on the use of multiple objective linear programming algorithms to solve problems with fractional objectives abstract multiple objective linear fractional programming can be used in a wide range of problem formulations in private and public enterprises under certain restrictions a simpler multiple objective linear programming approach can be used for problem solution in this note we analyse the use of this simplified approach to molfp and the practical effect of the restrictions implied

on symplectic cobordisms in this note we make several observations concerning symplectic cobordisms among other things we show that every contact 3 manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3 manifolds are symplectic cobordism equivalent

null surgery on knots in l spaces let k be a knot in an l space y with a dehn surgery to a surface bundle over s 1 we prove that k is rationally fibered that is the knot complement admits a fibration over s 1 as part of the proof we show that if k c y has a dehn surgery to s 1 x s 2 then k is rationally fibered in the case that k admits some s 1 x s 2 surgery k is floer simple that is the rank of hfk y k is equal to the order of h 1 y by combining the latter two facts we deduce that the induced contact structure on the ambient manifold y is tight in a different direction we show that if k is a knot in an l space y then any thurston norm minimizing rational seifert surface for k extends to a thurston norm minimizing surface in the manifold obtained by the null surgery on k i e the unique surgery on k with b 1 0

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

local search for the probabilistic traveling salesman problem correction to the 2 p opt and 1 shift algorithms abstract the probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space where each customer requires a visit only with some known probability a solution to this problem is an a priori tour which visits all customers and the objective is to minimize the expected length of the a priori tour over all customer subsets assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour this problem belongs to the class of stochastic vehicle routing problems a class which has received increasing attention in recent years and which is of major importance in real world applications several heuristics have been proposed and tested for the probabilistic traveling salesman problem many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem in particular two local search algorithms 2 p opt and 1 shift were introduced by bertsimas in a previous report we have shown that the expressions for the cost evaluation of 2 p opt and 1 shift moves as proposed by bertsimas are not correct in this paper we derive the correct versions of these expressions and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions

a heuristic algorithm for the traveling salesman location problem on networks in this paper we present a heuristic for the traveling salesman location problem on a network each day the salesman e g a repair vehicle must visit all the calls that are registered in a service list each call is generated with a given probability and the service list contains at most n calls the heuristic requires on3 time to find the location that minimizes the expected distance traveled a worst case analysis of the heuristic indicates that it will produce a solution which is at most 50 worse than the optimal solution the paper also contains several asymptotic results for the problem in the plane

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

dynamic journeying under uncertainty abstract we introduce a journey planning problem in multi modal transportation networks under uncertainty the goal is to find a journey possibly involving transfers between different transport services from a given origin to a given destination within a specified time horizon due to uncertainty in travel times the arrival times of transport services at public transport stops are modeled as random variables if a transfer between two services is rendered unsuccessful the commuter has to reconsider the remaining path to the destination the problem is modeled as a markov decision process in which states are defined as paths in the transport network the main contribution is a backward induction method that generates an optimal policy for traversing the public transport network in terms of maximizing the probability of reaching the destination in time by assuming history independence and independence of successful transfers between services we obtain approximate methods for the same problem analysis and numerical experiments suggest that while solving the path dependent model requires the enumeration of all paths from the origin to the destination the proposed approximations may be useful for practical purposes due to their computational simplicity in addition to on time arrival probability we show how travel and overdue costs can be taken into account making the model applicable to freight transportation problems

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

almost contact 5 manifolds are contact the existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5 dimensional manifold

right veering diffeomorphisms of compact surfaces with boundary we initiate the study of the monoid of right veering diffeomorphisms on a compact oriented surface with nonempty boundary the monoid strictly contains the monoid of products of positive dehn twists we explain the relationship to tight contact structures and open book decompositions

tightness in contact metric 3 manifolds this paper begins the study of relations between riemannian geometry and global properties of contact structures on 3 manifolds in particular we prove an analog of the sphere theorem from riemannian geometry in the setting of contact geometry specifically if a given three dimensional contact manifold m admits a complete compatible riemannian metric of positive 4 9 pinched curvature then the underlying contact structure is tight in particular the contact structure pulled back to the universal cover is the standard contact structure on s3 we also describe geometric conditions in dimension three for to be universally tight in the nonpositive curvature setting

a priori optimization for the probabilistic maximum independent set problem we first propose a formal definition for the concept of probabilistic combinatorial optimization problem under the a priori method next we study the complexity of optimally solving probabilistic maximum independent set problem under several a priori optimization strategies as well as the complexity of approximating optimal solutions for the different strategies studied we present results about the restriction of probabilistic independent set on bipartite graphs

tight contact structures with no symplectic fillings we exhibit tight contact structures on 3 manifolds that do not admit any symplectic fillings

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

inclusion theorems for non explosive and strongly exposed cones in normed spaces abstract this paper is composed of five sections in the first one we classify cones into pseudonuclear intermediate and explosive the notion of strongly exposed cone and the equivalence between pseudonuclearity and strong exposition for cones in a normed space are recalled in the second one it is shown that several properties for a cone in a normed space exist each one being equivalent to strong exposition of the cone furthermore 2 5 an estimate by chung is improved in the third one through projections of rank one some cones that are an immediate extension of elliptic cones in elementary geometry are defined and it is established 3 3 that a cone is strongly exposed if and only if it can be contained in one of that type in the fourth one it is shown 4 2 that also the non explosivity of a cone is expressible in terms of metric properties more precisely a cone is non explosive if and only if it can be contained in a balanced cone of elliptic kind but in which the axis becomes a finite dimensional subspace it may be interesting to observe that the result does not depend on particular hypotheses e g reflexivity on the space in the fifth one it is put in evidence a property of a cone in a reflexive normed space which excludes that the cone is intermediate an orientative essential bibliography concludes the paper

on the classification of tight contact structures ii we present complete classification results for tight contact structures on two classes of 3 manifolds i torus bundles which fiber over the circle and ii circle bundles which fiber over closed surfaces

normal cr structures on s we classify the normal cr structures on s3 and their automorphism groups together with 2 this closes the classification of normal cr structures on compact 3 manifolds we give a criterion to compare two normal cr structures and we show that the underlying contact structure is up to diffeomorphism unique mathematics subject classification 2000 53a40 53c25 53d10

on the one commodity pickup and delivery traveling salesman problem with stochastic demands this paper studies how to set the vehicle capacity for traveling salesman problems where some of the customer demands are stochastic the analyses are done for the one commodity pickup and delivery tsp as this problem also includes the setting of the initial load the paper first considers feasibility issues this includes finding the smallest vehicle capacity and some initial load such that a given tour is feasible for all scenarios different variants are considered as a function of the time when information becomes available the paper then analyzes the case where some penalties are paid for routing a tour unable to handle customer demands various types of penalties are considered the paper studies properties of the minimal expected penalty of a given tour which are then used to provide approaches to find near optimal tours computational results are presented

classification of engel knots let m mathcal d be an engel 4 manifold we show that the scanning map from the space of engel knots to the space of formal engel knots is a weak homotopy equivalence when restricted to the complement of the closed ker mathcal d orbits this is a relative parametric and c 0 close h principle

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

geometric criteria for overtwistedness we establish geometric criteria to decide whether a contact manifold is over twisted starting with the original denition we rst relate the dierent overtwisted disks d 2n ot in each dimension and show that a manifold is overtwisted if the legendrian un knot is loose then we characterize overtwistedness in terms of open book decompositions and provide several applications

compact stein surfaces with boundary as branched covers of b 4 we prove that stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of b 4 whose branch set is a positive braided surface as a consequence we have that a smooth oriented 3 manifold is stein fillable iff it has a positive open book decomposition

tight not semi fillable contact circle bundles extending our earlier results we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi fillable thus confirming a conjecture of ko honda

contact topology and hydrodynamics iii knotted orbits we employ the relationship between contact structures and beltrami fields derived in part i of this series to construct a steady nonsingular solution to the euler equations on a riemannian s3 whose flowlines trace out closed curves of all possible knot and link types using careful contacttopological controls we can make such vector fields real analytic and transverse to the tight contact structure on s3 sufficient review of concepts is included to make this paper independent of the previous works in this series

on efficient sets in vector maximum problems a brief survey the notion of efficient nondominated noninferior pareto optimal functional efficient set to a vectormaximum problem vmp has been analysed and developed in several directions during the last 30 years starting with the basic notion of efficiency given by pareto 1896 formal descriptions of the efficient properly efficient locally proper efficient weak or strong efficient set have been developed beginning in the 50th of this century based on these notions various characteristics and properties of the efficient set have been studied whereas the feasible set x and the functions zk x k 1 k constituting the vector valued criterion z x of vmp have various properties e g convexity of x concavity of zk x for all k differentiability etc based on such properties the structure of the efficient set and the existence of efficient solutions have been analysed a part of the corresponding publications are rather of a pure theoretical character others try to develope theories serving as the basis for working out methods for determining the efficient set or for interactive methods determining some compromise solutions duality theories for more or less general cases have been developed and various aspects of stability of vmp have been investigated a brief survey is given

a priori optimization consider a complete graph g v e in which each node is present with probability p we are interested in solving combinatorial optimization problems on subsets of nodes which are present with a certain probability we introduce the idea of a priori optimization as a strategy competitive to the strategy of reoptimization under which the combinatorial optimization problem is solved optimally for every instance we consider four problems the traveling salesman problem tsp the minimum spanning tree vehicle routing and traveling salesman facility location we discuss the applicability of a priori optimization strategies in several areas and show that if the nodes are randomly distributed in the plane the a priori and reoptimization strategies are very close in terms of performance we characterize the complexity of a priori optimization and address the question of approximating the optimal a priori solutions with polynomial time heuristics with provable worst case guarantees finally we use the tsp as an example to find practical solutions based on ideas of local optimality

weak and strong fillability of higher dimensional contact manifolds for contact manifolds in dimension three the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent we extend these facts to higher dimensions in particular we define a natural generalization of weak fillings and prove that it is indeed weaker at least in dimension five while also being obstructed by all known manifestations of overtwistedness we also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense these depend on a higher dimensional analogue of giroux torsion which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary

existence of engel structures we develop a construction of engel structures on 4 manifolds based on decompositions of manifolds into round handles this allows us to show that all parallelizable 4 manifolds admit an engel structure we also show that given two engel manifolds m1 m2 satisfying a certain condition on the characteristic foliations there is an engel structure on m1 m2 s2 s2 which is closely related to the original engel structures

adaptive sample size and importance sampling in estimation based local search for the probabilistic traveling salesman problem the probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem for this problem recently an estimation based local search algorithm using delta evaluation has been proposed in this paper we adopt two well known variance reduction procedures in the estimation based local search algorithm the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in monte carlo evaluation the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation we investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation based local search

different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is a topic of theoretical and practical importance in the study of stochastic network problems it provides researchers with a modeling framework for exploring the stochastic effects in routing problems this paper proposed three initial solution generators nn1 nn2 ran under a genetic algorithm ga framework for solving the ptsp a set of numerical experiments based on heterogeneous and homogeneous ptsp instances were conducted to test the effectiveness and efficiency of the proposed algorithms the results from the heterogeneous ptsp show that the average e t values obtained by the three generators under a ga framework are similar to those obtained by the previous best but with an average computation time saving of 50 2 as for the homogeneous ptsp instances nn1 is a relatively better generator among the three examined while ran consistently performs worse than the other two generators in terms of average e t values additionally as compared to previously reported studies no one single algorithm consistently outperformed the others across all homogeneous ptsp instances in terms of the best e t values the fact that no one initial solution generator consistently performs best in terms of the e t value obtained across all instances in heterogeneous cases and that the performance of each examined algorithm is dependent on the number of nodes n and probability p for homogeneous cases suggest the possibility of context dependent phenomenon finally to obtain valid results researchers are advised to include at least a certain amount of test instances with the same combination of n and p when conducting ptsp experiments

probabilistic combinatorial optimization problems on graphs a new domain in operational research abstract we present a new research domain in operational research probabilistic combinatorial optimization problems we survey and comment a number of results and we discuss several problems dealing with this domain as well as future research issues

a priori solution of a travelling salesman problem in which a random subset of the customers are visited consider a problem of routing through a set of n points on any given instance of the problem only a subset consisting of k out of n points 0 k n has to be visited with the number k random with known probability distribution we wish to find a priori a tour through all n points on any given instance the k points present will then be visited in the same order as they appear in the a priori tour the problem of finding such a tour of minimum length in the expected value sense is defined as a probabilistic traveling salesman problem ptsp what distinguishes one ptsp from another is the probability distribution or more generally the probability law that specifies the number k and the identity of the points that need to be visited on any given instance of the problem after motivating the problem by applications we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions we then provide in a unified way

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem this paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem ptsp different combinations of three commonly used local search methods in the ptsp i e 1 shift 2 opt and 3 opt were used to investigate its effects a set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances the numerical results and the permutation test showed that the diversified local search strategy especially by combining 1 shift and 2 opt algorithms can most effectively solve the homogeneous and heterogeneous ptsp in most of the tested instances in comparison with the single local search strategy under scatter search framework

fractional programming applications and algorithms abstract the paper deals with decision problems that give rise to the optimization of ratios subject to constraints these so called fractional programs have been treated in a considerable number of papers it is attempted to survey applications as well as solution methods in linear quadratic and concave convex fractional programming in the first part we give a detailed outline on the major areas of applications of fractional programming following a brief review on some basic theoretical results like duality relations in the second part we then discuss different algorithmic approaches a primal a dual and a parametric solution method is considered here we try to determine for each method the kind of fractional programs that are solved by it suitably

solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation abstract this paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer the solution to this vehicle routing problem with stochastic demand vrpsd involves the optimization of complete routing schedules with minimum travel distance driver remuneration and number of vehicles subject to a number of constraints such as time windows and vehicle capacity to solve such a multiobjective and multi modal combinatorial optimization problem this paper presents a multiobjective evolutionary algorithm that incorporates two vrpsd specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions a new way of assessing the quality of solutions to the vrpsd on top of comparing their expected costs is also proposed it is shown that the algorithm is capable of finding useful tradeoff solutions for the vrpsd and the solutions are robust to the stochastic nature of the problem the developed algorithm is further validated on a few vrpsd instances adapted from solomon s vehicle routing problem with time windows vrptw benchmark problems

analysis of probabilistic combinatorial optimization problems in euclidean spaces probabilistic combinatorial optimization problems are generalized versions of deterministic combinatorial optimization problems with explicit inclusion of probabilistic elements in the problem definitions based on the probabilistic traveling salesman problem ptsp and on the probabilistic minimum spanning tree problem pmstp the objective of this paper is to give a rigorous treatment of the probabilistic analysis of these problems in the plane more specifically we present general finite size bounds and limit theorems for the objective functions of the ptsp and pmstp we also discuss the practical implications of these results and indicate some open problems

contractibility of efficient point sets in normed spaces l e a topological vector space over reals c a convex cone and x a set in e c specifies a partial order in e as follows x y e x c y if x y c with this order one can define the set of efficient points of x with respect to c as the set min x c x x if x c y for some y x then y c x whenever the relative interior of c ric is nonempty the cone 0 u ric gives another order defining the set of weakly efficient points of x with respect to c denoted w min x c one of the most important problems of vector optimization is to investigate topological properties of the sets min x c and w min x c our aim is to develop a new method which makes it possible to prove most important properties of efficient point sets such as contractibility and connectedness in general normed spaces

connectedness of efficient solutions in multiple criteria combinatorial optimization abstract in multiple criteria optimization an important research topic is the topological structure of the set x e of efficient solutions of major interest is the connectedness of x e since it would allow the determination of xe without considering non efficient solutions in the process we review general results on the subject including the connectedness result for efficient solutions in multiple criteria linear programming this result can be used to derive a definition of connectedness for discrete optimization problems we present a counterexample to a previously stated result in this area namely that the set of efficient solutions of the shortest path problem is connected we will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization the spanning tree problem

two generalizations of a theorem of arrow barankin and blackwell in 1953 arrow barankin and blackwell proved that if r n is equipped with its natural ordering and if a is a closed convex subset of r n then the set of points in a that can be supported by strictly positive linear functionals is dense in the set of all efficient maximal points of a in this note two generalizations of this result are given the first of these is in the setting of a dual system and requires relatively weak assumptions on the ordering cone but a rather strong compactness assumption on the set a the second generalization which is in the setting of a locally convex space relaxes the compactness assumption on the set a but demands more stringent assumptions on the ordering cone this second result was recently obtained by petschke for normed spaces m petschke on a theorem of arrow barankin and blackwell siam j control optim 28 1990 pp 395 401 the proof given here is substantially different from that given by petschke

connectedness of the efficient point sets in quasiconcave vector maximization one of the key questions of vector optimization theory is to investigate the structure of efficient point sets among the topological properties of these sets the connectedness and contractibility are of interest as they provide a possibility of continuous moving from one optimal solution to any other along optimal alternatives only and they have a close relation to the fixed point property that is a useful argument in economic equilibrium theory see 1 21 and there cited references motivated by this in 2 the structure of the sets of points which are efficient with respect to cones has been studied for some classes of convex sets several topological properties as closedness compactness contractibility etc have been obtained although the research has been carried out in the space of outcomes rather than in the space of alternatives in 11 the author has pointed out his investigation in the space of alternatives for quasiconcave maximization problems only the preference orders are assumed to be defined by the nonnegative orthant of the space as noted in 1 the connectivity results for quasiconcave criteria do not extend to more general preference order however they may be expected under a modified quasiconcavity assumption in the present paper we shall establish the connectedness of the set of weakly efficient alternatives for cone quasiconcave criteria and the contractibility of the set of efficient alternatives for the strictly conequasiconcave case under a weakened assumption on the compactness of the upper level sets the results then will be carried into the space of outcomes suppose that x is a nonempty closed convex set in the n dimensional euclidean space r c is a pointed closed convex cone in r with a nonempty interior int c andf is a map from x into r we say that x e x is an efficient alternative with respect to c if there is no y e x such that

existence and classification of overtwisted contact structures in all dimensions we establish a parametric extension h principle for overtwisted contact structures on manifolds of all dimensions which is the direct generalization of the 3 dimensional result from 12 it implies in particular that any closed manifold admits a contact structure in any given homotopy class of almost contact structures

pareto optimization in topological vector spaces to identify the efficient point of a set is an important problem in vector optimization the existence of efficient points in infinite dimensional space has been studied by many authors such as 2 10 12 15 17 19 21 23 25 etc to study pareto efficiency in locally convex topological vector space in 1983 the second author isac 14 has introduced an important notion of nuclear cone which has also other applications as for example in the fixed point theory in the best approximation theory in the study of vector optimization for multivalued functions in the study of conically bounded sets and in the study of nuclearity of topological vector space about these one can refer to hyers isac rassias book 11 and references therein in 1990 pontini 20 defined a more general notion of pseudonuclear in general topological vector space may not be locally convex pontini remarked that in a normed vector space a cone is pseudo nuclear if and only if it is nuclear but in a general topological vector space this result is not true in 11 13 isac hyers and rassias discussed the existence of efficient points in locally convex topological space ordered by pointed nuclear cone and got some general results in book 11 and paper 13 they also posed the following open problem

a priori optimization of the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is defined on a graph g v e where v is the vertex set and e is the edge set each vertex vi has a probability pi of being present with each edge vi vj is associated a distance or cost cij in a first stage an a priori hamiltonian tour on g is designed the list of present vertices is then revealed in a second stage the a priori tour is followed by skipping the absent vertices the ptsp consists of determining a first stage solution that minimizes the expected cost of the second stage tour the problem is formulated as an integer linear stochastic program and solved by means of a branch and cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function problems involving up to 50 vertices are solved to optimality

symplectic homology of some brieskorn manifolds this paper consists of two parts in the first part we use symplectic homology to distinguish the contact structures on the brieskorn manifolds varsigma 2 ell 2 2 2 which contact homology cannot distinguish this answers a question from kwon and van koert brieskorn manifolds in contact topology preprint 2013 arxiv 1310 0343 in the second part we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on s 7 distinguished by the mean euler characteristic of s 1 equivariant symplectic homology apart from various connected sum constructions these contact structures can be taken from the brieskorn manifolds varsigma 78k 1 13 6 3 3 we end with some considerations about extending this result to higher dimensions

new invariants of open symplectic and contact manifolds the first specifically symplectic invariant of open symplectic manifolds the width was defined by m gromov in gr gromov s width was generalized by h hofer i ekeland e zehnder and a floer see eh hz fh they defined numerous new invariants i e symplectic capacities all these invariants are monotonic with respect to the inclusion and serve as obstructions for symplectic embeddings for contact manifolds so far no invariants are known except overtwisting phenomenon in dimension three see bel ell it turns out that much more rich obstructions for symplectic see 2 2 and what is especially interesting for contact embeddings see 3 2 and 3 5 can be defined if we consider embeddings in a prescribed homotopy class this paper is inspired by a recent discovery by benci and sikorav see bc sil ls and 2 4 1 below recently j c sikorav see si2 independently developed some of the ideas discussed in this paper his interests are exclusively in the symplectic field while i am interested mainly in applications to the contact topology the most interesting result of this paper is theorem 3 5 1 which shows in particular the existence of nonisomorphic open solid tori in the standard contact s3

bounds and heuristics for capacitated routing problems in a capacitated routing problem the objective is to minimize the total distance travelled by vehicles of limited capacity to serve a set of customers that are located in the euclidean plane we develop asymptotically optimal bounds and heuristics for this problem under the assumption that the capacity of a vehicle is expressed in terms of an upper bound on the number of customers that it can serve the analysis culminates in an algorithm that for a given capacity and given will find a solution with relative error at most in time polynomial in the number of customers

reducibility in sasakian geometry the purpose of this paper is to study reducibility properties in sasakian geometry first we give the sasaki version of the de rham decomposition theorem however we need a mild technical assumption on the sasaki automorphism group which includes the toric case next we introduce the concept of it cone reducible and consider s 3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2 torus of reeb type in particular we can classify all such sasakian structures up to contact isotopy on s 3 bundles over a riemann surface of genus greater than zero finally we show that in the toric case an extremal sasaki metric on a sasaki join always splits

existence h principle for engel structures in this article we prove that the inclusion of the space of engel structures of a smooth 4 manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups in particular we construct engel structures representing any given full flag

taubes s proof of the weinstein conjecture in dimension three does every smooth vector field on a closed three manifold for example the three sphere have a closed orbit no according to counterexamples by k kuperberg and others on the other hand there is a special class of vector fields called reeb vector fields which are associated to contact forms the three dimensional case of the weinstein conjecture asserts that every reeb vector field on a closed oriented three manifold has a closed orbit this conjecture was recently proved by taubes using seiberg witten theory we give an introduction to the weinstein conjecture the main ideas in taubes s proof and the bigger picture into which it fits taubes s proof of the weinstein conjecture is the culmination of a large body of work both by taubes and by others in an attempt to make this story accessible to nonspecialists much of the present article is devoted to background and context and taubes s proof itself is only partially explained hopefully this article will help prepare the reader to learn the full story from taubes s paper 62 more exposition of this subject which was invaluable in the preparation of this article can be found in the online video archive from the june 2008 msri hot topics workshop 44 and in the article by auroux 5 below in 1 3 we introduce the statement of the weinstein conjecture and discuss some examples in 4 6 we discuss a natural strategy for approaching the weinstein conjecture which proves it in many but not all cases and provides background for taubes s work in 7 we give an overview of the big picture surrounding taubes s proof of the weinstein conjecture readers who already have some familiarity with the weinstein conjecture may wish to start here in 8 9 we recall necessary material from seiberg witten theory in 10 we give an outline of taubes s proof and in 11 we explain some more details of it to conclude in 12 we discuss some further results and open problems related to the weinstein conjecture 1 statement of the weinstein conjecture the weinstein conjecture asserts that certain vector fields must have closed orbits before stating the conjecture at the end of this section we first outline its origins this is discussion is only semi historical because only a sample of the relevant works will be cited and not always in chronological order 1 1 closed orbits of vector fields let y be a closed manifold in this article all manifolds and all objects defined on them are smooth unless otherwise stated 2000 mathematics subject classification 57r17 57r57 53d40 partially supported by nsf grant dms 0806037

on the contact geometry of nodal sets in the 3 dimensional riemannian geometry contact structures equipped with an adapted riemannian metric are divergence free nondegenerate eigenforms of the laplace beltrami operator we trace out a 2 d analogue of this fact there is a close relationship between the topology of the contact structure on a convex surface in the 3manifold the dividing curves and the nodal curves of laplacian eigenfunctions on that surface motivated by this relationship we consider a topological version of payne s conjecture for the free membrane problem we construct counterexamples to payne s conjecture for closed riemannian surfaces in light of the correspondence between the nodal lines and dividing curves we interpret payne s conjecture in terms of the tight versus overtwisted dichotomy for contact structures

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

contact homology and virtual fundamental cycles we give a construction of contact homology in the sense of eliashberg givental hofer specifically we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo holomorphic curves the aim of this work is to provide a rigorous construction of contact homology an invariant of contact manifolds and exact symplectic cobordisms due to eliashberg givental hofer eli98 egh00 the contact homology of y is defined in terms of pseudo holomorphic curve counts in the sense of gromov gro85 in the symplectization r y unfortunately the moduli spaces of such curves often suffer from a severe lack of transversality and for this reason it has until now remained an important open problem to define the relevant curve counts in full generality we will use the framework developed in par15 to construct coherent virtual fundamental cycles on the moduli spaces in question thus giving rise to the desired curve counts our methods are quite general and should apply equally well to many other moduli spaces of interest previously a number of important special cases of contact homology and closely related invariants have been constructed rigorously using generic and or automatic transversality techniques cylindrical contact homology of some three manifolds was constructed by bao

tight contact structures on solid tori in this paper we study properties of tight contact structures on solid tori in particular we discuss ways of distinguishing two solicl tori with tight contact structures we also give examples of unusual tight contact structures on solid tori we prove the existence of a 2 valued and a llt 27re valued invaria nt of a closed solid torus we call them the self linking number and the rotation number respectively we then extend these definitions to the case of an open solid torus we show that these invariants exhibit certain monotonicity properties with respect to inclusion we also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic at the same time we discuss various ways of constructing a tight contact structure on a solid torus we then produce examples of solid tori with tight contact structures and calculate self linking and rotation numbers for these tori these examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori at the end we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite sheeted cover of that solid torus is no longer tight this answers negatively a question asked by eliashberg in 1990 we also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure another example o1 unexpected behavior

characterization of the probabilistic traveling salesman problem we show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem the probabilistic traveling salesman must decide on an a priori order in which to visit n cities randomly distributed over a unit square before learning that some cities can be omitted we find the optimized average length of the pruned tour follows e l over bar pruned rootnp 0 872 0 105p f np where p is the probability of a city needing to be visited and f np 1 as np infinity the average length of the a priori tour before omitting any cities is found to follow e l a priori rootn pbeta p where beta p 1 1 25 0 82 ln p is measured for 0 05less than or equal topless than or equal to0 6 scaling arguments and indirect measurements suggest that beta p tends towards a constant for p 0 03 our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths exploiting the sampling error to provide the analog of thermal fluctuations in simulated thermal annealing the method has general application to the optimization of functions whose cost to evaluate rises with the precision required

an index for closed orbits in beltrami fields abstract we consider the class of beltrami fields eigenfields of the curl operator on three dimensional riemannian solid tori such vector fields arise as steady incompressible inviscid fluids and plasmas using techniques from contact geometry we construct an integer valued index for detecting closed orbits in the flow which are topologically inessential they have winding number zero with respect to the solid torus the most important feature of the index is its computability it can be rigorously determined from a c 1 approximation to the vector field on any meridional disc of the solid torus as a result we obtain a test for detecting the non existence of a global cross section to the 3d flow via purely 2d information

new closedness results for efficient sets in multiple objective mathematical programming abstract there are various theoretical algorithmic and practical reasons for developing necessary and sufficient conditions for the efficient set of a multiple objective mathematical programming problem p to be closed yet only a small number of results of this type limited to special cases of problem p have been developed in this article we present a necessary condition and several sufficient conditions for the closedness of the efficient sets of more general cases of problem p than have heretofore been studied our approach relies in part upon generalizing the concepts of quasi concavity and strict quasi concavity for real valued functions to vector valued functions our approach also relies upon some new characterizations of an efficient solution for problem p that we develop in the article

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

a characterization of the tight 3 sphere ii recall that every closed oriented connected 3 manifold m admits a contact form we shall give conditions on a periodic orbit p0 of the reeb vector field defined by that allows the construction of an open book decomposition of m p0 into embedded planes asymptotic to p0 such that p0 is the binding orbit of the decomposition it follows that m is the tight 3 sphere as a by product a new dynamical criterion for the tightness of a contact structure is derived in the proof the results extend earlier results 1999 john wiley sons inc

connectedness in multiple linear fractional programming the geometric properties of the sets of efficient and weakly efficient solutions of multiple linear fractional programming problems are investigated weakly efficient solutions are path connected by finitely many linear line segments when the constrained region is compact