Showing papers in "International Journal of Computer Mathematics in 2012"
TL;DR: The r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and the corresponding optimal solutions are determined and classified.
Abstract: The r-component connectivity κ r (G) of the non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. So, κ2 is the usual connectivity. In this paper, we determine the r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and we classify all the corresponding optimal solutions.
87 citations
TL;DR: It is proved that the dynamic behaviour of the novel computer virus propagation model, known as the SLBS model, is determined by a threshold R 0, and some effective strategies for eradicating computer viruses distributed in the Internet are suggested.
Abstract: By taking into account the fact that, in general, a computer immediately possesses infectivity as soon as it is infected, a novel computer virus propagation model, known as the SLBS model, is established. It is proved that the dynamic behaviour of the model is determined by a threshold R 0. Specifically, the virus-free equilibrium is globally asymptotically stable if R 0≤1, whereas the virulent equilibrium is globally asymptotically stable if 1 R 0≤4. It is conjectured that the virulent equilibrium is also globally asymptotically stable if R 0>4. These results suggest some effective strategies for eradicating computer viruses distributed in the Internet.
85 citations
TL;DR: It is shown that if the authors apply a minimum level of insecticide, it is possible to maintain the basic reproduction number below unity, and it has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold is less than one.
Abstract: Dengue is one of the major international public health concerns. Although progress is underway, developing a vaccine against the disease is challenging. Thus, the main approach to fight the disease is vector control. A model for the transmission of dengue disease is presented. It consists of eight mutually exclusive compartments representing the human and vector dynamics. It also includes a control parameter (insecticide) in order to fight the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) and another endemic equilibrium. It has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold, known as the basic reproduction number, is less than one. We show that if we apply a minimum level of insecticide, it is possible to maintain the basic reproduction number below unity. A case study, using data of the outbreak that occurred in 2009 in Cape Verde, is presented.
85 citations
TL;DR: In this paper, the concept of multilevel network is introduced to embody some topological properties of complex systems with structures in the mesoscale, which are not completely captured by the classical models.
Abstract: The new concept of multilevel network is introduced in order to embody some topological properties of complex systems with structures in the mesoscale, which are not completely captured by the classical models. This new model, which generalizes the hyper-network and hyper-structure models, fits perfectly with several real-life complex systems, including social and public transportation networks. We present an analysis of the structural properties of the multilevel network, including the clustering and the metric structures. Some analytical relationships amongst the efficiency and clustering coefficient of this new model and the corresponding parameters of the underlying network are obtained. Finally, some random models for multilevel networks are given to illustrate how different multilevel structures can produce similar underlying networks and therefore that the mesoscale structure should be taken into account in many applications.
64 citations
TL;DR: A multi-authority attribute-based encryption scheme in which only the set of recipients defined by the encrypting party can decrypt a corresponding ciphertext, which is secure in the selective ID model and can tolerate an honest-but-curious central authority.
Abstract: An attribute-based encryption scheme capable of handling multiple authorities was recently proposed by Chase. The scheme is built upon a single-authority attribute-based encryption scheme presented earlier by Sahai and Waters. Chase's construction uses a trusted central authority that is inherently capable of decrypting arbitrary ciphertexts created within the system. We present a multi-authority attribute-based encryption scheme in which only the set of recipients defined by the encrypting party can decrypt a corresponding ciphertext. The central authority is viewed as ‘honest-but-curious’: on the one hand, it honestly follows the protocol, and on the other hand, it is curious to decrypt arbitrary ciphertexts thus violating the intent of the encrypting party. The proposed scheme, which like its predecessors relies on the Bilinear Diffie–Hellman assumption, has a complexity comparable to that of Chase's scheme. We prove that our scheme is secure in the selective ID model and can tolerate an honest-but-cur...
62 citations
TL;DR: The approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method and the overall work decreases in the optimal case to O(h −2) if h is the error of the approximation.
Abstract: In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo MLMC method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O h −2 if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.
59 citations
TL;DR: The spatial regularity of semilinear parabolic parabolic stochastic partial differential equations on bounded Lipschitz domains is studied to determine the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes.
Abstract: We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains 𝒪⊆ ℝ d in the scale , 1/τ=α/ d +1/ p, p ≥2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing weighted Sobolev estimates and combining them with wavelet characterizations of Besov spaces.
59 citations
TL;DR: The generalized finite difference method is applied to solve the problem of thin and thick elastic plates to solve second- order partial differential equation systems and fourth-order partial differential equations.
Abstract: This paper describes the generalized finite difference method to solve second-order partial differential equation systems and fourth-order partial differential equations. This method is applied to solve the problem of thin and thick elastic plates.
57 citations
TL;DR: The auxiliary model-based recursive least-squares algorithm is used to estimate the parameters of one-step state-delay systems and the convergence of the proposed algorithm is studied by using the stochastic process theory.
Abstract: Based on the input–output representation of one-step state-delay systems, we use the auxiliary model-based recursive least-squares algorithm to estimate the parameters of the systems and study the convergence of the proposed algorithm by using the stochastic process theory. A simulation example is provided.
55 citations
TL;DR: This paper presents the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs) using an implicit time-stepping scheme.
Abstract: In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method meshfree approximation method to estimate the solution of high-dimensional stochastic partial differential equations SPDEs. Using an implicit time-stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centred at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.
54 citations
TL;DR: The main feature of this method is that it computes the closure using solely the inference system of the SL FD logic, which is used in the design of automated deduction methods to solve the closure problem.
Abstract: In this paper, a method for computing the closure of a set of attributes according to a specification of functional dependencies of the relational model is described. The main feature of this method is that it computes the closure using solely the inference system of the SL FD logic. For the first time, logic is used in the design of automated deduction methods to solve the closure problem. The strong link between the SL FD logic and the closure algorithm is presented and an SL FD simplification paradigm emerges as the key element of our method. In addition, the soundness and completeness of the closure algorithm are shown. Our method has linear complexity, as the classical closure algorithms, and it has all the advantages provided by the use of logic. We have empirically compared our algorithm with the Diederich and Milton classical algorithm. This experiment reveals the best behaviour of our method which shows a significant improvement in the average speed.
TL;DR: A piecewise rational function in a cubic/cubic form is proposed, which, in each interval, involves four free parameters in its construction that are constrained to preserve the shape of convex, monotone and positive data.
Abstract: This work is a contribution towards the graphical display of 2D data when they are convex, monotone and positive. A piecewise rational function in a cubic/cubic form is proposed, which, in each interval, involves four free parameters in its construction. These four free parameters have a direct geometric interpretation, making their use straightforward. Illustrations of their effect on the shape of the rational function are given. Two of these free parameters are constrained to preserve the shape of convex, monotone and positive data, while the other two parameters are utilized for the modification of positive, monotone and convex curves to obtain a visually pleasing curve. The problem of shape preservation of data lying above a line is also discussed. The method that is presented applies equally well to data or data with derivatives. The developed scheme is computationally economical and pleasing. The error of rational interpolating function is also derived when the arbitrary function being interpolated ...
TL;DR: This paper proposes a stochastic grid method for estimating the optimal exercise policy and uses this policy to obtain a low-biased estimator for high-dimensional Bermudan options.
Abstract: This paper considers the problem of pricing options with early-exercise features whose pay-off depends on several sources of uncertainty. We propose a stochastic grid method for estimating the optimal exercise policy and use this policy to obtain a low-biased estimator for high-dimensional Bermudan options. The method has elements of the least-squares method LSM of Longstaff and Schwartz [ Valuing American options by simulation: A simple least-squares approach , Rev. Finan. Stud. 3 2001, pp. 113–147], the stochastic mesh method of Broadie and Glasserman [ A stochastic mesh method for pricing high-dimensional American option , J. Comput. Finance 7 2004, pp. 35–72], and stratified state aggregation along the pay-off method of Barraquand and Martineau [ Numerical valuation of high-dimensional multivariate American securities , J. Financ. Quant. Anal. 30 1995, pp. 383–405], with certain distinct advantages over the existing methods. We focus on the numerical results for high-dimensional problems such as max option and arithmetic basket option on several assets, with basic error analysis for a general one-dimensional problem.
TL;DR: A new second-order exponential time differencing (ETD) method based on the Cox and Matthews approach is developed and applied for pricing American options with transaction cost and is seen to be strongly stable and highly efficient for solving the nonlinear Black–Scholes model.
Abstract: In this paper, a new second-order exponential time differencing ETD method based on the Cox and Matthews approach is developed and applied for pricing American options with transaction cost The method is seen to be strongly stable and highly efficient for solving the nonlinear Black–Scholes model Furthermore, it does not incur unwanted oscillations unlike the ETD–Crank–Nicolson method for exotic path-dependent American options The computational efficiency and reliability of the new method are demonstrated by numerical examples and by comparing it with the existing methods
TL;DR: A new point of view in numerical approximation of stochastic differential equations is proposed by using Ito–Taylor expansions, and it is proved that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments.
Abstract: In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito–Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values.
TL;DR: The quaternion matrix equation X−AXF=C and X−A[Xtilde] F=C is investigated and based on the Kronecker map and complex representation of a quaternions matrix, the solution expressions of the quaternia Stein matrix equation and quaternian Stein-conjugate matrix equation are given.
Abstract: In the present paper, we investigate the quaternion matrix equation X − AXF = C and X − A[Xtilde] F = C . For convenience, we named the quaternion matrix equations X − AXF = C and X − A[Xtilde] F = C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method.
TL;DR: A nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation is studied by means of the radial basis function (RBF) collocation method and results agree well with those by the numerical method, which verifies the validity of the present work.
Abstract: In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation ODE is investigated by means of the radial basis function RBF collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.
TL;DR: A variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains is investigated using the boundary element method coupled with the conjugate gradient method and it is proved the convergence of this scheme with and without Tikhonov regularization.
Abstract: In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme.
TL;DR: A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described, based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way.
Abstract: A new radial basis functions RBFs algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.
TL;DR: A modification of the Zoutendijk method of feasible directions and a strategy of active inequalities are applied to find the local maxima of the problem of packing a maximal number of identical circles of a given radius into a multiconnected domain.
Abstract: The paper considers the problem of packing a maximal number of identical circles of a given radius into a multiconnected domain. The domain is a circle with prohibited areas to be finite unions of circles of given radii. We construct a mathematical model of the problem and investigate its characteristics. The starting points are constructed in a random way or on the ground of the hexagonal lattice. To find the local maxima, a modification of the Zoutendijk method of feasible directions and a strategy of active inequalities are applied. We compare our results with the benchmark instances of packing circles into circular and annular containers. A number of numerical examples are given.
TL;DR: This paper illustrates a computable matrix technique that can be used to derive explicit expressions for the transient state probabilities of a finite waiting space single-server queue, namely (M/M/1/N), having discouraged arrivals and reneging.
Abstract: This paper illustrates a computable matrix technique that can be used to derive explicit expressions for the transient state probabilities of a finite waiting space single-server queue, namely (M/M/1/N), having discouraged arrivals and reneging. The discipline is the classical first-come, first-served (FCFS). We obtain the transient solution of the system, with results in terms of the eigenvalues of a symmetric tridiagonal matrix. Finally, numerical calculations are given to illustrate the effectiveness of this technique and system behaviour.
TL;DR: Numerical methods are proposed for solving the one-dimensional fractional Schrödinger differential equation with Dirichlet condition in the space variable.
Abstract: The first and second orders of accuracy difference schemes for the mixed problem for the multidimensional fractional Schrodinger differential equation with dependent coefficients are considered. Stability estimates for solutions of these difference schemes are obtained. Numerical methods are proposed for solving the one-dimensional fractional Schrodinger differential equation with Dirichlet condition in the space variable.
TL;DR: This paper presents an algorithm for embedding special class of circulant networks into their optimal hypercubes with dilation 2 and proves its correctness.
Abstract: Hypercubes are a very popular model for parallel computation because of their regularity and the relatively small number of interprocessor connections. In this paper, we present an algorithm for embedding special class of circulant networks into their optimal hypercubes with dilation 2 and prove its correctness. Also, we embed special class of circulant networks into special class of generalized Petersen graphs with dilation 2 and vice versa.
TL;DR: This paper introduces some key exchange protocols over noncommutative rings that use some polynomials with coefficients in the centre of the ring as part of the private keys and concludes that the only possible attack is by brute force.
Abstract: In this paper we introduce some key exchange protocols over noncommutative rings. These protocols use some polynomials with coefficients in the centre of the ring as part of the private keys. We give some examples over the ring , where p is a prime number. We also give a security analysis of the proposed protocols and conclude that the only possible attack is by brute force.
ETSI1
TL;DR: In this paper, a sufficient condition which guarantees the existence of stable models for a normal residuated logic program interpreted on the truth-space is introduced, namely the continuity of the connectives involved in the program.
Abstract: We introduce a sufficient condition which guarantees the existence of stable models for a normal residuated logic program interpreted on the truth-space [0, 1] n . Specifically, the continuity of the connectives involved in the program ensures the existence of stable models. Then, we study conditions which guarantee the uniqueness of stable models in the particular case of the product t-norm, its residuated implication and the standard negation.
TL;DR: The Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved approximately.
Abstract: The Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved approximately. This class of equations includes as a special case the Euler equations of gasdynamics. It has been observed that in contrast to the gasdynamic case, the pressure varies across the contact discontinuity. The iterative procedure is used to find the densities between the left acoustic wave and the right contact discontinuity and between the right contact discontinuity and the right acoustic wave, respectively. All other quantities follow directly throughout the (x, t)-plane, except within rarefaction waves, where an extra iterative procedure is used along with a Gaussian quadrature rule to find particle velocity; indeed, the determination of the particle velocity involves numerical integration when the magneto-acoustic wave is a rarefaction wave. Lastly, we discuss numeric...
TL;DR: A highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options using a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time.
Abstract: In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided.
TL;DR: A simple mathematical model of interaction between the sugarcane borer and its egg parasitoid Trichogramma galloi is proposed and Linear feedback control strategy is proposed to indicate how the natural enemies should be introduced into the environment.
Abstract: In this paper, we propose a simple mathematical model of interaction between the sugarcane borer (Diatraea saccharalis) and its egg parasitoid Trichogramma galloi. In this model, the sugarcane borer is represented by the egg and the larval stages, and the parasitoid is considered in terms of the parasitized eggs. Linear feedback control strategy is proposed to indicate how the natural enemies should be introduced into the environment.
TL;DR: A front-fixing finite element method for the free boundary problems is proposed and implemented and its stability is established under reasonable assumptions.
Abstract: American option problems under regime-switching model are considered in this paper. The conjectures in [H. Yang, A numerical analysis of American options with regime switching , J. Sci. Comput. 44 2010, pp. 69–91] about the position of early exercise prices are proved, which generalize the results in [F. Yi, American put option with regime-switching volatility finite time horizon – Variational inequality approach , Math. Methods. Appl. Sci. 31 2008, pp. 1461–1477] by allowing the interest rates to be different in two states. A front-fixing finite element method for the free boundary problems is proposed and implemented. Its stability is established under reasonable assumptions. Numerical results are given to examine the rate of convergence of our method and compare it with the usual finite element method.
TL;DR: The original Schellekens method is extended in order to yield asymptotic upper bounds for a certain class of recursive algorithms whose running time of computing cannot be discussed following the techniques given by Cerdà-Uguet et al.
Abstract: Schellekens [ The Smyth completion: A common foundation for denotational semantics and complexity analysis , Electron. Notes Theor. Comput. Sci. 1 1995, pp. 211–232.] introduced the theory of complexity quasi-metric spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms in 1995. The applicability of this theory to the asymptotic complexity analysis of divide and conquer algorithms was also illustrated by Schellekens in the same paper. In particular, he gave a new formal proof, based on the use of the Banach fixed-point theorem, of the well-known fact that the asymptotic upper bound of the average running time of computing of Mergesort belongs to the asymptotic complexity class of n log 2 n . Recently, Schellekens’ method has been shown to be useful in yielding asymptotic upper bounds for a class of algorithms whose running time of computing leads to recurrence equations different from the divide and conquer ones reported in Cerda-Uguet et al. [ The Baire partial quasi-metric space: A mathematical tool for the asymptotic complexity analysis in Computer Science , Theory Comput. Syst. 50 2012, pp. 387–399.]. However, the variety of algorithms whose complexity can be analysed with this approach is not much larger than that of algorithms that can be analysed with the original Schellekens method. In this paper, on the one hand, we extend Schellekens’ method in order to yield asymptotic upper bounds for a certain class of recursive algorithms whose running time of computing cannot be discussed following the techniques given by Cerda-Uguet et al. and, on the other hand, we improve the original Schellekens method by introducing a new fixed-point technique for providing, contrary to the case of the method introduced by Cerda-Uguet et al. , lower asymptotic bounds of the running time of computing of the aforementioned algorithms and those studied by Cerda-Uguet et al. We illustrate and validate the developed method by applying our results to provide the asymptotic complexity class asymptotic upper and lower bounds of the celebrated algorithms Quicksort, Largetwo and Hanoi.