algorithms for checking rational roots of b functions and their applications abstract the bernstein sato polynomial of a hypersurface is an important object with many applications however its computation is hard as a number of open questions and challenges indicate in this paper we propose a family of algorithms called checkroot for optimized checking whether a given rational number is a root of bernstein sato polynomial and in the affirmative case computing its multiplicity these algorithms are used in the new approach to compute the global or local bernstein sato polynomial and b function of a holonomic ideal with respect to a weight vector they can be applied in numerous situations where a multiple of the bernstein sato polynomial can be established namely a multiple can be obtained by means of embedded resolution for topologically equivalent singularities or using the formula of a campo and spectral numbers we also present approaches to the logarithmic comparison problem and the intersection homology d module several applications are presented as well as solutions to some challenges which were intractable with the classical methods one of the main applications is the computation of a stratification of affine space with the local b function being constant on each stratum notably the algorithm we propose does not employ primary decomposition our results can be also applied for the computation of bernstein sato polynomials for varieties the examples in the paper have been computed with our implementation of the methods described in singular plural

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 weak solutions of perturbated systems with critical growth sur un domaine borne de r n on considere le probleme de dirichlet pour le systeme elliptique a u g u f sur et u 0 ou a u est un operateur quasi lineaire et la perturbation g u depend de u et de d u on etudie des conditions suffisantes sur a u et sur g u pour l existence de solutions faibles

time reversal in stochastic processes and the dirac equation we consider the motion of a classical particle in 1 1 dimensional space time four probability distributions govern the trajectory of the particle these give the probability of moving to the left or right in space while moving backwards or forwards in time if these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner then the master equations for these probabilities give rise to the dirac equation without recourse to direct analytic continuation in contrast when a particle always moves forward in time an analytic continuation is required to recover the dirac equation

on b function spectrum and rational singularity by duality 18 because r tr ccy 0 for i 0 by 6 31 this follows also from 13 21 23 where rc is assumed projective here coy denotes the dualizing sheaf i e the dualizing complex 18 shifted by the dimension to the right the trace morphism 0 2 is injective and its image is independent of the choice of resolution because 0 2 is an isomorphism if y is smooth we will denote by csy the image of 0 2 see 2 4 8 below now assume y is a reduced divisor d on a complex manifold x of dimension n let f be a reduced defining equation of d on a neighborhood of a c d the b function i e bernstein polynomial of f is a monic polynomial bf s in s with rational coefficients and is a generator of the ideal whose element b s satisfies the relation

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

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

jump control of probability densities with applications to autonomous vehicle motion we investigate the problem of controlling the probability density of the state of a process that is observed by the controller via a fixed but unknown scalar non negative function of the state the goal is to control the process so that its probability density at a point in the state space becomes proportional to the value of the function observed at that point our solution inspired by bacterial chemotaxis involves a randomized controller that switches among different deterministic modes we show that under appropriate existence conditions this controller guarantees convergence of the probability density to the desired function the results can be applied to the problem of in loco optimization of a measurable signal using a team of autonomous vehicles that measure the signal but do not have access to position measurements alternative applications in the area of mobile robotics include deployment and environmental monitoring

quasilinear elliptic problems with singular and homogeneous lower order terms abstract we deal with singular quasilinear elliptic equations namely u u x u q u q 1 f x in u 0 in u 0 on where is a bounded smooth domain of r n n 3 r 1 q 2 0 l and 0 f l p for some p n 2 we completely describe the set of values of the parameter for which the problem admits solution thus we study existence nonexistence and uniqueness of bounded weak solutions in both cases f 0 and f 0

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

solvability of perturbed elliptic equations with critical growth exponent for the gradient sur un domaine borne cr n on considere le probleme de dirichlet a u g u f u 0 ou a u t 1 n 1 d i a i x u d u definit un operateur pseudomonotone d un espace de sobolev h 0 1 p dans son espace dual et la perturbation g u g x u d u satisfait une condition de croissance

existence results for some quasilinear parabolic equations on considere le probleme de cauchy pour une equation dont la partie principale est un operateur parabolique quasilineaire le terme non lineaire additionnel a au plus une croissance quadratique par rapport au gradient de la solution on demontre l existence d une solution bornee faible sous des hypotheses de regularite minimales en supposant l existence d une sous solution et d une supersolution pour le probleme

kinetics of self induced aggregation in brownian particles we study a model of interacting random walkers that proposes a simple mechanism for the emergence of cooperation in a group of individuals each individual represented by a brownian particle experiences an interaction produced by the local unbalance in the spatial distribution of the other individuals this interaction results in a nonlinear velocity driving the particle trajectories in the direction of the nearest more crowded region the competition among different aggregating centers generates nontrivial dynamical regimes our simulations show that for sufficiently low randomness the system evolves through a coalescence behavior characterized by clusters of particles growing with a power law in time in addition the typical scaling properties of the general theory of stochastic aggregation processes are verified

adaptation and optimal chemotactic strategy for e coli extending the classic works of berg and purcell on the biophysics of bacterial chemotaxis we find the optimal chemotactic strategy for the peritrichous bacterium ital e coli in the high and low signal to noise ratio limits the optimal strategy depends on properties of the environment and properties of the individual bacterium and is therefore highly adaptive we review experiments relevant to testing both the form of the proposed strategy and its adaptability and propose extensions of them which could test the limits of the adaptability in this simplest sensory processing system copyright ital 1998 ital the american physical society

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

macroscopic model of self propelled bacteria swarming with regular reversals periodic reversals in the direction of motion in systems of self propelled rod shaped bacteria enable them to effectively resolve traffic jams formed during swarming and maximize the swarming rate of the colony in this paper a connection is established between a microscopic one dimensional cell based stochastic model of reversing nonoverlapping bacteria and a macroscopic nonlinear diffusion equation describing the dynamics of cellular density boltzmann matano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency stochastic dynamics averaged over an ensemble is shown to be in very good agreement with the numerical solutions of this nonlinear diffusion equation critical density p0 is obtained such that nonlinear diffusion is dominated by the collisions between cells for the densities p p0 an analytical approximation of the pairwise collision time and semianalytical fit for the total jam time per reversal period are also obtained it is shown that cell populations with high reversal frequencies are able to spread out effectively at high densities if the cells rarely reverse then they are able to spread out at lower densities but are less efficient at spreading out at higher densities

on steenbrink s conjecture let f x c be a holomorphic function on a complex manifold using the monodromy and mixed hodge structure on the cohomology of milnor fiber we can associate the spectrum sp f x of f at x e xo f 1 0 it is a fractional laurent polynomial n t with n z this notion is first introduced by steenbrink 15 cteq in the isolated singularity case cf also 19 and 16 in general if f has isolated singularity n are nonnegative and 0 n 0 counted with multiplicity are called the exponents or spectra 19 of f assume f has one dimensional singular locus and consider a deformation of the form f g where g is a linear form i e a function such that g x 0 and dg 0 at x then f g has isolated singularity if g is generic let zk be a local irreducible component of singf at x and mk its multiplicity along z zk x f is viewed as constant deformation of isolated singularity let k j be its exponents which are constant on z 20 we can also define ilk j e 0 1 in a compatible way with k j so that exp 2ni 3k j are the eigenvalues of the monodromy along z then steenbrink conjectured

multiplier ideals v filtrations and transversal sections we show that the restriction to a smooth transversal section commutes to the computation of multiplier ideals and v filtrations as an application we prove the constancy of the jumping numbers and the spectrum along any stratum of a whitney regular stratification

existence and non existence of positive solutions for nonlinear elliptic singular equations with natural growth abstract in this paper we analyze the existence non existence and uniqueness of positive solutions of some nonlinear elliptic equations containing singular terms and natural growth in the gradient term we use an adequate sub super solution method to prove the existence of solutions the characterization of eigenvalues and the integrability of the term u 2 u 2 for the non existence and results from arcoya segura de leon for the uniqueness

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

regularizing effect for some p laplacian systems abstract we study existence and regularity of weak solutions for the following p laplacian system p u a 1 u r 2 u f u w 0 1 p p u r w 0 1 p where is an open bounded subset of r n n 2 p v div v p 2 v is the p laplacian operator for 1 p n a 0 r 1 0 p 1 and f belongs to a suitable lebesgue space in particular we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions

strongly nonlinear parabolic equations with natural growth terms in orlicz spaces abstract we prove approximation and compactness results in inhomogeneous orlicz sobolev spaces and look at as an application the cauchy dirichlet equation u a u g x t u u f w 1 x e m where a is a leray lions operator having a growth not necessarily of polynomial type we also give a trace result allowing to deduce the continuity of the solutions with respect to time

residues and filtered d modules for an embedding of sufficiently high degree of a smooth projective variety x into projective space we use residues to define a filtered holonomic mathcal d module mathcal m f on the dual projective space this gives a concrete description of the intermediate extension to a hodge module on p of the variation of hodge structure on the middle dimensional cohomology of the hyperplane sections of x we also establish many results about the sheaves f k mathcal m such as positivity vanishing theorems and reflexivity

asymptotic behaviour of linear dirichlet parabolic problems with variable operators depending on time in varying domains abstract we study the asymptotic behaviour of the solutions of linear parabolic dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously in the limit problem it appear the h limit of the operators and as it is usual in the homogenization of dirichlet problems a new term of order zero we also obtain a corrector result

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

spectral hirzebruch milnor classes of singular hypersurfaces we introduce spectral hirzebruch milnor classes for singular hypersurfaces these can be identified with steenbrink spectra in the isolated singularity case and may be viewed as their global analogues in general their definition uses vanishing cycles of mixed hodge modules and the todd class transformation these are compatible with the pushforward by proper morphisms and the classes can be calculated by using resolutions of singularities formulas for hirzebruch milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1 these formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases here we use the thom sebastiani theorem for the underlying filtered d modules of vanishing cycles from which we can deduce the thom sebastiani type theorem for spectral hirzebruch milnor classes for the chern classes after specializing to y 1 we can give a relatively simple formula for the localized milnor classes which implies a new formula for the euler numbers of projective hypersurfaces using iterated hyperplane sections applications to log canonical thresholds and du bois singularities are also explained for instance the latter can be detected by using hirzebruch milnor classes in the projective hypersurface case

variational inequalities in orlicz sobolev spaces on etudie l existence de solutions d inegalites variationnelles associees a des operateurs differentiels elliptiques

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

existence results for some nonlinear elliptic system with lack of compactness abstract we are interested in existence of solutions for some typical elliptic systems governed by two pseudo laplacian operators with critical sobolev exponents we show this result by a local minimization for an adapted variational problem a concentration compactness principle allows us to obtain some convergence results

effects of cross diffusion on turing bifurcations in two species reaction transport systems we study the turing bifurcation in general two species reaction transport systems where particle dispersal is governed by diffusion including cross diffusion by performing a linear stability analysis we find conditions for the turing instability and compare the results with the standard turing conditions we apply our results to two model systems the lengyel epstein model and the brusselator and find strong effects of cross diffusion on the turing bifurcation

non autonomous perturbations for a class of quasilinear elliptic equations on r abstract this paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on r involving the p laplacian with a non autonomous perturbation the first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain pass argument the special features of the problem here is the complex structure of the linear part which in particular oblige to work into the space w 1 p r then one faces problems in the convergence of the sequences of derivatives u n u

on some class of nonlinear parabolic inequalities in orlicz spaces abstract the paper deals with the existence of solutions of some unilateral problems in the framework of orlicz spaces

rate laws of the self induced aggregation kinetics of brownian particles abstract in this paper we have studied the self induced aggregation kinetics of brownian particles in the presence of both multiplicative and additive noises in addition to the drift due to the self aggregation process the environment may induce a drift term in the presence of a multiplicative noise then there would be an interplay between the two drift terms it may account qualitatively the appearance of the different laws of aggregation process at low strength of white multiplicative noise the cluster number decreases as a gaussian function of time if the noise strength becomes appreciably large then the variation of cluster number with time is fitted well by the mono exponentially decaying function of time for additive noise driven case the decrease of cluster number can be described by the power law but in case of multiplicative colored driven process cluster number decays multi exponentially however we have explored how the rate constant in the mono exponentially cluster number decaying case depends on strength of interference of the noises and their intensity we have also explored how the structure factor at long time depends on the strength of the cross correlation cc between the additive and the multiplicative noises

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

randomly curved runs interrupted by tumbling a model for bacterial motion received 23 february 2005 published 24 august 2005small bacteria are strongly buffeted by brownian forces that make completely straight runs impossible amodel for bacterial motion is formulated in which the effects of uctuational forces and torques on the runphase are taken into account by using coupled langevin equations an integrated description of the motion including runs and tumbles is then obtained by the use of convolution and laplace transforms the propertiesof the velocity velocity correlation function of the mean displacement and of the two relevant diffusioncoef cients are examined in terms of the bacterial sizes and of the magnitude of the propelling forces forbacteria smaller than

optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n p 1 0 and f 0 u 0 0 are in a suitable class of functions there are deep differences with respect to the heat equation 0 the main features in the paper are the following if 0 there exists a critical exponent p such that for p p there is no nontrivial local solution p is optimal in the sense that if p p there exists solution for suitable data if we consider the cauchy problem i e r n we find the same phenomenon about the critical power p as above moreover there exists a fujita type exponent f p in the sense that independently of the initial datum for 1 p f any solution blows up in a finite time respect to an integral norm this is a major difference with respect to the heat equation 0

a cellular automata model of flow in ant trails non monotonic variation of speed with density generically in models of driven interacting particles the average speed of the particles decreases monotonically with increasing density we propose a counterexample motivated by the motion of ants in a trail where the average speed of the particles varies non monotonically with their density because of the coupling of their dynamics with another dynamical variable these results in principle can be tested experimentally

the p dirichlet to neumann operator with applications to elliptic and parabolic problems abstract in this paper we investigate the dirichlet to neumann operator associated with second order quasi linear operators of p laplace type for 1 p which acts on the boundary of a bounded lipschitz domain in r d for d 2 we establish well posedness and holder continuity with uniform estimates of weak solutions of some elliptic boundary value problems involving the dirichlet to neumann operator by employing these regularity results of weak solutions of elliptic problems we show that the semigroup generated by the negative dirichlet to neumann operator on l q enjoys an l q c 0 smoothing effect and the negative dirichlet to neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order preserving semigroup moreover we establish convergence in large time with decay rates of all trajectories of the semigroup and in the singular case 1 e 2 d d 2 p 2 for some e 0 we give upper estimates of the finite time of extinction

on hodge spectrum and multiplier ideals we describe a relation between two invariants which measure the complexity of a hypersurface singularity one is the hodge spectrum which is related to the monodromy and the hodge filtration on the cohomology of the milnor fiber the other is the multiplier ideal having to do with log resolutions

individual based and mean field modeling of direct aggregation we introduce two models of biological aggregation based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood in the first order model the location of each individual is subject to a density dependent random walk while in the second order model the density dependent random walk acts on the velocity variable together with a density dependent damping term the main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals instead aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density we formally derive the corresponding mean field limits leading to nonlocal degenerate diffusions then we carry out the mathematical analysis of the first order model in particular we prove the existence of weak solutions and show that it allows for measure valued steady states we also perform linear stability analysis and identify conditions for pattern formation moreover we discuss the role of the nonlocality for well posedness of the first order model finally we present results of numerical simulations for both the first and second order model on the individual based and continuum levels of description