zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

from weak to strong coupling in abjm theory the partition function of mathcal n 6 supersymmetric chern simons matter theory known as abjm theory on mathbb s 3 as well as certain wilson loop observables are captured by a zero dimensional super matrix model this super matrix model is closely related to a matrix model describing topological chern simons theory on a lens space we explore further these recent observations and extract more exact results in abjm theory from the matrix model in particular we calculate the planar free energy which matches at strong coupling the classical iia supergravity action on rm ads 4 times mathbb c mathbb p 3 and gives the correct n3 2 scaling for the number of degrees of freedom of the m2 brane theory furthermore we find contributions coming from world sheet instanton corrections in mathbb c mathbb p 3 we also calculate non planar corrections both to the free energy and to the wilson loop expectation values this matrix model appears also in the study of topological strings on a toric calabi yau manifold and an intriguing connection arises between the space of couplings of the planar abjm theory and the moduli space of this calabi yau in particular it suggests that in addition to the usual perturbative and strong coupling ads expansions a third natural expansion locus is the line where one of the two t hooft couplings vanishes and the other is finite this is the conifold locus of the calabi yau and leads to an expansion around topological chern simons theory we present some explicit results for the partition function and wilson loop observables around this locus

multi instantons and multicuts we discuss various aspects of multi instanton configurations in generic multicut matrix models explicit formulas are presented in the two cut case and in particular we obtain general formulas for multi instanton amplitudes in the one cut matrix model case as a degeneration of the two cut case these formulas show that the instanton gas is ultradilute due to the repulsion among the matrix model eigenvalues we exemplify and test our general results in the cubic matrix model where multi instanton amplitudes can be also computed with orthogonal polynomials as an application we derive general expressions for multi instanton contributions in two dimensional quantum gravity verifying them by computing the instanton corrections to the string equation the resulting amplitudes can be interpreted as regularized partition functions for multiple zz branes which take into full account their backreaction on the target geometry finally we also derive structural properties of the trans series solution to the painleve i equation

superstrings and topological strings at large n we embed the large n chern simons topological string duality in ordinary superstrings this corresponds to a large n duality between generalized gauge systems with n 1 supersymmetry in four dimensions and superstrings propagating on noncompact calabi yau manifolds with certain fluxes turned on we also show that in a particular limit of the n 1 gauge theory system certain superpotential terms in the n 1 system including deformations if spacetime is noncommutative are captured to all orders in 1 n by the amplitudes of noncritical bosonic strings propagating on a circle with self dual radius we also consider d brane anti d brane system wrapped over vanishing cycles of compact calabi yau manifolds and argue that at large n they induce a shift in the background to a topologically distinct calabi yau which we identify as the ground state system of the brane anti brane system

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

duality of orthogonal and symplectic matrix integrals and quaternionic feynman graphs we present an asymptotic expansion for quaternionic self adjoint matrix integrals the feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non orientable counterparts we show that the 2n 2n gaussian orthogonal ensemble goe and n n gaussian symplectic ensemble gse have exactly the same expansion term by term except that the contributions from graphs on a non orientable surface with odd euler characteristic carry the opposite sign as an application we give a new topological proof of the known duality for correlations of characteristic polynomials demonstrating that this duality is equivalent to poincare duality of graphs drawn on a compact surface another consequence of our graphical expansion formula is a simple and simultaneous re derivation of the central limit theorem for goe gue gaussian unitary ensemble and gse the three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

support and approximation properties of hermite splines abstract in this paper we formally investigate two mathematical aspects of hermite splines that are relevant to practical applications we first demonstrate that hermite splines are maximally localized in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties then we precisely quantify the approximation power of hermite splines for the reconstruction of functions and their derivatives it is known that the hermite and b spline approximation schemes have the same approximation order more precisely their approximation error vanishes as o t 4 when the step size t goes to zero in this work we show that they actually have the same asymptotic approximation error constants too therefore they have identical asymptotic approximation properties hermite splines combine optimal localization and excellent approximation power while retaining interpolation properties and closed form expression in contrast to existing similar functions these findings shed a new light on the convenience of hermite splines in the context of computer graphics and geometrical design

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

summing graphs for random band matrices a method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices the topological classification of feynman diagrams which was actively used in last years for matrix model regularization of 2d gravity turns out to be very useful for band matrices the critical behavior at the edge of spectrum and the asymptotics of energy level correlation function are considered this correlation function together with the hypothesis about universality of spectral correlations allows to estimate easily the localization length for eigen vectors a smoothed two point correlation function of local density of states overline rho e 1 i rho e 2 j c as well as the energy level correlation for finite size band matrices are also found as d dimensional generalization of band matrices lattice hamiltonians with long range random hopping are considered as well

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

a geometric parametrization for the virtual euler characteristics of the moduli spaces of real and complex algebraic curves we determine an expression 9 1x for the virtual euler characteristics of the moduli spaces of s pointed real y 1 2 and complex y 1 algebraic curves in particular for the space of real curves of genus g with a fixed point free involution we find that the euler characteristic is 2 s 1 1 29 1 g s 2 bg g where bg is the gth bernoulli number this complements the result of harer and zagier that the euler characteristic of the moduli space of complex algebraic curves is 1 s g s 2 bg g 1 g 1 the proof uses strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells the approach involves a parameter y that permits specialization of the formula to the real and complex cases this suggests that 4 y itself may describe the euler characteristics of some related moduli spaces although we do not yet know what these spaces might be

the refined bps index from stable pair invariants a refinement of the stable pair invariants of pandharipande and thomas for non compact calabi yau spaces is introduced based on a virtual bialynicki birula decomposition with respect to a mathbb c action on the stable pair moduli space or alternatively the equivariant index of nekrasov and okounkov this effectively calculates the refined index for m theory reduced on these calabi yau geometries based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of morrison mozgovoy nagao and szendroi for local mathbb p 1 we explicitly compute refined invariants in low degree for local mathbb p 2 and local mathbb p 1 times mathbb p 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula the modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined pt invariants on appropriate geometries to nekrasov s partition function and a refinement of chern simons theory on a lens space we also relate our product formula to wall crossing

region based strategies for active contour models the variational method has been introduced by kass et al 1987 in the field of object contour modeling as an alternative to the more traditional edge detection edge thinning edge sorting sequence since the method is based on a pre processing of the image to yield an edge map it shares the limitations of the edge detectors it uses in this paper we propose a modified variational scheme for contour modeling which uses no edge detection step but local computations instead only around contour neighborhoods as well as an anticipating strategy that enhances the modeling activity of deformable contour curves many of the concepts used were originally introduced to study the local structure of discontinuity in a theoretical and formal statement by leclerc zucker 1987 but never in a practical situation such as this one the first part of the paper introduces a region based energy criterion for active contours and gives an examination of its implications as compared to the gradient edge map energy of snakes then a simplified optimization scheme is presented accounting for internal and external energy in separate steps this leads to a complete treatment which is described in the last sections of the paper 4 and 5 the optimization technique used here is mostly heuristic and is thus presented without a formal proof but is believed to fill a gap between snakes and other useful image representations such as split and merge regions or mixed line labels image fields

topological open strings on orbifolds we use the remodeling approach to the b model topological string in terms of recursion relations to study open string amplitudes at orbifold points to this end we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest we exemplify this procedure for the mathbb c 3 mathbb z 3 orbifold point of local mathbb p 2 where we present results for topological string amplitudes for genus zero and up to three holes and for the one holed torus these amplitudes can be understood as generating functions for either open orbifold gromov witten invariants of mathbb c 3 mathbb z 3 or correlation functions in the orbifold cft involving insertions of both bulk and boundary operators

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

topological strings and almost modular forms the b model topological string theory on a calabi yau threefold x has a symmetry group generated by monodromies of the periods of x this acts on the topological string wave function in a natural way governed by the quantum mechanics of the phase space h3 x we show that depending on the choice of polarization the genus g topological string amplitude is either a holomorphic quasi modular form or an almost holomorphic modular form of weight 0 under moreover at each genus certain combinations of genus g amplitudes are both modular and holomorphic we illustrate this for the local calabi yau manifolds giving rise to seiberg witten gauge theories in four dimensions and local ip2 and ip1 ip1 as a byproduct we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space which we use to give predictions for gromov witten invariants of the orbifold mathbb c 3 mathbb z 3

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

quantum hitchin systems via beta deformed matrix models we study the quantization of hitchin systems in terms of beta deformations of generalized matrix models related to conformal blocks of liouville theory on punctured riemann surfaces we show that in a suitable limit corresponding to the nekrasov shatashvili one the loop equations of the matrix model reproduce the hamiltonians of the quantum hitchin system on the sphere and the torus with marked points the eigenvalues of these hamiltonians are shown to be the epsilon 1 deformation of the chiral observables of the corresponding mathcal n 2 four dimensional gauge theory moreover we find the exact wave functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions

extreme multiplicity in cylindrical rayleigh benard convection ii bifurcation diagram and symmetry classification a large number of flows with distinctive patterns have been observed in experiments and simulations of rayleigh benard convection in a water filled cylinder whose radius is twice the height we have adapted a time dependent pseudospectral code first to carry out newton s method and branch continuation and second to carry out the exponential power method and arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states the resulting bifurcation diagram represents a compromise between the tendency in the bulk toward parallel rolls and the requirement imposed by the boundary conditions that primary bifurcations be toward states whose azimuthal dependence is trigonometric the diagram contains 17 branches of stable and unstable steady states these can be classified geometrically as roll states containing two three and four rolls axisymmetric patterns with one or two tori threefold symmetric patterns called mercedes mitsubishi marigold and cloverleaf trigonometric patterns called dipole and pizza and less symmetric patterns called co and asymmetric three rolls the convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points in order to better understand this complicated bifurcation diagram we have partitioned it according to azimuthal symmetry we have been able to determine the bifurcation theoretic origin from the conductive state of all the branches observed at high rayleigh number

extreme multiplicity in cylindrical rayleigh benard convection i time dependence and oscillations rayleigh benard convection in a cylindrical container can take on many different spatial forms motivated by the results of hof phys fluids 11 2815 1999 who observed coexistence of several stable states at a single set of parameter values we have carried out simulations at the same prandtl number that of water and a radius to height aspect ratio of two we have used two kinds of thermal boundary conditions perfectly insulating sidewalls and perfectly conducting sidewalls in both cases we obtain a wide variety of coexisting steady and time dependent flows

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

characteristic polynomials of real symmetric random matrices the correlation functions of the random variables det x in which x is an hermitian n n random matrix are known to exhibit universal local statistics in the large n limit we study here the correlation of those same random variables for real symmetric matrices goe the derivation relies on an exact dual representation of the problem the k point functions are expressed in terms of finite integrals over quaternionic k k matrices however the control of the dyson limit in which the distance of the various parameters s is of the order of the mean spacing requires an integration over the symplectic group it is shown that a generalization of the itzykson zuber method holds for this problem but contrary to the unitary case the semi classical result requires a finite number of corrections to be exact we have also considered the problem of an external matrix source coupled to the random matrix and obtain explicit integral formulae which are useful for the analysis of the large n limit

multiplicity of steady states in cylindrical rayleigh benard convection three dimensional steady rayleigh b enard convection in a vertical cylinder is investigated by numerical simulation and bifurcation analysis the complex pattern formation beyond the onset of the convection is presented by a bifurcation diagram the coexistence of multiple stable states is observed near the threshold of the first bifurcation and two group symmetries are summarized for the corresponding primary branches the first stable target pattern originates through a subcritical bifurcation a multiplicity of flow states for the rayleigh number of 14 phantom rule 0 2em 0ex 200 is validated numerically in comparison with the experiment and a four spoke pattern is observed

snakes with an ellipse reproducing property we present a new class of continuously defined parametric snakes using a special kind of exponential splines as basis functions we have enforced our bases to have the shortest possible support subject to some design constraints to maximize efficiency while the resulting snakes are versatile enough to provide a good approximation of any closed curve in the plane their most important feature is the fact that they admit ellipses within their span thus they can perfectly generate circular and elliptical shapes these features are appropriate to delineate cross sections of cylindrical like conduits and to outline bloblike objects we address the implementation details and illustrate the capabilities of our snake with synthetic and real data

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

the uses of the refined matrix model recursion we study matrix models in the ensemble by building on the refined recursion relation proposed by chekhov and eynard we present explicit results for the first deformed corrections in the one cut and the two cut cases as well as two applications to supersymmetric gauge theories the calculation of superpotentials in n 1 gauge theories and the calculation of vevs of surface operators in superconformal n 2 theories and their liouville duals finally we study the deformation of the chern simons matrix model our results indicate that this model does not provide an appropriate description of the deformed topological string on the resolved conifold and therefore that the deformation might provide a different generalization of topological string theory in toric calabi yau backgrounds

sampling of planar curves theory and fast algorithms we introduce a continuous domain framework for the recovery of a planar curve from a few samples we model the curve as the zero level set of a trigonometric polynomial we show that the exponential feature maps of the points on the curve lie on a low dimensional subspace we show that the null space vector of the feature matrix can be used to uniquely identify the curve given a sufficient number of samples the worst case theoretical guarantees show that the number of samples required for unique recovery depends on the bandwidth of the underlying trigonometric polynomial which is a measure of the complexity of the curve we introduce an iterative algorithm that relies on the low rank property of the feature maps to recover the curves when the samples are noisy or when the true bandwidth of the curve is unknown we also demonstrate the preliminary utility of the proposed curve representation in the context of image segmentation

active contour external force using vector field convolution for image segmentation snakes or active contours have been widely used in image processing applications typical roadblocks to consistent performance include limited capture range noise sensitivity and poor convergence to concavities this paper proposes a new external force for active contours called vector field convolution vfc to address these problems vfc is calculated by convolving the edge map generated from the image with the user defined vector field kernel we propose two structures for the magnitude function of the vector field kernel and we provide an analytical method to estimate the parameter of the magnitude function mixed vfc is introduced to alleviate the possible leakage problem caused by choosing inappropriate parameters we also demonstrate that the standard external force and the gradient vector flow gvf external force are special cases of vfc in certain scenarios examples and comparisons with gvf are presented in this paper to show the advantages of this innovation including superior noise robustness reduced computational cost and the flexibility of tailoring the force field

oscillatory matrix model in chern simons theory and jacobi theta determinantal point process the partition function of the chern simons theory on the three sphere with the unitary group u n provides a one matrix model the corresponding n particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the stieltjes wigert orthogonal polynomials the matrix model and the point process are regarded as q extensions of the random matrix model in the gaussian unitary ensemble and its eigenvalue point process respectively we prove the convergence of the n particle system to an infinite dimensional determinantal point process in n in which the correlation kernel is expressed by jacobi s theta functions we show that the matrix model obtained by this limit realizes the oscillatory matrix model in chern simons theory discussed by de haro and tierz

matrix integration and combinatorics of modular groups integration over gaussian matrix ensembles was used to obtain the virtual euler characteristics of mapping class groups we present some simplifications in the combinatorial part of the calculation

regularized laplacian zero crossings as optimal edge integrators we view the fundamental edge integration problem for object segmentation in a geometric variational framework first we show that the classical zero crossings of the image laplacian edge detector as suggested by marr and hildreth inherently provides optimal edge integration with regard to a very natural geometric functional this functional accumulates the inner product between the normal to the edge and the gray level image gradient along the edge we use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models we also incorporate a 2d geometric variational explanation to the haralick edge detector into the geometric active contour framework

hermite snakes with control of tangents we introduce a new model of parametric contours defined in a continuous fashion our curve model relies on hermite spline interpolation and can easily generate curves with sharp discontinuities it also grants direct access to the tangent at each location with these two features the hermite snake distinguishes itself from classical spline snake models and allows one to address certain bioimaging problems in a more efficient way more precisely the hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image using the formalism of spline theory the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties finally the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our hermite snake model and its usefulness is illustrated on real biological images acquired using brightfield phase contrast differential interference contrast and scanning electron microscopy

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

explicit multi matrix topological expansion for quaternionic random matrices we present an explicit formula for the expected value of a product of several independent symplectically invariant matrices in which the trace and real part function may be applied possibly to different subexpressions this takes the form of a topological expansion however each term has two topologies one for the trace and another for the real part the traces and real parts can always be written in terms of index contraction but in some cases it is possible to write the expression as a product in which the two functions are applied to bracketed intervals in a legal bracket diagram we present the conditions under which this may be done and an algorithm to construct such an expression given the contracted indices when possible the summands in the topological expansion are written in terms of matrix cumulants we compute the matrix cumulants of quaternionic ginibre gaussian symplectic quaternionic wishart and haar distributed symplectic matrices which allow direct computation of an expression constructed from several independent ensembles of any of these matrices

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

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

ellipse preserving hermite interpolation and subdivision we introduce a family of piecewise exponential functions that have the hermite interpolation property our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses we prove that the proposed hermite functions form a riesz basis and that they reproduce prescribed exponential polynomials we present a method based on green s functions to unravel their multi resolution and approximation theoretic properties finally we derive the corresponding vector and scalar subdivision schemes which lend themselves to a fast implementation the proposed vector scheme is interpolatory and level dependent but its asymptotic behavioris the same as the classical cubic hermite spline algorithm the same convergence properties i e fourth order of approximation are hence ensured

wall crossing as seen by matrix models the number of bps bound states of d branes on a calabi yau manifold depends on two sets of data the bps charges and the stability conditions for d0 and d2 branes bound to a single d6 brane wrapping a calabi yau 3 fold x both are naturally related to the kahler moduli space mathcal m x we construct unitary one matrix models which count such bps states for a class of toric calabi yau manifolds at infinite t hooft coupling the matrix model for the bps counting on x turns out to give the topological string partition function for another calabi yau manifold y whose kahler moduli space mathcal m y contains two copies of mathcal m x one related to the bps charges and another to the stability conditions the two sets of data are unified in mathcal m y the matrix models have a number of other interesting features they compute spectral curves and mirror maps relevant to the remodeling conjecture for finite t hooft coupling they give rise to yet more general geometry widetilde y containing y

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

global minimum for active contour models a minimal path approach a new boundary detection approach for shape modeling is presented it detects the global minimum of an active contour model s energy between two end points initialization is made easier and the curve is not trapped at a local minimum by spurious edges we modify the snake energy by including the internal regularization term in the external potential term our method is based on finding a path of minimal length in a riemannian metric we then make use of a new efficient numerical method to find this shortest path it is shown that the proposed energy though based only on a potential integrated along the curve imposes a regularization effect like snakes we explore the relation between the maximum curvature along the resulting contour and the potential generated from the image the method is capable to close contours given only one point on the objects boundary by using a topology based saddle search routine we show examples of our method applied to real aerial and medical images

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

unsupervised contour representation and estimation using b splines and a minimum description length criterion this paper describes a new approach to adaptive estimation of parametric deformable contours based on b spline representations the problem is formulated in a statistical framework with the likelihood function being derived from a region based image model the parameters of the image model the contour parameters and the b spline parameterization order i e the number of control points are all considered unknown the parameterization order is estimated via a minimum description length mdl type criterion a deterministic iterative algorithm is developed to implement the derived contour estimation criterion the result is an unsupervised parametric deformable contour it adapts its degree of smoothness complexity number of control points and it also estimates the observation image model parameters the experiments reported in the paper performed on synthetic and real medical images confirm the adequate and good performance of the approach

finsler active contours in this paper we propose an image segmentation technique based on augmenting the conformal or geodesic active contour framework with directional information in the isotropic case the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest typically edges is small the conformal factor that is chosen depends only upon position and is in this sense isotropic although directional information has been studied previously for other segmentation frameworks here we show that if one desires to add directionality in the conformal active contour framework then one gets a well defined minimization problem in the case that the factor defines a finsler metric optimal curves may be obtained using the calculus of variations or dynamic programming based schemes finally we demonstrate the technique by extracting roads from aerial imagery blood vessels from medical angiograms and neural tracts from diffusion weighted magnetic resonance imagery

large n expansions for the laguerre and jacobi β ensembles from the loop equations the ensembles of random matrix theory with classical weights have many special properties one is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at the general order of the correlator via aomoto s method from the theory of the selberg integral we use aomoto s method to derive the full hierarchy of loop equations for laguerre and jacobi ensembles and use these to systematically construct the explicit form of the 1 n expansion at low orders this allows us to give the explicit form of corrections to the global density and allows various moments to be computed complementing results available in the literature motivated by problems in quantum transport

wkb method and quantum periods beyond genus one we extend topological string methods in order to perform wkb approximations for quantum mechanical problems with higher order potentials efficiently this requires techniques for the evaluation of the relevant quantum periods for riemann surfaces beyond genus one the basis of these quantum periods is fixed using the leading behaviour of the classical periods the full expansion of the quantum periods is obtained using a system of picard fuchs like operators for a sequence of integrals of meromorphic forms of the second kind discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve in this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of sl as we check for a symmetric sextic potential

e8 spectral curves i provide an explicit construction of spectral curves for the affine mathrm e 8 relativistic toda chain their closed form expression is obtained by determining the full set of character relations in the representation ring of mathrm e 8 for the exterior algebra of the adjoint representation this is in turn employed to provide an explicit construction of both integrals of motion and the action angle map for the resulting integrable system i consider two main areas of applications of these constructions on the one hand i consider the resulting family of spectral curves in the context of the correspondences between toda systems 5d seiberg witten theory gromov witten theory of orbifolds of the resolved conifold and chern simons theory to establish a version of the b model gopakumar vafa correspondence for the mathrm sl n l e murakami ohtsuki invariant of the poincar e integral homology sphere to all orders in 1 n on the other i consider a degenerate version of the spectral curves and prove a 1 dimensional landau ginzburg mirror theorem for the frobenius manifold structure on the space of orbits of the extended affine weyl group of type mathrm e 8 introduced by dubrovin zhang equivalently the orbifold quantum cohomology of the type mathrm e 8 polynomial mathbb c p 1 orbifold this leads to closed form expressions for the flat co ordinates of the saito metric the prepotential and a higher genus mirror theorem based on the chekhov eynard orantin recursion i will also show how the constructions of the paper lead to a generalisation of a conjecture of norbury scott to ade mathbb p 1 orbifolds and a mirror of the dubrovin zhang construction for all weyl groups and choices of marked roots

boundary finding with parametrically deformable models segmentation using boundary finding is enhanced both by considering the boundary as a whole and by using model based global shape information the authors apply flexible constraints in the form of a probabilistic deformable model to the problem of segmenting natural 2 d objects whose diversity and irregularity of shape make them poorly represented in terms of fixed features or form the parametric model is based on the elliptic fourier decomposition of the boundary probability distributions on the parameters of the representation bias the model to a particular overall shape while allowing for deformations boundary finding is formulated as an optimization problem using a maximum a posteriori objective function results of the method applied to real and synthetic images are presented including an evaluation of the dependence of the method on prior information and image quality