interim analysis on survival data its potential bias and how to repair it we consider interim analyses in clinical trials or observational studies with a time to event outcome variable where the survival curves are compared using the hazard ratio resulting from a proportional hazards ph model or tested with the logrank test or another two sample test we show and illustrate with an example that if the ph assumption is violated the results of interim analyses can be heavily biased this is due to the fact that the censoring pattern in interim analyses can be completely different from the final analysis we argue that when the ph assumption is violated interim analyses are only sensible if a fixed time horizon for the final analysis is specified and at the time of the interim analysis sufficient information is available over the whole time interval up to the horizon we show how the bias can then be remedied by introducing in the estimation and testing procedures an appropriate weighting that reflects the weights to be expected in the final analysis the consequences for design and analysis are discussed and some practical recommendations are given copyright 2005 john wiley sons ltd

a numerical approach to the generalized nonlinear fractional fokker planck equation in this paper we propose a fully discrete galerkin finite element method to solve the generalized nonlinear fractional fokker planck equation which has a multi fractional spatial operator characteristic that describes the levy flight in the time direction we use the finite difference method and in the spatial direction we use the fractional finite element method in the framework of the fractional sobolev spaces we derive a fully discrete scheme for the considered equation we prove the existence and uniqueness of the discrete solution and give the error estimates the numerical examples are also included which support the theoretical analysis

random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

meta analysis of pairs of survival curves under heterogeneity a poisson correlated gamma frailty approach we address the problem of meta analysis of pairs of survival curves under heterogeneity starting point for the meta analysis is a set of studies each comparing the same two treatments containing information about multiple survival outcomes under heterogeneity we model the number of events using an extension of the poisson correlated gamma frailty model with serial within arm and positive between arm correlations the parameters of the models are estimated following a two stage estimation procedure in the first stage the underlying hazards and between study variance are estimated using the marginals while a second stage is used to estimate both within arm and between arm correlations the methodology is illustrated with an observational study on breast cancer

two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem abstract in this paper we tackle the conflict resolution problem using a new variant of the minimum weight maximum clique model the problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs we design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict free maneuvers a maximum clique of minimal weight yields a conflict free situation that involves all the aircraft and minimizes the costs induced the model uses a different cost structure compared to classical clique search problems the costs of the vertices cannot be determined a priori since they depend on the vertices in the clique we formulate the problem as a mixed integer linear program since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process our mathematical framework is valid for any hypotheses on the aircraft dynamics and any choice of the available maneuvers in particular the aircraft can perform dynamic velocity heading and flight level changes to solve instances involving a large number of aircraft spread over several flight levels we introduce two decomposition algorithms the first is a sequential mixed integer linear programming procedure that iteratively refines the discretization of the maneuvers to yield a trade off between computational time and cost the second is a large neighborhood search heuristic that uses the first procedure as a subroutine the best solutions for the available set of maneuvers are obtained in less than ten seconds for instances with up to 250 aircraft randomly allocated to bisten flight levels

two stage meta analysis of survival data from individual participants using percentile ratios methods for individual participant data meta analysis of survival outcomes commonly focus on the hazard ratio as a measure of treatment effect recently siannis et al 2010 statistics in medicine 29 3030 3045 proposed the use of percentile ratios as an alternative to hazard ratios we describe a novel two stage method for the meta analysis of percentile ratios that avoids distributional assumptions at the study level

a complementary pivoting approach to the maximum weight clique problem given an undirected graph with positive weights on the vertices the maximum weight clique problem mwcp is to find a subset of mutually adjacent vertices i e a clique having largest total weight the problem is known to be np hard even to approximate motivated by a recent quadratic programming formulation which generalizes an earlier remarkable result of motzkin and straus in this paper we propose a new framework for the mwcp based on the corresponding linear complementarity problem lcp we show that generically all stationary points of the mwcp quadratic program exhibit strict complementarity despite this regularity result however the lcp turns out to be inherently degenerate and we find that lemke s well known pivoting method equipped with standard degeneracy resolution strategies yields unsatisfactory experimental results we exploit the degeneracy inherent in the problem to develop a variant of lemke s algorithm which incorporates a new and effective look ahead pivot rule the resulting algorithm is tested extensively on various instances of random as well as dimacs benchmark graphs and the results obtained show the effectiveness of our method

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

a serially correlated gamma frailty model for longitudinal count data a poisson gamma model is introduced to account for between subjects heterogeneity and within subjects serial correlation occurring in longitudinal count data the model extends the usual time constant shared frailty approach to allow time varying serially correlated gamma frailty whilst retaining standard marginal assumptions a composite likelihood approach to estimation and testing for serial correlation is proposed the work is motivated by a clinical trial on patient controlled analgesia where the number of analgesic doses taken by hospital patients in successive time intervals following abdominal surgery is recorded

second order approximation scheme combined with h1 galerkin mfe method for nonlinear time fractional convection diffusion equation abstract in this article a second order approximation scheme combined with an h 1 galerkin mixed finite element mfe method for solving nonlinear convection diffusion equation with time fractional derivative is proposed and analyzed by introducing an auxiliary variable a coupled system is formulated then the spatial direction is approximated by h 1 galerkin mfe method and the temporal fractional derivative and integer derivative are discretized by second order weighted and shifted grunwald difference wsgd formula and linearized second order difference scheme respectively the optimal priori error estimates in l 2 and h 1 norm for the unknown function and the auxiliary variable with second order convergent rate in time are obtained compared to the commonly used l1 approximation with 2 th order convergence rate our method can arrive at the order 2 in time what is more compared with the standard finite element method our method can well approximate the auxiliary variable finally the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis

local search with edge weighting and configuration checking heuristics for minimum vertex cover the minimum vertex cover mvc problem is a well known combinatorial optimization problem of great importance in theory and applications in recent years local search has been shown to be an effective and promising approach to solve hard problems such as mvc in this paper we introduce two new local search algorithms for mvc called ewls edge weighting local search and ewcc edge weighting configuration checking the first algorithm ewls is an iterated local search algorithm that works with a partial vertex cover and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima nevertheless ewls has an instance dependent parameter further we propose a strategy called configuration checking for handling the cycling problem in local search this is used in designing a more efficient algorithm that has no instance dependent parameters which is referred to as ewcc unlike previous vertex based heuristics the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution a detailed experimental study is carried out using the well known dimacs and bhoslib benchmarks the experimental results conclude that ewls and ewcc are largely competitive on dimacs benchmarks where they outperform other current best heuristic algorithms on most hard instances and dominate on the hard random bhoslib benchmarks moreover ewcc makes a significant improvement over ewls while both ewls and ewcc set a new record on a twenty year challenge instance further ewcc performs quite well even on structured instances in comparison to the best exact algorithm we know we also study the run time behavior of ewls and ewcc which shows interesting properties of both algorithms

a fourth order extrapolated compact difference method for time fractional convection reaction diffusion equations with spatially variable coefficients abstract this paper is concerned with numerical methods for a class of time fractional convection reaction diffusion equations the convection and reaction coefficients of the equation may be spatially variable based on the weighted and shifted grunwald letnikov formula for the time fractional derivative and a compact finite difference approximation for the spatial derivative we establish an unconditionally stable compact difference method the local truncation error and the solvability of the resulting scheme are discussed in detail the stability of the method and its convergence of third order in time and fourth order in space are rigorously proved by the discrete energy method combining this method with a richardson extrapolation we present an extrapolated compact difference method which is fourth order accurate in both time and space a rigorous proof for the convergence of the extrapolation method is given numerical results confirm our theoretical analysis and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

iterative generalized least squares for meta analysis of survival data at multiple times a method is presented for joint analysis of survival proportions reported at multiple times in published studies to be combined in a meta analysis generalized least squares is used to fit linear models including between trial and within trial covariates using current fitted values iteratively to derive correlations between times within studies multi arm studies and nonrandomized historical controls can be included with no special handling the method is illustrated with data from two previously published meta analyses in one an early treatment difference is detected that was not apparent in the original analysis

a bivariate approach to meta analysis the usual meta analysis of a sequence of randomized clinical trials only considers the difference between two treatments and produces a point estimate and a confidence interval for a parameter that measures this difference the usual parameter is the log odds ratio linked to mantel haenszel methodology inference is made either under the assumption of homogeneity or in a random effects model that takes account of heterogeneity between trials this paper has two goals the first is to present a likelihood based method for the estimation of the parameters in the random effects model which avoids the use of approximating normal distributions the second goal is to extend this method to a bivariate random effects model in which the effects in both groups are supposed random in this way inference can be made about the relationship between improvement and baseline effect the method is demonstrated by a meta analysis dataset of collins and langman

a branch and bound algorithm for the maximum diversity problem this article begins with a review of previously proposed integer formulations for the maximum diversity problem mdp this problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized we propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the mdp empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium sized instances with 50 elements as well as some large sized instances with 100 elements we compare our method with the best previous linear integer formulation solved with the well known software cplex the comparison favors the proposed procedure

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

maximum stable set formulations and heuristics based on continuous optimization the stability number g for a given graph g is the size of a maximum stable set in g the lovasz theta number provides an upper bound on g and can be computed in polynomial time as the optimal value of the lovasz semidefinite program in this paper we show that restricting the matrix variable in the lovasz semidefinite program to be rank one and rank two respectively yields a pair of continuous nonlinear optimization problems each having the global optimal value g we propose heuristics for obtaining large stable sets in g based on these new formulations and present computational results indicating the effectiveness of the heuristics

numerical and analytical solutions of new generalized fractional diffusion equation our current paper is devoted to studying the numerical and analytical solutions for a class of generalized fractional diffusion equations gfdes with new generalized time fractional derivative gtfd the gtfd we propose here is defined in the caputo sense we consider the gfdes on a bounded domain the numerical solutions are obtained by using the finite difference method fdm of full discretization the stability of fdm is discussed and the order of convergence is evaluated numerically numerical experiments are given which illustrate that the fdm is simple and effective for solving the gfdes with different coefficients and source functions an interesting phenomenon is that we can observe the period like solution in gfdes with a particular positive periodic weight function using the method of separation of variables we convert the homogeneous gfde into two ordinary differential equations and solve them via the help of solutions of the initial value problem with caputo derivative in the analytical solution we observe that the weight function in the denominator and scale function mapping the response domain differently since the derivative considered in this article is new many existing results of fdes are generalized

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

finite element method for a space fractional anti diffusive equation the numerical solution of a nonlinear and space fractional anti diffusive equation used to model dune morphodynamics is considered spatial discretization is effected using a finite element method whereas the crank nicolson scheme is used for temporal discretization the fully discrete scheme is analyzed to determine stability condition and also to obtain error estimates for the approximate solution numerical examples are presented to illustrate convergence results

numerical approximations for fractional diffusion equations via splines a one dimensional fractional diffusion model is considered where the usual second order derivative gives place to a fractional derivative of order a with 1 email protected 2 we consider the caputo derivative as the space derivative which is a form of representing the fractional derivative by an integral operator an implicit numerical method is derived which uses a spline approximation for the caputo derivative the consistency and stability of the method are examined and numerical results are presented

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

boundary layer flows of viscoelastic fluids over a non uniform permeable surface abstract this study investigates viscoelastic fluid which possesses both viscous and elastics properties by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance the spatial fractional derivative used in the momentum conservation equation is the riemann liouville type derivative further we use non uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi infinite permeable flat surface owing to the fractional derivative model and non uniform boundary conditions this problem is complex thus a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized the accuracy convergence and stability of the numerical method are presented it is shown that non uniform mass transfer through a permeable surface considerably affects the velocity boundary layer by illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface this paper predicts the velocity distributions with varying permeable surface and also provides the possibility of changing the velocity fields by altering the permeable sheet the results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

a hybrid metaheuristic method for the maximum diversity problem abstract the maximum diversity problem mdp consists in selecting a subset of m elements from a given set of n elements n m in such a way that the sum of the pairwise distances between the m chosen elements is maximized we present a hybrid metaheuristic algorithm denoted by mamdp for mdp the algorithm uses a dedicated crossover operator to generate new solutions and a constrained neighborhood tabu search procedure for local optimization mamdp applies also a distance and quality based replacement strategy to maintain population diversity extensive evaluations on a large set of 120 benchmark instances show that the proposed approach competes very favorably with the current state of art methods for mdp in particular it consistently and easily attains all the best known lower bounds and yields improved lower bounds for 6 large mdp instances the key components of mamdp are analyzed to shed light on their influence on the performance of the algorithm

application of modified decomposition method for the analytical solution of space fractional diffusion equation abstract spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow finance and others areas of application this paper presents the analytical solutions of the space fractional diffusion equations by modified decomposition method mdm by using initial conditions the explicit solutions of the equations have been presented in the closed form the decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self cancelling noise phenomenon two examples the first one is one dimensional and the second one is two dimensional fractional diffusion equation are presented to show the application of the present technique the present method performs extremely well in terms of efficiency and simplicity

a review on algorithms for maximum clique problems the maximum clique problem mcp is to determine in a graph a clique i e a complete subgraph of maximum cardinality the mcp is notable for its capability of modeling other combinatorial problems and real world applications as one of the most studied np hard problems many algorithms are available in the literature and new methods are continually being proposed given that the two existing surveys on the mcp date back to 1994 and 1999 respectively one primary goal of this paper is to provide an updated and comprehensive review on both exact and heuristic mcp algorithms with a special focus on recent developments to be informative we identify the general framework followed by these algorithms and pinpoint the key ingredients that make them successful by classifying the main search strategies and putting forward the critical elements of the most relevant clique methods this review intends to encourage future development of more powerful methods and motivate new applications of the clique approaches

an efficient numerical method for a two point boundary value problem with a caputo fractional derivative abstract in this paper a two point boundary value problem with a caputo fractional derivative is considered by using a shooting method based on the secant iterative method the boundary value problem is turned into an initial value problem then the initial value problem is transformed into an equivalent integral differential equation with a weakly singular kernel an integral discretization scheme on the uniform mesh is developed to approximate the integral differential equation by applying the truncation error estimate techniques and a discrete analogue of gronwall s inequality it is proved that the numerical scheme is first order convergent in the discrete maximum norm numerical experiments verify the theoretical results

parallel in time multigrid for space time finite element approximations of two dimensional space fractional diffusion equations abstract the paper investigates a non intrusive parallel time integration with multigrid for space fractional diffusion equations in two spatial dimensions which is discretized by the space time finite element method to propagate solutions we develop a multigrid reduction in time mgrit algorithm with time dependent time grid propagators and provide its two level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time grid propagators numerical results show that the proposed method possesses the saturation error order theoretical results of the two level variant deliver good predictions for our model problems and significant speedups of the mgrit can be achieved when compared to the two level variant with f relaxation an equivalent version of the parareal algorithm and the sequential time stepping approach

an unstructured mesh finite difference finite element method for the three dimensional time space fractional bloch torrey equations on irregular domains abstract in this paper we use the finite element method fem to solve the time space fractional bloch torrey equation on irregular domains in r 3 based on linear lagrange basis functions a space semi discrete fem scheme is given by adopting the l 2 1 approximation for the caputo fractional derivative a fully discrete scheme is presented furthermore we provide the details on how to implement our fem for the space fractional bloch torrey equation also the stability and convergence of the fully discrete scheme is investigated the error estimations with respect to the l 2 and energy norms are given in addition some numerical examples are presented to verify the efficiency of our method

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

towards optimal lower bounds for clique and chromatic number it was previously known that neither max clique nor min chromatic number can be approximated in polynomial time within n1 e for any constant e 0 unless np zpp in this paper we extend the reductions used to prove these results and combine the extended reductions with a recent result of samorodnitsky and trevisan to show that unless np zptime 2o log n log log n 3 2 neither max clique nor min chromatic number can be approximated in polynomial time within n1 e n where e o log log n 1 2 since there exists polynomial time algorithms approximating both problems within n1 e n where e n log log n log n our result shows that the best possible ratio we can hope for is of the form n1 o 1 for some yet unknown value of o 1 between o log log n 1 2 and log log n log n

iterated greedy for the maximum diversity problem abstract the problem of choosing a subset of elements with maximum diversity from a given set is known as the maximum diversity problem many algorithms and methods have been proposed for this hard combinatorial problem including several highly sophisticated procedures by contrast in this paper we present a simple iterated greedy metaheuristic that generates a sequence of solutions by iterating over a greedy construction heuristic using destruction and construction phases extensive computational experiments reveal that the proposed algorithm is highly effective as compared to the best so far metaheuristics for the problem under consideration

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

new facets and a branch and cut algorithm for the weighted clique problem abstract we consider a polyhedral approach to the weighted maximal b clique problem given a node and edge weighted complete graph the problem is to find a complete subgraph clique with no more than b nodes such that the sum of the weights of all nodes and edges in the clique is maximal we introduce four new classes of facet defining inequalities for the associated b clique polytope one of these inequality classes constitutes a generalization of the well known tree inequalities the other classes are associated with multistars we use these inequalities together with other classes of facet defining inequalities in a branch and cut algorithm for the problem we give a description of this algorithm including some separation procedures and present the computational results for different sets of test problems the computation times that are obtained indicate that this algorithm is more efficient than previously described algorithms for the problem

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

central difference approximation of convection in caputo fractional derivative two point boundary value problems a rigorous numerical analysis is given for a fractional derivative two point boundary value problem the highest order term of the differential operator is a caputo fractional derivative of order 1 2 the second and higher order derivatives of the solution of the problem are in general unbounded at one end of the interval which creates difficulties for the analysis the problem is discretized on an equidistant mesh of diameter h using a standard central difference approximation of the convection term and nodal convergence of order o h 1 is proved provided that h satisfies a stability condition that depends on and the convective coefficient this condition may be restrictive when is near 1 when it is violated the solution may exhibit small oscillations but these can be removed by a simple and inexpensive postprocessing technique numerical results are given to display the performance of the method

heuristic and special case algorithms for dispersion problems the dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities the problem also arises in selecting nondominated solutions for multiobjective decision making it is known to be np hard under two objectives maximizing the minimum distance max min between any pair of facilities and maximizing the average distance max avg we consider the question of obtaining near optimal solutions for max min we show that if the distances do not satisfy the triangle inequality there is no polynomial time relative approximation algorithm unless p np when the distances satisfy the triangle inequality we analyze an efficient heuristic and show that it provides a performance guarantee of two we also prove that obtaining a performance guarantee of less than two is np hard for max avg we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality we also present a polynomial ti

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

iterated tabu search for the maximum diversity problem in this paper we deal with the maximum diversity problem mdp which asks to select a specified number of elements from a given set so that the sum of distances between the selected elements is as large as possible we develop an iterated tabu search its algorithm for solving this problem we also present a steepest ascent algorithm which is well suited in application settings where solutions of satisfactory quality are required to be provided very quickly computational results for problem instances involving up to 5000 elements show that the its algorithm is a very attractive alternative to the existing approaches in particular we demonstrate the outstanding performance of its on the mdp instances taken from the literature for 69 such instances new best solutions were found

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

a note on the finite element method for the space fractional advection diffusion equation in this paper a note on the finite element method for the space fractional advection diffusion equation with non homogeneous initial boundary condition is given where the fractional derivative is in the sense of caputo the error estimate is derived and the numerical results presented support the theoretical results