on nonlinear maps preserving the reduced minimum modulus on differences of matrices abstract we prove that if m n c m n c is a bijective map which preserves the reduced minimum modulus on differences of matrices then is automatically bicontinuous and therefore by a result of mashreghi and stepanyan of a standard form

nonlinear maps preserving the reduced minimum modulus of operators abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe bijective bicontinuous maps from b x to b y satisfying s t s t for all s t b x where t is the reduced minimum modulus of an operator t an analogue result for the finite dimensional case is obtained

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

spectrum nonincreasing maps on matrices abstract maps which do not increase the spectrum on complex matrices in a sense that sp a b sp a b are classified

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

nonlinear maps preserving the minimum and surjectivity moduli abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe maps from b x onto b y satisfying c s t c s t for all s t b x where c stands either for the minimum modulus or the surjectivity modulus or the maximum modulus we also obtain analog results for the finite dimensional case

breaking records in the evolutionary race abstract we explore some aspects of the relationship between biological evolution processes and the mathematical theory of records for eigen s quasispecies model with an uncorrelated fitness landscape we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics with two complications first the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent identically distributed random variables second fitness records can be bypassed which strongly reduces the number of genotypes that take part in an evolutionary trajectory for exponential and gaussian fitness distributions this number scales with sequence length n as n and it is of order unity for distributions with a power law tail this is in strong contrast to the number of records which is of order n for any fitness distribution

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

maps on matrices compressing the local spectrum in the spectrum abstract let m n c be the algebra of all complex n n matrices let x 0 be a nonzero vector in c n we describe maps on m n c not linear or surjective satisfying t s x 0 t s for all t s m n c we also obtain a similar description by supposing that is surjective and t s x 0 t s for all t s m n c

depth functions of symbolic powers of homogeneous ideals this paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal our investigation is focused on the behavior of the function mathrm depth r i t dim r mathrm pd i t 1 where i t denotes the t th symbolic power of a homogeneous ideal i in a noetherian polynomial ring r and mathrm pd denotes the projective dimension it has been an open question whether the function mathrm depth r i t is non increasing if i is a squarefree monomial ideal we show that mathrm depth r i t is almost non increasing in the sense that mathrm depth r i s ge mathrm depth r i t for all s ge 1 and t in e s where begin aligned e s bigcup i ge 1 t in mathbb n i s 1 1 le t le is end aligned which contains all integers t ge s 1 2 1 the range e s is the best possible since we can find squarefree monomial ideals i such that mathrm depth r i s mathrm depth r i t for t not in e s which gives a negative answer to the above question another open question asks whether the function mathrm depth r i t is always constant for t gg 0 we are able to construct counter examples to this question by monomial ideals on the other hand we show that if i is a monomial ideal such that i t is integrally closed for t gg 0 e g if i is a squarefree monomial ideal then mathrm depth r i t is constant for t gg 0 with begin aligned lim t rightarrow infty mathrm depth r i t dim r dim oplus t ge 0 i t mathfrak m i t end aligned our last result which is the main contribution of this paper shows that for any positive numerical function phi t which is periodic for t gg 0 there exist a polynomial ring r and a homogeneous ideal i such that mathrm depth r i t phi t for all t ge 1 as a consequence for any non negative numerical function psi t which is periodic for t gg 0 there is a homogeneous ideal i and a number c such that mathrm pd i t psi t c for all t ge 1

convex hull of n planar brownian motions exact results and an application to ecology we compute exactly the mean perimeter and area of the convex hull of n independent planar brownian paths each of duration t both for open and closed paths we show that the mean perimeter alpha n sqrt t and the mean area beta n t for all t the prefactors alpha n and beta n computed exactly for all n increase very slowly logarithmically with increasing n this slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size n

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

linear preservers of polynomial numerical hulls of matrices abstract let m n be the algebra of all n n complex matrices 1 k n 1 be an integer and m n m n be a linear operator in this paper it is shown that preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix u m n such that either a u a u for all a m n or a u a t u for all a m n

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

mean perimeter of the convex hull of a random walk in a semi infinite medium we study various properties of the convex hull of a planar brownian motion defined as the minimum convex polygon enclosing the trajectory in the presence of an infinite reflecting wall recently in a rapid communication phys rev e textbf 91 050104 r 2015 we announced that the mean perimeter of the convex hull at time t rescaled by sqrt dt is a non monotonous function of the initial distance to the wall in the present article we first give all the details of the derivation of this mean rescaled perimeter in particular its value when starting from the wall and near the wall we then determine the physical mechanism underlying this surprising non monotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull finally we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the brownian motion as a function of the initial distance to the wall

reductions and special parts of closures abstract we provide an axiomatic framework for working with a wide variety of closure operations on ideals and submodules in commutative algebra including notions of reduction independence spread and special parts of closures this framework is applied to tight frobenius and integral closures applications are given to evolutions and special briancon skoda theorems

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

integrally closed and componentwise linear ideals in a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is cohen macaulay in higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do we restrict our attention to 0 dimensional homogeneous ideals in polynomial rings r of arbitrary dimension we identify a class of integrally closed ideals the goto class mathcal g which is closed under product and it has a suitable unique factorization property ideals in mathcal g have a cohen macaulay associated graded ring if either they are monomial or dim r 3 our approach is based on the study of the relationship between the notions of integrally closed contracted full and componentwise linear ideals

punctuated evolution for the quasispecies model abstract biological evolution in a sequence space with random fitnesses is studied within eigen s quasispecies model a strong selection limit is employed in which the population resides at a single sequence at all times evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps the distribution of the total evolution time displays a universal power law tail with exponent 2 simulations show that the evolutionary dynamics is very well represented by a simplified shell model in which the sub populations at local fitness maxima grow independently the shell model allows for highly efficient simulations and provides a simple geometric picture of the evolutionary trajectories

on the structure of the fiber cone of ideals with analytic spread one for a given a local ring a m we study the fiber cone of ideals in a with analytic spread one in this case the fiber cone has a structure as a module over its noether normalization which is a polynomial ring in one variable over the residue field one may then apply the structure theorem for modules over a principal domain to get a complete description of the fiber cone as a module we analyze this structure in order to study and characterize in terms of the ideal itself the arithmetical properties and other numerical invariants of the fiber cone as multiplicity reduction number or castelnuovo mumford regularity

intrinsic volumes of sobolev balls with applications to brownian convex hulls a formula due to sudakov relates the first intrinsic volume of a convex set in a hilbert space to the maximum of the isonormal gaussian process over this set using this formula we compute the first intrinsic volumes of infinite dimensional convex compact sets including unit balls with respect to sobolev type seminorms and ellipsoids in the hilbert space we relate the distribution of the random one dimensional projections of these sets to the distributions s1 s2 c1 c2 studied by biane pitman yor bull ams 38 2001 we show that the k th intrinsic volume of the set of all functions on 0 1 which have lipschitz constant bounded by 1 and which vanish at 0 respectively which have vanishing integral is given by vk k 2 3 2 k 1 respectively vk k 1 2 2 3 2 k 3 2 this is related to the results of gao and vitale discrete comput geom 26 2001 elect comm probab 8 2003 who considered a similar question for functions with a restriction on the total variation instead of the lipschitz constant using the results of gao and vitale we give a new proof of the formula for the expected volume of the convex hull of the d dimensional brownian motion which is due to eldan elect j probab to appear additionally we prove an analogue of eldan s result for the brownian bridge similarly we show that the results on the intrinsic volumes of the lipschitz balls can be translated into formulae for the expected volumes of zonoids aumann integrals generated by the brownian motion and the brownian bridge also these results have discrete versions for gaussian random walks and bridges our proofs exploit sudakov s and tsirelson s theorems which establish a connection between the intrinsic volumes and the isonormal gaussian process

universality and time scale invariance for the shape of planar levy processes for a broad class of planar markov processes viz l evy processes satisfying certain conditions valid e g in the case of brownian motion and l evy flights we establish an exact universal formula describing the shape of the convex hull of sample paths we show indeed that the average number of edges joining paths points separated by a time lapse ensuremath delta ensuremath tau ensuremath in left ensuremath delta ensuremath tau 1 ensuremath delta ensuremath tau 2 right is equal to 2 phantom rule 0 16em 0ex text ln left ensuremath delta ensuremath tau 2 ensuremath delta ensuremath tau 1 right regardless of the specific distribution of the process s increments and regardless of its total duration t the formula also exhibits invariance when the time scale is multiplied by any constant apart from its theoretical importance our result provides insights regarding the shape of two dimensional objects e g polymer chains modeled by the sample paths of stochastic processes generally more complex than brownian motion in particular for a total time or parameter duration t the average number of edges on the convex hull cut off to discard edges joining points separated by a time lapse shorter than some ensuremath delta ensuremath tau lt will be given by 2 phantom rule 0 16em 0ex text ln frac t ensuremath delta ensuremath tau thus it will only grow logarithmically rather than at some higher pace

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

persistence of a continuous stochastic process with discrete time sampling we introduce the concept of discrete time persistence which deals with zero crossings of a continuous stochastic process x t measured at discrete times t n delta t for a gaussian markov process with relaxation rate mu we show that the persistence no crossing probability decays as rho a n for large n where a exp mu delta t and we compute rho a to high precision we also define the concept of alternating persistence which corresponds to a 1 corresponding to motion in an unstable potential mu 0 there is a nonzero probability of having no zero crossings in infinite time and we show how to calculate it

dimension of families of determinantal schemes a scheme x p n c of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t x t c 1 matrix and x is said to be good determinantal if it is standard determinantal and a generic complete intersection given integers a 0 a 1 a t c 2 and b 1 b t we denote by w b a c hilb p p n c resp w s b a the locus of good resp standard determinantal schemes x c p n c of codimension c defined by the maximal minors of a t x t c 1 matrix f ij i 1 t j 0 t c 2 where f ij k x 0 x 1 x n c is a homogeneous polynomial of degree a j b i in this paper we address the following three fundamental problems to determine 1 the dimension of w b a resp w s b a in terms of a j and b i 2 whether the closure of w b a is an irreducible component of hilb p p n c and 3 when hilb p p n c is generically smooth along w b a concerning question 1 we give an upper bound for the dimension of w b a resp w s b a which works for all integers a 0 a 1 a t c 2 and b 1 b t and we conjecture that this bound is sharp the conjecture is proved for 2 2 and for c 5 under certain numerical assumptions

time at which the maximum of a random acceleration process is reached we study the random acceleration model which is perhaps one of the simplest yet nontrivial non markov stochastic processes and is key to many applications for this non markov process we present exact analytical results for the probability density p tm t of the time tm at which the process reaches its maximum within a fixed time interval 0 t we study two different boundary conditions which correspond to the process representing respectively i the integral of a brownian bridge and ii the integral of a free brownian motion our analytical results are also verified by numerical simulations

algebraic cycles modular forms and euler systems fix a squarefree integer n and let f be a newform of weight 2 for 0 n we assume that f does not have complex multiplication it was shown in 14 and 15 that for a set of primes l of density 1 the naive deformation theory of the mod l galois representation associated to f is unobstructed in the sense that the universal deformation ring is a power series ring over the witt vectors in 31 these methods were modified to obtain results on the deformation problems studied by taylor wiles in this paper we extend the results of flach and mazur to the case of newforms f of weight 2 for 1 n we now state our results more precisely fix l max 5 1 let f be as above and let h be the associated l adic representation h is a free module of rank 2 over a certain hecke algebra a which itself is a finite flat local gorenstein zl algebra let t be the tate twist endah 1 of the module of trace zero endomorphisms of h using techniques of flach we construct a collection of cohomology classes c in h q t with tightly controlled ramification with some mild additional hypotheses applying the methods of kolyvagin to these classes yields a certain annihilator a of the selmer group h f q t of the cartier dual of t this selmer group is dual to the differentials r ra where r is the universal minimally ramified deformation ring of the residual representation of h in the case that is a unit this then implies that both r and a are isomorphic to the ring of witt vectors over the residue field of a in the general case following mazur we show that our construction yields a derivation from a to the selmer group h f q t t it follows by a formal argument that the natural surjection r a induces an isomorphism r ra a although not the strongest possible result this does provide a great deal of information on the structure of the ring r it is possible that any such map r a must be an isomorphism although as far as i know this question remains open we also show that the isomorphism r ra a is characterized by the fact that a r ra homzl h f q t ql zl

extreme value statistics and traveling fronts various applications abstract an intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems in this short review we outline a few such problems and consider their various applications

maximum of n independent brownian walkers till the first exit from the half space we consider the one dimensional target search process that involves an immobile target located at the origin and n searchers performing independent brownian motions starting at the initial positions all on the positive half space the process stops when the target is first found by one of the searchers we compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power law tail for large m thus all moments of m up to the order n 1 are finite while the higher moments diverge the prefactor bn increases with n faster than exponentially our solution gives the exit probability of a set of n particles from a box 0 l through the left boundary incidentally it also provides an exact solution of laplace s equation in an n dimensional hypercube with some prescribed boundary conditions the analytical results are in excellent agreement with monte carlo simulations

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

a convex hull based estimator of home range size this paper considers the problem of estimating the home range size of an animal from locational data which are usually collected by use of radio telemetry with convex hull based estimators it is shown that the widely used minimum convex polygon has some undesirable characteristics as an estimator and thus should be used with caution a method based on convex hull peeling of the data to estimate convex density contours is studied as an alternative simple home range size estimator monte carlo simulations show that this estimator has very similar statistical properties if the point fixes are generated from a bivariate normal distribution a uniform on a disc distribution and two other distributions with intermediate tail behaviour these standard distributions are used as a reference to define a sample size bias corrected estimator of home range size and to provide an approximate method for calculating the precision of this estimator the technique is applied to the estimation of the home range sizes of deer

symbolic squares in regular local rings in this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal our techniques come from residual intersections integral closures of ideals and differentials

the alpha problem line count configurations abstract motivated by the work of chudnovsky and the eisenbud mazur conjecture on evolutions harbourne and huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in projective space p n the conjectures involve both containment statements and bounds for the initial degree in which there is a non zero form in an ideal working with initial degrees we verify two of these conjectures for special line count configurations in projective 2 space over an algebraically closed field of characteristic 0

lipschitz spectrum preserving mappings on algebras of matrices abstract it is proved that for any lipschitz mapping t on the algebra m n of n n matrices over the complex numbers satisfying t 0 0 and t a t b a b a b m n there exists an invertible matrix u m n such the t a uau 1 for all a m n or t a ua t u 1 for all a m n

large deviations of convex hulls of self avoiding random walks a global picture of a random particle movement is given by the convex hull of the visited points we obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self avoiding random walks namely the classical self avoiding walk the smart kinetic self avoiding walk and the loop erased random walk to obtain a comprehensive description of the measured random quantities we applied sophisticated large deviation techniques which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than p 10 ensuremath 100 we give an approximate closed form of the so called large deviation rate function mathrm ensuremath phi which generalizes above the upper critical dimension to the previously studied case of the standard random walk further we show correlations between the two observables also in the limits of atypical large or small values

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

convex hull of n planar brownian paths an exact formula for the average number of edges we establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent brownian paths in the plane this requires the introduction of a counting criterion which amounts to cutting off edges that are in a specific sense small the main argument consists in a mapping between planar brownian convex hulls and configurations of constrained independent linear brownian motions this new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of brownian convex hulls in the plane

joint distributions of partial and global maxima of a brownian bridge we analyze the joint distributions and temporal correlations between the partial maximum m and the global maximum m achieved by a brownian bridge on the subinterval and on the entire interval respectively we determine three probability distribution functions the joint distribution of both maxima the distribution p m of the partial maximum and the distribution of the gap between the maxima we present exact results for the moments of these distributions and quantify the temporal correlations between m and m by calculating the pearson correlation coefficient

symbolic powers of perfect ideals of codimension 2 and birational maps abstract this work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field assuming that the entries of the corresponding presentation matrix are general linear forms the main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal a full description of the corresponding symbolic rees algebra is given in some cases one application is an affirmative solution of a conjecture of eisenbud mazur in 11 section 2

inelastic collapse of a ball bouncing on a randomly vibrating platform a theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform of interest are the distributions of the number of flights n f and the total time ensuremath tau c until the ball has effectively collapsed i e coalesced with the platform in the strictly elastic case both distributions have power law tails characterized by exponents that are universal i e independent of the detail of the platform noise distribution however in the inelastic case both distributions have exponential tails p n f ensuremath sim mathrm exp ensuremath ensuremath theta 1 n f and p ensuremath tau c ensuremath sim mathrm exp ensuremath ensuremath theta 2 ensuremath tau c the decay exponents ensuremath theta 1 and ensuremath theta 2 depend continuously on the coefficient of restitution and are nonuniversal however as one approaches the elastic limit they vanish in a manner which turns out to be universal an explicit expression for ensuremath theta 1 is provided for a particular case of the platform noise distribution

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

hilbert series of fiber cones of ideals with almost minimal mixed multiplicity abstract let r m be a cohen macaulay local ring and let i be an m primary ideal we introduce ideals of almost minimal mixed multiplicity which are analogues of ideals studied by j sally compositio math 40 1980 167 175 the main theorem describes the hilbert series of fiber cones of these ideals

flux to a trap the flux of particles to a single trap is investigated for two systems 1 particles in 3d space which jump a fixed step lengthl the rayleigh flight and are adsorbed by a spherical surface and 2 particles on a lattice jumping to nearest neighbor sites with a single adsorbing site initially the particles are uniformly distributed outside the traps when the jump length goes to zero both processes go over to regular diffusion and the first case yields the diffusive flux to a sphere as solved by smoluchowski for nonzero step length the flux for large times is given by a modified form of smoluchowski s result with the effective radius replaced byr cl wherec 0 29795219 andcl is the milne extrapolation length for this problem for the second problem a similar expression for the flux is found with the effective trap radius a function of the lattice sc bcc fcc being considered

persistence exponents and the statistics of crossings and occupation times for gaussian stationary processes we consider the persistence probability the occupation time distribution and the distribution of the number of zero crossings for discrete or equivalently discretely sampled gaussian stationary processes gsps of zero mean we first consider the ornstein uhlenbeck process finding expressions for the mean and variance of the number of crossings and the partial survival probability we then elaborate on the correlator expansion developed in an earlier paper g c m a ehrhardt and a j bray phys rev lett 88 070602 2002 to calculate discretely sampled persistence exponents of gsps of known correlator by means of a series expansion in the correlator we apply this method to the processes d n x dt n ensuremath eta t with ng 3 incorporating an extrapolation of the series to the limit of continuous sampling we then extend the correlator method to calculate the occupation time and crossing number distributions as well as their partial survival distributions and the means and variances of the occupation time and number of crossings we apply these general methods to the d n x dt n ensuremath eta t processes for n 1 random walk n 2 random acceleration and larger n and to simple diffusion from random initial conditions in one to three dimensions the results for discrete sampling are extrapolated to the continuum limit where possible

spectrum preserving additive maps abstract let x and y be complex banach spaces we show that a spectrum preserving surjective additive map from b x to b y is either of the form f ata 1 for a linear isomorphism a of x onto y or of the form t bt b 1 for a linear isomorphism b of x onto y

on a random area variable arising in discrete time queues and compact directed percolation a well known discrete time single server queueing system with mean arrival rate and mean departure rate is considered from the perspective of the area a swept out by the queue occupation process during a busy period we determine the exact form of the tail of the distribution pr a x in particular we show that pr a x cx 1 4 exp dx1 2 for all 1 where and expressions for c and d are given for the critical case 1 we show that pr a x c x 1 3 with c also given a simple mapping used in the derivation establishes a connection with compact directed percolation on a square lattice as a corollary therefore we are also able to specify the large area asymptotic behaviour of this model at all points in the phase diagram this extends previous scaling results which are only valid close to the percolation threshold

maps on matrices that preserve the spectrum abstract let t be a continuous map of the space of complex n n matrices into itself satisfying t 0 0 such that the spectrum of t x t y is always a subset of the spectrum of x y there exists then an invertible n n matrix u such that either t a uau 1 for all a or t a ua t u 1 for all a we arrive at the same conclusion by supposing that the spectrum of x y is always a subset of the spectrum of t x tt y without the continuity assumption on t

estimation of a convex density contour in two dimensions abstract if a density in two dimensions has a convex contour containing probability the contour may be estimated from a sample by finding the convex polygon of smallest area containing a proportion of the sample points an algorithm for finding a particular contour is given that takes o n 2 space and o n 3 time for n sample points

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

normal cones of monomial primes we explicitly calculate the normal cones of all monomial primes which define the curves of the form tl tl 1 tl n where n 4 all of these normal cones are reduced and cohen macaulay and their reduction numbers are independent of the reduction these monomial primes are new examples of integrally closed ideals for which the product with the maximal homogeneous ideal is also integrally closed substantial use was made of the computer algebra packages maple and macaulay2