block jacobi matrices and zeros of multivariate orthogonal polynomials a commuting family of symmetric matrices are called the block jacobi matrices if they are block tridiagonal they are related to multivariate orthogonal polynomials we study their eigenvalues and joint eigenvectors the joint eigenvalues of the truncated block jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials

a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

on leja sequences some results and applications in this paper we collect and review some results concerning leja sequences on one and two dimensional domains then we present two applications and discuss some open problems on the asymptotic distribution of leja points on two dimensional domains

coupled lattice boltzmann method for generalized keller segel chemotaxis model in this work an efficient and stable lattice boltzmann method lbm for generalized keller segel k s chemotaxis model is proposed through the chapman enskog analysis the proposed lbm can correctly recover to the k s model the stability of the proposed lbm has been improved through adding correction terms in the evolution equations moreover a local computational scheme for the gradient operator which is included in the evolution equation is developed making the proposed lbm be implemented locally hence both 2d and 3d problems with arbitrary geometries can be processed easily in the numerical experiments several representative chemotaxis problems are studied including the blow up problem in square and circle domains two species chemotaxis blow up problem chemotactic bacteria pattern formation in semi solid medium in circle domain 3d pattern formation in liquid medium and the tumor invasion into surrounding healthy tissue the numerical results demonstrate the high efficiency stability and robustness of the proposed lbm furthermore the capability of the proposed lbm in handling both 2d and 3d problems with complex domain is also illustrated

on the design of general purpose flux limiters for finite element schemes i scalar convection the algebraic flux correction afc paradigm is extended to finite element discretizations with a consistent mass matrix it is shown how to render an implicit galerkin scheme positivity preserving and remove excessive artificial diffusion in regions where the solution is sufficiently smooth to this end the original discrete operators are modified in a mass conserving fashion so as to enforce the algebraic constraints to be satisfied by the numerical solution a node oriented limiting strategy is employed to control the raw antidiffusive fluxes which consist of a convective part and a contribution of the consistent mass matrix the former offsets the artificial diffusion due to upwinding of the spatial differential operator and lends itself to an upwind biased flux limiting the latter eliminates the error induced by mass lumping and calls for the use of a symmetric flux limiter the concept of a target flux and a new definition of upper lower bounds make it possible to combine the advantages of algebraic fct and tvd schemes introduced previously by the author and his coworkers unlike other high resolution schemes for unstructured meshes the new algorithm reduces to a consistent high order galerkin scheme in smooth regions and is designed to provide an optimal treatment of both stationary and time dependent problems its performance is illustrated by application to the linear advection equation for a number of 1d and 2d configurations

an improved multi step gradient type method for large scale optimization in this paper we propose an improved multi step diagonal updating method for large scale unconstrained optimization our approach is based on constructing a new gradient type method by means of interpolating curves we measure the distances required to parameterize the interpolating polynomials via a norm defined by a positive definite matrix by developing on implicit updating approach we can obtain an improved version of hessian approximation in diagonal matrix form while avoiding the computational expenses of actually calculating the improved version of the approximation matrix the effectiveness of our proposed method is evaluated by means of computational comparison with the bb method and its variants we show that our method is globally convergent and only requires o n memory allocations

commuting family of block jacobi matrices abstract the block jacobi matrices considered in this paper are a family of block tridiagonal matrices which are natural extensions of a singular jacobi matrix in the sense that they are associated with orthogonal polynomials in several variables we present the basic properties of these matrices

zero sets of bivariate hermite polynomials abstract we establish various properties for the zero sets of three families of bivariate hermite polynomials special emphasis is given to those bivariate orthogonal polynomials introduced by hermite by means of a rodrigues type formula related to a general positive definite quadratic form for this family we prove that the zero set of the polynomial of total degree n m consists of exactly n m disjoint branches and possesses n m asymptotes a natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property the results show that the properties of the zero sets considered as affine algebraic curves in r 2 are completely different for the three families analyzed

three step fixed point quasi newtonmethods for unconstrained optimisation multistep quasi newton methods were introduced by ford and moghrabi 1 they address the problem of the unconstrained minimisation of a function f n whose gradient and hessian are denoted by g and g respectively these methods generalised the standard construction of quasi newton methods and were based on employing interpolatory polynomials to utilise information from more than one previous step in a series of papers ford and moghrabi 2 5 have developed various techniques for determining the parametrisation of the interpolating curves in 2 they introduced two step metric based methods which determine the set of parameter values required through measuring distances between various pairs of the iterates employed in the current interpolation one of the most successful methods in 2 was found to be in the fixed point class in which the parametrisation of the interpolating curve is determined at each iteration by reference to distances measured from a fixed iterate as suggested in 1 multistep quasi newton methods can be constructed for any number of steps in this paper we therefore extend the previous work by describing the development of some three step methods which use the fixed point approach and data derived from the latest four iterates the experimental results provide evidence that the new methods offer a significant improvement in performance when compared with the standard bfgs method and the unit spaced three step method particularly as the dimension of the test problems grows

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

uniqueness and symmetry of equilibria in a chemotaxis model we consider in a disc of r a class of parameter dependent nonlocal elliptic boundary value problems that describe the steady states of some chemotaxis systems if the appearing parameter is less than an explicit critical value we establish several uniqueness results for solutions that are invariant under a group of rotations furthermore we discuss the associated consequences for the time asymptotic behavior of the solutions to the corresponding time dependent chemotaxis systems our results also provide optimal constants in some moser trudinger type inequalities mathematics subject classification 2000 35j60 35j50 92c17

new implicit updates in multi step quasi newton methods for unconstrained optimisation multi step quasi newton methods for optimisation using data from more than one previous step to revise the current approximate hessian were introduced by ford and moghrabi in j comput appl math 50 1994 305 where they showed how to construct such methods by means of interpolating curves these methods also utilise standard quasi newton formulae but with the vectors normally employed in the formulae replaced by others determined from a multi step version of the secant equation some methods the accumulative and fixed point approaches for defining the parameter values which correspond to the iterates on the interpolating curve were presented by ford and moghrabi in optim methods software 2 1993 357 both the accumulative and the fixed point methods measure the distances required to parameterise the interpolating polynomials via a norm defined by a positive definite matrix m the fixed point algorithm which takes m to be the current approximate hessian was found experimentally to be the best of the six multi step methods studied in ford and moghrabi 1993 all of which exhibited improved numerical performance by comparison with the standard single step bfgs method to produce a better parameterisation of the interpolation ford comput math appl 42 2001 1083 developed the idea of implicit update methods the fundamental concept here is to determine an improved version of the hessian approximation to be used in computing the metric while avoiding the computational expense of actually calculating the improved version two implicit methods denoted by i2 and i3 were developed from f2 in ford 2001 the method i2 employed parameter values generated from an implicit single step bfgs update while i3 used values from an implicit two step update in this paper we describe the derivation of new implicit updates which are similar to i3 the experimental results we present show that one of the new implicit methods produces markedly better performance than the existing implicit methods particularly as the dimension of the test problem grows

commuting extensions and cubature formulae based on a novel point of view on 1 dimensional gaussian quadrature we present a new approach to d dimensional cubature formulae it is well known that the nodes of 1 dimensional gaussian quadrature can be computed as eigenvalues of the so called jacobi matrix the d dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices these are obtained by extending adding rows and columns to certain noncommuting matrices a1 ad related to the coordinate operators x1 xd in rd we prove a correspondence between cubature formulae and commuting extensions of a1 ad satisfying a compatibility condition which in appropriate coordinates constrains certain blocks in the extended matrices to be zero thus the problem of finding cubature formulae can be transformed to the problem of computing and then simultaneously diagonalizing commuting extensions we give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them

the use of alternation and recurrences in two step quasi newton methods of the multistep quasi newton methods introduced by the authors in 1 the mostsuccessful was the so called fixed point method using the existing hessian approximation to compute at each iteration the parameters required in the interpolation in order to avoid the burden of computing the additional matrix vector products required by this approach approximations based on the secant equation were proposed in 2 a different approach to dealing with this difficulty was proposed in which standard single step quasi newton updates were alternated on successive iterations with two step updates so that approximations were no longer necessary recent work has shown that the quantities required to compute the parameters referred to above may be computed exactly by means of a recurrence so that the technique of alternation is no longer the only alternative if we wish to avoid approximations in this paper we describe the derivation of this recurrence we present the results of a range of numerical experiments to compare and evaluate the three approaches of approximation alternation and recurrence finally we show how the use of recurrences may be extended to multistep methods employing three or more steps

a positivity preserving finite element method for chemotaxis problems in 3d we present an implicit finite element method for a class of chemotaxis models in three spatial dimensions the proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density to enforce the discrete maximum principle the standard galerkin discretization is constrained using a local extremum diminishing flux limiter to demonstrate the efficiency and robustness of this approach we solve blow up problems in a 3d chemostat domain to give a flavor of more complex and realistic chemotactic applications we investigate the pattern dynamics and aggregating behavior of the bacteria escherichia coli and salmonella typhimurium the obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature

analysis for a system of coupled reaction diffusion parabolic equations arising in biology we study a system where the equations are coupled by transport and reaction terms in the general case we prove local existence and uniqueness of the solution and in a particular case for a priori fixed time t global existence on 0 t

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

cubature formulae and orthogonal polynomials abstract the connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 yr the article j radon published about 50 yr ago j radon zur mechanischen kubatur monatsh math 52 1948 286 300 has been very influential in this text we describe some of the results that were obtained during the search for answers to questions raised by his article

new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model we develop a family of new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model this model is described by a system of two nonlinear pdes a convection diffusion equation for the cell density coupled with a reaction diffusion equation for the chemoattractant concentration it has been recently shown that the convective part of this system is of a mixed hyperbolic elliptic type which may cause severe instabilities when the studied system is solved by straightforward numerical methods therefore the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original keller segel model in the form of a convection diffusion reaction system with a hyperbolic convective part we then design interior penalty discontinuous galerkin methods for the rewritten keller segel system our methods employ the central upwind numerical fluxes originally developed in the context of finite volume methods for hyperbolic systems of conservation laws in this paper we consider cartesian grids and prove error estimates for the proposed high order discontinuous galerkin methods our proof is valid for pre blow up times since we assume boundedness of the exact solution we also show that the blow up time of the exact solution is bounded from above by the blow up time of our numerical solution in the numerical tests presented below we demonstrate that the obtained numerical solutions have no negative values and are oscillation free even though no slope limiting technique has been implemented

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

a class of bivariate orthogonal polynomials and cubature formula the existence of gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros we examine a class of orthogonal polynomials of two variables generated from that of one variable

hyperinterpolation on the square we show that hyperinterpolation at near minimal cubature points for the product chebyshev measure along with xu compact formula for the corresponding reproducing kernel provide a simple and powerful polynomial approximation formula in the uniform norm on the square the lebesgue constant of the hyperinterpolation operator grows like log 2 of the degree as that of quasi optimal interpolation sets recently proposed in the literature moreover we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points and we compare it with interpolation formulas at the same set of points

minimum curvature multistep quasi newton methods abstract we consider multistep quasi newton methods for unconstrained optimization problems such methods were introduced by the authors 1 3 and employ interpolating polynomials to utilize data from the m most recent iterations for the case m 2 we observe that there is a free parameter used in constructing the interpolation which is essentially at our disposal we propose as a criterion for determining this parameter the minimization of a measure of the curvature of the interpolating polynomial thus producing a smooth interpolant we show how this minimum curvature problem may be solved cheaply and effectively at each iteration the performance of an algorithm which employs this technique is compared numerically with that of the standard bfgs method and of previously introduced multistep methods 2 3

studies of bacterial branching growth using reaction diffusion models for colonial development abstract various bacterial strains exhibit colonial branching patterns during growth on poor substrates these patterns reflect bacterial cooperative self organization and cybernetic processes of communication regulation and control employed during colonial development one method of modeling is the continuous or coupled reaction diffusion approach in which continuous time evolution equations describe the bacterial density and the concentration of the relevant chemical fields in the context of branching growth this idea has been pursued by a number of groups we present an additional model which includes a lubrication fluid excreted by the bacteria we also add fields of chemotactic agents to the other models we then present a critique of this whole enterprise with focus on the models potential for revealing new biological features

galerkin finite element method for cancer invasion mathematical model abstract a finite element scheme for the solution of a cancer invasion model is proposed the cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density extra cellular matrix and the matrix degrading enzymes the model incorporates proliferation and haptotaxis effect of cancer cells their interaction with extracellular matrix the production of matrix degrading enzymes and consequent degradation of the extracellular matrix the coupled partial differential equations are discretized in space with the standard galerkin finite elements and in time with the crank nicolson method moreover the nonlinear terms in the coupled equations are treated semi implicitly in the finite element scheme the numerical scheme is validated with numerical results taken from the literature in addition to the mesh convergence study the effects of haptotactic rate proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated

alternating multi step quasi newton methods for unconstrained optimization we consider multi step quasi newton methods for unconstrained optimization these methods were introduced by the authors ford and moghrabi 5 6 8 who showed how an interpolating curve in the variable space could be used to derive an appropriate generalization of the secant equation normally employed in the construction of quasi newton methods one of the most successful of these multi step methods employs the current approximation to the hessian to determine the parametrization of the interpolating curve and hence the derivatives which are required in the generalized updating formula however certain approximations were found to be necessary in the process in order to reduce the level of computation required which must be repeated at each iteration to acceptable levels in this paper we show how a variant of this algorithm which avoids the need for such approximations may be obtained this is accomplished by alternating on successive iterations a single step and a two step method the results of a series of experiments which show that the new algorithm exhibits a clear improvement in numerical performance are reported

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a second order positivity preserving numerical method for gamma equation in this work we consider cauchy problem for the so called gamma equation which can be derived by transforming the fully nonlinear black scholes equation for option price into a quasi linear parabolic equation for the second derivative greek c v s s of the option price v we develop an efficient positivity preserving explicit numerical method for solving the model problem concerning different volatility terms we prove that the obtained semi discretization is positive and the corresponding full approximation also preserves this property if the time step is restricted the stability of the difference scheme in l 1 norm is shown and the existence of interface curves is investigated numerically results of numerical simulations are given and discussed

operator splitting combined with positivity preserving discontinuous galerkin method for the chemotaxis model the advection diffusion reaction adr systems are often used to describe the chemotaxis models arising in biology in general the stiffness from the reaction and diffusion terms often requires very restricted time step size and the advection term depending on the concentration gradients of another component the chemoattractant may lead to sharp peaks in localized spatial regions it is challenging to design numerical methods that can efficiently handle both difficulties in this paper we apply the operator splitting approach to solve the advection diffusion reaction systems in particular for advection term we use the positivity preserving dg method with strong stability preserving ssp high order time discretizations for reaction diffusion term direct discontinuous galerkin ddg method is used in spatial discretization and krylov iif method is applied in time discretization numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

a multi iterate method to solve systems of nonlinear equations we propose an extension of secant methods for nonlinear equations using a population of previous iterates contrarily to classical secant methods where exact interpolation is used we prefer a least squares approach to calibrate the linear model we propose an explicit control of the numerical stability of the method we show that our approach can lead to an update formula in that case we prove the local convergence of the corresponding undamped quasi newton method finally computational comparisons with classical quasi newton methods highlight a significant improvement in terms of robustness and number of function evaluations we also present numerical tests showing the robust behavior of our method in the presence of noise

an analysis of polynomial chaos approximations for modeling single fluid phase flow in porous medium systems we examine a variety of polynomial chaos motivated approximations to a stochastic form of a steady state groundwater flow model we consider approaches for truncating the infinite dimensional problem and producing decoupled systems we discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature it would be necessary to find new multivariate cubature rules finally we use the acceleration of monte carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than monte carlo for the relatively small domains considered in this work a curse of dimensionality in the series expansion of the log normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work other than the monte carlo method

self similar blow up for a reaction diffusion system abstract this work is concerned with the following system which is a model to describe several phenomena in which aggregation plays a crucial role as for instance motion of bacteria by chemotaxis and equilibrium of self attracting clusters when the space dimension n is equal to three we show here that s has radial solutions with finite mass that blow up in finite time in a self similar manner when n 2 however no radial solution with finite mass may give rise to self similar blow up

multi step quasi newton methods for optimization abstract quasi newton methods update at each iteration the existing hessian approximation or its inverse by means of data deriving from the step just completed we show how multi step methods employing in addition data from previous iterations may be constructed by means of interpolating polynomials leading to a generalization of the secant or quasi newton equation the issue of positive definiteness in the hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original single step methods the results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach by comparison with single step methods particularly as the dimension of the problem increases

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

efficient accurate and flexible finite element solvers for chemotaxis problems in the framework of finite element discretizations we introduce a fully nonlinear newton like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities a chemical agent and the reacting population of certain biological organisms species we investigate the benefit of a corresponding monolithic approach and the decoupled variant in particular we analyze accuracy efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems

discontinuous galerkin methods for the chemotaxis and haptotaxis models in this work first we formulate and compare three different discontinuous interior penalty galerkin methods for the 2d keller segel chemotaxis model keller segel chemotaxis model is the important starting step in the modeling of the real biological system we show in the numerical tests that two of the proposed methods fail to give accurate oscillation free solutions next we consider the application of the successful method for the keller segel model to the simulation of the more realistic and closely related haptotaxis model of tumor invasion into healthy tissues

a new two step gradient type method for large scale unconstrained optimization in this paper we propose some improvements on a new gradient type method for solving large scale unconstrained optimization problems in which we use data from two previous steps to revise the current approximate hessian the new method which we considered resembles to that of barzilai and borwein bb method the innovation features of this approach consist in using approximation of the hessian in diagonal matrix form based on the modified weak secant equation rather than the multiple of the identity matrix in the bb method using this approach we can obtain a higher order accuracy of hessian approximation when compares to other existing bb type method by incorporating a simple monotone strategy the global convergence of the new method is achieved practical insights into the effectiveness of the proposed method are given by numerical comparison with the bb method and its variant

recurrence formulas for multivariate orthogonal polynomials in this paper necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula as a consequence orthogonal polynomials of total degree n in d variables that have dim i id common zeros can now be constructed recursively the result is important to the construction of gaussian cubature formulas

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

fast calculation of scattering by 3 d inhomogeneities in uniaxial anisotropic multilayers a fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them the uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure the generalized poisson summation formula is used to efficiently calculate the discrete fourier transform of spatial responses of the layeredmedia from its continuous fourier spectrum using the windowing technique the integral involved in constructing the discrete fourier spectrum can be performed in a small domain without compromising the accuracy of the result furthermore a fast numerical integration method based on the recently proposed padua points is used which together with the windowing technique yields a fast solution to such scattering problems numerical simulations illustrate the accuracy and efficiency of the proposed algorithm

multivariate orthogonal polynomials quantum decomposition deficiency rank and support of measure abstract in this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting fock spaces our framework is on the same stream line of the recent paper by accardi barhoumi and dhahri 1 the classical coordinate variables are decomposed into non commuting quantum operators called creation annihilation and preservation operators in the interacting fock spaces getting the commutation relations which follow from the commuting property of the coordinate variables between themselves we can develop the reconstruction theory of the measure namely the favard s theorem we then further develop some related problems including the marginal distributions and the rank theory of the jacobi operators we will see that the deficiency rank of the jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa we will provide with some examples

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

an afc stabilized implicit finite element method for partial differential equations on evolving in time surfaces in this article we present a new implicit numerical scheme for reaction diffusion advection equations on an evolving in time hypersurface t the partial differential equations are solved on a stationary quadrilateral resp hexahedral mesh the zero level set of the time dependent indicator function t implicitly describes the position of t the dominating convective like terms which are due to the presence of chemotaxis transport of the cell density and surface evolution may lead to the non positiveness of a given numerical scheme and in such a way cause appearance of negative values and give rise of nonphysical oscillations in the numerical solution the proposed finite element method is constructed to avoid this problem implicit treatment of corresponding discrete terms in combination with the algebraic flux correction afc techniques make it possible to obtain a sufficiently accurate solution for reaction diffusion advection pdes on evolving surfaces

stability of operator splitting methods for systems with indefinite operators advection diffusion reaction systems this brief paper presents an a stability result for operator splitting type time integration methods applied to advection diffusion reaction equations with possibly indefinite source terms these results extend our earlier work on diffusion reaction systems d l ropp j n shadid stability of operator splitting methods for systems with indefinite operators reaction diffusion systems j comput phys 203 2 2005 449 466 the a stability result presents sufficient conditions that control both low and high wave number instabilities a corollary shows that if l stable methods are used for the diffusion term the high wave number instability will be controlled more easily numerical results are presented that verify second order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an l stable diffusion integrator

a petrov galerkin finite element method for simulating chemotaxis models on stationary surfaces abstract in this paper we present a petrov galerkin finite element method for a class of chemotaxis models defined on surfaces which describe the movement by one community in reaction to one chemical or biological signal on manifolds it is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property which is a challenge due to the singular behavior of numerical solution thus a petrov galerkin method is combined with an effective mass conservation factor to overcome the challenge furthermore we prove two facts this method maintains positivity and discrete mass conservation property in addition decoupled approach is applied based on the gradient and laplacian recoveries to solve the coupling system the relevant stability analyses is provided finally numerical simulations of blowing up problems and pattern formulations demonstrate the effectiveness of the proposed method

the keller segel model with logistic sensitivity function and small diffusivity the keller segel model is the classical model for chemotaxis of cell populations it consists of a drift diffusion equation for the cell density coupled to an equation for the chemoattractant here a variant of this model is studied in one dimensional position space where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi stationary balance of reaction and diffusion with production of the chemoattractant by the cells the case of small cell diffusivity is studied by asymptotic and numerical methods on a time scale characteristic for the convective effects convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity close to each of these p

numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method abstract in this paper the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces the models and the numerical method concerned are in terms of a generic form of chemotaxis models we develop a new semi implicit finite element scheme based on the gradient and laplacian recoveries to linearize the equations meanwhile we modify the proposed semi implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface the lumped mass modification is employed for positivity preservation and the related analysis results are provided we investigate the accuracy and convergence of the proposed method by numerical tests and the simulations of blowing up solution pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods

gaussian cubature and bivariate polynomial interpolation gaussian cubature is used to study bivariate polynomial interpolation based on the common zeros of quasi orthogonal polynomials

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

implicit updates in multistep quasi newton methods abstract we consider multistep quasi newton methods for unconstrained optimization these methods were introduced by ford and moghrabi 1 2 who showed how interpolating curves could be used to derive a generalization of the secant equation the relation normally employed in the construction of quasi newton methods one of the most successful of these multistep methods makes use of the current approximation to the hessian to determine the parametrization of the interpolating curve in the variable space and hence the generalized updating formula in this paper we investigate the use of implicit updates to the approximate hessian in an attempt to determine a better parametrization of the interpolation while avoiding the computational burden of actually carrying out the update and thus improve the numerical performance of such algorithms

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation