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

search games with multiple hidden objects we consider a class of zero sum search games in which a searcher seeks to minimize the expected time to find several objects hidden by a hider we begin by analyzing a game in which the searcher wishes to find k balls hidden among n k boxes there is a known cost of searching each box and the searcher seeks to minimize the total expected cost of finding all the objects in the worst case we show that it is optimal for the searcher to begin by searching a k subset h of boxes with probability nu h which is proportional to the product of the search costs of the boxes in h the searcher should then search the n k remaining boxes in a random order a worst case hider distribution is the distribution nu we distinguish between the case of a smart searcher who can change his search plan as he goes along and a normal searcher who has to set out his plan from the beginning we show that a smart searcher has no advantage we then show how the game can be formulated in terms of a more general ne

a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

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

multiplex pi control for consensus in networks of heterogeneous linear agents in this paper we propose a multiplex proportional integral approach for solving consensus problems in networks of heterogeneous nodes dynamics affected by constant disturbances the proportional and integral actions are deployed on two different layers across the network each with its own topology sufficient conditions for convergence are derived that depend upon the structure of the network the parameters and topologies characterizing the control layers and the node dynamics the effectiveness of the theoretical results is illustrated numerically on a set of representative examples including a power network model

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

rendezvous search on the labeled line the rendezvous search problem is the problem of finding optimal search strategies for two people who are placed randomly on a known search region and want to meet each other in minimal expected time we focus on initial location distributions that are centrally symmetric and nonincreasing as one moves away from the center including the discretized and or truncated gaussian densities when the search region is a discrete or a continuous interval and the interval is labeled so that the searchers know their own location at all times we prove that the optimal strategy for both searchers is to go directly to the center and wait there the same result also holds for rendezvous search on the infinite line

online searching with turn cost we consider the problem of searching for an object on a line at an unknown distance opt from the original position of the searcher in the presence of a cost of d for each time the searcher changes direction this is a generalization of the well studied linear search problem we describe a strategy that is guaranteed to find the object at a cost of at most 9 opt 2d which has the optimal competitive ratio 9 with respect to opt plus the minimum corresponding additive term our argument for upper and lower bound uses an infinite linear program which we solve by experimental solution of an infinite series of approximating finite linear programs estimating the limits and solving the resulting recurrences for an explicit proof of optimality we feel that this technique is interesting in its own right and should help solve other searching problems in particular we consider the star search or cowpath problem with turn cost where the hidden object is placed on one of m rays emanating from the original position of the searcher for this problem we give a tight bound of 1 2mm m 1 m 1 opt m m m 1 m 1 1 d we also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line

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

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

contraction analysis for a class of nondifferentiable systems with applications to stability and network synchronization in this paper we extend to a generic class of piecewise smooth dynamical systems a fundamental tool for the analysis of convergence of smooth dynamical systems contraction theory we focus on switched nondifferentiable systems satisfying caratheodory conditions for the existence and uniqueness of a solution after generalizing the classical definition of contraction to this class of dynamical systems we give sufficient conditions for global convergence of their trajectories the theoretical results are then applied to solve a set of representative problems such as proving global asymptotic stability of switched linear systems giving conditions for incremental stability of piecewise smooth systems and analyzing the convergence of networked switched systems

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

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

search for an immobile hider in a known subset of a network abstract a unit speed searcher constrained to start in a given closed set s wishes to quickly find a point x known to be located in a given closed subset h of a metric network q this defines a game g g q h s where the payoff to the maximizing hider is the time for the searcher path to reach x lengths on q are defined by a measure which then defines distance as least length of connecting path for trees q we find that the minimax search time value v of g is given by v h d h s 2 where d h s is what we call the h diameter of s and equals the usual diameter d s of s in the case h q for the classical case of gal where the s is a singleton and h q our formula reduces to his result v q if s h q our formula gives dagan and gal s result v q d q 2 in all other cases our result is new optimal searches consist of minimum length paths covering h which start and end at points of s traversed equiprobably in either direction

robustness of synchronization of heterogeneous agents by strong coupling and a large number of agents this technical note studies robustness of synchronization against heterogeneity of multi agent systems here heterogeneity also includes uncertainties and external disturbances in each agent in order to effectively deal with the heterogeneous agents we introduce the concept of the averaged dynamics which is the average of all agents dynamics and then claim that two sources enhance the robustness of synchronized behavior against heterogeneity of agents dynamics first in spite of the heterogeneity of agents dynamics we show that strong coupling among agents makes the trajectories of all agents arbitrarily close to the averaged dynamics second in the averaged dynamics the effect of variations in each agent gets smaller as the number of agents increases simulation results confirm the claim that a large number of agents with strong couplings have robust synchronization

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

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

optimal symmetric rendezvous search on three locations in the symmetric rendezvous search game played on n locations two players are initially placed at two distinct locations the game is played in discrete steps at each of which each player can either stay where he is or move to a different location the players share no common labelling of the locations we wish to find a strategy such that if both players follow it independently then the expected number of steps until they are in the same location is minimized informal versions of the rendezvous problem have received considerable attention in the popular press the problem was proposed by alpern in 1976 alpern s 1976 hide and seek games seminar at the institut fur hohere studien vienna and it has proved notoriously difficult to analyse in this paper we prove a 20 year old conjecture that the following strategy is optimal for the game on three locations in each block of two steps stay where you are or search the other two locations in random order doing these with probabilities 1 3 and 2 3 respectively this is now the first nontrivial symmetric rendezvous game of any type to be fully solved

brief paper on the v stability of complex dynamical networks this paper introduces the concept of lyapunov v stability for complex dynamical networks under the new framework each dynamical node is associated with a passivity degree which indicates to what extent an effort is required for stabilizing the node from this approach the network stability problem is converted to measuring the negative definiteness of one simple matrix that characterizes the topology of the network pinning control is then suggested and investigated based on the new v stability formulation as an illustrative example a network with different node dynamics and non uniform coupling strengths is simulated to verify the analytic results moreover a comparison study for three different kinds of networks is provided to further illustrate the novelty and efficacy of the proposed v stability criterion and stabilization scheme

global bounded synchronization of general dynamical networks with nonidentical nodes this note addresses the problem of synchronization for general dynamical networks with nonidentical nodes the coupling strength outer coupling configuration and inner connection in such networks are all time varying neither an equilibrium for each node nor a synchronization manifold is assumed to exist an estimate of the convergence domain for a general class of time varying nonlinear systems is given by introducing the average dynamics of all nodes and based on this estimate a criterion of global synchronization in the sense of boundedness of the maximum state deviation between nodes is developed an explicit bound of the maximum state deviation between nodes is obtained by the maximum difference between each node dynamics and the average dynamics the proposed criterion is an extension of several related synchronization criteria for the case of identical nodes to the case of nonidentical nodes

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

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

search for an immobile entity on a network we consider the problem of searching for a single uniformly distributed immobile entity on an undirected network this problem differs from edge covering problems e g the chinese postman problem cpp since the objective here is not to find the minimum length tour that covers all the edges at least once but instead to minimize the expected time to find the entity we introduce a heuristic algorithm to deal with the search process given that the entity is equally likely to be at any point on the network computational results are presented

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

an evasion game on a finite tree the paper considers the following two person zero sum multistage game an evader starts at a given vertex of a tree and at discrete intervals of time chooses either to move to one of the vertices adjacent to him or stay where he is a gunner with a single bullet may at each of the same discrete intervals of time either fire the bullet at any of the vertices of the tree or hold his fire the gunner always hits the vertex at which he aims and the bullet takes one unit of time to reach its target the payoff to the gunner is 1 if he hits the evader mu where mu 1 if he fires and misses and 0 if he never fires it is shown that whatever vertex the evader starts at the value of the game is 1 mu 2

on the approximation ratio of the random chinese postman tour for network search abstract we consider a classic search problem first proposed by gal in which a searcher randomizes between unit speed paths on a network aiming to find a hidden point in minimal expected time in the worst case this can be viewed as a zero sum game between the searcher and a time maximizing hider it is a natural model for many search problems such as search and rescue operations the search for an enemy a bomb or weapons in a military context or predator prey search a chinese postman tour cpt is a minimal time tour of the network that searches all the arcs and a random chinese postman tour rcpt is an equiprobable choice of any given cpt and its reverse the full class of networks for which a rcpt is optimal is known but otherwise little is known about the solution of the game except in some special cases that have complicated optimal strategies that would be impractical to implement the question of how well a rcpt or any other search strategy performs for general networks has never been analyzed we show that a rcpt has an approximation ratio of 4 3 that is the maximum expected time it takes to find a point on the network is no greater than 4 3 times that of the optimal search strategy we then examine the performance of a rcpt in a related search game recently proposed by alpern in which the searcher must return to his starting point after finding the hider

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

lebesgue measure and integration theory on non archimedean real closed fields with archimedean value group given a non archimedean real closed field with archimedean value group which contains the reals we establish for the category of semialgebraic sets and functions a full lebesgue measure and integration theory such that the main results from the classical setting hold the construction involves methods from model theory o minimal geometry and valuation theory we set up the construction in such a way that it is determined by a section of the valuation if the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism the range of the measure and integration is obtained in a controlled and tame way from the real closed field we start with the main example is given by the case of the field of puiseux series where the range is the polynomial ring in one variable over this field

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

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

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

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

find and fetch search on a tree we introduce a new type of search game called the find and fetch game f q o the hider simply picks any point h in the network q the searcher starts at time zero at a given point o of q moving at unit speed until he reaches h finds the hider then he returns at a given speed along the shortest path back to o arriving at time r the payoff this models the problem faced in many types of search including search and rescue problems and foraging problems of animals where food must be found and returned to the lair when q is a binary tree we derive optimal probabilities for the searcher to branch at the nodes these probabilities give a positive bias towards searching longer branches first we show that the minimax value of the return time r the game value of f q o is d where is the total length of q and d is the mean distance from the root o to the leaves terminal nodes of q where the mean is taken with respect to what is known as the equal branch density distribution as goes to infinity our problem reduces to the search game model where the payoff is simply the time to reach the hider and our results tend to those obtained by gal gal s 1979 search games with mobile and immobile hider siam j control optim 17 1 99 122 and anderson and gal anderson e j s gal 1990 search in a maze probab engrg inform sci 4 3 311 318 for that model we also apply our return time formula d to determine the ideal location for the root lair or rescue center o assuming it can be moved in the traditional find only model the location of o does not matter

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

on search games that include ambush we present a stochastic game that models ambush search in a finite region q which has area but no other structure the searcher can search a unit area of q in unit time or adopt an ambush mode for a certain period the searcher captures the hider when the searched region contains the hider s location or if the hider moves while the searcher is in ambush mode the payoff in this zero sum game is the capture time our game is motivated by the still unsolved princess and monster game on a star graph with a large number of leaves

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

stability under integration of sums of products of real globally subanalytic functions and their logarithms we study lebesgue integration of sums of products of globally subanalytic functions and their logarithms called constructible functions our first theorem states that the class of constructible functions is stable under integration the second theorem treats integrability conditions in fubini type settings and the third result gives decay rates at infinity for constructible functions further we give preparation results for constructible functions related to integrability conditions

synchronization and dynamic consensus of heterogeneous networked systems we present an analysis framework for the study of synchronization of heterogeneous nonlinear systems interconnected over networks described by directed graphs heterogeneous systems may have totally different dynamical models albeit of the same dimension or may possess equal models with different lumped parameters we show that their behavior when network interconnected is fully characterized in terms of two properties whose study may be recasted in terms of the stability analysis of two corresponding interconnected dynamical systems that evolve in orthogonal spaces on one hand we have the so called emergent dynamics and on the other the synchronization error dynamics based on this premise we introduce the concept of dynamic consensus and we present results on robust stability which assess the conditions for practical asymptotic synchronization of networked systems and characterize their collective behavior to illustrate our main theoretical findings we broach a brief case study on mutual synchronization of heterogeneous chaotic oscillators

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

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

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

convergence and synchronization in heterogeneous networks of smooth and piecewise smooth systems this paper presents a framework for the study of convergence in networks where the nodes dynamics may be both piecewise smooth and or nonidentical sufficient conditions are derived for global convergence of all node trajectories towards the same bounded region in the synchronization error space the analysis is based on the use of set valued lyapunov functions and bounds are derived on the minimum coupling strength required to make all nodes in the network converge towards each other we also provide an estimate of the asymptotic bound on the mismatch between the node state trajectories the analysis is performed both for linear and nonlinear coupling protocols the theoretical analysis is extensively illustrated and validated via its application to a set of representative numerical 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

rendezvous evasion search in two boxes an agent who may or may not want to be found is located in one of two boxes at time 0 suppose that he is in box b with probability p he wishes to be found in which case he has been asked to stay in box b with probability 1 p he tries to evade the searcher in which case he may move between boxes a and b the searcher looks into one of the boxes at times 1 2 3 between each search the agent may change boxes if he wants the searcher is trying to minimise the expected time to discovery the agent is trying to minimise this time if he wants to be found but trying to maximise it otherwise this paper finds a solution to this game in a sense defined in the paper associated strategies for the searcher and each type of agent and a continuous value function v p giving the expected time for the agent to be discovered the solution method which is to solve an associated zero sum game would apply generally to this type of game of incomplete information

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

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

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

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

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

integration of semialgebraic functions and integrated nash functions when integrating semialgebraic functions one has to leave the semialgebraic setting for example one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent we show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions to realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives we analyze these rings profoundly in particular we show that the weierstrass division theorem and the weierstrass preparation theorem hold this allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions finally we investigate the structure generated by the integrated algebraic power series

a mixed strategy minimax theorem without compactness minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables this paper proves the following theorem let x be a compact hausdorfl space and let y a be a measurable space let f x times y be a measurable function which is bounded below and lower semicontinuous on x for all fixed y in y let m be any convex set of probability measures mixed strategies on y a then mathop min limits mu in b x mathop sup limits rho in m iint f x y d mu d rho mathop sup limits rho in m mathop min limits mu in b x iint f x y d mu d rho where b x denotes the regular borel probability measures on x

generalized best first search strategies and the optimality of a this paper reports several properties of heuristic best first search strategies whose scoring functions depend on all the information available from each candidate path not merely on the current cost g and the estimated completion cost h it is shown that several known properties of a retain their form with the minmax of f playing the role of the optimal cost which helps establish general tests of admissibility and general conditions for node expansion for these strategies on the basis of this framework the computational optimality of a in the sense of never expanding a node that can be skipped by some other algorithm having access to the same heuristic information that a uses is examined a hierarchy of four optimality types is defined and three classes of algorithms and four domains of problem instances are considered computational performances relative to these algorithms and domains are appraised for each class domain combination we then identify the strongest type of optimality that exists and the algorithm for achieving it the main results of this paper relate to the class of algorithms that like a return optimal solutions i e admissible when all cost estimates are optimistic i e h h on this class a is shown to be not optimal and it is also shown that no optimal algorithm exists but if the performance tests are confirmed to cases in which the estimates are also consistent then a is indeed optimal additionally a is also shown to be optimal over a subset of the latter class containing all best first algorithms that are guided by path dependent evaluation functions

technical note trading off quick versus slow actions in optimal search we consider the search for a target whose precise location is uncertain the search region is divided into grid cells and the searcher decides which cell to visit next and whether to search it quickly or slowly a quick search of a cell containing the target may damage it resulting in a failed search or it may locate the target safely if the target is not in the cell the search continues over the remaining cells if a slow search is performed on a cell then the search ends in failure with a fixed probability regardless of whether or not the target is in that cell e g because of enemy fire while performing the slow search if the slow search survives this failure possibility then the search ends in success if the target is in that cell otherwise the search continues over the remaining cells we seek to minimize the probability of the search ending in failure and consider two types of rules for visiting cells the unconstrained search in which the searcher may visit cells in any order and the constrained search in which the searcher may only visit adjacent cells e g up down left or right of cells already visited we prove that the optimal policy for the unconstrained search is to search quickly some initial set of cells with the lowest probabilities of containing the target before slowly searching the remaining cells in decreasing order of probabilities for the special case in which a quick search on a cell containing the target damages it with certainty the optimal policy is to search all cells slowly in decreasing order of probabilities we use the optimal solution of the unconstrained search in a branch and bound optimal solution algorithm for the constrained search for larger instances we evaluate heuristics and approximate dynamic programming approaches for finding good solutions

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

practical synchronization with diffusive couplings we investigate the problem of synchronizing nonidentical or perturbed nonlinear systems in the considered setup the systems are incapable to synchronize under diffusive couplings instead assuming the quad property for each system we derive conditions under which the synchronization error can be kept arbitrarily small by a proper choice of the interconnection structure this motivates the definition of practical synchronization as an alternative synchronization notion for nonidentical or perturbed dynamical systems the presented results are intimately related to synchronization of passive systems but it is shown that the stronger quad assumption is essential in our framework the proposed concept of practical synchronization translates directly into a notion of robust synchronization beyond that the results open the way for an investigation of synchronization phenomena on unbalanced graphs leading to the concept of cluster synchronization