compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

commutator theory without join distributivity we develop commutator theory for congruences of general alge braic systems henceforth called algebras assuming only the existence of a ternary term d such that d a b b a a a a a d b b a whenever a is a congruence and aab our results apply in particular to congruence modular and n permutable varieties to most locally finite varieties and to inverse semigroups we obtain results concerning permutability of congruences abelian and solv able congruences connections between congruence identities and commutator identities we show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis for other lattices some intervals are forced to be abelian and others are forced to be nonabelian we give simplified proofs of some results about the commutator in modu lar varieties and generalize some of them to single algebras having a modular congruence lattice

congruence modularity implies the arguesian law for single algebras with a difference term abstract recently a generalization of commutator theory has been developed for algebraic systems belonging to a congruence modular variety this general commutator theory is used here both to provide a very simple proof of a classical result by r freese and b jonsson and to solve a problem raised by s tschantz some other known results are improved or are given a simple proof moreover we pursue the goal of extending commutator theory to a somewhat larger setting thus we give a more general version of the result by freese and jonsson partially solving a problem we raised in p lipparini 1994 trans amer math soc 346 177 202

metrics and smoothing of translation invariant radon transforms along curves abstract in this paper new l p l q estimates are proved for translation invariant radon transforms along curves for and p q for a fixed and if p is sufficiently close to 2 the best possible q is obtained up to e the method is related to that of greenblatt ph d thesis princeton university 1998

commutator theory for relatively modular quasivarieties we develop a commutator theory for relatively modular quasivarieties that extends the theory for modular varieties we characterize relatively modular quasivarieties prove that they have an almost equational axiomatization and we investigate the lattice of subquasivarieties we derive the result that every finitely generated relatively modular quasivariety of semigroups is finitely based

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

self dual regularization of monotone operators via the resolvent average we give two self dual regularizations of maximal monotone operators on hilbert spaces these regularizations and their set valued inverses are strongly monotone single valued and lipschitz with full domain moreover these regularizations graphically converge to the original monotone operator if a maximal monotone operator has nonempty zeros these self dual regularizations can be used to find its least norm solution when the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with nonempty minimizers this translates to finding the least norm minimizer

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

the resolvent average of monotone operators dominant and recessive properties within convex analysis a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago when one considers the subdifferential of the proximal average a natural averaging operation of the subdifferentials of the averaged functions emerges in the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in hilbert spaces transforming it into the resolvent average the theory of resolvent averages contains many desirable properties in particular we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average as a consequence we not only recover a significant part of the theory of proximal averages but also obtain results and properties which have not been previously available

sobolev estimates for fractional and singular radon transforms abstract we prove sobolev inequalities for singular and fractional radon transforms which are defined as in a paper of phong and stein under certain hypothesis on the corresponding lagrangian n c which does not necessarily have to be a canonical graph in the proof we use oscillatory integrals the cotlar stein almost orthogonality theorem a sort of littlewood paley decomposition for a certain operator some basic facts about fourier integral operators and pseudodifferential operators the main ideas come from papers by phong and stein acta math 157 1986 99 157 and sogge and stein j analyse math 54 1990 165 188

a finite basis theorem for difference term varieties with a finite residual bound we prove that if v is a variety of algebras i e an equationally axiomatizable class of algebraic structures in a finite language v has a difference term and v has a finite residual bound then v is finitely axiomatizable this provides a common generalization of r mckenzie s finite basis theorem for congruence modular varieties with a finite residual bound and r willard s finite basis theorem for congruence meet semidistributive varieties with a finite residual bound this paper is a contribution to an old problem in logic from the schools of a tarski and a maltsev which finite algebraic structures a algebras for short have a finite basis for their identities equivalently for which finite algebras a is the variety v a the smallest equational class containing a finitely axiomatizable on the one hand every finite group 38 finite ring 22 28 finite commutative semigroup 40 finite lattice 33 or two element algebra in a finite language 30 is known to be finitely based on the other hand the list of finite algebras which are not finitely based includes in addition to pathological examples e g 31 37 some finite semigroups 40 some finite non associative k algebras 41 29 12 and even a finite group with one non identity element named by a constant 5 in 1996 r mckenzie 36 proved that the problem of determining whether a finite algebra is finitely based is undecidable settling tarski s finite basis problem the evidence suggests that a full classification of finitely based finite algebras is beyond reach however there are some remarkable partial results in particular in the early 1970s k baker 1 2 proved the following if a is a finite algebra in a finite language and v a is congruence distributive i e for every b v a the lattice of congruence relations of b is a distributive lattice then a is finitely based two important ingredients in the proof were provided by b jonsson 13 1 a characterization in terms of identities of the condition that a variety be congruence distributive and 2 a proof that if a is finite and v a is congruence distributive then every subdirectly date july 8 2014 2010 mathematics subject classification primary 03c05 secondary 08b05

asplund decomposition of monotone operators we establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a remainder monotone mapping where the remainder is acyclic in the sense that it contains no nontrivial subdifferential component this is the nonlinear analogue of a skew linear operator examples of indecomposable and acyclic operators are given in particular we present an explicit nonlinear acyclic operator

boundedness of singular radon transforms on l p spaces under a finite type condition in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger are proven 1 introduction in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger in 3 are proved singular radon transforms are a type of singular integral operator a prototype for which is the hilbert transform along curves let 1 1 r n be a curve with 0 0 and let f be a schwarz function on r n 1

an explicit example of a maximal 3 cyclically monotone operator with bizarre properties abstract subdifferential operators of proper convex lower semicontinuous functions and more generally maximal monotone operators are ubiquitous in optimization and nonsmooth analysis in between these two classes of operators are the maximal n cyclically monotone operators these operators were carefully studied by asplund who obtained a complete characterization within the class of positive semidefinite not necessarily symmetric matrices and by voisei who presented extension theorems a la minty all previous explicit examples of maximal n cyclically monotone operators are maximal monotone thus they inherit the known good properties of maximal monotone operators in this paper we construct an explicit maximal 3 cyclically monotone operator with quite bizarre properties this construction builds upon a recent nonconstructive and zorn s lemma based example our operator possesses two striking properties that sets it far apart from both the maximal monotone operator and the subdifferential operator case it is not maximal monotone and its domain which is closed fails to be convex indeed the domain is the boundary of the unit diamond in the euclidean plane the path leading to this operator requires some new results that are interesting in their own right

the lp lq estimates for degenerate radon transforms in c2 abstract let p be a polynomial in c 2 of bounded degree we obtain some l p l q estimates for the semi translation invariant radon like transform defined by r f z c f p z z d a where is a cutoff function supported in a small neighborhood of the origin here d a d x d y provided that x i y where x y r

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

autoconjugate representers for linear monotone operators monotone operators are of central importance in modern optimization and nonlinear analysis their study has been revolutionized lately due to the systematic use of the fitzpatrick function pioneered by penot and svaiter a topic of recent interest has been the representation of maximal monotone operators by so called autoconjugate functions two explicit constructions were proposed the first by penot and z linescu in 2005 and another by bauschke and wang in 2007 the former requires a mild constraint qualification while the latter is based on the proximal average we show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces the continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate furthermore we also construct an infinite family of autoconjugate representers for the identity operator on the real line

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

an alternating projection that does not converge in norm this paper proves that the method of alternating projections between two closed convex intersecting sets does not always converge in norm weak convergence was established by bregman soviet math dokl 6 1965 688 but the status of norm convergence was undetermined an explicit counterexample is provided

nearly convex sets fine properties and domains or ranges of subdifferentials of convex functions nearly convex sets play important roles in convex analysis optimization and theory of monotone operators we give a systematic study of nearly convex sets and construct examples of subdifferentials of lower semicontinuous convex functions whose domains or ranges are nonconvex

a variational method using fractional order hilbert spaces for tomographic reconstruction of blurred and noised binary images we provide in this article a refined functional analysis of the radon operator restricted to axisymmetric functions and show that it enjoys strong regularity properties in fractional order hilbert spaces this study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects for which we have available one single blurred and noised snapshot we propose a variational approach to handle this problem consisting in solving a minimization problem settled in adapted fractional order hilbert spaces we show the existence of solutions and then derive first order necessary conditions for optimality in the form of optimality systems

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by b f svaiter abstract in this paper we give two explicit examples of unbounded linear maximal monotone operators the first unbounded linear maximal monotone operator s on l 2 is skew we show its domain is a proper subset of the domain of its adjoint s and s is not maximal monotone this gives a negative answer to a recent question posed by svaiter the second unbounded linear maximal monotone operator is the inverse volterra operator t on l 2 0 1 we compare the domain of t with the domain of its adjoint t and show that the skew part of t admits two distinct linear maximal monotone skew extensions these unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened and they are simpler than the example given by phelps simons interesting consequences on fitzpatrick functions for sums of two maximal monotone operators are also given

the brezis browder theorem in a general banach space during the 1970s brezis and browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces in this paper we extend and refine their result to a general banach space we also provide an affirmative answer to a problem posed by phelps and simons

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

projection and proximal point methods convergence results and counterexamples recently hundal has constructed a hyperplane h a cone k and a starting point y0 in 2 such that the sequence of alternating projections pk ph n y0 n n converges weakly to some point in h k but not in norm we show how this construction results in a counterexample to norm convergence for iterates of averaged projections hence we give an a rmative answer to a question raised by reich two decades ago furthermore new counterexamples to norm convergence for iterates of rmly nonexpansive maps a la genel and lindenstrauss and for the proximal point algorithm al a g uler are provided we also present a counterexample along with some weak and norm convergence results for the new framework of string averaging projection methods introduced by censor et al extensions to banach spaces and the situation for the hilbert ball are discussed as well 2003 elsevier ltd all rights reserved msc 47h09 47j25 90c25

minimal antiderivatives and monotonicity abstract we consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope given certain partial data regarding a subdifferential we consider the family of all convex antiderivatives that comply with the given data we prove that this family is not empty and in particular contains a minimal antiderivative under a fairly general assumption on the given data it turns out that the representation of monotone operators by convex functions fits naturally in these settings duality properties of representing functions are also captured by these settings and the gap between the fitzpatrick function and the fitzpatrick family is filled by this broader sense of minimality of the fitzpatrick function

the sum of a maximal monotone operator of type fpv and a maximal monotone operator with full domain is maximal monotone abstract the most important open problem in monotone operator theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that rockafellar s constraint qualification holds in this paper we prove the maximal monotonicity of a b provided that a and b are maximal monotone operators such that dom a int dom b a n dom b is of type fpv and dom a dom b dom b the proof utilizes the fitzpatrick function in an essential way

the proximal average basic theory the recently introduced proximal average of two convex functions is a convex function with many useful properties in this paper we introduce and systematically study the proximal average for finitely many convex functions the basic properties of the proximal average with respect to the standard convex analytical notions domain fenchel conjugate subdifferential proximal mapping epi continuity and others are provided and illustrated by several examples

monotone operator theory in convex optimization several aspects of the interplay between monotone operator theory and convex optimization are presented the crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized we review the properties of subdifferentials as maximally monotone operators and in tandem investigate those of proximity operators as resolvents in particular we study new transformations which map proximity operators to proximity operators and establish connections with self dual classes of firmly nonexpansive operators in addition new insights and developments are proposed on the algorithmic front

twosetsofnewcharacterizationsfornormaland ep matrices two sets of new characterizations for normal matrices and ep matrices are presented which are derived through ranks and generalized inverses of matrices

primal dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators a fundamental result due to rockafellar states that every cyclically monotone operator a admits an antiderivative f in the sense that the graph of a is contained in the graph of the subdifferential operator partial f given a method mathfrak m that assigns every finite cyclically monotone operator a some antiderivative mathfrak m a we say that the method is primal dual symmetric if mathfrak m applied to the inverse of a produces the fenchel conjugate of mathfrak m a rockafellar s antiderivatives do not possess this property utilizing fitzpatrick functions and the proximal average we present novel primal dual symmetric intrinsic methods the antiderivatives produced by these methods provide a solution to a problem posed by rockafellar in 2005 the results leading to this solution are illustrated by various examples