two component kp hierarchy and the classical boussinesq equation it has been shown that solutions to the kp equation and other equations in the kp hierarchy take a wronskian form the authors extend these ideas to two component wronskians and the hirota equations 1986 which they satisfy by a reduction procedure they show how dependence on some of the variables can be removed and look in detail at the classical boussinesq equation and its modifications and show how these equations fit into the two component kp hierarchy

3 d image denoising by local smoothing and nonparametric regression three dimensional 3 d images are becoming increasingly popular in image applications such as magnetic resonance imaging mri functional mri fmri and other image applications observed 3 d images often contain noise that should be removed beforehand for improving the reliability of subsequent image analyses in the literature most existing image denoising methods are for 2 d images their direct extensions to 3 d cases generally cannot handle 3 d images efficiently because the structure of 3 d images is often substantially more complicated than that of 2 d images for instance edge locations are surfaces in 3 d cases which are much more challenging to handle compared to edge curves in 2 d cases in this article we propose a novel 3 d image denoising procedure based on nonparametric estimation of a 3 d jump surface from noisy data one important feature of this method is its ability to preserve edges and major edge structures such as intersections of two edge surfaces pyramids pointed corn

pseudo parabolic regularization of forward backward parabolic equations power type nonlinearities we study a quasilinear parabolic equation of forward backward type under assumptions on the nonlinearity which hold for a wide class of mathematical models using a pseudo parabolic regularization of power type we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures it is shown that these solutions satisfy suitable entropy inequalities we also study their qualitative properties in particular proving that the singular part of the solution with respect to the lebesgue measure is constant in time

computing optical flow via variational techniques defined as the apparent motion in a sequence of images the optical flow is very important in the computer vision community where its accurate estimation is necessary for many applications it is o

finite rank constraints on linear flows and the davey stewartson equation finite rank constraints for the right derivatives of certain automorphisms solving the heat equation imply the davey stewartson system solutions comprising wronskian and grammian determinants are found and the real reductions for the davey stewartson i and davey stewartson ii are considered

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

plethystic algebras and vector symmetric functions an isomorphism is established between the plethystic hopf algebra pleth super l and the algebra of vector symmetric functions the hall inner product of symmetric function theory is extended to the hopf algebra pleth super l

on a class of forward backward parabolic equations existence of solutions abstract we study the initial boundary value problem u t u x x u t x x in 0 t u u t 0 in 0 t u u 0 in 0 where is an interval and u 0 is a nonnegative radon measure on the map is increasing in 0 and decreasing in for some 0 and satisfies 0 0 the regularizing map is increasing and bounded we prove existence of suitably defined nonnegative radon measure valued solutions the solution class is natural since smooth initial data may generate solutions which become measure valued after finite time

bilinear covariants and spinor fields duality in quantum clifford algebras classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum clifford algebras on minkowski spacetime once the bilinear covariants are expressed in terms of algebraic spinor fields the duality between spinor and quantum spinor fields can be discussed thus by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric quantum algebraic spinor fields and deformed bilinear covariants can be constructed they are thus compared to the classical non quantum ones classes of quantum spinor fields classes are introduced and compared with lounesto s spinor field classification a physical interpretation of the deformed parts and the underlying z grading is proposed the existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity s w hawking the unpredictability of quantum gravity commun

artistic edge and corner enhancing smoothing two important visual properties of paintings and painting like images are the absence of texture details and the increased sharpness of edges as compared to photographic images painting like artistic effects can be achieved from photographic images by filters that smooth out texture details while preserving or enhancing edges and corners however not all edge preserving smoothers are suitable for this purpose we present a simple nonlinear local operator that generalizes both the well known kuwahara filter and the more general class of filters known in the literature as ldquocriterion and value filter structure rdquo this class of operators suffers from intrinsic theoretical limitations which give rise to a dramatic instability in presence of noise especially on shadowed areas such limitations are discussed in the paper and overcome by the proposed operator a large variety of experimental results shows that the output of the proposed operator is visually similar to a painting comparisons with existing techniques on a large set of natural images highlight conditions on which traditional edge preserving smoothers fail whereas our approach produces good results in particular unlike many other well established approaches the proposed operator is robust to degradations of the input image such as blurring and noise contamination

edge preserving filtering of images with low photon counts edge preserving filters such as local m smoothers or bilateral filtering are usually designed for gaussian noise this paper investigates how these filters can be adapted in order to efficiently deal with poissonian noise in addition the issue of photometry invariance is addressed by changing the way filter coefficients are normalized the proposed normalization is additive instead of being multiplicative and leads to a strong connection with anisotropic diffusion experiments show that ensuring the photometry invariance leads to comparable denoising performances in terms of the root mean square error computed on the signal

strongly anisotropic diffusion problems asymptotic analysis the subject matter of this paper concerns anisotropic diffusion equations we consider heat equations whose diffusion matrices have disparate eigenvalues we determine first and second order approximations we study the well posedness of them and establish convergence results the analysis relies on averaging techniques which have been used previously for studying transport equations whose advection fields have disparate components

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

the davey stewartson equation and constrained linear flows abstract the davey stewartson system is shown to appear when finite rank constraints are imposed over the right derivatives of certain automorphisms solving the heat equation families of solutions encompassing wronskian and zakharov shabat nakamura type solutions are found for the davey stewartson system and the reduction to the davey stewartson i is discussed

nonlinear multiscale analysis of three dimensional echocardiographic sequences introduces a new model for multiscale analysis of space time echocardiographic sequences the proposed nonlinear partial differential equation representing the multiscale analysis filters the sequence while keeping the space time coherent structures it combines the ideas of regularized perona malik anisotropic diffusion and the galilean invariant movie multiscale analysis of alvarez et al arch rat mech anal vol 123 p 200 57 1993 a numerical method for solving the proposed partial differential equation is suggested and its stability is shown computational results on synthesized and real sequences are provided a qualitative and quantitative evaluation of the accuracy of the method is presented

a problem of cayley from 1857 and how he could have solved it abstract in a paper of 1857 cayley suggests the problem of determining all the symmetric functions of the common solutions of a linear cubic homogeneous system of ternary equations as functions of the coefficients of the equations we show how cayley could have solved the problem himself if he had taken another look at it all the facts necessary for the solution were known to cayley including a crucial formula known as cayley s identity we study the problem by mixing techniques from vector symmetric function theory with techniques from the study of determinants and permanents we develop some new formulas for determinants of matrices which arise by the hadamard or pointwise product from other matrices

two phase entropy solutions of a forward backward parabolic equation this article deals with the cauchy problem for a forward backward parabolic equation which is of interest in physical and biological models considering such an equation as the singular limit of an appropriate pseudoparabolic third order regularization we consider the framework of entropy solutions namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation moreover we restrict the attention to two phase solutions that is solutions taking values in the intervals where the parabolic equation iswell posed proving existence and uniqueness of such solutions

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

bounded self weights estimation method for non local means image denoising using minimax estimators a non local means nlm filter is a weighted average of a large number of non local pixels with various image intensity values the nlm filters have been shown to have powerful denoising performance excellent detail preservation by averaging many noisy pixels and using appropriate values for the weights respectively the nlm weights between two different pixels are determined based on the similarities between two patches that surround these pixels and a smoothing parameter another important factor that influences the denoising performance is the self weight values for the same pixel the recently introduced local james stein type center pixel weight estimation method ljs outperforms other existing methods when determining the contribution of the center pixels in the nlm filter however the ljs method may result in excessively large self weight estimates since no upper bound is assumed and the method uses a relatively large local area for estimating the self weights which may lead to a strong bias in this paper we investigated these issues in the ljs method and then propose a novel local self weight estimation methods using direct bounds lmm db and reparametrization lmm rp based on the baranchik s minimax estimator both the lmm db and lmm rp methods were evaluated using a wide range of natural images and a clinical mri image together with the various levels of additive gaussian noise our proposed parameter selection methods yielded an improved bias variance trade off a higher peak signal to noise psnr ratio and fewer visual artifacts when compared with the results of the classical nlm and ljs methods our proposed methods also provide a heuristic way to select a suitable global smoothing parameters that can yield psnr values that are close to the optimal values

robust estimation approach for blind denoising this work develops a new robust statistical framework for blind image denoising robust statistics addresses the problem of estimation when the idealized assumptions about a system are occasionally violated the contaminating noise in an image is considered as a violation of the assumption of spatial coherence of the image intensities and is treated as an outlier random variable a denoised image is estimated by fitting a spatially coherent stationary image model to the available noisy data using a robust estimator based regression method within an optimal size adaptive window the robust formulation aims at eliminating the noise outliers while preserving the edge structures in the restored image several examples demonstrating the effectiveness of this robust denoising technique are reported and a comparison with other standard denoising filters is presented

stable patterns in a viscous diffusion equation we consider a pseudoparabolic regularization of a forward backward nonlinear diffusion equation ut a f u vut motivated by the problem of phase separation in a viscous binary mixture the function f is nonmonotone so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases we find that any bounded measurable steady state solution u x satisfying f u constant f u x 0 a e is dynamically stable to perturbations in the sense of convergence in measure in particular smooth solutions may achieve discontinuous asymptotic states furthermore stable states need not correspond to absolute minimizers of free energy thus violating gibbs principle of stability for phase mixtures

singularity theory and phantom edges in scale space the process of detecting edges in a one dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal it is shown that the second derivative of the contrast function at the critical point is related to the classification of the associated edge as being phantom or authentic the contrast of authentic edges decreases with filter scale while the contrast of phantom edges are shown to increase with scale as the filter scale increases an authentic edge must either turn into a phantom edge or join with a phantom edge and vanish the points in the scale space at which these events occur are seen to be singular points of the contrast function using ideas from singularity or catastrophy theory the scale map contours near these singular points are found to be either vertical or parabolic

plethystic vertex operators and boson fermion correspondences we study the algebraic properties of plethystic vertex operators introduced in 2010 j phys a math theor 43 405202 underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group defined to stabilize tensors of young symmetry type characterized by a partition of arbitrary shape here we establish an extension of the well known boson fermion correspondence involving schur functions and their associated bernstein vertex operators for each the modes generated by the plethystic vertex operators and their suitably constructed duals satisfy the anticommutation relations of a complex clifford algebra the combinatorial manipulations underlying the results involve exchange identities exploiting the hopf algebraic structure of certain symmetric function series and their plethysms

stochastic differential equations and geometric flows in previous years curve evolution applied to a single contour or to the level sets of an image via partial differential equations has emerged as an important tool in image processing and computer vision curve evolution techniques have been utilized in problems such as image smoothing segmentation and shape analysis we give a local stochastic interpretation of the basic curve smoothing equation the so called geometric heat equation and show that this evolution amounts to a tangential diffusion movement of the particles along the contour moreover assuming that a priori information about the shapes of objects in an image is known we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise we also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features in contrast to the original geometric heat flow which tends to circularize any closed curve

geometrically induced force interaction for three dimensional deformable models in this paper we propose a novel 3 d deformable model that is based upon a geometrically induced external force field which can be conveniently generalized to arbitrary dimensions this external force field is based upon hypothesized interactions between the relative geometries of the deformable model and the object boundary characterized by image gradient the evolution of the deformable model is solved using the level set method so that topological changes are handled automatically the relative geometrical configurations between the deformable model and the object boundaries contribute to a dynamic vector force field that changes accordingly as the deformable model evolves the geometrically induced dynamic interaction force has been shown to greatly improve the deformable model performance in acquiring complex geometries and highly concave boundaries and it gives the deformable model a high invariancy in initialization configurations the voxel interactions across the whole image domain provide a global view of the object boundary representation giving the external force a long attraction range the bidirectionality of the external force field allows the new deformable model to deal with arbitrary cross boundary initializations and facilitates the handling of weak edges and broken boundaries in addition we show that by enhancing the geometrical interaction field with a nonlocal edge preserving algorithm the new deformable model can effectively overcome image noise we provide a comparative study on the segmentation of various geometries with different topologies from both synthetic and real images and show that the proposed method achieves significant improvements against existing image gradient techniques

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

a global sparse gradient based coupled system for image denoising abstract we consider a coupled system for image denoising which consists of an anisotropic diffusion equation and a global sparse gradient model gsg the global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly the anisotropic diffusion process is guided by the edge map furthermore we prove the existence and uniqueness of the solutions of the coupled system with dirichlet initial boundary value problem finally comparing with the edge indicators based on local methods experimental results demonstrate that our proposed method has the state of the art performance among the pde based anisotropic diffusion approaches both in objective measurement and visual evaluation

nonuniqueness of solutions for a class of forward backward parabolic equations abstract we study the initial boundary value problem u t u x x e u t x x in 0 u e u t 0 in 0 u u 0 0 in 0 with measure valued initial data here is a bounded open interval 0 0 is increasing in 0 and decreasing in and the regularising term is increasing but bounded it is natural to study measure valued solutions since singularities may appear spontaneously in finite time nonnegative radon measure valued solutions are known to exist and their construction is based on an approximation procedure until now nothing was known about their uniqueness in this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions whence uniqueness fails in addition we classify the steady state solutions

existence and multiplicity of positive solutions for elliptic equation with critical weighted hardy sobolev exponents and boundary singularities abstract we study a semilinear elliptic equation involving critical weighted hardy sobolev exponents with boundary singularities the existence and multiplicity of positive solutions are established our method relies upon ekeland s variational principle and mountain pass lemma

degenerate regularization of forward backward parabolic equations the regularized problem we study a quasilinear parabolic equation of forward backward type in one space dimension under assumptions on the nonlinearity which hold for a number of important mathematical models for example the one dimensional perona malik equation using a degenerate pseudoparabolic regularization proposed in barenblatt et al siam j math anal 24 1414 1439 1993 which takes time delay effects into account we prove existence and uniqueness of positive solutions of the regularized problem in a space of radon measures we also study qualitative properties of such solutions in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the lebesgue measure in this respect the existence of a family of viscous entropy inequalities plays an important role

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

passage to the limit over small parameters in the viscous cahn hilliard equation abstract we study singular passage to the limit over different small parameters for the viscous cahn hilliard equation under weak growth assumptions on the nonlinearity a rigorous proof of convergence to solutions of either the cahn hilliard equation or of the allen cahn equation or of the sobolev equation depending on the choice of the parameter is provided we also study the singular limit of the cahn hilliard equation as the parameter in the fourth order term goes to zero in particular we show that a radon measure valued solution of the limiting ill posed problem can arise depending on the behavior of the nonlinearity at infinity

sobolev regularization of a class of forward backward parabolic equations we address existence and asymptotic behaviour for large time of young measure solutions of the dirichlet initial boundary value problem for the equation ut u where the function need not satisfy monotonicity conditions under suitable growth conditions on these solutions are obtained by a vanishing viscosity method from solutions of the corresponding problem for the regularized equation ut u ut the asymptotic behaviour as t of young measure solutions of the original problem is studied by limit set techniques relying on the tightness of sequences of time translates of the limiting young measure when n 1 this measure is characterized as a linear combination of dirac measures with support on the branches of the graph of

on a pseudoparabolic regularization of a forward backward forward equation abstract we consider an initial boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward backward forward parabolic equation with a bounded nonlinearity which is increasing at infinity we prove existence of suitably defined nonnegative solutions of the problem in a space of radon measures solutions satisfy several monotonicity and regularization properties in particular their singular part is nonincreasing and may disappear in finite time the problem is of intrinsic mathematical interest but also arises naturally when studying by time reversal the spontaneous appearance of singularities in a specific application

hopf algebras distributive laplace pairings and hash products a unified approach to tensor product decompositions of group characters we show for bicommutative graded connected hopf algebras that a certain distributive laplace subgroup of the convolution monoid of 2 cochains parameterizes certain well behaved hopf algebra deformations using the laplace group or its frobenius subgroup we define higher derived hash products and develop a general theory to study their main properties applying our results to the universal bicommutative graded connected hopf algebra of symmetric functions we show that classical tensor product and character decompositions such as those for the general linear group mixed co and contravariant or rational characters orthogonal and symplectic group characters thibon and reduced symmetric group characters are special cases of higher derived hash products in the appendix we discuss a relation to formal group laws

the dressing method and the solutions with constant asymptotic values at infinity of ds ii equation several classes of exact solutions with constant asymptotic values at infinity of ds ii equation are constructed via the dressing method among these solutions are the solutions with functional parameters multi line solitons and breathers and pure rational solutions

interpreting line drawings of curved objects in this thesis we study the problem of interpreting line drawings of scenes composed or opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them it is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint that the line drawing is error free and that there are no lines due to shadows or specularities our definition implicitly excludes laminae wires and the apices of cones a major component of the interpretation of line drawings is line labelling by line labelling we mean a classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene and b further subclassification of each kind of discontinuity for a depth discontinuity we determine whether it is a limb a locus of points on the surface where the line of sight is tangent to the surface or an occluding edge a tangent plane discontinuity of the surface for an orientation discontinuity we determine whether it corresponds to a convex or concave edge this thesis presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described by analysing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing an algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing a local minimum complexity rule at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex is used in order to prune highly counter intuitive interpretations the labelling scheme was implemented and tested on a number of line drawings the labellings obtained are few and by and large in accordance with human interpretations

radon measure valued solutions for a class of quasilinear parabolic equations initial value problems for quasilinear parabolic equations having radon measures as initial data have been widely investigated looking for solutions which for positive times take values in some function space in contrast it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the radon measures of the initial data we call such solutions measure valued in contrast to function valued solutionspreviously considered in the literature we first show that there is a natural notion of measure valued solution of problem p below in spite of its nonlinear character a major consequence of our definition is that if the space dimension is greater than one the concentrated part of the solution with respect to the newtonian capacity is constant in time subsequently we prove that there exists exactly one solution of the problem such that the diffuse part with respect to the newtonian capacity of the singular part of the solution with respect to the lebesgue measure is concentrated for almost every positive time on the set where the regular part with respect to the lebesgue measure is large moreover using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time finally the regularity problem is addressed as we give conditions depending on the space dimension the initial data and the rate of convergence at infinity of the nonlinearity to ensure that the measure valued solution of problem p is in fact function valued

energy exchange of interacting coherent structures in multidimensions abstract the formalism recently developed by fokas and the author for the solution of the initial boundary value problem for the davey stewartson equation is used to study the time evolution of the multidromion solution showing in particular that interacting coherent structures in multidimensions not only exhibit a two dimensional phase shift but also change amplitude and exchange energy several examples of physically relevant boundary conditions which generate other kinds of explicit finite energy solutions are also presented

existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two hardy sobolev critical exponents abstract in this paper by applying ekeland s variational principle and strong maximum principle we investigate the existence and multiplicity of positive solutions for a perturbed semilinear elliptic equation with two critical hardy sobolev exponents and boundary singularities

on a class of forward backward parabolic equations formation of singularities abstract we study the formation of singularities for the problem u t u x x e u t x x in 0 t u e u t 0 in 0 t u u 0 0 in 0 where and t are positive constants a bounded interval u 0 a nonnegative radon measure on a nonmonotone and nonnegative function with 0 0 and an increasing bounded function we show that if u 0 is a bounded or continuous function singularities may appear spontaneously the class of singularities which can arise in finite time is remarkably large and includes infinitely many dirac masses and singular continuous measures

degenerate regularization of forward backward parabolic equations the vanishing viscosity limit we address the vanishing viscosity limit of the regularized problem studied in smarrazzo and tesei arch rat mech anal 2012 in press we show that the limiting points in a suitable topology of the family of solutions of the regularized problem can be regarded as suitably defined weak measure valued solutions of the original problem in general these solutions are the sum of a regular term which is absolutely continuous with respect to the lebesgue measure and a singular term which is a radon measure singular with respect to the other by using a family of entropy inequalities we prove that the singular term is nondecreasing in time we also characterize the disintegration of the young measure associated with the regular term proving that it is a superposition of two dirac masses with support on the branches of the graph of the nonlinearity varphi

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given