statistics of collapsing lattice animals we consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal in general the thermodynamic properties of the polymer are determined by monomer monomer and monomer solvent interactions we examine a two variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated in particular we discuss the collapse transition in these models

the structured gerstenhaber problem iii abstract let b be a symmetric or alternating bilinear form on a finite dimensional vector space v when the characteristic of the underlying field is not 2 we determine the greatest dimension for a linear subspace of nilpotent b symmetric or b alternating endomorphisms of v expressing it as a function of the dimension the rank and the witt index of b similar results are obtained for subspaces of nilpotent b hermitian endomorphisms when b is a hermitian form with respect to a non identity involution in three situations b symmetric endomorphisms when b is symmetric b alternating endomorphisms when b is alternating and b hermitian endomorphisms when b is hermitian and the underlying field has more than 2 elements we also characterize the linear subspaces with the maximal dimension our results are wide generalizations of results of meshulam and radwan 7 who tackled the case of a non degenerate symmetric bilinear form over the field of complex numbers and recent results of kokol bukovsek and omladic 4 in which the spaces with maximal dimension were determined when the underlying field is the one of complex numbers the bilinear form b is symmetric and non degenerate and one considers b symmetric endomorphisms

renormalized field theory of collapsing directed randomly branched polymers we present a dynamical field theory for directed randomly branched polymers and in particular their collapse transition we develop a phenomenological model in the form of a stochastic response functional that allows us to address several interesting problems such as the scaling behavior of the swollen phase and the collapse transition for the swollen phase we find that by choosing model parameters appropriately our stochastic functional reduces to the one describing the relaxation dynamics near the yang lee singularity edge this corroborates that the scaling behavior of swollen branched polymers is governed by the yang lee universality class as has been known for a long time the main focus of our paper lies on the collapse transition of directed branched polymers we show to arbitrary order in renormalized perturbation theory with ensuremath epsilon expansion that this transition belongs to the same universality class as directed percolation

metropolis monte carlo simulation of lattice animals we propose and study a metropolis monte carlo algorithm for the simulation of weighted lattice animals in in the canonical ensemble with a fixed number of edges we examine the implementation and performance of the algorithm data obtained by sampling uniformly weighted animals and animals weighted as critical percolation clusters are reported and analysed in particular we estimate autocorrelation times and dynamical exponents to study the efficiency of the algorithm and estimate the metric and branch exponents of lattice animals in two and three dimensions

spaces of matrices with few eigenvalues abstract let k be a commutative field with characteristic not 2 and v be a linear subspace of matrices of m n k that have at most two eigenvalues in k respectively at most one non zero eigenvalue in k we prove that dim v n 2 2 provided that n 3 respectively dim v n 2 1 we also classify up to similarity the linear subspaces of m n k in which every matrix has at most two eigenvalues respectively at most one non zero eigenvalue in an algebraic closure of k and which have the maximal dimension among such spaces

ϵ expansion for directed animals introduces the problem of directed lattice animals and shows that they are highly anisotropic with two different correlation lengths xi sub and xi perpendicular to parallel and perpendicular to the privileged direction a field theory is introduced and the critical exponents calculated to first order in epsilon dc d where the upper critical dimension dc is 7

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

on gerstenhaber s theorem for spaces of nilpotent matrices over a skew field abstract let k be a skew field and k 0 be a subfield of the central subfield of k such that k has finite dimension q over k 0 let v be a k 0 linear subspace of n n nilpotent matrices with entries in k we show that the dimension of v is bounded above by q n 2 and that equality occurs if and only if v is similar to the space of all n n strictly upper triangular matrices over k this generalizes famous theorems of gerstenhaber and serezhkin which cover the special case k k 0

equation of state for directed percolation using field theoretic renormalization group methods we calculate the equation of state for non equilibrium systems belonging to the universality class of directed percolation to second order in by introducing a parametric representation the result can be written to this order in a very simple form we use our result to obtain a universal amplitude ratio to second order in

linear spaces of nilpotent matrices abstract we consider several questions on spaces of nilpotent matrices we present sufficient conditions for triangularizability and give examples of irreducible spaces we give a necessary and sufficient condition in terms of the trace for all linear combinations of a given set of operators to be nilpotent we also consider the question of the dimension of a space l of nilpotents on f n in particular we give a simple new proof of a theorem due to m gerstenhaber concerning the maximal dimension of such spaces

large affine spaces of non singular matrices let k be an arbitrary commutative field with at least three elements it was recently proven that an affine subspace of mn k consisting only of non singular matrices must have a dimension lesser than or equal to n here we classify up to equivalence the subspaces whose dimension equals n 2 this is done by classifying up to similarity all the n 2 dimensional linear subspaces of mn k consisting of matrices with no non zero invariant vector reinforcing a classical theorem of gerstenhaber both classifications only involve the quadratic structure of the field k

spaces of matrices without non zero eigenvalues in their field of definition and a question of szechtman abstract let v be a vector space of dimension n over any field f extreme values for the possible dimension of a linear subspace of end f v with a particular property are considered in two specific cases it is shown that if e 1 is a subspace of end f v and there exists an endomorphism g of v not in e 1 such that for every hyperplane h of v some element of e 1 agrees with g on h then e 1 has dimension at least n n 1 2 1 this answers a question that was posed by szechtman in 2003 it is also shown that a linear subspace of m n f in which no element possesses a non zero eigenvalue in f may have dimension at most n n 1 2 the connection between these two properties which arises from duality considerations is discussed

cluster size distribution above the percolation threshold discusses how the cluster size distribution above the percolation threshold is related to the existence of metastable states of the potts model the authors thus find that the average number of clusters per site containing n sites is given by c n approximately n theta exp constant n1 1 d where d is the dimensionality of the lattice the exponent theta is given for a range of values of d

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

flory theory for directed lattice animals and directed percolation the free energies of directed lattice animals in good and theta solvents and the free energy of directed percolation are found by the use of a simple flory type approximation which accounts for the inherent anisotropy of these systems from these free energies the authors obtain the upper critical dimension below which mean field theory breaks down they also calculate closed form dimension dependent expressions for the parallel and perpendicular correlation length exponents which characterise the asymptotic cluster shapes these exponents are in excellent agreement with existing numerical data

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

solvable hopf algebras and their twists abstract we show that for any solvable group g and a drinfel d twist j k g j is solvable in the sense of the intrinsic definition of solvability given in 2 more generally if a hopf algebra h has a normal solvable series so does h j furthermore while solvable groups are defined as having certain commutative quotients quasitriangular normally solvable hopf algebras have appropriate quantum commutative quotients we end with a detailed example

on the matrices of given rank in a large subspace abstract let v be a linear subspace of m n p k with codimension lesser than n where k is an arbitrary field and n p in a recent work of the author it was proven that v is always spanned by its rank p matrices unless n p 2 and k f 2 here we give a sufficient condition on codim v for v to be spanned by its rank r matrices for a given r 1 p 1 this involves a generalization of the gerstenhaber theorem on linear subspaces of nilpotent matrices

projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

on directed interacting animals and directed percolation we study the phase diagram of fully directed lattice animals with nearest neighbour interactions on the square lattice this model comprises several interesting ensembles directed site and bond trees bond animals strongly embeddable animals as special cases and its collapse transition is equivalent to a directed bond percolation threshold precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n 17 the crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent is determined directly from the singular part of the free energy we show using scaling arguments and an exact relation due to dhar that is equal to the fisher exponent governing the size distribution of large directed percolation clusters

non universality in the collapse of two dimensional branched polymers in this paper we study the complete phase diagram of a model of interacting branched polymers the model we consider is a lattice animal one where the collapse transition can be driven both by a contact fugacity between two occupied nearest neighbours and by a fugacity related to each occupied edge using a potts model formulation of the problem we conjecture the existence of two different universality classes for the theta transitions with thermal exponents nu and phi equal to 1 2 2 3 and 8 15 8 15 separated by a higher order percolation point we also present convincing numerical evidence for these exponent values using a transfer matrix approach we discuss the possibility of a collapse collapse transition and we predict the behaviour of our model when an adsorbing surface is included

comment on scaling in reaction diffusion systems it is shown that the claims and results of elderfield and wilby on critical behaviour in reaction diffusion models of the form px from mx to qx nx to sx are incorrect on a number of points

the singular linear preservers of non singular matrices given an arbitrary field k we reduce the determination of the singular endomorphisms f of mn k such that f gln k gln k to the classification of n dimensional division algebras over k our method which is based upon dieudonne s theorem on singular subspaces of mn k also yields a proof for the classical non singular case

generalized epidemic process and tricritical dynamic percolation the renowned general epidemic process describes the stochastic evolution of a population of individuals which are either susceptible infected or dead a second order phase transition belonging to the universality class of dynamic isotropic percolation lies between the endemic and pandemic behavior of the process we generalize the general epidemic process by introducing a fourth kind of individuals viz individuals which are weakened by the process but not yet infected this weakening gives rise to a mechanism that introduces a global instability in the spreading of the process and therefore opens the possibility of a discontinuous transition in addition to the usual continuous percolation transition the tricritical point separating the lines of first and second order transitions constitutes an independent universality class namely the universality class of tricritical dynamic isotropic percolation using renormalized field theory we work out a detailed scaling description of this universality class we calculate the scaling exponents in an epsilon expansion below the upper critical dimension d c 5 for various observables describing tricritical percolation clusters and their spreading properties in a remarkable contrast to the usual percolation transition the exponents beta and beta governing the two order parameters viz the mean density and the percolation probability turn out to be different at the tricritical point in addition to the scaling exponents we calculate for all our static and dynamic observables logarithmic corrections to the mean field scaling behavior at d c 5

the collapse of directed animals directed animals on a square lattice with attractive nearest neighbour and next nearest neighbour interactions are considered using an equivalence between the collapse problem and the directed site bond percolation problem the bulk free energy per site is calculated exactly in the collapsed phase it shows no singularity as the transition is approached from the collapsed side for the three dimensional directed animals bulk free energy per site in the collapsed phase is calculated exactly if the interaction weights are related by an algebraic equation