topological bf theories in 3 and 4 dimensions in this paper we discuss topological bf theories in 3 and 4 dimensions observables are associated to ordinary knots and links in 3 dimensions and to 2 knots in 4 dimensions the vacuum expectation values of such observables give a wide range of invariants here we consider mainly the 3 dimensional case where these invariants include alexander polynomials homfly polynomials and kontsevich integrals

combinatorial quantization of the hamiltonian chern simons theory ii motivated by a recent paper of fock and rosly 6 we describe a mathematically precise quantization of the hamiltonian chern simons theory we introduce the chern simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly the lattice model enjoys the symmetry with respect to a quantum gauge group using this fact we construct the algebra of observables of the hamiltonian chern simons theory equipped with a operation and a positive inner product

shape from focus the shape from focus method presented here uses different focus levels to obtain a sequence of object images the sum modified laplacian sml operator is developed to provide local measures of the quality of image focus the operator is applied to the image sequence to determine a set of focus measures at each image point a depth estimation algorithm interpolates a small number of focus measure values to obtain accurate depth estimates a fully automated shape from focus system has been implemented using an optical microscope and tested on a variety of industrial samples experimental results are presented that demonstrate the accuracy and robustness of the proposed method these results suggest shape from focus to be an effective approach for a variety of challenging visual inspection tasks

combinatorial formulas for vassiliev invariants from chern simons gauge theory we analyze the perturbative series expansion of the vacuum expectation value of a wilson loop in chern simons gauge theory in the temporal gauge from the analysis emerges the notion of the kernel of a vassiliev invariant the kernel of a vassiliev invariant of order n is not a knot invariant since it depends on the regular knot projection chosen but it differs from a vassiliev invariant by terms that vanish on knots with n singular crossings we conjecture that vassiliev invariants can be reconstructed from their kernels we present the general form of the kernel of a vassiliev invariant and we describe the reconstruction of the full primitive vassiliev invariants at orders two three and four at orders two and three we recover known combinatorial expressions for these invariants at order four we present new combinatorial expressions for the two primitive vassiliev invariants present at this order

wilson surfaces and higher dimensional knot invariants an observable for nonabelian higher dimensional forms is introduced its properties are discussed and its expectation value in bf theory is described this is shown to produce potential and genuine invariants of higher dimensional knots

strassen type functional laws for strong topologies letw denote a wiener process the functional law of the iterated logarithm due to strassen 1964 establishes that the sequence 2nlln 1 2w ns of functions ofs 0 1 is almost surely compact in the uniform topology and gives a simple description of the corresponding limit set in this paper we obtain a general characterization of the topologies under which this statement remains valid

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

some applications of isoperimetric methods to strong limit theorems for sums of independent random variables we develop several applications to almost sure limit theorems for sums of independent vector valued random variables of an isoperimetric inequality due to talagrand a general treatment of the classical laws of large numbers of kolmogorov and prokorov and laws of the iterated logarithm of kolmogorov and hartman and wintner is described new results as well as simpler new proofs of known ones illustrate the usefulness of isoperimetric methods in this context we show further how this approach can be used in the study of limit theorems for trimmed sums of independent and identically distributed random variables 1 introduction and theoretical statement isoperimetric methods already appeared in the study of strong limit theorems for sums of independent vector valued random variables through concentration inequalities mainly of gaussian type these inequalities are commonly used with the tool of randomization and applied conditionally in particular this was the basic idea of the main result of 24 which characterizes banach space valued random variables satisfying the classical hartman wintner strassen law of the iterated logarithm in a recent work 32 a new isoperimetric inequality for product measures was obtained and applied to the integrability of sums of independent vector valued random variables providing at the same time new perspectives in the study of strong limit theorems this inequality was indeed further applied in 25 to get extensions of kolmogorov s law of the iterated logarithm and prokorov s law of large numbers in banach space the purpose of this article is to demonstrate further the usefulness of this approach we study a general statement section 2 on sums of independent banach space valued random

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

wheeled pro p file of batalin vilkovisky formalism using a technique of wheeled props we establish a correspondence between the homotopy theory of unimodular lie 1 bialgebras and the famous batalin vilkovisky formalism solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1 1 correspondence with representations of a wheeled dg prop which on the one hand is isomorphic to the cobar construction of the prop of unimodular lie 1 bialgebras and on the other hand is quasi isomorphic to the dg wheeled prop of unimodular poisson structures these results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular lie 1 bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology it is proven that in the category of quantum bv manifolds associated with the homotopy theory of unimodular lie 1 bialgebras quasi isomorphisms are equivalence relations it is shown that losev mnev s bf theory for unimodular lie algebras can be naturally extended to the case of unimodular lie 1 bialgebras and eventually to the case of unimodular poisson structures using a finite dimensional version of the batalin vilkovisky quantization formalism it is rigorously proven that the feynman integrals computing the effective action of this new bf theory describe precisely homotopy transfer formulae obtained within the wheeled properadic approach to the quantum master equation quantum corrections which are present in our bf model to all orders of the planck constant correspond precisely to what are often called higher massey products in the homological algebra

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

curvature augmented tensor voting for shape inference from noisy 3d data improves the basic tensor voting formalism to infer the sign and direction of principal curvatures at each input site from noisy 3d data unlike most previous approaches no local surface fitting partial derivative computation nor oriented normal vector recovery is performed in our method these approaches are known to be noise sensitive since accurate partial derivative information is often required which is usually unavailable from real data also unlike approaches that detect signs of gaussian curvature we can handle points with zero gaussian curvature uniformly without first localizing them in a separate process the tensor voting curvature estimation is non iterative does not require initialization and is robust to a considerable amount of outlier noise as its effect is reduced by collecting a large number of tensor votes qualitative and quantitative results on synthetic and real complex data are presented

characterization of lil behavior in banach space in a recent paper by the authors a general result characterizing two sided lil behavior for real valued random variables has been established in this paper we look at the corresponding problem in the banach space setting we show that there are analogous results in this more general setting in particularly we provide a necessary and sufficient condition for lil behavior with respect to sequences of the form p nh n where h is from a suitable subclass of the positive nondecreasing slowly varying functions to prove these results we have to use a different method one of our main tools is an improved fuk nagaev type inequality in banach space which should be of independent interest

stereo using monocular cues within the tensor voting framework we address the fundamental problem of matching in two static images the remaining challenges are related to occlusion and lack of texture our approach addresses these difficulties within a perceptual organization framework considering both binocular and monocular cues initially matching candidates for all pixels are generated by a combination of matching techniques the matching candidates are then embedded in disparity space where perceptual organization takes place in 3d neighborhoods and thus does not suffer from problems associated with scanline or image neighborhoods the assumption is that correct matches produce salient coherent surfaces while wrong ones do not matching candidates that are consistent with the surfaces are kept and grouped into smooth layers thus we achieve surface segmentation based on geometric and not photometric properties surface overextensions which are due to occlusion can be corrected by removing matches whose projections are not consistent in color with their neighbors of the same surface in both images finally the projections of the refined surfaces on both images are used to obtain disparity hypotheses for unmatched pixels the final disparities are selected after a second tensor voting stage during which information is propagated from more reliable pixels to less reliable ones we present results on widely used benchmark stereo pairs

old and new link polynomials from the theory of exactly solvable models abstract a general theory is presented to construct link polynomials topological invariants for knots and links from exactly solvable integrable models representations of the braid group and the markov traces on the representations are made through the general theory which is based on fundamental properties of the models various examples leading to alexander jones kauffman and new link polynomials are explicitly shown in a word the soliton theory contains an essence of the knot theory

exactly soluble diffeomorphism invariant theories a class of diffeomorphism invariant theories is described for which the hilbert space of quantum states can be explicitly constructed these theories can be formulated in any dimension and include witten s solution to 2 1 dimensional gravity as a special case higher dimensional generalizations exist which start with an action similar to the einstein action inn dimensions many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories one is a version of yang mills theory in which the only quantum states ons3 r are the vacua finally it is shown that the three dimensional chern simons theory which witten has shown is intimately connected with knot theory arises naturally from a four dimensional topological gauge theory

higher dimensional bf theories in the batalin vilkovisky formalism the bv action and generalized wilson loops this paper analyzes in detail the batalin vilkovisky quantization procedure for bf theories on an n dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables in particular generalized wilson loops for bf theories with additional polynomial b interactions are discussed in any dimensions the paper also contains the explicit proofs to the theorems stated in 16

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

a novel algorithm for estimation of depth map using image focus for 3d shape recovery in the presence of noise three dimensional shape recovery from one or multiple observations is a challenging problem of computer vision in this paper we present a new focus measure for the estimation of a depth map using image focus this depth map can subsequently be used in techniques and algorithms leading to the recovery of a three dimensional structure of the object a requirement of a number of high level vision applications the proposed focus measure has shown robustness in the presence of noise as compared to the earlier focus measures this new focus measure is based on an optical transfer function implemented in the fourier domain the results of the proposed focus measure have shown drastic improvements in estimation of a depth map with respect to the earlier focus measures in the presence of various types of noise including gaussian shot and speckle noises the results of a range of focus measures are compared using root mean square error and correlation metric measures

four dimensional yang mills theory as a deformation of topological bf theory the classical action for pure yang mills gauge theory can be formulated as a deformation of the topological bf theory where beside the two form field b one has to add one extra field given by a one form which transforms as the difference of two connections the ensuing action functional gives a theory that is both classically and quantistically equivalent to the original yang mills theory in order to prove such an equivalence it is shown that the dependency on the field can be gauged away completely this gives rise to a field theory that for this reason can be considered as semi topological or topological in some but not all the fields of the theory the symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections a mathematical analysis of this group action and of the relevant brst complex is discussed in detail

on chern simons and wzw partition functions direct analysis of the path integral reduces partition functions in chern simons theory on a three manifold m with group g to partition functions in a wzw model of maps from a riemann surface to g in particular chern simons theory on s 3 s 1 b 3 and the solid torus correspond respectively to the wzw model of maps from s 2 to g the g g model for and witten s gauged wzw path integral ansatz for chern simons states using maps from s 2 and from the torus to g the reduction hinges on the characterization of the space of connections modulo those gauge transformations which are the identity at a point n as itself a principal fiber bundle with affine linear fiber

algebraic invariants of knots and disordered potts model the authors propose the reformulation of the kauffman bracket invariant of the knot in terms of statistical mechanics of the 2d disordered potts model this allows one to put the question of the determination of knot entropy or of the probability of an arbitrary knot formation in terms of usual statistical mechanics to demonstrate the possibilities of their approach they give a constructive estimation for the trivial knot formation probability for a long strongly contracted closed random path confined in a thin slit they use the mean field approximation for the free energy of the potts system at the point of the transition from the paramagnetic phase to the spin glass one

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

a direct derivation of polynomial invariants from perturbative chern simons gauge theory there have been several methods to show that the expectation values of wilson loop operators in the su n chern simons gauge theory satisfy the homfly skein relation we shall give another method from the perturbative method of the su n chern simons gauge theory in the light cone gauge which is more direct than already known methods

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

topological field theory interpretation of string topology the string bracket introduced by chas and sullivan is reinterpreted from the point of view of topological field theories in the batalin vilkovisky or brst formalisms namely topological action functionals for gauge fields generalizing chern simons and bf theories are considered together with generalized wilson loops the latter generate a poisson or gerstenhaber algebra of functionals with values in the s 1 equivariant cohomology of the loop space of the manifold on which the theory is defined it is proved that in the case of gl n with standard representation the poisson or bv bracket of two generalized wilson loops applied to two cycles is the same as the generalized wilson loop applied to the string bracket of the cycles generalizations to other groups are briefly described

quantization of the lie bialgebra of string topology let m be a smooth simply connected closed oriented manifold and lm the free loop space of m using a poincare duality model for m we show that the reduced equivariant homology of lm has the structure of a lie bialgebra and we construct a hopf algebra which quantizes the lie bialgebra

dimensionality estimation manifold learning and function approximation using tensor voting we address instance based learning from a perceptual organization standpoint and present methods for dimensionality estimation manifold learning and function approximation under our approach manifolds in high dimensional spaces are inferred by estimating geometric relationships among the input instances unlike conventional manifold learning we do not perform dimensionality reduction but instead perform all operations in the original input space for this purpose we employ a novel formulation of tensor voting which allows an n d implementation tensor voting is a perceptual organization framework that has mostly been applied to computer vision problems analyzing the estimated local structure at the inputs we are able to obtain reliable dimensionality estimates at each instance instead of a global estimate for the entire data set moreover these local dimensionality and structure estimates enable us to measure geodesic distances and perform nonlinear interpolation for data sets with varying density outliers perturbation and intersections that cannot be handled by state of the art methods quantitative results on the estimation of local manifold structure using ground truth data are presented in addition we compare our approach with several leading methods for manifold learning at the task of measuring geodesic distances finally we show competitive function approximation results on real data

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

optimal depth estimation by combining focus measures using genetic programming three dimensional 3d shape reconstruction is a fundamental problem in machine vision applications shape from focus sff is one of the passive optical methods for 3d shape recovery that uses degree of focus as a cue to estimate 3d shape in this approach usually a single focus measure operator is applied to measure the focus quality of each pixel in the image sequence however the applicability of a single focus measure is limited to estimate accurately the depth map for diverse type of real objects to address this problem we develop optimal composite depth ocd function through genetic programming gp for accurate depth estimation the ocd function is constructed by optimally combining the primary information extracted using one or more focus measures the genetically developed composite function is then used to compute the optimal depth map of objects the performance of the developed nonlinear function is investigated using both the synthetic and the real world image sequences experimental results demonstrate that the proposed estimator is more useful in computing accurate depth maps as compared to the existing sff methods moreover it is found that the heterogeneous function is more effective than homogeneous function

the cayley hamilton theorem for supermatrices starting from the expression for the superdeterminant of xi m where m is an arbitrary supermatrix we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix some particular cases and examples are discussed

knot theory and the heuristics of functional integration abstract this paper is an exposition of the relationship between the heuristics of witten s functional integral and the theory of knots and links in three dimensional space

on the law of the iterated logarithm for independent banach space valued random variables in this paper we establish some general forms of the law of the iterated logarithm for independent random variables xn with banach space values where xn is not necessarily identically distributed our results include the kolmogorov law of the iterated logarithm lil in both finite and infinite dimensional cases and they improve the wittmann lil as well as extend it to the vector setting the ledoux talagrand lil for an i i d sequence is also a simple corollary of our results

characterization of the law of the iterated logarithm in banach spaces on utilise une technique de randomisation gaussienne pour demontrer qu une variable aleatoire a valeurs dans un espace de banach satisfait la loi du logarithme itere sous une certaine condition necessaire et suffisante

link invariants for flows in higher dimensions linking numbers in higher dimensions and their generalization including gauge fields are studied in the context of bf theories the linking numbers associated with n manifolds with smooth flows generated by divergence free p vector fields endowed with an invariant flow measure are computed in the context of quantum field theory they constitute invariants of smooth dynamical systems for nonsingular flows and generalize previous proposals of invariants in particular they generalize arnold s asymptotic hopf invariant from three to higher dimensions this invariant is generalized by coupling with a non abelian gauge flat connection with nontrivial holonomy the computation of the asymptotic jones witten invariants for flows is naturally extended to dimension n 2p 1 finally we give a possible interpretation and implementation of these issues in the context of 11 dimensional supergravity and string theory

classical bv theories on manifolds with boundary in this paper we extend the classical bv framework to gauge theories on spacetime manifolds with boundary in particular we connect the bv construction in the bulk with the bfv construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners we present several examples including electrodynamics yang mills theory and topological field theories coming from the aksz construction in particular the chern simons theory the bf theory and the poisson sigma model this paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing

kontsevich integral for vassiliev invariants from chern simons perturbation theory in the light cone gauge we analyze the structure of the perturbative series expansion of chern simons gauge theory in the light cone gauge after introducing a regularization prescription that entails the consideration of framed knots we present the general form of the vacuum expectation value of a wilson loop the resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges it also contains the kontsevich integral for vassiliev invariants of framed knots

un critere sur les petites boules dans le theoreme limite central we show that a banach space valued random variablex such that mathop lim limits x to infty t 2 mathbb p left left x right t right 0 satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled mathop lim limits x to infty t 2 mathbb p left left x right t right 0

almost sure behaviour of f valued random fields letx n n d be a field of independent random variables taking values in a semi normed measurable vector spacef for a broad class of fields n d of positive numbers the almost sure behaviour of k nxk n n d is studied the main result allows us to deduce some new and well known theorems for fields of independentf random variables from related results for fields of independent real random variables

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

an almost sure invariance principle for stationary ergodic sequences of banach space valued random variables in this paper we establish an almost sure invariance principle with an error termo t log logt 1 2 ast for partial sums of stationary ergodic martingale difference sequences taking values in a real separable banach space as partial sums of weakly dependent random variables can often be well approximated by martingales this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as o mixing and mixing sequences and functionals of such sequences compared with previous related work for vector valued random variables starting with an article by kuelbs and philipp 27 the present approach leads to a unification of the theory at least for stationary sequences moment conditions required by earlier authors are relaxed only second order weak moments are needed and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself

integral invariants of 3 manifolds ii this note is a sequel to our earlier paper of the same title 4 and describes invariants of rational homology 3 spheres associated to acyclic orthogonal local systems our work is in the spirit of the axelrod singer papers 1 generalizes some of their results and furnishes a new setting for the purely topological implications of their work

moore and seiberg s equations and 3d topological field theory we prove that any solution to moore and seiberg s equations defines a projective rational topological field theory prtft using surgery presentations for decorated three manifolds and kirby s calculus

loop and path spaces and four dimensional bf theories connections holonomies and observables we study the differential geometry of principal g bundles whose base space is the space of free paths loops on a manifold m in particular we consider connections defined in terms of pairs a b where a is a connection for a fixed principal bundle p m g and b is a 2 form on m the relevant curvatures parallel transports and holonomies are computed and their expressions in local coordinates are exhibited when the 2 form b is given by the curvature of a then the so called non abelian stokes formula follows

simultaneous two view epipolar geometry estimation and motion segmentation by 4d tensor voting we address the problem of simultaneous two view epipolar geometry estimation and motion segmentation from nonstatic scenes given a set of noisy image pairs containing matches of n objects we propose an unconventional efficient and robust method 4d tensor voting for estimating the unknown n epipolar geometries and segmenting the static and motion matching pairs into n independent motions by considering the 4d isotropic and orthogonal joint image space only two tensor voting passes are needed and a very high noise to signal ratio up to five can be tolerated epipolar geometries corresponding to multiple rigid motions are extracted in succession only two uncalibrated frames are needed and no simplifying assumption such as affine camera model or homographic model between images other than the pin hole camera model is made our novel approach consists of propagating a local geometric smoothness constraint in the 4d joint image space followed by global consistency enforcement for extracting the fundamental matrices corresponding to independent motions we have performed extensive experiments to compare our method with some representative algorithms to show that better performance on nonstatic scenes are achieved results on challenging data sets are presented

on rates of convergence for sample average approximations in the almost sure sense and in mean we study the rates at which optimal estimators in the sample average approximation approach converge to their deterministic counterparts in the almost sure sense and in mean to be able to quantify these rates we consider the law of the iterated logarithm in a banach space setting and first establish under relatively mild assumptions almost sure convergence rates for the approximating objective functions which can then be transferred to the estimators for optimal values and solutions of the approximated problem by exploiting a characterisation of the law of the iterated logarithm in banach spaces we are further able to derive under the same assumptions that the estimators also converge in mean at a rate which essentially coincides with the one in the almost sure sense this in turn allows to quantify the asymptotic bias of optimal estimators as well as to draw conclusive insights on their mean squared error and on the estimators for the optimality gap finally we address the notion of convergence in probability to derive rates in probability for the deviation of optimal estimators and weak rates of error probabilities without imposing strong conditions on exponential moments we discuss the possibility to construct confidence sets for the optimal values and solutions from our obtained results and provide a numerical illustration of the most relevant findings

2 d physics and 3 d topology invariants of three dimensional manifolds and of framed oriented labeled links in them are rigorously defined using any solution to the moore seiberg axioms for a rational conformal field theory these invariants are generalizations of witten s chern simons path integrals connections are explored with supersymmetry four dimensional manifolds and quantum gravity

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

shape from focus by tensor voting in this correspondence we address the task of recovering shape from focus sff as a perceptual organization problem in 3 d using tensor voting depth hypotheses from different focus operators are validated based on their likelihood to be part of a coherent 3 d surface thereby exploiting scene geometry and focus information to generate reliable depth estimates the proposed method is fast and yields significantly better results compared with existing sff methods

a relationship between the jones and kauffman polynomials a simple relationship is presented between the kauffman polynomial of a framed link l and the jones polynomial of a derived link l the link is l obtained by splitting each component of l into two parallel strands using the framing to determine the linking number of the strands the relation is checked in several nontrivial examples and a proof of the general result is given

abelian bf theories and knot invariants in the context of the batalin vilkovisky formalism a new observable for the abelian bf theory is proposed whose vacuum expectation value is related to the alexander conway polynomial the three dimensional case is analyzed explicitly and it is proved to be anomaly free moreover at the second order in perturbation theory a new formula for the second coefficient of the alexander conway polynomial is obtained an account on the higher dimensional generalizations is also given

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

inferring segmented dense motion layers using 5d tensor voting we present a novel local spatiotemporal approach to produce motion segmentation and dense temporal trajectories from an image sequence a common representation of image sequences is a 3d spatiotemporal volume x y t and its corresponding mathematical formalism is the fiber bundle however directly enforcing the spatiotemporal smoothness constraint is difficult in the fiber bundle representation thus we convert the representation into a new 5d space x y t vx vy with an additional velocity domain where each moving object produces a separate 3d smooth layer the smoothness constraint is now enforced by extracting 3d layers using the tensor voting framework in a single step that solves both correspondence and segmentation simultaneously motion segmentation is achieved by identifying those layers and the dense temporal trajectories are obtained by converting the layers back into the fiber bundle representation we proceed to address three applications tracking mosaic and 3d reconstruction that are hard to solve from the video stream directly because of the segmentation and dense matching steps but become straightforward with our framework the approach does not make restrictive assumptions about the observed scene or camera motion and is therefore generally applicable we present results on a number of data sets

chern simons theory on r3 in axial gauge a rigorous approach we study pure chern simons models on m r3 using a functional integral quantization approach which is based on axial gauge fixing it is well known see e g comm math phys 126 1989 167 comm math phys 186 1997 563 that in axial gauge the chern simons action function is quadratic and that the faddeev popov determinant of this gauge fixing procedure is a constant function this means that the wilson loop observables wlos of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of gaussian type we demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from white noise analysis with certain regularization techniques like loop smearing and framing for the special case g u 1 g being the group of the model we carry out the details of this approach which is also applicable to the case of non abelian g and find well known linking number expressions for the wlos