brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

exact ground state energy of the strong coupling polaron the polaron has been of interest in condensed matter theory and field theory for about half a century especially the limit of large coupling constant a it was not until 1983 however that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals here we derive the same asymptotic result e 0 c 2 and with explicit error bounds by simple rigorous methods applied directly to the hamiltonian our method is easily generalizable to other settings e g the excitonic and magnetic polarons

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

a many body rage theorem we prove a generalized version of the rage theorem for n body quantum systems the result states that only bound states of systems with 0 leqslant n leqslant n particles persist in the long time average the limit is formulated by means of an appropriate weak topology for many body systems which was introduced by the second author in a previous work and is based on reduced density matrices this topology is connected to the weak topology of states on the algebras of canonical commutation or anti commutation relations and we give a formulation of our main result in this setting

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

the thermodynamic limit of quantum coulomb systems part ii applications abstract in a previous paper c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue we have developed a general theory of thermodynamic limits we apply it here to three different coulomb quantum systems for which we prove the convergence of the free energy per unit volume the first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles we recover and generalize a previous result of fefferman in the second example both the nuclei and the electrons are quantum particles submitted to a periodic magnetic field we thereby extend a seminal result of lieb and lebowitz finally in our last example we take again classical nuclei but optimize their position to our knowledge such a system was never treated before the verification of the assumptions introduced in c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue uses several tools which have been introduced before in the study of large quantum systems in particular an electrostatic inequality of graf and schenker is one main ingredient of our new approach

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

the exponential homomorphism for the second syntomic cohomology abstract let ok be a complete discrete valuation ring with perfect residue field f of characteristic p 3 and m a free filtered dieudonne module the second relative syntomic cohomology h2 ok f s m 2 with coefficients in m is an object related to arithmetic geometry such as the albanese kernel in this article we study h2 ok f s m 2 by constructing the exponential homomorphism we determine the structure of h2 ok f s m 2 under the assumption that m is of hodge witt type and that the absolute ramification index of ok is prime to p

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

a continuous time dynamic fuzzy system ii fuzzy potentials abstract this paper is a sequel of yoshida 9 fuzzy potentials are introduced in a continuous time dynamic fuzzy system and a fuzzy relational differential equation is given to characterize the fuzzy potentials

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

absolute retracts and a general fixed point theorem for fuzzy sets abstract the space of fuzzy sets on a compact metric space with the sendograph metric for fuzzy sets is shown to have the fixed point property the relationship between absolute retracts and spaces of fuzzy sets is demonstrated

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

syntomic cohomology as a p adic absolute hodge cohomology the purpose of this paper is to interpret rigid syntomic cohomology defined by amnon besser bes as a p adic absolute hodge cohomology this is a p adic analogue of a work of beilinson be1 which interprets beilinson deligne cohomology in terms of absolute hodge cohomology in the process we will define a theory of p adic absolute hodge cohomology with coefficients which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

minimizers for the hartree fock bogoliubov theory of neutron stars and white dwarfs we prove the existence of minimizers for hartree fock bogoliubov hfb energy functionals with attractive two body interactions given by newtonian gravity this class of hfb functionals serves as a model problem for self gravitating relativistic fermi systems which are found in neutron stars and white dwarfs furthermore we derive some fundamental properties of hfb minimizers such as a decay estimate for the minimizing density a decisive feature of the hfb model in gravitational physics is its failure of weak lower semicontinuity this fact essentially complicates the analysis compared to the well studied hartree fock theories in atomic physics

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

the mean field approximation and the non linear schrodinger functional for trapped bose gases we study the ground state of a trapped bose gas starting from the full many body schrodinger hamiltonian and derive the nonlinear schrodinger energy functional in the limit of large particle number when the interaction potential converges slowly to a dirac delta function our method is based on quantitative estimates on the discrepancy between the full many body energy and its mean field approximation using hartree states these are proved using finite dimensional localization and a quantitative version of the quantum de finetti theorem our approach covers the case of attractive interactions in the regime of stability in particular our main new result is a derivation of the 2d attractive nonlinear schrodinger ground state

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

fuzzy homoclinic orbits and commuting fuzzifications fuzzy homoclinic orbits and commuting fuzzifications are defined and studied we study sufficient conditions involving commuting fuzzifications under which a crisp homoclinic orbit is preserved after fuzzification we show the usual fuzzification is commuting and therefore preserves homoclinic orbits several examples are given regarding fuzzy homoclinic orbits and commuting fuzzifications

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

stability in possibilistic linear programming with continuous fuzzy number parameters abstract we prove that possibilistic linear programming problems introduced by buckley in 2 with continuous fuzzy number parameters are well posed i e small changes of the membership function of the parameters may cause only a small deviation in the possibilistic distribution of the objective function

the multiconfiguration equations for atoms and molecules charge quantization and existence of solutions we prove the existence of ground state solutions for the multiconfiguration self consistent field equations for atoms and molecules whenever the total nuclear charge z exceeds n 1 where n is the number of electrons moreover we show that for arbitrary values of z and n the scattering charge i e the asymptotic amount of charge lost by an energy minimizing sequence is integer quantized our analysis applies to the mc equations of arbitrary rank as special cases we recover in a new and unified way the existence theorems of zhislin zh60 for the n body schrodinger equation infinite rank mc and of lieb simon ls77 for the hartree fock equations rank n mc our approach is a direct study of an invariant orbital free formulation in n body space of the underlying variational principle proofs involve i the geometric n body localization methods for the linear schrodinger equation first introduced by enss en77 and developed in sim77 sig82 which can be adapted to become powerful tools in nonlinear many body theory as well ii weak convergence methods from the theory of nonlinear partial differential equations iii careful analysis of the structure of the one and two body density matrices of the bound and scattering fragments delivered by geometric localization which allows us to overcome the fact that the rank of the fragments is not reduced by localization

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

solutions of the multiconfiguration equations in quantum chemistry the multiconfiguration methods are the natural generalization of the well known hartree fock theory for atoms and molecules by a variational method we prove the existence of a minimum of the energy and of infinitely many solutions of the multiconfiguration equations a finite number of them being interpreted as excited states of the molecule our results are valid when the total nuclear charge z exceeds n 1 n is the number of electrons and cover most of the methods used by chemists the saddle points are obtained with a min max principle we use a palais smale condition with morse type information and a new and simple form of the euler lagrange equations

on fuzzifications of discrete dynamical systems let x denote a locally compact metric space and f x x be a continuous map in the 1970s zadeh presented an extension principle helping us to fuzzify the dynamical system x f i e to obtain a map f for the space of fuzzy sets on x we extend an idea mentioned in p diamond a pokrovskii chaos entropy and a generalized extension principle fuzzy sets syst 61 1994 277 283 to generalize zadeh s original extension principle in this paper we study basic properties of so called g fuzzifications such as their continuity properties we also show that for any g fuzzification i a uniformly convergent sequence of uniformly continuous maps on x induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and ii a conjugacy resp a semi conjugacy between two discrete dynamical systems can be extended to a conjugacy resp a semi conjugacy between fuzzified dynamical systems throughout this paper we consider different topological structures in the space of fuzzy sets namely the sendograph endograph and levelwise topologies

on a topological fuzzy fixed point theorem and its application to non ejective fuzzy fractals abstract a topological fuzzy fixed point theorem is given when generalizing and improving the main result by diamond kloeden and pokrovskii 1997 1 apart from the sole existence a weak local stability property called non ejectivity in the sense of browder of fuzzy fixed points is established this theorem is then applied for obtaining non ejective fuzzy fractals an alternative approach via the knaster tarski theorem is also presented

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

membership induced on manifolds by vector fields and flows abstract in many engineering and scientific applications the need arises to perform differential calculations on manifolds if such calculations involve imprecise data methods for computing with such data must be determined definitions and results are given for memberships on manifolds induced by differentiation along vector fields and by flows of vector fields with uncertain parameters examples are presented which illustrate the concepts

a note on the sendograph metric of fuzzy numbers in this paper by studying the attainable properties of sendograph metric in fuzzy number space we generalize monotone convergence theorem and nested theorem of intervals from the space of real numbers to the space of fuzzy numbers

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

scattering theory for a class of fermionic pauli fierz models abstract the scattering theory for a class of fermionic pauli fierz models is considered we give a proof of the asymptotic completeness of the dynamics in the case of massive fermions the result applied to the hamiltonian of a quantized spin 1 2 dirac particle interacting with an external field through a cutoff yukawa interaction and to the hamiltonian of a system of finitely many confined particles coupled to a fermionic field with a quadratic interaction

geometric methods for nonlinear many body quantum systems abstract geometric techniques have played an important role in the seventies for the study of the spectrum of many body schrodinger operators in this paper we provide a formalism which also allows to study nonlinear systems we start by defining a weak topology on many body states which appropriately describes the physical behavior of the system in the case of lack of compactness that is when some particles are lost at infinity we provide several important properties of this topology and use them to write a simple proof of the famous hvz theorem in the repulsive case in the second step we recall the method of geometric localization in fock space as proposed by derezinski and gerard and we relate this tool to our weak topology we then provide several applications we start by studying the so called finite rank approximation which consists in imposing that the many body wavefunction can be expanded using finitely many one body functions we thereby emphasize geometric properties of hartree fock states and prove nonlinear versions of the hvz theorem in the spirit of works of friesecke in the last section we study translation invariant many body systems comprising a nonlinear term which effectively describes the interactions with a second system as an example we prove the existence of the multi polaron in the pekar tomasevich approximation for certain values of the coupling constant

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings this paper generalizes a classical result about the space of bounded closed sets with the hausdorff metric and establishes the completeness of cb x with respect to the completeness of the metric space x where cb x is the class of fuzzy sets with nonempty bounded closed a cut sets equipped with the supremum metric d which takes the supremum on the hausdorff distances between the corresponding a cut sets in addition some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space