regular star complements in strongly regular graphs abstract we prove that aside from the complete multipartite graphs and graphs of steiner type there are only finitely many connected strongly regular graphs with a regular star complement of prescribed degree s n we investigate the possible parameters when s 5

graphs with least eigenvalue 2 a historical survey and recent developments in maximal exceptional graphs abstract we survey the main results of the theory of graphs with least eigenvalue 2 starting from late 1950s papers by a j hoffman et al via important results p j cameron et al j algebra 43 1976 305 involving root systems to the recent approach by the star complement technique which culminated in finding and characterizing maximal exceptional graphs some novel results on maximal exceptional graphs are included as well in particular we show that all exceptional graphs except for the cone over l k 8 can be obtained by the star complement technique starting from a unique exceptional star complement for the eigenvalue 2

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

on the characterization of graphs by star complements abstract let g be a graph of order n and be an adjacency eigenvalue of g with multiplicity k 1 a star complement for in g is an induced subgraph of g of order n k with no eigenvalue in this paper all the regular graphs with k 1 1 t as a star complement are determined also the maximal graphs with k 1 1 t t 8 9 as a star complement for the eigenvalue 1 and k 7 as a star complement for the eigenvalue 2 are described

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

an improved spectral meshless radial point interpolation for a class of time dependent fractional integral equations 2d fractional evolution equation abstract in this paper a kind of spectral meshless radial point interpolation smrpi technique is applied to the fractional evolution equation in two dimensional for arbitrary fractional order the applied approach is based on erudite combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of smrpi in the current work the thin plate splines tps are used as the basis functions and also the discretization technique employed is patterned after an idea of ch lubich it is proved the scheme is unconditional stable with respect to the time variable in l 2 norm and convergent by the order of convergence o t 1 0 1 the results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme three numerical examples show that the smrpi has reliable accuracy in irregular domains

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

analysis of the spectral meshless radial point interpolation for solving fractional reaction subdiffusion equation abstract the present paper is devoted to the development of spectral meshless radial point interpolation smrpi technique for solving fractional reaction subdiffusion equation in one and two dimensional cases the time fractional derivative is described in the riemann liouville sense the applied approach is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is unconditionally stable with respect to the time variable in h 1 and we show convergence order of the time discrete scheme is o t 0 1 in the current work the thin plate splines tps are used to construct the basis functions the results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme

some characterizations of graphs by star complements abstract let be an eigenvalue of the graph g with multiplicity k a star complement for in g is an induced subgraph h g x such that x k and is not an eigenvalue of g x various graphs related to generalized line graphs or their complements are characterized by star complements corresponding to eigenvalues 2 or 1

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

star complements and exceptional graphs let g be a finite graph of order n with an eigenvalue of multiplicity k thus the eigenspace of a 0 1 adjacency matrix of g has dimension k astar complement for in g is an induced subgraph g x of g such that x k and g x does not have as an eigenvalue an exceptional graph is a connected graph other than a generalized line graph whose eigenvalues lie in 2 we establish some properties of star complements and of eigenvectors of exceptional graphs with least eigenvalue 2

on graphs whose second largest eigenvalue equals 1 the star complement technique the star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts called star complements here we first identify among trees and complete graphs those graphs which can be star complements for 1 as the second largest eigenvalue using the graphs just obtained we next search for their maximal extensions either by theoretical means or by computer aided search

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

letter to the editor convergence of fem with interpolated coefficients for semilinear hyperbolic equation to solve spatially semidiscrete approximative solution of a class of semilinear hyperbolic equations the finite element method fem with interpolated coefficients is discussed by use of semidiscrete finite element for linear problem as comparative function the error estimate in l norm is derived by the nonlinear argument in chen structure theory of superconvergence of finite elements hunan press of science and technology changsha 2001 in chinese this indicates that convergence of fems with interpolated coefficients for a semilinear equation is similar to that of classical fems

star complements and switching in graphs star complements are used to construct switching equivalent graphs which share an eigenvalue of the same multiplicity a variant of the construction is used to obtain a characterization of the mclaughlin graph

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

an improved numerical scheme for the sine gordon equation in 2 1 dimensions a rational approximant of order 4 which is applied to a three time level recurrence relation is used to transform the initial boundary value problem associated with the two dimensional sine gordon sg equation arising in the josephson junctions problem the resulting non linear system which is analyzed for stability is solved using an appropriate predictor corrector p c scheme in which an explicit scheme of order 2 is used as predictor for the implementation of the corrector in order to avoid extended matrix evaluations an auxiliary vector was successfully introduced in this p c scheme a modification in the corrector has been proposed according to which the already evaluated corrected values are considered the behavior of this p c scheme is tested numerically on line and ring solitons known from the bibliography regarding the sg equation and conclusions for both undamped and damped problems are derived copyright 2008 john wiley sons ltd

an extension of the star complement technique for regular graphs abstract we extend the means by which a regular graph can sometimes be identified from a star complement in two applications we determine i the regular graphs with a path as a star complement for the eigenvalue 1 ii the extremal regular graphs which have another type of tree as a star complement for an eigenvalue 1 0

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

stable and convergent unsymmetric meshless collocation methods in the theoretical part of this paper we introduce a simplified proof technique for error bounds and convergence of a variation of kansa s well known unsymmetric meshless collocation method for a numerical implementation of the convergent variation a previously proposed greedy technique is coupled with linear optimization this algorithm allows a fully adaptive on the fly data dependent meshless selection of test and trial spaces the new method satisfies the assumptions of the background theory and numerical experiments demonstrate its stability

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

star complements and edge connectivity in finite graphs abstract let g be a finite graph with h as a star complement for an eigenvalue other than 0 or 1 let g g denote respectively the vertex connectivity and minimum degree of g we prove that g is controlled by g and h in particular for each k n there exists a smallest non negative integer f k such that g k whenever h k and g f k we show that f 1 0 f 2 2 f 3 3 f 4 5 and f 5 7

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

assessing differential gene expression with small sample sizes in oligonucleotide arrays using a mean variance model the identification of the genes that are differentially expressed in two sample microarray experiments remains a difficult problem when the number of arrays is very small we discuss the implications of using ordinary t statistics and examine other commonly used variants for oligonucleotide arrays with multiple probes per gene we introduce a simple model relating the mean and variance of expression possibly with gene specific random effects parameter estimates from the model have natural shrinkage properties that guard against inappropriately small variance estimates and the model is used to obtain a differential expression statistic a limiting value to the positive false discovery rate pfdr for ordinary t tests provides motivation for our use of the data structure to improve variance estimates our approach performs well compared to other proposed approaches in terms of the false discovery rate

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

on bipartite graphs with complete bipartite star complements abstract let g be a bipartite graph with as an eigenvalue of multiplicity k 1 we show that if g has k r s 1 r s as a star complement for then k s 1 moreover if is non main then k s 2 for large enough s we provide examples of graphs in which various bounds on k or s are attained we also describe the bipartite graphs with k 1 s as a star complement for a non main eigenvalue of multiplicity s 1 1

eigenvalue multiplicity in triangle free graphs abstract let g be a connected triangle free graph of order n 5 with 1 0 as an eigenvalue of multiplicity k 1 we show that if d is the maximum degree in g then k n d 1 moreover if k n d 1 then either a g is non bipartite and k 2 3 1 2 2 1 with equality only if g is strongly regular or b g is bipartite and k d 1 with equality only if g is a bipolar cone in each case we discuss the extremal graphs that arise

characterizing line graphs by star complements abstract in this paper it is shown that for any odd integer t 3 the line graph l k t is the unique maximal graph having the cycle c t as a star complement for the eigenvalue 2 this result yields a characterization of l g for hamiltonian graphs g with an odd number of vertices we also show that if t r s where r and s are odd integers 1 then provided that t 8 l k t is the unique maximal graph having c r c s as a star complement for the eigenvalue 2

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

there is no 75 32 10 16 strongly regular graph abstract we show that there is no 75 32 10 16 strongly regular graph the result is obtained by a mix of algebraic and computational approaches the main idea is to build large enough induced structure and apply the star complement technique our result implies that there is no partial geometry pg 4 7 2 and no regular two graph with 76 vertices from which it follows that there also exists no strongly regular graph with parameters 76 35 18 14 in order to solve this classification problem we also develop an efficient algorithm for the problem of finding a maximal clique in a graph

spectral results on graphs with regularity constraints abstract graphs with k regular sets and equitable partitions are examples of graphs with regularity constraints a k regular set of a graph g is a subset of vertices s v g inducing a k regular subgraph and such that each vertex not in s has neighbors in s the existence of such structures in a graph provides some information about the eigenvalues and eigenvectors of its adjacency matrix for example if a graph g has a k1 1 regular set s1 and a k2 2 regular set s2 such that k1 1 k2 2 then is an eigenvalue of g with a certain eigenvector additionally considering primitive strongly regular graphs a necessary and sufficient condition for a particular subset of vertices to be k regular is introduced another example comes from the existence of an equitable partition in a graph if a graph g has an equitable partition then its line graph l g also has an equitable partition induced by and the adjacency matrix of the quotient graph l g is obtained from the adjacency matrix of g

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

line star sets for laplacian eigenvalues abstract let g be a mixed graph with a nonzero laplacian eigenvalue of multiplicity k a line star set for in g is a set y of k edges of g such that is not a laplacian eigenvalue of g y it is shown that line star set exists for any nonzero laplacian eigenvalue of any mixed graph some basic properties for line star sets are given we also obtain some results on line star sets in undirected graphs

rational radial basis function interpolation with applications to antenna design functions with poles occur in many branches of applied mathematics which involve resonance phenomena such functions are challenging to interpolate in particular in higher dimensions in this paper we develop a technique for interpolation with quotients of two radial basis function rbf expansions to approximate such functions as an alternative to rational approximation since the quotient is not uniquely determined we introduce an additional constraint the sum of the rbf norms of the numerator and denominator squared should be minimal subjected to a norm condition on the function values the method was designed for antenna design applications and we show by examples that the scattering matrix for a patch antenna as a function of some design parameters can be approximated accurately with the new method in many cases e g in antenna optimization the function evaluations are time consuming and therefore it is important to reduce the number of evaluations but still obtain a good approximation a sensitivity analysis of the new interpolation technique is carried out and it gives indications how efficient adaptation methods could be devised a family of such methods are evaluated on antenna data and the results show that much performance can be gained by choosing the right method

on graphs whose star complement for 2 is a path or a cycle abstract it was proved recently by one of the authors that if h is a path p t t 2 with t 7 or 8 or an odd cycle c t t 3 then there is a unique maximal graph having h as a star complement for 2 the methods employed were analytical in nature making use of the reconstruction theorem for star complements here we offer an alternative approach based on the forbidden subgraph technique in addition we resolve the exceptional situations arising when h p 7 or p 8

on nested split graphs whose second largest eigenvalue is less than 1 abstract we determine all nested split graphs i e graphs having no induced subgraphs equal to 2 k 2 p 4 or c 4 having the second largest eigenvalue less than 1 and give some data and observations regarding obtained results

mixed spline function method for reaction subdiffusion equations in this paper we develop two classes of finite difference schemes for the reaction subdiffusion equations by using a mixed spline function in space direction forward and backward difference in time direction respectively it has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters by fourier method we prove that one class of difference scheme is unconditionally stable and convergent the other is conditionally stable and convergent finally some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes

on independent star sets in finite graphs abstract let g be a finite graph with as an eigenvalue of multiplicity k a star set for is a set x of k vertices in g such that is not an eigenvalue of g x we investigate independent star sets of largest possible size in a variety of situations we note connections with symmetric designs codes strongly regular graphs and graphs with least eigenvalue 2

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

regular graphs with a prescribed complete multipartite graph as a star complement abstract let g be a simple graph of order n and let be an adjacency eigenvalue with multiplicity k 1 a star complement h for in g is an induced subgraph of order n k with no eigenvalue in this paper the regular graphs with the complete multipartite graph s k 1 k t s 2 as a star complement are investigated more precisely when is a main eigenvalue we prove that g k s 2 when is a non main eigenvalue we determine the regular graphs with s k 1 k t as a star complement for t 1 or s 3 we show that the complete multipartite graph s 1 k 2 s 2 is the only possible graph under a certain condition and we raise a conjecture for the general case

star complements in regular graphs old and new results we survey results concerning star complements in finite regular graphs and note the connection with designs and strongly regular graphs in certain cases we include improved proofs along with new results on stars and windmills as star complements

bayesian models for gene expression with dna microarray data two of the critical issues that arise when examining dna microarray data are 1 determination of which genes best discriminate among the different types of tissue and 2 characterization of expression patterns in tumor tissues for 1 there are many genes that characterize dna expression and it is of critical importance to try and identify a small set of genes that best discriminate between normal and tumor tissues for 2 it is critical to be able to characterize the dna expression of the normal and tumor tissue samples and develop suitable models that explain patterns of dna expression for these types of tissues toward this goal we propose a novel bayesian model for analyzing dna microarray data and propose a model selection methodology for identifying subsets of genes that show different expression levels between normal and cancer tissues in addition we propose a novel class of hierarchical priors for the parameters that allow us to borrow strength across genes for making inference the pro

on graphs with prescribed star complements abstract let be an eigenvalue of a simple graph g with multiplicity k 1 a star complement for in g is an induced subgraph of g of order n k with no eigenvalue in this paper we study the maximal graphs with the star s m as a star complement for 2 the maximal graphs with s 3 s 4 s 13 and s 21 as a star complement for 2 are described we also describe the regular graphs with k 2 s s 2 as a star complement for an eigenvalue

star partitions and regularity in graphs abstract we investigate regular graphs with a star partition x 1 x 2 x m such that g xi is regular for some i the cubic graphs with this property are explored in detail

a highly accurate trivariate spectral collocation method of solution for two dimensional nonlinear initial boundary value problems abstract in this paper we propose a new numerical method namely the trivariate spectral collocation method for solving two dimensional nonlinear partial differential equations pdes arising from unsteady processes the problems considered are nonlinear pdes defined on regular geometries in the current solution approach the quasi linearization method is used to simplify the nonlinear pdes the solutions of the linearized pdes are assumed to be trivariate lagrange interpolating polynomials constructed using chebyshev gauss lobatto cgl points a purely spectral collocation based discretization is employed on the two space variables and the time variable to yield a system of linear algebraic equations that are solved by iteration the numerical scheme is tested on four typical examples of nonlinear pdes reported in the literature as a single equation or system of equations numerical results confirm that the proposed solution approach is highly accurate and computationally efficient when applied to solve two dimensional initial boundary value problems defined on small time intervals and hence it is a reliable alternative numerical method for solving this class of problems the new error bound theorems and proofs on trivariate polynomial interpolation that we present support findings from the numerical simulations

reaction subdiffusion equations to analyze possible generalizations of reaction diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion a ensuremath rightarrow b we derive the corresponding kinetic equations for the local a and b concentrations their form is rather unusual the parameters of the reaction influence the diffusion term in the equation for a component a a consequence of the non markovian nature of subdiffusion the equation for the product contains a term which depends on the concentration of a at all previous times our discussion shows that reaction subdiffusion equations may not resemble the corresponding reaction diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one

new concepts in meshless methods meshless methods have been extensively popularized in literature in recent years due to their flexibility in solving boundary value problems two kinds of truly meshless methods the meshless local boundary integral equation mlbie method and the meshless local petrov galerkin mlpg approach are presented and discussed both methods use the moving least squares approximation to interpolate the solution variables while the mlbie method uses a local boundary integral equation formulation and the mlpg employs a local symmetric weak form the two methods are truly meshless ones as both of them do not need a finite element or boundary element mesh either for purposes of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regularly shaped domains in general spheres in three dimensional problems and their boundaries numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable in essence the present meshless method based on the lswf is found to be a simple efficient and attractive method with a great potential in engineering applications copyright 2000 john wiley sons ltd

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

on graphs with complete bipartite star complements abstract let be an eigenvalue of the graph g with multiplicity m a star complement for in g is an induced subgraph g x such that x m and is not an eigenvalue of g x some general observations concerning graphs with the complete bipartite graph k r s r s 2 as a star complement are followed by a complete analysis of the case r 2 s 5 the results include a characterization of the schlafli graph and the construction of all the regular graphs which have k 2 5 as a star complement

coupling of the improved singular boundary method and dual reciprocity method for multi term time fractional mixed diffusion wave equations abstract the main purpose of this work is to present an impressive numerical scheme to solve two dimensional multi term time fractional mixed diffusion wave differential equations tfmdwe the proposed method is based on the compact dual reciprocity method and the meshless improved singular boundary method isbm the most significant privilege of the proposed method is mathematically simple efficient quite accurate which requires relatively less computational cost singular boundary method is one of truly meshless methods and it does not need any element or mesh for both field interpolation and background integration for the considered problems time derivatives are approximated via the time stepping method several numerical results are provided to show the accuracy of the present method for multi term time fractional mixed diffusion wave equations

co cliques and star complements in extremal strongly regular graphs abstract suppose that the positive integer is the eigenvalue of largest multiplicity in an extremal strongly regular graph g by interlacing the independence number of g is at most 4 2 4 2 star complements are used to show that if this bound is attained then either a 1 and g is the schlafli graph or b 2 and g is the mclaughlin graph

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

meshless spectral method for solution of time fractional coupled kdv equations abstract in this article an efficient and accurate meshless spectral interpolation method is formulated for the numerical solution of time fractional coupled kdv equations that govern shallow water waves meshless shape functions constructed via radial basis functions rbfs and point interpolation are used for discretization of the spatial operator approximation of fractional temporal derivative is obtained via finite differences of order o 2 and a quadrature formula the formulated method is applied to various test problems available in the literature for its validation approximation quality and efficiency of the method is measured via discrete error norms e2 e and erms convergence analysis of the proposed method in space and time is numerically determined by varying nodal points m and time step size respectively stability of the proposed method is discussed and affirmed computationally which is an important ingredient of the current study

spectral characterization of unicyclic graphs whose second largest eigenvalue does not exceed 1 abstract connected graphs with the same number of vertices and edges are called unicyclic graphs all unicyclic graphs whose second largest eigenvalue does not exceed 1 denoted by u 2 are determined by xu 10 let g t s denote the graph obtained by attaching t 1 pendent paths of length 2 and s 0 pendent edges to a vertex of a triangle in this paper we first prove that the graph g t s is determined by its adjacency spectrum then we prove that the graphs in u 2 are determined by their adjacency spectra except for four graphs

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

on graphs with multiple eigenvalues let e be an eigenspace of a finite graph g with dimension m and codimension t 1 it is shown that if 1 0 then m 12 t 1 t 4 a necessary and sufficient condition for to be a multiple eigenvalue of g is established and used to construct examples from intersecting families of sets

meshless local petrov galerkin mlpg approximation to the two dimensional sine gordon equation during the past few years the idea of using meshless methods for numerical solution of partial differential equations pdes has received much attention throughout the scientific community and remarkable progress has been achieved on meshless methods the meshless local petrov galerkin mlpg method is one of the truly meshless methods since it does not require any background integration cells the integrations are carried out locally over small sub domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions in this paper the mlpg method for numerically solving the non linear two dimensional sine gordon sg equation is developed a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of proposed method to deal with the unsteady non linear problems in large domains

on tricyclic graphs whose second largest eigenvalue does not exceed 1 a tricyclic graph of order n is a connected graph with n vertices and n 2 edges in this paper all tricyclic graphs whose second largest eigenvalue does not exceed 1 are identified

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis