makar limanov s conjecture on free subalgebras abstract it is proved that over every countable field k there is a nil algebra r such that the algebra obtained from r by extending the field k contains noncommutative free subalgebras of arbitrarily high rank it is also shown that over every countable field k there is an algebra r without noncommutative free subalgebras of rank two such that the algebra obtained from r by extending the field k contains a noncommutative free subalgebra of rank two this answers a question of makar limanov lenny makar limanov private communication beijing june 2007

local tomography with nonsmooth attenuation ii we continue investigation of local tomography with nonsmooth attenuation which was proposed earlier by the author our main result is a new and simple characterization of the wave front of the local tomography function tilde f lambda left phi right which holds under fairly general conditions on the density function f and the attenuation coefficient

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

local tomography for the generalized radon transform let f be piecewise smooth with only jump discontinuities across s singsupp f the asymptotic behavior of cal b f near s is obtained here cal b is an elliptic pseudodifferential operator which satisfies certain not very restrictive assumptions an efficient numerical scheme for computing local tomography lt functions for the generalized radon transform grt is proposed the effect of mollification of the two lt functions is investigated an algorithm for finding values of jumps of f knowing local tomographic data corresponding to the grt is developed and tested on the simulated data

modifications of the newton raphson method for finite element simulations in ferroelectroelasticity abstract the system of equations arising in finite element simulations of components made of ferroelectroelastic materials is non linear if the loading is sufficiently high the newton raphson method represents a widely used iterative technique to solve this system of non linear equations however if the scalar potential formulation is utilised convergence difficulties may occur this circumstance can be primarily attributed to the specific form of the non linear response of typical ferroelectroelastic materials being subjected to electrical loading the present paper is devoted to modifications of the newton raphson method which are capable of improving the convergence behaviour experienced in the finite element iteration we extend an existing modification to the fully coupled ferroelectroelastic case additionally a new modification of the newton raphson method is proposed this method applies an iteration algorithm which is virtually equivalent to the iteration algorithm of the unmodified newton raphson method combined with the vector potential formulation an important feature of both modifications is that they are applied on the integration point level therefore the global non linear finite element iteration scheme remains unchanged finally the practicability of the modifications discussed in the paper is shown in a numerical example

differential morphology and image processing image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain we provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems this includes ideas on using partial differential or difference equations pdes to model distance propagation or nonlinear multiscale processes in images we briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics we also review some nonlinear pdes that model the evolution of multiscale morphological operators and use morphological derivatives among the new ideas presented we develop some general 2 d max min sum difference equations that model the space dynamics of 2 d morphological systems including the distance computations and some nonlinear signal transforms called slope transforms that can analyze these systems in a transform domain in ways conceptually similar to the application of fourier transforms to linear systems thus distance transforms are shown to be bandpass slope filters we view the analysis of the multiscale morphological pdes and of the eikonal pde solved via weighted distance transforms as a unified area in nonlinear image processing which we call differential morphology and briefly discuss its potential applications to image processing and computer vision

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

entropy like proximal methods in convex programming we study an extension of the proximal method for convex programming where the quadratic regularization kernel is substituted by a class of convex statistical distances called divergences which are typically entropy like in form after establishing several basic properties of these quasi distances we present a convergence analysis of the resulting entropy like proximal algorithm applying this algorithm to the dual of a convex program we recover a wide class of nonquadratic multiplier methods and prove their convergence

semi lagrangian schemes for hamilton jacobi equations discrete representation formulae and godunov methods we study a class of semi lagrangian schemes which can be interpreted as a discrete version of the hopf lax oleinik representation formula for the exact viscosity solution of first order evolutive hamilton jacobi equations that interpretation shows that the scheme is potentially accurate to any prescribed order we discuss how the method can be implemented for convex and coercive hamiltonians with a particular structure and how this method can be coupled with a discrete legendre trasform we also show that in one dimension the first order semi lagrangian scheme coincides with the integration of the godunov scheme for the corresponding conservation laws several test illustrate the main features of semi lagrangian schemes for evolutive hamilton jacobi equations

on the relation between u hessians and second order epi derivatives abstract we consider second order objects for a convex function defined as the maximum of a finite number of c2 functions variational analysis yields explicit formulae for the second order epi derivatives of such max functions another second order object can be defined by means of a space decomposition that allows us to identify a subspace on which a lagrangian related to a max function is smooth this decomposition yields an explicit expression for the so called u hessian of the function defined as the hessian of the related lagrangian we show that the second order epi derivative relative to the u subspace and the u hessian are equivalent second order objects thus the u lagrangian of a max function captures the function s second order epi differential behavior with ordinary second derivatives

primitive algebras with arbitrary gelfand kirillov dimension abstract we construct for every real 2 a primitive affine algebra with gelfand kirillov dimension unlike earlier constructions there are no assumptions on the base field in particular this is the first construction over or l given a recursive sequence v n of elements in a free monoid we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in v n we bound the dimension of the resulting algebra in terms of the growth of v n in particular if n is less than doubly exponential then the dimension is 2 this also answers affirmatively a conjecture of salwa 1997 comm algebra 25 3965 3972

examples in finite gel fand kirillov dimension for any field f we give an example of a prime finitely generated f algebra of gk dimension 2 for which gk dimension is not finitely partitive for each m 0 1 2 3 we give an example

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

inversion of limited angle tomographic data abstract a new method is given for analytical inversion of the radon transform given limited angle data numerical aspects of this inversion problem are discussed

properties of the moreau yosida regularization of a piecewise c2 convex function in this paper we discuss second order properties of the moreau yosida regularization f of a piecewise twice continuously differentiable convex function f we introduce a new constraint qualification in order to prove that the gradient of f is piecewise continuously differentiable in addition we discuss conditions depending on the hessians of the pieces that guarantee positive definiteness of the generalized jacobians of the gradient of f

an infinite dimensional affine nil algebra with finite gelfand kirillov dimension the famous 1960 s construction of golod and shafarevich yields infinite dimensional nil but not nilpotent algebras however these algebras have exponential growth here we construct an infinite dimensional nil but not locally nilpotent algebra which has polynomially bounded growth

numerical computation of rank one convex envelopes we describe an algorithm for the numerical computation of the rank one convex envelope of a function f mm m times n rightarrow rr we prove its convergence and an error estimate in l infty

stable calculation of the legendre transform abstract given x such that x x l is derived such that formula l c the constant c depends on sup d 2 where d 2 is an arbitrary second derivative of x x r n an example is given which shows that the critical point of x cannot be calculated stably from the noisy data x the notion of the generalized legendre transform is introduced and a stable method for its calculation given the noisy data is described

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

zero preserving iso spectral flows based on parallel sums abstract driessel k r driessel computing canonical forms using flows linear algebra appl 379 2004 353 379 introduced the notion of quasi projection onto the range of a linear transformation from one inner product space into another inner product space here we introduce the notion of quasi projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space as an application we design a new family of iso spectral flows on the space of symmetric matrices that preserves zero patterns we discuss the equilibrium points of these flows we conjecture that these flows generically converge to diagonal matrices we perform some numerical experiments with these flows which support this conjecture we also compare our zero preserving flows with the toda flow

enlargements and sums of monotone operators in this paper we study two important notions related to monotone operators one is the concept of enlargement of a given monotone operator which has turned out to be a useful tool in the analysis of approximate solutions to problems involving monotone operators the second one is the notion of sum of monotone operators for the latter we introduce and study a kind of extended sum of two monotone operators which in several important cases turns out to be a maximal monotone operator

variable metric bundle methods from conceptual to implementable forms to minimize a convex function we combine moreau yosida regularizations quasi newton matrices and bundling mechanisms first we develop conceptual forms using reversal quasi newton formulae and we state their global and local convergence then to produce implementable versions we incorporate a bundle strategy together with a curve search no convergence results are given for the implementable versions however some numerical illustrations show their good behaviour even for large scale problems

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

singularities of the radon transform the singular support of the radon transform rf of a function f with a compact support is described in terms of the generalized legendre transform the asymptotic behavior of rf near the singular support modulo smooth functions is described in terms of a geometrical invariant introduced in the paper

how to transform one convex function continuously into another the proximal average operator provides a parametric family of convex functions that continuously transform one convex function into another even when the domains of the two functions do not intersect we prove that the proximal average operator is a homotopy with respect to the epi topology study its properties and present several explicit formulas for specific classes of functions the parametric family inherits desirable properties such as differentiability and strict convexity from the given functions the results illustrate the powerful tools available in convex and variational analysis from both a theoretical and a computational point of view

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

a question of kaplansky abstract we show using affinization that over any field f there exists a primitive algebraic affine f algebra that is not locally finite answering a question of kaplansky in the affirmative

on the kurosh problem for algebras of polynomial growth over a general field lenagan and smoktunowicz 2007 ls see also lenagan smoktunowicz and young in press lsy gave an example of a nil algebra of finite gelfand kirillov dimension their construction requires a countable base field however we show that for any field k and any monotonically increasing function f n which grows super polynomially but subexponentially there exists an infinite dimensional finitely generated nil k algebra whose growth is asymptotically bounded by f n this construction gives the first examples of nil algebras of subexponential growth over uncountable fields

a quasi second order proximal bundle algorithm this paper introduces an algorithm for convex minimization which includes quasi newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm the method uses the hessian of a certain outer function which depends on the jacobian of a proximal point mapping which in turn depends on the preconditioner matrix and on a lagrangian hessian relative to a certain tangent space convergence is proved under boundedness assumptions on the preconditioner sequence

polynomial rings over nil rings need not be nil abstract we construct a nil algebra over a countable field the polynomial ring over which is not nil this answers a question of amitsur

on mathcalvu theory for functions with primal dual gradient structure we consider a general class of convex functions having what we call primal dual gradient structure it includes finitely determined max functions and maximum eigenvalue functions as well as other infinitely defined max functions for a function in this class we discuss a space decomposition that allows us to identify a subspace on which the function appears to be smooth moreover using the special structure of such a function we compute smooth trajectories along which certain second order expansions can be obtained we also give an explicit expression for the hessian of a related lagrangian

fast legendre fenchel transform and applications to hamilton jacobi equations and conservation laws we are interested in the study of a fast algorithm introduced by brenier computing the discrete legendre fenchel transform of a real function we present convergence results and show how the order of convergence grows with the regularity of the function to be transformed applications to hamilton jacobi equations for front propagation problems and conservation laws are presented

some aspects of macroscopic phenomenological material models for ferroelectroelastic ceramics abstract macroscopic phenomenological constitutive models are used in order to describe the behavior of ferroelectroelastic materials a representative example for such a model is the one proposed by landis during recent years the authors of the present paper applied this model for practical problems in this context several problems occurred which lead to failure of the computations the purpose of the paper is to draw attention to these difficulties and to provide solution strategies where possible three different issues are discussed in detail these include 1 the occurrence of indefinite linear moduli 2 the occurrence of negative switching multipliers due to the particular choice of the functional form for the tensor of piezoelectricity and 3 the occurrence of negative switching multipliers due to the particular choice of the functional form for the switching surface moreover the paper is intended to serve as a motivation for further research

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

return mapping algorithms and consistent tangent operators in ferroelectroelasticity return mapping algorithms for a rather general class of phenomenological rate independent models for ferroelectroelastic materials are presented the fully coupled thermodynamically consistent three dimensional constitutive model with two internal variables remanent polarization vector and remanent strain tensor proposed by c m landis in 2002 is used for the simulation of electromechanical hysteresis effects in polycrystalline ferroelectric ceramics based on the operator splitting methodology the return mapping algorithm employs the closest point projection scheme to obtain an efficient and robust integration of the constitutive model the consistent tangent operator is obtained in closed form by linearizing the return mapping algorithm and is found to be non symmetric in the general case due to the dependence of the switching criterion on internal variables conditions that provide the symmetry of the consistent tangent matrix are analyzed the compactness and generality of the received relations are achieved by means of using the thermodynamically based compact notations combining mechanical and electrical values both the cases scalar potential finite element fe formulation primary variables strain and electric field and vector potential fe formulation primary variables strain and electric displacement are considered the accuracy and robustness of the algorithms are assessed through numerical examples copyright 2009 john wiley sons ltd

on the superlinear convergence of the variable metric proximal point algorithm using broyden and bfgs matrix secant updating in previous work the authors provided a foundation for the theory of variable metric proximal point algorithms in hilbert space in that work conditions are developed for global linear and super linear convergence this paper focuses attention on two matrix secant updating strategies for the finite dimensional case these are the broyden and bfgs updates the bfgs update is considered for application in the symmetric case e g convex programming applications while the broyden update can be applied to general monotone operators subject to the linear convergence of the iterates and a quadratic growth condition on the inverse of the operator at the solution super linear convergence of the iterates is established for both updates these results are applied to show that the chen fukushima variable metric proximal point algorithm is super linearly convergent when implemented with the bfgs update

uniform prox regularity of functions and epigraphs in hilbert spaces abstract we study the local uniform prox regularity of functions make the connection with the lower c 2 property and essentially with the prox regularity of epigraphs

minimizing within convex bodies using a convex hull method we present numerical methods to solve optimization problems on the space of convex functions or among convex bodies hence convexity is a constraint on the admissible objects whereas the functionals are not required to be convex to deal with this our method mixes geometrical and numerical algorithms we give several applications arising from classical problems in geometry and analysis alexandrov s problem of finding a convex body of prescribed surface function cheeger s problem of a subdomain minimizing the ratio surface area on volume newton s problem of the body of minimal resistance in particular for the latter application the minimizers are still unknown except in some particular classes we give approximate solutions better than the theoretical known ones hence demonstrating that the minimizers do not belong to these classes

entropic proximal mappings with applications to nonlinear programming we introduce a family of new transforms based on imitating the proximal mapping of moreau and the associated moreau yosida proximal approximation of a function the transforms are constructed in terms of the a divergence functional a generalization of the relative entropy and of bregman s measure of distance an analogue of moreau s theorem associated with these entropy like distances is proved we show that the resulting entropic proximal maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems applications of the results to the construction of generalized augmented lagrangians for nonlinear programs and the minimax problem are presented