jordan blocks in a one dimensional markov map abstract a simple one dimensional chaotic map whose spectral decomposition of the frobenius perron operator in the space of polynomials has jordan block structure is given and analyzed the jordan block structure implies that some correlation functions have exponential decay modified by coefficients which are polynomials in t such a correlation is given for this system and a numerical simulation which agrees with the theoretical prediction is presented

structure of some noetherian si rings abstract we describe the structure of rings over which every cyclic or finitely generated right module is a direct sum of a projective module and an injective module

kinetic theory of area preserving maps application to the standard map in the diffusive regime the evolution of the distribution function of a dynamical system governed by a general two dimensional area preserving iterative map is studied by the methods of nonequilibrium statistical mechanics a closed non markovian master equation determines the angle averaged distribution function the density profile the complementary angle dependent part the fluctuations is expressed as a non markovian functional of the density profile whenever there exist two widely separated intrinsic time scales the master equation can be markovianized yielding an asymptotic kinetic equation the general theory is applied to the standard map in the diffusive regime i e for large stochasticity parameter and large scale length the non markovian master equation can be written and solved analytically in this approximation the two characteristic time scales are exhibited this permits the thorough study of the evolution of the density profile its tendency toward the markovian approximation and eventually toward a diffusive gaussian packet the evolution of the fluctuations is also described in detail the various relaxation processes are governed asymptotically by a single diffusion coefficient which is calculated analytically this model appears as a testing bench for the study of kinetic equations the various previous approaches to this problem are reviewed and critically discussed

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

spectral decomposition of chebyshev maps abstract we construct a spectral decomposition of the frobenius perron operator for the chebyshev polynomials of the first kind we defined a suitable dual pair or rigged hilbert space which provides mathematical meaning to the spectral decomposition the spectra of the even chebyshev maps do not contain the odd powers of 1 m and the odd eigenfunctions are in the null space of the frobenius perron operator moreover the odd chebyshev maps have degenerate spectra without jordan blocks the eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of tent maps

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

snakes and spiders brownian motion on ℝ trees we consider diffusion processes on a class of trees the processes are defined in a manner similar to that of le gall s brownian snake each point in the tree has a real valued height or generation and the height of the diffusion process evolves as a brownian motion when the height process decreases the diffusion retreats back along a lineage whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages the class of trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation

characterizing rings in terms of the extent of the injectivity and projectivity of their modules abstract given a ring r we define its right i profile resp right p profile to be the collection of injectivity domains resp projectivity domains of its right r modules we study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and vice versa we show that the i profile is isomorphic to an interval of the lattice of linear filters of right ideals of r and is therefore modular and coatomic in particular we give a practical characterization of the profile of a right artinian ring and offer an example of a ring without injective left middle class for with the same is not true on the right hand side we characterize the p profile of a right perfect ring and show through an example that the right p profile of a ring is not necessarily a set in addition we use our results to provide a characterization of a special class of qf rings in which the injectivity and projectivity domains of all modules coincide the study of rings in terms of their i or p profile was inspired by the study of rings with no right i or p middle class initiated in recent papers by er lopez permouth and sokmez and by holston lopez permouth and orhan ertas

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

exit times for an increasing levy tree valued process we give an explicit construction of the increasing tree valued process introduced by abraham and delmas using a random point process of trees and a grafting procedure this random point process will be used in companion papers to study record processes on levy trees we use the poissonian structure of the jumps of the increasing tree valued process to describe its behavior at the first time the tree grows higher than a given height using a spinal decomposition of the tree similar to the classical bismut and williams decompositions we also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree valued process

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

spines skeletons and the strong law of large numbers for superdiffusions consider a supercritical superdiffusion x t t 0 on a domain d r d with branching mechanism x z x z x z 2 0 e zy 1 zy x dy the skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion we use this decomposition to derive the strong law of large numbers slln for a wide class of superdiffusions from the corresponding result for branching particle diffusions that is we show that for suitable test functions f and starting measures f x t p f x t w p almost surely as t where w is a finite non deterministic random variable characterized as a martingale limit our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the slln for superdiffusions the result covers many of the key examples of interest and in particular proves a conjecture by fleischmann and swart stochastic process appl 106 2003 141 165 for the super wright fisher diffusion

on artinian rings with restricted class of injectivity domains abstract in a recent paper of alahmadi alkan and lopez permouth a ring r is defined to have no simple middle class if the injectivity domain of any simple r module is the smallest or largest possible er lopez permouth and sokmez use this idea of restricting the class of injectivity domains to classify rings and give a partial characterization of rings with no middle class in this work we continue the study of the property of having no simple middle class we give a structural description of right artinian right nonsingular rings with no right middle class we also give a characterization of right artinian rings that are not si to have no middle class which gives rise to a full characterization of rings with no middle class furthermore we show that commutative rings with no middle class are those artinian rings which decompose into a sum of a semisimple ring and a ring of composition length two also artinian rings with no simple middle class are characterized we demonstrate our results with several examples

an alternative perspective on injectivity of modules abstract given modules m and n m is said to be n subinjective if for every extension k of n and every homomorphism n m there exists a homomorphism k m such that n for a module m the subinjectivity domain of m is defined to be the collection of all modules n such that m is n subinjective as an opposite to injectivity a module m is said to be indigent if its subinjectivity domain is smallest possible namely consisting of exactly the injective modules properties of subinjectivity domains and of indigent modules are studied in particular the existence of indigent modules is determined for some families of rings including the ring of integers and artinian serial rings it is also shown that some rings e g artinian chain rings have no middle class in the sense that all modules are either injective or indigent for various classes of modules such as semisimple singular and projective necessary and sufficient conditions for the existence of indigent modules of those types are studied indigent modules are analog to the so called poor modules an opposite of injectivity in terms of injectivity domains recently studied in papers by alahmadi alkan and lopez permouth and by er lopez permouth and sokmez relations between poor and indigent modules are also investigated here

branching brownian motion in a strip survival near criticality we consider a branching brownian motion with linear drift in which particles are killed on exiting the interval 0 k and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible we combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near critical survival probability this allows us to deduce the existence of a quasi stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality this paper is motivated by recent work on survival of near critical branching brownian motion with absorption at the origin by aidekon and harris near critical survival probability of branching brownian motion with an absorbing barrier 2010 unpublished manuscript as well as the work of berestycki et al ann probab 41 2013 527 618 j stat phys 143 2011 833 854

growth of the brownian forest trees in brownian excursions have been studied since the late 1980s forests in excursions of brownian motion above its past minimum are a natural extension of this notion in this paper we study a forest valued markov process which describes the growth of the brownian forest the key result is a composition rule for binary galton watson forests with i i d exponential branch lengths we give elementary proofs of this composition rule and explain how it is intimately linked with williams decomposition for brownian motion with drift

smaller population size at the mrca time for stationary branching processes we consider an elementary model of random size varying population governed by a stationary continuous state branching process we compute the distributions of various variables related to the most recent common ancestor mrca the time to the mrca the size of the current population and the size of the population just before the mrca in particular we observe a natural mild bottleneck effect as the size of the population just before the mrca is stochastically smaller than the size of the current population we also compute the number of individuals involved in the last coalescent event of the genealogical tree that is the number of individuals at the time of the mrca who have descendants in the current population by studying more precisely the genealogical structure of the population we get asymptotics for the number of ancestors just before the current time we give explicit computations in the case of the quadratic branching mechanism in this case the size of the population at the mrca is in mean 2 3 of the size of the current population we also provide in this case the fluctuations for the renormalized number of ancestors

polynomial shift states of a chaotic map we present a piecewise linear map of the unit interval in which the resolvent of the frobenius perron operator considered in a polynomial basis has an essen tial singularity at the origin associated with the essential singularity are polynomial shift states which are obtained from creation and annihilation operators in non self dual function spaces correlation functions of general polynomial observables have decay components that vanish in a finite time

the extension of classical dynamics for unstable hamiltonian systems abstract classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the liouville equation it is shown that for the class of large nonintegrable poincare systems lps the two descriptions are not equivalent practically all dynamical systems studied in statistical mechanics belong to this class the basic step is the extension of the liouville operator l h outside the hilbert space to functions singular in their fourier transformation this function space plays an important role in statistical mechanics as functions of the hamiltonian and therefore equilibrium distribution functions belong to this class physically these functions correspond to situations characterized by persistent interactions as they are realized in macroscopic physics persistent interactions are introduced in contrast to transient interactions studied in quantum mechanics by the s matrix approach asymptotically free in and out states the eigenvalue problem for the liouville operator l h is solved in this generalized function space for lps we obtain a complex irreducible spectral representation complex means that the eigenvalues are complex numbers whose imaginary parts refer to the various irreversible processes such as relaxation times diffusion irreducible means that these representations cannot be implemented by trajectory theory as the result the dynamical group of evolution splits into two semigroups moreover the laws of classical dynamics take a new form as they have to be formulated on the statistical level they express possibilities and no more certitudes two examples of typical classical systems i e interacting particles and anharmonic lattices are studied

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

poor modules with no proper poor direct summands abstract as a mean to provide intrinsic characterizations of poor modules the notion of a pauper module is introduced a module is a pauper if it is poor and has no proper poor direct summand we show that not all rings have pauper modules and explore conditions for their existence in addition we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule the second condition holds for the ring of integers and is just as significant as the first one for noetherian rings since in that context modules having poor pure submodules must themselves be poor it is shown that the existence of paupers is equivalent to the noetherian condition for rings with no middle class as indecomposable poor modules are pauper we study rings with no indecomposable right middle class i e the ring whose indecomposable right modules are pauper or injective we show that semiartinian v rings satisfy this property and also that a commutative noetherian ring r has no indecomposable middle class if and only if r is the direct product of finitely many fields and at most one ring of composition length 2 structure theorems are also provided for rings without indecomposable middle class when the rings are artinian serial or right artinian rings for which not having an indecomposable middle class suffices not to have a middle class include commutative noetherian and artinian serial rings the structure of poor modules is completely determined over commutative hereditary noetherian rings pauper abelian groups with torsion free rank one are fully characterized

rings and modules characterized by opposites of injectivity abstract in a recent paper aydo du and lopez permouth have defined a module m to be n subinjective if every homomorphism n m extends to some e n m where e n is the injective hull of n clearly every module is subinjective relative to any injective module their work raises the following question what is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules namely injectives we show using a dual opposite injectivity condition that such a ring r is isomorphic to the direct product of a semisimple artinian ring and an indecomposable ring which is i a hereditary artinian serial ring with j 2 0 or ii a qf ring isomorphic to a matrix ring over a local ring each case is viable and conversely i is sufficient for the said property and a partial converse is proved for a ring satisfying ii using the above mentioned classification it is also shown that such rings coincide with the fully saturated rings of trlifaj except possibly when von neumann regularity is assumed furthermore rings and abelian groups which satisfy these opposite injectivity conditions are characterized

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

spectral decomposition of tent maps using symmetry considerations the spectral decomposition of the frobenius perron operator of maps composed of many tents is determined from symmetry considerations the eigenstates involve euler as well as bernoulli polynomials

on the relaxation time of gauss continued fraction map ii the banach space approach transfer operator method the spectrum of the transfer operator for the maptx 1 x 1 x when restricted to a certain banach space of holomorphic functions is shown to coincide with the spectrum of the adjointu of koopman s isometric operatoruf x f t x when the former is restricted to the hilbert space introduced in part i of this work ifn denotes the operator p1 withp1 the projector onto the eigenfunction to the dominant eigenvalue 1 1 of then n is au0 positive operator with respect to some cone and therefore has a dominant positive simple eigenvalue 2 a minimax principle holds giving rigorous upper and lower bounds both for 2 and the relaxation time of the mapt

r adic one dimensional maps and the euler summation formula for the r adic one dimensional maps the author explicitly constructs the decaying eigenstates and adjoint eigenstates associated with the ruelle resonances it is shown that the eigenfunctions of the corresponding frobenius perron operator are the well known bernoulli polynomials the adjoint eigendistributions are obtained as derivatives of the dirac distributions at the end points of the unit interval the resulting expansion of the initial density in terms of the decaying eigenstates is given by the euler summation formula

intrinsic irreversibility and the validity of the kinetic description of chaotic systems irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities the generalized representation is valid for one dimensional systems when the initial probability density satisfies certain physical conditions of smoothness the formalism is first applied to the one dimensional multi bernoulli map which is a simple map displaying deterministic diffusion the two dimensional invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered the generalized representation is constructed using a resolvent formalism the eigenstates of the diffusive systems are seen to be of a fractal nature

rings whose modules have maximal or minimal injectivity domains in a recent paper alahmadi alkan and lopez permouth defined a module m to be poor if m is injective relative only to semisimple modules and a ring to have no right middle class if every right module is poor or injective we prove that every ring has a poor module and characterize rings with semisimple poor modules next a ring with no right middle class is proved to be the ring direct sum of a semisimple artinian ring and a ring t which is either zero or of one of the following types i morita equivalent to a right pci domain ii an indecomposable right si ring which is either right artinian or a right v ring and such that soc tt is homogeneous and essential in tt and t has a unique simple singular right module or iii an indecomposable right artinian ring with homogeneous right socle coinciding with the jacobson radical and the right singular ideal and with unique non injective simple right module in case iii either tt is poor or t is a qf ring with j t 2 0 converses of these cases are discussed it is shown in particular that a qf ring r with j r 2 0 and homogeneous right socle has no middle class

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

a continuum tree valued markov process we present a construction of a levy continuum random tree crt associated with a super critical continuous state branching process using the so called exploration process and a girsanov theorem we also extend the pruning procedure to this super critical case let be a critical branching mechanism we set let or be the set of values of for which is a conservative branching mechanism the pruning procedure allows to construct a decreasing levy crt valued markov process mathcal t theta theta in theta such that mathcal t theta has branching mechanism it is sub critical if 0 and super critical if 0 we then consider the explosion time a of the crt the smallest negative time for which the continuous state branching process cb associated with mathcal t theta has finite total mass i e the length of the excursion of the exploration process that codes the crt is finite we describe the law of a as well as the distribution of the crt just after this explosion time the crt just after explosion can be seen as a crt conditioned not to be extinct which is pruned with an independent intensity related to a we also study the evolution of the crt valued process after the explosion time this extends results from aldous and pitman on galton watson trees for the particular case of the quadratic branching mechanism we show that after explosion the total mass of the cb behaves like the inverse of a stable subordinator with index 1 2 this result is related to the size of the tagged fragment for the fragmentation of aldous s crt

regenerative tree growth binary self similar continuum random trees and poisson dirichlet compositions we use a natural ordered extension of the chinese restaurant process to grow a twoparameter family of binary self similar continuum fragmentation trees we provide an explicit embedding of ford s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of ford s trees in general the markov branching trees induced by the two parameter growth rule are not sampling consistent so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case we develop here a new approach to establish such limits based on regenerative interval partitions and the urn model description of sampling from dirichlet random distributions ams 2000 subject classifications 60j80

rings whose cyclic modules have restricted injectivity domains abstract a poor module is one that is injective relative only to semisimple modules and a module is maximally injective if its domain of injectivity is a coatom in the lattice of domains of injectivity the so called injective profile of a ring a ring is said to have no right middle class if every right module is injective or poor we consider two families of rings characterized by their cyclic right modules those whose nonzero cyclic modules are poor property p and those whose nonzero cyclic modules are maximally injective property q we show that rings satisfying property q are precisely those rings that have no right middle class and property p structural properties of both classes of rings are obtained and it is shown in particular that a ring with property q is isomorphic to a matrix ring over a nonsemisimple local right artinian ring and is such that its right socle right singular ideal and jacobson radical coincide

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion