on the approximation of largest common subtrees and largest common point sets this paper considers the approximability of the largest common subtree and the largest common subgraph problems which have applications in molecular biology it is shown that approximating the problems within a factor of ne is np complete while a general search algorithm which approximates both problems within a factor of o n log n is presented moreover several variants of the largest common subtree problem are studied

kernelization and parameterized algorithms for covering a tree by a set of stars or paths two restricted versions of the subforest isomorphism problem the covering a tree by a set of stars ctss and the covering a tree by a set of paths ctsp problems are studied both problems are np complete the problems are closely related to a number of well studied problems including the problems subgraph isomorphism tree editing and graph packing it is shown that the problems ctss and ctsp are fixed parameter tractable thorough development of parameterized algorithms and kernelization algorithms for these problems are presented

exact algorithms for computing the tree edit distance between unordered trees this paper presents a fixed parameter algorithm for the tree edit distance problem for unordered trees under the unit cost model that works in o 2 62 email protected poly n time and o n 2 space where the parameter k is the maximum bound of the edit distance and n is the maximum size of input trees this paper also presents polynomial time algorithms for the case where the maximum degree of the largest common subtree is bounded by a constant

ordered and unordered tree inclusion the following tree matching problem is considered given labeled trees p and t can p be obtained from t by deleting nodes deleting a node u entails removing all edges incident to u and if u has a parent v replacing the edge from v to u by edges from v to the children of u the problem is motivated by the study of query languages for structured text databases simple solutions to this problem require exponential time for ordered trees an algorithm is presented that requires o p t time and space the corresponding problem for unordered trees is also considered and a proof of its np completeness is given an algorithm is presented for the unordered problem this algorithm works in o p t time if the out degrees of the nodes in p are bounded by a constant and in polynomial time if they are o log t

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

a tree matching algorithm based on node splitting and merging an algorithm that computes the best matching of two trees is described the degree of mismatch i e the distance is measured in terms of the number of node splitting and merging operations required the proposed tree distance is a more appropriate measurement of structural defonnation than the tree distance measure in terms of the number of insertions deletions and substitutions of tree nodes as defined in previous studies an algorithm that uses a divide and conquer strategy is presented the analysis shows that the time complexity is o nm2 where n and al are the number of nodes of the two trees respectively the algorithm has been implemented on a vax 11 780

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

classical particles with internal structure ii second order internal spaces a systematic study of classical relativistic particles with internal structure initiated in a previous paper is continued and a study of second order internal spaces sos is presented within the framework of the lagrangian form of constrained dynamics such internal spaces q are those for which a phase space treatment must necessarily use the cotangent bundle t q the large variety of possible sos s ten discrete cases and two one parameter families is separated into those capable of a manifestly covariant description and those for which special methods based on the transitive action of sl 2 c on a coset space are needed the concept of the isotopy representation plays an important role in this context seven of the possible discrete sos s are shown to be describable in a manifestly covariant way two discrete and one one parameter family of sos s are shown to be unphysical in the sense that no lagrangians can be written in which the internal and the space time position variables are nontrivially coupled and the remaining single discrete and one one parameter family are shown to be physical though not describable in a manifestly covariant way general phase space methods uniformly applicable to all sos s are developed and as an illustrative example a lagrangian model for an sos in which the internal space is spanned by two orthonormal spacelike unit vectors is presented

on invariants of graphs with applications to knot theory to each weighted graph two invariants a polynomial p x y z and the signature are defined the various partial degress of p x y z and are expressed in terms of maximal spanning graphs of furthermore one unexpected property of tutte s dichromate is proved these results are applied to knots or links in s 3

on the complexity of finding a largest common subtree of bounded degree the largest common subtree problem is to find a bijective mapping between subsets of nodes of two input rooted trees of maximum cardinality or weight that preserves labels and ancestry relationship the problem is known to be np hard for unordered trees in this paper we consider a restricted unordered case in which the maximum outdegree of a common subtree is bounded by a constant d we present an o nd time algorithm where n is the maximum size of two input trees which improves a previous o n2d time algorithm we also present an o h222h1d2h d1poly n time algorithm where h is the maximum height of two input trees

matroid theory and chern simons it is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum yang mills physics but also for m theory our discussion is focused in an action consisting purely of the chern simons term but in principle the main ideas can be applied beyond such an action in our treatment the theorem due to thistlethwaite which gives a relationship between the tutte polynomial for graphs and jones polynomial for alternating knots and links plays a central role before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the fano matroid and d 11 supergravity our approach also seems to be related to loop solutions of quantum gravity based in an ashtekar formalism

nonlinear pattern matching in trees tree pattern matching is a fundamental operation that is used in a number of programming tasks such as mechanical theorem proving term rewriting symbolic computation and nonprocedural programming languages in this paper we present new sequential algorithms for nonlinear pattern matching in trees our algorithm improves upon know tree pattern matching algorithms in important aspects such as time performance ease of integration with several reduction strategies and ability to avoid unnecessary computation steps on match attempts that fail the expected time complexity of our algorithm is linear in the sum of the sizes of the two trees

tree structure for contractible regions in r 3 for contractible regions in 3 with generic smooth boundary we determine the global structure of the blum medial axis m we give an algorithm for decomposing m into irreducible components which are attached to each other along fin curves the attaching cannot be described by a tree structure as in the 2d case however a simplified but topologically equivalent medial structure m with the same irreducible components can be described by a two level tree structure the top level describes the simplified form of the attaching and the second level tree structure for each irreducible component specifies how to construct the component by attaching smooth medial sheets to the network of y branch curves the conditions for these structures are complete in the sense that any region whose blum medial axis satisfies the conditions is contractible

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

lifting kk elements asymptotic unitary equivalence and classification of simple c algebras let a and c be two unital simple c algebras with tracial rank zero suppose that c is amenable and satisfies the universal coefficient theorem denote by kke c a the set of those in kk c a for which k0 c 0 k0 a 0 and 1c 1a suppose that kke c a we show that there is a unital monomorphism c a such that suppose that c is a unital ah algebra and is a continuous affine map for which t p t p for all projections p in all matrix algebras of c and any tt a where t a is the simplex of tracial states of a and is the convex set of faithful tracial states of c we prove that there is a unital monomorphism c a such that induces both and suppose that h c a is a unital monomorphism and hom k1 c aff a we show that there exists a unital monomorphism c a such that h in kk c a t t h for all tracial states t and the associated rotation map can be given by denote by kkt c a the set of compatible pairs where kle c a and is a continuous affine map from t a to together with a result on asymptotic unitary equivalence in h lin asymptotic unitary equivalence and asymptotically inner automorphisms arxiv math 0703610 2007 this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from c to a to where is a subgroup related to vanishing rotation maps as an application combining these results with a result of w winter w winter localizing the elliott conjecture at strongly self absorbing c algebras arxiv 0708 0283v3 2007 we show that two unital amenable simple stable c algebras are isomorphic if they have the same elliott invariant and the tensor products of these c algebras with any uhf algebra have tracial rank zero in particular if a and b are two unital separable simple stable c algebras with unique tracial states which are inductive limits of c algebras of type i then they are isomorphic if and only if they have isomorphic elliott invariants

classical particles with internal structure general formalism and application to first order internal spaces group theoretic methods are used to systematically classify all possible internal structures for an elementary classical relativistic particle in terms of coset spaces of sl 2 c with respect to its continuous subgroups the allowed internal spaces q are separated into first and second order ones depending on whether a canonical description can be given using q itself or if it needs the cotangent bundle t q three of the former are found one corresponding to the use of a majorana spinor as the internal variable the other two related to orbits in the lie algebra of so 3 1 under the adjoint action for the latter two a lagrangian description of an elementary object with the corresponding internal space is set up and the dynamics studied

approximation and parameterized algorithms for common subtrees and edit distance between unordered trees given two rooted labeled unordered trees the common subtree problem is to find a bijective matching between subsets of nodes of the trees of maximum cardinality which preserves labels and ancestry relationship the tree edit distance problem is to determine the least cost sequence of insertions deletions and substitutions that converts a tree into another given tree both problems are known to be hard to approximate within some constant factor in general we tackle these problems from two perspectives giving exact algorithms either for special cases or in terms of some parameters and approximation algorithms and hardness of approximation we present a parameterized algorithm in terms of the number of branching nodes that solves both problems and yields polynomial algorithms for several special classes of trees this is complemented with a tighter apx hardness proof that holds when the trees are of height one and two respectively furthermore we present the first approximation algorithms for both problems in particular for the common subtree problem for t trees we present an algorithm achieving a tlog 2 b o p t 1 ratio where b o p t is the number of branching nodes in the optimal solution we also present constant factor approximation algorithms for both problems in the case of bounded height trees

simulation relations for pattern matching in directed graphs we consider the problem of finding the occurrences of a pattern tree t in a directed graph g and propose two algorithms one for preprocessing and one for searching for t in g it is assumed that

soldered bundle background for the de sitter top we prove that the mathematical framework for the de sitter top system is the de sitter fiber bundle in this context the concept of soldering associated with a fiber bundle plays a central role we comment on the possibility that our formalism may be of particular interest in different contexts including macdowell mansouri theory two time physics and oriented matroid theory

on the stable rank of simple c algebras in 16 m a rieffel introduced the notion of stable rank for c algebras for a a unital c algebra the stable rank denoted by sr a is the least integer n such that the set of n tuples over a which generate a as a left ideal is dense in an if no such n exists we set sr a oo for unital c algebras the stable rank was shown to coincide with the bass stable rank which is defined for rings see 12 the stable rank can also be defined for nonunital c algebras but we shall only consider the unital case stable rank is related to nonstable k theory in 16 theorem 10 12 and 3 corollary 7 14 it was shown that if a is a simple unital c algebra of stable rank one then the natural map u a u a o 4 ki a is an isomorphism see also 17 2 and 18 it is a natural question which values occur as the stable rank of simple c algebras suppose that a is simple and infinite then by 7 proposition 1 5 a contains two isometries with orthogonal ranges and so by 16 proposition 6 5 sr a oo in the case of finite simple c algebras the following is known whenever a is simple and stably finite and b is a uhf algebra the tensor product a0 b has stable rank one see 19 this is also the case for every simple approximately homogeneous ah algebra with slow dimension growth see 4 and 8 so in short sr oo for every infinite simple c algebra and sr 1 for various classes of finite simple c algebras in this paper we answer the question above by showing that every n c n u oo is the stable rank of some simple finite c algebra more precisely it is shown that for every n 2 3 4 oo there exists a simple separable and unital ah algebra a of stable rank n in particular the group of invertible elements in a is not dense and like the examples in 20 a does not have slow dimension growth and is not approximately divisible see 5 i thank marius dadarlat for suggesting that i study ah algebras with connecting homomorphisms as described in the following section and juliana erlijman for drawing my attention to some of the powerful tools of differential topology and i thank george elliott for my stay at the fields institute where this paper was written

wigner rotations bargmann invariants and geometric phases the concept of the wigner rotation familiar from the composition law of pure lorentz transformations is described in the general setting of lie group coset spaces and the properties of coset representatives examples of abelian and non abelian wigner rotations are given the lorentz group wigner rotation occurring in the coset space sl 2 r so 2 simeq so 2 1 so 2 is shown to be an analytic continuation of a wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations belonging to the coset space su 2 u 1 simeq so 3 so 2 the possibility of interpreting these two wigner rotations as geometric phases is shown in detail essential background material on geometric phases bargmann invariants and null phase curves all of which are needed for this purpose is provided

a short survey of noncommutative geometry we give a survey of selected topics in noncommutative geometry with some emphasis on those directly related to physics including our recent work with dirk kreimer on renormalization and the riemann hilbert problem we discuss at length two issues the first is the relevance of the paradigm of geometric space based on spectral considerations which is central in the theory as a simple illustration of the spectral formulation of geometry in the ordinary commutative case we give a polynomial equation for geometries on the four sphere with fixed volume the equation involves an idempotent e playing the role of the instanton and the dirac operator d it is of the form e 12 d e 4 5 and determines both the sphere and all its metrics with fixed volume form the expectation x is the projection on the commutant of the algebra of 4 by 4 matrices we also show using the noncommutative analog of the polyakov action how to obtain the noncommutative metric in spectral form on the noncommutative tori fr

scheduling split intervals we consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line each job j j is associated with an interval i j which consists of up to t segments for some t geq 1 and a weight profit w j two jobs are in conflict if their intervals intersect such jobs show up in a wide range of applications including the transmission of continuous media data allocation of linear resources e g bandwidth in linear processor arrays and computational biology geometry the objective is to schedule a subset of nonconflicting jobs of maximum total weight our problem can be formulated as the problem of finding a maximum weight independent set in a t interval graph the special case of t 1 is an ordinary interval graph we show that for t geq 2 this problem is apx hard even for highly restricted instances our main result is a 2t approximation algorithm for general instances this is based on a novel fractional version of the local ratio technique one implication of this result is the first constant factor approximation for nonoverlapping alignment of genomic sequences we also derive a bicriteria polynomial time approximation scheme for a restricted subclass of t interval graphs

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

searching for a connection between matroid theory and string theory we make a number of observations about matter ghost string phase which may eventually lead to a formal connection between matroid theory and string theory in particular in order to take advantage of the already established connection between matroid theory and chern simons theory we propose a generalization of string theory in terms of some kind of kahler metric we show that this generalization is closely related to the kahler chern simons action due to nair and schiff in addition we discuss matroid string connection via matroid bundles and a schild type action and we add new information about the relationship between matroid theory d 11 supergravity and chern simons formalism

alignment of trees an alternative to tree edit abstract in this paper we propose the alignment of trees as a measure of the similarity between two labeled trees both ordered and unordered trees are considered an algorithm is designed for ordered trees the time complexity of this algorithm is o t 1 t 2 deg t 1 deg t 2 2 where t 1 is the number of nodes in t 1 and deg t 1 is the degree of t 1 i 1 2 the algorithm is faster than the best known algorithm for tree edit when deg t 1 and deg t 2 are smaller than the depths of t 1 and t 2 for unordered trees we show that the alignment problem can be solved in polynomial time if the trees have a bounded degree and becomes max snp hard if one of the trees is allowed to have an arbitrary degree in contrast the edit problem for unordered trees is max snp hard even if both trees have a bounded degree zhang and jiang 1994 finally multiple alignment of trees is discussed

a survey on tree edit distance and related problems we survey the problem of comparing labeled trees based on simple local operations of deleting inserting and relabeling nodes these operations lead to the tree edit distance alignment distance and inclusion problem for each problem we review the results available and present in detail one or more of the central algorithms for solving the problem

simple fast algorithms for the editing distance between trees and related problems ordered labeled trees are trees in which the left to right order among siblings is significant the distance between two ordered trees is considered to be the weighted number of edit operations insert delete and modify to transform one tree to another the problem of approximate tree matching is also considered specifically algorithms are designed to answer the following kinds of questions 1 what is the distance between two trees 2 what is the minimum distance between t 1 and t 2 when zero or more subtrees can be removed from t 2 3 let the pruning of a tree at node n mean removing all the descendants of node n the analogous question for prunings as for subtrees is answered a dynamic programming algorithm is presented to solve the three questions in sequential time o t 1 times t 2 times min textit depth t 1 textit leaves t 1 times min textit depth t 2 textit leaves t 2 and space o t 1 times t 2 compared with o t 1 times t 2 t

algorithms for the constrained editing distance between ordered labeled trees and related problems abstract this paper considers the problem of computing a constrained editing distance between ordered labeled trees this distance metric can be applied to pattern recognition syntactic tree comparison classification tree comparison and other applications the problem of approximate ordered tree matching is also considered we present optimal algorithms for solving these problems in sequential time o t1 t2