density functional theory and optimal transportation with coulomb cost we present here novel insight into exchange correlation functionals in density functional theory based on the viewpoint of optimal transport we show that in the case of two electrons and in the semiclassical limit the exact exchange correlation functional reduces to a very interesting functional of novel form which depends on an optimal transport map t associated with a given density since the above limit is strongly correlated the limit functional yields insight into electron correlations we prove the existence and uniqueness of such an optimal map for any number of electrons and each and determine the map explicitly in the case when is radially symmetric

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

optimal maps for the multidimensional monge kantorovich problem let 1 n be borel probability measures on d denote by 1 n the set of all n tuples t t1 tn such that ti d d i 1 n are borel measurable and satisfy 1 i v for all borel v d the multidimensional monge kantorovich problem investigated in this paper consists of finding s s1 sn 1 n minimizing over the set 1 n we study the case where the i s have finite second moments and vanish on d 1 rectifiable sets we prove existence and uniqueness of optimal maps s when we impose that s1 x x and give an explicit form of the maps si the result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality as a consequence we obtain uniqueness and characterization of an optimal measure for the multidimensional kantorovich problem 1998 john wiley sons inc

random vortex simulation of transient wall driven flow in a rectangular enclosure abstract two dimensional transient flow in square cavity following impulsive acceleration of the sliding wall is simulated by random vortex calculations of the type initiated by a chorin comparison calculations are made using a transient finite difference method that is first order accurate in time and second order accurate in space the reynolds number r based on the sliding wall length and speed is 2000 5000 or 10 000 the simulated history starts when the moving wall starts ends when the wall has moved three times its length and shows the development of primary and secondary eddies in the corner where the boundary layer on the moving wall is peeled off by the stationary wall special features of this random vortex simulation include 1 the use of the closed form expression for the velocity induced by a line vortex in a rectangular enclosure 2 special treatment of regions very close to the corners where the sheet like elements ordinarily used in a very thin wall layer are inappropriate and 3 the generation of a small three member ensemble of simulations at each reynolds number comparisons between the two numerical schemes were made at reynolds numbers of 2000 and 5000 a comparison with experimental data of koseff is given for r 2000 both numerical methods exhibited the same qualitative flow features and were in fair quantitaive agreement the random vortex calculations show somewhat better agreement with the laboratory experiment

on a generalization of cyclic monotonicity and distances among random vectors abstract we generalize an investigation of distances among random vectors begun by olkin and pukelsheim and continued by olkin and rachev

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

generation of vorticity near the boundary in planar navier stokes flows we construct the solutions of the planar navier stokes flow for a viscous incompressible fluid in the half plane by means of a boundary layer equation describing the production of vorticity on the boundary regularity properties are also discussed

convergence of euler stokes splitting of the navier stokes equations we consider approximation by partial time steps of a smooth solution of the navier stokes equations in a smooth domain in two or three space dimensions with no slip boundary condition for small k 0 we alternate the solution for time k of the inviscid euler equations with tangential boundary condition and the solution of the linear stokes equations for time k with the no slip condition imposed we show that this approximation remains bounded in h2 p and is accurate to order k in lp for p the principal difficulty is that the initial state for each stokes step has tangential velocity at the boundary generated during the euler step and thus does not satisfy the boundary condition for the stokes step the validity of such a fractional step method or splitting is an underlying principle for some computational methods 1994 john wiley sons inc

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

viscous splitting approximation of mixed hyperbolic parabolic convection diffusion equations we first analyse a semi discrete operator splitting method for nonlinear possibly strongly degenerate convection diffusion equations due to strong degeneracy solutions can be discontinuous and are in general not uniquely determined by their data hence weak solutions satisfying an entropy condition are sought we then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic parabolic convection diffusion equations

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

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

a class of multi marginal c cyclically monotone sets with explicit c splitting potentials abstract multi marginal optimal transport plans are concentrated on c splitting sets it is known that similar to the two marginal case c splitting sets are c cyclically monotone within a suitable framework the converse implication was very recently established by griessler however for an arbitrary cost c given a multi marginal c cyclically monotone set the question whether there exists an analogous explicit construction to the one from the two marginal case of c splitting potentials is still open when the margins are one dimensional and the cost belongs to a certain class carlier proved that the two marginal projections of a c splitting set are monotone for arbitrary products of sets equipped with cost functions which are sums of two marginal costs we show that the two marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of c splitting potentials our condition is in principle easier to verify than the one of multi marginal c cyclic monotonicity various examples illustrate our results we show that in general our condition is sufficient however it is not necessary on the other hand we conclude that when the margins are one dimensional equipped with classical cost functions our condition is a characterization of c splitting sets and extends classical convex analysis

space time block codes a maximum snr approach in tarokh et al 1999 space time block codes were introduced to obtain coded diversity for a multiple antenna communication system in this work we cast space time codes in an optimal signal to noise ratio snr framework and show that they achieve the maximum snr and in fact they correspond to a generalized maximal ratio combiner the maximum snr framework also helps in calculating the distribution of the snr and in deriving explicit expressions for bit error rates we bring out the connection between the theory of amicable orthogonal designs and space time codes based on this we give a much simpler proof to one of the main theorems on space time codes for complex symbols we present a rate 1 2 code for complex symbols which has a smaller delay than the code already known we also present another rate 3 4 code which is simpler than the one already known in the sense it does not involve additions or multiplications we also point out the connection between generalized real designs and generalized orthogonal designs

wasserstein barycenters over riemannian manifolds we study barycenters in the space of probability measures on a riemannian manifold equipped with the wasserstein metric under reasonable assumptions we establish absolute continuity of the barycenter of general measures p p m p p m on wasserstein space extending on one hand results in the euclidean case for barycenters between finitely many measures of agueh and carlier 1 to the riemannian setting and on the other hand results in the riemannian case of cordero erausquin mccann schmuckenschlager 12 for barycenters between two measures to the multi marginal setting our work also extends these results to the case where is not finitely supported as applications we prove versions of jensen s inequality on wasserstein space and a generalized brunn minkowski inequality for a random measurable set on a riemannian manifold

mmse interference suppression for direct sequence spread spectrum cdma we consider interference suppression for direct sequence spread spectrum code division multiple access cdma systems using the minimum mean squared error mmse performance criterion the conventional matched filter receiver suffers from the near far problem and requires strict power control typically involving feedback from receiver to transmitter for acceptable performance multiuser detection schemes previously proposed mitigate the near far problem but are complex and require explicit knowledge or estimates of the interference parameters in this paper we present and analyze several new mmse interference suppression schemes which have the advantage of being near far resistant to varying degrees depending on their complexity and can be implemented adaptively when interference parameters are unknown and or time varying numerical results are provided that show that these schemes offer significant performance gains relative to the matched filter receiver we conclude that mmse detectors can alleviate the need for stringent power control in cdma systems and may be a practical alternative to the matched filter receiver

convergence of chorin marsden product formula in the half plane consider a viscous incompressible fluid in the half plane and letu t be a solution of the navier stokes equation in this paper we prove that the product formula e t n g t n u n u0 wheree t is the euler flow g t is the heat flow and is a suitable operator describing the vorticity production due to the boundary converges uniformly tou t in the limitn

amplitude and phase variation of point processes we develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping known as the problem of separation of amplitude and phase variation the amplitude variation of a real random function y x x is an element of 0 1 corresponds to its random oscillations in the y axis typically encapsulated by its co variation around a mean level in contrast its phase variation refers to fluctuations in the x axis often caused by random time changes we formalise similar notions for a point process and nonparametrically separate them based on realisations of i i d copies pi i of the phase varying point process a key element in our approach is to demonstrate that when the classical phase variation assumptions of functional data analysis fda are applied to the point process case they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure we demonstrate that these induce a natural wasserstein geometry tailored to the warping problem including a formal notion of bias expressing over registration within this framework we construct nonparametric estimators that tend to avoid over registration in finite samples we show that they consistently estimate the warp maps consistently estimate the structural mean and consistently register the warped point processes even in a sparse sampling regime we also establish convergence rates and derivev root n consistency and a central limit theorem in the cox process case under dense sampling showing rate optimality of our structural mean estimator in that case

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

multi marginal optimal transport on riemannian manifolds we study a multi marginal optimal transportation problem on a riemannian manifold with cost function given by the average distance squared from multiple points to their barycenter under a standard regularity condition on the first marginal we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable thus inducing a monge solution this result generalizes mccann s polar factorization theorem on manifolds from two to several marginals in the same sense that a well known result of gangbo and swiech generalizes brenier s polar factorization theorem on bbb r n

iterative detection in block orthogonal and spatially multiplexed multisensor signaling schemes the impact of iterative minimum mean square error mmse detection on multisensor signaling schemes in frequency selective media is considered the suitability of a decision feedback mmse df mmse front end relative to a linear mmse lmmse front end is examined using two case studies namely layered spatial multiplexing and block orthogonal space time signaling iterative mmse detection is observed to afford near optimal performance further in channels with large delay spreads a combination of the lmmse and df mmse front ends seems appropriate the former in earlier iterations and the latter in concluding iterations

turbo space time processing to improve wireless channel capacity by deriving a generalized shannon capacity formula for multiple input multiple output rayleigh fading channels and by suggesting a layered space time architecture concept that attains a tight lower bound on the capacity achievable foschini see wireless pers commun vol 6 no 3 p 311 35 1998 has shown a potential enormous increase in the information capacity of a wireless system employing multiple element antenna arrays at both the transmitter and receiver the layered space time architecture allows signal processing complexity to grow linearly rather than exponentially with the promised capacity increase this paper includes two important contributions first we show that foschini s lower bound is in fact the shannon bound when the output signal to noise ratio snr of the space time processing in each layer is represented by the corresponding matched filter bound this proves the optimality of the layered space time concept second we present an embodiment of this concept for a coded system operating at a low average snr and in the presence of possible intersymbol interference this embodiment utilizes the already advanced space time filtering coding and turbo processing techniques to provide yet a practical solution to the processing needed performance results are provided for quasi static rayleigh fading channels with no channel estimation errors we see for the first time that the shannon capacity for wireless communications can be both increased by n times where n is the number of the antenna elements at the transmitter and receiver and achieved within about 3 db in average snr about 2 db of which is a loss due to the practical coding scheme we assume the layered space time processing itself is nearly information lossless

smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the hohenberg kohn functional we prove rigorously that the exact n electron hohenberg kohn density functional converges in the strongly interacting limit to the strictly correlated electrons sce functional and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi marginal optimal transport problem with coulomb cost associated to the sce functional this extends our previous work for n 2 cotar etal in commun pure appl math 66 548 599 2013 the correct limit problem has been derived in the physics literature by seidl phys rev a 60 4387 4395 1999 and seidl gorigiorgi and savin phys rev a 75 042511 1 12 2007 in these papers the lack of a rigorous proofwas pointed out we also give amathematical counterexample to this type of result by replacing the constraint of given one body density an infinite dimensional quadratic expression in the wavefunction by an infinite dimensional quadratic expression in the wavefunction and its gradient connections with the lawrentiev phenomenon in the calculus of variations are indicated

an iterative multiuser decoder for near capacity communications the combination of forward error correction fec coding and random interleaving is shown to overcome the limitations of multiuser detectors decoders when the user cross correlations are high in particular one can asymptotically achieve single user performance in a highly correlated multiuser system in addition an optimal iterative multiuser detector is derived from iterative techniques for cross entropy minimization a practical suboptimal implementation of this algorithm is presented and simulations demonstrate that even with highly correlated users it achieves optimal asymptotic efficiency the effects of the theoretical limits on channel capacity are evident in many of the simulation results the complexity of the suboptimal algorithm is approximately o 2 sup k o 2 sup spl kappa per bit per iteration where k is the number of users and spl kappa is the code constraint length

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

optimal transportation with infinitely many marginals abstract we formulate and study an optimal transportation problem with infinitely many marginals this is a natural extension of the multi marginal problem studied by gangbo and wi ch 1998 15 we prove results on the existence uniqueness and characterization of the optimizer which are natural extensions of the results in gangbo and wi ch 1998 15 the proof relies on a relationship between this problem and the problem of finding barycenters in the wasserstein space a connection first observed for finitely many marginals by agueh and carlier 2011 1

viscous splitting for the unbounded problem of the navier stokes equations the viscous splitting for the exterior initial boundary value problems of the navier stokes equations is considered it is proved that the approximate solutions are uniformly bounded in the space l 0 t hs l s 2 and converge with a rate of 0 k in the space l 0 t hi q where k is the length of the time steps

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

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