caratheodory helly and radon numbers for sublattice and related convexities the caratheodory helly and radon numbers are three main invariants in convexity theory these invariants have been determined exactly or approximately for a number of different convexity structures we consider convexity structures defined by the sublattices and by the convex sublattices of finite dimensional euclidian integer and boolean spaces such sublattices arise in submodular optimization lattice programming and in monotone comparative statics of optimization and fixed point problems we also consider integral l natural convexities induced by dual network flow constraint systems we determine the exact caratheodory helly and radon numbers of most of these convexities and very close upper and lower bounds for the other caratheodory numbers our results imply for example that if a set can be obtained with unions and intersections from a given family of subsets of a finite set then it can be obtained with unions and intersections from a small subfamily we also show that finding the cara

a general class of greedily solvable linear programs a greedy algorithm solves a dual pair of linear programs where the primal variables are associated to the elements of a sublattice b of a finite product lattice and the cost coefficients define a submodular function on b this approach links and generalizes two well known classes of greedily solvable linear programs the primal problem generalizes the ordinary and multi index transportation problems satisfying a monge condition hoffman 1963 bein et al 1995 to the case of forbidden cells where the nonforbidden cells form a sublattice the dual problem generalizes to an arbitrary finite product lattice the linear optimization problem over submodular polyhedra lovasz 1983 fujishige and tomizawa 1983 which stemmed from the work of edmonds 1970 on polymatroids our model and results also encompass the dual pairs of linear programs and their greedy solutions defined by lovasz 1983 for the special case of the boolean algebra and by faigle and kern 1996 for the case of so called rooted forests we also discuss relationships between monge properties and submodularity and present a class of problems with submodular costs arising in production and logistics

shift operators on banach spaces abstract j r holub has obtained several results for shift operators on c x in this paper we answer some questions of holub and obtain extensions of many of his results in particular we show that c x r does not admit a shift operator if x has only countably many components and each component is infinite we show that c x c does not admit a shift operator for certain special compact hausdorff spaces x we show that there exists a compact hausdorff space x which is not totally disconnected and both c x c and c x r admit shift operators if 1 p x is a finite non atomic measure space then l p r does not admit a disjointness preserving shift operator we also show that l p for 1 p is the only l p r space which admits a disjointness preserving shift operator

on the absence of infinite ab percolation clusters in bipartite graphs the authors give a rigorous proof that infinite ab percolation does not occur for any parameter value in a certain class of bipartite planar graphs in particular infinite ab percolation cannot occur on the square lattice

counterexamples in ab percolation halley 1983 conjectured that the ab percolation probability function attains its maximum at p 1 2 and that consequently infinite ab percolation is impossible on a bipartite graph whose standard site percolation critical probability is greater than 1 2 the authors produce counterexamples that demonstrate that both conjectures are false in fact for any positive integer n a graph may be constructed which has at least n ab percolation transitions

weighted composition operators ofc0 x s abstract in this paper we prove that into isometries and disjointness preserving linear maps from c 0 x into c 0 y are essentially weighted composition operators tf h f for some continuous map and some continuous scalar valued function h

some remarks on ab percolation in high dimensions in this paper we consider the ab percolation model on z d and zd let phalt zd be the critical probability for ab percolation on zd we show that phalt zd 1 2d2 if the probability of a site to be in state a is 2d2 for some fixed 1 then the probability that ab percolation occurs converges as d to the unique strictly positive solution y of the equation y 1 exp y we also find the limit for the analogous quantities for oriented ab percolation on z d in particular phalt z d 2 d2 we further obtain a small extension to the two parameter problem in which even vertices of zd have probability pa of being in state a and odd vertices have probability pb of being in state b but without relation between pa and pb the principal tools in the proofs are a method of penrose 1993 for asymptotics of percolation on graphs with vertices of high degree and the second moment method

substitutes complements and ripples in network flows we study the qualitative variation of minimum cost network flows and their associated costs with arc parameters demands are given at each node flow is conserved each arc s parameter lies in a lattice and the flow cost is real or infinity valued our main results are roughly as follows if the flow cost is arc additive the problem can be decomposed into independent problems on each biconnected component of the graph if also the flow cost is convex in the flow the magnitude of the change in the optimal flow in arc a resulting from changing arc b s parameter diminishes the less biconnected a is to b arc a is less biconnected to b than is arc d if every simple cycle containing a and b also contains d this relation is a quasi order with two distinct arcs being equivalent if and only if deleting them disconnects the graph hie hasse diagram of the partial order of the induced equivalence classes is a tree with all arcs directed towards the class containing b two arcs are complements resp substitute

infinite ab percolation clusters exist on the triangular lattice infinite ab percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values it was not previously known that infinite ab percolation occurred with positive probability on any planar lattice

disjointness preserving shifts on c0 x abstract we study disjointness preserving quasi n shift operators on c 0 x where x is locally compact and hausdorff when c 0 x admits a quasi n shift t there is a countable subset of x x equipped with a tree like structure called tree with exactly n joints such that the action of t on c 0 x can be implemented as a shift on the tree if t is an n shift then the tree is dense in x and thus x is separable by analyzing the structure of the tree we show that every quasi n shift on c 0 can always be written as a product of n quasi 1 shifts although it is not the case for general c 0 x as shown by our counter examples we can do so after dilation

on ab percolation on bipartite graphs a method is presented for determining lower bounds for the ab percolation critical probability of bipartite graphs for some graphs the method shows that ab percolation is impossible in an interval containing p 1 2 the results improve those of appel and wierman 1987 by removing symmetry and periodicity restrictions and lend support to a conjecture of halley 1983

antipercolation on bethe and triangular lattices the antipercolation problem is solved on the bethe lattice the critical exponents are identical to the percolation exponents when the coordination number z is greater than a critical value zc 3 for which the problem has new exponents satisfying extended universality and below which there is no transition for alternate lattices the problem may be transformed into a percolation problem with different occupation probabilities on the two sublattices this allows a connection with an s state potts model with different z spin interactions on the two sublattices in two dimensions there is no transition on the alternate square and honeycomb lattices whereas a transition exists on the triangular lattice using the phenomenological renormalisation group method the critical concentration is found to be pc a 0 21 whereas the correlation length exponent nu a seems to converge towards the accepted percolation value nu p 1 333

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

bi partite percolation and gelation bi partite percolation is discussed in the context of gelation the partial generating function method is used to derive expansions for the mean number of clusters and for the weight average molecular weight in terms of the reactant concentrations a and b renormalisation group and other arguments imply that the critical behaviour in bi partite percolation is determined by the ordinary percolation fixed point at a b pc

minimum aggregate concave cost multicommodity flows in strong series parallel networks this work develops a polynomial time dynamic programming algorithm for minimum aggregate concave cost multicommodity flow problems in strong series parallel networks many important problems from production and inventory management capacity planning network design and transportation can be formulated in these terms our results include a characterization of extreme flows in strong series parallel networks and an algorithm based on this characterization that searches extreme flows efficiently to find an optimal one the algorithm runs in time proportional to a n k where a n and k are respectively the numbers of arcs nodes and commodities in the network and appears to be the first to solve the problem in polynomial time when applied to the dynamic economic order quantity problem the algorithm matches the performance of that of wagner and whitin 1958 moreover our algorithm has broader applications including the multi division capacity expansion problem and the generalization of the dynamic economic order quantity problem to series parallel production processes in which subassemblies are assembled into a finished product

isometric shifts onc k abstract several problems regarding isometric shifts proposed by gutek et al j funct anal 101 1991 97 119 are solved

on some network flow games we analyze three subclasses of cooperative games arising from network optimization problems in which the resources such as arcs or nodes in the network are controlled by individuals who have conflicting objectives the first subclass of cooperative games is induced by network optimization problems over directed augmented trees we show that for this subclass of games the kernel coincides with the nucleolus and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range we further give a linear characterization of the core of this subclass of games which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it the second subclass of cooperative games is induced by maximum flow problems in simple undirected networks we provide a useful parametric representation of the core of this subclass of games which is used to characterize the nucleolus and the intersection of the core and the kernel explicitly we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners we further demonstrate that among the core vectors the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense the third cooperative game that we study is the assignment game introduced by shapley and shubik 1972 this game is induced by the assignment problem which can be cast as a network optimization problem we investigate the relationship between the kernel and the core of the assignment game and provide a necessary and sufficient condition for the core to be contained in the kernel we further show that in general the intersection of the kernel and the core is not a convex set we also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense finally we consider the horse market example of bohm bawerk 1923 for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core

send and split method for minimum concave cost network flows many problems from inventory production and capacity planning and from network design exhibit scale economies and can be formulated in terms of finding minimum additive concave cost nonnegative network flows we reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic programming method called the send and split method to solve it the main work of the method entails repeatedly solving set splitting and minimum cost chain problems in uncapacitated networks with n nodes a arcs and d 1 demand nodes i e nodes with nonzero exogenous demand the algorithm requires up to n2 13d s2d operations additions and comparisons where s n log2n 3a is the number of operations required to solve a minimum cost chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph if also the network is k planar i e the graph is planar with all demand nodes lying on the boundary of k faces the method requires at most n2 kd3k sd2k operations the algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones these results unify significantly generalize e g to cyclic problems and sometimes improve upon e g for tandem facilities known polynomial time dynamic programming algorithms for wagner and whitin s wagner h m whitin t m 1958 dynamic version of the economic lot size model management sci 5 89 96 dynamic economic order quantity problem zangwill s zangwill w i 1969 a backlogging model and a multi echelon model of a dynamic economic lot size production system a network approach management sci 15 506 527 generalization to tandem facilities and veinott s veinott jr a f 1969 minimum concave cost solution of leontief substitution models of multi facility inventory systems oper res 17 262 291 increasing capacity warehousing problem the networks for the finite period versions of these problems are each 1 planar the method improves upon zangwill s zangwill w i 1968 minimum concave cost flows in certain networks management sci 14 429 450 related oand running time dynamic programming method for finding minimum additive concave cost non negative flows in circuitless single source networks we also implement the method to solve in polynomial time the d 1 demand node and k planar versions of the minimum cost forest and steiner problems in graphs the running time for the d 1 demand node steiner problem in graphs is comparable to that of dreyfus and wagner s dreyfus s e wagner r a 1971 the steiner problem in graphs networks1 195 207 method

limits on relief through constrained exchange on random graphs abstract agents are represented by nodes on a random graph e g small world each agent is endowed with a zero mean random value that may be either positive or negative all agents attempt to find relief i e to reduce the magnitude of that initial value to zero if possible through exchanges the exchange occurs only between the agents that are linked a constraint that turns out to dominate the results the exchange process continues until pareto equilibrium is achieved only 40 90 of the agents achieved relief on small world graphs with mean degree between 2 and 40 even fewer agents achieved relief on scale free like graphs with a truncated power law degree distribution the rate at which relief grew with increasing degree was slow only at most logarithmic for all of the graphs considered viewed in reverse the fraction of nodes that achieve relief is resilient to the removal of links

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length