size distribution of clusters in irreversible kinetic aggregation the size distribution of clusters is derived analytically at a given time in a process of irreversible kinetic aggregation of diffusive clusters which starts from a collection of a great number of individual particles a general reduced shape is given in the scaling regime where gelation does not occur the time evolution is derived by use of smoluchowski s equation then a scaling reasoning allows one to find the exponents in real cases and a quantitative comparison is made with numerical simulations

on a thin film model with insoluble surfactant abstract this paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational capillary and van der waals forces we prove the existence of global weak solutions for medium sized initial data in large function spaces moreover exponential decay towards the flat equilibrium state is established where an estimate on the decay rate can be computed explicitly

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

smoluchowski s equation and the θ exponent for branched polymers bonding of large clusters by surface reactions can be modelled by smoluchowski s coagulation equation with coagulation rates kij ij omega with omega d 1 d in d dimensional systems it is shown that the cluster size distribution for large clusters well below the gelation transition has the form ck approximately k theta xi k k to infinity where theta 2 omega results are compared with those from lattice theories and from the flory stockmayer theory for the models kij 1 2 imu jnu inu jmu with mu nu 1 one finds theta mu nu for kij ijomega jiomega with 1 or omega 1 one finds theta 1 2 1 omega and for kij ij one has theta 5 2

intersection cuts with infinite split rank we consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables when this model only has two integer variables dey and louveaux characterized the intersection cuts that have infinite split rank we show that for any number of integer variables the split rank of an intersection cut generated from a rational lattice free polytope l is finite if and only if the integer points on the boundary of l satisfy a certain 2 hyperplane property the dey louveaux characterization is a consequence of this more general result

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

computer simulation of a model for irreversible gelation a random mixture of bifunctional and tetrafunctional units is placed on a simple cubic lattice permanent bonds between these units are then formed by the random motion of active centres radicals resulting in a model for the gelation of polyacrylamide and similar processes the largest of the clusters formed by this kinetic percolation process is identified with the gel fraction the mean cluster size with the weight average degree of polymerisation the critical exponent gamma of the mean cluster size is roughly the same as for random percolation that is different from that of the classical theory of flory 1941 and stockmayer 1943 but the corresponding amplitude ratio that is the ratio of average molecular weight on both sides of the gel point differs strongly from its value in random percolation thus this kinetic gelation model seems to belong to a universality class of its own different from both that of random percolation and that of classical gelation theories

reverse split rank the reverse split rank of an integral polyhedron p p is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is p p already in mathbb r 3 r3 there exist polyhedra with infinite reverse split rank we give a geometric characterization of the integral polyhedra in mathbb r n rn with infinite reverse split rank

maximal lattice free convex sets in linear subspaces we consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice free convex sets in an affine subspace we also show that these sets are polyhedra the latter result extends a theorem of lovasz characterizing maximal lattice free convex sets in rn

clustering of clusters processes above their upper critical dimensionalities in contrast with the diffusion limited particle cluster aggregation p cl the cluster cluster aggregation model cl cl has a finite upper critical dimensionality above which the fractal dimension of the clusters is trivial this exponent is derived analytically for a whole set of models including p cl and cl cl the results clarify the physical differences between these two models

a survey of numerical solutions to the coagulation equation we present the results of a systematic survey of numerical solutions to the coagulation equation for a rate coefficient of the form aij i j i j and monodisperse initial conditions the results confirm that there are three classes of rate coefficients with qualitatively different solutions for 1 and 1 the numerical solution evolves in an orderly fashion and tends towards a self similar solution at large time t the properties of the numerical solution in the scaling limit agree with the analytic predictions of van dongen and ernst in particular for the subset with 0 and 1 the numerical solution tends towards a self similar solution as t approaches a finite time t0 the mass spectrum nk develops at t0 a power law tail nk k at large masses that violates mass conservation and runaway growth gelation is expected to start at tcrit t0 in the limit the initial number of particles n0 the exponent is in general less than the analytic prediction 3 2 and t0 k 1 n0a11 with k 1 2 if 1 1 for 1 the behaviours of the numerical solution are similar to those found in a previous paper by us they strongly suggest that there are no self consistent solutions at any time and that runaway growth is instantaneous in the limit n0 they also indicate that the time tcrit for the onset of runaway growth decreases slowly towards zero with increasing n0

scaling and crossover properties of a new solvable model of aggregation kinetics in a previous paper a new reaction kernel for the smoluchowski equations of aggregation was solved exactly this kernel k j k 2 qj qk for 0 q 1 a real positive quantity interpolates between two well understood exactly solved cases namely that of k j k 2 and that of k j k j k this new model however shows a number of unexpected features not found in either of the two limiting cases it is shown that this model has a remarkable behaviour with respect to the commonly accepted scaling theory on the one hand it satisfies a rigorous form of the scaling hypothesis but on the other hand it clearly violates some relations which are ordinarily assumed to follow from it these issues are discussed as well as the nature of the singular limit in which q is very close to one for which our kernel becomes close to the sum kernel mentioned above in particular the form of the crossover between two kernels with different degrees of homogeneity can be discussed here in an exact way

the motion of a thin liquid film driven by surfactant and gravity we investigate wave solutions of a lubrication model for surfactant driven flow of a thin liquid film down an inclined plane we model the flow in one space dimension with a system of nonlinear pdes of mixed hyperbolic parabolic type in which the effects of capillarity and surface diffusion are neglected numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves some with jumps in height and surfactant concentration gradient the various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane jump conditions admit new shock waves sustained by a linear surfactant wave traveling upstream the stability of these waves is investigated analytically and numerically for initial value problems a critical ratio of upstream to downstream height separates two distinct long time wave patterns below the critical ratio there is also an exact solution in which the height is piecewise constant and

split rank of triangle and quadrilateral inequalities a simple relaxation consisting of two rows of a simplex tableau is a mixed integer set with two equations two free integer variables and nonnegative continuous variables recently andersen et al and cornuejols and margot showed that the facet defining inequalities of this set are either split cuts or intersection cuts obtained from lattice free triangles and quadrilaterals from an example given by cook et al it is known that one particular class of facet defining triangle inequality does not have finite split rank in this paper we show that all other facet defining triangle and quadrilateral inequalities have finite split rank

size distribution in the polymerisation model afrbg the polymer size distribution in flory s polymerisation model afrbg is in equilibrium and non equilibrium of the general form ck ank xi k both below and above the gelation transition here explicit expressions are derived for a alpha and xi alpha as a function of the extent of reaction alpha the combinatorial factors nk are calculated from the recursion relation 2 k 1 nk sigma i j kkijninj with kij si f sj g si g sj f and sk f f 1 k 1 using a generating function technique the authors express nk in terms of laguerre polynomials g to infinity and jacobi polynomials g finite whose large k behaviour is of the form nk approximately bk tau xi c k with tau 3 2 no gelation occurs if f or g equals 1 and with tau 5 2 gelation occurs if f 1 and g 1

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

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

on the rank of disjunctive cuts let l be a family of lattice free polyhedra in rm containing the splits given a polyhedron p in rm n we characterize when a valid inequality for p zm rn can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l we also characterize the lattice free polyhedra m such that all the disjunctive cuts corresponding to m can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l for every polyhedron p our results imply interesting consequences related to split rank and to integral lattice free polyhedra that extend recent research findings

scaling theory for systems with instantaneous gelation partial results we study a system of kinetic equations describing irreversible aggregation the so called smoluchowski equations for a peculiar case in which even starting from initial conditions for which only a monomer is present at arbitrarily small times an infinite cluster appears and causes the total mass of the system to decrease this phenomenon known as instantaneous gelation is known to arise in cases where the reaction rates between large clusters increase strongly as the cluster size goes to infinity we resolve two puzzles linked to this behaviour first from general results it is well known that a power law behaviour must form instantly at large masses for arbitrarily small times we show that the small time behaviour can be analysed for these systems but this must be done in an essentially different manner than in the regular case the correct treatment provides a mechanism for the instantaneous appearance of a power law tail at large masses in the aggregate size distribution we additionally show that instantaneous gelation is connected to the presence of an essential singularity in the solution and also show how from the numerical study of the formal power series describing the solution the nature of the singularity can be guessed and the time dependence of the mass scale for the onset of the power law behaviour conjectured

on convergence in mixed integer programming let p subseteq mathbb r m n be a rational polyhedron and let pi be the convex hull of p cap mathbb z m times mathbb r n we define the integral lattice free closure of p as the set obtained from p by adding all inequalities obtained from disjunctions associated with integral lattice free polyhedra in mathbb r m we show that the integral lattice free closure of p is again a polyhedron and that repeatedly taking the integral lattice free closure of p gives pi after a finite number of iterations such results can be seen as a mixed integer analogue of theorems by chvatal and schrijver for the pure integer case one ingredient of our proof is an extension of a result by owen and mehrotra in fact we prove that for each rational polyhedron p the split closures of p yield in the limit the set pi

the asymptotic behavior and self similar solutions for disperse systems with coagulation and fragmentation the paper analyzes the asymptotic behavior of disperse systems with coagulation and fragmentation of particles the possible types of self similarity regimes have been analyzed and conditions required for their existence have been set the generalized approximation method ga method numerical simulation is used to determine the actual behavior of moments l t the examples of ga method application show its suitability for use in research problems in general the obtained results show that binary breakage coagulation is a wide and non trivial scope for investigation a number of regimes are represented such as steady state coagulation winning gelation collapsing self similarity and spectrum singularity the existence of collapsing accumulating in zero self similar spectra is illustrated in terms of a particular example of the coagulation kernel k g n gn and breakage rate f g n a

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

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