a differential complex for cat 0 cubical spaces abstract in the 1980 s pierre julg and alain valette and also tadeusz pytlik and ryszard szwarc constructed and studied a certain fredholm operator associated to a simplicial tree the operator can be defined in at least two ways from a combinatorial flow on the tree similar to the flows in forman s discrete morse theory or from the theory of unitary operator valued cocycles there are applications of the theory surrounding the operator to c algebra k theory to the theory of completely bounded representations of groups that act on trees and to the selberg principle in the representation theory of p adic groups the main aim of this paper is to extend the constructions of julg and valette and pytlik and szwarc to cat 0 cubical spaces a secondary aim is to illustrate the utility of the extended construction by developing an application to operator k theory and giving a new proof of k amenability for groups that act properly on finite dimensional cat 0 cubical spaces

a data based enumeration technique for fully correlated signals presents a novel method for estimating the number of signals impinging on a uniform linear array using observed sensor data unlike other algorithms that apply rissanen s minimum description length mdl principle to the observed data for source enumeration this method applies it to the prediction errors of a linear model that has been fitted to an appropriate data matrix it is a 1d method that achieves improved performance even for fully correlated signals over contemporary approaches particularly with short data records and closely spaced signals asymptotic consistency is shown and substantiating simulation examples are included

the mobius function of generalized subword order let p be a poset and let p be the set of all finite length words over p generalized subword order is the partial order on p obtained by letting u w if and only if there is a subword u of w having the same length as u such that each element of u is less than or equal to the corresponding element of u in the partial order on p classical subword order arises when p is an antichain while letting p be a chain gives an order on compositions for any finite poset p we give a simple formula for the mobius function of p in terms of the mobius function of p this permits us to rederive in an easy and uniform manner previous results of bjorner sagan and vatter and tomie we are also able to determine the homotopy type of all intervals in p for any finite p of rank at most 1

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

proximal methods in vector optimization we consider the vector optimization problem of finding weakly efficient points for maps from a hilbert space x to a banach space y with respect to the partial order induced by a closed convex and pointed cone c subset y with a nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization in this extension the subproblems consist of finding weakly efficient points for suitable regularizations of the original map we present both an exact and an inexact version in which the subproblems are solved only approximately within a constant relative tolerance in both cases we prove weak convergence of the generated sequence to a weakly efficient point assuming convexity of the map with respect to c and c completeness of the initial section in cases where this last assumption fails we still establish that the generating sequence is in a suitable sense a minimizing one we also exhibit a particular instance of the algorithm for which under a mild hypothesis on c the weak limit of the generated sequence is an efficient rather than a weakly efficient point

homotopy groups of suspended classifying spaces an experimental approach when the results of a computer program are confronted to some theorems proved on a theoretical basis three situations can occur there can be an agreement between both approaches the computer program can obtain calculations not covered by the theorems or a discrepancy can be found between both methods in this note we report on a work where the three above mentioned situations happen we have enhanced the computer algebra called kenzo to deal with the computation of homotopy groups of suspended classifying spaces a problem tackled by mikhailov and wu in a paper published in the journal algebraic and geometric topology our experimental approach based on completely different methods from those by mikhailov and wu has allowed us in particular to detect an error in one of their published theorems 1 motivation and statement of the problem it is well known that the suspension functor applied to a topological space x shifts its homology groups that is to say hn x hn 1 x however in the case of homotopy groups the situation is not so favorable and in general there is not a direct relation between x and x let us consider for instance a circle s whose homotopy type is very simple the homology groups of the suspension s s are obvious but the problem of computing the homotopy groups of s is a difficult problem in algebraic topology given a group g its classifying space that is to say the eilenberg maclane spacek g 1 10 has trivial homotopy groups 1 k g 1 g and n k g 1 0 for each n 6 1 but when applying the suspension functor the new homotopy groups k g 1 are in general unknown in 11 several groups n k g 1 are obtained for some particular cases of g and n making use of different results and techniques from group theory and homotopy theory more concretely the main results of the article by mikhailov and wu are descriptions of the groups 4 k a 1 and 5 k a 1 when a is any finitely generated abelian group as applications they also determine n k g 1 with n 4 and 5 for some nonabelian groups as g 3 the 3 th symmetric group and g sl z the standard linear group and 4 k a4 1 for a4 the 4 th alternating group the goal of this work consists in trying to obtain a computer program calculating k g 1 for a general group g which will allow one to obtain as particular computations some theoretical results of 11 to this aim we have developed 2010 mathematics subject classification 68w05 55 04 55q99 partially supported by ministerio de ciencia e innovacion spain project mtm2009 13842c02 01 c xxxx american mathematical society

the assignment problem with seniority and job priority constraints consider an assignment problem in which persons are qualified for some but usually not all of the jobs moreover assume persons belong to given seniority classes and jobs have given priority levels seniority constraints impose that the solution be such that no unassigned person can be given a job unless an assigned person with the same or higher seniority becomes unassigned priority constraints specify that the solution must be such that no unassigned job can become assigned without a job with the same or higher priority becoming unassigned it is shown that i adding such constraints does not reduce and may even increase the number of assigned persons in the optimal solution ii using a greedy heuristic for constrained assignment as often done in practice may reduce the number of assigned persons by half and iii an optimal solution to the assignment problem with both types of constraints can be obtained by solving a classical assignment problem with adequately modified coefficients

solving group steiner problems as steiner problems abstract the generalized spanning tree or group steiner problem gsp is a generalization of the steiner problem in graphs spg one requires a tree spanning at least one vertex of each subset given in a family of vertex subsets while minimizing the sum of the corresponding edge costs specialized solution procedures have been developed for this problem in this paper we investigate the performance of a known but so far neglected transformation to the undirected steiner problem in graphs when combined with a recent metaheuristic for the spg this straightforward approach compares favorably with specialized gsp heuristics thus we set a standard for future algorithms

torsion in the matching complex and chessboard complex abstract topological properties of the matching complex were first studied by bouc in connection with quillen complexes and topological properties of the chessboard complex were first studied by garst in connection with tits coset complexes bjorner lovasz vrecica and ivaljevic established bounds on the connectivity of these complexes and conjectured that these bounds are sharp in this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds moreover we show that for sufficiently large n the bottom nonvanishing homology of the matching complex m n is an elementary 3 group improving a result of bouc and that the bottom nonvanishing homology of the chessboard complex m n n is a 3 group of exponent at most 9 when n 2 mod 3 the bottom nonvanishing homology of m n n is shown to be z 3 our proofs rely on computer calculations long exact sequences representation theory and tableau combinatorics

non abelian quantum statistics on graphs we show that non abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces in particular we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space x the framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of x which can be achieved by determining its homology groups we apply this methodology for configuration spaces of graphs as a conclusion we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non abelian anyons on networks that are used in quantum computing these conclusions are based on our solution of the so called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs

box constrained multi objective optimization a gradient like method without a priori scalarization abstract the aim of this paper is the development of an algorithm to find the critical points of a box constrained multi objective optimization problem the proposed algorithm is an interior point method based on suitable directions that play the role of gradient like directions for the vector objective function the method does not rely on an a priori scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box the key tool to define the mentioned vector field is the notion of vector pseudogradient we prove that the limit points of the solutions of the system satisfy the karush kuhn tucker kkt first order necessary condition for the box constrained multi objective optimization problem these results allow us to develop an algorithm to solve box constrained multi objective optimization problems finally we consider some test problems where we apply the proposed computational method the numerical experience shows that the algorithm generates an approximation of the local optimal pareto front representative of all parts of optimal front

an entropy based persistence barcode in persistent homology the persistence barcode encodes pairs of simplices meaning birth and death of homology classes persistence barcodes depend on the ordering of the simplices called a filter of the given simplicial complex in this paper we define the notion of minimal barcodes in terms of entropy starting from a given filtration of a simplicial complex k an algorithm for computing a proper filter a total ordering of the simplices preserving the partial ordering imposed by the filtration as well as achieving a persistence barcode with small entropy is detailed by way of computation and subsequent modification of maximum matchings on subgraphs of the hasse diagram associated to k examples demonstrating the utility of computing such a proper ordering on the simplices are given highlightswe define the notion of minimal barcodes in terms of entropy given a simplicial complex an algorithm for computing a proper filter f is detailed f preserves the partial ordering imposed by the filtration f achieves a persistence barcode with small entropy examples demonstrating the utility of computing such a proper filter are given

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

discrete morse theory and classifying spaces abstract the aim of this paper is to develop a refinement of forman s discrete morse theory to an acyclic partial matching on a finite regular cw complex x forman introduced a discrete analogue of gradient flows although forman s gradient flow has been proved to be useful in practical computations of homology groups it is not sufficient to recover the homotopy type of x forman also proved the existence of a cw complex which is homotopy equivalent to x and whose cells are in one to one correspondence with the critical cells of but the construction is ad hoc and does not have a combinatorial description by relaxing the definition of forman s gradient flows we introduce the notion of flow paths which contains enough information to reconstruct the homotopy type of x while retaining a combinatorial description the critical difference from forman s gradient flows is the existence of a partial order on the set of flow paths from which a 2 category c is constructed it is shown that the classifying space of c is homotopy equivalent to x by using homotopy theory of 2 categories this result can be also regarded as a discrete analogue of the unpublished work of cohen jones and segal on morse theory in early 90 s

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

on the local behavior of spaces of natural images in this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images to this end we use a space of 3 by 3 high contrast patches we develop a theoretical model for the high density 2 dimensional submanifold of showing that it has the topology of the klein bottle using our topological software package plex we experimentally verify our theoretical conclusions we use polynomial representation to give coordinatization to various subspaces of we find the best fitting embedding of the klein bottle into the ambient space of our results are currently being used in developing a compression algorithm based on a klein bottle dictionary

solving the k cardinality assignment problem by transformation the k cardinality linear assignment problem k lap with k a given integer is a generalization of the linear assignment problem one wants to assign k rows a free choice out of more rows to k columns a free choice out of more columns minimizing the sum of the corresponding costs for this polynomially solvable problem special algorithms are known based on transformation to min cost flow or on shortest augmenting paths we describe a transformation that enables to solve the k lap by any standard linear assignment algorithm the transformation can be modified to solve the group assignment problem as a standard problem computational results for random test instances up to size n 500 with various k values and randomly drawn cost coefficients show that the transformation approach is suited to solve the k lap within short computer times on a standard personal computer

convergence of the projected gradient method for quasiconvex multiobjective optimization we consider the projected gradient method for solving the problem of finding a pareto optimum of a quasiconvex multiobjective function we show convergence of the sequence generated by the algorithm to a stationary point furthermore when the components of the multiobjective function are pseudoconvex we obtain that the generated sequence converges to a weakly efficient solution

n particle quantum statistics on graphs we develop a full characterization of abelian quantum statistics on graphs we explain how the number of anyon phases is related to connectivity for 2 connected graphs the independence of quantum statistics with respect to the number of particles is proven for non planar 3 connected graphs we identify bosons and fermions as the only possible statistics whereas for planar 3 connected graphs we show that one anyon phase exists our approach also yields an alternative proof of the structure theorem for the first homology group of n particle graph configuration spaces finally we determine the topological gauge potentials for 2 connected graphs

pruned cellular free resolutions of monomial ideals abstract using discrete morse theory we give an algorithm that prunes the excess of information in the taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal the pruned resolution is not simplicial in general but we can slightly modify our algorithm in order to obtain a simplicial resolution we also show that the lyubeznik resolution fits into our pruning strategy the pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the taylor and the lyubeznik resolutions as we will see in some examples we finally use our methods to give a different approach to the theory of splitting of monomial ideals we deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

a shortest augmenting path algorithm for the semi assignment problem the objective of this study is to develop a shortest augmenting path algorithm for solving the semi assignment problem and conduct an extensive computational comparison with the best alternative approaches the algorithm maintains dual feasibility and complementary slackness and works toward satisfying primal feasibility effective heuristics arc used to achieve an excellent advanced start and convergence is assured via the use of the shortest augmenting path procedure using reduced costs for arc lengths unlike other algorithms such as the primal simplex or the auction algorithm each iteration during the final phase of the procedure also known as the end game achieves one additional assignment the software implementations of our algorithm are fastest for the semi assignment problems that we tested our dense code is also faster than the best software for assignment problems in addition the algorithm has the best polynomial worst case running time bound that we have seen and the memory requirements are relatively small

discrete morse functions from lexicographic orders this paper shows how to construct a discrete morse function with a relatively small number of critical cells for the order complex of any finite poset with o and i from any lexicographic order on its maximal chains specifically if we attach facets according to the lexicographic order on maximal chains then each facet contributes at most one new face which is critical and at most one betti number changes facets which do not change the homotopy type also do not contribute any critical faces dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets as one application the mobius function may be computed as the alternating sum of morse numbers the above construction enables us to prove that the poset n s of partitions of a set 1 1 k k with repetition is homotopy equivalent to a wedge of spheres of top dimension when a is a hook shaped partition it is likely that the proof may be extended to a larger class of a and perhaps to all a despite a result of ziegler 1986 which shows that n s is not always cohen macaulay

combinatorial vector fields and dynamical systems in this paper we introduce the notion of a combinatorial dynamical system on any cw complex earlier in fo3 and fo4 we presented the idea of a combinatorial vector field see also fo1 for the one dimensional case and studied the corresponding morse theory equivalently we studied the homological properties of gradient vector fields these terms were defined precisely in fo3 see also sect 2 of this paper in this paper we broaden our investigation and consider general combinatorial vector fields we first study the homological properties of such vector fields generalizing the morse inequalities of fo3 we then introduce various zeta functions which keep track of the closed orbits of the corresponding flow and prove that these zeta functions initially defined only on a half plane can be analytically continued to meromorphic functions on the entire complex plane lastly we review the notion of reidemeister torsion of a cw complex introduced in re fr and show that the torsion is equal to the value at z 0 of one of the zeta functions introduced earlier much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in p p fra fri1 2 and elsewhere

the morse complex of a line arrangement abstract let m a be the complement in c 2 of a complexified line arrangement we provide compact formulas for a morse complex which computes the co homology of m a with coefficients in an abelian local system this refines and simplifies in the two dimensional case a general construction appeared in m salvetti s settepanella combinatorial morse theory and minimality of hyperplane arrangements geom topol 11 2007 1733 1766 giving also a direct geometrical interpretation

on the golod property of stanley reisner rings abstract recently in m jollenbeck on the multigraded hilbert and poincare series of monomial rings j pure appl algebra 207 2 2006 261 298 the second author made a conjecture about the structure of ext a k k as an n n n graded vector space where a is a monomial ring over a field k that is the quotient of a polynomial ring p k x 1 x n by a monomial ideal and he verified this conjecture for several classes of such rings using the results of a berglund poincare series and homotopy lie algebras of monomial rings licentiate thesis stockholm university http www math su se reports 2005 6 2005 by the first author we are able to prove this conjecture in general in particular we get a new explicit formula for the multigraded hilbert series of ext a k k a surprising consequence of our results is that a monomial ring a is golod if and only if the product on tor p a k is trivial for stanley reisner rings of flag complexes we get a complete combinatorial characterization of golodness we introduce the concept of minimally non golod complexes and show that boundary complexes of stacked polytopes are minimally non golod finally we discuss the relation between minimal non golodness and the gorenstein property for simplicial complexes

derivations of leavitt path algebras abstract in this paper we describe the k module h h 1 l k of outer derivations of the leavitt path algebra l k of a row finite graph with coefficients in an associative commutative ring k with unit we explicitly describe a set of generators of h h 1 l k and relations among them we also describe a lie algebra structure of outer derivation algebra of the toeplitz algebra we prove that every derivation of a leavitt path algebra can be extended to a derivation of the corresponding c algebra

the nonlinear statistics of high contrast patches in natural images recently there has been a great deal of interest in modeling the non gaussian structures of natural images however despite the many advances in the direction of sparse coding and multi resolution analysis the full probability distribution of pixels values in a neighborhood has not yet been described in this study we explore the space of data points representing the values of 3 3 high contrast patches from optical and 3d range images we find that the distribution of data is extremely sparse with the majority of the data points concentrated in clusters and non linear low dimensional manifolds furthermore a detailed study of probability densities allows us to systematically distinguish between images of different modalities optical versus range which otherwise display similar marginal distributions our work indicates the importance of studying the full probability distribution of natural images not just marginals and the need to understand the intrinsic dimensionality and nature of the data we believe that object like structures in the world and the sensor properties of the probing device generate observations that are concentrated along predictable shapes in state space our study of natural image statistics accounts for local geometries such as edges in natural scenes but does not impose such strong assumptions on the data as independent components or sparse coding by linear change of bases

a lefschetz fixed point theorem for multivalued maps of finite spaces we prove a version of the lefschetz fixed point theorem for multivalued maps f x multimap x in which x is a finite t 0 space

convergence of a projected gradient method variant for quasiconvex objectives we present a version of the projected gradient method for solving constrained minimization problems with a competitive search strategy an appropriate step size rule through an armijo search along the feasible direction thereby obtaining global convergence properties when the objective function is quasiconvex or pseudoconvex in contrast to other similar step size rules this one requires only one projection onto the feasible set per iteration rather than one projection for each tentative step during the search for the step size which represents a considerable saving when the projections are computationally expensive

discrete morse theory and the cohomology ring in morse theory for cell complexes we presented a discrete morse theory that can be applied to general cell complexes in particular we defined the notion of a discrete morse function along with its associated set of critical cells we also constructed a discrete morse cocomplex built from the critical cells and the gradient paths between them which has the same cohomology as the underlying cell complex in this paper we show how various cohomological operations are induced by maps between morse cocomplexes for example given three discrete morse functions we construct a map from the tensor product of the first two morse cocomplexes to the third morse cocomplex which induces the cup product on cohomology all maps are constructed by counting certain configurations of gradient paths this work is closely related to the corresponding formulas in the smooth category as presented by betz and cohen and by fukaya

barcodes the persistent topology of data this article surveys recent work of carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high dimensional data the primary mathematical tool considered is a homology theory for point cloud data sets persistent homology and a novel representation of this algebraic characterization barcodes we sketch an application of these techniques to the classification of natural images 1 the shape of data when a topologist is asked how do you visualize a four dimensional object the appropriate response is a socratic rejoinder how do you visualize a threedimensional object we do not see in three spatial dimensions directly but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended we spend a significant portion of our first year of life learning how to infer three dimensional spatial data from paired planar projections years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension the inference of global structure occurs on much finer scales as well with regards to converting discrete data into continuous images dot matrix printers scrolling led tickers televisions and computer displays all communicate images via arrays of discrete points which are integrated into coherent global objects this also is a skill we have practiced from childhood no adult does a dot to dot puzzle with anything approaching anticipation 1 1 topological data analysis problems of data analysis share many features with these two fundamental integration tasks 1 how does one infer high dimensional structure from low dimensional representations and 2 how does one assemble discrete points into global structure the principal themes of this survey of the work of carlsson de silva edelsbrunner harer zomorodian and others are the following 1 it is beneficial to replace a set of data points with a family of simplicial complexes indexed by a proximity parameter this converts the data set into global topological objects 2 it is beneficial to view these topological complexes through the lens of algebraic topology specifically via a novel theory of persistent homology adapted to parameterized families 3 it is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a betti number a barcode the author gratefully acknowledges the support of darpa hr0011 07 1 0002 the work reviewed in this article is funded by the darpa program tda topological data analysis

connectivity calculus of fractal polyhedrons the paper analyzes the connectivity information more precisely numbers of tunnels and their homological co cycle classification of fractal polyhedra homology chain contractions and its combinatorial counterparts called homological spanning forest hsf are presented here as an useful topological tool which codifies such information and provides an hierarchical directed graph based representation of the initial polyhedra the menger sponge and the sierpinski pyramid are presented as examples of these computational algebraic topological techniques and results focussing on the number of tunnels for any level of recursion are given experiments performed on synthetic and real image data demonstrate the applicability of the obtained results the techniques introduced here are tailored to self similar discrete sets and exploit homology notions from a representational point of view nevertheless the underlying concepts apply to general cell complexes and digital images and are suitable for progressing in the computation of advanced algebraic topological information of 3 dimensional objects highlightsthis paper analyzes the connectivity information of fractal polyhedra this paper recalls homology strong deformation retracts as an appropriate tool for doing so this paper illustrates the relevance of this tool for menger sponge and sierpinski pyramids this paper provides examples and detailed proofs

a steepest descent method for vector optimization in this work we propose a cauchy like method for solving smooth unconstrained vector optimization problems when the partial order under consideration is the one induced by the nonnegative orthant we regain the steepest descent method for multicriteria optimization recently proposed by fliege and svaiter we prove that every accumulation point of the generated sequence satisfies a certain first order necessary condition for optimality which extends to the vector case the well known gradient equal zero condition for real valued minimization finally under some reasonable additional hypotheses we prove global convergence to a weak unconstrained minimizer as a by product we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the so called abstract equilibrium problem

theory and algorithms for constructing discrete morse complexes from grayscale digital images we present an algorithm for determining the morse complex of a two or three dimensional grayscale digital image each cell in the morse complex corresponds to a topological change in the level sets i e a critical point of the grayscale image since more than one critical point may be associated with a single image voxel we model digital images by cubical complexes a new homotopic algorithm is used to construct a discrete morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets we make use of discrete morse theory and simple homotopy theory to prove correctness of this algorithm the resulting morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology

homological and finiteness properties of picture groups picture groups are a class of groups introduced by guba and sapir known examples include thompson s groups f t and v in this paper a large class of picture groups is proved to be of type f a morse theoretic argument shows that for a given picture group the rational homology vanishes in almost all dimensions

the self regulation problem as an inexact steepest descent method for multicriteria optimization abstract in this paper we study an inexact steepest descent method for multicriteria optimization whose step size comes with armijo s rule we show that this method is well defined moreover by assuming the quasi convexity of the multicriteria function we prove full convergence of any generated sequence to a pareto critical point as an application we offer a model for the psychology s self regulation problem using a recent variational rationality approach

co homology of lie algebras via algebraic morse theory abstract the fundamental theorem of cancellation amt 4 and 11 which is the algebraic generalization of discrete morse theory 2 for simplicial complexes and smooth morse theory 10 for differentiable manifolds is discussed in the context of general chain complexes of free modules the chevalley co homology table of a lie algebra is often a tremendous beast using amt we compute the homology of the lie algebra of all triangular matrices sol n over q or z p for large enough primes p we determine the column and row in the table of h k sol n z where the p torsion first appears module h k sol n z p is expressed by the homology of a chain subcomplex for the lie algebra of all strictly triangular matrices nil n using the kunneth formula all conclusions are accompanied by computer experiments then we generalize some results to lie algebras of strictly triangular matrices gl n and gl n with respect to any partial ordering on n we determine the multiplicative structure of h gl n w r t the cup product over fields of zero or sufficiently large characteristic the result being the exterior algebra matchings used here can be analogously defined for other lie algebra families and in other co homology theories we collectively call them normalization matchings they are useful for theoretical as well as computational purposes

a two objective fuzzy k cardinality assignment problem this paper investigates a two objective k cardinality assignment problem as a result a chance constrained goal programming model is constructed for the problem also tabu search algorithm based on fuzzy simulation is designed to solve the problem finally a numerical example is presented to show the application of the algorithm

reeb graphs for shape analysis and applications reeb graphs are compact shape descriptors which convey topological information related to the level sets of a function defined on the shape their definition dates back to 1946 and finds its root in morse theory reeb graphs as shape descriptors have been proposed to solve different problems arising in computer graphics and nowadays they play a fundamental role in the field of computational topology for shape analysis this paper provides an overview of the mathematical properties of reeb graphs and reconstructs its history in the computer graphics context with an eye towards directions of future research

a relaxed projection method for solving multiobjective optimization problems abstract in this paper we propose an algorithm for solving multiobjective minimization problems on nonempty closed convex subsets of the euclidean space the proposed method combines a reflection technique for obtaining a feasible point with a projected subgradient method under suitable assumptions we show that the sequence generated using this method converges to a pareto optimal point of the problem we also present some numerical results

discrete morse theory for manifolds with boundary we introduce a version of discrete morse theory specific for manifolds with boundary the idea is to consider morse functions for which all boundary cells are critical we obtain relative morse inequalities relating the homology of the manifold to the number of interior critical cells we also derive a ball theorem in analogy to forman s sphere theorem the main corollaries of our work are 1 for each d 3 and for each k 0 there is a pl d sphere on which any discrete morse function has more than k critical d 1 cells this solves a problem by chari 2 for fixed d and k there are exponentially many combinatorial types of simplicial d manifolds counted with respect to the number of facets that admit discrete morse functions with at most k critical interior d 1 cells this connects discrete morse theory to enumerative combinatorics discrete quantum gravity 3 the barycentric subdivision of any simplicial constructible d ball is collapsible this almost solves a problem by hachimori 4 every constructible ball collapses onto its boundary minus a facet this improves a result by the author and ziegler 5 any 3 ball with a knotted spanning edge cannot collapse onto its boundary minus a facet this strengthens a classical result by bing and a recent result by the author and ziegler

discrete morse theory and extended l2 homology abstract a brief overview of forman s discrete morse theory is presented from which analogues of the main results of classical morse theory can be derived for discrete morse functions these being functions mapping the set of cells of a cw complex to the real numbers satisfying some combinatorial relations the discrete analogue of the strong morse inequality was proved by forman for finite cw complexes using a witten deformation technique this deformation argument is adapted to provide strong morse inequalities for infinite cw complexes which have a finite cellular domain under the free cellular action of a discrete group the inequalities derived are analogous to the l 2 morse inequalities of novikov and shubin and the asymptotic l 2 morse inequalities of an inexact morse 1 form as derived by mathai and shubin we also obtain quantitative lower bounds for the morse numbers whenever the spectrum of the laplacian contains zero using the extended category of farber

homology representations arising from the half cube abstract we construct a cw decomposition c n of the n dimensional half cube in a manner compatible with its structure as a polytope for each 3 k n the complex c n has a subcomplex c n k which coincides with the clique complex of the half cube graph if k 4 the homology of c n k is concentrated in degree k 1 and furthermore the k 1 st betti number of c n k is equal to the k 2 nd betti number of the complement of the k equal real hyperplane arrangement these betti numbers which also appear in theoretical computer science numerical analysis and engineering are the coefficients of a certain pascal like triangle sloane s sequence a119258 the coxeter groups of type d n act naturally on the complexes c n k and thus on the associated homology groups

homotopy type of the boolean complex of a coxeter system in any coxeter group the set of elements whose principal order ideals are boolean forms a simplicial poset under the bruhat order this simplicial poset defines a cell complex called the boolean complex in this paper it is shown that for any coxeter system of rank n the boolean complex is homotopy equivalent to a wedge of n 1 dimensional spheres the number of such spheres can be computed recursively from the unlabeled coxeter graph and defines a new graph invariant called the boolean number specific calculations of the boolean number are given for all finite and affine irreducible coxeter systems as well as for systems with graphs that are disconnected complete or stars one implication of these results is that the boolean complex is contractible if and only if a generator of the coxeter system is in the center of the group

topology of posets with special partial matchings abstract special partial matchings spms are a generalisation of brenti s special matchings let a pircon be a poset in which every non trivial principal order ideal is finite and admits an spm thus pircons generalise marietti s zircons we prove that every open interval in a pircon is a pl ball or a pl sphere it is then demonstrated that bruhat orders on certain twisted identities and quasiparabolic w sets constitute pircons together these results extend a result of can cherniavsky and twelbeck prove a conjecture of hultman and confirm a claim of rains and vazirani

topology of bounded degree graph complexes abstract the bounded degree graph complexes were first introduced by reiner and roberts j algebraic combin 11 2000 135 154 they arise from the finite free resolution of quadratic veronese rings and modules we prove various results about the homotopy types of these complexes and deduce corresponding characteristic free results about the quadratic veronese resolutions in particular we characterize the set of multidegrees which support at least one higher syzygy in this resolution the answer turns out to be independent of the field characteristic

homology groups for particles on one connected graphs we present a mathematical framework for describing the topology of configuration spaces for particles on one connected graphs in particular we compute the homology groups over integers for different classes of one connected graphs our approach is based on some fundamental combinatorial properties of the configuration spaces mayer vietoris sequences for different parts of configuration spaces and some limited use of discrete morse theory as one of the results we derive the closed form formulae for ranks of the homology groups for indistinguishable particles on tree graphs we also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles our motivation is the search for new kinds of quantum statistics

morse theory from an algebraic viewpoint forman s discrete morse theory is studied from an algebraic viewpoint and we show how this theory can be extended to chain complexes of modules over arbitrary rings as applications we compute the homologies of a certain family of nilpotent lie algebras and show how the algebraic morse theory can be used to derive the classical anick resolution as well as a new two sided anick resolution

newton s method for multiobjective optimization we propose an extension of newton s method for unconstrained multiobjective optimization multicriteria optimization this method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions newton s direction at each iterate is obtained by minimizing the max ordering scalarization of the variations on the quadratic approximations of the objective functions the objective functions are assumed to be twice continuously differentiable and locally strongly convex under these hypotheses the method as in the classical case is locally superlinear convergent to optimal points again as in the scalar case if the second derivatives are lipschitz continuous the rate of convergence is quadratic our convergence analysis uses a kantorovich like technique as a byproduct existence of optima is obtained under semilocal assumptions