critical behaviour of directed self avoiding walks a new directed self avoiding walk directed saw is proposed and studied on a square lattice the critical behaviour of such directed saws in two dimensions is established and shown to be of different universality class from that of an ordinary saw

an α 1 2 singularity in the vicinity of a disorder variety and its random walk interpretation the authors exhibit a new solution of the anisotropic triangular ising model it is based on diagrammatics well suited to the study of the vicinity of the so called disorder varieties a systematic expansion in this vicinity reveals an unexpected alpha 1 2 singularity this hidden singularity is in turn interpreted as the genuine critical behaviour of a related random walk process

poincare and sobolev inequalities in the monge ampere quasi metric structure abstract we establish a number of poincare and sobolev inequalities in the monge ampere quasi metric structure that are either new or significant improvements upon known ones as applications we derive a host of new weighted poincare and sobolev inequalities

hat interpolation wavelet based multi scale galerkin method for thin walled box beam analysis the objective of the present work is to propose a new adaptive wavelet galerkin method based on the lowest order hat interpolation wavelets the specific application of the present method is made on the one dimensional analysis of thin walled box beam problems exhibiting rapidly varying local end effects higher order interpolation wavelets have been used in the wavelet collocation setting but the lowest order hat interpolation is applied here first and a hat interpolation wavelet based galerkin method is newly formulated unlike existing orthogonal or biorthogonal wavelet based galerkin methods the present method does not require special treatment in dealing with general boundary conditions furthermore the present method directly works with nodal values and does not require special formula for the evaluation of system matrices though interpolation wavelets do not have any vanishing moment an adaptive scheme based on multi resolution approximations is possible and a preconditioned conjugate gradient method can be used to enhance numerical efficiency copyright 2001 john wiley sons ltd

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

holder regularity of the 2d dual semigeostrophic equations via analysis of linearized monge ampere equations we obtain the holder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity our main tool is an interior holder estimate in two dimensions for an inhomogeneous linearized monge ampere equation with right hand side being the divergence of a bounded vector field as a further application of our holder estimate we prove the holder regularity of the polar factorization for time dependent maps in two dimensions with densities bounded away from zero and infinity our applications improve previous work by g loeper who considered the cases of densities sufficiently close to a positive constant

multiscale galerkin method using interpolation wavelets for two dimensional elliptic problems in general domains one major hurdle in developing an efficient wavelet based numerical method is the difficulty in the treatment of general boundaries bounding two or three dimensional domains the objective of this investigation is to develop an adaptive multiscale wavelet based numerical method which can handle general boundary conditions along curved boundaries the multiscale analysis is achieved in a multi resolution setting by employing hat interpolation wavelets in the frame of a fictitious domain method no penalty term or the lagrange multiplier need to be used in the present formulation the validity of the proposed method and the effectiveness of the multiscale adaptive scheme are demonstrated by numerical examples dealing with the dirichlet and neumann boundary value problems in quadrilateral and quarter circular domains copyright 2003 john wiley sons ltd

on certain degenerate and singular elliptic pdes ii divergence form operators harnack inequalities and applications abstract by means of the monge ampere real analysis and pde techniques associated to certain convex functions an approach towards harnack inequalities is developed that simultaneously extends the one for uniformly elliptic operators from the de giorgi nash moser theory and the one for the linearized monge ampere operator from the caffarelli gutierrez theory applications include regularity properties for solutions to divergence form elliptic equations with power like singularities and c 2 estimates for solutions to the monge ampere equation

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

sobolev and isoperimetric inequalities with monomial weights abstract we consider the monomial weight x 1 a 1 x n a n in r n where a i 0 is a real number for each i 1 n and establish sobolev isoperimetric morrey and trudinger inequalities involving this weight they are the analogue of the classical ones with the lebesgue measure dx replaced by x 1 a 1 x n a n d x and they contain the best or critical exponent which depends on a 1 a n more importantly for the sobolev and isoperimetric inequalities we obtain the best constant and extremal functions when a i are nonnegative integers these inequalities are exactly the classical ones in the euclidean space r d with no weight when written for axially symmetric functions and domains in r d r a 1 1 r a n 1

properties of the solutions to the monge ampere equation abstract we consider solutions to the equation det d 2 when has a doubling property we prove new geometric characterizations for this doubling property by means of the so called engulfing property and deduce the quantitative behaviour of also a constructive approach to the celebrated c1 estimates proved by l caffarelli is presented settling one of the open questions posed by villani amer math soc 58 2003

adaptive multiscale wavelet galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization the multiscale galerkin formulation of two dimensional elasticity problems is presented for easy interpolation and boundary handlings as well as efficient adaptive analysis two dimensional interpolation wavelets are used as the multiscale trial functions in the galerkin formulation after the validity of the present multiscale adaptive method is verified with some benchmark problems the present wavelet based method is applied to the multiscale topology optimization that progresses design resolution levels dyadically from low to high levels by this application we show the potential of the multiscale method and the possibility of developing a fully integrated analysis and topology design optimization in the multiscale multiresolution setting

on geometric characterizations for monge ampere doubling measures abstract in this article we prove a theorem on the size of the image of sections of a convex function under its normal mapping when the sections satisfy a geometric property we apply this result to get new geometric characterizations for monge ampere doubling measures

shape optimization of 2d structures using simulated annealing the goal of this study is to obtain globally optimum shapes for two dimensional structures subject to quasi static loads and restraints for this purpose a technique is proposed using which the volume or weight of a structure can be minimized the emphasis is on how one can define the shape precisely and find a shape that accurately reflects the globally optimum shape as design constraints stresses developed in the structure should not exceed the maximum allowable stress and connectivity of the structure should not be lost during shape changes optimization is achieved by a stochastic search algorithm called direct simulated annealing dsa which seeks the global minimum through randomly generated configurations in order to obtain random configurations a boundary variation technique is used in this technique a set of key points is chosen and connected by cubic splines to describe the boundary of the structure whenever the positions of the key points are changed in random directions a new shape is obtained thus coordinates of the key points serve as design variables in order to apply the optimization procedure a general computer code was developed using ansys parametric design language a number of cases were examined to test its effectiveness the results show that this technique can be applied to two dimensional shape optimization problems with high reliability even for cases where the entire free boundary is allowed to vary 2007 elsevier b v all rights reserved

eulerian shape design sensitivity analysis and optimization with a fixed grid conventional shape optimization based on the finite element method uses lagrangian representation in which the finite element mesh moves according to shape change while modern topology optimization uses eulerian representation in this paper an approach to shape optimization using eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed a continuum geometric model is defined on the fixed grid of finite elements an active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization in which each element has a unique shape density the shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements using this transformation it has been shown that the shape design problem can be treated as a parameter design problem which is a much easier method than the former a detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary the accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement two design optimization problems are presented to show the feasibility of the proposed design approach

wavelet methods for second order elliptic problems preconditioning and adaptivity wavelet methods allow us to combine high order accuracy efficient preconditioning techniques and adaptive approximations in order to solve efficiently elliptic operator equations many difficulties remain in particular related to the adaptation of wavelet decompositions to bounded domains with prescribed boundary conditions leading to possibly high constants in the cal o 1 preconditioning in this paper we consider the framework of conforming domain decomposition to generate our wavelet bases and second order operators we emphasize the choice of the wavelets near the boundary of the tensor product reference domain in order to optimize the efficiency of the diagonal preconditioning of elliptic operators in order to improve the constants obtained by such diagonal preconditionings we propose to take into account interactions between the scales through the computation of a sparse approximate inverse spai on a set of nonzero entries obtained from the compression of the operator itself in the wavelet basis the efficiency of these methods is illustrated by solving elliptic second order problems with variable or constant coefficients and homogeneous boundary conditions on a uniform discretization finally we propose a coupling of the iterative solver with an adaptive space refinement technique on the laplacian model problem our experiments show that this algorithm generates an optimal nonlinear approximation of the solution

a note on interior w 2 1 varepsilon estimates for the monge ampere equation by a variant of the techniques introduced by the first two authors in de philippis and figalli invent math 2012 to prove that second derivatives of solutions to the monge ampere equation are locally in l log l we obtain interior w 2 1 varepsilon estimates

w2 1 regularity for solutions of the monge ampere equation in this paper we prove that a strictly convex alexandrov solution u of the monge ampere equation with right hand side bounded away from zero and infinity is w 2 1 mathrm loc this is obtained by showing higher integrability a priori estimates for d 2 u namely d 2 u llog k l for any k

from equilibrium spin models to probabilistic cellular automata the general equivalence betweend dimensional probabilistic cellular automata pca and d 1 dimensional equilibrium spin models satisfying a disorder condition is first described in a pedagogical way and then used to analyze the phase diagrams the critical behavior and the universality classes of some automata diagrammatic representations of time dependent correlation functions of pca are introduced two important classes of pca are singled out for which these correlation functions simplify 1 quasi hamiltonian automata which have a current carrying steady state and for which some correlation functions are those of ad dimensional static model pca satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes 2 linear and more generally affine pca for which the diagrammatics reduces to a random walk problem closely related to d 1 dimensional directed saws both problems display a critical behavior with mean field exponents in any dimension the correlation length and effective velocity of propagation of excitations can be calculated for affine pca as is shown on an explicitd 1 example we conclude with some remarks on nonlinear pca for which the diagrammatics is related to reaction diffusion processes and which belong in some cases to the universality class of reggeon field theory

a wavelet galerkin scheme for analysis of large scale problems on simple domains a wavelet galerkin scheme tailored to address the numerical solution of large scale boundary value problems defined on domains of simple geometry is presented the variation of parameters e g material properties within the domain is arbitrary but the method is specifically designed to solve problems where parameters vary in raster like fashion boundary conditions are imposed via lagrange multipliers using a fictitious domain approach a preconditioner specially designed for this problem is developed to guarantee that convergence of conjugate gradient algorithms is quick and insensitive to problem size the strategy is applied to the solution of steady state heat conduction problems in 2 d but it can be generalized without conceptual changes to 3 d problems and to problems in linear elasticity copyright 1999 john wiley sons ltd

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

remesh free shape optimization using the wavelet galerkin method abstract the objective of this investigation is to develop a new remesh free shape optimization method based on the adaptive wavelet galerkin analysis to avoid cumbersome remeshing processes we embed the original analysis design domain inside a simple fictitious domain and use the predetermined analysis grid points for efficient adaptive analysis we utilize the multiscale multiresolution characteristics of the hat interpolation wavelets that are associated with predetermined grid points furthermore we approximate the design boundary curve by oblique lines that do not necessarily pass through the analysis grid points and develop an accurate stress evaluation technique based on the proposed boundary approximation the validity of the proposed remesh free shape optimization formulation based on the wavelet galerkin method is verified through benchmark shape optimization problems

on certain degenerate and singular elliptic pdes i nondivergence form operators with unbounded drifts and applications to subelliptic equations abstract we prove a harnack inequality for nonnegative strong solutions to degenerate and singular elliptic pdes modeled after certain convex functions and in the presence of unbounded drifts our main theorem extends the harnack inequality for the linearized monge ampere equation due to caffarelli and gutierrez and it is related although under different hypotheses to a recent work by n q le since our results are shown to apply to the convex functions x p with p 2 and their tensor sums the degenerate elliptic operators that we can consider include subelliptic grushin and grushin like operators as well as a recent example by a montanari of a nondivergence form subelliptic operator arising from the geometric theory of several complex variables in the light of these applications it follows that the monge ampere quasi metric structure can be regarded as an alternative to the usual carnot caratheodory metric in the study of certain subelliptic pdes