solution of population balance equations in applications with fine particles the accurate description and robust simulation at relatively low cost of global quantities e g number density or volume fraction as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications for this purpose two types of methods are investigated for solving the population balance equation with aggregation continuous particle size change growth and size reduction and nucleation the extended quadrature method of moments eqmom based on the work of yuan et al 52 and a hybrid method tsm between the sectional and moment methods considering two moments per section based on the work of laurent et al 30 for both methods the closure employs a continuous reconstruction of the number density function of the particles from its moments thus allowing evaluation of all the unclosed terms in the moment equations including the negative flux due to the disappearance of particles here new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method moreover robust and accurate numerical methods are developed ensuring the realizability of the moments the robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy eqmom and tsm are compared to a sectional method for various simple but relevant test cases showing their ability to describe accurately the fine particle population with a much lower number of variables these results demonstrate the efficiency of the modeling and numerical choices and their potential for the simulation of real world applications

mass based finite volume scheme for aggregation growth and nucleation population balance equation in this paper a new mass based numerical method is developed using the notion of forestier coste mancini forestier coste mancini 2012 siam j sci comput 34 b840b860 doi 10 1137 11084799

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

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

invariant manifolds of admissible classes for semi linear evolution equations abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of invariant manifolds of this equation these manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory the existence of such manifolds is obtained in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to certain classes of admissible function spaces

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

on brauer stars this paper contains some results on how representations in the principal p block of a finite group with a cyclic sylow p subgroup yield information on the structure of the group throughout the paper g denotes a finite group p a fixed odd prime dividing 1 gi p a sylow p subgroup of g with p p and b p the principal p block of g if p is cyclic then the ordinary and modular irreducible characters in any p block of g can be described in terms of a graph called the bruuer tree belonging to the block as in 5 vii 61 a tree is called a star if there are no paths with more than two edges i e

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

on the mumford tate group of an abelian variety with complex multiplication let k be a primitive cm type with k o 2n for definitions and previous results see section 1 1 fix n and consider the collection s n rank where rank counts the number of independent translates of qi under the galois action and k ranges over all primitive types the smallest element of s n denoted by b n is referred to as the sharp lower bound for the rank in dimension n as was brought to the author s attention by ribet bounds on b n of the form p 1 for p a prime dividing n follow directly from the proof of ribet s nondegeneracy theorem 21 this is recorded as theorem 1 4 ribet s method also gives bounds of the form 2q for q a prime with q2 dividing n as is observed in theorem 1 12 when combined with the author s constructions of abelian varieties in s 91 we obtain the precise value of b n for many values of n corollaries 1 5 to 1 8 and 1 13 we recall that these constructions use the analytic method of weil and shimura together with new results on the reflex field from an investigation suggested to the author by shim ura the interest of the rank comes from the theory of complex multiplication if a is an abelian variety of cm type k then the kubota rank of a is rank and controls properties of the classfields constructed from a as in kubota 161 and ribet 20 this is connected with the fact that the rank of is also the dimension of the mumford tate group of a and with the relation of this group to the i adic representations of a as in serre 24 251 the main body of the paper contains results that provide information on s n the main result theorem 2 5 asserts that when n is odd there is a computable subset s n of s n that accounts for the ranks of many cm types on most cm fields more precisely let k be the maximal totally real subfield of k and kg be the galois closure of k we consider the per

on the approximate solutions of fragmentation equations a numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems initially it is considered that the particle fragments are characterized by a single property i e particle s volume we then investigate the extension of the proposed model for solving breakage problems considering two properties of particles the efficiency to estimate the different moments with good accuracy and simple extension for multi variable problems are the key features of the proposed method moreover the mathematical convergence analysis is performed for one dimensional problems all mathematical findings and numerical results are validated over several test problems for numerical validation we propose the extension of bourgade filbet 2008 math comput 77 851 882 doi 10 1090 s0025 5718 07 02054 6 model for solving two dimensional pure breakage problems in this aspect numerical treatment of the two dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature

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

on a class of continuous coagulation fragmentation equations abstract a model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered each particle being assumed to be fully identified by its size existence of solutions to the corresponding evolution integral partial differential equation is shown for product type coagulation kernels with a weak fragmentation the failure of density conservation or gelation is also investigated in some particular cases

a differentially weighted monte carlo method for two component coagulation the direct simulation monte carlo dsmc method for population balance modeling is capable of retaining the history of each simulation particle and is thus able to deal with multivariate properties in a simple and straightforward manner as opposed to conventional dsmc approaches that track equally weighted simulation particles a differentially weighted monte carlo method is extended to simulate two component coagulation processes and is thereby able to simulate the micromixing of the components a new feature of the method for this bivariate population balance modeling is that it is possible to specify how the simulation particles are distributed over the compositional axis this allows us to obtain information about particles in those regions of the size and composition distribution functions where the non weighted mc methods place insufficient simulation particles to obtain an inaccurate solution the new feature results in lower statistical noise for simulating two component coagulation which is validated by using two component coagulation cases for which analytical solutions exist a discrete process with sum kernel for initial monodisperse populations and a process with constant kernel for initial polydisperse populations

exact solutions of population balance equations for breakage and growth processes using group analysis abstract population balance equations have been applied to model a wide range of processes including breakage crystal nucleation growth agglomeration comminution of ores and other solids but the lack of exact solutions necessitates the use of numerical methods a class of population balance equations with time independent but size dependent growth rate and breakage rate is investigated analytically all invariant solutions explicit exact solutions reduced equations and optimal system of subalgebras for the corresponding system of equations are presented in this paper

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

quasiprimitive groups containing a transitive alternating group abstract this paper classifies quasiprimitive permutation groups with a transitive subgroup which is isomorphic to a n for some n 5

numerical solution of the smoluchowski kinetic equation and asymptotics of the distribution function we obtain the numerical solution of the smoluchowski kinetic equation for model kernels u m1 m2 varies as m1 m2 lambda and u m1 m2 varies as m1m2 lambda 2 0 lambda or 2 we show that the behaviour of the solution for the kernel u varies as m1 m2 lambda at 0 lambda or 1 and u varies as m1m2 lambda 2 at 0 lambda or 2 becomes self similar after some time the shape of the scaling function is analysed in particular a simple approximate expression for it at u varies as m1 m2 lambda is found an interesting result is obtained for u varies as m1m2 lambda 2 0 lambda 1 the asymptotic behaviour of the scaling function proved to be non power we develop the procedure for determining tcr the critical indices and the exponent of the power law asymptotics of the smoluchowski equation solution the concrete values of these quantities for the model kernels are obtained the stability conditions of the algorithm for numerical solving are analysed the possibilities afforded by numerical solution for investigation of the smoluchowski equation are discussed

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

nontrivial polydispersity exponents in aggregation models we consider the scaling solutions of smoluchowski s equation of irreversible aggregation for a non gelling collision kernel the scaling mass distribution f s diverges as s tau when s 0 tau is non trivial and could until now only be computed by numerical simulations we develop here new general methods to obtain exact bounds and good approximations of tau for the specific kernel kdd x y x 1 d y 1 d d describing a mean field model of particles moving in d dimensions and aggregating with conservation of mass s r d r is the particle radius perturbative and nonperturbative expansions are derived for a general kernel we find exact inequalities for tau and develop a variational approximation which is used to carry out the first systematic study of tau d d for kdd the agreement is excellent both with the expansions we derived and with existing numerical values finally we discuss a possible application to 2d decaying turbulence

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

similarity solution of coagulation equation with an inverse kernel abstract we analyse a model for the aggregation of polystyrene particles which arises in an electrorheology system in which linear clusters grow upon the application of an alternating electromagnetic field we consider a coagulation kernel involving negative powers of cluster sizes and investigate the reduction of the governing equations to a similarity solution in the large time limit comparison between the experimental results and the theory presented here shows a good collapse of the data onto a single curve which matches the theoretical results particularly well at the larger cluster sizes

exponential dichotomy of evolution equations and admissibility of function spaces on a half line consider an evolution family u u t s ts0 on a half line r and an integral equation u t u t s u s t s u t f d we characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of lp type the lorentz spaces lp q and many other function spaces occurring in interpolation theory we then apply our results to study the robustness of the exponential dichotomy of evolution families on a half line under small perturbations

a finite version of smoluchowski s coagulation equation the authors formulate a finite version of smoluchowski s coagulation equation in which molecules of size n react in the normal way whereas molecules of size n to 2n can be produced but cannot react they prove that the solution converges to the solution of the infinite system of equations in the limit n to infinity

on the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear schrodinger equation we study the mathematical properties of a kinetic equation derived in escobedo and velazquez arxiv 1305 5746v1 math ph which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear schrodinger equation in particular we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass we also prove that any nontrivial initial datum yields the instantaneous onset of a condensate by which we mean that for any nontrivial solution the mass of the origin is strictly positive for any positive time furthermore we show that the only stationary solutions with finite total measure are dirac masses at the origin we finally construct solutions with finite energy where the energy is transferred to infinity in a self similar manner

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

exponential dichotomy of difference equations and applications to evolution equations on the half line abstract for a sequence of bounded linear operator a n n 0 on a banach space x we investigate the characterization of exponential dichotomy of the difference equations v n 1 a n v n we characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations v n 1 a n v n f n in l space then we apply the results to study the exponential dichotomy of evolution families generated by evolution equations

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

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

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

input output admissibility and exponential trichotomy of difference equations abstract we propose a new method for the study of the asymptotic behavior of difference equations in infinite dimensional spaces providing characterizations for the property of uniform exponential trichotomy we deduce the structure of the stable unstable and bounded subspace and prove the uniqueness of the projection families we introduce a new admissibility concept with respect to a discrete input output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy throughout the paper there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations this paper presents a finite volume scheme fvs for solving general breakage population balance equations bpbes in particular the number density based bpbe is transformed to the form of a mass conservation law then it becomes easy to apply the well known fvss that have an important property of mass conservation following kumar and warnecke for fixed pivot fp method 16 and cell average technique cat 15 the stability and the convergence analysis of the semi discretized fvs are studied unlike the cat and the fp method the fvs is second order consistent independently of the type of meshes we also observe that fvs gives second order convergence rate on four different types of uniform and non uniform meshes with non decreasing behavior of mesh width nevertheless one order better accuracy than the fp method is achieved on locally uniform meshes it is also noticed that on non uniform random meshes the fvs shows one order and two orders higher accuracy than the cat and the fp method respectively the mathematical results of convergence analysis are validated numerically by taking three test problems the numerical simulations are also compared with the results obtained by the cat and the fp method

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

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 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

an efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena a new class of efficient stochastic algorithms for the numerical treatment of coagulation processes is proposed the algorithms are based on the introduction of fictitious jumps combined with an acceptance rejection technique for distributions depending on particle size the increased efficiency is demonstrated by numerical experiments in particular gelation phenomena are studied

groups whose prime graphs have no triangles abstract let g be a finite group and let cd g be the set of all complex irreducible character degrees of g let g be the set of all primes which divide some character degree of g the prime graph g attached to g is a graph whose vertex set is g and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of g in this paper we show that if g is a finite group whose prime graph g has no triangles then g has at most 5 vertices we also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups finally we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices

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

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

admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line abstract we give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation in our approach the input space i r x is an intersection of spaces that can be successively minimized and the output space c r x can be one of some well known spaces of continuous functions using computational arguments we show that the admissibility of c r x i r x leads to a nonuniform exponential dichotomy we expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences moreover we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input output operator finally we present a direct application of the main result in the case of evolution families which admit uniform exponential growth

cluster size distribution in irreversible aggregation at large times it is assumed that the size distribution ck t satisfies smoluchowski s coagulation equation with rate coefficients k i j behaving as k i j approximately imu jnu i 0 the general solution ck t approaches for t to infinity the exact solution cbk t k 1 2 where the bk s are independent of the initial conditions ck 0 and can be determined from a recursion relation in class ii systems mu 0 ck t c1 t to bk t to infinity k 1 2 but the bk s depend on ck 0 only in the scaling limit k to infinity s t to infinity with k s t finite s t is the mean cluster size does ck t approach a form independent of the initial distribution class iii where ck t c1 t to infinity t to infinity k 2 3 has not been considered here

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

the probabilistic zeta function of finite simple groups abstract let g be a finite group there exists a uniquely determined dirichlet polynomial p g s such that if t n then p g t gives the probability of generating g with t randomly chosen elements we show that it may be recognized from the knowledge of p g s whether g frat g is a simple group

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

on finite factorizable groups in this paper we use freely the recent classification theorem for finite simple groups from the early stages of finite group theory finite factorizable groups satisfying h were studied by many authors in 1904 burnside 6 proved the solvability of groups factorizable by two prime power subgroups many years later wielandt 29 proved the solvability of finite groups factorizable by two nilpotent subgroups of relatively prime orders kegel extended wielandt s result in the case that both subgroups a and b are nilpotent without restrictions the second author in 121 classified finite factorizable groups of type h where a and b are solvable subgroups results on groups satisfying h can be found in 3 4 10 12 19 20 291 and in other papers on finite factorizable groups theorem 1 1 and corollary 1 2 of this paper state the final solution to the problem of classifying groups of type h we prove the following

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

a finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence in this paper we present a deterministic numerical approximation of the coalescence or smoluchowski equation the proposed numerical scheme conserves the first order momentum and deals with nonuniform grids the generalization to a multidimensional framework is also described we validate the scheme considering some classical tests in both one and two dimensions and discuss its behavior when gelation occurs our numerical results and code are compared with those already existent in the literature

stable manifolds for semi linear evolution equations and admissibility of function spaces on a half line abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of stable manifolds of solutions to this equation in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

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 primitive overgroups of quasiprimitive permutation groups a permutation group is said to be quasiprimitive if all its non trivial normal subgroups are transitive we investigate pairs g h of permutation groups of degree n such that g h sn with g quasiprimitive and h primitive an explicit classification of such pairs is obtained except in the cases where the primitive group h is either almost simple or the blow up of an almost simple group the theory in these remaining cases is investigated in separate papers the results depend on the finite simple group classification

approximation of coalescence integrals in population balance models with local mass conservation the solution of population balance equations is a function f t r x describing a population density of particles of the property x at time t and space r for instance the additional independent variable x may denote the mass of the particle the describing equation contains additional sink and source terms involving integral operators since the coordinate x adds at least one further dimension to the spatial directions and time coordinate an efficient numerical treatment of the integral terms is crucial one of the more involved integral terms appearing in population balance models is the coalescence integral which is of the form int 0 x kappa x y y f y f x y mathrm d y the discretisation may use a locally refined grid in this paper we describe an algorithm which i is efficient the cost is mathcal o n log n n data size and ii ensures local mass conservation