some model theory of polish structures we introduce a notion of polish structure and in doing so provide a setting which allows the application of ideas and techniques from model theory descriptive set theory topology and the theory of profinite groups we define a topological notion of independence in polish structures and prove that it has some nice properties using this notion we prove counterparts of some basic results from geometric stability theory in the context of small polish structures then we prove some structural theorems about compact groups regarded as polish structures each small nm stable compact g group is solvable by finite each small compact g group of finite n m rank is nilpotent by finite examples of small polish structures and groups are also given

coordinated search for an object hidden on the line we consider the coordinated search problem faced by two searchers who start together at zero and can move at speed one to find an object symmetrically distributed on the line in particular we fully analyze the case of the negative exponential distribution given by the density f x e x 2 0 the searchers wish to minimize the expected time to find the object and meet back together with the object at zero we give necessary and sufficient conditions for the existence of an optimal search strategy when the target density is continuous and decreasing we show that for the negative exponential distribution the optimal time is between 4 728 and 4 729 a strategy with expected time in this interval begins with the searchers going in opposite directions and returning to the origin after searching up to successive distances 0 745 2 11 3 9 6 8 4 these results extend the theory of coordinated search to unbounded regions it has previously been studied for objects hidden on a circle by thomas and on an interval by the author

equilibria of two sided matching games with common preferences problems of matching have long been studied in the operations research literature assignment problem secretary problem stable marriage problem all of these consider a centralized mechanism whereby a single decision maker chooses a complete matching which optimizes some criterion this paper analyzes a more realistic scenario in which members of the two groups buyers sellers employers workers males females randomly meet each other in pairs interviews dates over time and form couples if there is mutual agreement to do so we assume members of each group have common preferences over members of the other group generalizing an earlier model of alpern and reyniers alpern s reyniers d j 2005 strategic mating with common preferences j theor biol 237 337 354 we assume that one group called males is r times larger than the other r 1 thus all females but only 1 r of the males end up matched unmatched males have negative utility c we analyze equilibria of this matching game depending on the parameters r and c in a region of r c space with multiple equilibria we compare these and analyze their efficiency in several respects this analysis should prove useful for designers of matching mechanisms who have some control over the sex ratio e g by capping numbers of males at a singles event or by having ladies free nights or the nonmating cost c e g tax benefits to married couples

patrolling a perimeter abstract in this paper we study the problem of patrolling a perimeter the general situation considered here can correspond to different tactical problems and it is studied from the point of view of game theory to put the ideas in a context we describe it as follows an intruder seeks to carry out a sabotage on the perimeter of a protected zone he has to perform the action along n consecutive days and has to position himself each day at one of m strategic points placed on this border the first day he can take his place at any of the m points but on successive days he can move only to adjacent points furthermore the perimeter is protected by a patroller who will select each day one of the m points to inspect the strategic situation is modeled as a two person zero sum game which is developed on a cyclic set of m points over n time units we prove some interesting properties of the strategies solve the game in closed form under certain constraints and obtain bounds for the value of the game in several non solved cases

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

searching a variable speed network a point lies on a network according to some unknown probability distribution starting at a specified root of the network a searcher moves to find this point at speeds that depend on his location and direction he seeks the randomized search algorithm that minimizes the expected search time this is equivalent to modeling the problem as a zero sum hide and seek game whose value is called the search value of the network we make a new and direct derivation of an explicit formula for the search value of a tree proving that it is equal to half the sum of the minimum tour time of the tree and a quantity called its incline the incline of a tree is an average over the leaf nodes of the difference between the time taken to travel from the root to a leaf node and the time taken to travel from a leaf node to the root this difference can be interpreted as height of a leaf node assuming uphill is slower than downhill we then apply this formula to obtain numerous results for general networks we also introduce a new general method of comparing the search value of networks that differ in a single arc some simple networks have very complicated optimal strategies that require mixing of a continuum of pure strategies many of our results generalize analogous ones obtained for constant velocity in both directions by s gal but not all of those results can be extended

rotational chaos and strange attractors on the 2 torus we construct a 2 torus homeomorphism h homotopic to the identity with an attracting r h bing s pseudocircle c such that the rotation set of h c is not a unique vector the only known examples of such an attractor were birkhoff s attractors arising for dissipative maps with a twist our construction relies heavily on the barge martin method for constructing attractors as inverse limits of graphs

taking advantage of symmetries gathering of many asynchronous oblivious robots on a ring one of the recently considered models of robot based computing makes use of identical memoryless mobile units placed in nodes of an anonymous graph the robots operate in look compute move cycles in one cycle a robot takes a snapshot of the current configuration look takes a decision whether to stay idle or to move to one of the nodes adjacent to its current position compute and in the latter case makes an instantaneous move to this neighbor move cycles are performed asynchronously for each robot in such a restricted scenario we study the influence of symmetries of the robot configuration on the feasibility of certain computational tasks more precisely we deal with the problem of gathering all robots at one node of the graph and propose a solution based on a symmetry preserving strategy when the considered graph is an undirected ring and the number of robots is sufficiently large more than 18 such an approach is proved to solve the problem for all starting situations as long as gathering is feasible in this way we also close the open problem of characterizing symmetric situations on the ring which admit a gathering r klasing e markou a pelc gathering asynchronous oblivious mobile robots in a ring theoret comput sci 390 1 2008 27 39 the proposed symmetry preserving approach which is complementary to symmetry breaking techniques found in related work appears to be new and may have further applications in robot based computing

rendezvous search on the line with indistinguishable players alpern introduced a problem in which two players are placed on the real line at a distance drawn from a bounded distribution f known to both they can move at maximum velocity one and wish to meet as soon as possible neither knows the direction of the other nor do they have a common notion of a positive direction on the line it is required to find the symmetric rendezvous value r s f which is the minimum expected meeting time achievable by players using the same mixed strategy this corresponds to the case where the players are indistinguishable they both take directions from a controller who does not know their names in this paper we give a mixed strategy which has an expected meeting time of 1 78d mu 2 where d is the maximum of f and mu its mean this leads to an upper bound r s f le 1 78d mu 2 on the symmetric rendezvous value which is better than the upper bound r s f le 2d mu 2 obtained by alpern

lines in space times we construct a complete timelike maximal geodesic line in a timelike geodesically complete spacetimem containing a compact acausal spacelike hypersurfaces which lies in the past of somes ray ans ray is a future complete geodesic starting ons which maximizes lorentzian distance froms to any of its points if the timelike convergence condition strong energy condition holds a line exists only ifm is static i e it splits geometrically as space time so timelike completeness must fail for a nonstatic spacetime with strong energy condition which contains a closed universe s with the above properties

the lorentzian splitting theorem without the completeness assumption on demontre le theoreme de decomposition lorentzien pour des espace temps globalement hyperboliques obeissant a la condition d energie forte sans hypothese de completude geodesique du genre temps

asynchronous deterministic rendezvous in graphs two mobile agents robots having distinct labels and located in nodes of an unknown anonymous connected graph have to meet we consider the asynchronous version of this well studied rendezvous problem and we seek fast deterministic algorithms for it since in the asynchronous setting meeting at a node which is normally required in rendezvous is in general impossible we relax the demand by allowing meeting of the agents inside an edge as well the measure of performance of a rendezvous algorithm is its cost for a given initial location of agents in a graph this is the number of edge traversals of both agents until rendezvous is achieved if agents are initially situated at a distance d in an infinite line we show a rendezvous algorithm with cost o d lmin 2 when d is known and o d lmax 3 if d is unknown where lmin and lmax are the lengths of the shorter and longer label of the agents respectively these results still hold for the case of the ring of unknown size but then we also give an optimal algorithm of cost o n lmin if the size n of the ring is known and of cost o n lmax if it is unknown for arbitrary graphs we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost o d lmin if the topology of the graph and the initial positions are known to agents

stable extensions of homeomorphisms on the pseudo arc we prove the following theorem if p is a proper subcontinuum of the pseudoarc p h is a homeomorphism from p onto itself and 0 is an open set in p that contains p then there is a homeomorphism h from p onto itself such that h p h and h x x for x 0 a pseudoarc can be characterized as a chainable hereditarily indecomposable continuum although the main result here extends a result of w lewis lw and answers in the affirmative a question he asked in that same paper stable homeomorphisms of the pseudoarc this paper was motivated by a conversation the author had with professor marcy barge who works in the area of topological dynamics professor barge is interested in knowing what sort of self homeomorphisms the pseudoarc admits and in particular what sort of homeomorphisms it admits with the property that some of the points of the space have periodic orbits that pseudoarcs admit many such homeomorphisms will follow from a much earlier result of g lehner lg and the result here it will also follow by combining the theorem proven here g lehner s theorem and a theorem in kj that any continuum with the property that all its proper nondegenerate subcontinua are pseudoarcs admits many such homeomorphisms examples of continua with this property include pseudocircles b pseudosolenoids f some examples due to t ingram i and h cook c as well as the pseudoarc itself we will prove the following theorem theorem if p is a proper nondegenerate subcontinuum of the pseudoarc p p c o an open set in p and h is a homeomorphism from p onto itself then h can be extended to a homeomorphism h of p onto itself such that h x x for x o a continuum is indecomposable if no proper subcontinuum of it has interior and it is hereditarily indecomposable if each subcontinuum is indecomposable a chain is a collection of sets c co cy cn such that ci f c3 0 if and only if i j 1 a continuum is chainable if it has open chain covers of arbitrarily small mesh as mentioned previously the pseudoarc p is a hereditarily indecomposable chainable continuum kb this continuum also has the fixed point property h received by the editors june 9 1987 this research was presented at the annual spring topology conference march 1 1987 university of alabama at birmingham this conference was supported by the nsf 1980 mathematics subject classification 1985 revision primary 54f20 54f50

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

improved bounds for the symmetric rendezvous value on the line a notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line when the initial distance between the two players is two we show that the symmetric rendezvous value is within the interval 4 1520 4 2574 which considerably improves the previous best known result 3 9546 4 3931 to achieve the improved bounds we call upon results from absorbing markov chain theory and mathematical programming theory particularly fractional quadratic programming and semidefinite programming moreover we also establish some important properties of this problem which could be of independent interest and useful for resolving this problem completely finally we conjecture that the symmetric rendezvous value is asymptotically equal to 4 25 based on our numerical calculations

on the multichannel rendezvous problem fundamental limits optimal hopping sequences and bounded time to rendezvous one of the fundamental problems in a cognitive radio network known as the multichannel rendezvous problem is for two secondary users to find a common channel that is not blocked by primary users the basic idea for solving such a problem in most works in the literature is for the two users to select their own channel hopping sequences and then rendezvous when they both hop to a common unblocked channel at the same time in this paper we focus on the fundamental limits of the multichannel rendezvous problem and formulate such a problem as a constrained optimization problem where the selection of the random hopping sequences of the two secondary users must satisfy certain constraints we derive various lower bounds for the expected respectively maximum time to rendezvous under certain constraints for some of these lower bounds we are also able to construct optimal channel hopping sequences that achieve the lower bounds inspired by the constructions of quorum systems and relative difference sets our constructions of the channel hopping sequences are based on the mathematical theories of finite projective planes orthogonal latin squares and sawtooth sequences the use of such theories in the constructions of channel hopping sequences appear to be new and better than other existing schemes in terms of minimizing the expected respectively maximum time to rendezvous

searching for an agent who may or may not want to be found there is an extensive theory regarding optimal continuous path search for a mobile or immobile target the traditional theory assumes that the target is one of three types i an object with a known distribution of paths ii a mobile or immobile hider who wants to avoid or delay capture or iii a rendezvouser who wants to find the searcher this paper introduces a new type of search problem by assuming that aims of the target are not known to the searcher the target may be either a type iii cooperator with a known cooperation probabilityc or a type ii evader this formulation models search problems like that for a lost teenager who may be a runaway or a lost intelligence agent who may be a defector in any given search context it produces a continuum of search problems g c 0 c 1 linking a zero sum searchgame withc 0 to a rendezvous problem withc 1 these models thus provide a theoretical bridge between two previously distinct parts of search theory namely searchgames and rendezvous search

rendezvous on the line when the players initial distance is given by an unknown probability distribution two players a and b are randomly placed on a line the distribution of the distance between them is unknown except that the expected initial distance of the two players does not exceed some constant mu the players can move with maximal velocity 1 and would like to meet one another as soon as possible most of the paper deals with the asymmetric rendezvous in which each player can use a different trajectory we find rendezvous trajectories which are efficient against all probability distributions in the above class it turns out that our trajectories do not depend on the value of mu we also obtain the minimax trajectory of player a if player b just waits for him this trajectory oscillates with a geometrically increasing amplitude it guarantees an expected meeting time not exceeding 6 8 mu we show that if player b also moves then the expected meeting time can be reduced to 5 7 mu the expected meeting time can be further reduced if the players use mixed strategies we show that if player b rests then the optimal strategy of player a is a mixture of geometric trajectories it guarantees an expected meeting time not exceeding 4 6 mu this value can be reduced even more below 4 42 mu if player b also moves according to a correlated mixed strategy we also obtain a bound for the expected meeting time of the corresponding symmetric rendezvous problem

classifying rendezvous tasks of arbitrary dimension the rendezvous is a type of distributed decision tasks including many well known tasks such as set agreement simplex agreement and approximation agreement an n dimensional rendezvous task n 1 allows n 2 distinct input values and each execution produces at most n 2 distinct output values a rendezvous task is said to implement another if an instance of its solution followed by a protocol based on shared read write registers solves the other the notion of implementation induces a classification of rendezvous tasks of every dimension two tasks belong to the same class if they implement each other previous work on classifying rendezvous tasks only focused on 1 dimensional ones this paper solves an open problem by presenting the classification of nice rendezvous of arbitrary dimension an n dimensional rendezvous task is said to be nice if the qth reduced homology group of its decision space is trivial for q n and free for q n well known examples are set agreement simplex agreement and approximation agreement each n dimensional rendezvous task is assigned an algebraic signature which consists of the nth homology group of the decision space as well as a distinguished element in the group it is shown that an n dimensional nice rendezvous task implements another if and only if there is a homomorphism from its signature to that of the other hence the computational power of a nice rendezvous task is completely characterized by its signature in each dimension there are infinitely many classes of rendezvous tasks and exactly countable classes of nice ones a representative is explicitly constructed for each class of nice rendezvous tasks

geometric analysis of lorentzian distance function on spacelike hypersurfaces some analysis on the lorentzian distance in a spacetime with controlled sectional or ricci curvatures is done in particular we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the omori yau maximum principle as a consequence and under appropriate hypotheses on the sectional or ricci curvatures of the ambient spacetime we obtain sharp estimates for the mean curvature of those hypersurfaces moreover we also give a sufficient condition for its hyperbolicity

two point one sided rendezvous abstract in a rendezvous search two or more teams called seekers try to minimize the time needed to find each other in this paper we consider two seekers in the plane this is a one sided problem since seeker 1 begins at a predetermined point o seeker 2 begins at one of a finite set of points x i with probability p i we first discuss the general situation and then consider the specific case when seeker 2 can begin from one of two points

a linear programming approach to the search game on a network with mobile hider this paper discusses a search game on a network q with two players a searcher and a hider who each move with continuous trajectories starting from different points subject to a maximal speed and termination time t this is a zero sum game with payoff given by the time elapsed until the searcher reaches a point that is occupied by the hider if this happens and t otherwise the problem is formulated as an infinite dimensional linear program and the extreme points and the reduced cost functional are studied an algorithm is derived for this problem and how it works on an example is demonstrated

rendezvous search on the line with more than two players suppose n blind speed one players are placed by a random permutation onto the integers 1 to n and each is pointed randomly to the right or left what is the least expected time required form m n of them to meet together at a single point if they must all use the same strategy we call this time the symmetric rendezvous value rn ms otherwise the asymmetric value rn ma we show that r3 2a 47 48 and that rn ns is asymptotic to n 2 these results respectively extend those for two players given by alpern and gal alpern s s gal 1995 rendezvous search on the line with distinguishable players siam j control optim 33 1270 1276 and anderson and essegaier anderson e j s essegaier 1995 rendezvous search on the line with indistinguishable players siam j control optim 33 1637 1642

pseudocircles in dynamical systems we construct an example of a coo map on a 3 manifold which has an invariant set with an uncountable number of components each of which is a pseudocircle furthermore any map which is sufficiently close in the c1metric to the constructed map has a similar set

infiltration games on arbitrary graphs the purpose of this note is to extend the work of lalley l and auger a on infiltration games on certain classes of graphs to an arbitrary graph by a simple application of menger s theorem h theorem 5 91 infiltration games as proposed by gal g sect 4 61 and generalized by auger a are determined by specifying two vertices a and b of a connected graph g and a capture probability 1 2 in the infiltration game r f g a b 2 the infiltrator chooses a path p a p o p l p t b from a to b in g of any length and the guard chooses a map g 1 2 v a b where v denotes the vertex set of g the interpretation is that at time t the infiltrator and guard are at the vertices p t and g t and the guard will capture the inliltrato i with probability 1 a if they coincide in the zero sum game r the payoff i7 is the probability that the infiltrator maximizer safely reaches the target vertex b z7 p g a where q is the number of times p and g coincide in some versions a time bound n is put on the length t of infiltrator paths p and the game is denoted f g a b 1 n related searchlight games are discussed in bb opl op2 r recently auger a has extended the work of lalley l by giving a complete solution for arbitrary time bounds n of the infiltration games on the graphs l lk mi m2 mk consisting of two vertices a and b joined by k vertex disjoint paths with m mk total m c mi interior vertices the value of the game t l a b a n is given by

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

linear time and space gathering of anonymous mobile agents in asynchronous trees we investigate the relation between the ideal time and space complexities for the gathering problem with k anonymous agents in asynchronous anonymous tree networks the gathering problem requires that all the agents in the network have to meet at a single node within a finite time although an asymptotically space optimal algorithm is known its time complexity is quite large in this paper we consider asymptotically ideal time optimal algorithms and investigate the minimum memory requirement per agent for asymptotically time optimal algorithms first we show that there exists a tree with n nodes in which w n bits of memory per agent is required to solve the gathering problem in o n time asymptotically time optimal then we present an asymptotically time optimal gathering algorithm this algorithm can be executed if each agent has o n bits of memory from this lower upper bound this algorithm is asymptotically space optimal on the condition that the time complexity is asymptotically optimal

on ruckle s conjecture on accumulation games in an accumulation game the hider secretly distributes his given total wealth h among n locations while the searcher picks r locations and confiscates the material placed there the hider wins if what is left at the remaining n r locations is at least 1 otherwise the searcher wins ruckle s conjecture says that an optimal hider strategy is to put an equal amount h k at k randomly chosen locations for some k we extend the work of kikuta and ruckle by proving the conjecture for several cases e g r 2 or n 2 n leq7 n 2r 1 h leq2 1 n r and n leq2r the last result uses the erdos ko rado theorem we establish a connection between ruckle s conjecture and the hoeffding problem of bounding tail probabilities of sums of random variables

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

move optimal partial gathering of mobile agents in asynchronous trees in this paper we consider the partial gathering problem of mobile agents in asynchronous tree networks the partial gathering problem is a new generalization of the total gathering problem which requires that all the agents meet at the same node the partial gathering problem requires for given input g that each agent should move to a node and terminate so that at least g agents should meet at the same node the requirement for the partial gathering problem is weaker than that for the well investigated total gathering problem and thus we clarify the difference on the move complexity between them we assume that n is the number of nodes and k is the number of agents we propose two algorithms to solve the partial gathering problem first we consider the strong multiplicity detection and non token model in this model we show that agents require kn total moves to solve the partial gathering problem and we propose an algorithm to achieve the partial gathering in o kn total moves second we consider the weak multiplicity detection and removable token model in this model we propose an algorithm to achieve the partial gathering in o gn total moves it is known that the partial gathering problem requires gn total moves hence the second algorithm is asymptotically optimal in terms of total moves

rendezvous in higher dimensions two players are placed on the integer lattice z n consisting of points in n dimensional space with all coordinates integers so that their vector difference is of length 2 and parallel to one of the axes their aim is to move to an adjacent node in each period so that they meet occupy same node in least expected time r left n right called the rendezvous value we assume they have no common notion of directions or orientations i e no common notion of clockwise we extend the known result r left 1 right 3 25 of alpern and gal to obtain r left 2 right 197 32 allowbreak 6 allowbreak16 and the bounds 2n leq r left n right leq left 32n 3 12n 2 2n 3 right 12n 2 for n 2 we characterize the set of all optimal strategies and show that none of them simultaneously maximizes the probability of meeting by time t for all t this behavior differs from that found by anderson and fekete and the authors for the related problem where the players are initially placed at diagonals of one of the squares of the lattice z 2

rendezvous search on the line with bounded resources expected time minimization abstract two players are placed on the line and want to meet neither knows the direction of the other but they know the distance between them or perhaps the distribution of this distance they can move with speed at most one and each has a resource constraint on the total distance he can travel we first consider the question of whether the two players can ensure that they meet when they can then we seek the least expected meeting time otherwise we maximize the probability of a meeting this generalizes the similar problem studied by the first author and s gal siam j control and optimization 33 4 1995 1270 without any resource constraint and indeed for sufficiently large resources gives the same rendezvous time i e least expected time to meet the paper may also be considered a generalization of the linear search problem studied by the second author and others which corresponds to the case when one of the resource constraints is zero so that player cannot move more specifically it generlaizes the bounded resource version of the linear search problem presented by foley hill and spruill naval research logistics 38 1991 555 565

static search games played over graphs and general metric spaces we define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure the game involves two players the searching player and the concealing player and is played over a metric space each player simultaneously chooses to deploy at a point in the space the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position the concealing player receiving a payoff of 1 otherwise the concepts of dominance and equivalence of strategies are examined in the context of this game before focusing on the more specific case of the game played over a graph methods are presented to simplify the analysis of such games both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs

a transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map a powerful theorem and construction of wayne lewis are used to build two homeomorphisms on the pseudoarc each of which is semiconjugate to the tent map on the unit interval the first homeomorphism is transitive thus answering a question of marcy barge as to whether such homeomorphisms exist the second homeomorphism admits wandering points also it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points a continuum is a compact connected metric space a continuum x is indecomposable if every proper subcontinuum of x is nowhere dense in x and x is hereditarily indecomposable if each subcontinuum of x is indecomposable a continuum x is chainable if for each e 0 there is a chain c c0 cn of open sets of diameter less than e that covers x c is a chain means that ci n ej 0 iff ii ii 2 or in the hilbert cube are pseudoarcs here we will show that this continuum admits a transitive homeomorphism thus answering a question of marcy barge in addition this homeomorphism has the property that it is semiconjugate to the tent map on the unit interval for more information on pseudoarcs and indecomposable continua see bil bi4 l1 l2 k km kr and ot exotic continua have been making their appearance in dynamical systems ever since the early part of this century with the work of c caratheodory c received by the editors may 20 1988 and in revised form july 25 1989 presented at the 1988 spring topology conference university of florida gainesville florida 32611 1980 mathematics subject classification 1985 revision primary 54f20 secondary 54h20

fugitive search games on graphs and related parameters abstract the goal of a fugitive search game on a graph is to trap a fugitive that hides on the vertices of the graph by systematically placing searchers on the vertices the fugitive is assumed to have complete knowledge of the graph and of the searchers moves but is restricted to move only along paths whose vertices are not guarded by searchers the search number of the graph is the least number of searchers necessary to trap the fugitive variants of the fugitive search game have been related to important graph parameters like treewidth and pathwidth in this paper we introduce a class of fugitive search games where the searchers do not see the fugitive and the fugitive can only move just before a searcher is placed on the vertex it occupies letting the fugitive s speed i e the maximum number of edges the fugitive can traverse at a single move vary we get different games we show that if the speed of the fugitive is unbounded then the search number minus 1 is equal to the treewidth of the graph while if the speed is 1 then the search number minus 1 is equal to a polynomially computable graph parameter which is called width or alternatively linkage and is studied in the context of the constraint satisfaction and boolean satisfiability problems we also show that in the above two cases the search number remains the same even if we consider only search strategies that at every step further restrict the fugitive s possible resorts this monotonicity property is usually expressed as recontamination does not help moreover we give an equivalent characterization of the search number for any given fugitive speed in terms of an elimination ordering of the vertices of the graph using this characterization we show that for any graph if the length of any chordless cycle is bounded by a constant s s 3 then the treewidth of the graph plus 1 is equal to the search number for fugitive speed s 2

splitting theorems for spatially closed space times a lorentzian splitting theorem is obtained for spatially closed spacetimes the proof employs and extends some recent results of bartnik and gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space times a splitting theorem for spatially closedtime periodic space times which generalizes a result first considered by avez is derived as a corollary

minimax rendezvous on the line suppose that n players are placed randomly on the real line at consecutive integers and faced in random directions each player has maximum speed one cannot see the others and doesn t know his relative position what is the minimum time m n required to ensure that all the players can meet together at a single point regardless of their initial placement we prove that m 2 3 m 3 4 and m n is asymptotic to n 2 we also consider a variant of the problem which requires players who meet to stick together and find in this case that three players require 5 time units to ensure a meeting this paper is thus a minimax version of the rendezvous search problem which has hitherto been studied only in terms of minimizing the expected meeting time

the princess and monster game on an interval a minimizing searcher s and a maximizing hider h move at unit speed on a closed interval until the first capture or payoff time t min t s t h t that they meet this zero sum princess and monster game or less colorfully search game with mobile hider was proposed by rufus isaacs for general networks q while the existence and finiteness of the value v v q has been established for such games only the circle network has been solved value and optimal mixed strategies it seems that the interval network q 1 1 had not been studied because it was assumed to be trivial with value 3 2 and obvious searcher mixed strategy going equiprobably from one end to the other we establish that this game is in fact nontrivial by showing that v 3 2 using a combination of continuous and discrete mixed strategies for both players we show that 15 11 leq v leq 13 9 the full solution of this very simple game is still open and appears difficult though many properties of the optimal strategies are derived here

the rendezvous search problem the author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time this time is called the rendezvous value of the region it is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise a general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players these concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network the discrete rendezvous problem on a cycle graph for players restricted to symmetric markovian strategies is then solved finally the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other

an optimal algorithm for the traveling salesman problem with time windows this paper presents the development of new elimination tests which greatly enhance the performance of a relatively well established dynamic programming approach and its application to the minimization of the total traveling cost for the traveling salesman problem with time windows the tests take advantage of the time window constraints to significantly reduce the state space and the number of state transitions these reductions are performed both a priori and during the execution of the algorithm the approach does not experience problems stemming from increasing problem size wider or overlapping time windows or an increasing number of states nearly as rapidly as other methods our computational results indicate that the algorithm was successful in solving problems with up to 200 nodes and fairly wide time windows when the density of the nodes in the geographical region was kept constant as the problem size was increased the algorithm was capable of solving problems with up to 800 nodes for these problems the cpu time increased linearly with problem size these problem sizes are much larger than those of problems previously reported in the literature

rendezvous search on the interval and the circle two people are placed randomly and independently on a street of unit length they attempt to find each other in the shortest possible expected time we solve this problem assuming each searcher knows where he or she is on the street for monotonic density functions for the initial placement this includes the uniform pdf as a special case this gives an example of a rendezvous search problem where there is no advantage in being allowed to use asymmetric strategies we also solve some corresponding problems for the circle when asymmetric strategies are permitted one of these shows that it can sometimes be optimal for one player to wait for the other to find him

rendezvous on a planar lattice we analyze the optimal behavior of two players who are lost on a planar surface and who want to meet each other in least expected time they each know the initial distribution of the other s location but have no common labeling of points and so cannot simply go to a location agreed to in advance they have no compasses so do not even have a common notion of north for simplicity we restrict their motions to the integer lattice z2 graph paper and their motions to horizontal and vertical directions as in the original work of anderson and fekete 2001

rendezvous search on the line with distinguishable players two players are placed on the real line at a distance d with a distribution f known to both neither knows the direction of the other nor do they have a common notion of a positive direction on the line we seek the least expected rendezvous time r r left f right in which they can meet given maximum speeds of one we consider the cases where f is a bounded point discrete or finite mean distribution we obtain upper bounds or exact values for r and in one case an optimality condition for search strategies a connection with beck s linear search problem is established

search hide games on trees abstract in this paper we consider search hide games on trees there are two players player h the hider or inspectee selects one of m vertices of the tree for hiding undeclared objects permanently player i the inspector is allowed to search some vertices of the tree trying to find the hidden objects however player i can only examine exactly k m vertices lying on a subtree with k vertices because of certain limitations player i wants to maximize the probability of detecting the undeclared objects and player h to minimize it this problem can be described by a fractional covering problem fc and its dual dfc an algorithm based on the revised simplex method with implicit column generation is proposed for solving fc and dfc the all possible strategies of the player i all possible subtree with k vertices is not necessary listed explicitly in order to find a profitable column a polynomial algorithm is developed by utilizing the structure of the underlying tree

raid games across a set with cyclic order abstract in this paper we shall deal with two person zero sum games where the strategies of the players are intervals on a set with cyclic order and the payoff function is a function depending on the intersection of the strategies for the players these games correspond to similar tactical problems we give a general method to obtain a solution of the discrete games of this kind this method is applied to solve the games with monotonic payoff function and a game with no monotonic one we also solve a continuous game

regularity of lorentzian busemann functions a general theory of regularity for lorentzian busemann functions in future timelike geodesically complete spacetimes is presented this treatment simplifies and extends the local regularity developed by eschenburg galloway and newman to prove the lorentzian splitting theorem criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of bartnik

ambush strategies in search games on graphs a blind searcher and a blind hider move at below unit speed along a finite length graph q known to both until the first time t when they meet a two person zero sum game arises if the searcher pays the hider t units we consider circumstances under which it may be optimal for the searcher to lie in wait at a node of q hoping the hider will come to him we also explicitly define a notion of equilibrium in distribution for such games which has been implicit in the literature we show that for the graph consisting of two nodes connected by three arcs of equal length there are optimal ambush strategies but there is no equilibrium in distribution

a search game when a search path is given abstract in this paper we investigate a search game in discrete time and space a searcher is given a search path in advance and his look on the path is determined by a randomized look strategy a target selects a path from some options the searcher gains a value on the detection of the target but expends search cost by the look a pay off function of the game for the searcher is the expected reward which is defined as the expected value minus the expected search cost first we show a recursive relation for the conditional optimal look strategy of the searcher given a target path we prove its np completeness though it looks simple and clarify some characteristics of the solution then our original continuous game is converted to a matrix game from these facts we consider a relationship between the game and the one sided optimizing problem and examine some examples

hide and seek games on a network using combinatorial search paths this paper introduces a new search paradigm to hide and seek games on networks the hider locates at any point on any arc the searcher adopts a combinatorial path when searching the network a sequence of arcs each adjacent to the last and traced out at unit speed in previous literature the searcher was allowed simple motion any unit speed path including ones that turn around inside an arc the new approach more closely models real problems such as search for improvised explosive devices using vehicles that can only turn around at particular locations on a road the search game is zero sum with the time taken to find the hider as the payoff using a lemma giving an upper bound for the expected search time on a semi eulerian network we solve the search game on a network q3 consisting of two nodes connected by three arcs of arbitrary lengths when two q3 networks with unit length arcs are linked by two small central arcs incident at the start node one of these arcs must be traversed at least th

a common notion of clockwise can help in planar rendezvous two players are lost in a grid of city streets and wish to meet as soon as possible knowing only the distribution of the other s initial location two nodes away in one of the four compass directions how do they move from intersection to intersection between nodes of the lattice z2 to achieve this we assume that they do not have common compass directions to coordinate on but that they can use their common notion of clockwise we show that the latter realistic assumption can aid them in expediting their meeting relative to a previous rendezvous problem which did not allow this we also solve the easier streets and avenues version of the problem in which the players can distinguish between the axes between streets and avenues we discover several new phenomenae which have not been seen before in planar rendezvous

orbits of the pseudocircle the following theorem is proved theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each orbit is the union of uncountably many composants a pseudocircle is a circularly chainable hereditarily indecomposable separating plane continuum r h bing 1 constructed a pseudocircle in 1951 and asked two questions are any two pseudocircles homeomorphic is a pseudocircle homogeneous in 1968 l fearnley 6 proved that the answer to the first question is yes and fearnley 5 and j t rogers jr 14 independently proved that the answer to the second question is no since the advent of the effros theorem several elegant proofs of the nonhomogeneity of the pseudocircle have appeared 7 10 and 13 in 1968 after seeing the proof that the pseudocircle is not homogeneous f b jones asked the second author how many orbits the pseudocircle had under the action of its homeomorphism group but the matter was not pursued further recently however the question has arisen again and wayne lewis 3 has asked if the pseudocircle has uncountably many orbits the purpose of this paper is to prove the following theorem which gives an affirmative answer to this question theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each of these orbits is the union of uncountably many composants an interesting sidelight of the proof is the construction of two homeomorphic sets one being an open set of the pseudocircle and the other being an open set of the pseudoarc during the proof we construct an uncountable abelian subgroup of the homeomorphism group of the pseudocircle the homeomorphism constructed by handel 8 might be an element of this group each orbit of the pseudocircle is a borel set 4 lewis has announced that no orbit can be a g6 in the course of our argument we prove this but we do not determine further restrictions on the type of the borel set for the orbit question what classes of borel sets occur as the orbits of the pseudocircle in particular are each two orbits of the same class received by the editors may 1 1984 and in revised form august 13 1985 1980 mathematics subject classification primary 54f20 54f50

a generalization of ruckle s results for an ambush game abstract this article considers a two person zero sum game in which a traveller player i chooses a point in the closed unit interval 0 1 so as to minimise the probability of being ambushed by two obstacles placed there by his adversary player ii the payoff to player i equals 1 if the point chosen by him is not in either of the intervals chosen by player ii and 0 otherwise this paper complements the results obtained by ruckle baston and bostock and lee

decomposition theorems for lorentzian manifolds with nonpositive curvature soit m g un espace temps globalement hyperbolique de dimension 2 avec partout des courbures sectionnelles du genre temps non positives k 0 qui contient une ligne du genre temps complete m g alors m g est isometrique a un produit r h dt 2 h ou h h est une variete de riemann complete le facteur r dt 2 est represente par et h h est represente par un ensemble de niveau d une fonction de busemann associee a

rendezvous search on the line we present two new results for the asymmetric rendezvous problem on the line we first show that it is never optimal for one player to be stationary during the entire search period in the two player rendezvous then we consider the meeting time of n players in the worst case and show that it has an asymptotic behavior of n 2 o log n

mining coal or finding terrorists the expanding search paradigm we show how to optimize the search for a hidden object terrorist or simply hider located at a point h according to a known or unknown distribution v on a rooted network q we modify the traditional pathwise search approach to a more general notion of expanding search when the hider is restricted to the nodes of q an expanding search s consists of an ordering a1 a2 of the arcs of a spanning subtree such that the root node is in a1 and every arc ai is adjacent to a previous arc aj j i if ak contains h the search time t is a1 ak5 where is length measure on q for more general distributions v an expanding search s is described by the nested family of connected sets s4t5 that specify the area of q that has been covered by time t s 0 is the root s t t and t min t h 2 s t for a known hider distribution v on a tree q the expected time minimizing strategy sn begins with the rooted subtree q maximizing the density v q q for arbitrary networks we use this criterion on all spanning subtrees the search sn can be interpreted as the optimal method of mining known coal seams when the time to move miners or machines is negligible compared to digging time when the hider distribution is unknown we consider the zero sum search game where the hider picks h the searcher s and the payoff is t for trees q the value is v q d 2 where d is a mean distance from root to leaf nodes if q is 2 arc connected v q 2 applications and interpretations of the expanding search paradigm are given particularly to multiple agent search

asymmetric rendezvous on the line is a double linear search problem we apply a new method of analysis to the asymmetric rendezvous search problem on the line arspl this problem previously studied in a paper of alpern and gal 1995 asks how two blind speed one players placed a distance d apart on the line can find each other in minimum expected time the distance d is drawn from a known cumulative probability distribution g and the players are faced in random directions we show that the arspl is strategically equivalent to a new problem we call the double linear search problem dlsp where an object is placed equiprobably on one of two lines and equiprobably at positions d a searcher is placed at the origin of each of these lines the two searchers move with a combined speed of one to minimize the expected time before one of them finds the object using results from a concurrent paper of the first author and j v howard 1998 we solve the dlsp and hence the arspl for the case where g is convex on its support and show that the solution is that conjectured in a paper of baston and gal 1998

optimal trade off between speed and acuity when searching for a small object a searcher seeks to find a stationary hider located at some point h not necessarily a node on a given network q the searcher can move along the network from a given starting point at unit speed but to actually find the hider she must pass it while moving at a fixed slower speed which may depend on the arc in this bimodal search game the payoff is the first time the searcher passes the hider while moving at her slow speed this game models the search for a small or well hidden object e g a contact lens improvised explosive device predator search for camouflaged prey we define a bimodal chinese postman tour as a tour of minimum time i which traverses every point of every arc at least once in the slow mode for trees and weakly eulerian networks networks containing a number of disjoint eulerian cycles connected in a tree like fashion the value of the bimodal search game is i 2 for trees the optimal hider strategy has full support on the network this differs from traditional search games where it is optimal for him to hide only at leaf nodes we then consider the notion of a lucky searcher who can also detect the hider with a positive probability q even when passing him at her fast speed this paper has particular importance for demining problems

search games on trees with asymmetric travel times a point h is hidden in a rooted tree q which is endowed with asymmetric distances travel times between nodes we determine the randomized search strategy starting from the root which minimizes the expected time to reach h in the worst case this is equivalent to a zero sum search game gamma left q right with minimizing searcher maximizing hider and payoff equal to the capture time the worst hiding distribution over the leaves from the searcher s viewpoint is one where at every node i the probability of each branch is proportional to the minimum time required to tour it from i the optimal randomized search is a mixture over depth first searches we also consider briefly some other networks and the possibility of a mobile hider our formulation with asymmetric travel times generalizes that of gal siam j control optim 17 1979 pp 99 122 for symmetric travel times and also the search games of kikuta j oper res 38 1995 pp 70 88 and kikuta and ruckle naval res logist 41 1994 pp 821 831 who posited search costs c i at each node i which were added to the travel time to obtain the payoff we also briefly consider what happens if we allow the searcher hider to start hide at any leaf node we determine when properties found by dagan and gal networks 52 2008 pp 156 161 for the symmetric version of such games hold in our asymmetric context

on the conjecture of wood and projective homogeneity abstract in 2005 kawamura and rambla independently constructed a metric counterexample to wood s conjecture from 1982 we exhibit a new nonmetric counterexample of a space l such that c 0 l c is almost transitive and show that it is distinct from a nonmetric space whose existence follows from the work of greim and rajagopalan in 1997 up to our knowledge this is only the third known counterexample to wood s conjecture we also show that contrary to what was expected if a one point compactification of a space x is r h bing s pseudo circle then c 0 x c is not almost transitive for a generic choice of points finally we point out close relation of these results on wood s conjecture to a work of irwin and solecki on projective fraisse limits and projective homogeneity of the pseudo arc and addressing their conjecture we show that the pseudo circle is not approximately projectively homogeneous

searching on a line we revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance d that separates the initial position of the searcher from the target prior to this work only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case we present the first tight bounds on the exact optimal competitive ratio achievable parameterized in terms of the given bounds on d along with an optimal search strategy that achieves this competitive ratio we prove that this optimal strategy is unique we characterize the conditions under which an optimal strategy can be computed exactly and when it cannot we explain how numerical methods can be used efficiently in addition we answer several related open questions including the maximal reach problem and we discuss how to generalize these results to m rays for any m 2

a brouwer translation theorem for free homeomorphisms we prove a generalization of the brouwer translation theorem which applies to a class of homeomorphisms free homeomorphisms which admit fixed points but retain a dynamical property of fixed point free orientation preserving homeomorphsims that is if h m2 m2 is a free homeomorphism where m2 is a surface then whenever d is a disc and h d n d 0 we have that hn d n d 0 for all n 0 theorem let h be a free homeomorphism of s2 the two sphere with finite fixed point set f then each p e s2 f lies in the image of an embedding xp r2 o s2 f p such that i hxp xpf where er z z 1 is the canonical tranalation of the plane and ii the image of each vertical line under xp is closed in s2 f the class of free homeomorphisms has been introduced and studied by m brown in connection with the dynamics of surface homeomorphisms introduction l e j brouwer from 1909 to 1919 formulated and presented the brouwer plane translation theorem 2 6 9 10 in modern terminology it can be stated thusly theorem let h be a fixed point free orientation preserving self homeomorphism of r2 the plane then each p e r2 lies in the image of an embedding xp r2 0 r2 p such that i hxp xpor where or z z 1 is the canonical translation of the plane and ii the image of each vertical line under xp is closed in r2 in his first proof of this theorem 3 6 he assumed that a simple curve bounded only two domains later 7 he discovered the lakes of wada continuum and realized the error in his proof the second proof 9 utilized the brouwer translation arc lemma which was proved using the no retraction theorem 8 and the concept of index of a fixed point however this proof contained serious gaps since 1920 a succession of authors 1 13 16 17 18 have presented proofs of this theorem each either correcting errors in the previous proof or embarking upon a simplification for the proof for a more thorough discussion of this problem the reader should consult 11 which presents the history of brouwer s work and 16 for the history of the plane translation theorem received by the editors august 18 1986 1980 mathematics subject classification 1985 revision primary 54h20 secondary 58f99

a general framework for searching on a line abstract consider the following classical search problem a target is located on a line at distance d from the origin starting at the origin a searcher must find the target with minimum competitive cost the classical competitive cost studied in the literature is the ratio between the distance travelled by the searcher and d note that when no lower bound on d is given no competitive search strategy exists for this problem therefore all competitive search strategies require some form of lower bound on d we develop a general framework that optimally solves several variants of this search problem our framework allows us to achieve optimal competitive search costs for previously studied variants such as 1 where the target is fixed and the searcher s cost at each step is a constant times the length of the step 2 where the target is fixed and the searcher s cost at each step is the length of the step plus a fixed constant often referred to as the turn cost 3 where the target is moving and the searcher s cost at each step is the length of the step our main contribution is that the framework allows us to derive optimal competitive search strategies for variants of this problem that do not have a solution in the literature such as 1 where the target is fixed and the searcher s cost at each step is 1 x 1 for moving distance x away from the origin and 2 x 2 for moving back with constants 1 2 1 2 2 where the target is moving and the searcher s cost at each step is a constant times the length of the step plus a fixed constant turn cost notice that the latter variant can have several interpretations depending on what the turn cost represents for example if the turn cost represents the amount of time for the searcher to turn then this has an impact on the position of the moving target on the other hand the turn cost can represent the amount of fuel needed to make an instantaneous turn thereby not affecting the target s position our framework addresses all of these variations