A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

Extended Gauss–Markov Theorem for Nonparametric Mixed-Effects Models

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.     

Dual simplex method for GUB problems

The dual simplex method for generalized upper bound (GUB) problems is presented. One of the major operations in the dual simplex method is to update the elements of therth row, wherer is the index for the leaving basic variable. Those updated elements are used for the ratio test to determine the entering basic variabble. A very simple formula for therth row update for the dual simplex method for a GUB problem is derived, which is similar to the formula for the standard linear program. This derivation is based on the change key operation, which is to exchange the key column and its counterpart in the nonkey section. The change key operation is possible because of a theorem that guarantees the existence of such a counterpart.     

Monotonicity of the search number in the Golovach problem

The Golovach problem, also known as the ɛ-search problem, is as follows. A team of pursuers pursues an evader on a topological graph. The objective of the pursuers is to catch the evader, that is, approach the evader to a distance not exceeding a given nonnegative number ɛ. It is assumed that the evader is invisible to the pursuers and is fully informed beforehand about the search program of the pursuers. The problem is to find the ɛ-search number, i.e., the least number of pursuers sufficient for capturing the evader. Graphs with monotone ɛ-search number are studied; the ɛ-search number of a graph G is said to be monotone if it is not exceeded by the ɛ-search numbers of all connected subgraphs H of G. It is known that the ɛ-search number of any tree is monotone for all nonnegative ɛ. The edgesearch number, which is equal to the 0-search number, is monotone for all connected subgraphs of an arbitrary graph. A sufficient monotonicity condition for the ɛ-search number of any graph is obtained. This result is improved in the case of complete subgraphs. The Golovach function is constructed for graphs obtained by removing one edge from complete graphs with unit edges.     

Numerical solution of complex modified Korteweg-de Vries equation by collocation method

The collocation method using quintic B-spline is derived for solving the complex modified Korteweg-de Vries (CMKdV). The method is based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The von Neumann stability is used to prove that the scheme is unconditionally stable. Newton’s method is used to solve the nonlinear block pentadiagonal system obtained. Numerical tests for single, two, and three solitons are used to assess the performance of the proposed scheme.     

Operations in A-theory

A construction for Segal operations for K-theory of categories with cofibrations, weak equivalences and a biexact pairing is given and coherence properties of the operations are studied. The model for K-theory, which is used, allows coherence to be studied by means of (symmetric) monoidal functors. In the case of Waldhausen A-theory it is shown how to recover the operations used in Waldhausen (Lecture Notes in Mathematics, Vol. 967, Springer, Berlin, 1982, pp. 390-409) for the A-theory Kahn-Priddy theorem. The total Segal operation for A-theory, which assembles exterior power operations, is shown to carry a natural infinite loop map structure. The basic input is the un-delooped model for K-theory, which has been developed from a construction by Grayson and Gillet for exact categories in Gunnarsson et al. (J. Pure Appl. Algebra 79 (1992) 255), and Grayson's setup for operations in Grayson (K-theory (1989) 247). The relevant material from these sources is recollected followed by observations on equivariant objects and pairings. Grayson's conditions are then translated to the context of categories with cofibrations and weak equivalences. The power operations are shown to be well behaved w.r.t. suspension and are extended to algebraic K-theory of spaces. Staying close with the philosophy of Waldhausen (1982) Waldhausen's maps are found. The Kahn-Priddy theorem follows from splitting the “free part” off the equivariant theory. The treatment of coherence of the total operation in A-theory involves results from Laplaza (Lecture Notes in Mathematics, Vol. 281, Springer, Berlin, 1972, pp. 29-65) and restriction to spherical objects in the source of the operation.     

A Predictive Model for the Police Response Function

The expected response time to a call for service (CFS) for a given configuration of police beats is developed. The effect of downtime calls on the response time to a CFS is determined. Consideration is given to both travel time and waiting time. Travel time and service time distributions are isolated. The model is valid for Poisson arrivals and arbitrary service time distributions. A probabilistic assignment policy is determined for each beat. The fraction of incoming calls arriving in beat k answered by unit l is obtained. Pre-emptive priorities are allowed. Application to the Aurora, Illinois, Police Department is shown.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.     

An integer linear programming formulation and heuristics for the minmax relative regret robust shortest path problem

The well-known Shortest Path problem (SP) consists in finding a shortest path from a source to a destination such that the total cost is minimized. The SP models practical and theoretical problems. However, several shortest path applications rely on uncertain data. The Robust Shortest Path problem (RSP) is a generalization of SP. In the former, the cost of each arc is defined by an interval of possible values for the arc cost. The objective is to minimize the maximum relative regret of the path from the source to the destination. This problem is known as the minmax relative regret RSP and it is NP-Hard. We propose a mixed integer linear programming formulation for this problem. The CPLEX branch-and-bound algorithm based on this formulation is able to find optimal solutions for all instances with 100 nodes, and has an average gap of 17 % on the instances with up to 1,500 nodes. We also develop heuristics with emphasis on providing efficient and scalable methods for solving large instances for the minmax relative regret RSP, based on Pilot method and random-key genetic algorithms. To the best of our knowledge, this is the first work to propose a linear formulation, an exact algorithm and metaheuristics for the minmax relative regret RSP.     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Infinite Divisibility of Random Objects in Locally Compact Positive Convex Cones

Random objects taking on values in a locally compact second countable convex cone are studied. The convex cone is assumed to have the property that the class of continuous additive positively homogeneous functionals is separating, an assumption which turns out to imply that the cone is positive. Infinite divisibility is characterized in terms of an analog to the Lévy–Khinchin representation for a generalized Laplace transform. The result generalizes the classical Lévy–Khinchin representation for non-negative random variables and the corresponding result for random compact convex sets inRⁿ. It also gives a characterization of infinite divisibility for random upper semicontinuous functions, in particular for random distribution functions with compact support and, finally, a similar characterization for random processes on a compact Polish space.     

Dynamic power control in a fading downlink channel subject to an energy constraint

A central controller chooses a state-dependent transmission rate for each user in a fading, downlink channel by varying transmission power over time. For each user, the state of the channel evolves over time according to an exogenous continuous-time Markov chain (CTMC), which affects the quality of transmission. The traffic for each user, arriving at the central controller, is modeled as a finite-buffer Markovian queue with adjustable service rates. That is, for each user data packets arrive to the central controller according to a Poisson process and packet size is exponentially distributed; an arriving packet is dropped if the associated buffer is full, which results in degradation of quality of service. The controller forwards (downlink) the arriving packets to the corresponding user according to an optimally chosen transmission rate from a fixed set A _i of available values for each user i, depending on the backlog in the system and the channel state of all users. The objective is to maximize quality of service subject to an upper bound on the long-run average power consumption. We show that the optimal transmission rate for each user is solely a function of his own packet queue length and channel state; the dependence among users is captured through a penalty rate. Further, we explicitly characterize the optimal transmission rate for each user. This project is partially supported by Motorola grant # 0970-350-AF24. The authors thank Phil Fleming,Randy Berry and Achal Bassamboo for helpful comments.     

Numerical convergence on adaptive grids for control volume methods

An adaptive technique for control‐volume methods applied to second order elliptic equations in two dimensions is presented. The discretization method applies to initially Cartesian grids aligned with the principal directions of the conductivity tensor. The convergence behavior of this method is investigated numerically. For solutions with low Sobolev regularity, the found L² convergence order is two for the potential and one for the flow density. The system of linear equations is better conditioned for the adaptive grids than for uniform grids. The test runs indicate that a pure flux‐based refinement criterion is preferable.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Long-wavelength thermal convection in a weak shear flow

The problem of thermal convection in an imposed shear flow isexamined for a horizontal layer of fluid between poorly conductingboundaries. The horizontal scale H of the convective motionnear its onset is much greater than the depth h of the fluidlayer, with h/H being proportional to the one-fourth power ofa Biot number appearing in the condition applied to the temperatureat the horizontal boundaries. It is known that an asymptoticexpansion in powers of h/H yields a nonlinear long-wavelengthevolution equation for the depth-averaged temperature fieldthat is spatially isotropic in the absence of an imposed shearflow, but is strongly anisotropic for ‘strong’ shear.We derive in this paper a nonlocal long-wavelength equationthat bridges these two cases, and that contains each case inthe zero-shear and large-shear limits. Using this evolutionequation, we show how the shear flow stabilizes the longitudinalrolls to the zigzag instability, and how a preference for asquare planfonn on a periodic square lattice gives way to apreference for longitudinal rolls near onset. The longitudinalrolls may then become unstable as the Rayleigh number is increased.The analytical work is illustrated by some numerical simulationsof the full three-dimensional Boussinesq Navier-Stokes equations.The problem of pattern selection on a hexagonal lattice is alsodiscussed, and some new results are presented.     

Semi-Conjugate Direction Methods for Real Positive Definite Systems

In this preliminary work, left and right conjugate direction vectors are defined for nonsymmetric, nonsingular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method has no breakdown for real positive definite systems. The method reduces to the usual conjugate gradient method when A is symmetric positive definite. A finite termination property of the semi-conjugate direction method is shown, providing a new simple proof of the finite termination property of conjugate gradient methods. The new method is well defined for all nonsingular M-matrices. Some techniques for overcoming breakdown are suggested for general nonsymmetric A. The connection between the semi-conjugate direction method and LU decomposition is established. The semi-conjugate direction method is successfully applied to solve some sample linear systems arising from linear partial differential equations, with attractive convergence rates. Some numerical experiments show the benefits of this method in comparison to well-known methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays

Existence of traveling wave front solutions is established for diffusive and cooperative Lotka-Volterra system with delays. The result is an extension of an existing result for delayed logistic scaler equation to systems, and is somewhat parallel to the existing result for diffusive and competitive Lotka-Volterra systems without delay. The approach used in this paper is the upper-lower solution technique and the monotone iteration recently developed by Wu and Zou (J. Dynam. Differential Equations 13 (2001) 651-687) for delayed reaction-diffusion systems without the so-called quasimonotonicity.     

The randomized algorithm for finding an eigenvector of the stochastic matrix with application to PageRank

The problem of finding the eigenvector corresponding to the largest eigenvalue of a stochastic matrix has numerous applications in ranking search results, multi-agent, consensus, networked control and data mining. The power method is a typical tool for its solution. However randomized methods could be competitors vs standard ones; they require much less calculations for one iteration and are well tailored for distributed computations. We propose a new randomized algorithm and provide upper bound for its rate of convergence which is O(lnN/n), where N is the dimension and n is the number of iterations. The bound looks promising because lnN is not large even for very high dimensions. The algorithm is based on the mirror-descent method for convex stochastic optimization. Applications to PageRank problem are discussed. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 426, No. 6, pp. 734–737. Presented by Academician S.N. Vasil’ev February 9, 2009 The article was translated by the authors.     

On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C ^k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.     

Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree

Given an undirected graph with edge weights, we are asked to find an orientation, that is, an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As in previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for graphs which are both planar and bipartite. This implies the NP-hardness for P₄-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the NP-hardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.     

Newton's method for convex nonlinear elliptic boundary value problems

Summary Newton's method is applied to solving the boundary value problem for the equationLu=f(x,u) whereL is a linear second order uniformly elliptic operator andf(x,u) is a convex monotone increasing function ofu for each pointx in the domainD. The Newton iterates are shown to converge uniformly, quadratically and monotonically downward to the solution of the problem. The convergence is independent of the choice for the initial Newton iterate. Numerical results are presented for several problems of physical interest.