uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

on the finite size scaling of clusters in compact directed percolation the exact finite size scaling properties of clusters in compact directed percolation on a square lattice are derived the results are implicit in previous work on the enumeration of staircase polygons but their explicit form has not been presented as such before the analysis provides important insights into the nature of this type of percolation transition

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

existence and stability of propagating fronts for an autocatalytic reaction diffusion system we study a one dimensional reaction diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic a b 2b and a cubic a 2b 3b autocatalysis the parameters of this system are in the ratio d d b d a of the diffusion constants of the reactant a and the autocatalyst b and the relative activity ii of the cubic reaction first for all values of d 0 and k greater than or equal to 0 we prove the existence of a family of propagating fronts or travelling waves describing the advance of the reaction in particular in the quadratic case k 0 we recover the results of billingham and needham phil trans r sec london a 334 1991 1 24 then if d is close to 1 and k is sufficiently small we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted sobolev spaces this extends part of the results that are known for the scalar equation to which our system reduces when d 1 c 1998 elsevier science b v

steady states and dynamics of an autocatalytic chemical reaction model with decay abstract the dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates upper lower solution method and bifurcation theory the effects of decay order decay rate and diffusion rates to the dynamical behavior are discussed

renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

diffusion approximation for self similarity of stochastic advection in burgers equation self similarity of burgers equation with stochastic advection is studied in self similar variables a stationary solution is constructed which establishes the existence of a stochastically self similar solution for the stochastic burgers equation the analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self similar variables furthermore by a diffusion approximation the long time convergence to the self similar solution is proved in the sense of distribution

proof of existence of global solutions for m component reaction diffusion systems with mixed boundary conditions via the lyapunov functional method to prove global existence for solutions of m component reaction diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy neumann boundary conditions while others satisfy nonhomogeneous dirichlet boundary conditions and nonhomogeneous robin boundary conditions the purpose of this paper is to prove the existence of a global solution using a single inequality for the polynomial growth condition of the reaction terms our technique is based on the construction of polynomial functionals this result generalizes those obtained recently by kouachi et al at press kouachi 2002 electron j diff eqns 2002 1 kouachi 2001 electron j diff eqns 2001 1 and independently by malham and xin 1998 commun math phys 193 287

global existence for reaction diffusion systems with mass control and critical growth with respect to the gradient abstract in this paper we study the existence of global weak solutions for 2 2 reaction diffusion systems for which two main properties hold the positivity of the solutions and the total mass of the components are preserved with time moreover we suppose that the non linearities have critical growth with respect to the gradient the technique we use here in order to prove global existence is in the same spirit of the method developed by boccardo murat and puel for a single equation

on the hulls of directed percolation clusters the properties of the hulls of directed percolation clusters are studied the scaling and finite size scaling of many quantities around the percolation threshold pc are derived and a novel monte carlo algorithm which is more than twice as fast as the standard algorithm at pc has been formulated to study these properties simulations have been conducted that enable an estimation of all the exponents involved in particular the central exponent x relating the average hull length of clusters to their mass has been estimated to be 0 773 4 at the percolation threshold this same exponent is estimated to be 0 905 5 for p p c thus this second value should hold for directed animals

scaling analysis for the adsorption transition in a watermelon network of n directed non intersecting walks the partition function for the problem of n directed non intersecting walks interacting via contact potentials with a wall parallel to the direction of the walks has previously been calculated as an n n determinant here we describe how to analyse the scaling behaviour of this problem using alternative representations of the solution in doing so we derive the asymptotics of the partition function of a watermelon network of n such walks for all temperatures and so calculate the associated network exponents in the three regimes desorbed adsorbed and at the adsorption transition furthermore we derive the full scaling function around the adsorption transition for all n at the adsorption transition we also derive a simple product form for the partition function these results have in part been derived using recurrence relations satisfied by the original determinantal solution

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

on qk teichmuller spaces abstract recently wulan and ye 36 introduced the q k teichmuller spaces and proved some theorems about the theory of q k teichmuller spaces in this note we continue to consider this theory and obtain some results on the q k teichmuller spaces in particular we establish some new characterizations of the little q k teichmuller spaces

universal self similarity of porous media equation with absorption the critical exponent case abstract in this paper we study the large time behavior of non negative solutions to the cauchy problem of u t u m u q in r n 0 where m 1 and q q c m 2 n is a critical exponent for non negative initial value u x 0 u 0 x l 1 r n we show that the solution converges if u 0 x 1 x k is bounded for some k n to a unique fundamental solution of u t u m independent of the initial value with additional logarithmic anomalous decay exponent in time as t

the brownian excursion area a numerical analysis abstract using an explicit expression for the laplace transform of the brownian excursion area generating function a numerical analysis gives moments density and distribution function an asymptotic formula is given for small argument

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

existence of solutions for a class of weakly coupled semilinear elliptic systems on etudie l existence des solutions de problemes aux limites pour des systemes elliptiques semi lineaires

a note on the bmo teichmuller space abstract astala zinsmeister 2 introduced a new topology on the set of all strongly quasisymmetric homeomorphisms of the unit circle denoted by sqs s 1 by means of bmo norm under this new topology we prove that sqs s 1 is a partial topological group and the characteristic topological subgroup of sqs s 1 consists of strongly symmetric homeomorphisms

parabolic systems in unbounded domains i existence and dynamics abstract in this paper we investigate the existence uniqueness and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space r n the exterior of a bounded domain and a half space in r n the asymptotic behavior of the solution is with respect to a pair of quasi solutions of the corresponding elliptic system and when these two quasi solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

the existence of minimum speed of traveling wave solutions to a non kpp isothermal diffusion system abstract the reaction diffusion system a t a x x a b n b t d b x x a b n where n 1 and d 0 arises from many real world chemical reactions whereas n 1 is the kpp type nonlinearity which is much studied and very important results obtained in literature not only in one dimensional spatial domains but also multi dimensional spaces but n 1 proves to be much harder one of the interesting features of the system is the existence of traveling wave solutions in particular for the traveling wave solution a x t a x v t b x t b x v t where v 0 if we fix lim x a b 0 1 it was proved by many authors with different bounds v n d 0 such that a traveling wave solution exists for any v v when n 1 for the latest progress see 7 that is the traveling wave problem exhibits the mono stable phenomenon for traveling wave of scalar equation u t u x x f u with f 0 f 1 0 f u 0 in 0 1 and u 0 is unstable and u 1 is stable a natural and significant question is whether like the scalar case there exists a minimum speed that is whether there exists a minimum speed v m i n 0 such that traveling wave solution of speed v exists iff v v m i n this is an open question in spite of many works on traveling wave of the system in last thirty years this is duo to the reason unlike the kpp case the minimum speed cannot be obtained through linear analysis at equilibrium points a b 0 1 and a b 1 0 in this work we give an affirmative answer to this question

universal teichmuller space and bmo abstract we first give some new characterizations on bmoa teichmuller space and various characterizations on vmoa teichmuller space as well in particular we prove that a quasisymmetric conformal welding h corresponds to an asymptotically smooth curve in the sense of pommerenke 1978 32 precisely when h is absolutely continuous with log h vmo we then show that these bmo teichmuller spaces have natural complex structures

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

nonlinear diffusion through large complex networks containing regular subgraphs transport through generalized trees is considered trees contain the simple nodes and supernodes either well structured regular subgraphs or those with many triangles we observe a superdiffusion for the highly connected nodes while it is brownian for the rest of the nodes transport within a supernode is affected by the finite size effects vanishing as n for the even dimensions of space d 2 4 6 the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat kernel expansion

quasiconformal mappings and chord arc curves given a quasiconformal mapping p on the plane what conditions on its dilatation guarantee that p r is rectifiable and pir is locally abso lutely continuous we show in this paper that if satisfies certain quadratic carleson measure conditions with small norm then p r is a chord arc curve with small constant and p x p 0 g0 ea t dt for x e r with a e bmo having small norm conversely given any such map from r c we show that it has an extension to c with the right kind of dilatation similar results hold with r replaced by a chord arc curve examples are given that show that it would be hard to improve these results applications are given to conformal welding and the theorem of coifman and meyer on the real analyticity of the riemann mapping on the manifold of chord arc curves let p be a quasiconformal map of the plane onto itself thus p is a homeomorphism with locally integrable distributional derivatives and ps upz where u e l c il ul100 1 here we use the notations fz af azf 2 z id f fz 2 aa ida f this function u is called the complex dilatation of p the mapping theorem for q c maps states that for each u e l c il ul100 1 there is a q c map p on c with dilatation u and p is unique up to normalization a basic problem is to understand how geometric properties of p are reflected in u for example one would like to have natural conditions on u which imply that p r is rectifiable and p ir is absolutely continuous this question arises naturally when considering problems in conformal mappings and conformal welding in this paper we obtain such estimates for the mapping theorem and we also give some applications our results involve bmo aoo weights chord arc curves and carleson measures and so we first review the appropriate definitions a locally integrable function f on r lies in bmo if llfil sup lil if x fil dx received by the editors june 1 1986 1980 mathematics subject classification 1985 revision primary 30g60 secondary 42b20 30e20

finite length scaling of collapsing directed walks we study the finite length scaling of self interacting partially directed self avoiding walks utilising enumeration data up to a total length of 6000 steps this facilitates the evaluation of the numerical techniques available for calculating exponents at the point and in the collapsed phase of walk type models another consequence is the conjecture of a new scaling theory for the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of undirected problems including interacting saw we provide a phenomenological picture of the phase transition in terms of the condensation of droplets that allows us to understand the various length scales involved in the problem pacs numbers 05 50 q 05 70 fh 61 41 e email prel aleks brak tonyg mundoe maths mu oz au

uniform asymptotics of area weighted dyck paths using the generalized method of steepest descents for the case of two coalescing saddle points we derive an asymptotic expression for the bivariate generating function of dyck paths weighted according to their length and their area in the limit of the area generating variable tending towards 1 the result is valid uniformly for a range of the length generating variable including the tricritical point of the model

area width scaling in generalised motzkin paths abstract we consider a generalised version of motzkin paths where horizontal steps have length l with l being a fixed positive integer we first give the general functional equation for the area width generating function of this model using a heuristic ansatz we then derive the area width scaling behaviour in terms of a scaling function in one variable for the special cases of dyck standard motzkin and schroder paths before generalising our approach to arbitrary l we then rigorously derive the tricritical scaling of schroder paths by applying the generalised method of steepest descents to the known exact solution for their area width generating function our results show that for dyck and schroder paths the heuristic scaling ansatz reproduces the rigorous results

qualitative theory for a model of laminar flames with arbitrary nonnegative initial data abstract we consider a system of nonlinear parabolic partial differential equations arising as a model of laminar flames in a premixed reactive gas the existence of traveling wave solutions has been established for positive ignition temperature by berestycki nicolaenko and scheurer and by terman for zero ignition temperature stability and instability of these solutions have been established in various situations by sivashinsky clavin and recently terman our goal is to study the equations with initial data that are bounded uniformly continuous and nonnegative but otherwise arbitrary we establish the existence of unique global strong solutions satisfying appropriate a priori estimates with a positivity condition imposed on the initial data for the temperature we show that the concentration decays exponentially this result while easy to obtain plays an important role in results that follow of greatest physical interest are the cases where ignition occurs precisely at one end our main result is that if the average of the initial temperature values at the ends of the chamber is above ignition temperature then on any ray coming from the ignition end the temperature is uniformly above ignition temperature and the concentration decays uniformly to zero we can continually advance the endpoint of this zone of ignition thus roughly mimicking the motion of traveling wave solutions we also point out the appropriateness of the averaging condition on the initial temperature and discuss two examples where this averaging condition is not satisfied in one we eventually have flame propagation and in the other example we have eventual flame extinguishment on any ray coming from the cold end of the chamber

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

the tricritical behavior of self interacting partially directed walks we present the thermodynamics of two variations of the interacting partially directed self avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values while the discrete model has been considered previously to varying degrees of success the continuous model we now define has not the examination of the continuous model leads to theexact derivation of several exponents for the discrete model some of these exponents can be calculated using a continued fraction representation for both models the crossover exponent is found to be 2 3 moreover we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function additionally the similarities noticed previously to other models but left unexplored are explained with the aid of necklacing arguments

group analysis and renormgroup symmetries an original regular approach to constructing special type symmetries for boundary value problems namely renormgroup symmetries is presented different methods of calculating these symmetries based on modern group analysis are described an application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

global large time self similarity of a thermal diffusive combustion system with critical nonlinearity we study the initial value problem of the thermal diffusive combustion system u 1 t u 1 x x u 1 u 2 2 u 2 t du 2 xx u 1 u 2 2 x is an element of r 1 for non negative spatially decaying initial data of arbitrary size and for any positive constant d we show that if the initial data decay to zero sufficiently fast at infinity then the solution u 1 u 2 converges to a self similar solution of the reduced system u 1 t u 1 xx u 1 u 2 2 u 2 t du 2 xx in the large time limit in particular u 1 decays to zero like o t 1 2 delta where delta 0 is an anomalous exponent depending on the initial data and u 2 decays to zero with normal rate o t 1 2 the idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self similarity of the solutions in the large time limit c 1996 academic press inc

higher bers maps and bmo teichmuller space abstract we prove that the higher bers maps induced by the higher schwarzian differential operators are holomorphic in bmo teichmuller space and their derivatives at the origin are linear bounded and surjective

higher order airy scaling in deformed dyck paths we introduce a deformed version of dyck paths ddp where additional to the steps allowed for dyck paths jumps orthogonal to the preferred direction of the path are permitted we consider the generating function of ddp weighted with respect to their half length area and number of jumps this represents the first example of an exactly solvable two dimensional lattice vesicle model showing a higher order multicritical point applying the generalized method of steepest descents we see that the associated two variable scaling function is given by the logarithmic derivative of a generalized higher order airy integral

the dilogarithm function for complex argument this paper summarizes the basic properties of the euler dilogarithm function often referred to as the spence function these include integral representations series expansions linear and quadrat

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

asymptotic behaviour of some reaction diffusion systems modelling complex combustion on bounded domains we consider three models of multiple step combustion processes on bounded spatial domains previously steady state convergence results have been established for these models with zero neumann boundary conditions imposed on the temperature as well as the mass fractions we retain here throughout the same boundary conditions on the mass fractions but in our first set of results we establish steady state convergence results with fixed dirichlet boundary conditions on the temperature next under certain physically reasonable assumptions we develop for two of the models estimates on the decay rates of both mass fractions to zero while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction these results hold under either set of boundary conditions but when the dirichlet conditions are imposed on the temperature we are able to obtain estimates on the rate of convergence of the temperature to its generally nonconstant steady state finally we improve the results of a previous paper by adding a temperature convergence result

cluster structure of collapsing polymers in order to better understand the geometry of the polymer collapse transition we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self avoiding walk we argue for this new correlated percolation problem that in two dimensions and possibly also in three dimensions a percolation transition takes place at a temperature lower than the collapse transition hence this novel transition should be governed by exponents unrelated to the point exponents this also implies that there is a temperature range in which the polymer has collapsed but has no long range cluster structure we use monte carlo to study the distribution of clusters on the simple cubic and manhattan lattices on the manhattan lattice where the data are most convincing we find that the percolation transition occurs at p 1 461 3 while the collapse transition is known to occur exactly at 1 414 we propose a finite size scaling form for the cluster distribution and estimate several of the critical exponents regardless of the value of p this percolation problem sheds new light on polymer collapse

scaling in the collapsed polymer phase exact results the author derives an exact scaling form for the low temperature partition function in a model of polymer collapse this confirms series work and so gives the exponents sigma 1 2 gamma 1 4 and chi 3 4 exactly the model considered is a variant of the self interacting partially directed self avoiding walk in two dimensions

traveling waves solutions of isothermal chemical systems with decay this article studies propagating traveling waves in a class of reaction diffusion systems which include a model of microbial growth and competition in a flow reactor proposed by smith and zhao 17 and isothermal autocatalytic systems in chemical reaction of order m with a decay order n where m and n are positive integers and m n a typical system in autocatalysis is a 2b 3b with rate k1ab2 and b c with rate k2b where m 2 and n 1 involving two chemical species a reactant a and an auto catalyst b whose diffusion coefficients da and db are unequal due to different molecular weights and or sizes here a is the concentration density of a b that of b and c an inert chemical species the two constants k1 and k2 are material constants measuring the relative strength of respective reactions it is shown that there exist traveling waves when m 1 and n 1 with suitable relation between the ratio db da traveling speed c and rate constants k1 k2 on the other hand it is proved that there exists no traveling wave when one of the chemical species is immobile db 0 or n m for all choices of other parameters

existence of traveling waves of auto catalytic systems with decay abstract this article establishes the existence of traveling waves of a class of reaction diffusion systems which model the pre mixed isothermal autocatalytic chemical reaction of order m m 1 between two chemical species a reactant and an auto catalyst and a linear decay moreover our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x this is in strong contrast to either the reaction diffusion systems of autocatalytic chemical reaction of the order m without decay or to the systems which have the same order of decay which were shown by various authors e g 8 13 17 26 that the set of traveling wave speeds contains c for some c 0 the same systems also appear in a mathematical model of microbial growth and competition in a flow reactor see 2 24

interior estimates for a class of reaction diffusion systems from l1 a priori estimates abstract we obtain interior estimates for a class of semilinear reaction diffusion systems from l 1 a priori estimates our results are applied to a predator prey model in which the species switch the role of predator and prey on given subsets of their domain of interaction and a one dimensional flame propagation model extensions of earlier results in morgan 14 15 follow from the analysis

rectangular vesicles in three dimensions the phase diagram of a model of rectangular vesicles in three dimensions is examined the scaling of the generating function is determined in the area perimeter volume area and volume perimeter ensembles the results are interpreted within the framework of tricritical scaling and the crossover exponents associated with the transitions are determined we identify three phases in the phase diagram a needle phase a disc phase and a cubical phase these phases are separated by three curves of transitions that meet in a multicritical point