Department of Mathematics

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End date: 2019

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Now showing 1 - 10 of 307
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    A new method for geometric interpretation of elliptic curve discrete logarithm problem
    (2019-09) Pal, Ankan
    In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These equations arise from the interesection of quadric hypersurfaces in an affine space of lower dimension. In cryptography, this interpretation can be used to design attacks on EC-DLP. Presently, the best known attack algorithm having a sub-exponential time complexity is through the implementation of Summation Polynomials and Weil Descent. It is expected that the proposed geometric interpretation can result in faster reduction of the problem into a system of equations. These overdetermined system of equations are hard to solve. We have used F4 (Faugere) algorithms and got results for primes less than 500,000. Quantum Algorithms can expedite the process of solving these over-determined system of equations. In the absence of fast algorithms for computing summation polynomials, we expect that this could be an alternative. We do not claim that the proposed algorithm would be faster than Shor's algorithm for breaking EC-DLP but this interpretation could be a candidate as an alternative to the 'summation polynomial attack' in the post-quantum era.
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    Legendre wavelet modified petrov–galerkin method in two-dimensional moving boundary problem
    (De Gruyter, 2017-12) Yadav, Sangita
    In this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x-axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.
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    Erratum to: Singular Adams inequality for biharmonic operator on Heisenberg Group and its applications
    (Springer, 2017-04) Dwivedi, Gaurav
    The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to where is a bounded domain, The special feature of this problem is that it contains an exponential nonlinearity and singular potential.
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    Dynamical behaviour of tuberculosis transmission
    (Biomath Communications, 2018) Das, Dhiraj Kumar
    Tuberculosis(TB) is a contagious disease in human caused by infection withВ Mycobacterium tuberculosis(Mtb). Most infections results a clinically asymptotic state termed as latent TB infection(LTBI) whereas a smaller portion ofВ infected individuals grow symptomatic active pulmonary TB. The main difference between TB and other infectious diseases is that, the disease progressionВ from primary infection(LTBI) to active pulmonary TB is signicantly time-consuming. We proposed and study an SEIR type mathematical model for TB transmission incorporating roles of both exogenous re-infection and endogenous reactivation. Our model possesses two kinds of steady states: infection free andВ endemic. The epidemiological threshold key that is, basic reproduction numberВ R0 has been obtained by using next-generation matrix. We observe that theВ disease transmission rate and exogenous re-infection level plays a signicantВ role in order to determine the qualitative behaviour of our proposed model system. Our results demonstrate that when exogenous re-infection level crosses aВ critical value our system undergoes backward bifurcation and hence a stable endemic equilibrium exists in spite of the fact R0 < 1. Therefore, reducing R0 lessВ than unity is not sucient to eradicate TB completely. We further investigateВ that proposed model experience stable periodic solutions as increases throughВ a critical value. Various numerical simulations have been conducted coveringВ the breadth of feasible parameter space to support analytical establishments.
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    Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation
    (Elsevier, 2018-05) Das, Dhiraj Kumar
    We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number . Our results demonstrate that the disease transmission rate and exogenous reinfection rate plays an important role to change the qualitative dynamics of TB. The disease transmission rate give rises to the possibility of backward bifurcation for , and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate and the exogenous reinfection rate . We conducted some numerical simulations to support our analytical findings.
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    Influence of multiple re-infections in tuberculosis transmission dynamics: A Mathematical Approach
    (IEEE, 2019) Das, Dhiraj Kumar
    This investigation accounts a TB transmission model with the possibility of both exogenous re-infections and recurrent TB. The qualitative characteristic of the model system has been analyzed covering stability of existing equilibrium points and bifurcation criteria. The basic reproduction number is obtained by using the next-generation matrix method. It has been observed that the system performs a backward bifurcation at Ro = 1 and hence Ro <; 1 can not guaranty the disease elimination. Several numerical simulations have been performed to support the analytical findings.
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    Extended Latin Hypercube Sampling for Integration and Simulation
    (Springer, 2013-01) Venkiteswaran, G.
    We analyze an extended form of Latin hypercube sampling technique that can be used for numerical quadrature and for Monte Carlo simulation. The technique utilizes random point sets with enhanced uniformity over the s-dimensional unit hypercube. A sample of N = n s points is generated in the hypercube. If we project the N points onto their ith coordinates, the resulting set of values forms a stratified sample from the unit interval, with one point in each subinterval [(k−1)/N,k/N). The scheme has the additional property that when we partition the hypercube into N subcubes ∏si=1[(ℓi−1)/n,ℓi/n), each one contains exactly one point. We establish an upper bound for the variance, when we approximate the volume of a subset of the hypercube, with a regular boundary. Numerical experiments assess that the bound is tight. It is possible to employ the extended Latin hypercube samples for Monte Carlo simulation. We focus on the random walk method for diffusion and we show that the variance is reduced when compared with classical random walk using ordinary pseudo-random numbers. The numerical comparisons include stratified sampling and Latin hypercube sampling.
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    Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice
    (De Gruyter, 2010-10) Venkiteswaran, G.
    We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.
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    Quasi-Monte Carlo Simulation of Diffusion in a Spatially Nonhomogeneous Medium
    (Springer, 2009-11) Venkiteswaran, G.
    We propose and test a quasi-Monte Carlo (QMC) method for solving the diffusion equation in the spatially nonhomogeneous case. For a constant diffusion coefficient, the Monte Carlo (MC) method is a valuable tool for simulating the equation: the solution is approximated by using particles and in every time step the displacement of each particle is drawn from a Gaussian distribution with constant variance. But for a spatially dependent diffusion coefficient, the straightforward extension using a spatially variable variance leads to biased results. A correction to the Gaussian steplength was recently proposed and provides satisfactory results. In the present work, we devise a QMC variant of this corrected MC scheme. We present the results of some numerical experiments showing that our QMC algorithm converges better than the corresponding MC method for the same number of particles.
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    Deterministic Particle Methods for High Dimensional Fokker-Planck Equations
    (Springer, 2006) Venkiteswaran, G.
    We consider a mathematical model for polymeric liquids which requires the solution of high-dimensional Fokker-Planck equations related to stochastic differential equations. While Monte-Carlo (MC) methods are classically used to construct approximate solutions in this context, we consider an approach based on Quasi- Monte-Carlo (QMC) approximations. Although QMC has proved to be superior to MC in certain integration problems, the advantages are not as pronounced when dealing with stochastic differential equations. In this article, we illustrate the basic difficulty which is related to the construction of QMC product measures.