easily the concept of scheduling and can contribute in developing more efficient scheduling algorithm. This will benefit interested researchers to carry out further work in this thrust area of research.

Keywords: Grid Computing, Job Scheduling, ResourceScheduling.

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Authors: Avadhesh Yadav, Manoj Kumar, Balram

Abstract:

A solar powered air heating system using parabolic trough collector was experimentally investigated. In this experimental setup, the reflected solar radiations were focused on absorber tube which was placed at focal length of the parabolic trough. In this setup, air was used as working fluid which collects the heat from absorber tube. To enhance the performance of parabolic trough, collector with different type of reflectors were used. It was observed For Aluminum sheet maximum temperature is 52.3ºC, which 24.22% more than steel sheet as reflector and 8.5% more than Aluminum foil as reflector, also efficiency by using Aluminum sheet as reflector compared to steel sheet as reflector is 61.18% more. Efficiency by using Aluminum sheet as reflector compared to Aluminum foil as reflector is 18.98% more.

Keywords: Parabolic trough collector, Reflectors, Air flow rates.

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Authors: Vedulla Manoj Kumar, B. Nageswara Rao, Sk. Farooq

Abstract:

Heat sinks are being considered in many advanced heat transfer applications including automotive and stationary fuel cells as well as cooling of electronic devices. However, there are innumerable fundamental issues in the fields of heat transfer and fluid mechanics perspectives which remains unresolved. The present review emphasizes on the progress of research in the field of pin fin heat sinks, while understanding the fluid dynamics and heat transfer characteristics with a detailed and sophisticated prediction of the temperature distribution, high heat flux removal and by minimizing thermal resistance. Lot of research work carried out across the globe to address this challenge and trying to come up with an economically viable and user friendly solution. The high activities for future pin fin heat sinks research and development to meet the current issue is recorded in this article.

Keywords: Heat sinks, heat transfer, heat flux, thermal resistance, electronic devices.

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Authors: N.S. Tomer, Phool Singh, Manoj Kumar

Abstract:

The flow and heat transfer characteristics for natural convection along an inclined plate in a saturated porous medium with an applied magnetic field have been studied. The fluid viscosity has been assumed to be an inverse function of temperature. Assuming temperature vary as a power function of distance. The transformed ordinary differential equations have solved by numerical integration using Runge-Kutta method. The velocity and temperature profile components on the plate are computed and discussed in detail for various values of the variable viscosity parameter, inclination angle, magnetic field parameter, and real constant (λ). The results have also been interpreted with the aid of tables and graphs. The numerical values of Nusselt number have been calculated for the mentioned parameters.

Keywords: Heat Transfer, Magnetic Field, Porosity, Viscosity

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Authors: Vishnu Kant Soni, Raksha Sharma, Manoj Kumar Mishra

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Grid computing is a high performance computing environment to solve larger scale computational applications. Grid computing contains resource management, job scheduling, security problems, information management and so on. Job scheduling is a fundamental and important issue in achieving high performance in grid computing systems. However, it is a big challenge to design an efficient scheduler and its implementation. In Grid Computing, there is a need of further improvement in Job Scheduling algorithm to schedule the light-weight or small jobs into a coarse-grained or group of jobs, which will reduce the communication time, processing time and enhance resource utilization. This Grouping strategy considers the processing power, memory-size and bandwidth requirements of each job to realize the real grid system. The experimental results demonstrate that the proposed scheduling algorithm efficiently reduces the processing time of jobs in comparison to others.

Keywords: Grid computing, Job grouping and Jobscheduling.

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Authors: Chaudhuri Manoj Kumar Swain, Susmita Das

Abstract:

This paper explores a detailed procedure of predicting a path loss (PL) model and its application in estimating the coverage probability in a WiMAX network. For this a hybrid approach is followed in predicting an empirical PL model of a 2.65 GHz WiMAX network deployed in a suburban environment. Data collection, statistical analysis, and regression analysis are the phases of operations incorporated in this approach and the importance of each of these phases has been discussed properly. The procedure of collecting data such as received signal strength indicator (RSSI) through experimental set up is demonstrated. From the collected data set, empirical PL and RSSI models are predicted with regression technique. Furthermore, with the aid of the predicted PL model, essential parameters such as PL exponent as well as the coverage probability of the network are evaluated. This research work may assist in the process of deployment and optimisation of any cellular network significantly.

Keywords: WiMAX, RSSI, path loss, coverage probability, regression analysis.

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Authors: Manas Ranjan Kabat, Manoj Kumar Patel, Chita Ranjan Tripathy

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Many multimedia communication applications require a source to transmit messages to multiple destinations subject to quality of service (QoS) delay constraint. To support delay constrained multicast communications, computer networks need to guarantee an upper bound end-to-end delay from the source node to each of the destination nodes. This is known as multicast delay problem. On the other hand, if the same message fails to arrive at each destination node at the same time, there may arise inconsistency and unfairness problem among users. This is related to multicast delayvariation problem. The problem to find a minimum cost multicast tree with delay and delay-variation constraints has been proven to be NP-Complete. In this paper, we propose an efficient heuristic algorithm, namely, Economic Delay and Delay-Variation Bounded Multicast (EDVBM) algorithm, based on a novel heuristic function, to construct an economic delay and delay-variation bounded multicast tree. A noteworthy feature of this algorithm is that it has very high probability of finding the optimal solution in polynomial time with low computational complexity.

Keywords: EDVBM, Heuristic algorithm, Multicast tree, QoS routing, Shortest path.

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Authors: Juhi Kaushik, Abhishek Agarwal, Manoj Sarda, Vatsal Sanjay, Arup Kumar Das

Abstract:

Fire accidents in trains bring huge disaster to human life and property. Evacuation becomes a major challenge in such incidents owing to confined spaces, large passenger density and trains moving at high speeds. The pantry car in Indian Railways trains carry inflammable materials like cooking fuel and LPG and electrical fittings. The pantry car is therefore highly susceptible to fire accidents. Numerical simulations have been done in a pantry car of Indian locomotive train using computational fluid dynamics based software. Different scenarios of a fire outbreak have been explored by varying Heat Release Rate per Unit Area (HRRPUA) of the fire source, introduction of exhaust in the cooking area, and taking a case of an air conditioned pantry car. Temporal statures of flame and soot have been obtained for each scenario and differences have been studied and reported. Inputs from this study can be used to assess casualties in fire accidents in locomotive trains and development of smoke control/detection systems in Indian trains.

Keywords: Fire propagation, flame contour, pantry fire, soot flow.

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Authors: Manoj Kumar, A.K. Verma, A. Srividya

Abstract:

Time varying network induced delays in networked control systems (NCS) are known for degrading control system-s quality of performance (QoP) and causing stability problems. In literature, a control method employing modeling of communication delays as probability distribution, proves to be a better method. This paper focuses on modeling of network induced delays as probability distribution. CAN and MIL-STD-1553B are extensively used to carry periodic control and monitoring data in networked control systems. In literature, methods to estimate only the worst-case delays for these networks are available. In this paper probabilistic network delay model for CAN and MIL-STD-1553B networks are given. A systematic method to estimate values to model parameters from network parameters is given. A method to predict network delay in next cycle based on the present network delay is presented. Effect of active network redundancy and redundancy at node level on network delay and system response-time is also analyzed.

Keywords: NCS (networked control system), delay analysis, response-time distribution, worst-case delay, CAN, MIL-STD-1553B, redundancy

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Authors: Raksha Sharma, Vishnu Kant Soni, Manoj Kumar Mishra, Prachet Bhuyan, Utpal Chandra Dey

Abstract:

Grid computing is a group of clusters connected over high-speed networks that involves coordinating and sharing computational power, data storage and network resources operating across dynamic and geographically dispersed locations. Resource management and job scheduling are critical tasks in grid computing. Resource selection becomes challenging due to heterogeneity and dynamic availability of resources. Job scheduling is a NP-complete problem and different heuristics may be used to reach an optimal or near optimal solution. This paper proposes a model for resource and job scheduling in dynamic grid environment. The main focus is to maximize the resource utilization and minimize processing time of jobs. Grid resource selection strategy is based on Max Heap Tree (MHT) that best suits for large scale application and root node of MHT is selected for job submission. Job grouping concept is used to maximize resource utilization for scheduling of jobs in grid computing. Proposed resource selection model and job grouping concept are used to enhance scalability, robustness, efficiency and load balancing ability of the grid.

Keywords: Agent, Grid Computing, Job Grouping, Max Heap Tree (MHT), Resource Scheduling.

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Authors: Fan-Yong Meng, Yan Wang

Abstract:

In this paper, we discuss the egalitarianism solution (ES) and center-of-gravity of the imputation-set value (CIV) for bicooperative games, which can be seen as the extensions of the solutions for traditional games given by Dutta and Ray [1] and Driessen and Funaki [2]. Furthermore, axiomatic systems for the given values are proposed. Finally, a numerical example is offered to illustrate the player ES and CTV.

Keywords: Bicooperative games, egalitarianism solution, center of- gravity of the imputation-set value.

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Authors: Shibesh Dutta, Sivaramakrishnan R., Sundar Gopalan, Balakrishnan Shankar

Abstract:

MOSCAPs of various combinations of Hafnium oxide and Titanium oxide of varying thickness with Aluminum as gate electrode have been fabricated and electrically characterized. The effects of voltage stress on the I-V characteristics for prolonged time durations have been studied and compared. Results showed hard breakdown and negligible degradation of reliability under stress.

Keywords: breakdown, MOSCAP, voltage stress.

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Authors: Manoj Singh, Mumtaz Ahmad Khan, Abdul Hakim Khan

Abstract:

The investigation in the present paper is to obtain certain types of relations for the well known hypergeometric functions by employing the technique of fractional derivative and integral.

Keywords: Fractional Derivatives and Integrals, Hypergeometric functions.

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Authors: Sachin Kumar, Vivek Tiwari, Goutam Dutta

Abstract:

The paper investigates parallel channel instabilities of natural circulation boiling water reactor. A thermal-hydraulic model is developed to simulate two-phase flow behavior in the natural circulation boiling water reactor (NCBWR) with the incorporation of ex-core components and recirculation loop such as steam separator, down-comer, lower-horizontal section and upper-horizontal section and then, numerical analysis is carried out for parallel channel instabilities of the reactor undergoing both in-phase and out-of-phase modes of oscillations. To analyze the relative effect on stability of the reactor due to inclusion of various ex-core components and recirculation loop, marginal stable point is obtained at a particular inlet enthalpy of the reactor core without the inclusion of ex-core components and recirculation loop and then with the inclusion of the same. Numerical simulations are also conducted to determine the relative dominance between two modes of oscillations i.e. in-phase and out-of-phase. Simulations are also carried out when the channels are subjected to asymmetric power distribution keeping the inlet enthalpy same.

Keywords: Asymmetric power distribution, Density wave oscillations, In-phase and out-of-phase modes of instabilities, Natural circulation boiling water reactor

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Authors: Abhishek Agarwal, Manoj Sarda, Juhi Kaushik, Vatsal Sanjay, Arup Kumar Das

Abstract:

Propagation of fire through a non-air conditioned railway compartment is studied by virtue of numerical simulations. Simultaneous computational fire dynamics equations, such as Navier-Stokes, lumped species continuity, overall mass and energy conservation, and heat transfer are solved using finite volume based (for radiation) and finite difference based (for all other equations) solver, Fire Dynamics Simulator (FDS). A single coupe with an eight berth occupancy is used to establish the numerical model, followed by the selection of a three coupe system as the fundamental unit of the locomotive compartment. Heat Release Rate Per Unit Area (HRRPUA) of the initial fire is varied to consider a wide range of compartmental fires. Parameters, such as air inlet velocity relative to the locomotive at the windows, the level of interaction with the ambiance and closure of middle berth are studied through a wide range of numerical simulations. Almost all the loss of lives and properties due to fire breakout can be attributed to the direct or indirect exposure to flames or to the inhalation of toxic gases and resultant suffocation due to smoke and soot. Therefore, the temporal stature of fire and smoke are reported for each of the considered cases which can be used in the present or extended form to develop guidelines to be followed in case of a fire breakout.

Keywords: Fire dynamics, flame propagation, locomotive fire, soot flow pattern.

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Authors: Sathinee Malee, Ronnason Chinram

Abstract:

The notion of k-fuzzy ideals of semirings was introduced by Kim and Park in 1996. In 2003, Dutta and Kar introduced a notion of ternary semirings. This structure is a generalization of ternary rings and semirings. The main purpose of this paper is to introduce and study k-fuzzy ideals in ternary semirings analogous to k-fuzzy ideals in semirings considered by Kim and Park.

Keywords: k-ideals, k-fuzzy ideals, fuzzy k-ideals, ternarysemirings

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Authors: S. Zith Dey Babu, S. Kour, S. Verma, C. Verma, V. Pathania, A. Agrawal, V. Chaudhary, A. Manoj Puthur, R. Goyal, A. Pal, T. Danti Dey, A. Kumar, K. Wadhwa, O. Ved

Abstract:

Background: Skin cancer is now is the buzzing button in the field of medical science. The cyst's pandemic is drastically calibrating the body and well-being of the global village. Methods: The extracted image of the skin tumor cannot be used in one way for diagnosis. The stored image contains anarchies like the center. This approach will locate the forepart of an extracted appearance of skin. Partitioning image models has been presented to sort out the disturbance in the picture. Results: After completing partitioning, feature extraction has been formed by using genetic algorithm and finally, classification can be performed between the trained and test data to evaluate a large scale of an image that helps the doctors for the right prediction. To bring the improvisation of the existing system, we have set our objectives with an analysis. The efficiency of the natural selection process and the enriching histogram is essential in that respect. To reduce the false-positive rate or output, GA is performed with its accuracy. Conclusions: The objective of this task is to bring improvisation of effectiveness. GA is accomplishing its task with perfection to bring down the invalid-positive rate or outcome. The paper's mergeable portion conflicts with the composition of deep learning and medical image processing, which provides superior accuracy. Proportional types of handling create the reusability without any errors.

Keywords: Computer-aided system, detection, image segmentation, morphology.

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Authors: Gaurav Kumar, Rakesh Kumar Bajaj

Abstract:

The applicability of Net Present Value (NPV) in an investment project is becoming more and more popular in the field of engineering economics. The classical NPV methodology involves only the precise and accurate data of the investment project. In the present communication, we give a new mathematical model for NPV which uses the concept of intuitionistic fuzzy set theory. The proposed model is based on triangular intuitionistic fuzzy number, which may be known as Intuitionistic Fuzzy Net Present Value (IFNPV). The model has been applied to an example and the results are presented.

Keywords: Net Present Value, Intuitionistic Fuzzy Set, Investment Projects.

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Authors: Manoj Sarda, Abhishek Agarwal, Juhi Kaushik, Vatsal Sanjay, Arup Kumar Das

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Railways in India remain primary mode of transport having one of the largest networks in the world and catering to billions of transits yearly. Catastrophic economic damage and loss to life is encountered over the past few decades due to fire to locomotives. Study of fire dynamics and fire propagation plays an important role in evacuation planning and reducing losses. Simulation based study of propagation of fire and soot inside an air conditioned coach of Indian locomotive is done in this paper. Finite difference based solver, Fire Dynamic Simulator (FDS) version 6 has been used for analysis. A single air conditioned 3 tier coupe closed to ambient surroundings by glass windows having occupancy for 8 people is the basic unit of the domain. A system of three such coupes combined is taken to be fundamental unit for the entire study to resemble effect to an entire coach. Analysis of flame and soot contours and concentrations is done corresponding to variations in heat release rate per unit volume (HRRPUA) of fire source, variations in conditioned air velocity being circulated inside coupes by vents and an alternate fire initiation and propagation mechanism via ducts. Quantitative results of fractional area in top and front view of the three coupes under fire and smoke are obtained using MATLAB (IMT). Present simulations and its findings will be useful for organizations like Commission of Railway Safety and others in designing and implementing safety and evacuation measures.

Keywords: Air-conditioned coaches, fire propagation, flame contour, soot flow, train fire.

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Authors: Sanjay Kumar Awasthi, Sunil Kumar Bajpai, Surendra Kumar Pandey, Ajay Utiye

Abstract:

The effect of gamma irradiation on micro-hardness of polymer blends of poly (ethyl methacrylate)(PEMA) and poly (ethylene oxide) (PEO) has been investigated to detect the radiation induced crosslinking. The blend system comprises a noncrystallizable polymer, PEMA and a crystallizable polymer, PEO. On irradiation, the overall hardness of the blend specimens for different dose levels infers occurrence of a crosslinking process. The radiation-induced crosslinking was greater for blends having lower concentration of PEO. However, increase in radiation dose causes softening of blend system due to radiation induced scissioning of the chains

Keywords: Microhardness, Radiation induced crosslinking, Solution cast technique, Vicker's hardness number.

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Authors: Devinder Singh, Arbind Kumar, Rajneesh Kumar

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The general solution of the equations for a homogeneous isotropic microstretch thermo elastic medium with mass diffusion for two dimensional problems is obtained due to normal and tangential forces. The Integral transform technique is used to obtain the components of displacements, microrotation, stress and mass concentration, temperature change and mass concentration. A particular case of interest is deduced from the present investigation.

Keywords: Normal and tangential force, Microstretch, thermoelastic, The Integral transform technique.

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Authors: Manoj Singh, Mumtaz Ahmad Khan, Abdul Hakim Khan

Abstract:

The object of the present paper is to investigate several general families of bilinear and bilateral generating functions with different argument for the Gauss’ hypergeometric polynomials.

Keywords: Appell’s functions, Gauss hypergeometric functions, Heat polynomials, Kampe’ de Fe’riet function, Laguerre polynomials, Lauricella’s function, Saran’s functions.

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Authors: Anil Kumar, C. Nagaraja Kumar

Abstract:

The position and momentum space information entropies of hydrogen atom are exactly evaluated. Using isospectral Hamiltonian approach, a family of isospectral potentials is constructed having same energy eigenvalues as that of the original potential. The information entropy content is obtained in position space as well as in momentum space. It is shown that the information entropy content in each level can be re-arranged as a function of deformation parameter.

Keywords: Information Entropy, BBM inequality, Isospectral Potential.

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