yang mills equations on the two dimensional sphere we construct a 1 1 correspondence between the equivalence classes of yang mills fields overs2 and the conjugacy classes of closed geodesics of the structure group furthermore we give an explicit isolation theorem for any yang mills field overs2

some rigorous results for yang mills theories on a cylinder using an approach based on the canonical formalism the yang mills theories on a cylinder are rigorously analyzed in this way the moduli space a g can be explicitly described with a being the space of connections and g the group of gauge transformations in particular a g0 g0 being the group of the pointed gauge transformations is diffeomorphic to the structure group of the theory g whereas a g is g modulo the group of inner automorphisms it is also proven that a g is a principal fiber bundle with structure group g0

regularized continuum yang mills process and feynman kac functional integral giving an ultraviolet regularization and volume cut off we construct a nuclear riemannian structure on the hilbert manifold mathfrak m of gauge orbits this permits us to define a regularized laplace beltrami operator on mathfrak m and an associated global diffusion in mathfrak m governed by this enables us to define via a feynman kac integral a euclidean continuum regularized yang mills process corresponding to a suitable regularization of the kinetic term of the classical yang mills lagrangian ont mathfrak m

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

canonical transformations theory for presymplectic systems we develop a theory of canonical transformations for presymplectic systems reducing this concept to that of canonical transformations for regular coisotropic canonical systems in this way we can also link these with the usual canonical transformations for the symplectic reduced phase space furthermore the concept of a generating function arises in a natural way as well as that of gauge group

on the holonomy of the coulomb connection over manifolds with boundary narasimhan and ramadas commun math phys 67 121 1979 showed that the restricted holonomy group of the coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle s3 su 2 s3 instead of a base manifold s3 we consider here a base manifold of dimension n 2 with a boundary and use dirichlet boundary conditions on connections as defined by marini commun pure appl math 45 1015 1992 a key step in the method of narasimhan and ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy lie algebra with respect to an appropriate sobolev norm our objective is to explore the effect of the presence of a boundary on this construction of the holonomy lie algebra fixing appropriate sobolev norms it will be shown that the space spanned linearly by the curvature form at any one connection is never dense in the holonomy lie algebra in contrast

learning generative models for multi activity body pose estimation we present a method to simultaneously estimate 3d body pose and action categories from monocular video sequences our approach learns a generative model of the relationship of body pose and image appearance using a sparse kernel regressor body poses are modelled on a low dimensional manifold obtained by locally linear embedding dimensionality reduction in addition we learn a prior model of likely body poses and a dynamical model in this pose manifold sparse kernel regressors capture the nonlinearities of this mapping efficiently within a recursive bayesian sampling framework the potentially multimodal posterior probability distributions can then be inferred an activity switching mechanism based on learned transfer functions allows for inference of the performed activity class along with the estimation of body pose and 2d image location of the subject using a rough foreground segmentation we compare binary pca and distance transforms to encode the appearance as a postprocessing step the globally optimal trajectory through the entire sequence is estimated yielding a single pose estimate per frame that is consistent throughout the sequence we evaluate the algorithm on challenging sequences with subjects that are alternating between running and walking movements our experiments show how the dynamical model helps to track through poorly segmented low resolution image sequences where tracking otherwise fails while at the same time reliably classifying the activity type

efficient morphological shape representation mathematical morphology is well suited to capturing geometric information hence morphology based approaches have been popular for object shape representation the two primary morphology based approaches the morphological skeleton and the morphological shape decomposition msd each represent an object as a collection of disjoint sets a practical shape representation scheme though should give a representation that is computationally efficient to use unfortunately little work has been done for the morphological skeleton and the msd to address efficiency we propose a flexible search based shape representation scheme that typically gives more efficient representations than the morphological skeleton and msd our method decomposes an object into a number of simple components based on homothetics of a set of structuring elements to form the representation the components are combined using set union and set difference operations we use three constituent component types and a thorough cost based search strategy to find efficient representations we also consider allowing object representation error which may yield even more efficient representations

regularisable and minimal orbits for group actions in infinite dimensions we introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like yang mills and string theory and which generalise finite dimensional riemannian principal fibre bundles induced by an isometric action we show that the orbits of regularisable bundles have well defined both heat kernel and zeta function regularised volumes we introduce a notion of minimality for these orbits which extend the finite dimensional one our approach uses heat kernel methods and yields both heat kernel obtained via heat kernel regularisation and zeta function obtained via zeta function regularisation minimality for specific values of the parameter for each of these notions we give an infinite dimensional version of hsiang s theorem which extends the finite dimensional case interpreting minimal orbits as orbits with extremal regularised volume

the geometry of the bundle of connections a generalized symplectic structure on the bundle of connections p colon c p rightarrow m of an arbitrary principal g bundle pi colon p rightarrow m is defined by means of a p ast mathrm ad p valued differential 2 form omega 2 on c p which is related to the generalized contact structure on j 1 p the hamiltonian properties of omega 2 are also analyzed

elliptic operators in the functional quantisation for gauge field theories given a gauge theory with gauge groupg acting on a path spacex g andx being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary it is shown that when the action ofg onx is smooth free and proper the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection x x g and the existence of the faddeev popov determinant arising in the functional quantisation of the gauge theory this holds for certain gauge theories with anomalies like bosonic closed string theory in non critical dimension and also holds for a class of gauge theories which includes yang mills theory

the structure of the space of homogeneous yang mills fields the structure of the space of solutions of yang mills equations is examined for solutions that are required to have a specified set of infinitesimal space time symmetries it is shown that when the set consists of killing vector fields which are tangent to a compact spacelike cauchy surface the space is a smooth ilh manifold near each solution that has only trivial gauge symmetries

gauge transformations and fiber bundle theory the relationship between the group of pure gauge transformations of atiyah hitchin and singer proc r soc london ser a 362 425 1978 of a principal fiber bundle and the group g of gauge transformations consisting of the direct product of the local gauge groups on the base space is studied is an invariant subgroup of g and the quotient g is identified with the group of inequivalent gauge transformations in the framework of the category of principal fiber bundles with connections a natural explanation for the relevance of the group in the classical and quantum theories of gauge fields is presented the paper is made self contained by an introductory discussion of the concepts of principal coordinate fiber bundle gauge fixed principal fiber bundle and principal fiber bundle and of the equivalence between the three different versions of the group of vertical automorphisms of the bundle

matching invariant to translations rotations and scale changes abstract an optimization approach to invariant matching is proposed in this approach an object or a pattern is invariantly represented by an object centred description called an attributed relational structure ars embedding invariant properties and relations between the primitives of the pattern such as line segments and points noise effect is taken into account such that a scene can consist of noisy sub parts of a model the matching is then to find the optimal mapping between the arss of the scene and the model a gain functional is formulated to measure the goodness of fit and is to be maximized by using the relaxation labelling method experiments are shown to illustrate the matching algorithm and to demonstrate that the approach is truly invariant to arbitrary translations rotations and scale changes under noise

an image understanding system using attributed symbolic representation and inexact graph matching this paper presents a powerful image understanding system that utilizes a semantic syntactic or attributed synibolic representation scheme in the form of attributed relational graphs arg s for comprehending the global information contents of images nodes in the arg represent the global image features while the relations between those features are represented by attributed branches between their corresponding nodes the extraction of arg representation from images is achieved by a multilayer graph transducer scheme this scheme is basically a rule based system that uses a combination of model driven and data driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial domain representation into a global representation further analysis and inter pretation of the imagery data is performed on the extracted arg representation a distance measure between images is defined in terms of the distance between their respective arg representations the distance between two arg s and the inexact matching of their respective components are calculated by an efficient dynamic programming technique the system handles noise distortion and ambiguity in real world images by two means namely through modeling and embedding them into the transducer s mapping rules as well as through the appropriate cost of error transformation for the inexact matching of the arg image representation two illustrative experiments are presented to demonstrate some capabilities of the proposed system experiment i deals with locating objects in multiobject scenes while experiment ii is concerned with target detection in sar images

quasi elliptic cohomology i abstract quasi elliptic cohomology is a variant of elliptic cohomology theories it is the orbifold k theory of a space of constant loops for global quotient orbifolds it can be expressed in terms of equivariant k theories thus the constructions on it can be made in a neat way this theory reflects the geometric nature of the tate curve in this paper we provide a systematic introduction of its construction and definition

the group of gauge transformations as a schwartz lie group the group of gauge transformations of a smooth principal bundle p m g over a not necessarily compact manifold m and with a not necessarily compact structure group g is proved to be a schwartz lie group its lie algebra and exponential map are discussed

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

the palais smale conditions for the yang mills functional on considere la fonctionnelle de yang mills definie sur des connexions sur un fibre principal sur une variete de riemann compacte de dimension 2 ou 3 on montre que si on considere la fonctionnelle de yang mills comme etant definie sur une variete de hilbert appropriee d orbites des connexions sous l action du groupe des automorphismes du fibre principal alors la fonctionnelle satisfait la condition de palais smale

3 d shape recovery using distributed aspect matching an approach to the recovery of 3 d volumetric primitives from a single 2 d image is presented the approach first takes a set of 3 d volumetric modeling primitives and generates a hierarchical aspect representation based on the projected surfaces of the primitives conditional probabilities capture the ambiguity of mappings between levels of the hierarchy from a region segmentation of the input image the authors present a formulation of the recovery problem based on the grouping of the regions into aspects no domain independent heuristics are used only the probabilities inherent in the aspect hierarchy are exploited once the aspects are recovered the aspect hierarchy is used to infer a set of volumetric primitives and their connectivity as a front end to an object recognition system the approach provides the indexing power of complex 3 d object centered primitives while exploiting the convenience of 2 d viewer centered aspect matching aspects are used to represent a finite vocabulary of 3 d parts from which objects can be constructed

geometrical aspects of operator ordering terms in gauge invariant quantum models finite dimensional quantum models with both boson and fermion degrees of freedom and which have a gauge invariance are studied here as simple versions of gauge invariant quantum field theories the configuration space of these finite dimensional models has the structure of a principal fibre bundle and has defined on it a metric which is invariant under the action of the bundle or gauge group when the gauge dependent degrees of freedom are removed thereby defining the quantum models on the base of the principal fibre bundle extra operator ordering terms arise by making use of dimensional reduction methods in removing the gauge dependence expressions are obtained here for the operator ordering terms which show clearly their dependence on the geometry of the principal fibre bundle structure

on the gribov problem for generalized connections the bundle structure of the space of ashtekar s generalized connections is investigated in the compact case it is proven that every stratum is a locally trivial fibre bundle the only stratum being a principal fibre bundle is the generic stratum its structure group equals the space of all generalized gauge transforms modulo the constant center valued gauge transforms for abelian gauge theories the generic stratum is globally trivial and equals the total space however for a certain class of non abelian gauge theories e g all su n theories the generic stratum is nontrivial this means there are no global gauge fixings the so called gribov problem nevertheless for many physical measures there is a covering of the generic stratum by trivializations each having total measure 1 finally possible physical consequences and the relation between fundamental modular domains and gribov horizons are discussed

yang mills and dirac fields in a bag constraints and reduction the structure of the constraint set in the yang mills dirac theory in a contractible bounded domain is analysed under the bag boundary conditions the gauge symmetry group is identified and it is proved that its action on the phase space is proper and admits slices the reduced phase space is shown to be the union of symplectic manifolds each of which corresponds to a definite mode of symmetry breaking

optimization in model matching and perceptual organization we introduce an optimization approach for solving problems in computer vision that involve multiple levels of abstraction our objective functions include compositional and specialization hierarchies we cast vision problems as inexact graph matching problems formulate graph matching in terms of constrained optimization and use analog neural networks to perform the optimization the method is applicable to perceptual grouping and model matching preliminary experimental results are shown

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

dependent coordinates in path integral measure factorization the transformation of the path integral measure under the reduction procedure in dynamical systems with a symmetry is considered investigation is carried out in the case of the wiener type path integrals that are used for description of the diffusion on a smooth compact riemannian manifold with the given free isometric action of the compact semisimple unimodular lie group the transformation of the path integral which factorizes the path integral measure is based on the application of the optimal nonlinear filtering equation from the stochastic theory the integral relation between the kernels of the original and reduced semigroups is obtained

on the gauge orbit space stratification a review first we review the basic mathematical structures and results concerning the gauge orbit space stratification this includes general properties of the gauge group action fibre bundle structures induced by this action basic properties of the stratification and the natural riemannian structures of the strata in the second part we study the stratification for theories with gauge group su n in spacetime dimension 4 we develop a general method for determining the orbit types and their partial ordering based on the 1 1 correspondence between orbit types and holonomy induced howe subbundles of the underlying principal su n bundle we show that the orbit types are classified by certain cohomology elements of spacetime satisfying two relations and that the partial ordering is characterized by a system of algebraic equations moreover operations for generating direct successors and direct predecessors are formulated which allow one to construct the set of orbit types starting from the principal type finally we discuss an application to nodal configurations in yang mills chern simons theory

recognition of shapes by attributed skeletal graphs in this paper we propose a framework to address the problem of generic 2 d shape recognition the aim is mainly on using the potential strength of skeleton of discrete objects in computer vision and pattern recognition where features of objects are needed for classification we propose to represent the medial axis characteristic points as an attributed skeletal graph to model the shape the information about the object shape and its topology is totally embedded in them and this allows the comparison of different objects by graph matching algorithms the experimental results demonstrate the correctness in detecting its characteristic points and in computing a more regular and effective representation for a perceptual indexing the matching process based on a revised graduated assignment algorithm has produced encouraging results showing the potential of the developed method in a variety of computer vision and pattern recognition domains the results demonstrate its robustness in the presence of scale reflection and rotation transformations and prove the ability to handle noise and occlusions

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

the hamiltonian structure of yang mills theories and instantons ii abstract the formalism of constraints reviewed in paper i is applied to yang mills theory to determine the physical phase space this turns out to be the cotangent bundle of orbit space the space of gauge inequivalent potentials self dual configurations are not hamiltonian with respect to the symplectic structure inherited from the general system

the evaluation map in field theory sigma models and strings ii in this paper we examine specifically the role of the evaluation map in sigma models and strings we discuss the difference between sigma models and field theory as far as anomaly cancellation is concerned the introduction of the wess zumino terms in different sigma models is considered anomalies in string theory are discussed with special attention to the conformal anomalies and to the sigma model anomalies for the imbedded or immersed world sheet of the string conformal anomalies in two dimensions are connected to holomorphic and gravitational anomalies in order to have the cancellation of the sigma model anomalies of the string certain topological conditions must be satisfied by the ambient manifold the role of the evaluation map in the calculations of global anomalies is also discussed both for field theories and for sigma models in particular global anomalies are connected with the differential characters of cheeger and simons we show that the absence of global anomalies in sigma models is guaranteed by the absence of torsion in suitable homology groups of the target space

the riemannian geometry of the configuration space of gauge theories we state some new results about the configuration space of pure yang mills theory these results come from the study of the kinetic energy term of the lagrangian of the theory this term defines a riemannian metric on the space of non equivalent gauge potentials we develop a riemannian calculus on the configuration space compute the riemannian connection the curvature tensor and solve for the geodesics etc we show that the gribov ambiguity is more than an artefact of the choice of a gauge condition and is related to the existence of conjugate points on the geodesics and is thus an intrinsic feature of the theory

the riemannian geometry of the yang mills moduli space the moduli space of self dual connections over a riemannian 4 manifold has a natural riemannian metric inherited from thel 2 metric on the space of connections we give a formula for the curvature of this metric in terms of the relevant green operators we then examine in great detail the moduli space 1 ofk 1 instantons on the 4 sphere and obtain an explicit formula for the metric in this case in particular we prove that 1 is rotationally symmetric and has finite geometry it is an incomplete 5 manifold with finite diameter and finite volume

flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes curve matching is one instance of the fundamental correspondence problem our flexible algorithm is designed to match curves under substantial deformations and arbitrary large scaling and rigid transformations a syntactic representation is constructed for both curves and an edit transformation which maps one curve to the other is found using dynamic programming we present extensive experiments where we apply the algorithm to silhouette matching in these experiments we examine partial occlusion viewpoint variation articulation and class matching where silhouettes of similar objects are matched based on the qualitative syntactic matching we define a dissimilarity measure and we compute it for every pair of images in a database of 121 images we use this experiment to objectively evaluate our algorithm first we compare our results to those reported by others second we use the dissimilarity values in order to organize the image database into shape categories the veridical hierarchical organization stands as evidence to the quality of our matching and similarity estimation

some remarks on brs transformations anomalies and the cohomology of the lie algebra of the group of gauge transformations we show that ghosts in gauge theories can be interpreted as maurer cartan forms in the infinite dimensional group of gauge transformations we examine the cohomology of the lie algebra of and identify the coboundary operator with the brs operator we describe the anomalous terms encountered in the renormalization of gauge theories triangle anomalies as elements of these cohomology groups

cohomology classes associated to anomalies one of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle g p m it is a bigraded algebra with two differentials which is invariantly defined i e independent of local coordinates of m i denote it by s m briefly s m p q consists of local p multilinear operators from the gauge algebra into q forms on m which depend on a connection a in a local manner local means that the operators decrease supports the gauge algebra is the lie algebra of the gauge group which consists of diffeomorphisms of p that respect the action of g and cover the identity diffeomorphism of m

non representability of cohomology classes by bi invariant forms gauge and kac moody groups we give a necessary topological condition on a cohomology class of any lie group modelled on a frechet space to be representable by a bi invariant form on as a corollary we show that if for somed 0 then there exists a cohomology class in which cannot be represented by any bi invariant form in particular we conclude that there are many cohomology generators in general in the case of gauge groups and also kac moody groups which cannot be represented by bi invariant forms although very often they are representable by left invariant forms

the geometrical interpretation of the faddeev popov determinant in polyakov s theory of random surfaces in the context of polyakov s theory the geometrical approach in which the functional measure is defined as a formal volume form is developed the special version of the fubini theorem on manifolds is derived which provides the geometrical interpretation of the faddeev popov method the conformal gauge is studied within this framework as a result the explicit form of functional measure in effective liouville quantum field theory is obtained

a poincare lemma for connection forms denote by p the space of piecewise smooth curves in rn beginning at the origin a path 2 form is a function h on p such that for each element in p h is a 2 form at the endpoint of with values in a lie algebra g for example if a is a smooth g valued connection form on rn with curvature f and parallel translation operator p then the equation la p 1 f 1 p defines la as a path 2 form a necessary and sufficient condition is given to characterize those path 2 forms which arise in this way by way of application it is shown that the birula mandelstam generalization of maxwell s equations to nonabelian gauge fields is equivalent to the yang mills equation