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

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

assessing priority weights from subsets of pairwise comparisons in multiple criteria optimization problems using a set covering concept we explore the possibility and flexibility of selecting a subset of all possible pairs of comparisons to assess priority weights of the decision criteria different techniques offer different priority weights with different properties they are however interrelated when special subsets such as basic coverings elementary cycles and complete covering are considered this paper provides a framework for unifying existing assessment methods and offers some fundamental theorems which reveal the intricate relationships among the methods over the subsets

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

error bounds for finite difference approximations for a class of nonlinear parabolic systems in this paper we establish error bounds for a finite difference approximation to solutions of certain parabolic systems of the form t f po u we assume that the cauchy data is of class bv and we show that the sup norm of the error is bounded by o axiln axi at positive times

local stability intervals in the analytic hierarchy process abstract this paper provides a sensitivity analysis of the judgements used in the analytic hierarchy process ahp in relation to the rank reversal produced in two different situations the selection of the best alternative p problem and the ranking of all the alternatives p problem in both cases under the supposition that we employ the row geometric mean method rgmm to determine the local priorities we obtain in the case of a single criterion a local stability interval for each judgement for each alternative and for the paired comparisons matrix which allow us to guarantee the best alternative and the ranking of all of them the p and p problems respectively with respect to these three situations judgement alternative and matrix and two problems p and p we also calculate a local stability index to detect the critical values of the resolution process both the local stability intervals and indexes are used as management tools in the final stage of the decision making process that is to say the exploitation phase especially in the negotiation process and the search for consensus between the actors

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

stability results for a diffusion equation with functional drift approximating a chemotaxis model a hyperbolic parabolic chemotaxis system modelling aggre gation of motile cells by production of a diffusible chemoattractant is ap proximated by a scalar diffusion equation for the cell density where the drift term is an explicit functional of the current density profile we prove the unique existence and using the hopf cole transformation the local stability of an equilibrium i e a steady aggregation state we also discuss the limit ing hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps 1 introduction and development of the model in this note we shall de velop and explore the behavior of a simple two speed one dimensional model which describes the motion of chemosensitive cells that is cells whose motions are influ enced by a chemical gradient dp dx where p p x t denotes the concentration of a diffusible substance produced by each cell a well known example is the mor phogenetic aggregation of slime mold ceils 5 which previously has been modelled by keller and segel 6 using a coupled system of two equations for p and the cell density n u x t the so called chemotaxis system for a recent mathematical analysis see 4 12 in most cases as in slime molds however parameter estimations reveal that the diffusion coefficient of the chemoattractant dp 10 7cm2 sec is about ten times larger than the motility of the cells du 10 8 cm2 sec compare for example 3 if also the production rate of the chemoattractant is relatively large then the quasi stationary gradient of the rapidly diffusing substance approx imately determines the chemotactic drift of the cells this finally leads to a scalar diffusion equation for the cell density u of type

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

well posedness of first order semilinear pdes by stochastic perturbation abstract we show that first order semilinear pdes by stochastic perturbation are well posed for globally holder continuous and bounded vector field with an integrability condition on the divergence this result extends the linear case presented in flandoli et al 2010 the proof is based on the stochastic characteristics method and a version of the commuting lemma

generalized integration and stochastic odes stochastic forward integrals for processes more general than semimartingales are shown to exist generalized forms of ito wentzell formula and covariation formula are proved and one dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved this generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

convergence results for an inhomogeneous system arising in various high frequency approximations this paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of hamilton jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the schrodinger equation we first state an existence and uniqueness result in the framework of viscosity and duality solutions then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence as an illustration some practical computations are provided

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

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 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

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves we consider a class of kinetic equations equipped with a single conservation law which generate l 1 contractions we discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a possibly degenerate parabolic equation the limits are obtained in the dissipative sense equivalent to the notion of entropy solutions for conservation laws which permits the use of the perturbed test function method and allows for simple proofs a general compactness framework is obtained for the diffusive scaling in l 1 the radiative transport equations satisfied by the wigner function for random acoustic waves present such a kinetic model that is endowed with conservation of energy the general theory is used to validate the diffusive approximation of the radiative transport equation

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

the density of the solution to the stochastic transport equation with fractional noise we consider the transport equation driven by the fractional brownian motion we study the existence and the uniqueness of the weak solution and by using the tools of the malliavin calculus we prove the existence of the density of the solution and we give gaussian estimates from above and from below for this density

estimating criterion weights using eigenvectors a comparative study abstract in addition to saaty s eigenvector there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios as the judgment of the ratios are dependent on personal experience learning situations and state of mind inconsistencies and degree of easiness or judgment on these individual ratios can be different we study the properties of the different eigen weight vectors including that of saaty and that recently proposed by cogger and yu a general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed

an application of incomplete pairwise comparison matrices for ranking top tennis players pairwise comparison is an important tool in multi attribute decision making pairwise comparison matrices pcm have been applied for ranking criteria and for scoring alternatives according to a given criterion our paper presents a special application of incomplete pcms ranking of professional tennis players based on their results against each other the selected 25 players have been on the top of the atp rankings for a shorter or longer period in the last 40 years some of them have never met on the court one of the aims of the paper is to provide ranking of the selected players however the analysis of incomplete pairwise comparison matrices is also in the focus the eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete pcms in our results the top three players of four decades were nadal federer and sampras some questions have been raised on the properties of incomplete pcms and remains open for further investigation

on the existence of a travelling wave solution for a model of actin based bacterial movement some bacteria like listeria monocytogenes shigella and rickettsia rickettsii can move inside the host cell thanks to an actin tail in this paper a one dimensional model for the motion of these bacteria introduced by b bazaliy y bazaliy and a friedman 2007 in 2 is studied in particular the model is a system of partial differential equation with two moving boundaries for which we prove the existence of a travelling wave solution

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

global existence of classical solutions to a combined chemotaxis haptotaxis model with logistic source abstract this paper deals with a mathematical model of cancer invasion of tissue the model consists of a system of reaction diffusion taxis partial differential equations describing interactions between cancer cells matrix degrading enzymes and the host tissue in two space dimensions we prove global existence and uniqueness of classical solutions to this model for any 0 where is the logistic growth rate of cancer cells the crucial point of proof is to raise the regularity estimate of a solution from l 1 to l 3 0 t where r 2 is some bounded domain and t 0 is some constant this paper develops new estimate techniques and improves greatly our previous results y tao m wang global solution for a chemotactic haptotactic model of cancer invasion nonlinearity 21 2008 2221 2238 in 2 dimensions

integration questions related to fractional brownian motion let b h u u be a fractional brownian motion fbm with index h 0 1 and b h be the closure in l 2 of the span sp b h of the increments of fbm b h it is well known that when b h b 1 2 is the usual brownian motion bm an element x b 1 2 can be characterized by a unique function f x l 2 in which case one writes x in an integral form as x f x u db 1 2 u from a different though equivalent perspective the space l 2 forms a class of integrands for the integral on the real line with respect to bm b 1 2 in this work we explore whether a similar characterization of elements of b h can be obtained when h 0 1 2 or h 1 2 1 since it is natural to define the integral of an elementary function f k 1 n f k 1 uk uk 1 by k 1 n f k b h u k 1 b h u k we want the spaces of integrands to contain elementary functions these classes of integrands are inner product spaces if the space of integrands is not complete then it characterizes only a strict subset of b h when 0 h 1 2 by using the moving average representation of fbm b h we construct a complete space of integrands when 1 2 h 1 however an analogous construction leads to a space of integrands which is not complete when 0 h 1 2 or 1 2 h 1 we also consider a number of other spaces of integrands while smaller and henceincomplete they form a natural choice and are convenient to workwith we compare these spaces of integrands to the reproducing kernel hilbert space of fbm