maximal quasimonotonicity and dense single directional properties of quasimonotone operators the concept of maximal quasimonotonicity of set valued map is introduced and studied regularity properties of this class of operators is investigated in particular the ccvc property an adaptation to the quasimonotone case of the classical notion of cusco map in an asplund space we provide sufficient conditions for a ccvc quasimonotone operator to be single directional on a g delta dense subset

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

on extensions of kenderov s single valuedness result for monotone maps and quasimonotone maps one of the most famous single valuedness results for set valued maps is due to kenderov fund math lxxxviii 1975 pp 61 69 and states that a monotone set valued operator is single valued at any point where it is lower semicontinuous this has been extended in christensen and kenderov math scand 54 1984 pp 70 78 to monotone maps satisfying a so called property our aim in this work is twofold first to prove that the property assumption can be weakened and second to emphasize that these classical single valuedness results for monotone operators can be obtained in very simple way as direct consequences of counterpart results proved for quasi monotone operators in terms of single directionality

when is the space of minimal usco cusco maps a topological vector space abstract we will define a vector structure on the space of minimal usco resp cusco maps from a locally compact space to a stegall locally convex space we will prove that these spaces equipped with the topology of uniform convergence on compacta are locally convex spaces

null sets and essentially smooth lipschitz functions in this paper we extend the notion of a lebesgue null set to a notion which is valid in any completely metrizable abelian topological group we then use this definition to introduce and study the class of essentially smooth functions these are roughly speaking those lipschitz functions which are smooth in each direction almost everywhere

non equilibrium dynamics of disordered systems understanding the broad continuum of relevant time scales via a strong disorder rg in configuration space we show that an appropriate description of the non equilibrium dynamics of disordered systems is obtained through a strong disorder renormalization procedure in configuration space that we define for any master equation with transitions rates between configurations the idea is to eliminate iteratively the configuration with the highest exit rate to obtain renormalized transition rates between the remaining configurations the multiplicative structure of the new generated transition rates suggests that for a very broad class of disordered systems the distribution of renormalized exit barriers defined as will become broader and broader upon iteration so that the strong disorder renormalization procedure should become asymptotically exact at large time scales we have checked numerically this scenario for the non equilibrium dynamics of a directed polymer in a two dimensional random medium

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

numerical approximation of phase field based shape and topology optimization for fluids we consider the problem of finding optimal shapes of fluid domains the fluid obeys the navier stokes equations inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow we formulate a corresponding optimization problem where flow outside the fluid domain is penalized the resulting formulation of the shape optimization problem is shown to be well posed hence there exists a minimizer and first order optimality conditions are derived for the numerical realization we introduce a mass conserving gradient flow and obtain a cahn hilliard type system which is integrated numerically using the finite element method an adaptive concept using reliable residual based error estimation is exploited for the resolution of the spatial mesh the overall concept is numerically investigated and comparison values are provided

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

a constructive proof of the existence of a utility in revealed preference theory within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence hence completely characterizes the associated preference structure by taking a dense demand sample this resolves the problem of revealed preferences under some very mild assumptions on the demand correspondence which are closely related to a number of established axioms in preference theory the proof establishes the existence of a limit of a sequences of indirect utilities that rationalise finite data sets where the sample size increases to infinity this limiting utility provides a rationalisation of the demand relation up to a rescaling this limiting indirect utility is unique and continuous on a set of full measure

critical behavior of an even offspringed branching and annihilating random walk cellular automaton with spatial disorder a stochastic cellular automaton exhibiting a parity conserving class transition has been investigated in the presence of quenched spatial disorder by large scale simulations numerical evidence has been found that weak disorder causes irrelevant perturbation for the universal behavior of the transition and the absorbing phase of this model this opens up the possibility for experimental observation of the critical behavior of a nonequilibrium phase transition to absorbing state for very strong disorder the model breaks up into blocks with exponential size distribution and continuously changing critical exponents are observed for strong disorder the randomly distributed diffusion walls introduce another transition within the inactive phase of the model in which residual particles survive the extinction the critical dynamical behavior of this transition has been explored

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

generalized subdifferentials a baire categorical approach we use baire categorical arguments to construct dramatically pathological locally lipschitz functions the origins of this approach can be traced back to banach and mazurkiewicz 1931 who independently used similar categorical arguments to show that almost every continuous real valued function defined on 0 1 is nowhere differentiable as with the results of banach and mazurkiewicz it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example among the most striking results contained in this paper are almost every 1 lipschitz function defined on a banach space has a clarke subdifferential mapping that is identically equal to the dual ball if t 1 t 2 ldots t n is a family of maximal cyclically monotone operators defined on a banach space x then there exists a real valued locally lipschitz function g such that partial 0 g x mbox co t 1 x t 2 x ldots t n x for each x in x in a separable banach space each non empty weak compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some lipschitz function and for locally lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

phase transitions and singularities in random quantum systems random quantum systems that exhibit unusual behavior associated with infinite randomness fixed points are discussed focusing on the random quantum ising model this system undergoes a transition at zero temperature from a phase with infinite susceptibility and continuously variable exponents to a ferromagnetic phase via a quantum critical point characterized by tunneling scaling with energy and length scales l related by 1n l exact results in one dimension and a scaling picture in higher dimensions are derived from a simple renormalization group other random quantum critical points and quantum disordered phases that can exhibit similar features are discussed briefly

adjusted sublevel sets normal operator and quasi convex programming a new notion of adjusted sublevel set of a function is introduced and studied these sets lie between the sublevel and strict sublevel sets of the function in contrast to the normal operators to sublevel or strict sublevel sets that were studied in the literature so far the normal operator to the adjusted sublevel sets is both quasi monotone and in the case of quasi convex functions cone upper semicontinuous this makes this new notion appropriate for all kinds of quasi convex functions and in particular for quasi convex functions whose graph presents a flat part application is given to quasi convex optimization through the study of an associated variational inequality problem

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

random walk in a two dimensional self affine random potential properties of the anomalous diffusion phase at small external force we study the dynamical response to an external force f for a particle performing a random walk in a two dimensional quenched random potential of hurst exponent h 1 2 we present numerical results on the statistics of first passage times that satisfy closed backward master equations we find that there exists a zero velocity phase in a finite region of the external force 0lfl f c where the dynamics follows the anomalous diffusion law x t ensuremath sim ensuremath xi f t ensuremath mu f the anomalous exponent 0l ensuremath mu f l1 and the correlation length ensuremath xi f vary continuously with f in the limit of vanishing force f ensuremath rightarrow 0 we measure the following power laws the anomalous exponent vanishes as ensuremath mu f ensuremath propto f a with a ensuremath simeq 0 6 instead of a 1 in dimension d 1 and the correlation length diverges as ensuremath xi f ensuremath propto f ensuremath ensuremath nu with ensuremath nu ensuremath simeq 1 29 instead of ensuremath nu 2 in dimension d 1 our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model where the traps are characterized by a typical length ensuremath xi f along the direction of the force and by a typical barrier 1 ensuremath mu f the fact that these traps are smaller in linear size and in depth than in dimension d 1 means that the particle uses the transverse direction to find lower barriers

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

the maximal normal operator space and integration of subdifferentials on nonconvex functions on definit des operateurs normaux maximaux et des fonctions primales sur des ensembles normaux on trouve que la normalite maximale est conservee dans plusieurs operations et que les operateurs normaux maximaux ont de bonnes proprietes de fermeture et de limitation

lipschitz functions with prescribed derivatives and subderivatives in general it is difficult to construct lipschitz functions which are not directly built up from either convex or distance functions one impediment to such constructions is that outside of the real line it is difficult to find anti derivatives the main result of this paper provides under suitable circumstances a technique for constructing such anti derivatives more precisely we show that if f 1 f 2 dots f n are continuously differentiable real valued locally lipschitz functions defined on a non empty open subset a of a separable banach space x then there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co nabla f 1 x nabla f 2 x dots nabla f n x this same construction also shows that for any finite family t 1 t 2 dots t n of maximal cyclically monotone mappings from a into non empty subsets of x there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co t 1 x t 2 x dots t n x moreover we show that g is convex if and only if t 1 t 2 cdots t n

a continuity property related to kuratowski s index of non compactness its relevance to the drop property and its implications for differentiability theory abstract we define and study the properties of upper semi continuity a new continuity property for set valued mappings from a topological space into subsets of a metric space expressed in terms of kuratowski s index of non compactness this upper semi continuity is related closely to the usual upper semi continuity but significantly upper semi continuous minimal weak cuscos from a baire space into subsets of the dual of any banach space are generically single valued kuratowski s index of non compactness has been used to study the drop property and upper semi continuity is dual to property studied there uniform upper semi continuity of the duality mapping is dual to nearly uniform rotundity properties importantly upper semi continuity has application in differentiability theory providing another characterisation for asplund spaces an examination of an associated concept denting point for sets yields still further advances concerning the differentiability of convex functions on a large class of banach spaces

a geometrical insight on pseudoconvexity and pseudomonotonicity generalised convexity is revisited from a geometric point of view a substitute to the subdifferential is proposed then generalised monotonicity is considered a representation of generalised monotone maps allows us to obtain a symmetry between maps and their inverses finally maximality of generalised monotone maps is analysed

single directional property of multivalued maps and variational systems dontchev and hager math oper res 19 1994 pp 753 768 have shown that a monotone set valued map defined from a banach space to its dual which satisfies the aubin property around a point x y of its graph is actually single valued in a neighborhood of x we prove a result which is the counterpart of the above for quasi monotone set valued maps based on the concept of single directional property as applications we provide sufficient conditions for this single valued property to hold for the solution map of general variational systems and quasi variational inequalities we also investigate the single directionality property for the normal operator to the sublevel sets of a quasi convex function

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

a phase field method for shape optimization of incompressible flows abstract in this paper we present a phase field method applied to the fluid based shape optimization the fluid flow is governed by the incompressible navier stokes equations a phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized the shape sensitivity analysis is presented in terms of phase field variable which is the main contribution of this paper it saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes numerical results on some benchmark problems are reported and it is shown that the phase field approach for fluid shape optimization is efficient and robust

a chain rule for essentially smooth lipschitz functions in this paper we introduce a new class of real valued locally lipschitz functions that are similar in nature and definition to valadier s saine functions which we call arcwise essentially smooth and we show that if g rm rightarrow r is arcwise essentially smooth on rm and each function f j r n rightarrow r 1 leq j leq m is strictly differentiable almost everywhere in rn then g circ f is strictly differentiable almost everywhere in rn where f equiv f 1 f 2 ldots f m we also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth thus we provide a large and robust lattice algebra of lipschitz functions whose generalized derivatives are well behaved

a phase field method for two phase fluid flow in arbitrary domains abstract we present a practical and efficient numerical method for two phase flow simulations in arbitrary domains a simple mathematical equation for the two phase fluid in arbitrary domains is used the proposed method is based on the ternary cahn hilliard ch system coupled with the modified navier stokes ns equation a complex domain is expressed by the third phase in the ternary system when we solve the ternary ch system we only solve the equation for one phase with the fixed third phase the second phase value is obtained from the local mass conservation a no slip condition was imposed for the velocity on the arbitrary domain boundaries the proposed method uses an unconditionally gradient stable type scheme y li et al 2016 for the convective ch equation and a projection method for the modified ns equation various numerical experiments such as the droplet movement in a cavity flow the rayleigh taylor instability in a porous media droplet passing through a triangle shaped solid structure and droplet movement in a circuitous channel were performed to demonstrate that the proposed method can deal with the two phase flow in arbitrary domains

integrability of subdifferentials of certain bivariate functions abstract we study integrability properties of bivariate functions defined on the product of banach spaces

shape and topology optimization based on the phase field method and sensitivity analysis this paper discusses a structural optimization method that optimizes shape and topology based on the phase field method the proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions the advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required structural shapes are represented by the phase field function defined in the design domain and optimization of this function is performed by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis the proposed method is applied to two dimensional linear elastic and vibration optimization problems such as the minimum compliance problem a compliant mechanism design problem and the eigenfrequency maximization problem the numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations

a derivative for complex lipschitz maps with generalised cauchy riemann equations a r t i c l e i n f o a b s t r a c t we introduce the lipschitz derivative or the l derivative of a locally lipschitz complex map it is a scott continuous compact and convex set valued map that extends the classical derivative to the bigger class of locally lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of cauchy riemann equations to these maps which form a continuous scott domain we show that a complex lipschitz map is analytic in an open set if and only if its l derivative is a singleton at all points in the open set the calculus of the l derivative for sum product and composition of maps is derived the notion of contour integration is extended to scott continuous non empty compact convex valued functions on the complex plane and by using the l derivative the fundamental theorem of contour integration is extended to these functions 2014 the author published by elsevier b v this is an open access article under the cc by license http creativecommons org licenses by 3 0

variational piecewise constant level set methods for shape optimization of a two density drum we apply the piecewise constant level set method to a class of eigenvalue related two phase shape optimization problems based on the augmented lagrangian method and the lagrange multiplier approach we propose three effective variational methods for the constrained optimization problem the corresponding gradient type algorithms are detailed the first uzawa type algorithm having applied to shape optimization in the literature is proven to be effective for our model but it lacks stability and accuracy in satisfying the geometry constraint during the iteration the two other novel algorithms we propose can overcome this limitation and satisfy the geometry constraint very accurately at each iteration moreover they are both highly initial independent and more robust than the first algorithm without penalty parameters the last projection lagrangian algorithm has less severe restriction on the time step than the first two algorithms numerical results for various instances are presented and compared with those obtained by level set methods the comparisons show effectiveness efficiency and robustness of our methods we expect our promising algorithms to be applied to other shape optimization and multiphase problems

correlation length bounds for disordered ising ferromagnets thed dimensional nearest neighbor disordered ising ferromagnet h sum j ij sigma i sigma j is studied as a function of both temperature t and a disorder parameter which measures the size of fluctuations of couplingsjij 0 a finite size scaling correlation length f t is defined in terms of the magnetic response of finite samples this correlation length is shown to be equivalent in the scaling sense to the quenched average correlation length t defined as the asymptotic decay rate of the quenched average two point function furthermore the magnetic response criterion which defines f is shown to have a scale invariant property at the critical point the above results enable us to prove that the quenched correlation length satisfies c left log xi t right xi t geqq left t t c right 2 mathord left vphantom 2 d right kern nulldelimiterspace d which implies the boundv 2 d for the quenched correlation length exponent

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems