cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

the asymptotic distributions of the size of the largest length of a reversible markov process of polymerization in this paper the subcritical near critical and supercritical asymptotic behaviour of a reversible markov process as a chemical model for polymerization was studied we establish the existence of three distinct stages subcritical near critical and supercritical stages of polymerization in the thermodynamic limit as n depending on the value of strength of the fragmentation reaction these three stages correspond to the size of the largest length of polymers of size n to be itself of order log n na 1 2 a 1 and n respectively especially a 2 2 1 when the coagulation reaction rate constant rij of the polymers satisfies rij i j with 1 2 3 2

spouge s conjecture on complete and instantaneous gelation we investigate the stochastic counterpart of the smoluchowski coagulation equation namely the marcus lushnikov coagulation model it is believed that for a broad class of kernels all particles are swept into one huge cluster in an arbitrarily small time which is known as a complete and instantaneous gelation phenomenon indeed spouge also domilovskii et al for a special case conjectured that k i j ij 1 are such kernels in this paper we extend the above conjecture and prove rigorously that if there is a function i j increasing in both i and j such that j 1 1 j i j 1 exhibit complete instantaneous gelation also we conjuncture the existence of a critical or metastable sol state if limi j ij k i j 0 and i j 1 1 k i j then gelation time tg satisfies 0 tg moreover the gelation is complete after tg

some analytical and numerical solutions for colloidal aggregation with fragmentation we study the kinetic equation which describes the evolution of cluster size distribution of the aggregation fragmentation problem the so called generalized smoluchowski equation for several dynamics of cluster growth by using the method of the generating function we find analytical solutions for some cases besides numerical solutions are found for the time evolution of clusters where the aggregation and fragmentation kernels do not allow a complete analytical solution

solutions and critical times for the monodisperse coagulation equation when aij a b i j cij the author shows that if aij a b i j cij the solutions of smoluchowski s coagulation equation monodisperse case can be expressed as equilibrium distributions conditioned on a deterministically changing and known parameter a partial converse is also given because the equilibrium distributions are already known this effectively solves the equation for the given aij if c not 0 then there is a finite critical time t tc when the moments cease to converge the author finds tc for all the given aij

pre and post gel size distributions in ir reversible polymerisation a class of irreversible coagulation processes can be modelled by smoluchowski s coagulation equation with rate constants kij a b i j cij non negative a b and c for c not 0 a gelation transition occurs the authors obtain explicit solutions for the size distribution ck t with ck 0 delta k1 next they construct and solve the equations for reversible polymerisation by incorporating break up processes in the kinetic equation with a unimolecular fragmentation rate fij lambda ninjkij ni j the degeneracy factors nk obey k 1 nk 1 2 sigma kijninj with i j k and n1 1 and the strength parameter lambda exp g kbt where the binding energy g to infinity for irreversible coagulation explicit results are only given for flory s polymerisation models raf and braf 1 in the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents

the evolution of the cluster size distribution in a coagulation system the coagulation equation with kernelkij a b i j cij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found for monodisperse initial conditions we recover the known size distribution expressed in terms of a degeneracy factornk which is determined by a recursion relation for polydisperse initial conditions a similar solution form is found which includes a degeneracy factornkl also determined by a recursion relation the physical meaning ofnkl and the recursion relation is given a method to get explicit expressions fornk andnkl is illustrated finally the pre gel solution is given explicitly and a general method to get the post gel solution is proposed

solutions and critical times for the polydisperse coagulation equation when a x y a b x y cxy gives solutions of the polydisperse smoluchowski coagulation equation delta f x t delta t 1 2 integral 0xf y t f x y t a y x y dy f x t integral 0infinity f y t a x y dy f x 0 g x for arbitrary g x when the coagulation kernel a x y a b x y cxy the solutions are given as recursions and infinite series and are practical for computation for the given kernels the author also gives the gelation times tg at which m2 t integral 0infinity x2f x t dx becomes infinite

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

scaling solutions of smoluchowski s coagulation equation we investigate the structure of scaling solutions of smoluchowski s coagulation equation of the formc k t s t k s t wherec k t is the concentration of clusters of sizek at timet s t is the average cluster size and x is a scaling function for the rate constantk i j in smoluchowski s equation we make the very general assumption thatk i j is a homogeneous function of the cluster sizesi andj k i j a k ai aj for alla 0 but we restrict ourselves to kernels satisfyingk i j j 0 asj we show that gelation occurs if 1 and does not occur if 1 for all gelling and nongelling models we calculate the time dependence ofs t and we derive an equation for x we present a detailed analysis of the behavior of x at large and small values ofx for all models we find exponential large x behavior x ax e x asx and for different kernelsk i j algebraic or exponential small x behavior x bx or x exp cx asx 0

exact solutions for the coagulation fragmentation equation the authors present new exact equilibrium and time dependent solutions to the two main versions of the coagulation fragmentation equation equilibria are first found for the pure coagulation and pure fragmentation equations equilibrium solutions for the full coagulation fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form time dependent solutions are also derived for the pure fragmentation equation

particle coagulation and the memory of initial conditions particle coagulation is mathematically described by an infinite set of coupled nonlinear differential equations a solution to these equations is derived for the case in which all particle clusters possess the same reactivity i e a constant kernel and where the initial conditions are bimodal consisting of monomers and any sized j mers properties of the solution are explored and it is shown that the scaling theory developed by swift and friedlander 1964 j colloid sci 19 621 and extended by van dongen and ernst 1984 54 1396 applies to all cluster sizes only in the limit as reported previously by kreer and penrose 1994 j stat phys 75 389 at finite times we find distinctly different scaling properties for the small and large ends of the size spectrum furthermore at all times the shape of the small end of the size spectrum retains a memory of the initial conditions these results may apply to other modes of coagulation so long as interactions between small clusters and between small and large clusters are as weak as the constant kernel employed here

on mass conserving solutions of the discrete coagulation equation the authors present the explicit pre gelation solution of the discrete coagulation equation for the general bilinear kernel and arbitrary initial conditions for the pure additional kernel they prove that gelation never takes place even if the second moment of the initial particle distribution is infinite

polymers and random graphs asymptotic equivalence to branching processes in 1974 falk and thomas did a computer simulation of flory s equireactive raf polymer model rings forbidden and rings allowed asymptotically the rings forbidden model tended to stockmayer s raf distribution in which the sol distribution sticks after gelation while the rings allowed model tended to the flory version of the raf distribution in 1965 whittle introduced the tree and pseudomultigraph models we show that these random graphs generalize the falk and thomas models by incorporating first shell substitution effects moreover asymptotically the tree model displays postgelation sticking hence this phenomenon results from the absence of rings and occurs independently of equireactivity we also show that the pseudomultigraph model is asymptotically identical to the branching process model introduced by gordon in 1962 this provides a possible basis for the branching process model in standard statistical mechanics

new results on a parity dependent model of aggregation kinetics a reaction kernel k j k k k j is studied for which the smoluchowski equations of aggregation j 1 2 k l 1 k k l ckcl k l j k j l j can be solved it takes only three values k j k k if j and k are both odd k j k l if j and k are both even and k j k m if j and k have different parities a considerable simplification over previous treatments is presented for the general case k l and m are three arbitrary positive numbers and the time evolution of the concentrations is exhibited in completely explicit form for the new special case l 4m in another special case k m the equation for the generating function of the concentrations can be reduced to quadratures the analysis of this case and of the general case is postponed to a future paper

gelation time in the discrete coagulation fragmentation equations with a bilinear coagulation kernel the discrete coagulation fragmentation equations with a bilinear coagulation kernel describing the evolution of clusters of particles undergoing binary reactions are studied we compute the gelation time and post gelation mass for the pure coagulation equation a method of characteristics is developed to solve numerically the partial differential equation satisfied by a moment generating function for a product coagulation kernel and a constant fragmentation rate the accuracy of the numerical method is verified by comparison of the numerical results with an exact solution for the number density of monomers for a given coagulation rate a critical value of the fragmentation rate values greater than which lead to mass conservation is identified

the uniqueness of cuntz krieger type algebras we introduce a class of c algebras which can be viewed as a generalization of the classical cuntz krieger algebras the feature is that it is a flexible generators and relations concept the main result is a canonical uniqueness theorem stating that the c algebras of this class are uniquely determined by their generators and relations we can show that rank one cuntz krieger algebras with infinitely large transition matrices fall in this class and this provides an alternative proof of exel and laca further we elaborate a subclass of rank two cuntz krieger algebras inspired by shifts of finite type in dimension two with an infinite set of generators and relations

hyperbolic octagons and teichmuller space in genus 2 in this paper we study different models of the teichmuller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems they are on the one hand complex enough to show the relevant features and on the other hand they possess a simple mathematical structure allowing practical implementations we give a new description of teichmuller space and the mapping class group in terms of geometric data of the octagons this provides modellings based on the vertices and also on the generators of the associated isometry group in addition we explicitly connect our approach with other models currently used in the literature helling matrices fenchel nielsen parameters the result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others

scaling and crossover properties of a new solvable model of aggregation kinetics in a previous paper a new reaction kernel for the smoluchowski equations of aggregation was solved exactly this kernel k j k 2 qj qk for 0 q 1 a real positive quantity interpolates between two well understood exactly solved cases namely that of k j k 2 and that of k j k j k this new model however shows a number of unexpected features not found in either of the two limiting cases it is shown that this model has a remarkable behaviour with respect to the commonly accepted scaling theory on the one hand it satisfies a rigorous form of the scaling hypothesis but on the other hand it clearly violates some relations which are ordinarily assumed to follow from it these issues are discussed as well as the nature of the singular limit in which q is very close to one for which our kernel becomes close to the sum kernel mentioned above in particular the form of the crossover between two kernels with different degrees of homogeneity can be discussed here in an exact way

exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition the coagulation fragmentation equation describes the concentration f i t of particles of size i in nn 0 at time t geq 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break up we show that when the rate of fragmentation is sufficiently stronger than that of coalescence f i t i in nn 0 tends to an unique equilibrium as t tends to infinity although we suppose that the initial datum is sufficiently small we do not assume a detailed balance or reversibility condition the rate of convergence we obtain is furthermore exponential

the k theory of certain c algebras endowed with gauge actions we compute the k theory of the toeplitz algebra of a finitely aligned higher rank graph and of a higher rank exel laca algebra under condition ii actually we deduce these results from a slightly more general technical theorem for c algebras endowed with gauge actions and fixed point algebra af among other requirements

analytic solutions to smoluchowski s coagulation equation a combinatorial interpretation many analytic solutions of the polydisperse coagulation equation have combinatorial interpretations the interpretations are based on random polycondensation models from polymer chemistry the notion of chemical bonding inherent in these polymer models introduces additional mathematical structure this structure allows combinatorial interpretation of the coagulation solutions when the coagulation kernel is bilinear

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

mutually catalyzed birth of population and assets in exchange driven growth we propose an exchange driven aggregation growth model of population and assets with mutually catalyzed birth to study the interaction between the population and assets in their exchange driven processes in this model monomer or equivalently individual exchange occurs between any pair of aggregates of the same species population or assets the rate kernels of the exchanges of population and assets are k k l kkl and l k l lkl respectively at which one monomer migrates from an aggregate of size k to another of size l meanwhile an aggregate of one species can yield a new monomer by the catalysis of an arbitrary aggregate of the other species the rate kernel of asset catalyzed population birth is i k l iklmu and that of population catalyzed asset birth is j k l jklnu at which an aggregate of size k gains a monomer birth when it meets a catalyst aggregate of size l the kinetic behaviors of the population and asset aggregates are solved based on the rate equations the evolution of the aggregate size distributions of population and assets is found to fall into one of three categories for different parameters mu and nu i population asset aggregates evolve according to the conventional scaling form in the case of mu 0 mu 0 and nu 0 and iii both population and asset aggregates undergo gelation transitions at a finite time in the case of mu nu 0

a direct simulation monte carlo method for cluster coagulation abstract a method for analyzing cluster coagulation is presented which relies on a monte carlo analysis of individual particles as they interact and form clusters from a homogeneous monodisperse medium four case studies are shown three of which compare the results of the code to the known analytic solutions of the smoluchowski equation and the fourth considers the cluster size spectrum obtained from a generalized analytic recurrence solution to the smoluchowski equation which allows in principle the generation of the entire cluster size distribution from the partial distribution given by the monte carlo code

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

size distribution in the polymerisation model afrbg the polymer size distribution in flory s polymerisation model afrbg is in equilibrium and non equilibrium of the general form ck ank xi k both below and above the gelation transition here explicit expressions are derived for a alpha and xi alpha as a function of the extent of reaction alpha the combinatorial factors nk are calculated from the recursion relation 2 k 1 nk sigma i j kkijninj with kij si f sj g si g sj f and sk f f 1 k 1 using a generating function technique the authors express nk in terms of laguerre polynomials g to infinity and jacobi polynomials g finite whose large k behaviour is of the form nk approximately bk tau xi c k with tau 3 2 no gelation occurs if f or g equals 1 and with tau 5 2 gelation occurs if f 1 and g 1

a new solvable model of aggregation kinetics a new reaction kernel k j k 2 qj qk with 0 q 1 is introduced for which the smoluchowski equations of aggregation can be solved the time evolution of the concentrations cj t and of their moments is analysed the cj t decay at large times as t 2 qj in striking contrast to the behaviour of the constant kernel k j k 2 for which cj t behaves as t 2 at large times on the other hand the moments behave in leading order at large times exactly like the moments of the constant kernel though differences appear at higher orders

a formula for the post gelation mass of a coagulation equation with a separable bilinear kernel abstract it is well known that solutions of the coagulation equation do not conserve mass if the coagulation kernel grows too rapidly the phenomenon whereby conservation of mass breaks down in finite time is known as gelation and is physically interpreted as being caused by the appearance of an infinite gel or superparticle in this paper we derive a formula for the post gelation mass for the case of a separable bilinear kernel k and investigate the asymptotic behaviour of the solution

the exact solution of the coagulation equation with kernel kij a i j b the smoluchowski coagulation equation with the kernel kij a i j b is solved exactly for arbitrary initial conditions the author obtains a compact form of the size distribution for monodisperse initial conditions for polydisperse initial conditions a simple form of the size distribution including a parameter nkl determined by a recursive relation is obtained in the special case with k sub ij i j the author obtains the compact form of the size distribution for arbitrary initial conditions

some results on the coagulation equation eq 1 represents the evolution of particles c t of size 0 at time t 0 undergoing a change in size governed by reaction kernel k a physical interpretation of the terms in 1 can be found in melzak 5 or the review article by drake 2 global existence and uniqueness of solutions to 1 has been proven by aizenman and bak 1 for constant kernels with suitable conditions imposed on the initial particle distribution c0 for the product kernel k mcleod 4 has shown again for a suitably restricted initial particle distribution that a mass conserving solution exists for some nite time that solutions in this case need not conserve mass for all time was demonstrated by stewart 6 the phenomenon whereby conservation of mass breaks down in nite time is known as gelation and is physically interpreted as being caused by the appearance of an in nite gel or superparticle 3

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

aggregation beyond the gel point a new class of exactly solvable models the existing models describing the kinetics of aggregation in the presence of an infinite cluster orgel are reviewed and a new class of post gel models is proposed in this new class of models clusters are assumed to be acyclic and the rate constant for reactions involving the gel can be varied the model is called thegeneralized acyclic model gam since it generalizes stockmayer s model and ziff s third model it is shown that the gam can be solved exactly in terms of standard hypergeometric functions the solutions are analyzed in detail both asymptotically in various limits and numerically