aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

diophantine geometry over groups viii stability this paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups projections of such sets diophantine sets and the structure of denable sets over free and hyperbolic groups in this eighth paper we use a modication of the sieve procedure which was used in proving quantier elimination in the theory of a free group to prove that free and torsion free gromov hyperbolic groups are stable in the

a multi scale vectorial l τ tv framework for color image restoration a general multi scale vectorial total variation model with spatially adapted regularization parameter for color image restoration is introduced in this paper this total variation model contains an l data fidelity for any c 1 2 the use of a spatial dependent regularization parameter improves the reconstruction of features in the image as well as an adequate smoothing for the homogeneous parts the automated adaptation of this regularization parameter is made according to local statistical characteristics of the noise which contaminates the image the corresponding multiscale vectorial total variation model is solved by fenchel duality and inexact semismooth newton techniques numerical results are presented for the cases 1 and 2 which reconstruct images contaminated with salt and pepper noise and gaussian noise respectively

analytical aspects of spatially adapted total variation regularisation abstract in this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem motivated by its application in signal recovery tasks we study in depth the relationship between the weight function and the creation of new discontinuities in the solution a partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed we prove that the weighted total variation minimisation problem is well posed even in the case of vanishing weight function despite the lack of coercivity this is based on the fact that the total variation of the solution is bounded by the total variation of the data a result that it also shown here finally the relationship to the corresponding weighted fidelity problem is explored showing that the two problems can produce completely different solutions even for very simple data functions

on the finite presentation of subdirect products and the nature of residually free groups we establish it virtual surjection to pairs vsp as a general criterion for the finite presentability of subdirect products of groups if gamma 1 ldots gamma n are finitely presented and s we use the vsp criterion to characterize the finitely presented residually free groups we prove that the class of such groups is recursively enumerable we describe an algorithm that given a finite presentation of a residually free group constructs a canonical embedding into a direct product of finitely many limit groups we solve the multiple conjugacy problem and membership problem for finitely presentable subgroups of residually free groups we also prove that there is an algorithm that given a finite generating set for such a subgroup will construct a finite presentation new families of subdirect products of free groups are constructed including the first examples of finitely presented subgroups that are neither rm fp infty nor of stallings bieri type

deciding context unification in first order term unification variables represent well formed terms over a given signature and we are to solve equations built using function symbols from the signature and such variables this problem is well known to be decidable in linear time in second order term unification the variables take arguments i e other terms and a substitution uses those arguments an arbitrary number of times for instance an equation f x c x c e x f c c has a solution x e b where b is a special symbol denoting the place in which the argument is substituted under this substitution both sides evaluate to f c c there are other solutions for instance x e f b b which evaluates both sides tof f c c f c c in general a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct second order unification is in general undecidable context unification is a natural problem in between first and second order unification we deal with equations over terms the variables take arguments but we restrict the set of solutions the argument is used exactly once formally contexts are terms with exactly one occurrence of the special symbol b and in context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation for instance when the aforementioned equation f x c x c e x f c c is treated as a context unification problem then it has exactly one solution x e b other substitutions that are solutions of it as an instance of the second unification problem say x e f b b are not valid as b is used more than once context unification also generalizes satisfiability of word equations which is decidable in pspace the decidability status of context unification remained unknown for almost two decades in this article we show that context unification is in pspace in exptime when tree regular constraints are also allowed those results are obtained by extending the recently developed recompression technique which was previously defined for strings and used to obtain a new pspace algorithm for satisfiability of word equations in this article the technique is generalized to trees and the corresponding algorithm is generalized from word equations to context unification the idea of recompression is to apply simple compression rules replacing pairs of neighboring function symbols to the solution of the context equation to this end we appropriately modify the equation without the knowledge of the actual solution so compressing the solution can be simulated by compressing parts of the equation it is shown that if the compression operations are appropriately chosen then the size of the instance is polynomial during the whole algorithm thus giving a pspace upper bound

total bounded variation regularization as a bilaterally constrained optimization problem it is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem existence of a lagrange multiplier and an optimality system are established this allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations here in particular the primal dual active set method considered as a semismooth newton method is analyzed and superlinear convergence is proved as a by product we obtain that the lagrange multiplier associated with the box constraints acts as an edge detector numerical results for image denoising and zooming resizing show the efficiency of the new approach

on ground state solutions for singular and semi linear problems including super linear terms at infinity abstract we establish a result concerning the existence of entire positive classical and bounded solutions which converge to zero at infinity for the semi linear equation u f x u x r n where f r n 0 0 is a suitable function and 0 is a real parameter this result completes the principal theorem of a mohammed a mohammed ground state solutions for singular semi linear elliptic equations nonlinear analysis 2008 doi 10 1016 j na 2008 11 080 mainly because his result does not address the super linear terms at infinity penalty arguments lower upper solutions and an approximation procedure will be used

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

non existence and existence of entire solutions for a quasi linear problem with singular and super linear terms abstract we establish results concerning non existence and existence of entire positive solutions for the nonlinear elliptic problem p u a x u m b x u n in r n u 0 in r n u x x 0 where m p 1 n with 1 p n a b 0 a b 0 are locally holder continuous functions and 0 is a real parameter the main purpose of this paper in short is to complement the principal theorem of b xu and z yang entire bounded solutions for a class of quasi linear elliptic equations boundary value problems 2007 art id 16407 8 pp showing existence and non existence of solutions for the above problem for 0 appropriately

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

ground state solutions for a semilinear elliptic problem with a convection term abstract by a sub supersolution method and a perturbed argument we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem u p x u q f x u u 0 x r n lim x u x 0 where q 1 2 p x c loc r n for some 0 1 p x 0 x r n and f r n 0 0 is a locally holder continuous function which may be singular at zero

viewing distance aware super resolution for high definition display in this paper we propose a novel algorithm for high definition displays to enlarge low resolution images while maintaining perceptual constancy i e the same field of view perceptual blur radius and the retinal image size in viewer s eyes we model the relationship between a viewer and a display by considering two main aspects of visual perception i e scaling factor and perceptual blur radius as long as we enlarge an image while adjust its image blur levels on the display we can maintain viewer s perceptual constancy we show that the scaling factor should be set in proportion to the viewing distance and the blur levels on the display should be adjusted according to the focal length of a viewer toward this we first refer to edge directions to interpolate a low resolution image with the increasing of viewing distance and the scaling factor after images are interpolated we utilize a local contrast to estimate the spatially varying image blur levels of the interpolated image we then further adjust the image blur levels using a parametric deblurring method which combines l1 as well as l2 reconstruction errors and tikhonov with total variation regularization terms by taking these factors into account high resolution images adaptive to viewing distance on a display can be generated experimental results on both natural image metric and user subjective studies across image scales demonstrate that the proposed super resolution algorithm for high definition displays performs favorably against the state of the art methods

canonical representatives and equations in hyperbolic groups we use canonical representatives in hyperbolic groups to reduce the theory of equations in torsion free hyperbolic groups to the theory in free groups as a result we get an effective procedure to decide if a system of equations in such groups has a solution for free groups this question was solved by makanin ma and razborov ra the case of quadratic equations in hyperbolic groups has already been solved by lysenok ly our whole construction plays an essential role in the solution of the isomorphism problem for torsion free hyperbolic groups se1 se2

the second order expansion of solutions to a singular dirichlet boundary value problem abstract in this paper we mainly study the second order expansion of classical solutions in a neighborhood of to the singular dirichlet problem u b x g u a x f u u 0 x u 0 where is a bounded domain with smooth boundary in r n 0 the weight functions b a c loc are positive in and both may be vanishing or be singular on the boundary the function g c 1 0 0 satisfies lim t 0 g t and f c 0 0 we show that the nonlinear term a x f u does not affect the second order expansion of solutions in a neighborhood of to the problem for some kinds of functions b and a

irreducible affine varieties over a free group ii systems in triangular quasi quadratic form and description of residually free groups abstract we shall prove the conjecture of myasnikov and remeslennikov 4 which states that a finitely generated group is fully residually free every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group if and only if it is embeddable in the lyndon s exponential groupfz x which is thez x completion of the free group herez x is the ring of polynomials of one variable with integer coefficients historically lyndon s attempts to solve tarski s famous problem concerning the elementary equivalence of free groups of different ranks led him to introducefz x an free group is a groupgsuch that the class of formulas true ing is the same as the class of formulas true in a nonabelian free group a finitely generated group is free if and only if it is fully residually free 22 our result gives an algebraic description of free groups we shall give an algorithm to represent a solution set of an arbitrary system of equations overfas a union of finite number of irreducible components in the zariski topology onfn the solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms the possibility of such a decomposition for a solution set was conjectured by razborov in 20 and also by rips we shall give a description of systems of equations determining irreducible components using methods developed in 13 19 it is possible to find some of these methods in 18 we are thankful to e rips for attracting our attention to these techniques

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

cubulating malnormal amalgams let g be a group splitting as a graph of groups where each vertex group acts properly on a cat 0 cube complex when g is hyperbolic relative to virtually abelian subgroups we examine conditions producing a proper action of g on a cat 0 cube complex when g is hyperbolic this action is also cocompact

a singular boundary value problem for a degenerate elliptic pde abstract we consider a singular boundary value problem associated with the laplacian and having nonseparable inhomogeneous term we establish the existence of ground state solutions to such a problem in bounded domains as well as in the whole euclidean space the investigation is based on subsolution and supersolution methods and existence of principal eigenfunctions to the eigenvalue problem of dirichlet type corresponding to the laplacian

complexity of makanin s algorithm the exponent of periodicity is an important factor in estimates of complexity of word unification algorithms we prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1 07d where d is the length of the equation we also give a lower bound 2 0 29d so our upper bound is almost optimal and exponentially better than the original bound 6d 22d4 2 consequently our result implies an exponential improvement of known upper bounds on complexity of word unification algorithms

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

elementary theory of free non abelian groups we prove that any two non abelian free groups have the same elementary theory and that this theory is decidable these results solve two questions that were raised by tarski in 1945

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

finding all solutions of equations in free groups and monoids with involution this paper presents a pspace algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids this became possible due to the recent recompression technique while this technique was successfully applied for pure word equations without involution or rational constraints it could not be used as a black box for free groups actually the presence of an involution and rational constraints complicates the situation and some additional analysis is necessary still the technique is general enough to accommodate both extensions in the end it simplifies proofs that satisfiability of word equations is in pspace and the corresponding result for equations in free groups with rational constraints as a byproduct we can decide in pspace whether the solution set is finite

an infeasible primal dual algorithm for total bounded variation based inf convolution type image restoration in this paper a primal dual algorithm for total bounded variation tv type image restoration is analyzed and tested analytically it turns out that employing a global boldsymbol l s regularization with 1 s leq 2 in the dual problem results in a local smoothing of the tv regularization term in the primal problem the local smoothing can alternatively be obtained as the infimal convolution of the ell r norm with r 1 s 1 1 and a smooth function in the case r s 2 this results in gauss tv type image restoration the globalized primal dual algorithm introduced in this paper works with generalized derivatives converges locally at a superlinear rate and is stable with respect to noise in the data in addition it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration a comprehensive numerical study ends the paper

on pspace generation of a solution set of a word equation and its applications abstract there are three approaches to solve satisfiability problem for word equations makanin s algorithm plandowski s algorithm jez s algorithm the approach by makanin and the approach by jez can be extended to generate a representation of all solutions of a word equation in case of makanin s algorithm the representation is usually an infinite tree in case of jez s approach the representation is a finite graph until now there was no extension of plandowski s algorithm to do the same we fill this gap we present extension of plandowski s algorithm which generates a representation of all solutions of a word equation the representation is a finite graph this allows to decide in p s p a c e whether a solution set of a word equation is finite our approach is generalized to solve in p s p a c e two problems dealing with expressible relations namely the emptiness of the relation and finiteness of the relation

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

a boundary value problem for normalized finsler infinity laplacian equations with singular nonhomogeneous terms abstract we study boundary value problems of finsler infinity laplacian equations with nonhomogeneous terms that may exhibit singularity when solutions vanish on the boundary

the existential theory of equations with rational constraints in free groups is pspace complete it is well known that the existential theory of equations in free groups is decidable this is a celebrated result of makanin which was published 1982 makanin did not discuss complexity issues but later it was shown that the scheme of his algorithm is not primitive recursive in this paper we present an algorithm that works in polynomial space this improvement is based upon an extension of plandowski s techniques for solving word equations we present a pspace algorithm in a more general setting where each variable has a rational constraint that is the solution has to respect a specification given by a regular word language we obtain our main result about the existential theory in free groups as a corollary of the corresponding statement in free monoids with involution

recompression a simple and powerful technique for word equations in this article we present an application of a simple technique of local recompression previously developed by the author in the context algorithms for compressed strings jez 2014a 2015b 2015a to word equations the technique is based on local modification of variables replacing x by ax or xa and iterative replacement of pairs of letters occurring in the equation by a fresh letter which can be seen as a bottom up compression of the solution of the given word equation or to be more specific building a straight line programme for the solution of the word equation using this technique we give new independent and self contained proofs of many known results for word equations to be more specific the presented nondeterministic algorithm runs in o n log n space and in time polynomial in n and log n where n is the size of the input equation and n the size of the length minimal solution of the word equation furthermore for o 1 variables the bound on the space consumption is in fact linear that is o m where m is the size of the space used by the input this yields that for each k the set of satisfiable word equations with k variables is context sensitive the presented algorithm can be easily generalised to a generator of all solutions of the given word equation without increasing the space usage furthermore a further analysis of the algorithm yields an independent proof of doubly exponential upper bound on the size of the length minimal solution the presented algorithm does not use exponential bound on the exponent of periodicity conversely the analysis of the algorithm yields an independent proof of the exponential bound on exponent of periodicity

a remark on the existence of entire solutions of a singular semilinear elliptic problem it is proved that the singular semilinear elliptic equation u p x g u 0 p x x rn 2 n lims 0 g s andg c1 0 0 which is strictly decreasing in 0 has a unique positivec2 loc rn solution that decays to zero near provided 0 t t dt where t imax i x tg x

a second order shape optimization approach for image segmentation the problem of segmentation of a given image using the active contour technique is considered an abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented the speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or newton type directions for the underlying optimization problem the newton type speed function is found by solving an elliptic problem on the current active contour in every time step numerical experiments comparing the classical gradient method with newton s method are presented