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 coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

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

some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

periodicity in bilinear lattices and the coxeter formalism abstract we introduce and study in detail so called circulant coxeter periodic elements and circulant families in a bilinear lattice k as well as their dual versions called anti circulant we show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the coxeter polynomial k of k and in consequence of coxeter polynomials of algebras of finite global dimension we discuss the properties of quadratic forms induced by circulant and anti circulant families moreover we interpret the results in the language of representation theory of algebras and point out applications facts concerning tubular families in auslander reiten quivers and quadratic forms of algebras abstract considerations in bilinear lattices are illustrated with a collection of non trivial examples arising from module and derived categories the results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts

representations for the first associated q classical orthogonal polynomials let pk x q be any system of the q classical orthogonal polynomials and let be the corresponding weight function satisfying the q difference equation dq where and are polynomials of degree at most 2 and exactly 1 respectively further let p 1 k x q be associated polynomials of the polynomials pk x q explicit forms of the coefficients bn k and cn k in the expansions p 1 n 1 x q k 0n 1bn k k x p 1 n 1 x q k 0n 1cn kpk x q are given in terms of basic hypergeometric functions here k x equals xk if 0 0 or x q k if 1 0 where x x q 1 x x the most important representatives of those two classes are the families of litte q jacobi and big q jacobi polynomials respectively writing the second order nonhomogeneous q difference equation satisfied by p 1 n 1 x q in a special form recurrence relations in k for bn k and cn k are obtained in terms of and

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x s q 2 abstract we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable on a non uniform lattice x s q2s i e the q analogues of charlier meixner krawtchouk and hahn polynomials in terms of the coefficients and of the pearson equation satisfied by the weight function and the coefficients of the three term recurrence relation and of two structure relations obeyed by these polynomials

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable i e associated with the names of charlier meixner krawtchouk and hahn also a recurrence relation is given for the coefficients in the formula expressing the nth associated polynomial in terms of the original polynomials

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a

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

on the coefficients of the coxeter polynomial of an accessible algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is basic connected with n pairwise non isomorphic simple modules we consider the coxeter polynomial a t of a one point extension algebra a b m and the polynomial of the extension p t 1 t 1 t b t a t if m is exceptional then p t 1 p 1 t p n 3 t n 3 t n 2 in that case we call s a b p 1 the linear index of the extension a b m we give conditions for s a b 0 for a tower t k a 1 a 2 a n a of access to a that is a i is a one point co extension of a i 1 by an exceptional module the index s t i 2 n s a i a i 1 n 1 a 2 is an invariant depending on the derived equivalence class of a where a 2 is the quadratic coefficient of a t we show that in the case a is piecewise hereditary then a 2 1 if and only if a is derived equivalent to a quiver algebra of type a n

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

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

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

construction of recurrences for the coefficients of expansions in q classical orthogonal polynomials an algorithmic approach is given to construct recurrence relations for the coefficients of the series expansion of a solution of a linear q difference equation in q classical orthogonal polynomials in the hahn s sense

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

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

upwinding of the source term at interfaces for euler equations with high friction we consider euler equations with a friction term that describe an isentropic gas flow in a porous domain more precisely we consider the transition between low and high friction regions in the high friction region the system is reduced to a parabolic equation the porous media equation in this paper we present a hyperbolic approach based on a finite volume technique to compute numerical solutions for the system in both regimes the upwind source at interfaces usi scheme that we propose satisfies the following properties firstly it preserves the nonnegativity of gas density secondly and this is the motivation the scheme is asymptotically consistent with the limit model porous media equation when the friction coefficient goes to infinity we show analytically and through numerical results that the above properties are satisfied we shall also compare results given with the use of usi hyperbolic parabolic coupling and classical centered sources schemes

wild canonical algebras canonical algebras p depending on a weight sequence p p1 pt of positive integers and a parameter sequence 3 t of pairwise distinct non zero elements from the base eld k were introduced and studied in 23 the nite dimensional representation theory mod is completely controlled by the category coh x of coherent sheaves on a non singular weighted projective line x x p since the derived categories d mod and d coh x are equivalent as triangulated categories 9 the curve x attached to has virtual genus gx 1 2 t 2 p t i 1 p pi where p l c m p1 pt the complexity of the classi cation problem for coh x and hence for mod is essentially determined by gx for gxi1 the algebra is concealed of extended dynkin type accordingly the classi cation problem for coh x and mod is equivalent to the classi cation of indecomposable modules over a tame hereditary algebra cf 18 9 or according to 10 to the classi cation problem for cohen macaulay modules over a simple surface singularity for gx 1 the algebra is of tubular type its representation theory is known from 23 while the classi cation problem for coh x 9 20 relates to atiyah s classi cation 1 of vector bundles over an elliptic curve very little was known on mod or coh x if gx 1 in this case 0 r is the one point extension of a wild hereditary algebra 0 accordingly the category mod 0 embeds into coh x as a full exact and extension closed subcategory implying that the classi cation problems for coh x and mod are wild from now on we assume gx 1 combining the sheaf theory 9 with the spectral investigation of wild hereditary algebras 6 we obtain a fairly complete picture on the structure of auslander reiten components for mod the asymptotic behaviour of the auslander reiten translations and

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

godunov type approximation for a general resonant balance law with large data abstract we consider the cauchy problem for the 2 2 nonstrictly hyperbolic system a t 0 u t f a u x g a u a x 0 a u t 0 a 0 u 0 for possibly large discontinuous and resonant data the generalized solution to the riemann problem is introduced interaction estimates are carried out using an original change of variables and the convergence of godunov approximations is shown uniqueness is addressed relying on a suitable extension of kru kov s techniques

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

the trace of the coxeter matrix and hochschild cohomology abstract for a basic finite dimensional k algebra of finite global dimension over an algebraically closed field k the coxeter matrix is defined the following formula is shown trm i 1 i dim k h i where tr is the trace and hi is the ith hochschild cohomology space

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

coxeter transformations of the derived categories of coherent sheaves abstract in this paper we study the coxeter transformations of the derived categories of coherent sheaves on smooth complete varieties we first obtain that if the rank of the grothendieck group is finite say m then its characteristic polynomials is 1 n m where n is dimension of the variety we then study the jordan canonical forms of the coxeter transformations for rational surfaces smooth complete toric varieties with ample canonical or anticanonical bundles and prove that their jordan canonical forms can determine and be determined by the betti numbers of these varieties as an application we compute the jordan canonical forms of tensor products of matrices

root systems and inflations of non negative quasi cartan matrices abstract cartan matrices quasi cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like lie theory representation theory and algebraic graph theory we study quasi cartan matrices by means of the inflation algorithm an idea used in ovsienko s classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results we prove that it is enough to perform linear number of inflations to reduce positive or non negative principal quasi cartan matrix to its canonical form i e to the cartan matrix of a dynkin or euclidean diagram respectively moreover we show that in the positive case the length of every sequence of inflations has quadratic bound and that the only other natural non negative class having such universal bound is the class of so called pos sincerely principal quasi cartan matrices and in this case the bound is linear we obtain such low bounds by applying among others some new observations on the properties of the reduced root systems in the sense of bourbaki associated with positive quasi cartan matrices

recursion formulae for basic hypergeometric functions abstract we show that the basic hypergeometric functions f k i 1 r a i q k q q k j 1 s b j q k k 1 k q k 2 s 1 r r 1 s 1 a 1 q k a r q k q k 1 b 1 q k b s q k q 2k 2 q q k s 1 r satisfy a recurrence relation of the form i 0 a i k 1 b i k f k i 0 max r 1 s 2 where a i k b i k are rational functions of qk and b0 k b k 0 when r s 1 and q this result can be refined namely we show that the functions f k q i 1 s 1 a i q k q q k j 1 s b j q k q k s 2 s 1 a 1 q k a s 1 q k q k 1 b 1 q k b s q k q 2k 2 q q satisfy a recurrence relation of order s 1 i 0 s 1 c i k f k i q 0 with rational coefficients in qk

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

accumulation points of real schur roots abstract let k be an algebraically closed field and q be an acyclic quiver with n vertices consider the category rep q of finite dimensional representations of q over k the exceptional representations of q that is the indecomposable objects of rep q without self extensions correspond to the so called real schur roots of the usual root system attached to q these roots are special elements of the grothendieck group z n of rep q when we identify the dimension vectors of the representations that is the non negative vectors of z n up to positive multiple we see that the real schur roots can accumulate in some directions of r n z n this paper is devoted to the study of these accumulation points after giving new properties of the canonical decomposition of dimension vectors we show how to use this decomposition to describe the rational accumulation points finally we study the irrational accumulation points and we give a complete description of them in case q is of weakly hyperbolic type

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric q polynomials

a chebysheff recursion formula for coxeter polynomials abstract a polynomial f t z t is represented by q t z t if f t 2 q t t deg q q t t 1 f t is graphically represented if f t 2 m t for m t the characteristic polynomial of a symmetric matrix m many instances of coxeter polynomials f a t for a a finite dimensional algebra are graphically representable we study the case of extended canonical algebras a see h lenzing j a de la pena extended canonical algebras and fuchsian singularities in press show that the corresponding polynomials f a t are representable and satisfy a chebysheff type recursion formula we get consequences for the eigenvalues of the coxeter transformation of a showing for instance that at most four eigenvalues may lie outside the unit circle

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

bounds related to coxeter spectral measures of graphs abstract let m be a real square matrix we give upper bounds for the sum of the absolute values of the real part of the eigenvalues of m quantity in some particular cases known as real energy of m from these results we obtain a combinatorial bound for the real energy of the coxeter matrix q of a tree digraph q with n vertices e r e q min n 2 a b c d 2 where a is the number of edges b and c are respectively the number of bifurcation and congregation paths of q as defined below d i 1 n i 1 i 2 with i the degree of a vertex i and where the minimum is taken over all possible orientations of edges in q as particular case we consider dynkin euclidean and star graphs obtaining practical bounds for the real coxeter energy of these classes of digraphs

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

methods to determine coxeter polynomials abstract a reduction formula for the characteristic polynomials of the coxeter matrices of certain split finite dimensional algebras is proved in the hereditary case this yields explicit expressions for the coxeter polynomials of large families of quivers moreover a combinatorial interpretation of the entries of the coxeter matrices of path algebras gives formulas for coxeter polynomials of some quivers which cannot be treated by the above mentioned reduction process

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

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 the coxeter polynomials of wild stars abstract the spectral radius of a coxeter transformation which plays an important role in the representation theory of hereditary algebras see v dlab c m ringel eigenvalues of coxeter transformations and the gelfand kirillov dimension of the preprojective algebras proc ams 83 1990 228 232 is its important invariant this paper provides both upper and lower bounds for the spectral radii of the coxeter transformations of wild stars i e trees that have a single branching point and are neither of dynkin nor of euclidean type in addition the paper determines limit of the spectral radii of particular infinite sequences of wild stars and shows different classes of graphs with the same limit the basic idea is to reduce the study of spectral radii of trees to the spectral radii of particular valued graphs with indefinite type of associated generalized cartan matrix

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

numerical high field limits in two stream kinetic models and 1d aggregation equations numerical resolution of two stream kinetic models in a strong aggregative setting is considered to illustrate our approach we consider a one dimensional kinetic model for chemotaxis in hydrodynamic scaling and the high field limit of the vlasov poisson fokker planck system a difficulty is that in this aggregative setting weak solutions of the limiting model blow up in finite time and therefore the scheme should be able to handle dirac measures it is overcome thanks to a careful discretization of the macroscopic velocity resulting of vol pert calculus accordingly a new well balanced and asymptotic preserving numerical scheme is provided numerical simulations confirm a good behavior of solutions

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

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

space localization and well balanced schemes for discrete kinetic models in diffusive regimes we derive and study well balanced schemes for quasi monotone discrete kinetic models by means of a rigorous localization procedure we reformulate the collision terms as nonconservative products and solve the resulting riemann problem whose solution is self similar the construction of an asymptotic preserving ap godunov scheme is straightforward and various compactness properties are established within different scalings finally some computational results are supplied to show that this approach is realizable and efficient on concrete 2 2 models

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

salem numbers defined by coxeter transformation abstract a real algebraic integer 1 is called a salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1 it is known j a de la pena coxeter transformations and the representation theory of algebras in v dlab et al eds finite dimensional algebras and related topics proceedings of the nato advanced research workshop on representations of algebras and related topics ottawa canada kluwer august 10 18 1992 pp 223 253 j f mckee p rowlinson c j smyth salem numbers and pisot numbers from stars number theory in progress in k gy ry et al eds proc int conf banach int math center diophantine problems and polynomials vol 1 de gruyter berlin 1999 pp 309 319 p lakatos on coxeter polynomials of wild stars linear algebra appl 293 1999 159 170 that the spectral radii of coxeter transformation defined by stars which are neither of dynkin nor of extended dynkin type are salem numbers we prove that the spectral radii of the coxeter transformation of generalized stars are also salem numbers a generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3

asymptotic high order schemes for 2 times2 dissipative hyperbolic systems we investigate finite difference schemes which approximate 2 times 2 one dimensional linear dissipative hyperbolic systems we show that it is possible to introduce some suitable modifications in standard upwinding schemes which keep into account the long time behavior of the solutions to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states respectively around stationary solutions and in the diffusion chapman enskog limit

on the linearization problem involving pochhammer symbols and their q analogues abstract in this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q polynomials in the exponential lattice x s c1qs c3 some simple examples are worked out in detail

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

conjugacy relation on coxeter elements abstract let w s be an irreducible finitely presented coxeter system the present paper is mainly concerned with a conjugacy relation on coxeter elements in the case where contains just one circle in particular when is itself a circle in the cases where is either a three multiple circle or a circle with three nodes we show that the ss equivalence relation on coxeter elements of w is the same as the w conjugacy relation an explicit formula is given for the characteristic polynomial of a coxeter element in the natural reflection representation of w when is a circle we also give the answers to some questions raised by coleman and extend some results of geck and pfeiffer concerning conjugacy relation in coxeter groups

tubes in derived categories and cyclotomic factors of the coxeter polynomial of an algebra let be a k algebra of finite global dimension we study tubular families in the auslander reiten quiver of the bounded derived category db satisfying certain natural axioms in particular we precisely describe their influence on the cyclotomic factors of the coxeter polynomial of and discuss several numerical limitations for their possible shapes moreover we show that our results provide an alternative relatively simple proof of non trivial classical facts concerning tubular families in module categories and also extend them slightly

recurrence relations for connection coefficients between q orthogonal polynomials of discrete variables in the non uniform lattice x s q2s we obtain the structure relations for q orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q orthogonal polynomials an explicit example is also given

on the q polynomials a distributed study in this paper we present a uni1ed distributional study of the classical discrete q polynomials in the hahn s sense from the distributional q pearson equation we will deduce many of their properties such as the three term recurrence relations structure relations etc also several characterizations of such q polynomials are presented c

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

the coxeter polynomial for a one point extension algebra let be a finite dimensional algebra over an algebraically closed field k of finite global dimension let m be a finitely generated module and let m be the one point extension algebra we show how to compute the coxeter polynomial for from the coxeter polynomial of and homological invariants of m

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

efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of al salam carlitz i polynomials two formulae expressing explicitly the derivatives and moments of al salam carlitz i polynomials of any degree and for any order in terms of al salam carlitz i themselves are proved two other formulae for the expansion coefficients of general order derivatives d p f x and for the moments xd p f x of an arbitrary function f x in terms of its original expansion coefficients are also obtained application of these formulae for solving q difference equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between al salam carlitz i polynomials and any system of basic hypergeometric orthogonal polynomials belonging to the q hahn class is described

hydrodynamic singular regimes in 1 1 kinetic models and spectral numerical methods classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models a model of chemotaxis involving a biased velocity redistribution integral term and a vlasov fokker planck vfp system both are coupled to an attractive elliptic equation producing corresponding mean field potentials spectral de compositions of stationary kinetic distributions are recalled based on a variation of case s elementary solutions for the first model and on a sturm liouville eigenvalue problem for the second one well balanced godunov schemes with strong stability properties are deduced moreover in the stiff hydrodynamical scaling an hybridized algorithm is set up for which asymptotic preserving properties can be established under mild restrictions on the computational grid several numerical validations are displayed including the consistency of the vfp model with burgers hopf dynamics on the velocity field after blowup of the macroscopic density into dirac masses

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

on the mahler measure of the coxeter polynomial of an algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is a basic connected and triangular algebra with n pairwise non isomorphic simple modules we consider the coxeter transformation a t as the automorphism of the grothendieck group k 0 a induced by the auslander reiten translation in the derived category der b mod a of the module category mod a of finite dimensional left a modules we say that a is of cyclotomic type if the characteristic polynomial a of a is a product of cyclotomic polynomials equivalently if the mahler measure m a 1 in 6 we have considered many examples of algebras of cyclotomic type in the representation theory literature in this paper we study the mahler measure of the coxeter polynomial of accessible algebras in 1933 d h lehmer found that the polynomial t 10 t 9 t 7 t 6 t 5 t 4 t 3 t 1 has mahler measure 0 1 176280 and he asked whether there exist any smaller values exceeding 1 in this paper we prove that for any accessible algebra a either m a 1 or m b 0 for some convex subcategory b of a we introduce interlaced tower of algebras a m a n with m n 2 satisfying a s 1 t 1 a s t a s 1 for m 1 s n 1 we prove that if spec a n s 1 r and a n is not of cyclotomic type then m a m m a n we construct several examples

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