exact combinatorial approach to finite coagulating systems this paper outlines an exact combinatorial approach to finite coagulating systems in this approach cluster sizes and time are discrete and the binary aggregation alone governs the time evolution of the systems by considering the growth histories of all possible clusters an exact expression is derived for the probability of a coagulating system with an arbitrary kernel being found in a given cluster configuration when monodisperse initial conditions are applied then this probability is used to calculate the time dependent distribution for the number of clusters of a given size the average number of such clusters and that average s standard deviation the correctness of our general expressions is proved based on the analytical and numerical results obtained for systems with the constant kernel in addition the results obtained are compared with the results arising from the solutions to the mean field smoluchowski coagulation equation indicating its weak points the paper closes with a brief discussion on the extensibility to other systems of the approach presented herein emphasizing the issue of arbitrary initial conditions

bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

mesoscopic simulation of ostwald ripening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening this phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange ising model this model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations the typical cluster size is observed to grow like t1 3 over range of times with faster growth at later times the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level

exact solution of a coagulation equation with a product kernel in the multicomponent case abstract in this paper we obtain the general solution for the continuous smoluchowski equation in the multicomponent case with a product kernel as a series expansion the solution of the problem involves the laplace transform in several dimensions we obtain a nonlinear partial differential equation pde of the advective kind generalizing the one previously given by other authors for the mono component case as in its relative mono component case gelation is produced at some point the conditions for its occurrence being the same as those for the mono component case though substituting a sum of derivatives by a derivative in the laplace transform field we demonstrate that for a multicomponent particle size distribution psd of multiplicative form it is sufficient for one of the marginal psds to generate instantaneous gelation for the occurrence of instantaneous gelation in the multicomponent psd the general solution is applied to several specific cases a discrete case that recovers a previously known solution and another two continuous cases which can be used to check numerical methods designed to directly solve the smoluchowski equation in more general cases we have compared the solutions for the multicomponent psd for constant additive and product kernels and we conjecture about the relation existing between the functional forms for the solutions both in the mono component and the multicomponent case finally we have analysed the shape of the solutions for multicomponent psd for constant additive and product kernels for very small masses of components obtaining a qualitatively different behaviour for the product kernel this has effects in the mixing state of the sol phase as time passes

local properties of self similar solutions to smoluchowski s coagulation equation with sum kernels the regularity of the scaling profiles to smoluchowski s coagulation equation is studied when the coagulation kernel k is given by k x y x y with 0 1 more precisely is c smooth on 0 and decays exponentially fast for large x furthermore the singular behaviour of x as x 0 is identified thus giving a rigorous proof of physical conjectures

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 becker doring equations at large times and their connection with the lsw theory of coarsening a systematic procedure is given for obtaining the asymptotic late time behavior of the becker doring equations describing the time evolution of a population of clusters of particles in lowest order of approximation the distribution of the sizes of the largest clusters satisfies the equations of the lifshitz slyozov wagner theory of coarsening

two species coagulation approach to consensus by group level interactions we explore the self organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole to this end we employ the widespread example of coagulation kinetics and characterize which interaction types lead to consensus formation and which do not as well as the corresponding different macroscopic patterns the crucial technical point is extending the usual one species coagulation dynamics to the two species one this is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning the corresponding solutions are calculated in the long time limit in which consensus may or may not be reached the lack of consensus is characterized by means of scaling limits of the solutions the possible applications of our results to some topics in which consensus reaching is fundamental such as collective animal motion and opinion spreading dynamics are also outlined

convergence of a finite volume scheme for coagulation fragmentation equations this paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation a time explicit finite volume scheme is developed based on a conservative formulation of the equation it is shown to converge under a stability condition on the time step while a first order rate of convergence is established and an explicit error estimate is given finally several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution

the l and gamma factors for generic representations of mathrm gsp 4 k times mathrm gl 2 k over a local non archimedean field k

upper bounds on coarsening rates we consider two standard models of surface energy driven coarsening a constant mobility cahn hilliard equation whose large time behavior corresponds to mullins sekerka dynamics and a degenerate mobility cahn hilliard equation whose large time behavior corresponds to motion by surface diffusion arguments based on scaling suggest that the typical length scale should behave as in the first case and in the second we prove a weak one sided version of this assertion showing roughly speaking that no solution can coarsen faster than the expected rate our result constrains the behavior in a time averaged sense rather than pointwise in time and it constrains not the physical length scale but rather the perimeter per unit volume the argument is simple and robust combining the basic dissipation relations with an interpolation inequality and an ode argument

bounds on eigenvalues of the hadamard product and the fan product of matrices we prove an upper bound for the spectral radius of the hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the fan product of m matrices these improve two existing results

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

a counterexample in coagulation theory two models of interacting polymers both within smoluchovski s theory are considered the first model has purely diagonal reaction rates and shows gelation the second model has the same diagonal reaction rates but also some nondiagonal ones the second model does not show gelation hence contrary to intuition gelation can be destroyed by an increase in the reaction rates the example also disproves a conjecture stating that the diagonal reaction rates are the determining factor for gelation

on global stability for lifschitz slyozov wagner like equations this paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations which includes the lifschitz slyozov wagner lsw system in the case when the initial data has compact support the main result of the paper is a proof of weak global asymptotic stability for lsw like systems previously strong local asymptotic stability results were obtained by niethammer and velazquez for the lsw system with initial data of compact support comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical the quadratic model extends the linear model of carr and penrose and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self similar solution of the lsw model

transient and self similar dynamics in thin film coarsening abstract we study coarsening in a simplified model of one dimensional thin films of viscous fluids on hydrophobic substrates lubrication theory shows that such films are unstable and dewet to form droplets that then aggregate over long timescales the masses and positions of the droplets can be described by a coarsening dynamical system cds consisting of odes and deletion rules we develop discrete and continuous mean field models that reproduce the well known n t o t 2 5 long time statistical power law for the number of drops a lifshitz slyozov wagner type lsw continuous model predicts the self similar distribution of drop masses matching with histograms produced by cds simulations and the discrete mean field model we also describe the distribution of drops following homogeneous versus heterogeneous dewetting and use these as initial conditions for the cds simulations that yield characteristic staircasing transients transients can also include recurring spike formation behavior in the mass distribution for idealized initial conditions we show that the transient dynamics can span the full coarsening process bypassing the power law regime entirely

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

global well posedness for an inhomogeneous lsw model in unbounded domains the lsw theory of domain coarsening describes the evolution of the size distribution of a system of particles evolving by diffusional mass exchange to reduce their total surface area we prove existence and uniqueness of solutions to an inhomogeneous extension of the lsw model in unbounded domains this model arises naturally as a homogenization limit of the underlying free boundary problem in the case of a system of particles for which the screening length the effective range of particle interactions is smaller than the system size the crucial ingredients in the analysis are first to establish the screening property by showing that the corresponding green s function decreases exponentially over the relevant distances second we have to control the mass fluxes between different regions to prevent aggregation of mass in few particles which would result in blow up of the solution

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

uniqueness of self similar solutions to smoluchowski s coagulation equations for kernels that are close to constant we consider self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 1 we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small which is the first such uniqueness result for kernels that are not solvable our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures

self similar solutions for the lsw model with encounters abstract the lsw model with encounters has been suggested by lifshitz and slyozov as a regularization of their classical mean field model for domain coarsening to obtain universal self similar long time behavior we rigorously establish that an exponentially decaying self similar solution to this model exist and show that this solutions is isolated in a certain function space our proof relies on setting up a suitable fixed point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem

self similar solutions to a coagulation equation with multiplicative kernel abstract existence of self similar solutions to the oort hulst safronov coagulation equation with multiplicative coagulation kernel is established these solutions are given by s t y s t for t y 0 t 0 where t is some arbitrary positive real number s t 3 t t 1 3 and the parameter ranges in a given interval c 3 in addition the second moment of these self similar solutions blows up at time t as for the profile it belongs to l 1 0 y 2 d y for each c 3 but its behaviour for small and large y varies with the parameter

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

weak selection and stability of localized distributions in ostwald ripening we support and generalize a weak selection rule predicted recently for the self similar asymptotics of the distribution function df in the zero volume fraction limit of ostwald ripening or an asymptotic perturbation theory is developed that when combined with an exact invariance property of the system yields the selection rule in terms of the initial condition predicts a power law convergence towards the selected self similar df and agrees well with our numerical simulations for the interface and diffusion controlled or

derivation of the lsw theory for ostwald ripening by homogenization methods the coarsening of a solid phase in an undercooled liquid is described by a stefan problem with surface tension the solid phase is characterized by a large number of balls with small volume fraction and small capacity we solve the quasi static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles the framework of young measures is used to pass to the limit of infinitely many particles we obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for ostwald ripening by lifshitz slyozov wagner lsw

dynamics and self similarity in min driven clustering we study a mean field model for a clustering process that may be described informally as follows at each step a random integer k is chosen with probability p k and the smallest cluster merges with k randomly chosen clusters we prove that the model determines a continuous dynamical system on the space of probability measures supported in 0 oo and we establish necessary and sufficient conditions for the approach to self similar form we also characterize eternal solutions for this model via a levy khintchine formula the analysis is based on an explicit solution formula discovered by gallay and mielke extended using a careful choice of time scale

some characterizations of m tensors via digraphs abstract for a tensor a a i 1 i m c n n the associated digraph of a has the vertex set v a 1 n and the arc set of a is e a i j a i i 2 i m 0 j i 2 i m i i in this paper we give some characterizations of m tensors and h tensors by using the associated digraph of a tensor

on some properties of three different types of triangular blocked tensors abstract we define and study three different types of upper and lower triangular blocked tensors which are all generalizations of the triangular blocked matrices and are also generalizations of reducible or weakly reducible tensors we study some common properties as well as different properties of these three types of triangular blocked tensors we obtain the formulas for the determinants characteristic polynomials and spectra of the first and second type triangular blocked tensors and give an example to show that these formulas no longer hold for the third type triangular blocked tensors we prove that the product of any two n 1 n r upper or lower triangular blocked tensors of the first or second or third type is still an n 1 n r upper or lower triangular blocked tensor of the same type we also prove that if an n 1 n r upper triangular blocked tensor of the first or second or third type has a left k inverse then its unique left k inverse is still an n 1 n r upper triangular blocked tensor of all the three types also if it has a right k inverse then all of its right k inverses are still n 1 n r upper triangular blocked tensors of all the three types by showing that the left k inverse if any of a weakly irreducible nonsingular m tensor is a positive tensor we show that the left k inverse if any of a first or second or third type normal n 1 n r upper triangular blocked nonsingular m tensor is an n 1 n r upper triangular blocked tensor of all the three types all of whose diagonal blocks are positive tensors we also show that every order m dimension n tensor is permutation similar to some third type normal upper triangular blocked tensor all of whose diagonal blocks are irreducible we give an example to show that this is not true for the first type normal upper triangular blocked tensor

universality classes in burgers turbulence we establish necessary and sufficient conditions for the shock statistics to approach self similar form in burgers turbulence with levy process initial data the proof relies upon an elegant closure theorem of bertoin and carraro and duchon that reduces the study of shock statistics to smoluchowski s coagulation equation with additive kernel and upon our previous characterization of the domains of attraction of self similar solutions for this equation

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

sticky particles and scalar conservation laws one dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line which stick under collision with conservation of mass and momentum

kinetic behaviours of aggregate growth driven by time dependent migration birth and death we propose a dynamic growth model to mimic some social phenomena such as the evolution of cities population in which monomer migrations occur between any two aggregates and monomer birth death can simultaneously occur in each aggregate considering the fact that the rate kernels of migration birth and death processes may change with time we assume that the migration rate kernel is ijf t and the self birth and death rate kernels are ig1 t and ig2 t respectively based on the mean field rate equation we obtain the exact solution of this model and then discuss semi quantitatively the scaling behaviour of the aggregate size distribution at large times the results show that in the long time limit i if t0g1 t dt t0g2 t dt 1 or exp t0 g2 t g1 t dt t0f t dt 0 the aggregate size distribution ak t can obey a generalized scaling form ii if t0g1 t dt t0g2 t dt 0 and exp t0 g2 t g1 t dt t0f t dt ak t can take a scale free form and decay exponentially in size k iii ak t will satisfy a modified scaling law in the remaining cases moreover the total mass of aggregates depends strongly on the net birth rate g1 t g2 t and evolves exponentially as exp t0 g1 t g2 t dt which is in qualitative agreement with the evolution of the total population of a country in real world

an eigenvalue method for testing positive definiteness of a multivariate form in this paper we present an eigenvalue method for testing positive definiteness of a multivariate form this problem plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control at first we apply the d andrea dickenstein version of the classical macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor by using the supersymmetry property we give detailed computation procedures for the bezoutians and specified ordering of monomials in this approach we then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables some numerical results of this method are reported

triangle percolation in mean field random graphs with pde we apply a pde based method to deduce the critical time and the size of the giant component of the triangle percolation on the erd s renyi random graph process investigated by derenyi palla and vicsek in phys rev lett 94 160202 2005 j stat phys 128 219 227 2007

self attracting poisson clouds in an expanding universe we consider the following elementary model for clustering by ballistic aggregation in an expanding universe at the initial time there is a doubly infinite sequence of particles lying in a one dimensional universe that is expanding at constant rate we suppose that each particle p attracts points at a certain rate a p 2 depending only on p and when two particles say p and q collide by the effect of attraction they merge as a single particle p q the main purpose of this work is to point at the following remarkable property of poisson clouds in these dynamics under certain technical conditions if at the initial time the system is distributed according to a spatially stationary poisson cloud with intensity 0 then at any time t 0 the system will again have a poissonian distribution now with intensity t where the family solves a generalization of smoluchowski s coagulation equation

some remarks on large time asymptotic of the lifshitz slyozov equations we obtain an entropy functional for the lifshitz slyozov system this can be used to investigate the time asymptotics of the system in particular we describe situations in which the monomers concentration either tend to 0 or saturate as time becomes large the latter situation can be excluded under assumptions on the support of the initial data

approach to self similarity in smoluchowski s coagulation equations we consider the approach to self similarity or dynamical scaling in smoluchowski s equations of coagulation for the solvable kernels k x y 2 x y and xy in addition to the known self similar solutions with exponential tails there are one parameter families of solutions with algebraic decay whose form is related to heavy tailed distributions well known in probability theory for k 2 the size distribution is mittag leffler and for k x y and k xyit is a power law rescaling of a maximally skewed stable levy distribution we characterize completely the domains of attraction of all self similar solutions under weak convergence of measures our results are analogous to the classical characterization of stable distributions in probability theory the proofs are simple relying on the laplace transform and a fundamental rigidity lemma for scaling limits c 2004 wiley periodicals inc

an analytical solution of coagulation processes with collision induced fragmentation we propose a single species coagulation model with collision induced fragmentation in which a pair of clusters can coagulate into a larger one if their encounter is a complete inelastic collision otherwise one of them will fragment into two smaller clusters due to a destructive collision consider a simple system with the constant coagulation kernel i i j i and the size dependent fragmentation kernel j l i j 2jlu we then investigate analytically the kinetic behavior of such a system by means of the rate equation approach it is indicated that the results are crucially dependent on the value of index u in the case of u 1 and i 2j u here x is the riemann zeta function the cluster size distribution ak t approaches the conventional scaling form and the system will evolve permanently in time while in other cases ak t always takes the modified scaling form moreover a balance can be established between coagulation and collision induced breakage processes and thus the system can eventually evolve to a steady state

asymptotic behaviour of solutions to a simplified lifshitz slyozov equation abstract for the system c u 1 c u 0 c d 0 c d 1 we prove for several classes of initial data c 0 that lim 2 x c x d x h x where h is a known function determined by the behaviour of c 0 near the largest values in its support if c 0 has compact support near whose supremum a it behaves like a q q 1 then h also has compact support near whose supremum it behaves like a q 1 if c 0 goes to zero exponentially fast as a or if its support is infinite and it decays exponentially fast for large then h x is e x if c 0 has infinite support and decays for large like a negative power of then so does h we also give examples of initial data for which the above limit does not exist

some inequalities on the minimum h eigenvalue of the fan product of z tensors abstract in this paper some bounds on the minimum h eigenvalue of a z tensor and the fan product of z tensors are given by applying the bounds we obtain some criteria for m tensors and the positive definiteness of the fan product of z tensors and give some properties for the determinant of the fan product of m tensors

stability of self similar solutions in a simplified lsw model abstract we study the large time asymptotics of solutions to a simplified lifshitz slyozov wagner lsw model of domain coarsening in earlier work it was shown that the asymptotics of solutions to the lsw model depend in a sensitive way on the behaviour of the initial data near the end of its support for the simplified lsw model we characterise the domains of attraction of all the self similar solutions

a survey of numerical solutions to the coagulation equation we present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form aij i j i j and monodisperse initial conditions the results confirm that there are three classes of rate coefficients with qualitatively different solutions for 1 and 1 the numerical solution evolves in an orderly fashion and tends towards a self similar solution at large time t the properties of the numerical solution in the scaling limit agree with the analytic predictions of van dongen and ernst in particular for the subset with 0 and 1 the numerical solution tends towards a self similar solution as t approaches a finite time t0 the mass spectrum nk develops at t0 a power law tail nk k at large masses that violates mass conservation and runaway growth gelation is expected to start at tcrit t0 in the limit the initial number of particles n0 the exponent is in general less than the analytic prediction 3 2 and t0 k 1 n0a11 with k 1 2 if 1 1 for 1 the behaviours of the numerical solution are similar to those found in a previous paper by us they strongly suggest that there are no self consistent solutions at any time and that runaway growth is instantaneous in the limit n0 they also indicate that the time tcrit for the onset of runaway growth decreases slowly towards zero with increasing n0

the modified becker doring system with aggregation dominated rate coefficients we consider a system of becker doring equations modified in the manner of dreyer duderstadt 2006 j stat phys 123 55 we analyse the case of size independent rate coefficients in the case of weak fragmentation aggregation dominated we use matched asymptotic expansions to construct approximations to the cluster size distribution as it evolves from purely monomeric initial data to the equilibrium solution in which the typical cluster size is extremely large we compare the results with earlier calculations of a similar limit in the standard becker doring system and find a similar sequence of timescales with similar profile shapes however there are notable differences in the kinetic behaviour within timescales due to the modified system having a fragmentation rate which depends on the total number of clusters present in the system

well posedness of smoluchowski s coagulation equation for a class of homogeneous kernels the uniqueness and existence of measure valued solutions to smoluchowski s coagulation equation are considered for a class of homogeneous kernels denoting by 2 0 the degree of homogeneity of the coagulation kernel a measure valued solutions are shown to be unique under the sole assumption that the moment of order of the initial datum is finite a similar result was already available for the kernels a x y 2 x y and xy and is extended here to a much wider class of kernels by a different approach the uniqueness result presented herein also seems to improve previous results for several explicit kernels furthermore a comparison principle and a contraction property are obtained for the constant kernel

self similar solutions with fat tails for smoluchowski s coagulation equation with locally bounded kernels the existence of self similar solutions with fat tails for smoluchowski s coagulation equation has so far only been established for the solvable kernels and the diagonal one in this paper we prove the existence of such self similar solutions for continuous kernels k that are homogeneous of degree gamma in infty 1 and satisfy k x y c x y more precisely for any rho in gamma 1 we establish the existence of a continuous weak nonnegative self similar profile with decay x 1 rho as x

self similar spreading in a merging splitting model of animal group size in a recent study of certain merging splitting models of animal group size degond et al in j nonlinear sci 27 2 379 424 2017 it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense corresponding to unbounded growth of group size in the present paper we show that for any such initial distribution with a power law tail the solution approaches a self similar spreading form a one parameter family of such self similar solutions exists with densities that are completely monotone having power law behavior in both small and large size regimes with different exponents

a uniqueness result for self similar profiles to smoluchowski s coagulation equation revisited in this note we indicate how to correct the proof of a uniqueness result in 6 for self similar solutions to smoluchowski s coagulation equation for kernels k k x y that are homogeneous of degree zero and close to constant in the sense that begin aligned varepsilon le k x y 2 le varepsilon big big frac x y big alpha big frac y x big alpha big end aligned for alpha in 0 frac 1 2 under the additional assumption in comparison to 6 that k has an analytic extension to mathbb c setminus infty 0 and that the precise asymptotic behaviour of k at the origin is prescribed we prove that self similar solutions with given mass are unique if varepsilon is sufficiently small the complete details of the proof are available in 4 in addition we give here the proof of a uniqueness result for a related but simpler problem that appears in the description of self similar solutions for x rightarrow infty

existence of self similar solutions to smoluchowski s coagulation equation the existence of self similar solutions to smoluchowski s coagulation equation has been conjectured for several years by physicists and numerical simulations have confirmed the validity of this conjecture still there was no existence result up to now except for the constant and additive kernels for which explicit formulae are available in this paper the existence of self similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

the inviscid burgers equation with brownian initial velocity the law of the hopf cole solution of the inviscid burgers equation with brownian initial velocity is made explicit as examples of applications we investigate the smoothness of the solution the statistical distribution of the shocks we determine the exact hausdorff function of the lagrangian regular points and investigate the existence of lagrangian regular points in a fixed borel set

non self similar behavior in the lsw theory of ostwald ripening the classical lifshitz slyozov wagner theory of domain coarsening predicts asymptotically self similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field here we consider the long time behavior of measure valued solutions for systems in which particle size is uniformly bounded i e for initial measures of compact support we prove that the long time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support we give a necessary criterion for convergence to other self similar solutions and conditional stability theorems for some such solutions for a dense set of initial data convergence to any self similar solution is impossible

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

instantaneous gelation in smoluchowski s coagulation equation revisited we study the solutions of the smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size m we focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization numerical simulations demonstrate that for such kernels with monodisperse initial data the regularized gelation time decreasesas m increases consistent with the expectation that the gelation time is zero in the unregularized system this decrease appears to be a logarithmically slow function of m indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff we also study the case when a source of monomers is introduced in the regularized system in this case a stationary state is reached we present a complete analytic description of this regularized stationary state for the model kernel k m 1 m 2 max m 1 m 2 ensuremath nu which gels instantaneously when m ensuremath rightarrow ensuremath infty if ensuremath nu g1 the stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ensuremath nu for large cluster sizes the total particle density in the stationary state slowly vanishes as ensuremath nu ensuremath 1 mathrm log m ensuremath 1 2 when m ensuremath rightarrow ensuremath infty the approach to the stationary state is nontrivial oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff m which decay very slowly when m is large a quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

numerical approximation of the lifshitz slyozov wagner equation the lifshitz slyozov wagner theory of coarsening ostwald ripening describes the late stages of the growth by diffusional mass transfer of the grains of a new phase from a supersaturated solution it results in a nonlinear transport equation with a nonlocal nonlinearity for the volume distribution function of the grains a time explicit finite volume numerical scheme is proposed to solve this equation in self similar variables and is shown to converge under a cfl condition numerical simulations are also presented

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

domain stability competition growth and selection in globally constrained bistable systems a general globally constrained reaction diffusion equation is suggested that describes formation of equilibrium domains and competition between domains in bistable media we study the stability properties of domains strips and perfect circlular or spherical drops in one two and three dimensions the dynamics of the distribution function df of many drops with respect to their radii is mapped into a mean field model of ostwald ripening a family of similarity solutions for the df is found and the long standing problem of selection rule for the correct asymptotic df is solved textcopyright 1996 the american physical society

bessel identities in the waldspurger correspondence over a p adic field in this paper we develop the local theory for a jacquet s relative trace formula the local theory is essential to the application of the trace formula it is an identity of bessel and relative bessel distributions we show that the relative bessel distribution attached to a distinguished that is self contragradient unitary representation of gl 2 k where k is a p adic field is given by a locally integrable function namely the relative bessel function we compute the relative bessel function for principal series complementary series and special representations we also show that the bessel distributions associated to unitary irreducible admissible representations of the double covers of gl 2 k and sl 2 k are given by a locally integrable bessel functions we compute these bessel functions for principal series complementary series and special representations finally we obtain the waldspurger correspondence via bessel identities between relative bessel functions on gl 2 k and bessel functions on the double cover of sl 2 k the bessel identity also implies the identity between bessel and relative bessel distributions

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

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

general solutions and scaling violation for fragmentation with mass loss explicit solutions of a linear rate equation lead to improved understanding of fragmenlatian with discrete and continuous mass loss discrete mass loss provides a general expression for the overall mass 10 rate which accounts for mass loss in the shattering regime where runaway fragmentation for small panicles producer a phase of zero mass panicles an explicit solution far the recession regime where small panicles lose large fractionsoftheir mass to surface recession shows that the total number ofpaniclesincreases with time until the typical panicles become small enough to lose all of their mass and disappear rather than to break a general series solution is presented for fragmentation with continuous and discrete mass loss a proof that continuous mass loss precludes dynamic scaling is presented

post gelation solutions to smoluchowski s coagulation equation it is proven how the post gelation behaviour originally suggested by flory 1993 can be obtained as a result of a limiting process passing from a finite to an infinite system in a previous paper by the authors it was shown how the post gelation behaviour first suggested by stockmayer 1943 can be obtained by passing to the limit of an infinite system in a different way it is thus demonstrated that different post gelation solutions of smoluchowski s coagulation equation can be obtained by different limiting processes

some properties of strong h tensors and general h tensors abstract h matrices matrices whose comparison matrix is an m matrix are well studied in matrix theory and have numerous applications e g linear complementarity problems and iterative methods for solving systems of linear equations in this article we establish some properties of their tensor counterparts strong h tensors and general h tensors

on the effect of stochastic fluctuations in the dynamics of the lifshitz slyozov wagner model in this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed corrections to the lifshitz slyozov wagner lsw model that take into account the stochastic character of the problem are computed it is proved under suitable smallness assumptions for the volume fraction filled by the particles that the effect of these corrections does not modify much the dynamics of the self similar solutions of the lsw system of equations

long time behavior of a modified becker doring system a modification based on asymptotic behavior of the becker doring system is introduced in which the concentration of monomers is slaved to the concentrations of the other clusters this modified system has the same continuum limit as the usual becker doring system for one member of the modified systems it is proved for compact initial data that all solutions will converge to the same self similar form as time tends to infinity

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

global stability for a class of nonlinear pde with non local term this paper is concerned with establishing global asymptotic stability results for a class of non linear pde in which the nonlinearity enters as a scalar multiple which is non local the method of proof does not involve a lyapounov function it is shown that stability for the pde is equivalent to stability for a differential delay equation stability for the delay equation is proven by exploiting certain monotonicity properties these are established by using the methods of optimal control theory

exponential decay towards equilibrium for the inhomogeneous aizenman bak model the aizenman bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non degenerate size dependent coefficient both the break up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels we demonstrate that the entropy entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non degenerate diffusion in dimension one we finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial up to our knowledge this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation fragmentation models

on bessel distributions for quasi split groups we show that the bessel distribution attached by gelfand and kazhdan and by shalika to a generic representation of a quasi split reductive group over a local field is given by a function when it is restricted to the open bruhat cell as in the case of the character distribution this function is real analytic for archimedean fields and locally constant for non archimidean fields

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

self similar behaviour for noncompactly supported solutions of the lsw model in this paper a method of approximating the classical lsw model near self similar solutions for initial data with infinite support is developed the resulting problem is an integrodifferential equation having two time scales that can be studied using multiple scale methods the analysis provides a detailed description of the precise manner in which the characteristics leak through the critical radius associated to the self similar solutions the analysis in this paper makes precise the meaning of the iterated logarithmic asymptotics in the dynamics of the lsw model that were already obtained in the original lifshitz slyozov paper examples of noncompactly supported solutions of the lsw model that do not behave in a self similar manner are also given

dynamical scaling in smoluchowski s coagulation equations uniform convergence smoluchowski s coagulation equation is a fundamental mean field model of clustering dynamics we consider the approach to self similarity or dynamical scaling of the cluster size distribution for the solvable rate kernels k x y 2 x y and xy in the case of continuous cluster size distributions we prove uniform convergence of densities to a self similar solution with exponential tail under the regularity hypothesis that a suitable moment have an integrable fourier transform for discrete size distributions we prove uniform convergence under optimal moment hypotheses our results are completely analogous to classical local convergence theorems for the normal law in probability theory the proofs rely on the fourier inversion formula and the solution for the laplace transform by the method of characteristics in the complex plane

on global asymptotic stability for the lsw model with subcritical initial data the main result of the paper is a global asymptotic stability result for solutions to the lifschitz slyozov wagner lsw system of equations this extends some local asymptotic stability results of niethammer velazquez commun partial differ equ 31 1679 1708 2006 the method of proof is along similar lines to the one used in a previous paper of the authors this previous paper proves global asymptotic stability for a class of infinite dimensional dynamical systems for which no lyapunov function is apparently available