exact cumulant kramers moyal like expansion abstract we derive an exact equation a cumulant kramers moyal equation ckme quite similar to the kramers moyal equation kme for the probability distribution of a markovian dynamical system it can be applied to any well behaved converging cumulants continuous time systems such as langevin equations or other models an interesting but significant difference with respect to the kme is that their jump moments are proportional to cumulants of the dynamical variables but not proportional to central moments as is the case for the kme in fact they still obey a weaker version of pawula s theorem namely marcinkiewicz s theorem we compare the results derived from the equations herein with the ones obtained by computing via gaussian and biased and unbiased poisson langevin dynamics and a poisson non langevin model we obtain the exact ckme time evolution equation for the systems and in several cases those are distinct from the fokker planck equation or the kme

new trace norm inequalities for 2 2 blocks of diagonal matrices abstract several new trace norm inequalities are established for 2 n 2 n block matrices in the special case where the four n n blocks are diagonal some of the inequalities are noncommutative analogs of hanner s inequality others describe the behavior of the trace norm under re ordering of diagonal entries of the blocks

monge s problem with a quadratic cost by the zero noise limit of h path processes we study the asymptotic behavior in the zero noise limit of solutions to schrodinger s functional equations and that of h path processes and give a new proof of the existence of the minimizer of monge s problem with a quadratic cost

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

a concentration inequality for product spaces abstract we prove a concentration inequality which asserts that under some mild regularity conditions every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior

brownian dynamics time averaging and colored noise we propose a method to obtain the equilibrium distribution for positions and velocities of a one dimensional particle via time averaging and laplace transformations we apply it to the case of a damped harmonic oscillator in contact with a thermal bath the present method allows us to treat among other cases a gaussian noise function exponentially correlated in time e g gaussian colored noise we obtain the exact equilibrium solution and study some of its properties

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

a proof of sudakov theorem with strictly convex norms the paper establishes a solution to the monge problem in mathbb r n for a possibly asymmetric norm cost function and absolutely continuous initial measures under the assumption that the unit ball is strictly convex but not necessarily differentiable nor uniformly convex the proof follows the strategy initially proposed by sudakov in 1976 found to be incomplete in 2000 the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem by strict convexity mass moves along rays and we also investigate the divergence of the vector field of rays

a study on the action of non gaussian noise on a brownian particle abstract we analytically address the non equilibrium problem of a brownian particle in contact with a thermal reservoir by means of a non gaussian langevin noise term t the presence of noise kurtosis is akin to a second temperature reservoir acting on the system and we exploit its consequences by means of studying a converging exact form for the stationary probability distribution

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

thermostatistics of small nonlinear systems gaussian thermal bath we discuss the statistical properties of small mechanothermodynamic systems one and two particle cases subject to nonlinear coupling and in contact with standard gaussian reservoirs we use a method that applies averages in the laplace fourier space which relates to a generalization of the final value theorem the key advantage of this method lies in the possibility of eschewing the explicit computation of the propagator traditionally required in alternative methods like path integral calculations which is hardly obtainable in the majority of the cases for one particle equilibrium systems we are able to compute the instantaneous equilibrium probability density functions of injected and dissipated power as well as the respective large deviation functions our thorough calculations explicitly show that for such models nonlinearities are irrelevant in the long term statistics which preserve the exact same values as computed for linear cases actually we verify that the thermostatistical effect of the nonlinearities is constricted to the transient towards equilibrium since it affects the average total energy of the system for the two particle system we consider each element in contact with a heat reservoir at different temperatures and focus on the problem of heat flux between them contrarily to the one particle case in this steady state nonequilibrium model we prove that the heat flux probability density function reflects the existence of nonlinearities in the system an important consequence of that it is the temperature dependence of the conductance which is unobserved in linear harmonic models our results are complemented by fluctuation relations for the injected power equilibrium case and heat flux nonequilibrium case

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

on regularity of transport density in the monge kantorovich problem we show that the optimal regularity result for the transport density in the classical monge kantorovich optimal mass transport problem with the measures having summable densities is a sobolev differentiability along transport rays

regularization techniques for learning with matrices there is growing body of learning problems for which it is natural to organize the parameters into a matrix as a result it becomes easy to impose sophisticated prior knowledge by appropriately regularizing the parameters under some matrix norm this work describes and analyzes a systematic method for constructing such matrix based regularization techniques in particular we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate our methodology is based on a known duality phenomenon a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm this result has already been found to be a key component in deriving and analyzing several learning algorithms we demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi task learning multi class learning and multiple kernel learning

the monge problem for supercritical mane potentials on compact manifolds abstract we prove the existence of optimal transport maps for the monge problem when the cost is a finsler distance on a compact manifold our point of view consists in considering the distance as a mane potential and to rely on recent developments in the theory of viscosity solutions of the hamilton jacobi equation

optimal transportation with traffic congestion and wardrop equilibria in the classical monge kantorovich problem the transportation cost depends only on the amount of mass sent from sources to destinations and not on the paths followed by this mass thus it does not allow for congestion effects using the notion of traffic intensity we propose a variant taking into account congestion this variant is a continuous version of a well known traffic problem on networks that is studied both in economics and in operational research the interest of this problem is in its relations with traffic equilibria of wardrop type what we prove in the paper is exactly the existence and the variational characterization of equilibria in a continuous space setting

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

variational evolution problems and nonlocal geometric motion we consider two variational evolution problems related to monge kantorovich mass transfer these problems provide models for collapsing sandpiles and for compression molding we prove the following connection between these problems and nonlocal geometric curvature motion the distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems thus we do the first step of the proof of heuristics developed in earlier works the main techniques we use are differential equation methods in the monge kantorovich theory

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

variational approximation for fokker planck equation on riemannian manifold under the bounded geometry assumption on riemannian manifold m a variational approximation for fokker planck equation on m is constructed by the scheme of jordan et al in siam j math anal 29 1 1 17 1998 moreover the uniqueness and global l p estimate of the solution for 1 p dim m dim m 1 are obtained for a broad class of potential

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

computing a non maxwellian velocity distribution from first principles we investigate a family of single particle anomalous velocity distribution by solving a particular class of stochastic liouville equations the stationary state is obtained analytically and the maxwell boltzmann distribution is reobtained in a particular limit we discuss the comparison with other different methods to obtain the stationary state extensions when the models cannot be solved in an exact way are also pointed out in connection with the one ficton approximation

hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm we develop the notions of hypercontractivity hc and the log sobolev ls inequality for completely bounded norms of one parameter semigroups of super operators acting on matrix algebras we prove the equivalence of the completely bounded versions of hc and ls under suitable hypotheses we also prove a version of the gross lemma which allows ls at general q to be deduced from ls at q 2

a large deviation analysis on the near equivalence between external and internal reservoirs within the spirit of van kampen s langevin approach we discuss the limits of validity of rephrasing the non equilibrium problem of a particle subject to an external work reservoir a system where the fluctuation dissipation relation is not verified into the simpler case with an internal heat reservoir for which the fluctuations and the dissipation arise from the same source using a convenient mapping of the thermomechanical parameters we show that counter intuitively such approach is not only valid for steady state time independent quantities but also for time dependent thermostatistical quantities namely the injected and dissipated fluxes we connect this result with the problem of large deviations and conclude that in this context we can only distinguish reservoirs by analysing the fluctuations of accumulated fluctuations as a by product we learn that the best reference approximation to the large deviation functions of a non markovian external reservoir system is not the respective internal reservoir limit as often assumed and suggested by the langevin approach but its internal reservoir analogue system obtained from the mapping of the original thermomechanical parameters

monge problem in metric measure spaces with riemannian curvature dimension condition abstract we prove the existence of solutions for the monge minimization problem addressed in a metric measure space x d m enjoying the riemannian curvature dimension condition rcd k n with n for the first marginal measure we assume that 0 m as a corollary we obtain that the monge problem and its relaxed version the monge kantorovich problem attain the same minimal value moreover we prove a structure theorem for d cyclically monotone sets neglecting a set of zero m measure they do not contain any branching structures that is they can be written as the disjoint union of the image of a disjoint family of geodesics

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

exact time average distribution for a stationary non markovian massive brownian particle coupled to two heat baths using a time averaging technique we obtain exactly the probability distribution for position and velocity of a brownian particle under the influence of two heat baths at different temperatures these baths are expressed by a white noise term representing the fast dynamics and a colored noise term representing the slow dynamics our exact solution scheme accounts for inertial effects that are not present in approaches that assume the brownian particle in the overdamped limit we are also able to obtain the contributions associated with the fast noise that are usually neglected by other approaches

a monge kantorovich mass transport problem for a discrete distance this paper is concerned with a monge kantorovich mass transport problem in which in the transport cost we replace the euclidean distance with a discrete distance we fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination for this problem we construct special kantorovich potentials and optimal transport plans via a nonlocal version of the pde formulation given by evans and gangbo for the classical case with the euclidean distance we also study how these problems when rescaling the step distance approximate the classical problem in particular we obtain taking limits in the rescaled nonlocal formulation the pde formulation given by evans gangbo for the classical problem

a weighted anisotropic sobolev type inequality and its applications to hardy inequalities in this paper we focus our attention on an embedding result for a weighted sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function f starting from this type of inequalities we prove some refined hardy type inequalities

on the relations between markovian master equations and stochastic differential equations abstract we have investigated the relationship between markovian master equations m e and the corresponding stochastic differential equations s d e for closed systems i e systems not subjected to external pumping we show that the form of the fluctuations in the s d e i e additive or multiplicative depends upon the properties of the kernel of the m e and the range of the state space of the stochastic variable s i e bounded or unbounded the knowledge of these two properties of the m e permits the determination of the way in which the fluctuations enter into the s d e i e additive or multiplicative and the calculation of their statistics several examples are presented to illustrate the general theory

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

existence of optimal transport maps for crystalline norms we show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in n assuming that the initial distribution of mass is absolutely continuous with respect to mathcal l n the proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the euclidean distance as cost function moreover improving a construction by larman we show the existence of a nikodym set in 3 having full measure in the unit cube intersecting each element of a family of pairwise disjoint open lines only in one point this example can be used to show that the regularity of the decomposition in transport rays plays an essential role in sudakov type arguments for proving the existence of optimal transport maps

an inequality for hilbert schmidt norm for the absolute value c c c 1 2 and the hilbert schmidt norm c hs trc c 1 2 of an operatorc the following inequality is proved for any bounded linear operatorsa andb on a hilbert space a b hs leqq 2 1 2 a b hs the corresponding inequality for two normal state and of a von neumann algebram is also proved in the following form d varphi psi leqq xi varphi xi psi leqq 2 1 2 d varphi psi here denotes the unique vector representative of a state in a natural positive conep form andd denotes the bures distance defined as the infimum which is also the minimum of the distance of vector representatives of and in particular xi varphi 1 xi varphi 2 leqq 2 1 2 xi 1 xi 2 for any vector representatives j of j j 1 2

exact nonequilibrium work generating function for a small classical system we obtain the exact nonequilibrium work generating function newgf for a small system consisting of a massive brownian particle connected to internal and external springs the external work is provided to the system for a finite time interval the jarzynski equality obtained in this case directly from the newgf is shown to be valid for the present model in an exact way regardless of the rate of external work

convergence rates for empirical barycenters in metric spaces curvature convexity and extendable geodesics this paper provides rates of convergence for empirical generalised barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry we study the validity of variance inequalities in spaces of non positive and non negative aleksandrov curvature in this last scenario we show that variance inequalities hold provided geodesics emanating from a barycenter can be extended by a constant factor we also relate variance inequalities to strong geodesic convexity while not restricted to this setting our results are largely discussed in the context of the 2 wasserstein space

evolution monge kantorovich equation abstract the paper deals with monge kantorovich equation mk for short in an open bounded domain with dirichlet boundary condition we study existence and uniqueness of a solution to the associated evolution problem emk for short and we prove the convergence to a solution of mk when time goes to a solution is a couple u where u is the potential and is the transportation flux we study the problem for a given radon measure source term and we show how to use the numerical method of dumont and igbida 2009 22 to provide numerical approximation of the solution of mk

harmonic potential driven by long range correlated noise the probability distribution of a particle in a quadratic potential driven by gaussian long range correlated noise has been obtained the long time asymptotic relaxation of the stochastic process has been characterized in terms of the long range correlated noise appearing in the corresponding stochastic differential equation defining the process the particular case when the particle is driven by a gaussian color noise has also been revisited in this way giving its exact probability distribution for all time by using a characteristic functional technique reported previously results for a non gaussian long range correlated noise are also shown emphasis has been placed on solving the plane rotator in the presence of arbitrary random torques having long range correlations in order to show strong and weak non markovian effects coming from different sources of noise the undamped free particle under the influence of arbitrary accelerations has been analyzed we also analyze in detail the structure of the trajectories of the overdamped and undamped free particle

embedding sn into rn 1 with given integral gauss curvature and optimal mass transport on sn in a d aleksandrov convex polyhedra gittl moscow ussr 1950 in russian english translation a d aleksandrov convex polyhedra springer berlin new york 2005 a d aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data the first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in euclidean space with prescribed integral gauss curvature our solution includes the case of a convex polytope this problem was also first considered by aleksandrov and below it is referred to as aleksandrov s problem the second goal of this paper is to show that in variational form the aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

the disintegration of the lebesgue measure on the faces of a convex function we consider the disintegration of the lebesgue measure on the graph of a convex function f rn r w r t the partition into its faces which are convex sets and therefore have a well defined linear dimension and we prove that each conditional measure is equivalent to the k dimensional hausdorff measure on the k dimensional face on which it is concentrated the remarkable fact is that a priori the directions of the faces are just borel and no lipschitz regularity is known notwithstanding that we also prove that a green gauss formula for these directions holds on special sets

brownian motion as a problem of eliminating fast variables abstract the hamilton equations for a brownian particle involve its mass as a large parameter as a consequence its motion is relatively slow and the fast motion of the surrounding fluid molecules can be eliminated by the standard method of eliminating fast variables the result is equivalent to the known langevin equation plus additional higher orders all coefficients are expressed in terms of correlation functions of the microscopic force on the particle

on 1 factorizations of bipartite kneser graphs abstract it is a challenging open problem to construct an explicit 1 factorization of the bipartite kneser graph h v t which contains as vertices all t element and v t element subsets of v 1 v where v 2 t and an edge between any two vertices when one is a subset of the other in this paper for the special case where t 2 and v is an odd prime power we construct such 1 factorizations using perpendicular arrays we also revisit two known 1 factorizations of h 2 t 1 t the lexical factorization and modular factorization among other results we give interesting alternative definitions of these 1 factorizations and design an optimal algorithm for computing the lexical factorization

constructions of optimal and near optimal multiply constant weight codes multiply constant weight codes mcwcs have been recently studied to improve the reliability of certain physically unclonable function response in this paper we give combinatorial constructions for the mcwcs which yield several new infinite families of optimal mcwcs furthermore we demonstrate that the johnson type upper bounds of the mcwcs are asymptotically tight for fixed hamming weights and distances finally we provide bounds and constructions of the 2 d mcwcs

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

optimal riemannian distances preventing mass transfer we consider an optimization problem related to mass transportation given two probabilities f and f on an open subset r we let vary the cost of the transport among all distances associated with conformally flat riemannian metrics on which satisfy an integral constraint precisely an upper bound on the l norm of the riemannian coefficient then we search for an optimal distance which prevents as much as possible the transfer of f into f higher values of the riemannian coefficient make the connection more difficult but the problem is non trivial due to the presence of the integral constraint in particular the existence of a solution is a priori guaranteed only on the relaxed class of costs which are associated with possibly non riemannian finsler metrics our main result shows that a solution does exist in the initial class of riemannian distances 2000msc 49j45 53c60

functional characterization of linear delay langevin equations we present an exact functional characterization of linear delay langevin equations driven by any noise structure defined through its characteristic functional this method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise general properties of the transient dissipative dynamics are analyzed the corresponding interplay with a color gaussian noise is presented as a full application of our functional method we study a model for population growth with non gaussian fluctuations the gompertz model driven by multiplicative white shot noise

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

partial localization lipid bilayers and the elastica functional partial localization is the phenomenon of self aggregation of mass into highdensity structures that are thin in one direction and extended in the others we give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes we demonstrate that this functional defined on a class of two dimensional spatial mass densities exhibits partial localization and displays the solid like behaviour of cell membranes specifically we show that density fields of moderate energy are partially localized that is resemble thin structures deviation from a specific uniform thickness creation of ends and the bending of such structures all carry an energy penalty of different orders in terms of the thickness of the structure these findings are made precise in a gamma convergence result we prove that a rescaled version of the energy functional converges in the zero thickness limit to a functional that is defined on a class of planar curves finiteness of the limit enforces both optimal thickness and non fracture if these conditions are met then the limit value is given by the classical elastica bending energy of the curve

new multiplicativity results for qubit maps let be a trace preserving positivity preserving but not necessarily completely positive linear map on the algebra of complex 2 2 matrices and let be any finite dimensional completely positive map for p 2 and p 4 we prove that the maximal p norm of the product map is the product of the maximal p norms of and restricting to the class of completely positive maps this settles the multiplicativity question for all qubit channels in the range of values p 4

optimal transport with convex obstacle we consider the monge transportation problem when the cost is the squared geodesic distance around a convex obstacle we show that there exists at least one and in general infinitely many optimal transport maps

monge s transport problem on a riemannian manifold monge s problem refers to the classical problem of optimally transporting mass given borel probability measures find the measurepreserving map s m m between them which minimizes the average distance transported set on a complete connected riemannian manifold m and assuming absolute continuity of an optimal map will be shown to exist aspects of its uniqueness are also established

a characterization for solutions of the monge kantorovich mass transport problem a measure theoretical approach is presented to study the solutions of the monge kantorovich optimal mass transport problems this approach together with kantorovich duality provide an effective tool to answer a long standing question about the support of optimal plans for the mass transport problem involving general cost functions we also establish a criterion for the uniqueness

continuity for the monge mass transfer problem in two dimensions in this paper we prove the continuity of the monotone optimal mapping of the monge mass transfer problem in two dimensions under certain conditions on the domains and the mass distributions

sharpening holder s inequality abstract we strengthen holder s inequality the new family of sharp inequalities we obtain might be thought of as an analog of the pythagorean theorem for the l p spaces our treatment of the subject matter is based on bellman functions of four variables

on a norm compression inequality for 2 n partitioned block matrices abstract we conjecture the following so called norm compression inequality for 2 n partitioned block matrices and the schatten p norms for p 2 a 1 a 2 a n b 1 b 2 b n p a 1 p a 2 p a n p b 1 p b 2 p b n p p while for 1 p 2 the ordering of the inequality is reversed this inequality includes hanner s inequality for matrices as a special case we prove several special cases of this inequality and show that the partitioning in 2 n blocks is essential by exhibiting counterexamples in the case of 3 3 and larger partitionings

convergence theorems of shrinking projection methods for equilibrium problem variational inequality problem and a finite family of relatively quasi nonexpansive mappings we introduce a w mapping for a finite family of relatively quasi nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem the solution set of the variational inequality problem for an inverse strongly monotone operator and set of common fixed points of a finite family of relatively quasi nonexpansive mappings strong convergence theorems are presented in a 2 uniformly convex and uniformly smooth banach space our results generalize and extend relative results

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

optimal affine invariant smooth minimization algorithms we formulate an affine invariant implementation of the accelerated first order algorithm in y nesterov dokl math 27 1983 pp 372 376 its complexity bound is proportional to an affine invariant regularity constant defined with respect to the minkowski gauge of the feasible set we extend these results to more general problems optimizing holder smooth functions using p uniformly convex prox terms and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best holder smoothness parameter and the best gradient lipschitz constant finally we detail matching complexity lower bounds when the feasible set is an ell p ball in this setting our upper bounds on iteration complexity for the algorithm in y nesterov dokl math 27 1983 pp 372 376 are thus optimal in terms of target precision smoothness and problem dimension

a noncommutative martingale convexity inequality let m be a von neumann algebra equipped with a faithful semifinite normal weight and n be a von neumann subalgebra of m such that the restriction of to n is semifinite and such that n is invariant by the modular group of let e be the weight preserving conditional expectation from m onto n as an application we show that there exists e0 0 such that for any free group fn and any q 4 e0 pt 2 q 1 t logq 1 where pt is the poisson semigroup defined by the natural length function of fn

strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems abstract in this paper we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method we obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem

fast transport optimization for monge costs on the circle consider the problem of optimally matching two measures on the circle or equivalently two periodic measures on the real line and suppose the cost of matching two points satisfies the monge condition we introduce a notion of locally optimal transport plan motivated by the weak kam aubry mather theory and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter this theory is applied to a transportation problem arising in image processing for two sets of point masses on the circle both of which have the same total mass find an optimal transport plan with respect to a given cost function satisfying the monge condition in the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations for the case of n real valued point masses we present an o n log epsilon algorithm that approximates the optimal cost within epsilon when all masses are integer multiples of 1 m the algorithm gives an exact solution in o n log m operations

low complexity two dimensional weight constrained codes two low complexity coding techniques are described for mapping arbitrary data to and from m n binary arrays in which the hamming weight of each row respectively column is at most n 2 respectively m 2 one technique is based on flipping rows and columns of an arbitrary binary array until the hamming weight constraint is satisfied in all rows and columns and the other is based on a certain explicitly constructed antipodal matching between layers of the boolean lattice both codes have a redundancy of roughly m n and may have applications in next generation resistive memory technologies