cayley bacharach schemes and their canonical modules a set of s points in p d is called a cayley bacharach scheme cb scheme if every subset of s 1 points has the same hilbert function we investigate the consequences of this weak uniformity the main result characterizes cb schemes in terms of the structure of the canonical module of their projective coordinate ring from this we get that the hilbert function of a cb scheme x has to satisfy growth conditions which are only slightly weaker than the ones given by harris and eisenbud for points with the uniform position property we also characterize cb schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points

on the solution of the coagulation equation with a time dependent source application to pulsed injection a general theoretical treatment is developed for the solution of the time dependent coagulation equation with constant coagulation kernel in the presence of a source term possessing arbitrary time dependence it is shown how the relevant nonlinear first order differential equation can be transformed into a linear second order equation which can then be used to obtain the general solution of the problem together with information about its asymptotic long term behaviour the technique is applied to a periodic source term where it is found that the long term behaviour of the solution exhibits the same periodicity as the source detailed results are derived for particular source terms

almost maximal growth of the hilbert function abstract let a s j be a standard artinian graded algebra over the polynomial ring s a theorem of macaulay dictates the possible growth of the hilbert function of a from any degree to the next and if this growth is the maximal possible then strong consequences have been given by gotzmann it can be phrased in terms of the base locus of the linear system defined by the relevant component s of j if j is the artinian reduction of the ideal of a finite set of points in projective space then this maximal growth for a was shown by bigatti geramita and the second author to imply strong geometric consequences for the points we now suppose that the growth of the hilbert function is one less than maximal this again has not as strong consequences for the base locus defined by the relevant component and when j is the artinian reduction of the ideal of a finite set of points in projective space we prove that almost maximal growth again forces geometric consequences

the equivalence of ensembles for classical systems of particles for systems of particles in classical phase space with standard hamiltonian we consider spatially averaged microcanonical gibbs distributions in finite boxes we show that infinite volume limits along suitable subsequences exist and are grand canonical gibbs measures on the way we establish a variational formula for the thermodynamic entropy density as well as a variational characterization of grand canonical gibbs measures

the smoluchowski coagulation equations with continuous injection we study a system of equations which models the formation of clusters by coagulation with particles of unit size being injected at a time dependent rate we observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases which have been studied previously we identify a variety of types of behaviour in the large time limit depending on the coagulation kernel and on the rate at which monomer is introduced into the system the results are obtained by means of exact generating function techniques matched asymptotic expansions and numerical simulations

curvature measures and random sets ii in choosing models of stochastic geometry three general problems play a role which are closely connected with each other in the present paper second order local geometric properties of random subsets of r d are of interest these properties are described by signed curvature measures in a measure geometric context the theory of point processes on general spaces here on the space of subsets with positive reach provides an appropriate framework for solving construction and measurability problems mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties ergodic interpretations of the curvature densities are also given the appendix provides auxiliary results for random signed radon measures in locally compact separable hausdorff spaces

euler characteristic and related measures for random geometric sets by an elementary calculation we obtain the exact mean values of minkowksi functionals for a standard model of percolating sets in particular a recurrence theorem for the mean euler characteristic recently put forward is shown to be incorrect related previous mathematical work is mentioned we also conjecture bounds for the threshold density of continuum percolation which are associated with the euler characteristic

fluctuations in a coupled population model we investigate a discrete markov process in which the immigration of individuals into one population is controlled by the fluctuations in another we examine the effect of coupling back the second population to the first through a similar mechanism and derive exact solutions for the generating functions of the population statistics we show that a stationary state exists over a certain parameter range and obtain expressions for moments and correlation functions in this regime when more than two populations are coupled cyclically transient oscillations and periodic behaviour of correlation functions are predicted we demonstrate that if the initial distribution of either population is stable or more generally has a power law tail that falls off like n 1 0 1 then for certain parameter values there exists a stationary state that is also power law but not stable this stationary state cannot be accessed from a single multiple immigrant population model but arises solely from the nonlinear interaction of the coupled system

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

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

hilbert functions and level algebras abstract the authors consider certain quotients a r i of the polynomial ring r k x1 xr over an arbitrary field k they first determine upper and lower bounds on the hilbert functions of any algebra having the form a r v where v is the largest ideal of r agreeing in degrees at least j with the ideal v generated by a vector subspace v rj of degree j forms these bounds extend the macaulay bounds theorem 1 4 a level artinian quotient of a r i has socle soc a 0 m m x1 xr in a single degree j the authors next determine the extremal hilbert functions hmax t j r and hmin t j r that occur for level graded artin algebra quotients of r having socle degree j and type t dimk soc a and they describe the extremal strata theorem 1 8 they next give a natural upper bound for the hilbert function of level algebra quotients of the coordinate ring o z of a punctual subscheme z p n n r 1 in terms of the hilbert function h z and j t this bound was known and known to be sharp in the case z is locally gorenstein and t 1 finally they show that there are no level algebras of hilbert function t 1 3 4 5 6 2 proposition 2 7 given that there are smooth punctual subschemes z of p 2 with hilbert function h z 1 3 4 5 6 6 this result shows that the sharpness of the natural upper bound in the case where z is gorenstein and t 1 does not extend to level algebra quotients of type 2 even when z is smooth the extremality results for the hilbert functions of level algebras and some counterexamples use the extremality theorem of f h s macaulay 1927 proc london math soc 26 531 555 and g gotzmann s 1978 math z 158 61 70 results on the hilbert scheme

large deviations and the maximum entropy principle for marked point random fields we establish large deviation principles for the stationary and the individual empirical fields of poisson and certain interacting random fields of marked point particles in d the underlying topologies are induced by a class of not necessarily bounded local functions and thus finer than the usual weak topologies our methods yield further that the limiting behaviour of conditional poisson distributions as well as certain distributions of gibbsian type is governed by the maximum entropy principle we also discuss various applications and examples

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

on support measures in minkowski spaces and contact distributions in stochastic geometry this paper is concerned with contact distribution functions of a random closed set n 1 n in r d where the n are assumed to be random nonempty convex bodies these distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body structuring element that contains the origin in its interior support measures with respect to such distances will be introduced and extended to sets in the local convex ring these measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions most of the results are obtained in a general nonstationary setting only the final section deals with the stationary case

stochastic flocculation of cohesive sediment analysis of floc mobility within the floc size spectrum 1 a stochastic lagrangian model is proposed to describe flocculation of suspended cohesive sediment flocs this model takes into account the floc population in a parametric fashion and predicts the time evolution of an individual floc by a sequence of stochastic aggregation and breakup events upon calibration on experimental data the model was used to investigate floc mobility within the population size spectrum bringing to light dynamical aspects not otherwise detectable with deterministic models our analysis showed that floc size fluctuations correlated with the population size distribution and breakup probability and that size dependent burst events repeated intermittently in time further analysis showed the effect of the floc population distribution on individual floc size transition and equilibrium and the effect of using either a constant or variable floc capacity dimension to describe floc geometry

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

on the genus of a maximal curve the upper limit and the first gap in the spectrum of genera of mathbf f q 2 maximal curves are known see 34 16 35 in this paper we determine the second gap both the first and second gaps are approximately constant times q 2 but this does not hold true for the third gap which is just 1 for q equiv 2 pmod 3 while at most constant times q for q equiv 0 pmod 3 this suggests that the problem of determining the third gap which is the object of current work on mathbf f q 2 maximal curves could be intricate here we investigate a relevant related problem namely that of characterising those mathbf f q 2 maximal curves whose genus is equal to the third or possible the forth largest value in the spectrum our results also provide some new evidence on mathbf f q 2 maximal curves in connection with castelnuovo s genus bound halphen s theorem and extremal curves

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

minkowski functionals of poisson processes this paper treats poisson processes n with the convex ring s as state space having a translation and rotation invariant intensity measure fixing a test set s 0 s denoting by qk the random set of all points covered by n at least k times and collecting the minkowski functionals in a generating function v a simple formula is derived for the expectation of v s 0 q k under an adequate moment condition on this formula reflects the poisson character of the process and extends to intersections of finite families of independent poisson processes

hydrodynamic limit of coagulation fragmentation type models of k nary interacting particles hydrodynamic limit of general k nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical smoluchovski s coagulation equations and many other models that are intensively studied in the current mathematical and physical literature existence and uniqueness theorems for these equations are proved at last for k nary mass exchange processes with k 2 an alternative nondeterministic measure valued limit diffusion approximation is discussed

combined nonparametric inference and state estimation for mixed poisson processes given independent identically distributed copies of a mixed poisson process n on a lccb space e i e a cox process whose directing measure is of the form m where 0 is a random variable with distribution and m is a measure on e we construct strongly consistent and asymptotically normal estimators of m and the laplace transform l methods are presented for estimating the directing measure of the n 1 st process by combining the data for that process with estimates of appropriate quantities the latter based on the first n processes the case where different processes are observed over different sets is addressed

a bound on the geometric genus of projective varieties verifying certain flag conditions fix integers n r s1 sl and let s n r s1 sl be the set of all integral projective and nondegenerate varieties v of degree s1 and dimension n in the projective space pr such that for all i 2 l v does not lie on any variety of dimension n i 1 and degree sl we determine an upper bound gh n r s1 sl depending only on n r s1 sl for the number g n r s1 sl max pg v v s n r s1 sl where pg v denotes the geometric genus of v in case n 1 and l 2 the study of an upper bound for the geometric genus has a quite long history and for n 1 l 2 and s2 r n it has been introduced by harris we exhibit sharp results for particular ranges of our numerical data n r s1 sl for instance we extend halphen s theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension n in pn 3 as the smooth varieties of maximal geometric genus with respect to appropriate flag condition this result applies to smooth surfaces in p5 next we discuss how far gh n r s1 sl is from g n r s1 sl and show a sort of lifting theorem which states that at least in certain cases the varieties v s n r s1 sl of maximal geometric genus g n r s1 sl must in fact lie on a flag such as v v n s1 v n 1 s2 v n l 1 sl p r where v j s denotes a subvariety of pr of degree s and dimension j we also discuss further generalizations of flag conditions and finally we deduce some bounds for castelnuovo s regularity of varieties verifying flag conditions 0 introduction fix integers n r s1 sl such that 1 n r 2 2 l r n and let s n r s1 sl be the set of all integral projective and nondegenerate varieties v of degree s1 and dimension n in the projective space p such that for all i 2 l v does not lie on any variety of dimension n i 1 and degree sl we determine an upper bound g n r s1 sl depending only on n r s1 sl for the number g n r s1 sl max pg v v s n r s1 sl 0 1 received by the editors october 23 1995 1991 mathematics subject classification primary 14j99 14m07 14m10 secondary 14f17

geometric consequences of extremal behavior in a theorem of macaulay f s macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer valued function which determine when such a function can be the hilbert function of a standard graded k algebra we investigate some algebraic and geometric consequences which arise from the extremal cases of macaulay s theorem our work also builds on the fundamental work of g gotzmann our principal applications are to the study of hilbert functions of zeroschemes with uniformity conditions as a consequence we have new strong limitations on the possible hilbert functions of the points which arise as a general hyperplane section of an irreducible curve