spectral shift functions and dirichlet to neumann maps the spectral shift function of a pair of self adjoint operators is expressed via an abstract operator valued titchmarsh weyl m function this general result is applied to different self adjoint realizations of second order elliptic partial differential operators on smooth domains with compact boundaries and schrodinger operators with compactly supported potentials in these applications the spectral shift function is determined in an explicit form with the help of energy parameter dependent dirichlet to neumann maps

on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

inverse problem for harmonic oscillator perturbed by potential characterization consider the perturbed harmonic oscillator ty y x2y q x y in l2 where the real potential q belongs to the hilbert space h q xq l2 the spectrum of t is an increasing sequence of simple eigenvalues n q 1 2n n n 0 such that n 0 as n let n x q be the corresponding eigenfunctions define the norming constants n q lim x log n x q n x q we show that open image in new window for some real hilbert space open image in new window and some subspace open image in new window furthermore the mapping q q n q 0 n q 0 is a real analytic isomorphism between h and open image in new window is the set of all strictly increasing sequences s s n 0 such that open image in new window the proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials we use ideas from the analysis of the inverse problem for the operator y py p l2 0 1 with dirichlet boundary conditions on the unit interval there is no literature about the spaces open image in new window we obtain their basic properties using their representation as spaces of analytic functions in the disk

on a trace formula of the buslaev faddeev type for a long range potential we propose an approach to obtaining new trace formulas of the gel fand levitan buslaev faddeev type valid for hilbert schmidt perturbations in this way we obtain a new trace formula for schrodinger operators on the half line with long range potentials

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

existence theorem to the cauchy problem of fuzzy differential equations under compactness type conditions kaleva fuzzy sets and systems 35 1990 389 396 has concluded that the peano theorem does not hold for fuzzy differential equations on en d this paper deals with the cauchy problem of fuzzy differential equations the existence theorems under compactness type conditions are obtained

on the stability of fuzzy dynamical systems abstract in this work we study the asymptotic behavior of fuzzy solutions obtained by using zadeh s extension at the deterministic solutions of initial value problems we obtain some result regarding the existence of fuzzy equilibrium points that generalize the already known results as show earlier the membership function of the fuzzy equilibrium points may be obtained in a relatively simple way also we study the projection of the fuzzy solution on the appropriate subspaces of the phase space several examples are given to illustrate the obtained results

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

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

trace formulas for non self adjoint periodic schrodinger operators and some applications abstract recently a trace formula for non self adjoint periodic schrodinger operators in l 2 r associated with dirichlet eigenvalues was proved in differential integral equations 14 2001 671 700 here we prove a corresponding trace formula associated with neumann eigenvalues in addition we investigate dirichlet and neumann eigenvalues of such operators in particular using the dirichlet and neumann trace formulas we provide detailed information on location of the dirichlet and neumann eigenvalues for the model operator with the potential k e 2 i x where k c

fuzzy quasilinear spaces and applications we introduce the concept of fuzzy quasilinear space and fuzzy quasilinear operator moreover we state some properties and give results which extend to the fuzzy context some results of linear functional analysis

on the m function and borg marchenko theorems for vector valued sturm liouville equations we will consider a vector valued sturm liouville equation of the form r u pu qu lambdawu xis an element of 0 b with p 1 w qis an element ofl loc 1 0 b mxm being hermitian and under some additional conditions on p 1 and w we give an elementary deduction of the leading order term asymptotics for the titchmarsh weyl m function corresponding to this equation in the special case of p w i qis an element ofl 1 0 infinity mxm and the neumann boundary conditions at 0 we will also prove that m 1 root lambda i r i r 1 where r lim n infinity r n sigma n 1 infinity q n for recursively defined sequences r n and q n if qis an element ofl loc 1 0 b mxm 0 bless than or equal toinfinity the same formula is valid with an exponentially small error for large lambda it is clear that expansions of this type are helpful in finding representatives of the kdv invariants for p w i we prove that the spectral measure corresponding to the equation r u lambdau uniquely determines q as well as b and the boundary conditions at 0 and b we finally give a new proof of a local form of the borg marchenko theorem cf gesztesy and simon on local borg marchenko uniqueness results commun math phys 211 273 287 2000 chap 3 a theorem which is due to simon see simon a new approach to inverse spectral theory i fundamental formalism ann math 150 1 29 1999 in the scalar case for applications to physics it is worth mentioning that vector valued sturm liouville equations appear in some problems in magneto hydro dynamics

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

global existence of solutions to fuzzy differential equations abstract the local existence and uniqueness theorems are investigated by wu et al j math anal appl 202 1996 629 644 for the cauchy problem of the fuzzy valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper we point out a variety of results which assure global existence of solutions to fuzzy differential equations

approximate solutions existence and uniqueness of the cauchy problem of fuzzy differential equations abstract the cauchy problem of fuzzy differential equations is investigated by the use of the concept of h differentiability due to m l puri and d a ralescu j math anal appl 91 1983 552 558 the existence and uniqueness theorem is obtained for the solution of the cauchy problem x t f t x t x t 0 x 0 for the fuzzy valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in r n where the function f satisfies the generalized lipschitz condition

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

spectral deformations of jacobi operators we extend recent work concerning isospectral deformations for one dimensional schrodinger operators to the case of jacobi operators we provide a complete spectral characterization of a new method that constructs isospectral deformations of a given jacobi operator hu n a n u n 1 a n 1 u n 1 b n u n our technique is connected to dirichlet data that is the spectrum of the operator h n0 on 2 n0 n0 with a dirichlet boundary condition at n0 the transformation moves a single eigenvalue of h n0 and perhaps flips which side of n0 the eigenvalue lives on the remainder of the spectrum the transformation is realized by a unitary operator

on a theorem of kac and gilbert we prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum

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 zero inflated ordered probit model with an application to modelling tobacco consumption abstract data for discrete ordered dependent variables are often characterised by excessive zero observations which may relate to two distinct data generating processes traditional ordered probit models have limited capacity in explaining this preponderance of zero observations we propose a zero inflated ordered probit model using a double hurdle combination of a split probit model and an ordered probit model monte carlo results show favourable performance in finite samples the model is applied to a consumer choice problem of tobacco consumption indicating that policy recommendations could be misleading if the splitting process is ignored

global solutions of fuzzy integro differential equations under generalized differentiability by the method of upper and lower solutions this paper is devoted to introduce a new concept of upper and lower solutions for fuzzy integro differential equations that is defined on integral equations associated with the main problem this plays an essential role in proof of existence and uniqueness results under weak conditions by the method of upper and lower solutions existence and uniqueness of global solutions of an initial value problem for first order nonlinear fuzzy integro differential equation under generalized differentiability are given our technique is illustrated by some examples

a bartlett adjustment to the likelihood ratio test for a system of equations abstract a bartlett adjustment to be applied to the likelihood ratio test statistic when testing parameter restrictions in a system of equations is derived as special cases results are also obtained for testing hypotheses on the mean of a normal variate linear restrictions in the linear model restrictions in a nonlinear model and linear restrictions in the multivariate linear model

asymptotic equilibrium and stability of fuzzy differential equations the local existence and uniqueness theorems and the global existence of solutions were investigated in 1 3 respectively for the cauchy problem of fuzzy valued functions of a real variable whose values are in the fuzzy number space e n d in this paper we first study the asymptotic equilibrium for fuzzy evolution equations then the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations are investigated by extending the lyapunov s direct method

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

higher order trace formulas of the buslaev faddeev type for the half line schrodinger operator with long range potentials we deal with trace formulas for half line schrodinger operators with long range potentials we generalize the buslaev faddeev trace formulas to the case of square integrable potentials the exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given the main results are optimal

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

asymptotic expansions of the information matrix test statistic the information matrix im test is shown to have a finite sample distribution which is poorly approximated by its asymptotic x 2 distribution in models and sample sizes commonly encountered in applied econometric research the quality of the x2 approximation depends upon the method chosen to compute the test failure to exploit restrictions on the covariance matrix of the test can lead to a test with appalling finite sample properties order o n 1 approximations to the exact distribution of an efficient form of the im test are reported these are developed from asymptotic expansions of the edgeworth and cornish fisher types they are compared with monte carlo estimates of the finite sample distribution of the test and are found to be superior to the usual x2 approximations in sample sizes of the magnitude found in applied micro econometric work the methods developed in the paper are applied to normal and exponential models and to normal regression models results are provided for the full im test and for heteroskedasticity and nonnormality diagnostic tests which are special cases of the im test in geieral the quality of alternative approximations is sensitive to covariate design however commonly used nonnormality tests are found to have distributions which to order o n 1 are invariant under changes in covariate design this leads to simple design and parameter invariant size corrections for nonnormality tests

a simple parametric model selection test abstractwe propose a simple model selection test for choosing among two parametric likelihoods which can be applied in the most general setting without any assumptions on the relation between the candidate models and the true distribution that is both one or neither is allowed to be correctly specified or misspecified they may be nested nonnested strictly nonnested or overlapping unlike in previous testing approaches no pretesting is needed since in each case the same test statistic together with a standard normal critical value can be used the new procedure controls asymptotic size uniformly over a large class of data generating processes we demonstrate its finite sample properties in a monte carlo experiment and its practical relevance in an empirical application comparing keynesian versus new classical macroeconomic models supplementary materials for this article are available online

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

neglected heterogeneity in moment condition models abstract the central concern of this paper is parameter heterogeneity in models specified by a number of unconditional or conditional moment conditions and thereby the provision of a framework for the development of apposite optimal m tests against its potential presence we initially consider the unconditional moment restrictions framework optimal m tests against moment condition parameter heterogeneity are derived with the relevant jacobian matrix obtained in terms of the second order own derivatives of the moment indicator in a leading case gmm and gel tests of specification based on generalized information matrix equalities appropriate for moment based models are described and their relation to optimal m tests against moment condition parameter heterogeneity examined a fundamental and important difference is noted between gmm and gel constructions the paper is concluded by a generalization of these tests to the conditional moment context and the provision of a limited set of simulation experiments to illustrate the efficacy of the proposed tests

oscillation property for fuzzy delay differential equations in this paper we obtain an oscillation criterion of solutions and the existence of nonoscillatory solutions to a class of fuzzy delay differential equation fdde of second order some examples are given to illustrate the main theorems

residual analysis in the grouped and censored normal linear model graphical and numerical analysis of residual can be informative about model misspecification even when data are censored or grouped this paper provides simple procedures for calculating diagnostic statistics to detect model misspecification when grouped or censored data are analysed using a normal linear model graphs can reveal the nature of the correlations that these statistics pick up but when grouping or censoring is severe they can be difficult to interpret we discuss the processing of graphs so that they are more easily interpreted and provide examples based on artificial data and on grouped data relating to unemployment durations

duration response measurement error the impact of response measurement error in duration data is investigated using small parameter asymptotic approximations and compared with the effect of hazard function heterogeneity the approximations lead to a specification test to detect measurement error which is shown to be related to the class of information matrix tests it is shown that in a commonly used class of models the test statistic is exactly pivotal the second order asymptotic properties of the alternative forms of the test statistic are derived and the quality of the approximations and the performance of the test are investigated via monte carlo experimentation

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

bootstrap based critical values for the information matrix test abstract in finite samples the true size of white s information matrix test often differs greatly from its nominal size based on asymptotic critical values this paper shows how the bootstrap can be used to obtain improved finite sample critical values the results of monte carlo experiments show that for the cases investigated the bootstrap largely eliminates the problem of incorrect finite sample size moreover when size corrected critical values are used forms of the test that have small size distortions with asymptotic critical values can have much lower power than forms that have large size distortions with asymptotic critical values

rank one perturbations at infinite coupling abstract we discuss rank one perturbations a a r a 0 self adjoint let d x be the spectral measure defined by a z 1 d x x z we prove there is a measure d which is the weak limit of 1 2 d x as if is cyclic for a then a the strong resolvent limit of a is unitarily equivalent to multiplication by x on l 2 r d this generalizes results known for boundary condition dependence of sturm liouville operators on half lines to the abstract rank one case

global existence and uniqueness of solutions for fuzzy differential equations under dissipative type conditions in this paper using the properties of a differential and integral calculus for fuzzy set valued mappings and completeness of metric space of fuzzy numbers the global existence uniqueness and the continuous dependence of a solution on a fuzzy differential equation are derived under the dissipative type conditions we also present the global existence and uniqueness of solutions for a fuzzy differential equation on a closed convex subset of fuzzy number space

generalized empirical likelihood non nested tests this paper examines non nested tests for competing moment condition models using a semi parametric generalized empirical likelihood gel framework the resultant gel estimators are first order asymptotically equivalent to those based on generalized method of moments gmm cox type moment encompassing and parametric encompassing non nested tests for competing moment condition models are proposed simulation experiments are conducted to examine the efficacy of the proposed gel statistics in terms of their size and power properties and to compare their properties with those of corresponding non nested test statistics based on gmm estimation

a trace formula for multidimensional schrodinger operators we prove multidimensional analogs of the trace formula obtained previously for one dimensional schrodinger operators for example letvbe a continuous function on 0 1 r fora 1 let abe the laplace operator on 0 1 with mixed dirichlet neumann boundary conditions formula let a number of points ina then we ll prove that formula with v the average ofvat the 2 corners of 0 1

practical stability boundedness criteria and lagrange stability of fuzzy differential systems in this paper we have investigated practical stability boundedness criteria and lagrange stability for fuzzy differential equations these properties are unified with lyapunov like functions to establish comparison results

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions abstract in this paper the existence and uniqueness of a fuzzy solution for the semilinear fuzzy integrodifferential equation is established via the banach fixed point analysis approach and using the fuzzy number whose values are normal convex upper semicontinuous and compactly supported interval in e n

traces and inverse nodal problems for dirac type integro differential operators on a graph abstract in this paper dirac type integro differential operators on the star graph are studied first we derive the asymptotic expressions for the large eigenvalues second the regularized trace formulae are obtained for these operators we also provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem

model specification tests a simultaneous approach abstract in econometrics specification tests have been constructed to verify the validity of one specification at a time it is argued that most of these tests are not in general robust in the presence of other misspecifications so their application may result in misleading conclusions using the lagrange multiplier principle we develop efficient test procedures that are capable of testing a number of specifications simultaneously these tests will confirm the validity or invalidity of a general model requiring the estimates of the restricted model only through an extensive monte carlo experiment we study the performance of these tests and some commonly used one directional tests we also suggest a multiple comparison procedure to identify different sources of errors this we hope will lead to a better specification of econometric models

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

likelihood ratio specification tests misspecification tests for parametric models f y that examine data for failure of moment conditions implied by the maintained parametric distribution are interpreted as score tests of h 0 0 in the context of a parametric family of distributions r y this family contains the maintained distribution as a special case 0 and has the property that only in that special case do the chosen moment conditions hold a likelihood ratio test of h 0 0 therefore constitutes an alternative test of the validity of the moment conditions this test admits a bartlett correction unlike conventional moment tests for which adjustments based on second order asymptotic theory may behave badly the dependence of the bartlett correction and of the o n 1 2 local power of the test on the way in which r y is constructed is studied in many cases the correction can be made to vanish leading to a specification test whose distribution is chi square to order o p n 2

initial value problem for fuzzy differential equations under dissipative conditions a new concept of inner product on the fuzzy space e n d is introduced studied and used to prove several theorems stating the existence uniqueness and boundedness of solutions of fuzzy differential equations a stability result is also proved in the same context

sharp phase transitions for the almost mathieu operator it is known that the spectral type of the almost mathieu operator amo depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency we study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place which solves jitomirskaya s conjecture together with a previous work by avila this gives the sharp description of phase transitions for the amo for the a e phase

trace formulas and inverse spectral theory for jacobi operators based on high energy expansions and herglotz properties of green and weyl m functions we develop a self contained theory of trace formulas for jacobi operators in addition we consider connections with inverse spectral theory in particular uniqueness results as an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit formulas for the coefficients in terms of minimal spectral data finally trace formulas are applied to scattering theory with periodic backgrounds

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

information theoretic approaches to inference in moment condition models one step efficient gmm estimation has been developed in the recent papers of back and brown 1990 imbens 1993 and qin and lawless 1994 these papers emphasized methods that correspond to using owen s 1988 method of empirical likelihood to reweight the data so that the reweighted sample obeys all the moment restrictions at the parameter estimates in this paper we consider an alternative klic motivated weighting and show how it and similar discrete reweightings define a class of unconstrained optimization problems which includes gmm as a special case such klic motivated reweightings introduce m auxiliary tilting parameters where m is the number of moments parameter and overidentification hypotheses can be recast in terms of these tilting parameters such tests when appropriately conditioned on the estimates of the original parameters are often startlingly more effective than their conventional counterparts this is apparently due to the local ancillarity of the original parameters for the tilting parameters

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

comparation between some approaches to solve fuzzy differential equations in this paper we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via zadeh s extension principle we get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented we also give some properties and we show the relationships with others interpretation finally we propose a numerical procedure for obtaining the fuzzy solution

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

numerical solutions for linear system of first order fuzzy differential equations with fuzzy constant coefficients in this paper a novel iterative method is proposed to obtain approximate analytical solutions for the linear systems of first order fuzzy differential equations fdes with fuzzy constant coefficients fccs while avoiding the complexities of eigen value computations a theorem for the convergence and the validity of the approach is also presented in detail numerical experiments and comparisons with exact solutions reveal that the proposed method is capable of generating accurate results

spectral properties of a class of reflectionless schrodinger operators abstract we prove that one dimensional reflectionless schrodinger operators with spectrum a homogeneous set in the sense of carleson belonging to the class introduced by sodin and yuditskii have purely absolutely continuous spectra this class includes all earlier examples of reflectionless almost periodic schrodinger operators in addition we construct examples of reflectionless schrodinger operators with more general types of spectra given by the complement of a denjoy widom type domain in c which exhibit a singular component

global existence of solutions for fuzzy second order differential equations under generalized h differentiability in this paper we study the global existence of solutions for second order fuzzy differential equations with initial conditions under generalized h differentiability second derivative of the h difference of two functions under generalized h differentiability is obtained two theorems which assure global existence of solutions for second order fuzzy differential equations are given and proved some examples are given to illustrate these results

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

eigenvalue estimates and trace formulas for the matrix hill s equation abstract estimates for floquet multipliers and periodic eigenvalues are developed for the matrix hill s equation basic estimates and a description of the periodic eigenvalues as roots of an entire function are adequate for establishing trace formulas with a residue computation refined estimates and power sum formulas are available for restricted pairs of hill s operators

on fuzzy differential equations in the quotient space of fuzzy numbers since addition possesses an inverse in the quotient space of fuzzy numbers based on mares equivalence relation in this paper we introduce a metric on the quotient space of fuzzy numbers and then deal with fuzzy mappings of a real variable whose values are equivalence classes of fuzzy numbers we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation finally some examples are given to illustrate the main theorems