partial regularity for holonomic minimisers of quasiconvex functionals we prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals over a class of sobolev mappings into a compact riemannian manifold to which such mappings are said to be holonomically constrained our approach uses the lifting of sobolev mappings to the universal covering space the connectedness of the covering space an application of ekeland s variational principle and a certain tangential mathbb a harmonic approximation lemma obtained directly via a lipschitz approximation argument this allows regularity to be established directly on the level of the gradient several applications to variational problems in condensed matter physics with broken symmetries are also discussed in particular those concerning the superfluidity of liquid helium 3 and nematic liquid crystals

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

time delay estimation using the cross bispectrum the cross bispectrum phase can be effectively used to estimate the time required for a nongaussian signal to propagate between a pair of spatially separated sensors in the presence of highly correlated gaussian noise the authors present a consistent estimator of the phase of the cross bispectrum derive the exact distribution of the phase of a complex gaussian sample bispectrum and show that in most cases the exact distribution can be approximated by a gaussian distribution using this gaussian approximation the authors derive the variance of the time delay estimate computed from the sample cross bispectrum of a signal in additive correlated noise these results allow the performance of time delay estimators based on the cross bispectrum phase to be quantified as a function of the sample size the skewness of the signal the signal to noise ratio snr and the noise correlation

a general comparison theorem for p harmonic maps in homotopy class abstract we prove a general comparison result for homotopic finite p energy c 1 p harmonic maps u v m n between riemannian manifolds assuming that m is p parabolic and n is complete and nonpositively curved in particular we construct a homotopy through constant p energy maps which turn out to be p harmonic when n is compact moreover we obtain uniqueness in the case of negatively curved n this generalizes a well known result in the harmonic setting due to r schoen and s t yau

fast algorithms for low level vision a recursive filtering structure is proposed that drastically reduces the computational effort required for smoothing performing the first and second directional derivatives and carrying out the laplacian of an image these operations are done with a fixed number of multiplications and additions per output point independently of the size of the neighborhood considered the key to the approach is first the use of an exponentially based filter family and second the use of the recursive filtering applications to edge detection problems and multiresolution techniques are considered and an edge detector allowing the extraction of zero crossings of an image with only 14 operations per output element at any resolution is proposed various experimental results are shown

a comment on the finite memory of stochastic processes it is shown that a proposed concept of finite memory for a zero mean strictly stationary stochastic process results in a stochastic process of random variables each of which is almost surely equal to zero

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

a holder continuity of minimizing symphonic maps abstract in the paper nakauchi and takenaka 2011 3 the second author introduced a stationary map for the functional of the l 2 norm of the pullbacks of metrics we call it a symphonic map in this paper using morrey s argument we give a regularity of locally minimizing symphonic maps into the euclidean space

connecting rational homotopy type singularities let n be a compact simply connected smooth riemannian manifold and for p 2 3 w 1 p r p 1 n be the sobolev space of measurable maps from r p 1 into n whose gradients are in l p the restriction of u to almost every p dimensional sphere s in r p 1 is in w 1 p s n and defines an homotopy class in p n white 1988 evaluating a fixed element z of hom p n r on this homotopy class thus gives a real number z u s the main result of the paper is that any w 1 p weakly convergent limit u of a sequence of smooth maps in c r p 1 n z u has a rectifiable poincare dual left gamma overrightarrow gamma theta right here is a a countable union of c 1 curves in r p 1 with hausdorff user1 mathcal h 1 measurable orientation overrightarrow gamma gamma to s p and density function r the intersection number between left gamma overrightarrow gamma theta right and s evaluates z u s for almost every p sphere s moreover we exhibit a non negative integer n z depending only on homotopy operation z such that int gamma left theta right p mathord left vphantom p left p n z right right kern nulldelimiterspace left p n z right d user1 mathcal h 1 infty even though the mass int gamma left theta right d user1 mathcal h 1 may be infinite we also provide cases of n p and z for which this rational power p p n z is optimal the construction of this poincare dual is based on 1 dimensional bubbling described by the notion of scans which was introduced in hardt and riviere 2003 we also describe how to generalize these results to r m for any m p 1 in which case the bubbling is described by an m p rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented euclidean p sphere

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

steerable wedge filters for local orientation analysis steerable filters have been used to analyze local orientation patterns in imagery such filters are typically based on directional derivatives whose symmetry produces orientation responses that are periodic with period spl pi independent of image structure we present a more general set of steerable filters that alleviate this problem

constancy of p harmonic maps of finite q energy into non positively curved manifolds we investigate p harmonic maps p 2 from a complete non compact manifold into a non positively curved target first we establish a uniqueness result for the p harmonic representative in the homotopy class of a constant map next we derive a caccioppoli inequality for the energy density of a p harmonic map and we prove a companion liouville type theorem provided the domain manifold supports a sobolev poincare inequality finally we obtain energy estimates for a p harmonic map converging with a certain speed to a given point

computerized flow field analysis oriented texture fields an approach to the solution of signal to symbol transformation in the domain of flow fields such as oriented texture fields and velocity vector fields is discussed the authors use the geometric theory of differential equations to derive a symbol set based on the visual appearance of phase portraits which are a geometric representation of the solution curves of a system of differential equations they also provide the computational framework to start with a given flow field and derive its symbolic representation specifically they segment the given texture derive its symbolic representation and perform a quantitative reconstruction of the salient features of the original texture based on the symbolic descriptors results of applying this technique to several real texture images are presented this technique is useful in describing complex flow visualization pictures defects in lumber processing defects in semiconductor wafer inspection and optical flow fields

precision edge contrast and orientation estimation the contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive gaussian noise the test image is an ideal step edge that has been sampled with a square aperture grid the effects of subpixel translations and rotations of the edge on the performance of the operators are studied it is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels methods with improved results are presented for sobel angle estimates and the nevatia babu operator and theoretical noise performance evaluations are also provided an edge operator based on two dimensional spatial moments is presented all methods are compared according to worst case and rms error in an ideal noiseless situation and rms error under various noise levels

density of lipschitz mappings in the class of sobolev mappings between metric spaces we prove that lipschitz mappings are dense in the newtonian sobolev classes n1 p x y of mappings from spaces x supporting p poincare inequalities into a finite lipschitz polyhedron y if and only if y is p connected 1 y 2 y p y 0 where p is the largest integer less than or equal to p

analysis of superimposed oriented patterns estimation of local orientation in images may be posed as the problem of finding the minimum gray level variance axis in a local neighborhood in bivariate images the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2times2 tensor for an ideal single orientation the tensor is rank deficient i e the smaller eigenvalue vanishes a large minimal eigenvalue signals the presence of more than one local orientation what may be caused by non opaque additive or opaque occluding objects crossings bifurcations or corners we describe a framework for estimating such superimposed orientations our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions unlike in the single orientation case the eigensystem analysis does not directly yield the orientations rather it provides so called mixed orientation parameters mops we therefore show how to decompose the mops into the individual orientations we also show how to use tensor invariants to increase efficiency and derive a new feature for describing local neighborhoods which is invariant to rigid transformations applications are e g in texture analysis directional filtering and interpolation feature extraction for corners and crossings tracking and signal separation

homotopy classification of minimizers of the ginzburg landau energy and the existence of permanent currents our objective is to explain the phenomenon of permanent currents within the context of the ginzburg landau model for superconductors using variational techniques we make a connection between the formation of permanent currents and the topology of the superconducting sample

topology and sobolev spaces abstract consider the sobolev class w 1 p m n where m and n are compact manifolds we present some sufficient conditions which guarantee that w 1 p m n is path connected we also discuss cases where w 1 p m n admits more than one component there are still a number of open problems especially concerning the values of p where a change in homotopy classes occurs

a comparison of optimized higher order spectral detection techniques for non gaussian signals using the gaussian noise rejection property of higher order spectra hos hos based detectors have been proposed that outperform conventional second order techniques in certain scenarios based on statistical tests proposed by suhha rao and gabr 1980 as well as hinich 1982 kletter and messer 1990 and hinich and wilson 1990 have developed similar bifrequency domain detectors that are dependent on bispectral estimates of the observation process formalizing the estimate consistency requirements and the asymptotics for these detectors we derive a new f test statistic we consider the detrimental effects of using spectral estimates in the denominator of hinich s test we determine refined conditional distributions for third and fourth order versions of his detector we also modify his test for colored scenarios extending the bispectral detectors to their kth order counterparts we calculate the optimal smoothing bandwidth to use in constructing the hos estimates producing the best detection performances for both our f test and hinich s test with our refined distributions these new bandwidths yield significant improvements in detector performance over previous results for the finite sample case our calculations characterize the tradeoff between the two detectors and demonstrate that a larger smoothing bandwidth than the one suggested by previous researchers should be used our calculations are verified using simulations for both white and colored cases

local minimizers with vortices to the ginzburg landau system in three dimensions we construct local minimizers to the ginzburg landau energy in certain three dimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak jacobians whenever there exists a collection of locally minimal line segments spanning the domain we can find local minimizers with arbitrarily assigned degrees with respect to each segment 2003 wiley periodicals inc

topology of sobolev mappings ii this paper addresses some topological and analytical issues concern ing sobolev mappings between compact riemannian manifolds among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of brezis and li in particular we solved two conjectures in bl we also give a topological obstruction for the weak sequential density of smooth maps in a given sobolev mapping space finally we show a necessary and suf ficient topological condition under which the smooth maps are strongly dense in the sobolev spaces the earlier result theorem 1 of b2 was shown to be not correct

metastable currents in loops with josephson junctions the problem of persistent currents in loops interrupted by josephson junctions is considered we prove that persistent currents are related to local minimizers of the ginzburg landau energy functional although the order parameter is discontinuous at the junction we show that the local minimizers are related to the homotopy types of the domain therefore persistent currents will occur in any multiply connected domain if the junction strength is weak enough

topology of sobolev mappings iii we establish a necessary and sufficient topological condition for maps that are in w1 p m n to be connected by continuous paths in w1 p m n to maps in w1 q m n q q 1 we also obtain a necessary and sufficient topological condition under which w1 q m n is strongly dense in w1 p m n several results concerning sequential weak density of smooth or w1 q m n maps in w1 p m n are also proven 2003 wiley periodicals inc

the jacobian the square root and the set abstract in these notes we wish to present a new approach to the problem of prescribing jacobian determinants in dimension two we restrict ourselves to the case the datum is a finite sum of dirac masses the main point is to show that we may relate this problem to the search of harmonic maps into a singular space shaped as the symbol the later problem in turn is closely linked to questions in complex analysis a large part of the paper is devoted to a presentation of these mathematical objects and their connections

a new adaptive framework for unbiased orientation estimation in textured images this paper focuses on directional texture analysis we propose a new approach for orientation estimation this approach hinges on two classes of convolution masks i e the gradient and the valleyness operators we provide a framework for their optimization regarding bias reduction and noise robustness as the gradient and the valleyness operators are complementary we propose a combination named gv joe this combination consists in using the gradient on inflexion pixels the valleyness on crests and valleys and a linear mixture of both elsewhere we implement an adaptive selection of the size of our operators in order to take into account the variations of the texture scale in the image we apply our approach both on synthetic and natural textures these experiments show that when used separately both classes of operators are more accurate than classical derivative approaches in noisy cases the gv joe implementation improves the robustness of our operators without affecting their accuracy moreover compared to well known orientation estimators it gives the best estimates in the most difficult cases i e for high frequency textures and low snr

local minimizers and quasiconvexity the impact of topology the aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals open image in new window open image in new window from a purely topological point of view the basic assumptions on open image in new window are sequential lower semicontinuity with respect to w1 p weak convergence and w1 p weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds open image in new window and open image in new window

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

fast nonparametric active contour adapted to quadratic inhomogeneous intensity fluctuations in the context of unsupervised segmentation of noisy images a minimum description length mdl polygonal active contour technique based on nonparametric modeling of the noise probability density function pdf has been proposed in 2011 this approach allows fast and efficient segmentation of an object without a priori knowledge on the intensity fluctuations nevertheless since the object and the background are assumed to be homogeneous degraded segmentation results are obtained when images present inhomogeneous intensity variations it is shown in this paper that this constraint of homogeneity can be removed still with minimizing a mdl criterion without undetermined parameters and adapted to nonparametric modeling of the noise pdf for that purpose the spatial inhomogeneity of the intensity is modeled with 2d quadratic functions moreover low computation times can be achieved approximately 60ms on 256x256 pixel images using a two step optimization strategy the efficiency and the robustness of this approach are then validated on various synthetic and real images acquired from different sensors

ginzburg landau minimizers with prescribed degrees capacity of the domain and emergence of vortices let omega subset mathbb r 2 be a simply connected domain let omega be a simply connected subdomain of omega and set a omega setminus overline omega let j be the class of complex valued maps on the annular domain a with degree 1 both on partial omega and on partial omega we consider the variational problem of minimizing the ginzburg landau energy e lambda among all maps in j because only the degrees of the map are prescribed on the boundary the set j is not closed under weak h 1 convergence we show that the attainability of the minimum of e lambda over j is determined by the value of cap a where cap a is the h 1 capacity of the domain a in contrast it is known that the existence of minimizers of e lambda among the maps with a prescribed dirichlet boundary data does not depend on this geometric characteristic when cap a ge pi such an a is denoted as either subcritical if cap a pi or critical if cap a pi we show that the global minimizers of e lambda exist for each lambda 0 and that they are vortexless when lambda is large assuming that lambda to infty we demonstrate that the minimizers of e lambda converge in h 1 a and even better to an mathbb s 1 valued harmonic map which we explicitly identify when cap a lambda 0 or ii the global minimizers exist for each lambda 0 we conjecture that the second case never occurs further for large lambda we establish that the minimizing sequences minimizers in supercritical domains develop exactly two vortices a vortex of degree 1 near partial omega and a vortex of degree 1 near partial omega

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

quasi minimizing surfaces in hyperbolic space let h 3 g denote 3 dimensional hyperbolic space with constant curvature 1 we are interested in surfaces f in h 3 with the following quasi minimality property if it f h 3 0 1 which does not depend on the homotopy if this condition is satisfied with q 1 then f is called homotopically minimizing a surface f c h 3 with this property is not necessarily area minimizing in contrast to the situation in higher dimensions cf 13 our main result is

deformable kernels for early vision early vision algorithms often have a first stage of linear filtering that extracts from the image information at multiple scales of resolution and multiple orientations a common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs a technique is presented that allows 1 computing the best approximation of a given family using linear combinations of a small number of basis functions and 2 describing all finite dimensional families i e the families of filters for which a finite dimensional representation is possible with no error the technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations the relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi scale 2d edge detection kernels the implementation issues are also discussed

on global singularities of sobolev mappings we define various invariants for sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong sobolev topology we show that these invariants classify various types of global singularities of sobolev maps these invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the sobolev space

the homological singularities of maps in trace spaces between manifolds we deal with mappings defined between riemannian manifolds that belong to a trace space of sobolev functions the homological singularities of any such map are represented by a current defined in terms of the boundary of its graph under suitable topological assumptions on the domain and target manifolds we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps moreover we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping

fractional sobolev spaces and topology abstract consider the sobolev class w s p m n where m and n are compact manifolds and p 1 s 0 1 1 p we present a necessary and sufficient condition for two maps u and v in w s p m n to be continuously connected in w s p m n we also discuss the problem of connecting a map u w s p m n to a smooth map f c m n

some mathematical challenges in materials science four challenges to mathematics research posed by the field of materials science are given plus an additional challenge purely from the field of geometric measure theory the problems all concern the effects of surface and grain boundary free energy motion by weighted mean curvature and or surface diffusion and or other kinetics proofs of minimality of soap bubble clusters and their anisotropic analogs shapes of crystals in a gravitational or other field incorporating data from simulations into mathematics and understanding flat chains modulo the figures are selected copies of transparencies presented at the lecture on which this paper is based materials science is both much older and much younger than mathematics ancient humans discovered various elemental metals like gold and copper at some point they discovered that if you mix a little tin into copper you get something quite different from either namely bronze there is a period of human history labeled the bronze age i don t think there is any labeled the arithmetic age on the other hand the term materials science only came into general use within the last 30 years or so as the subjects of metallurgy ceramics and polymers were collected into one discipline the materials research society was founded in 1973 and the international congress of materials research societies in 1991 and along with the old problems of making better structural materials such as steel came new issues unimaginable a century ago some of these include the ability to see and manipulate individual atoms with an atomic force microscope the study of carbon nanotubes the fabrication of electronic chips and the making of thin films for cd s and other storage media what are mathematicians doing in such a field first of all as rodney nichols said at the new york academy of sciences on june 5 2000 much more than a few decades ago there are mathematicians everywhere and that is for the good secondly there has been an effort by the national science foundation to encourage this interdisciplinary research a search of the nsf database under the division of mathematical sciences performed in the year 2000 revealed 85 grants since january 1 1997 with the words materials science in their abstracts there have been several multi million dollar awards co sponsored by darpa for such cooperative efforts targeted at specific problems part of the result is that the collection of mathematicians working on problems from materials science has grown from isolated individuals prior to the mid 1980 s to a small community and now received by the editors november 6 2000 and in revised form february 21 2002 2000 mathematics subject classification primary 74n20 secondary 49q20 49q15 35r99

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

multidimensional orientation estimation with applications to texture analysis and optical flow the problem of detection of orientation in finite dimensional euclidean spaces is solved in the least squares sense the theory is developed for the case when such orientation computations are necessary at all local neighborhoods of the n dimensional euclidean space detection of orientation is shown to correspond to fitting an axis or a plane to the fourier transform of an n dimensional structure the solution of this problem is related to the solution of a well known matrix eigenvalue problem the computations can be performed in the spatial domain without actually doing a fourier transformation along with the orientation estimate a certainty measure based on the error of the fit is proposed two applications in image analysis are considered texture segmentation and optical flow the theory is verified by experiments which confirm accurate orientation estimates and reliable certainty measures in the presence of noise the comparative results indicate that the theory produces algorithms computing robust texture features as well as optical flow