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

analyzing fractional horn constraint systems abstract in this paper we study linear constraint systems in which each constraint is a fractional horn constraint a constraint is fractional horn if it can be written in the form i 1 n a i x i c where each a i is integral c is integral at most one of the a i s is positive and all negative a i s are equal to 1 a conjunction of fractional horn constraints is called a fractional horn constraint systems fhs fhss generalize a number of specialized constraint systems such as difference constraint systems dcs and horn constraint systems hcs we know that the problem of checking lattice point feasibility in dcss and hcss is in p in contrast we show that the problem of checking lattice point feasibility in fhss is np complete we then study a sub class of fractional horn constraint systems called binary fractional horn constraint systems bfhs in a bfhs each constraint has at most two non zero coefficients in this case we show that the problem of checking integer feasibility is in p furthermore we establish several properties of the least integer point and the least linear point of a bfhs if they exist

bimonotone linear inequalities and sublattices of rn abstract a bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs if both different from zero a linear inequality defines a halfspace that is a sublattice of r n a subset closed with respect to componentwise maximum and minimum if and only if it is bimonotone veinott has shown that a polyhedron is a sublattice if and only if it can be defined by a finite system of bimonotone linear inequalities whereas topkis has shown that every sublattice of r n and of more general product lattices is the solution set of a system of nonlinear bimonotone inequalities in this paper we prove that a subset of r n is the solution set of a countable system of bimonotone linear inequalities if and only if it is a closed convex sublattice similarly we note that a subset of r n is closed and convex if and only if it is the solution set of a countable system of linear inequalities we also present necessary and or sufficient conditions for a sublattice to be the intersection of the cartesian product of its projections on the coordinate axes with the solution set of a possibly infinite system of bimonotone linear inequalities we provide explicit constructions of such systems of bimonotone linear inequalities under certain assumptions on the sublattice we obtain veinott s polyhedral representation theorem and a 0 1 version of birkhoff s representation theorem as corollaries we also point out a few potential pitfalls regarding properties of sublattices of r n

lattice like operations and isotone projection sets abstract by using some lattice like operations which constitute extensions of ones introduced by m s gowda r sznajder and j tao for self dual cones a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets the results of this paper are wide range generalizations of some results of the authors obtained for self dual cones the aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect to the order relation which give rise to the above mentioned lattice like operations the topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool for euclidean sublattices this approach was considered by g isac and respectively by h nishimura and e a ok

calculating hausdorff dimension of julia sets and kleinian limit sets we present a new algorithm for efficiently computing the hausdorff dimension of sets x invariant under conformal expanding dynamical systems by locating the periodic points of period up to n we construct approximations s n which converge to dim x super exponentially fast in n this method can be used to give rigorous estimates for important examples including hyperbolic julia sets and limit sets of schottky and quasifuchsian groups

representation of general and polyhedral subsemilattices and sublattices of product spaces abstract it is shown that each element of the lattice of meet resp join sublattices of a product s of n chains has a representation as the intersection of n subsets of s the i th of which is decreasing resp increasing for each fixed value of the i th coordinate for each i this result is applied to show that an arbitrary element of the lattice of sublattices of s has a representation as the intersection of n 2 subsets the ij th of which is decreasing for each fixed value of the i th and increasing for each fixed value of the j th coordinate for each i j irreducible representations are given in each case providing an alternative proof of an instance of hashimoto s 1952 representation of sublattices of a distributive lattice moreover irreducible representations are given for the polyhedral members of the lattice of closed convex subsets of n dimensional euclidean space that are at once subsemilattices or sublattices it is alsoshown that the polyhedral subsemilattices and sublattices can be represented as duals respectively of pre leontief substitution systems and generalized network flow problems finally the problems of checking whether a polyhedral set is a subsemilattice or sublattice are reduced to that of solving a system of linear inequalities thereby showing that these recognition problems can be solved in polynomial time

isotonic regression and isotonic projection abstract the note describes the cones in the euclidean space admitting isotonic metric projection with respect to the coordinate wise ordering as a consequence it is shown that the metric projection onto the isotonic regression cone the cone defined by the general isotonic regression problem admits a projection which is isotonic with respect to the coordinate wise ordering

hausdorff dimension and conformal dynamics iii computation of dimension this paper presents an eigenvalue algorithm for accurately computing the hausdorff dimension of limit sets of kleinian groups and julia sets of rational maps the algorithm is applied to schottky groups quadratic polynomials and blaschke products yielding both numerical and theoretical results dimension graphs are presented for a the family of fuchsian groups generated by reflections in 3 symmetric geodesics b the family of polynomials f c z z 2 c c 1 1 2 and c the family of rational maps f t z z t 1 z t 0 1 we also calculate h dim 1 305688 for the apollonian gasket and h dim j f 1 3934 for douady s rabbit where f z z 2 c satisfies f 3 0 0

the least element property of center location on tree networks with applications to distance and precedence constrained problems in the classicalp center location model on a network there is a set of customers and the primary objective is to selectp service centers that will minimize the maximum distance of a customer to a closest center suppose that thep centers receive their supplies from an existing central depot on the network e g a warehouse thus a secondary objective is to locate the centers that optimize the primary objective as close as possible to the central depot we consider tree networks and twop center models we show that the set of optimal solutions to the primary objective has a semilattice structure with respect to some natural ordering using this property we prove that there is ap center solution to the primary objective that simultaneously minimizes every secondary objective function which is monotone nondecreasing in the distances of thep centers from the existing central depot restricting the location models to a rooted path network real line we prove that the above results hold for the respective classicalp median problems as well

free vs locally free kleinian groups we prove that kleinian groups whose limit sets are cantor sets of hausdorff dimension 0 examples of non free purely hyperbolic kleinian groups whose limit set is a cantor set of hausdorff dimension 1 epsilon

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

invertibility preserving linear maps on j subspace lattice algebras it is shown that every unital surjective linear map between j subspace lattice algebras which preserves invertibility in both directions is a jordan isomorphism this result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras

surface subgroups of kleinian groups with torsion we prove that every finitely generated kleinian group that contains a finite non cyclic subgroup either is finite or virtually free or contains a surface subgroup hence every arithmetic kleinian group contains a surface subgroup

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

the linear complementarity problems with a few variables per constraint in this paper we consider the sparse linear complementarity problem denoted by k lcp the coefficient matrices are restricted to have at most k nonzero entries per row it is known that the 1 lcp is solvable in linear time and the 3 lcp is strongly np hard we show that the 2 lcp is strongly np hard and it can be solved in polynomial time if it is sign balanced i e each row of the matrix has at most one positive and one negative entry our second result matches the currently best known complexity bound for the corresponding sparse linear feasibility problem in addition we show that an integer variant of the sign balanced 2 lcp is weakly np hard and pseudo polynomially solvable and the generalized 1 lcp is strongly np hard

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

convergence and divergence of kleinian punctured torus groups in this paper we give a necessary and sufficient condition in which a sequence of kleinian punctured torus groups converges this result tells us that every exotically convergent sequence of kleinian punctured torus groups is obtained by the method due to anderson and canary invent math 1996 thus we obtain a complete description of the set of points at which the space of kleinian punctured torus groups self bumps we also discuss hausdorff limits of sequences of bers slices

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

relaxation based algorithms for minimax optimization problems with resource allocation applications we consider minimax optimization problems where each term in the objective function is a continuous strictly decreasing function of a single variable and the constraints are linear we develop relaxation based algorithms to solve such problems at each iteration a relaxed minimax problem is solved providing either an optimal solution or a better lower bound we develop a general methodology for such relaxation schemes for the minimax optimization problem the feasibility tests and formulation of subsequent relaxed problems can be done by using phase i of the simplex method and the farkas multipliers provided by the final simplex tableau when the corresponding problem is infeasible such relaxation based algorithms are particularly attractive when the minimax optimization problem exhibits additional structure we explore special structures for which the relaxed problem is formulated as a minimax problem with knapsack type constraints efficient algorithms exist to solve such problems the relaxation schemes are also adapted to solve certain resource allocation problems with substitutable resources there instead of phase i of the simplex method a max flow algorithm is used to test feasibility and formulate new relaxed problems

the hausdorff dimension of the limit set of a geometrically finite kleinian group we consider in this paper groups of mobius transformations of l r tj zc such a group is called kleinian if it acts discontinuously somewhere in 1 the action of g extends to the n d dimensional hyperbolic space h i r zc and g is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of g in h t for a more precise definition see 15 1 b we prove in this paper that the hausdor f dimension dim l g of the limit set l g of a geometrically finite kleinian group g of 1 is less than n theorem d our proof of this theorem is based on the following obser ation assume that g is of compact t3 pe i e if izi i h iui then i l g g is compact then there is an integer q such that if we divide any n cube q of r into q equal subcubes then at least one of these subcubes does not touch l g let l4 q be the family of these subcubes which touch l g then the n measures of q e le q do not add up to the n measure of q and we get