searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

connected searching of weighted trees in this paper we consider the problem of connected edge searching of weighted trees barriere et al claim in l barriere p flocchini p fraigniaud n santoro capture of an intruder by mobile agents in spaa 02 proceedings of the fourteenth annual acm symposium on parallel algorithms and architectures acm new york ny usa 2002 pp 200 209 that there exists a polynomial time algorithm for finding an optimal search strategy that is a strategy that minimizes the number of used searchers however due to some flaws in their algorithm the problem turns out to be open it is proven in this paper that the considered problem is strongly np complete even for node weighted trees the weight of each edge is 1 with one vertex of degree greater than 2 it is also shown that there exists a polynomial time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices this is an fpt algorithm with respect to the maximum degree of a tree

markov property and strong additivity of von neumann entropy for graded quantum systems the quantum markov property is equivalent to the strong additivity of von neumann entropy for graded quantum systems the additivity of von neumann entropy for bipartite graded systems implies the statistical independence of states however the structure of markov states for graded systems is different from that for tensor product systems which have trivial grading for three composed graded systems we have u 1 gauge invariant markov states whose restriction to the marginal pair of subsystems is nonseparable

markovian quasifree states on canonical anticommutation relation algebras the characterization of quasifree product states on car algebras is given we also prove that the quasifree states on car algebra which saturate the strong additivity of von neumann entropy with equality are product states

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

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

hardy spaces and the szegő projection of the non smooth worm domain dβ we define hardy spaces hp d p 1 on the non smooth worm domain d z1 z2 c2 imz1 log z2 2 2 log z2 2 2 and we prove a series of related results such as the existence of boundary values on the distinguished boundary d of the domain and a fatou type theorem i e pointwise convergence to the boundary values thus we study the szeg projection operator s and the associated szeg kernel kd more precisely if hp d denotes the space of functions which are boundary values for functions in hp d we prove that the operator s extends to a bounded linear operator s lp d hp d for every p 1 and s wk p d wk p d for every k 0 here wk p denotes the sobolev space of order k and underlying lp norm p 1 as a consequence of the lp boundedness of s we prove that hp d c d is a dense subspace of hp d

standard potentials for non even dynamics the notion of standard potentials is introduced for a general dynamics this is a generalization of earlier works of araki and moriya which is restricted to even dynamics

lower and upper competitive bounds for online directed graph exploration abstract we study the problem of exploring all nodes of an unknown directed graph a searcher has to construct a tour that visits all nodes but only has information about the parts of the graph it already visited analogously to the traveling salesman problem the goal is to minimize the cost of such a tour in this article we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs our bounds are sharp or sharp up to a small constant depending on the specific model as it turns out restricting the diameter the incoming outgoing degree or randomly choosing a starting point does not improve lower bounds beyond a small constant factor even supplying the searcher in a planar euclidean graph with the nodes coordinates does not help essentially exploring a directed graph has a multiplicative overhead linear in the number of nodes furthermore if one wants to search for a specific node in unweighted directed graphs a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well

super derivations and associated standard super potentials abstractly defined super derivations on fermionic systems on a lattice are studied the existence and uniqueness of the associated standard super potential are shown for every super derivation with the subalgebra of all local operators as its domain the relation between the standard super potential of a super derivation and the standard potential for the square of the super potential which is shown to be a derivation in the case of finite range super potentials is obtained by use of local super hamiltonian for the super derivation and local hamiltonian for the square as a consequence a necessary and sufficient condition for a super derivation to be nilpotent is obtained in terms of the corresponding standard super potential examples of translation invariant nilpotent super derivations are given in the case of super potentials of finite ranges on a one dimensional lattice

non deterministic graph searching in trees non deterministic graph searching was introduced by fomin et al to provide a unified approach for pathwidth treewidth and their interpretations in terms of graph searching games given q 0 the q limited search number s q g of a graph g is the smallest number of searchers required to capture an invisible fugitive in g when the searchers are allowed to know the position of the fugitive at most q times the search parameter s 0 g corresponds to the pathwidth of a graph g and s g to its treewidth determining s q g is np complete for any fixed q 0 in general graphs and s 0 t can be computed in linear time in trees however the complexity of the problem on trees has been unknown for any q 0 we introduce a new variant of graph searching called restricted non deterministic the corresponding parameter is denoted by rs q and is shown to be equal to the non deterministic graph searching parameter s q for q 0 1 and at most twice s q for any q 2 for any graph g our main result is a polynomial time algorithm that computes rs q t for any tree t and any q 0 this provides a 2 approximation of s q t for any tree t and shows that the decision problem associated to s 1 is polynomial in the class of trees our proofs are based on a new decomposition technique for trees which might be of independent interest

approximate search strategies for weighted trees the problems of classical searching and connected searching of weighted trees are known to be computationally hard in this work we give a polynomial time 3 approximation algorithm that finds a connected search strategy of a given weighted tree this in particular yields constant factor approximation algorithms for the non connected classical searching problems and for the weighted pathwidth problem for this class of graphs

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

inequalities for quantum entropy a review with conditions for equality this article presents self contained proofs of the strong subadditivity inequality for von neumann s quantum entropy s and some related inequalities for the quantum relative entropy most notably its convexity and its monotonicity under stochastic maps moreover the approach presented here which is based on klein s inequality and lieb s theorem that the function a tr ek log a is concave allows one to obtain conditions for equality in the case of strong subadditivity which states that s 123 s 2 s 12 s 23 where the subscripts denote subsystems of a composite system equality holds if and only if log 123 log 12 log 2 log 23 using the fact that the holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy we show that equality can be attained for that bound only when the states in the ensemble commute the article concludes with an appendix giving a short description of epstein s elegant proof of lieb s the

monotony properties of connected visible graph searching search games are attractive for their correspondence with classical width parameters for instance the invisible search number a k a node search number of a graph is equal to its pathwidth plus 1 and the visible search number of a graph is equal to its treewidth plus 1 the connected variants of these games ask for search strategies that are connected i e at every step of the strategy the searched part of the graph induces a connected subgraph we focus on monotone search strategies i e strategies for which every node is searched exactly once the monotone connected visible search number of an n node graph is at most o logn times its visible search number first we prove that this logarithmic bound is tight precisely we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is w logn second we prove that as opposed to the non connected variant of visible graph searching recontamination helps for connected visible search precisely we prove that for any k 4 there exists a graph with connected visible search number at most k and monotone connected visible search number k

conditional expectations relative to a product state and the corresponding standard potentials for a lattice system with a finite number of fermions and spins on each lattice point conditional expectations relative to an even product state such as fermion fock vacuum are introduced and the corresponding standard potential for any given dynamics or more generally for any given time derivative at time 0 of strictly local operators is defined with the case of the tracial state previously treated as a special case the standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states where one can compare states satisfying the variational principle for translation invariant states or the local thermodynamical stability or the gibbs condition all in terms of the standard potential relative to different even product states

on the complexity of covering vertices by faces in a planar graph the pair g d consisting of a planar graph g v e with n vertices together with a subset of d special vertices d subseteq v is called k planar if there is an embedding of g in the plane so that at most k faces of g are required to cover all of the vertices in d checking 1 planarity can be done in linear time since it reduces to a problem of checking planarity of a related graph we present an algorithm which given a graph g and a value k either determines that g is not k planar or generates an appropriate embedding and associated minimum cover in o c k n time where c is a constant hence the algorithm runs in linear time for any fixed k the fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k the associated decision problem is strongly np complete even when the planar graph has essentially a unique planar embedding d theta n and all facial cycles have bounded length these results provide a polynomial t

diagonalization uniformity and fixed point theorems abstract we derive new fixed point theorems for subrecursive classes together with a theorem on the uniformity of certain reductions from a general formulation of the technique of delayed diagonalization this formulation extends the main theorem of u schoning theoret comput sci 18 1982 95 103 to cases which involve infinitely many diagonal classes c k and which allow each c k to contain uncountably many members the main technical work ties the familiar concept of a witness function directly to the often studied cantor set topology on languages and provides a delay construction which refines those due to schoning s breidtbart and d schmidt our a e fixed point theorems do not require that the programming system for the subrecursive class in question be well behaved we compare them to results which do the other theorem is similar to the uniform boundedness theorem of classical analysis and extends work of j grollmann and a selman

recontamination does not help to search a graph this paper is concerned with a game on graphs called graph searching the object of this game is to clear all edges of a contaminated graph clearing is achieved by moving searchers a kind of token along the edges of the graph according to clearing rules certain search strategies cause edges that have been cleared to become contaminated again megiddo et al 9 conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring that is without clearing any edge twice in this paper this conjecture is proved this places the graph searching problem in np completing the proof by megiddo et al that the graph searching problem is np complete furthermore by eliminating the need to consider recontamination this result simplifies the analysis of searcher requirements with respect to other properties of graphs

the bergman projection on hartogs domains in estimates in l 2 sobolev norms are proved for the bergman projection in certain smooth bounded hartogs domains in c 2 in particular 1 if the domain is pseudoconvex and nonwormlike the normal vector does not wind on a critical set in the boundary then the bergman projection is regular and 2 barrett s counter example domains with irregular bergman projection nevertheless admit a priori estimates

connected graph searching in the graph searching game the opponents are a set of searchers and a fugitive in a graph the searchers try to capture the fugitive by applying some sequence of moves that include placement removal or sliding of a searcher along an edge the fugitive tries to avoid capture by moving along unguarded paths the search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive in this paper we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean i e non accessible to the fugitive remain connected we give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement we also settle the question on the price of connectivity that is how many searchers more are required for searching a graph when the connectivity demand is imposed we make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees in particular for an n vertex graph the ratio between the connected searching number and the non connected one is o logn while for trees this ratio is always at most 2 we also conjecture that this constant ratio upper bound for trees holds also for all graphs our combinatorial results imply a complete characterization of connected graph searching on trees it is based on a forbidden graph characterization of the connected search number we prove that the connected search game is monotone for trees i e restricting search strategies to only those where the clean territories increase monotonically does not require more searchers a consequence of our results is that the connected search number can be computed in polynomial time on trees moreover we show how to make this algorithm distributed finally we reveal connections of this parameter to other invariants on trees such as the horton strahler number

nonconstructive tools for proving polynomial time decidability recent advances in graph theory and graph algorithms dramatically alter the traditional view of concrete complexity theory in which a decision problem is generally shown to be in p by producing an efficient algorithm to solve an optimization version of the problem nonconstructive tools are now available for classifying problems as decidable in polynomial time by guaranteeing only the existence of polynomial time decision algorithms in this paper these new methods are employed to prove membership in p for a number of problems whose complexities are not otherwise known powerful consequences of these techniques are pointed out and their utility is illustrated a type of partially ordered set that supports this general approach is defined and explored

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

validity and failure of some entropy inequalities for car systems basic properties of von neumann entropy such as the triangle inequality and what we call mono ssa are studied for car systems we show that both inequalities hold for every even state by using symmetric purification which is applicable to such a state we construct a certain class of noneven states giving examples of the nonvalidity of those inequalities

de finetti theorem on the car algebra the symmetric states on a quasi local c algebra on the infinite set of indices j are those invariant under the action of the group of the permutations moving only a finite but arbitrary number of elements of j the celebrated de finetti theorem describes the structure of the symmetric states i e exchangeable probability measures in classical probability in the present paper we extend the de finetti theorem to the case of the car algebra that is for physical systems describing fermions namely after showing that a symmetric state is automatically even under the natural action of the parity automorphism we prove that the compact convex set of such states is a choquet simplex whose extremal i e ergodic w r t the action of the group of permutations previously described are precisely the product states in the sense of araki moriya in order to do that we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self containing interest

joint extension of states of subsystems for a car system the problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a fermion system where a novel feature is non commutativity between algebras of subsystems for an arbitrary finite or infinite number of given subsystems a product state extension is shown to exist if and only if all states of subsystems except at most one are even with respect to the fermion number if the states of all subsystems are pure then the same condition is shown to be necessary and sufficient for the existence of any joint extension if the condition holds the unique product state extension is the only joint extension for a pair of subsystems with one of the given subsystem states pure a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions unique for almost all cases are given for a pair of subsystems with non pure subsystem states some classes of examples of joint extensions are given where non uniqueness of joint extensions prevails

on fermion grading symmetry for quasi local systems we discuss fermion grading symmetry for quasi local systems with graded commutation relations we introduce a criterion of spontaneously symmetry breaking ssb for general quasi local systems it is formulated based on the idea that each pair of distinct phases appeared in spontaneous symmetry breaking should be disjoint not only for the total system but also for every complementary outside system of a local region specified by the given quasi local structure under a completely model independent setting we show the absence of ssb for fermion grading symmetry in the above sense we obtain some structural results for equilibrium states of lattice systems if there would exist an even kms state for some even dynamics that is decomposed into noneven kms states then those noneven states inevitably violate our local thermal stability condition

generalized hypertree decompositions np hardness and tractable variants the generalized hypertree width ghw h of a hypergraph h is a measure of its cyclicity classes of conjunctive queries or constraint satisfaction problems whose associated hypergraphs have bounded ghw are known to be solvable in polynomial time however it has been an open problem for several years if for a fixed constant k and input hypergraph h it can be determined in polynomial time whether ghw h k here this problem is settled by proving that even for k e 3 the problem is already np hard on the way to this result another long standing open problem originally raised by goodman and shmueli 1984 in the context of join optimization is solved it is proven that determining whether a hypergraph h admits a tree projection with respect to a hypergraph g is np complete our intractability results on generalized hypertree width motivate further research on more restrictive tractable hypergraph decomposition methods that approximate generalized hypertree decomposition ghd we show that each such method is dominated by a tractable decomposition method definable through a function that associates a set of partial edges to a hypergraph by using one particular such function we define the new component hypertree decomposition method which is tractable and strictly more general than other approximations to ghd published so far

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

exclusive graph searching vs pathwidth in graph searching a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph equivalently the searchers try to clear a contaminated network the problem is to compute the minimum number of searchers required to accomplish this task several variants of graph searching have been studied mainly because of their close relationship with the pathwidth of a graph in this paper we study the complexity of the exclusive graph searching variant we show that the problem is np hard in planar graphs and it can be solved in linear time in the class of cographs we also show that monotone exclusive graph searching is np complete in split graphs where pathwidth is known to be solvable in polynomial time moreover we prove that monotone exclusive graph searching is in p in a subclass of star like graphs where pathwidth is known to be np hard