loss of spectral peaks in autoregressive spectral estimation on montre par une analyse asymptotique comprenant les termes du second ordre que les estimateurs des moindres carres doivent etre preferes aux estimateurs de yule walker

on the estimation of residual variance and order in autoregressive time series summary we study the bias of yule walker least squares and burg type estimates of the residual variance of autoregressive processes both simulations and theory indicate that yulewalker estimates are inferior to least squares and burg type estimates the effect on order determination is also studied and we extend the results on overestimation of the aic criterion to the general multivariate case for strongly autocorrelated processes yule walker estimates of residual variance and order may be severely biased even for comparatively large sample sizes

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

a method for the estimation and identification of transfer function models summary this paper proposes an approach to the simultaneous identification and estimation of transfer function models an estimation procedure is advanced consisting of a sequence of four ordinary least squares calculations and the asymptotic properties of each step are presented consistency and asymptotic efficiency being shown a strategy for the identification of the model structure based on a selection criterion that arises naturally from the last step of the estimation algorithm is also investigated consideration is given to the numerical implementation of the techniques discussed and finally a practical example is employed to illustrate the methods

a convergence of geometric mean for t related fuzzy numbers we study a convergence of geometric mean for t related fuzzy numbers where t is an archimedian t norm in particular we show that the class of archimedian t norms can be characterized by the validity of a general convergence of geometric mean for sequences of l r fuzzy numbers

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

law of large numbers for the possibilistic mean value abstract a law of large numbers for the possibilistic mean value of a variable in a possibility space is presented an example shows that the convergence in distribution under a definition involving the possibilistic mean value of the sample average to a variable with a certain distribution cannot be replaced in general by convergence either almost surely or in necessity even so the usual presentation of the law of large numbers as a statement that holds in necessity follows from this result

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

vector autoregressive moving average identification for macroeconomic modeling a new methodology this paper develops a new methodology for identifying the structure of varma time series models the analysis proceeds by examining the echelon form and presents a fully automatic strongly consistent data driven approach to model specification a novel feature of the inferential procedures developed here is that they work in terms of a canonical representation based on the kronecker invariants in which the variables are expressed in the form of scalar dynamic structural equations derived from the varma system this feature facilitates the construction of procedures which from the perspective of macroeconomic modeling can be efficacious in that they do not rely on var approximations techniques that are applicable to both asymptotically stationary and unit root partially nonstationary cointegrated time series models are presented the inferential potential of the techniques is illustrated via simulation experiments that use data generating mechanisms based on real world examples drawn from the time series literature aspects of the kronecker invariants that impinge on practical application and that have not hitherto been discussed in the literature are explored

granger causality in multiple time series abstract the concept of granger causality is formulated for a finite dimensional multiple time series special attention is given to causality patterns in autoregressive series and it is shown how these patterns can be tested under quite general assumptions using a 2 statistic the power of the test is discussed and it is shown that the 2 statistic results from a lagrange multiplier test in the gaussian case the causality test is tried both on artificial data and some economic time series finally we consider the problem of constrained estimation in models with a known causality structure

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

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions

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

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

on cournot nash equilibrium distributions for games with a nonmetrizable action space and upper semi continuous payoffs we report results on the existence of a cournot nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous our work relies on the theory of radon measures as developed by schwartz topsoe and on the epitopology as developed by dolecki salinetti wets

a t sum bound of lr fuzzy numbers abstract in this paper we provide an upper bound and a lower bound of t sum of lr fuzzy numbers with different spreads where t is archimedean t norm and also show in three examples how close they are to actual membership functions furthermore we study when the membership function of t sum achieves the upper bound

equivalent conditions for laws of large numbers for t related l r fuzzy numbers in this paper we give some necessary and sufficient conditions for laws of large numbers for sequence of mutually t related l r fuzzy numbers when t is an continuous archimedean t norm and diameter of support of the fuzzy numbers is not uniformly bounded we also consider some necessary and sufficient conditions for laws of large numbers for l r fuzzy random numbers and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 303 triesch fuzzy sets and systems 58 1993 339 342 and hong and lee fuzzy sets and systems 121 2001 537 543

renewal process with t related fuzzy inter arrival times and fuzzy rewards in this paper we consider a renewal process in which the inter arrival times and rewards are characterized as fuzzy variables under t norm based fuzzy operations a t related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables

effects of not knowing the order of an autoregressive process on the mean squared error of prediction i abstract given a realization of t consecutive observations of an autoregressive process x of unknown order m a process of an arbitrary order p is assumed to be fitted for predicting the future values of another process xtilde t where xtilde t is independent of xt but otherwise has identical probabilistic structure the mean squared error of prediction when the autoregressive coefficients are estimated by least squares is obtained to terms of order t 1 the bias in estimating the autoregressive coefficients is examined the results are illustrated for three processes with m 2 by considering fitting of a process with p 1

the identification of arma models summary we present a new powerful method for determining the order of arma p q models having small sets of observations the procedure is based on an autoregressive order determination criterion and on linear estimation methods simulated data are used to demonstrate the capabilities of the approach for a survey nevertheless the reliability and adaptability of the methods remain at best tenuous in this paper we propose a simple but accurate procedure based on an autoregressive order determination criterion and linear estimation methods to identify the order of arma processes from a finite set of observations the paper is organized as follows in 2 we describe our approach in 3 we demonstrate the performance of our identification procedure based on simulations of several model structures with varying number of observations and 4 contains concluding remarks it is important that in reality the order of any process based on finite samples may never be correctly identified by any procedure all models are simplifications of reality their value and strength lie in their ability to explain the current situation and to predict the future our procedure therefore should be viewed as one which parsimoniously identifies the order of the arma model which reasonably depicts the underlying process governing the behaviour of the available data box jenkins 1976 ch 1 for brevity we assume that the reader is familiar with the characteristics and properties of arma models unless otherwise stated we use the notation and conventions of box

a note on law of large numbers for fuzzy numbers in a banach space abstract in this paper we study a law of large numbers for sequences of mutually t related fuzzy numbers in banach space where t is an archimedean t norm and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 triesch fuzzy sets and systems 58 1993 339 and hong and kim fuzzy sets and systems 77 1996 349

the statistical convergence for sequences of fuzzy number valued functions based on the concept of statistical convergence of sequences of fuzzy numbers the statistical convergence uniformly statistical convergence and equi statistical convergence of sequences of fuzzy number valued functions are defined and investigated in this paper the relationship among statistical convergence uniformly statistical convergence equi statistical convergence of sequences of fuzzy number valued functions and their representations of sequences of level cuts are discussed in addition the egorov and lebesgue theorems for the statistical convergence of sequences of fuzzy number valued functions are obtained in a finite measure set finally the statistical convergence in measure for sequences of fuzzy number valued functions is investigated and it is proved that the outer and inner statistical convergence in measure are equivalent in a finite measure set for a sequence of fuzzy number valued functions

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

law of large numbers for uncertain random variables the law of large numbers in probability theory states that the average of random variables converges to its expected value in some sense under some conditions sometimes random factors and human uncertainty exist simultaneously in complex systems and a concept of uncertain random variable has been proposed to study this type of complex systems this paper aims to provide a law of large numbers for uncertain random variables which states that the average of uncertain random variables converges in distribution to an uncertain variable as a byproduct the convergence of a sequence of uncertain variables is also studied

characterisation of archimedian t norms and a law of large numbers abstract we study a law of large numbers for mutually t related fuzzy numbers where t is an archimedian t norm and extend earlier results of fuller in this area in particular we show that the class of archimedian t norms can be characterised by the validity of a very general law of large numbers for sequences of l r fuzzy numbers

on the jordan kinderlehrer otto scheme abstract in this paper we prove that the jordan kinderlehrer otto scheme for a family of linear parabolic equations on the flat torus converges uniformly in space

a law of large numbers for fuzzy numbers abstract we study the problem if 1 2 are fuzzy numbers with modal values m1 m2 then what is the strongest t norm for which where mn m1 mn n the arithmetic mean 1 n n i is defined via sup t norm convolution and nes denotes necessity

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

revisiting simple linear regression with autocorrelated errors this paper studies properties of ordinary and generalised least squares estimators in a simple linear regression with stationary autocorrelated errors explicit expressions for the variances of the regression parameter estimators are derived for some common time series autocorrelation structures including a first order autoregression and general moving averages applications of the results include confidence intervals and an example where the variance of the trend slope estimator does not increase with increasing autocorrelation

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

the bias of autoregressive coefficient estimators abstract this article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary but known finite order the results include models both with and without a constant term the effects of overspecification of the model order on the bias are described the emphasis is on least squares and yule walker estimators but the methods extend to other estimators of similar design although only the order t 1 component of the bias is captured where t is the series length this asymptotic approximation is shown to be very accurate for least squares estimators through some numerical simulations the simulations examine fourth order autoregressions chosen to resemble some data series from the literature the order t 1 bias approximations for yule walker estimators need not be accurate especially if the zeros of the associated polynomial have moduli near 1 examples are given where the approximation is accurate and where it is useless the bias expressions are

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

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

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

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

on the law of large numbers for mutually t related l r fuzzy numbers abstract we study a law of large numbers for sequences of mutually t related l r fuzzy numbers where t is an archimedean t norm and the diameter of the support of the fuzzy numbers is not uniformly bounded earlier results of fuller 1992 and triesch 1993 in this area are extended some conditions concerning the shapes of f r and f l and spreads of l r fuzzy numbers are given where f is the additive generator of the archimedean t norm t several examples are given to illustrate the results

the burg algorithm for segments in many applications the duration of an uninterrupted measurement of a time series is limited however it is often possible to obtain several separate segments of data the estimation of an autoregressive model from this type of data is discussed a straightforward approach is to take the average of models estimated from each segment separately in this way the variance of the estimated parameters is reduced however averaging does not reduce the bias in the estimate with the burg algorithm for segments both the variance and the bias in the estimated parameters are reduced by fitting a single model to all segments simultaneously as a result the model estimated with the burg algorithm for segments is more accurate than models obtained with averaging the new weighted burg algorithm for segments allows combining segments of different amplitudes

mean estimation bias in least squares estimation of autoregressive processes abstract estimation of the parameters of an autoregressive process with a mean that is a function of time is considered approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived for the case of a mean function that is a polynomial in time a reparameterization that isolates the bias is given using the approximate expressions a method of modifying the least squares estimator is proposed a monte carlo study of the second order autoregressive process is presented the monte carlo results agree well with the approximate theory and generally speaking the modified least squares estimators performed better than the least squares estimator for the second order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one two and three periods in the future

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

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

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

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

optimal transport and cournot nash equilibria we study a class of games with a continuum of players for which a cournot nash equilibria can be obtained by the minimisation of some cost related to optimal transport this cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases this enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria

a law of large numbers for fuzzy numbers in a banach space abstract we study a law of large numbers for mutually t related fuzzy number in a banach space where t is an archimedean t norm and extend earlier results of fuller and triesch in this area

numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms we propose a numerical algorithm for solving nonlinear continuity equations written in lagrangian coordinates this transformation is intimately related to variational approaches for the well posedness of gradient flows of energy functionals with respect to the quadratic transportation distance in optimal transport theory these schemes allow the numerical approximation of both diffusive and aggregation regimes of different models positivity energy decreasing and mesh adaptation are built in in the numerical scheme and thus we are capable of capturing blow up densities and of dealing with vacuum regions and merging of mass patches in a natural way

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

the law of large numbers for fuzzy numbers with unbounded supports we study laws of large numbers for mutually t related fuzzy numbers with unbounded supports where t is an archimedean t norm and generalize earlier result of badard

bias of some commonly used time series estimates summary we study the bias of yule walker and least squares estimates for univariate and multivariate autoregressive processes we obtain explicit formulae for the large sample bias of yule walker estimates in the scalar first and second order cases and for least squares estimates in the general case both simulations and theory indicate that yulewalker estimates are inferior to least squares estimates for strongly autocorrelated processes yule walker estimates can be severely biased even for comparatively largesample sizes

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

series of fuzzy sets series of fuzzy sets with weakly closed weakly compact or compact a levels are considered the basic space is banach space the subject of this paper is investigation of infinite addition of fuzzy sets and condition when the sum of a levels is equal to the a level of the sum the results in this paper complement and complete some previous results proved by the same authors in stojakovic and stojakovic addition and series of fuzzy sets fuzzy sets and systems 83 1996 341 346 for the series with compact a levels

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

fuzzy vector spaces under triangular norms abstract the object of this paper is to redefine fuzzy subspaces using the notion of a t norm and to obtain several results some of which are generalisations of the results of katsaras and liu 3

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