generic singularities of nilpotent orbit closures according to a theorem of brieskorn and slodowy the intersection of the nilpotent cone of a simple lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity at the opposite extremity of the poset of nilpotent orbits the closure of the minimal nilpotent orbit is also an isolated symplectic singularity called a minimal singularity for classical lie algebras kraft and procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures specifically any such singularity is either a simple surface singularity a minimal singularity or a union of two simple surface singularities of type a 2k 1 in the present paper we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional lie algebras up to normalization in a few cases we summarize the results in some graphs at the end of the paper in most cases we also obtain simple surface singularities or minimal singularities though often with more complicated branching than occurs in the classical types there are however six singularities that do not occur in the classical types three of these are unibranch non normal singularities an sl 2 c variety whose normalization is a 2 an sp 4 c variety whose normalization is a 4 and a two dimensional variety whose normalization is the simple surface singularity a 3 in addition there are three 4 dimensional isolated singularities each appearing once we also study an intrinsic symmetry action on the singularities extending slodowy s work for the singularity of the nilpotent cone at a point in the subregular orbit

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

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

gradings on semisimple algebras abstract the classification of gradings by abelian groups on finite direct sums of simple finite dimensional nonassociative algebras over an algebraically closed field is reduced by means of the use of loop algebras to the corresponding problem for simple algebras this requires a good definition of free products of group gradings

normality of nilpotent varieties in e6 abstract we determine which nilpotent orbits in e6 have closures which are normal varieties and which do not at the same time we are able to verify a conjecture in e sommers comm math univ sancti pauli 49 1 2000 101 104 concerning functions on non special nilpotent orbits for e6

on the coordinate ring of spherical conjugacy classes let g be a simple algebraic group over an algebraically closed field k of characteristic zero and mathcal o be a spherical conjugacy class of g we determine the decomposition of the coordinate ring k mathcal o of mathcal o into simple g modules

kinetics of gelation and universality the exact solution size distribution ck t and moments m t of smoluchowski s coagulation equation s model and of a modified equation f model with a coagulation rate k ij for i and j clusters is obtained for arbitrary ck 0 in the sol t phases where tc is the gel point the behaviour of ck t and m t is given for k co t co and t i the critical exponents critical amplitudes and scaling function that characterise the singularities near the non equilibrium phase transition are calculated for short range ck 0 i e all m a the f model belongs to the universality class of classical gelation theories and of bond percolation on cayley trees the s model does not

on the possible occurrence of instantaneous gelation in smoluchowski s coagulation equation considers possible solutions of smoluchowski s coagulation equation if the rate constants k i j behave as k i j approximately imu jv as j to infinity with an exponent nu satisfying nu 1 the author finds that for such rate constants smoluchowski s equation predicts the instantaneous occurrence of a gelation transition thus the gel time tc 0 in such models this result confirms recent speculation in the literature the author also studies the structure of post gel solutions of smoluchowski s equation if they exist for a given value of nu the results depend on the value of the exponent mu if mu nu 1 one finds that the cluster size distribution ck t approaches a universal form at large times t to infinity no solutions exist if mu or nu 1 physically this means that the sol phase is depleted instantaneously

new exactly solvable models of smoluchowski s equations of coagulation the smoluchowski equations of coagulation are solved analytically in two cases involving a finite cut off of the system the constant kernel set to zero for any j n on the one hand and the general three particle case on the other both are seen to exhibit rather unusual large time behaviour the first model can be used to account for large particles precipitating out of a system and its behaviour is therefore of particular interest

solutions of smoluchowski s coagulation equation at large cluster sizes abstract in this paper we determine the behavior of solutions c k t of smoluchowski s coagulation equation for cluster sizes much larger than the mean cluster size s t we consider in general the homogeneous rate constants k i j behaving as k i j i j v as j where special attention is paid to models with an exponent v 1 the behavior of c k t is studied in three different limits i the short time limit t 0 with k 1 ii the limit k with t 0 fixed and iii the scaling limit with k s t the two most important conclusions of this paper are first that the detailed behavior of c k t at large cluster sizes k s t may be drastically different for different rate constants k i j and secondly that the results for c k t obtained in the limits i ii and iii are closely related

more non semigroup lie gradings this note is devoted to the construction of two very easy examples of respective dimensions 4 and 6 of graded lie algebras whose grading is not given by a semigroup the latter one being a semisimple algebra it is shown that 4 is the minimal possible dimension for such graded lie algebras

an exact solution of the discrete smoluchowski equation and its correspondence to the solution of the continuous equation an exact general solution of the discrete smoluchowski equation for the kernel kk a b k j is derived rules are given for taking the continuous limit of the discrete solution the limit obtained is an exact general solution of the continuous smoluchowski equation examples of limits are given for specific initial distributions for the case of constant kernel

domain number distribution in the nonequilibrium ising model we study domain distributions in the one dimensional ising model subject to zero temperature glauber and kawasaki dynamics the survival probability of a domain s t t and an unreacted domain q1 t t are characterized by two independent nontrivial exponents we develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents

nilpotent orbits normality and hamiltonian group actions the theory of coadjoint orbits of lie groups is central to a number of areas in mathematics a list of such areas would include 1 group representation theory 2 symmetry related hamiltonian mechanics and attendant physical theories 3 symplectic geometry 4 moment maps and 5 geometric quantization from many points of view the most interesting cases arise when the group g in question is semisimple for semisimple g the most familiar of the orbits are orbits of semisimple elements in that case the associated representation theory is pretty much understood the borel weil bott theorem and noncompact analogs e g zuckerman functors the isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations a totally different experience is encountered with nilpotent orbits of semisimple groups here the associated representation theory unipotent representations is poorly understood and there is a loss of reductivity of isotropy subgroups to make matters worse or really more interesting orbits are no longer closed and there can be a failure of normality for orbit closures in addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations the interest in nilpotent orbits of semisimple lie groups has increased sharply over the last two decades this perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits e g the springer correspondence with representations of the weyl group this paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry the starting point is the recognition made also by others that the ring r of regular functions on any g homogeneous cover m of 0 is not only a poisson algebra the case for any coadjoint orbit but that r is also naturally graded the key theme is that the same nilpotent orbit may be shared by more than one simple group the key result is the determination of all pairs of simple lie groups having a shared nilpotent orbit furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

gradings on tensor products of composition algebras and on the smirnov algebra abstract we give classifications of group gradings up to equivalence and up to isomorphism on the tensor product of a cayley algebra c and a hurwitz algebra over a field of characteristic different from 2 we also prove that the automorphism group schemes of c n and c n are isomorphic on the other hand we prove that the automorphism group schemes of a smirnov algebra t c a 35 dimensional simple exceptional structurable algebra constructed from a cayley algebra c and c are isomorphic this is used to obtain classifications up to equivalence and up to isomorphism of the group gradings on smirnov algebras

cyclization in bipartite random graphs in this paper the time evolution of a finite bipartite graph initially comprising two sorts of isolated vertices is considered the graph is assumed to evolve by adding edges one at a time each new edge connects either two linked components and forms a new component of a larger order coalescence of graphs or increases by one the number of edges in a given linked component cycling any state of the graph is thus characterized by the set of occupation numbers the numbers of linked components comprising a given numbers of vertexes of the both sorts and a given number of edges once the rate of appearance of an extra edge in the graph being known the master equation governing the time evolution of the probability to find the random graph in a given state is reformulated in terms of the functional generating the probability to find the evolving graph in a given state the exact solution of the evolution equation for the generating functional applies for analyzing the average population numbers of linked components in the limit of large order of the graph the distribution factorizes into two multipliers one of which is just the spectrum of linked components in the infinite bipartite graph the second multiplier includes the dependence on the total size of the graph both these multipliers contain information on the emergence of the giant component that forms at a critical time

coarsening in the 1d ising model evolving with swendsen wang dynamics an unusual scaling we consider a simple model of domain growth the zero temperature 1d ising model evolving according to the swendsen wang dynamics we find that in the long time limit the pair correlation function scales with a characteristic length increasing as the square of the average domain size in that limit a few large domains occupy almost all the space with many small domains between them in contrast to the usual picture of coarsening the average domain size here is not a characteristic length of the growth problem instead one finds a power law distribution for the sizes of large domains with a cut off at a length which grows as the square of the average size of the domains

scaling and multiscaling in models of fragmentation we introduce a simple geometric model which describes the kinetics of fragmentation of d dimensional objects in one dimension our model coincides with the randon scission model and shows a simple scaling behavior in the long time limit for dg1 the volume of the fragments is characterized by a single scale 1 t while other geometric properties such as the length are characterized by an infinite number of length scales and thus exhibit multiscaling

fluctuations in coagulating systems ii in this paper we consider coagulation processes in large but finite systems and study the time dependent behavior of the nonequilibrium fluctuations in the cluster size distribution for this purpose we apply van kampen s expansion to a master equation describing coagulation processes and derive an approximate fokker planck equation for the probability distribution of the fluctuations first we consider two exactly soluble models corresponding to the choicesk i j i jandk i j 1 for the rate constants in the fokker planck equation for these models and monodisperse initial conditions we calculate the probability distribution of the fluctuations and the equal time and two time correlation functions for general initial conditions we study the behavior of the fluctuations at large cluster sizes and in the scaling limit next we consider in general homogeneous rate constants with the propertyk i j a k ai aj for alla 0 and we give asymptotic expressions for the equal time correlation functions at large cluster sizes and in the scaling limit in the scaling limit we find that the fluctuations show relatively simple scaling behavior for all homogeneous rate constantsk i j

a solvable two species catalysis driven aggregation model we study the kinetics of a two species catalysis driven aggregation system in which an irreversible aggregation between any two clusters of one species occurs only with the catalytic action of another species by means of a generalized mean field rate equation we obtain the asymptotic solutions of the cluster mass distributions in a simple process with a constant rate kernel for the case without any consumption of the catalyst the cluster mass distribution of either species always approaches a conventional scaling law however the evolution behaviour of the system in the case with catalyst consumption is complicated and depends crucially on the relative data of the initial concentrations of the two species

kinetic theory of random graphs from paths to cycles the structural properties of evolving random graphs are investigated treating linking as a dynamic aggregation process rate equations for the distribution of node to node distances paths and of cycles are formulated and solved analytically at the gelation point the typical length of paths and cycles l scales with the component size k as l approximately k 1 2 dynamic and finite size scaling laws for the behavior at and near the gelation point are obtained finite size scaling laws are verified using numerical simulations

unicyclic components in random graphs the distribution of unicyclic components in a random graph is obtained analytically the number of unicyclic components of a given size approaches a self similar form in the vicinity of the gelation transition at the gelation point this distribution decays algebraically uk 4k 1 for k 1 as a result the total number of unicyclic components grows logarithmically with the system size

on the irreducibility of associated varieties of w algebras abstract we investigate the irreducibility of the nilpotent slodowy slices that appear as the associated variety of w algebras furthermore we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves

nonequilibrium fluctuations in μ space the properties of fluctuations in space in or outside thermal equilibrium are obtained by solving hierarchies of equations derived either from the liouville or the master equation in particular we study the one two etc time correlation functions that describe the spatial and temporal behavior of the fluctuations in space explicit solutions are obtained for a dilute gas the langevin approach is briefly discussed our results are compared with those obtained in the extensive literature which is reviewed in some detail

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman 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 they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

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

kinetics of ring formation we study reversible polymerization of rings in this stochastic process two monomers bond and as a consequence two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings this aggregation fragmentation process exhibits a percolation transition with a finite ring phase in which all rings have microscopic length and a giant ring phase where macroscopic rings account for a finite fraction of the entire mass interestingly while the total mass of the giant rings is a deterministic quantity their total number and their sizes are stochastic quantities the size distribution of the macroscopic rings is universal although the span of this distribution increases with time moreover the average number of giant rings scales logarithmically with system size we introduce a card shuffling algorithm for efficient simulation of the ring formation process and we present numerical verification of the theoretical predictions

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

random planar lattices and integrated superbrownian excursion in this paper a surprising connection is described between a specific brand of random lattices namely planar quadrangulations and aldous integrated superbrownian excursion ise as a consequence the radius r n of a random quadrangulation with n faces is shown to converge up to scaling to the width r r l of the support of the one dimensional ise or precisely more generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ise shifted to set the minimum of its support in zero the first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half line reminiscent of cori and vauquelin s well labelled trees the second step relates these trees to embedded discrete trees in the sense of aldous via the conjugation of tree principle an analogue for trees of vervaat s construction of the brownian excursion from the bridge from probability theory we need a new result of independent interest the weak convergence of the encoding of a random embedded plane tree by two contour walks to the brownian snake description of ise our results suggest the existence of a continuum random map describing in term of ise the scaled limit of the dynamical triangulations considered in two dimensional pure quantum gravity

stochastic aggregation scaling properties we study the scaling properties of stochastic aggregation processes in one dimension numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent non trivial exponents corresponding to the survival probability of particles and monomers the overall behaviour agrees qualitatively with the mean field theory this theory also provides a useful approximation for the decay exponents as well as the limiting mass distribution

a general algorithm for exact simulation of multicomponent aggregation processes a monte carlo mc algorithm is presented for the simulation of the time evolution of aggregation processes featuring multiple components properties or conservation laws instead of using deterministic differential population balance equations the mc algorithm utilizes a stochastic approach to aggregation kinetics as a result exact simulation of spatially independent aggregation processes is possible without the need for numerical approximations furthermore simulations exactly predict all moments of the size and composition distributions of aggregating particles for both nongelling and gelling kernels and extend these results to the postgelation period the algorithm is shown to require at most o 1 2 1 1 rate limiting operations per time step for a component aggregation process featuring i monomers of each component i a substantial performance improvement over the potential of previous methods simulation results are presented for bivariate sum product and constant kernels and for the perikinetic brownian kernel

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

fluctuations in reaction diffusion systems a new exactly soluble growth model the method of compounding moments devised by van kampen is used to study the spatial fluctuations in a model describing the irreversible formation of clusters the reaction and diffusion constants in this model are chosenindependent of the cluster sizes for a monodisperse initial distribution explicit expressions are calculated for the equal time and two time correlation functions of the concentrations ofm andn mers for general initial conditions the fluctuations in the mass density are considered and a scaling theory is presented for the fluctuations at large times extensions to more general models are discussed

universal distribution of masses in a one dimensional sticky gas abstract agglomeration of particles by perfectly inelastic collisions is studied in the case of a one dimensional gas particles move freely between collisions both mass and momentum are conserved at binary encounters it is shown that under a weak mean field like hypothesis admitting the existence of correlations between the velocities of colliding particles the dynamics drives the system towards a scaling regime with a universal exponential distribution of masses

a characterization of nilpotent orbit closures among symplectic singularities we prove that a conical symplectic variety with maximal weight 1 is isomorphic to i an affine space with the standard symplectic form or ii a normal nilpotent orbit closure of a complex semisimple lie algebra

aggregation with multiple conservation laws aggregation processes with an arbitrary number of conserved quantities are investigated on the mean field level an exact solution for the size distribution is obtained the asymptotic form of this solution exhibits nontrivial double scaling while processes with one conserved quantity are governed by a single scale processes with multiple conservation laws exhibit an additional diffusion like scale the theory is applied to ballistic aggregation with mass and momentum conserving collisions and to diffusive aggregation with multiple species

polymers and random graphs we establish a precise connection between gelation of polymers in lushnikov s model and the emergence of the giant component in random graph theory this is achieved by defining a modified version of the erdos renyi process when contracting to a polymer state space this process becomes a discrete time markov chain embedded in lushnikov s process the asymptotic distribution of the number of transitions in lushnikov s model is studied a criterion for a general markov chain to retain the markov property under the grouping of states is derived we obtain a noncombinatorial proof of a theorem of erdos renyi type