conforming approximation of convex functions with the finite element method we consider the interior approximation of convex functions with convex finite element functions the main motivation for this study is the investigation of a novel discretization of optimization problems with convexity constraints by the finite element method under a mild assumption on the family of meshes we show that the conforming approximation is convergent if the finite elements are at least piecewise quadratic we further provide similar results under additional constraints on the function values or on the gradient the theoretical findings are illustrated by numerical examples

surface states and spectra let d 1 let h 0 be the discrete laplacian on the hilbert space l 2 d 1 with a dirichlet boundary condition and let v be a potential supported on the boundary d 1 we introduce the notions of surface states and surface spectrum of the operator h h 0 v and explore their properties our main result is that if the potential v is random and if the disorder is either large or small enough then in dimension two h has no surface spectrum on h 0 with probability one to prove this result we combine aizenman molchanov theory with techniques of scattering theory

anderson localization and lifshits tails for random surface potentials we consider schrodinger operators on l2 rd with a random potential concentrated near the surface rd1 0 rd we prove that the integrated density of states of such operators exhibits lifshits tails near the bottom of the spectrum from this and the multiscale analysis by boutet de monvel and stollmann arch math 80 2003 87 97 we infer anderson localization pure point spectrum and dynamical localization for low energies our proof of lifshits tails relies on spectral properties of schrodinger operators with partially periodic potentials in particular we show that the lowest energy band of such operators is parabolic

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

ironing sweeping and multidimensional screening we provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of mussa and rosen 1978 these results are also directly applicable to the multidimensional nonlinear pricing problems studied by wilson 1993 and armstrong 1996 we establish that bunching is robust in these multidimensional screening problems even with very regular distributions of types this comes from a strong conflict between participation constraints and second order incentive compatibility conditions we consequently design a new technique the sweeping procedure for dealing with bunching in multidimensional contexts this technique extends the ironing procedure of mussa and rosen 1978 to several dimensions we illustrate it on two examples we first solve a linear quadratic version of the bidimensional nonlinear pricing problem where consumers types are exponentially distributed the solution involves pure bundling for consumers with low demands the second example is the bidimensional version of the mussa and rosen problem when consumers types are uniformly distributed on a square the solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum but only limited choice for lower qualities this seems to be a quite general pattern for multidimensional screening problems the sweeping procedure is potentially applicable to other multidimensional screening problems

approximating optimization problems over convex functions many problems of theoretical and practical interest involve finding an optimum over a family of convex functions for instance finding the projection on the convex functions in hk and optimizing functionals arising from some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and functions with positive semidefinite discrete hessian need not be convex in a discrete sense previous work has concentrated on non local descriptions of convexity making the number of constraints to grow super linearly with the number of nodes even in dimension 2 and these descriptions are very difficult to extend to higher dimensions in this paper we propose a finite difference approximation using positive semidefinite programs and discrete hessians and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to problems in two and three dimensions

handling convexity like constraints in variational problems we provide a general framework to construct finite dimensional approximations of the space of convex functions which also applies to the space of c convex functions and to the space of support functions of convex bodies we give precise estimates of the distance between the approximation space and the admissible set this framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite dimensional spaces such as tensor product splines we show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints our implementation relies on proximal algorithms and can be easily parallelized thus making it applicable to large scale problems in dimension two and three we illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation three dimensional denoising the principal agent problem and optimizat

regularity of solutions for some variational problems subject to a convexity constraint we first study the minimizers in the class of convex functions of an elliptic functional with nonhomogeneous dirichlet boundary conditions we prove c1 regularity of the minimizers under the assumption that the upper envelope of admissible functions is c1 this condition is optimal at least when the functional depends only on the gradient 3 we then give various extensions of this result in particular we consider a problem without boundary conditions arising in an economic model introduced by rochet and chone in 4

billiards and two dimensional problems of optimal resistance a body moves in a medium composed of noninteracting point particles the interaction of the particles with the body is completely elastic the problem is find the body s shape that minimizes or maximizes resistance of the medium to its motion this is the general setting of the optimal resistance problem going back to newton here we restrict ourselves to the two dimensional problems for rotating generally non convex bodies the main results of the paper are the following first to any compact connected set with piecewise smooth boundary b subset mathbb r 2 we assign a measure b on conv b 2 2 generated by the billiard in mathbb r 2 setminus b and characterize the set of measures b second using this characterization we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special monge kantorovich problems

uniqueness of the cheeger set of a convex body abstract we prove that if c r n is an open bounded convex set then there is only one cheeger set inside c and it is convex a cheeger set of c is a set which minimizes the ratio perimeter over volume among all subsets of c

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

on waring s problem for cubes and smooth weyl sums non trivial estimates for fractional moments of smooth cubic weyl sums are developed complemented by bounds for such sums of use on both the major and minor arcs in a hardy littlewood dissection these estimates are applied to derive an upper bound for the s th moment of the smooth cubic weyl sum of the expected order of magnitude as soon as s ge 7 691 related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers all of whose prime divisors are at most exp c log n log log n 1 2 for a suitable positive number c this conclusion improves a previous result of g harcos in which nine cubes are required 1991 mathematics subject classification 11p05 11l15 11p55

on convex functions and the finite element method many problems of theoretical and practical interest involve finding a convex or concave function for instance optimization problems such as finding the projection on the convex functions in h k omega or some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and an adequate discrete version of the hessian must be given in this paper we propose a finite element description of the hessian and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to optimization problems

localization of surface spectra we study spectral properties of the discrete laplacian h on the half space with random boundary condition the v n are independent random variables on a probability space and is the coupling constant it is known that if the v n have densities then on the interval 2 d 1 2 d 1 h 0 the spectrum of the dirichlet laplacian the spectrum of h is p a s absolutely continuous for all jl1 here we show that if the random potential p satisfies the assumption of aizenman molchanov am then there are constants d and d such that for d the spectrum of h outside h 0 is p a s pure point with exponentially decaying eigenfunctions

spectral structure of anderson type hamiltonians we study self adjoint operators of the form h h 0 n n where the n s are a family of orthonormal vectors and the n s are independently distributed random variables with absolutely continuous probability distributions we prove a general structural theorem saying that for each pair n m if the cyclic subspaces corresponding to the vectors n and m are not completely orthogonal then the restrictions of h to these subspaces are unitarily equivalent with probability one this has some consequences for the spectral theory of such operators in particular we show that well behaved absolutely continuous spectrum of anderson type hamiltonians must be pure and use this to prove the purity of absolutely continuous spectrum in some concrete cases

discretization of functionals involving the monge ampere operator gradient flows in the wasserstein space have become a powerful tool in the analysis of diffusion equations following the seminal work of jordan kinderlehrer and otto jko the numerical applications of this formulation have been limited by the difficulty to compute the wasserstein distance in dimension geqslant 2 one step of the jko scheme is equivalent to a variational problem on the space of convex functions which involves the monge ampere operator convexity constraints are notably difficult to handle numerically but in our setting the internal energy plays the role of a barrier for these constraints this enables us to introduce a consistent discretization which inherits convexity properties of the continuous variational problem we show the effectiveness of our approach on nonlinear diffusion and crowd motion models

two dimensional body of maximum mean resistance a two dimensional body exhibiting a slight rotational movement moves in a rarefied medium of particles which collide with it in a perfectly elastic way in previously realized investigations by the first two authors alexander yu plakhov paulo d f gouveia problems of maximal mean resistance on the plane nonlinearity 20 2007 2271 2287 shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement giving continuity to this study new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier this result now presented consists of a two dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value but its interest does not lie solely in the maximization of newtonian resistance on regarding its characteristics other areas of application are seen to begin to appear which are thought to be capable of having great utility the optimal shape which has been encountered resulted from numerical studies thus it is the object of additional study of an analytical nature where it proves some important properties which explain in great part its effectiveness

the hasse principle for pairs of diagonal cubic forms by means of the hardy littlewood method we apply a new mean value theorem for exponential sums to confirm the truth over the rational numbers of the hasse principle for pairs of diagonal cubic forms in thirteen or more variables

an algorithm for computing solutions of variational problems with global convexity constraints we present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions the main motivation behind the numerical method is to compute solutions to adverse selection problems within a principal agent framework problems such as product lines design optimal taxation structured derivatives design etc can be studied through the scope of these models we develop a method to estimate their optimal pricing schedules

regularity of the surface density of states abstract we prove that the integrated surface density of states of continuous or discrete anderson type random schrodinger operators is a measurable locally integrable function rather than a signed measure or a distribution this generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of a chahrour in the discrete case the proof uses the new l p bound on the spectral shift function recently obtained by combes hislop and nakamura also we provide a simple proof of their result on the holder continuity of the integrated density of bulk states

adaptive anisotropic and hierarchical cones of discrete convex functions we introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions among the various mathematical problems which possess such a formulation the monopolist problem rochet and chone econometrica 66 783 826 1998 ekeland and moreno bromberg numer math 115 45 69 2010 arising in economics is our main motivation consider a two dimensional domain omega sampled on a grid x x of n points we show that the cone mathrm conv x conv x of restrictions to x x of convex functions on omega is typically characterized by approx n 2 n2 linear inequalities a direct computational use of this description therefore has a prohibitive complexity we thus introduce a hierarchy of sub cones mathrm conv mathcal v conv v of mathrm conv x conv x associated to stencils mathcal v v which can be adaptively locally and anisotropically refined we show using the arithmetic structure of the grid that the trace u x u x of any convex function u on omega is contained in a cone mathrm conv mathcal v conv v defined by only mathcal o n ln 2 n o nln2n linear constraints in average over grid orientations numerical experiments for the monopolist problem based on adaptive stencil refinement strategies show that the proposed method offers an unrivaled accuracy complexity trade off in comparison with existing methods we also obtain as a side product of our theory a new average complexity result on edge flipping based mesh generation

wave operators for the surface maryland model we study scattering properties of the discrete laplacian h on the half space z d 1 zd z with the boundary condition n 1 tan n n 0 where 0 1 d we denote by h0 the dirichlet laplacian on z d 1 khoruzenko and pastur phys rep 288 109 126 1997 have shown that if has typical diophantine properties then the spectrum of h on r h0 is pure point and that corresponding eigenfunctions decay exponentially we demonstrated in an earlier paper lett math phys 45 185 1998 that for every independent over the rationals the spectrum of h on h0 is purely absolutely continuous in this paper we continue the analysis of h on h0 and prove that whenever is independent over the rationals the wave operators h h0 exist and are complete on h0 moreover we show that under the same conditions h has no surface states on h0

the numerical solution of newton s problem of least resistance in this paper we consider newton s problem of finding a convex body of least resistance this problem could equivalently be written as a variational problem over concave functions in mathbb r 2 r 2 we propose two different methods for solving it numerically first we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions the discretized problem could be solved by standard optimization software efficiently second we conjecture that the optimal body has a certain structure we exploit this structure and obtain a variational problem in mathbb r 1 r 1 deriving its euler lagrange equation yields a program with two unknowns which can be solved quickly

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

a numerical approach to variational problems subject to convexity constraint we describe an algorithm to approximate the minimizer of an elliptic functional in the form int omega j x u nabla u on the set cal c of convex functions u in an appropriate functional space x such problems arise for instance in mathematical economics 4 a special case gives the convex envelope u 0 of a given function u 0 let t n be any quasiuniform sequence of meshes whose diameter goes to zero and i n the corresponding affine interpolation operators we prove that the minimizer over cal c is the limit of the sequence u n where u n minimizes the functional over i n cal c we give an implementable characterization of i n cal c then the finite dimensional problem turns out to be a minimization problem with linear constraints

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

a numerical method for variational problems with convexity constraints we consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex this problem is at the interface between convex analysis convex optimization variational problems and partial differential equation techniques the approach is to approximate the nonpolyhedral cone of convex functions by a polyhedral cone which can be represented by linear inequalities this approach leads to an optimization problem with linear constraints which can be computed efficiently hundreds of times faster than existing methods

cantor spectrum of graphene in magnetic fields this research was partially supported by the nsf dms 1401204 support by the epsrc grant ep l016516 1 for the university of cambridge cdt the cca is gratefully acknowledged s b r h would like to thank the institute for advanced study princeton for its hospitality during the 2017 18 academic year r h was supported in part by nsf dms 1638352

problems of optimal transportation on the circle and their mechanical applications abstract we consider a mechanical problem concerning a 2d axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry the body moves through a medium of non interacting particles at rest and collisions of particles with the body s boundary are perfectly elastic billiard like the body has a blunt nose a line segment orthogonal to the symmetry axis it is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance this problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transportation problem on the circle with the transportation cost being the squared euclidean distance we find the explicit solution for this problem when the amplitude of rotation is smaller than a fixed critical value and give a numerical solution otherwise as a by product we get explicit description of the solution for a class of optimal transportation problems on the circle

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