determination of eigenstates via lanczos based forward substitution and filter diagonalization a filter diagonalization scheme based on the lanczos tridiagonalization algorithm is proposed for the determination of highly excited eigenstates in a prespecified energy range filtered states localized in the energy domain can be generated during the lanczos iteration by forward substitution which formally corresponds to a green function filter eigenstates in the energy range of interest are then obtained by diagonalizing a small hamiltonian matrix in a subspace spanned by the filtered states it is shown that the accuracy of this approach is affected neither by the loss of orthogonality among the lanczos states nor by the instability in the forward substitution since the filtered states are assembled while the lanczos states are generated no storage of the lanczos states is needed the computational savings can be significant in calculating high energy eigenstates of large systems the proposed scheme is shown numerically to have comparable accuracy with the basic lanczos method

exchangeability and convergence for random sets abstract in many statistical analysis some kind of dependency of random variables may be required and exchangeability as an alternative to the random sample with independent identically distributed random variables gives the study of asymptotic properties of random variables on the other hand statistical estimators are expressible of the linear form of random variables which involves possibly random weights and possibly functions of dependent random variables which are permutation invariant with respect to distributions this paper attempts to provide some properties useful to the limit theorems and laws of large numbers for arrays of banach spaced valued fuzzy random sets are examined with emphasis on permutation invariance among the random sets

the computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue in this paper we study the existence of a nontrivial solution of the elliptic problem which is a resonant type both at the origin and at infinity with the same higher eigenvalue and the nonlinear term is permitted to be unbounded our main ingredient is the computation of the critical groups

approximating spectral densities of large matrices in physics it is sometimes desirable to compute the so called density of states dos also known as the spectral density of a real symmetric matrix a the spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line the most straightforward way to obtain this density is to compute all eigenvalues of a but this approach is generally costly and wasteful especially for matrices of large dimension there exist alternative methods that allow us to estimate the spectral density function at much lower cost the major computational cost of these methods is in multiplying a with a number of vectors which makes them appealing for large scale problems where products of the matrix a with arbitrary vectors are relatively inexpensive this article defines the problem of estimating the spectral density carefully and discusses how to measure the accuracy of an approximate spectral density it then surveys a few kno

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

convex integral functionals we study nonlinear integral functionals determined by normal convex integrands first we obtain expressions for their convex conjugate their e subdifferential e 0 and their directional derivative then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution we also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators finally we examine the conditional expectation of random sets

reconstruction of thermally symmetrized quantum autocorrelation functions from imaginary time data in this paper i propose a technique for recovering quantum dynamical information from imaginary time data via the resolution of a one dimensional hamburger moment problem it is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin a general class of reconstruction algorithms is then identified according to theorem 3 the technique is advocated as especially effective for a certain class of quantum problems in continuum space for which only a few moments are necessary for such problems it is argued that the derivatives at origin can be evaluated by monte carlo simulations via estimators of finite variances in the limit of an infinite number of path variables finally a maximum entropy inversion algorithm for the hamburger moment problem is utilized to compute the quantum rate of reaction for a one dimensional symmetric eckart barrier

approximating stationary points of stochastic optimization problems in banach space in this paper we present a uniform strong law of large numbers for random set valued mappings in separable banach space and apply it to analyze the sample average approximation of clarke stationary points of a nonsmooth one stage stochastic minimization problem in separable banach space moreover under hausdorff continuity we show that with probability approaching one exponentially fast with the increase of sample size the sample average of a convex compact set valued mapping converges to its expected value uniformly the result is used to establish exponential convergence of stationary sequence under some metric regularity conditions

on fatou s lemma and parametric integrals for set valued functions abstract in this paper we present new versions of the set valued fatou s lemma for sequences of measurable multifunctions and their conditional expectations then we use them to study the continuity and measurability properties of parameterized set valued integrals

measurability and integrability of the weak upper limit of a sequence of multifunctions we provide several properties of the weak upper limit of a sequence of subsets of a separable banach space such as a criterion of non vacuity of closedness etc we also examine the measurability of the multifunction defined as the weak upper limit of a sequence of multifunctions at last applications to the existence of a measurable and bochner integrable selector for this multifunction are presented

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

convergence of conditional expectations for unbounded random sets integrands and integral functionals given a sequence of unbounded convex random sets we study under which conditions fatou s lemma for the weak upper limit of their conditional expectations holds we also give multivalued versions of dominated and monotone convergence theorems and we discuss the special case of the integral finally applications to epigraphic convergence of integrands and to mosco convergence of certain integral functionals are provided

a bounded mountain pass lemma without the ps condition and applications we present a version of the mountain pass lemma which does not require the ps condition we apply this version to problems where the ps condition is not satisfied

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

regularity and stopping theorem for fuzzy martingales with continuous parameters the notion of graph convergence for fuzzy valued stochastic processes is introduced regularity for fuzzy super martingales with continuous parameters in the sense of gr km convergence is discussed finally we establish doob s stopping theorem for continuous parameter super sub martingales whose cut sets may be unbounded

a representation theorem for aumann integrals in recent years the study of set valued functions has been developed extensively by many authors with applications to mathematical economics and control theory see refs 5 11 15 161 in those papers three approaches can be distinguished according to whether the range space values of setvalued functions is z a banach space or a locally convex topological space the purpose of this paper is to establish properties of aumann s integrals of set valued functions f t 2 whose values are nonempty subsets of a real separable reflexive banach space x and to continue the work due to aumann 2 and datko 7 81 while previous analysis has always treated the case of special finite nonatomic measure spaces we focus here on the case of general a finite nonatomic measure spaces in this last situation moreover the analogous results we establish hold under less stringent hypotheses more precisely all through the paper we consider a measure space t c p where p is supposed to be positive nonatomic and u finite and we give the following statements

fatou s lemmas for the set of integrable selections abstract in this paper we prove fatou type results for the set of bochner integrable selections from a set valued function also we obtain some new dominated convergence theorems for the aumann integral our results have applications in economic theory

law of large numbers for random sets and allocation processes in this paper we establish a strong law of large numbers for unbounded random sets and then apply it to an optimization problem arising in allocation processes under uncertainty

weak convergence of random sets in banach spaces abstract in this paper we introduce the notion of weak convergence for random sets in a separable banach space we study its properties we prove some weak completeness results a dunford pettis type compactness theorem and an analog of mazur s lemma then we present some applications on set valued submartingales on approximation theory and on infinite dimensional control systems

computations of critical groups and periodic solutions for asymptotically linear hamiltonian systems the purpose of this paper is two fold firstly we will give some parabolic like conditions which improve the well known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity as an application we then consider the second order hamiltonian systems u t h t u t 0 t r where h r rn r is t periodic in its first variable and h is asymptotically linear both at origin and at infinity based on the computations of the critical groups and the morse theory we obtain the existence and multiplicity results for periodic solutions under new classes of conditions it turns out that our main results improve sharply some known results in the literature

weak sequential convergence in l1 μ x and an exact version of fatouʼs lemma abstract the class of nonatomic finite measure spaces with the saturation property as developed in maharam 1942 and hoover keisler 1984 is characterized by the fatou and lebesgue property of a well dominated sequence of multifunctions taking values in a banach space with multifunctions reduced to functions this fatou characterization also extends to a variant of the closure property found in optimal control theory the results are developed through a considered overview of the relevant literature on the exact and approximate fatou lemma phrased in terms of bochner integration

convergence theorems for pettis integrable functions and regular methods of summability abstract we characterize the relatively sequentially compact subsets of p 1 x the space of all x valued pettis integrable functions where x is a separable banach space for the weak topology of p 1 x by using the regular methods of summability these characterizations are alternative descriptions of the results already done by amrani and castaing in a amrani c castaing weak compactness in pettis integration bull pol acad sci math 45 2 1997 139 150 we also study the theorem of komlos in p 1 x which is a generalization of a result of e j balder in e j balder infinite dimensional extension of a theorem of komlos probab theory related fields 81 1989 185 188 theorem b we also prove some convergence theorems by applying the theorem we also prove convergence theorems in p 1 x analogous to the results of a amrani a amrani lemme de fatou pour l integrale de pettis publ math 42 1998 67 79 and h ziat h ziat convergence theorems for pettis integrable multifunctions bull pol acad sci math 45 2 1997 123 137 finally we prove some convergence theorems in p 1 x which are generalizations of some results of n c yannelis n c yannelis weak sequential convergence in l p x j math anal appl 141 1989 72 83 and a ulger a ulger weak compactness in l 1 x proc amer math soc 113 1991 143 149

decomposition and representation of fuzzy valued measure abstract fuzzy valued measure is a mapping which maps a algebra to a set of fuzzy sets with convex compact level defined on a banach space x decomposition and representation of fuzzy valued measure by a sequence of single valued measures is given the case when the set of images is the set of fuzzy numbers that is when x r is considered

polynomially filtered exact diagonalization approach to many body localization polynomially filtered exact diagonalization method polfed for large sparse matrices is introduced the algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix the memory requirements scale better with system size than in the state of the art shift invert approach the potential of polfed is demonstrated examining many body localization transition in 1d interacting quantum spin 1 2 chains we investigate the disorder strength and system size scaling of thouless time system size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite size effects we discuss possible scenarios regarding the many body localization transition obtaining estimates for the critical disorder strength

fuzzy valued measure abstract the main purpose of this paper is to introduce and develop the notion of a fuzzy valued measure in separable banach space this definition of fuzzy valued measure is a natural generalization of the set valued measure radon nikod m theorems for fuzzy valued measure are established

on convergence and closedness of multivalued martingales in this paper various convergence theorems and criteria of closedness of multivalued martingales submartingales and supermartingales are proved

central limit theorems for generalized set valued random variables abstract we give central limit theorems for generalized set valued random variables whose level sets are compact both in r d or in a banach space under milder conditions than those obtained recently by the latter two authors

on set valued stochastic integrals in an m type 2 banach space in a separable banach space for set valued martingale several equivalent conditions based on the measurable selections are discussed and then in an m type 2 banach space at first we define single valued stochastic integral by the differential of a real valued brownian motion after that extend it to set valued case we prove that the set valued stochastic integral becomes a set valued submartingale which is different from single valued case and obtain the castaing representation theorem for the set valued stochastic integral which is applicable for set valued stochastic differential equations

high performance implementation of chebyshev filter diagonalization for interior eigenvalue computations we study chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices in this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from chebyshev expansions of window functions after the discussion of the conceptual foundations of chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel search space size and filter polynomial degree on the computational accuracy and effort before we describe the necessary steps towards a parallel high performance implementation because chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers to demonstrate the potential of chebyshev filter diagonalization for large scale problems of this kind we include as an example the computation of the 102 innermost eigenpairs of a topological insulator matrix with dimension 109 derived from quantum physics applications

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

radon nikody m theorem and conditional expectation of fuzzy valued measures and variables abstract the notions of fuzzy valued measure and conditional expectation of a fuzzy valued variable are defined the main result of the paper is a proof of the existence of the radon nikod m derivative of the fuzzy valued measure a convergence theorem for conditional expectations is proved as well

quantitative stability analysis of stochastic generalized equations we consider the solution of a system of stochastic generalized equations sge where the underlying functions are mathematical expectation of random set valued mappings sge has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem we derive quantitative continuity of expected value of the set valued mapping with respect to the variation of the underlying probability measure in a metric space this leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the sge under some metric regularity conditions we derive aubin s property of the solution set mapping with respect to the change of probability measure the established results are applied to stability analysis of stochastic variational inequality stationary points of classical one stage and two stage stochastic minimization problems two stage stochastic mathematical programs with equilibriu

linear scaling calculation of an n type gaas quantum dot a linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation the method is based on the chebyshev polynomial expansion and the time dependent method which is tested on the calculation of the electronic structure of a model n type gaas quantum dot

support functions for fuzzy sets the main purpose of this paper is to introduce and develop the notion of support functions for fuzzy sets this approach gives the opportunity to investigate a lot of properties of fuzzy valued functions functions with values in space of fuzzy sets defined on banach space using rich apparatus of functionals in this paper we examine some aspects of measurability integrability and measure theory

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

on the conditional expectation and convergence properties of random sets in this paper we study random sets with values in a separable banach space first we establish several useful properties of the set valued conditional expectation and then prove some convergence theorems for set valued amarts and uniform amarts using the weak kuratowski mosco and hausdorff modes of set convergence

interior eigenvalues from density matrix expansions in quantum mechanical molecular dynamics an accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed the scheme is based on recursive density matrix expansions e g a m n niklasson phys rev b 66 2002 155115 which are accelerated by a scale and fold technique e h rubensson j chem theory comput 7 2011 pp 1233 1236 the acceleration scheme requires interior eigenvalue estimates which may be expensive and cumbersome to come by here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost our method is illustrated with density functional tight binding born oppenheimer molecular dynamics simulations where the computational effort is dominated by the density matrix construction in our analysis we identify two different phases of the recursive polynomial expansion the conditioning and purification phases and we show that the acceleration repres

representation of fuzzy valued mappings by the sequence of single valued mappings abstract in this paper the analysis focusses on the important classes of fuzzy valued mappings measurable fuzzy valued mappings integrable fuzzy valued mappings our aim is to discuss the problem of representation of such mappings by countable family of single valued mappings this is a most useful result since it provides us with a remarkable list of alternative definitions criteria and proofs of theorems related to this type of fuzzy mappings

mountain impasse theorem and spectrum of semilinear elliptic problems this paper studies a minimax problem for functionals in hilbert space in the form of g u 1 2p u 2 g u where g u is frechet differentiable with weakly continuous derivative if g has a mountain pass geometry it does not necessarily have a critical point such a case is called in this paper a mountain impasse this paper states that in a case of mountain impasse there exists a sequence u j h such that g u j p j u j p j p u j and g u j approximates the minimax value from above

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

equilibrium points of nonatomic games over a banach space schmeidler s results on the equilibrium points of nonatomic games with strategy sets in euclidean space are generalized to nonatomic games with strategy sets in a banach space our results also extend previous work of the author which assumed the banach space to be separable and its dual to possess the radon nikodym property our proofs use recent results in functional analysis 1 introduction in this paper we study cournot nash equilibria of games with a continuum of players each of whom has a strategy set in a banach space the importance of such games for economic theory has been recently underscored by dubey mas colell and shubik 7 see their example 2 in particular however 7 is primarily concerned with the equivalence of walrasian and cournot nash equilibria our focus here is on the existence theory the results reported here can be seen as a continuation of research initiated by schmeidler 19 who formulated and studied nonatomic games over zz dimensional euclidean space they also extend earlier work of the author in at least three important respects to begin with 14 assumed that each player s strategy set lies in a separable banach space whose dual has the radon nikodym property this condition is not satisfied for the infinite dimensional spaces typically studied in economics see bewley 2 and mas colell 18 it is of interest that such an assumption can be relaxed at the cost of a rather mild uniformity assumption on the distribution of strategy sets secondly it seems desirable to have results for con jugate banach spaces which are not separable and for which the strategy sets are not weakly compact but only norm bounded and weak closed we also present such results here they are especially relevant for commodity spaces studied in 2 and 18 finally unlike 14 our results on the existence of mixed strategy equilibria rely only on preference orderings rather than on payoff functions it is not known to us whether our conditions on preferences are sufficient for them to be represented as jointly measurable payoff functions such theorems are difficult to obtain even for a finite dimensional setting see wesley 21

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

weak sequential convergence in lp μ x we provide some new results on the weak convergence of sequences or nets lying in lp t x lp x 1 p i e the space of equivalence classes of x valued x is a banach space bochner integrable functions on the finite measure space t our theorems generalize in several directions recent resuls on weak sequential convergence in l1 x obtained by m a khan and m majumdar j math anal appl 114 1986 569 573 and z artstein j math econ 6 1979 277 282 and they can be used to obtain dominated convergence results for the aumann integral our results have useful applications in economics and game theory

doob s stopping theorem for fuzzy super sub martingales with discrete time fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets the concepts of fuzzy conditional expectation and fuzzy super sub martingales are given in this paper we study doob s stopping theorem on fuzzy super sub martingales with discrete parameters

kernel polynomial approximations for densities of states and spectral functions chebyshev polynomial approximations are an efficient and numerically stable way to calculate properties of the very large hamiltonians important in computational condensed matter physics the present paper derives an optimal kernel polynomial which enforces positivity of density of states and spectral estimates achieves the best energy resolution and preserves normalization this kernel polynomial method kpm is demonstrated for electronic structure and dynamic magnetic susceptibility calculations for tight binding hamiltonians of si we show how to achieve high precision and rapid convergence of the cohesive energy and vacancy formation energy by careful attention to the order of approximation for disordered xxz magnets we show that the kpm provides a simpler and more reliable procedure for calculating spectral functions than lanczos recursion methods polynomial approximations to fermi projection operators are also proposed

essential convex closure of a family of random sets and its applications abstract in this paper the essential convex closure of a family of random sets is defined and its properties are studied by the essential convex closure a more general optional sampling theorem for set valued super sub martingales is proved and the k m convergence of the unbounded set valued superpramarts is studied

a functional version of the birkhoff ergodic theorem for a normal integrand a variational approach in this paper we prove a new version of the birkhoff ergodic theorem bet for random variables depending on a parameter alias integrands this involves variational convergences namely epigraphical hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands we also have to establish the measurability of the epigraphical lower and upper limits with respect to the field of invariant subsets from the main result applications to uniform versions of the bet to sequences of random sets and to the strong consistency of estimators are briefly derived

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

jensen s inequality for random elements in metric spaces and some applications abstract jensen s inequality is extended to metric spaces endowed with a convex combination operation applications include a dominated convergence theorem for both random elements and random sets a monotone convergence theorem for random sets and other results on set valued expectations in metric spaces and on random probability measures

convexification in limit laws of random sets in banach spaces a strong law of large numbers for random compact sets in a banach space with the minkowski addition is provided by puri and ralescu 6 and among other results by gine hahn and zinn 2 and hess 3 this follows the finite dimensional version introduced in 1 the latter however allows nonconvex sets while the generalizations to the banach space setting were proved for convex sets a convexification argument stated in the lemma below shows that the convexity assumption can be dropped the proof reveals information about the nature of the convergence since writing this note we learned that f hiai 4 has independently established this result using a different method some notations let b be a banach space with norm 1111 the minkowski addition of two compact subsets k and l is k l x y x e k y e li and ak is the set tax x e k the closed convex hull of k is denoted cok it is a linear operation the hausdorff distance h k l is defined as max maxxekminyel 11 x y 11 maxyelminxek 11 x y y convergence of sets in this paper is meant in the hausdorff metric