existence of weak solutions for a hyperbolic model of chemosensitive movement a hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered in contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account this qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment the model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation we assume that the speed depends on the external stimulus only and not on its gradients in that case solutions with steep gradients are expected which have the interpretation of moving swarms a notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown the proof relies on the vanishing viscosity method i e we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

fixed points of weak contractions in cone metric spaces abstract cone metric spaces are generalizations of metric spaces where the metric is banach space valued weak contractions are generalizations of the banach s contraction mapping which have been studied by several authors in the present work we establish a unique fixed point result for weak contractions in cone metric spaces our result is supported by an example

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

a note on a family of two variable polynomials the main object of this paper is to construct a two variable analogue of jacobi polynomials and to give some properties of these polynomials we show that these polynomials are orthogonal then we obtain various recurrence formulas for them furthermore we give some integral representations for these polynomials

common fixed points for maps on cone metric space the purpose of this paper is to generalize and to unify fixed point theorems of das and naik ciric jungck huang and zhang on complete cone metric space

coincidence and fixed point theorems for generalized ψ φ weak nonlinear contraction in ordered k metric spaces in this paper we establish coincidence and fixed point theorems for a pair of mappings satisfying a new generalized j f weak nonlinear contraction type condition in ordered k metric spaces the presented results generalize and extend the very recent results of choudhury and metiya b s choudhury and n metiya the point of coincidence and common fixed point for a pair of mappings in cone metric spaces computers and mathematics with applications 60 2010 1686 1695

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

cone metric spaces and fixed point theorems of contractive mappings in this paper we introduce cone metric spaces prove some fixed point theorems of contractive mappings on cone metric spaces

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

fredholmness and smooth dependence for linear time periodic hyperbolic systems abstract this paper concerns n n linear one dimensional hyperbolic systems of the type t u j a j x x u j k 1 n b j k x u k f j x t j 1 n with periodicity conditions in time and reflection boundary conditions in space we state conditions on the data a j and b j k and the reflection coefficients such that the system is fredholm solvable moreover we state conditions on the data such that for any right hand side there exists exactly one solution that the solution survives under small perturbations of the data and that the corresponding data to solution map is smooth with respect to appropriate function space norms in particular those conditions imply that no small denominator effects occur we show that perturbations of the coefficients a j lead to essentially different results than perturbations of the coefficients b j k in general our results cover cases of non strictly hyperbolic systems as well as systems with discontinuous coefficients a j and b j k but they are new even in the case of strict hyperbolicity and of smooth coefficients

mesenchymal motion models in one dimension mesenchymal motion denotes a form of cell movement through tissue which can be observed for certain cancer metastases in t hillen j math biol 53 2006 pp 585 616 a mathematical model for this form of movement was introduced in the current paper we present a comprehensive analysis of the one dimensional mesenchymal motion model we establish the global existence of classical solutions and rigorously carry out the parabolic limit of the model we discuss the stationary solutions prove the existence of traveling wave solutions and use numerical simulations to illustrate the results finally we discuss the biological implications of the results

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

exponential dichotomy for hyperbolic systems with periodic boundary conditions abstract we investigate evolution families generated by general linear first order hyperbolic systems in one space dimension with periodic boundary conditions we state explicit conditions on the coefficient functions that are sufficient for the existence of exponential dichotomies on r in the space of continuous periodic functions

maxwellian decay for well balanced approximations of a super characteristic chemotaxis model we focus on the numerical simulation of a one dimensional model of chemotaxis dynamics proposed by greenberg and alt trans amer math soc 300 1987 pp 235 258 in a bounded domain by means of a previously introduced well balanced wb and asymptotic preserving ap scheme l gosse j math anal appl 2011 we are especially interested in studying the decay onto numerical steady states for two reasons 1 conventional upwind schemes have been shown to stabilize onto spurious non maxwellian states with a very big mass flux see e g f guarguaglini et al discrete contin dyn syst ser b 12 2009 pp 39 76 and 2 the initial data lead to a dynamic which is mostly super characteristic in the sense of s jin and m katsoulakis siam j appl math 61 2000 pp 273 292 thus the stability results of gosse do not apply a reflecting boundary condition which is compatible with the well balanced character is presented a mass preservation property is proved and some results on super characteristic relaxation are recalled numerical experiments with coarse computational grids are presented in detail they illustrate the bifurcation diagrams of guarguaglini et al which relate the total initial mass of cells with the time asymptotic values of the chemoattractant substance on each side of the domain it is shown that the wb scheme stabilizes correctly onto zero mass flow rate hence maxwellian steady states which agree with the aforementioned bifurcation diagrams the evolution in time of residues is commented for every considered test case

on the derivative of the legendre function of the first kind with respect to its degree we study the derivative of the legendre function of the first kind p z with respect to its degree at first we provide two contour integral representations for p z then we proceed to investigate the case of p z n with n being an integer this case is met in some physical and engineering problems since it holds that we focus on the sub case of n being a non negative integer we show that where rn z is a polynomial in z of degree n we present alternative derivations of several known explicit expressions for rn z and also add some new a generating function for rn z is also constructed properties of the polynomials vn z rn z 1 nrn z 2 and wn 1 z rn z 1 nrn z 2 are also investigated it is found that wn 1 z is the christoffel polynomial well known from the theory of the legendre function of the second kind qn z as examples of applications of the results obtained we present non standard derivations of some representations of qn z sum to closed forms some legendre series evaluate some definite integrals involving legendre polynomials and also derive an explicit representation of the indefinite integral of the legendre polynomial squared

fredholm solvability of time periodic boundary value hyperbolic problems abstract we investigate a large class of linear boundary value problems for the general first order one dimensional hyperbolic systems in the strip 0 1 r we state rather broad natural conditions on the data under which the operators of the problems satisfy the fredholm alternative in the spaces of continuous and time periodic functions a crucial ingredient of our analysis is a non resonance condition which is formulated in terms of the data responsible for the bijective part of the fredholm operator in the case of 2 2 systems with reflection boundary conditions we provide a criterium for the non resonant behavior of the system

coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations abstract in this paper coincidence and common fixed point results are established in a partially ordered cone metric space an application of our results obtained to prove the existence of a common solution to integral equations is presented

invariance principles for hyperbolic random walk systems abstract reaction random walk systems are hyperbolic models for the description of spatial motion in one dimension and reaction of particles in contrast to reaction diffusion equations particles have finite propagation speed for parabolic systems invariance results and maximum principles are well known a convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix and if the vector field defined by the pure reaction equation points inward at the boundary here we show a corresponding result for random walk systems the model parameters are the particle speeds the rates of change in direction and the reaction vector field a convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the speed matrix and if a vector field given by the reaction equation combined with the turning rates points inward finally a positivity result is shown

common fixed point results for noncommuting mappings without continuity in cone metric spaces the existence of coincidence points and common fixed points for mappings satisfying certain contractive conditions without appealing to continuity in a cone metric space is established these results generalize several well known comparable results in the literature

a branching random evolution and a nonlinear hyperbolic equation particles moving at constant speed along the real line and reversing direction as a poisson process constitute a random evolution if the particles simultaneously undergo an independent branching p

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

asymptotic high order mass preserving schemes for a hyperbolic model of chemotaxis we introduce a new class of finite difference schemes for approximating the solutions to an initial boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term which arises as a simple model of chemotaxis since the solutions to this problem may converge to nonconstant asymptotic states for large times standard schemes usually fail to yield a good approximation therefore we propose a new class of schemes which use an asymptotic higher order correction second and third order in our examples to balance the effects of the source term and the influence of the asymptotic solutions special care is needed to deal with boundary conditions to avoid harmful loss of mass convergence results are proved for these new schemes and several numerical tests are presented and discussed to verify the effectiveness of their behavior

on cone metric spaces a survey abstract using an old m krein s result and a result concerning symmetric spaces from s radenovic z kadelburg quasi contractions on symmetric and cone symmetric spaces banach j math anal 5 1 2011 38 50 we show in a very short way that all fixed point results in cone metric spaces obtained recently in which the assumption that the underlying cone is normal and solid is present can be reduced to the corresponding results in metric spaces on the other hand when we deal with non normal solid cones this is not possible in the recent paper m a khamsi remarks on cone metric spaces and fixed point theorems of contractive mappings fixed point theory appl 2010 7 pages article id 315398 doi 10 1115 2010 315398 the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant also that underlying banach space and the associated cone subset are not necessary in fact khamsi s approach includes a small class of results and is very limited since it requires only normal cones so that all results with non normal cones which are proper extensions of the corresponding results for metric spaces cannot be dealt with by his approach

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

endpoints and fixed points of set valued contractions in cone metric spaces abstract in this paper we introduce the concept of set valued contractions in cone metric spaces and establish new endpoint and fixed point theorems for such contractions the paper includes examples

the basin of attraction of the steady states for a chemotaxis model in mathbb r 2 with critical mass we consider the cauchy problem for a parabolic elliptic system in mathbb r 2 which is amathematical model of chemotaxis and also amodel of self attracting particles in the critical mass case we determine the basin of attraction of the steady states for the cauchy problem through a lyapunov functional

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

approximation of fixed points of weakly contractive nonself maps in banach spaces abstract let k be a closed convex subset of a real uniformly smooth banach space e suppose k is a nonexpansive retract of e with p as the nonexpansive retraction let t k e be a d weakly contractive map such that a fixed point x int k of t exists it is proved that a descent like approximation sequence converges strongly to x furthermore if k is a nonempty closed convex subset of an arbitrary real banach space and t k k is a uniformly continuous d weakly contractive map with f t x k tx x it is proved that a descent like approximation sequence converges strongly to x f t

interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics motivated by observations of the dynamics of myxococcus xanthus we present a self interacting random walk model that describes the competition between chemokinesis and chemotaxis cells are constrained to move in one dimension but release a chemical chemoattractant at a steady state the bacteria sense the chemical that they produce the probability of direction reversals is modeled as a function of both the absolute level of chemoattractant sensed directly under each cell as well as the gradient sensed across the length of the cell if the chemical does not degrade or diffuse rapidly the one dimensional trajectory depends on the entire past history of the trajectory we derive the corresponding fokker planck equations use an iterative mean field approach that we solve numerically for short times and perform extensive monte carlo simulations of the model cell positional distributions and the associated moments are computed in this feedback system average drift and mean squared displacements are found crossover behaviors among different diffusion regimes are found

a note on parameter derivatives of the jacobi polynomials on the triangle the aim of the present paper is to obtain derivatives of the jacobi polynomials with two variables on the triangle with respect to their parameters by using the parameter derivatives of the classical jacobi polynomials p n x furthermore we give orthogonality properties of the parametric derivatives on the triangle

addendum to on the derivative of the legendre function of the first kind with respect to its degree in our recent paper szmytkowski 2006 j phys a math gen 39 15147 2007 j phys a math theor 40 7819 corrigendum we have investigated the derivative of the legendre function of the first kind p z with respect to its degree in this addendum we derive some further representations of p z n with the results are used to obtain in an elementary way two formulae for the legendre function of the second kind of integer degree qn z

asymptotic behavior of equilibrium states of reaction diffusion systems with mass conservation abstract we deal with a stationary problem of a reaction diffusion system with a conservation law under the neumann boundary condition it is shown that the stationary problem turns to be the euler lagrange equation of an energy functional with a mass constraint when the domain is the finite interval 0 1 we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d the ratio of the diffusion coefficient of the system tends to zero in view of a logarithmic function in the leading term of the potential we get to a scaling parameter satisfying the relation e d log 2 the main result shows that a sequence of minimizers converges to a dirac mass multiplied by the total mass and that by a scaling with the asymptotic profile exhibits a parabola in the nonvanishing region we also prove the existence of an unstable monotone solution when the mass is small

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

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a patlak keller segel model this paper is concerned with a chemotaxis aggregation model for cells more precisely with a parabolic elliptic semilinear patlak keller segel system in a ball of rn exhibiting a critical mass phenomenon for n 2 the main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case we stress that this covers in particular the classical keller segel system with n 2 well known for its critical mass 8 and that the result improves on the known results even for this most studied problem a key tool is an associated one dimensional degenerate parabolic problem the proof exploits its formal gradient flow structure ut f u t on an infinite dimensional riemannian manifold in particular we show a new hardy type inequality equivalent to the strict convexity of f at any steady state of subcritical mass which heuristically explains the exponential speed of convergence

fixed point theorems for generalized weakly contractive mappings we prove a fixed point theorem for a generalized weakly contractive mapping and a fixed point theorem for a pair of weakly contractive mappings we also show that these mappings satisfy properties p and q

the point of coincidence and common fixed point for a pair of mappings in cone metric spaces in this paper the existence of a point of coincidence and a common fixed point for two weakly compatible maps on a cone metric space has been established the two mappings are assumed to satisfy certain weak inequalities supporting examples are also given

on parameter derivatives of a family of polynomials in two variables the purpose of the present paper is to give the parameter derivative representations of the form p n k x y m 0 n 1 j 0 m d n j m p m j x y j 0 k e n j k p n j x y for a family of orthogonal polynomials of variables x and y with being a parameter and 0 k n n k 0 1 2 first we shall present the representations of the parameter derivatives of the generalized gegenbauer polynomials c n x with the help of the parameter derivatives of the classical jacobi polynomials p n x i e p n x and p n x then by using these derivatives we investigate the parameter derivatives for two variable analogues of the generalized gegenbauer polynomials furthermore we discuss orthogonality properties of the parametric derivatives of these polynomials

on parameter derivatives of the associated legendre function of the first kind with applications to the construction of the associated legendre function of the second kind of integer degree and order abstract a relationship between partial derivatives of the associated legendre function of the first kind with respect to its degree p m z n and to its order p n z m is established for m n n 0 this relationship is used to deduce four new closed form representations of p m z n from those found recently for p n z m by the author r szmytkowski on the derivative of the associated legendre function of the first kind of integer degree with respect to its order with applications to the construction of the associated legendre function of the second kind of integer degree and order j math chem 46 2009 231 several new expressions for the associated legendre function of the second kind of integer degree and order q n m z suitable for numerical purposes are also derived