on fuzzy valued operations and fuzzy valued fuzzy sets this paper is devoted to investigate fuzzy valued operations and the lattice structures of the algebra of fuzzy values the algebra of fuzzy values is proven to be a complete completely distributive lattice and the list of all join irreducible elements is provided fuzzy valued t norms and t conorms are induced by arbitrary t norms and t conorms regardless of their continuity respectively moreover fuzzy valued fuzzy implications are constructed from fuzzy valued t norms induced by left continuous t norms and residuated lattices are constructed on the algebra of fuzzy values as a consequence fuzzy valued fuzzy rough sets and fuzzy valued fuzzy complete lattices are proposed and investigated in fuzzy valued fuzzy sets

estimates on green s functions localization and the quantum kicked rotor model and phase space l2 1r it has been conjectured by a number of authors cf fgp bel that for typical parameter values for a b the wave function t satisfying equations 0 1 0 2 will be almost periodic in time hence f t the fourier transform remains localized we will prove this here assuming the parameter s in 0 2 small thus we have theorem for k stnall and a b o tside a set of small rneasure o for k 0 the following holds let 4 w t x solve 0 1 0 2 and let to 4 0 x be a sujgciently smooth function on t then is an alrnost periodic function of timbe say as an hl t valued map in particular supt ll t llhl

anderson transition in a three dimensional kicked rotor we investigate anderson localization in a three dimensional 3d kicked rotor by a finite size scaling analysis we identify a mobility edge for a certain value of the kicking strength k k c for k k c dynamical localization does not occur all eigenstates are delocalized and the spectral correlations are well described by wigner dyson statistics this can be understood by mapping the kicked rotor problem onto a 3d anderson model am where a band of metallic states exists for sufficiently weak disorder around the critical region k approximately k c we carry out a detailed study of the level statistics and quantum diffusion in agreement with the predictions of the one parameter scaling theory opt and with previous numerical simulations the number variance is linear level repulsion is still observed and quantum diffusion is anomalous with proportional t 2 3 we note that in the 3d kicked rotor the dynamics is not random but deterministic in order to estimate the differences between these two situations we have studied a 3d kicked rotor in which the kinetic term of the associated evolution matrix is random a detailed numerical comparison shows that the differences between the two cases are relatively small however in the deterministic case only a small set of irrational periods was used a qualitative analysis of a much larger set suggests that deviations between the random and the deterministic kicked rotor can be important for certain choices of periods heuristically it is expected that localization effects will be weaker in a nonrandom potential since destructive interference will be less effective to arrest quantum diffusion however we have found that certain choices of irrational periods enhance anderson localization effects

subsolutions supersolutions and asymptotic behavior of an evolution problem with unilateral constraint on etudie l inegalite variationnelle parabolique suivante f u v u t 0 t t 0 u x t x sur 0 t u x 0 u 0 x dans u 0 sur 0 t v v x t x sur 0 t v 0 sur 0 t 0 ou a est un operateur differentiel d ordre 2 coercif la non linearite f est non negative croissante et reguliere l obstacle est positif et regulier sur domaine borne de r n on etudie l existence des solutions et leur comportement asymptotique

categories with fuzzy sets and relations we define a 2 category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions we enrich this category with additional monoidal and involutive structure coming from t norms and negations on the unit interval we develop the basic properties of this category and consider its relation to other familiar categories a discussion is made of extending these results to the setting of type 2 fuzzy sets

localization for the anderson model on a strip with singular potentials abstract we prove exponential localization in the anderson model on a one dimensional strip for any potential whose probability distribution is not concentrated on a single point and possesses some finite moment in particular we obtain localization for bernoulli potentials on a strip

a note on the lattice structure for subalgebras of the algebra of truth values of type 2 fuzzy sets abstract the algebra of truth values of type 2 fuzzy sets is not a lattice but some of its subalgebras may be lattices in this paper we give a necessary and sufficient condition under which subalgebras of the truth value algebra of type 2 fuzzy sets form lattices further we prove that if a certain subalgebra is a lattice then it is isomorphic to an appropriate subalgebra whose all elements are convex functions with the same height based on these results if a certain subalgebra is a lattice the equivalent characterizations of the partial order in it are given

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

spectral flow and level spacing of edge states for quantum hall hamiltonians we consider a non relativistic particle on the surface of a semi infinite cylinder of circumference l submitted to a perpendicular magnetic field of strength b and to the potential of impurities of maximal amplitude w this model is of importance in the context of the integer quantum hall effect in the regime of strong magnetic field or weak disorder b w it is known that there are chiral edge states which are localized within a few magnetic lengths close to and extended along the boundary of the cylinder and whose energy levels lie in the gaps of the bulk system these energy levels have a spectral flow uniform in l as a function of a magnetic flux which threads the cylinder along its axis through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by 2 l 1 with 0 independent of l and of the configuration of impurities this implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual anderson model and their statistics cannot obey one of the gaussian ensembles our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum hall systems

aggregation operators on type 2 fuzzy sets abstract cubillo et al in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset partially ordered set in particular on m set of fuzzy membership degrees of t2fss which are the functions from 0 1 to 0 1 previously takac in 2014 had applied zadeh s extension principle to obtain a set of operators on m which are under some conditions aggregation operators on l the set of strongly normal and convex functions of m in this paper we introduce a more general set of operators on m than were given by takac and we study among other properties the conditions required to satisfy the axioms of the aggregation operator on l set of normal and convex functions on m which includes the set l

lattices of convex normal functions the algebra of truth values of type 2 fuzzy sets is the set of all functions from the unit interval into itself with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min this algebra has been studied rather extensively both from a theoretical and from a practical point of view it has a number of interesting subalgebras and this paper is about the subalgebra of all convex normal functions and closely related ones these particular algebras are de morgan algebras and our concern is principally with their completeness as lattices a special feature of our treatment is a representation of these algebras as monotone functions with pointwise order making the operations more intuitive

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

note on the absorption laws in the algebra of truth values of type 2 fuzzy sets abstract this note gives a necessary and sufficient condition under which the absorption laws in the algebra of truth values of type 2 fuzzy sets are true

type 2 fuzzy rough sets based on extended t norms a type 2 fuzzy rough set based on extended t norms is investigated from constructive and axiomatic approaches an approximation algorithm of type 2 fuzzy rough sets is listed based on cut sets weak type 2 fuzzy topologies are proposed to investigate the topological properties of type 2 fuzzy rough sets an interval valued type 2 fuzzy rough set is proposed and investigated this paper is devoted to investigate type 2 fuzzy rough sets based on extended t norms with respect to type 2 fuzzy relations with convex normal fuzzy truth values type 2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches in the constructive approach different classes of type 2 fuzzy rough sets are investigated an approximate algorithm of type 2 fuzzy rough sets based on cut sets is given with numerical examples in the axiomatic approach the axiomatic characterizations of type 2 fuzzy rough approximation operators are studied moreover the topological properties of type 2 fuzzy rough sets are discussed in the end this paper introduces interval valued type 2 fuzzy rough sets and discusses their properties

a lower wegner estimate and bounds on the spectral shift function for continuum random schrodinger operators abstract we prove a strictly positive locally uniform lower bound on the density of states dos of continuum random schrodinger operators on the entire spectrum i e we show that the dos does not have a zero within the spectrum this follows from a lower wegner estimate for finite volume continuum random schrodinger operators we assume throughout iid random variables and the single site distribution having a lebesgue density bounded from below on its support the main mathematical novelty in this paper are pointwise in energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from dirichlet to neumann boundary conditions along the boundary of a cube we show that the bound scales with the area of the hypersurface where the boundary conditions are changed we also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators

transient response of inhomogeneous transversely isotropic thermoelastic cylinder to a dynamic input elliptic boundary value problems with discontinuous nonlinearities are embedded into a family of smooth problems first families of nonlinear regularizations are discussed that guarantee that the solutions of the generated auxiliary problems form uniform lower and upper bounds of solutions of the original discontinuous elliptic boundary value problem later these regularizations are approximated by inner iterations using linear boundary value problems unlike the regular case here the linear problems are not collectively bounded thus a new stopping rule for the inner iteration is introduced finally the rate of convergence of the outer approximation is estimated and the proposed smoothing technique is combined with some finite element discretization a numerical example illustrates the proposed monotone smoothing and iteration

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

localization and universality of poisson statistics for the multidimensional anderson model at weak disorder we prove anderson localization with the mean field lyapunov exponent and poisson statistics for eigenvalue spacing for the multi dimensional anderson model at weak disorder these results are obtained by developing the supersymmetric formalism initiated in w1 see also sjw rid

the spectrum of skew shift schrodinger operators contains intervals abstract i prove that the spectrum of a skew shift schrodinger operator contains large intervals in the semi classical regime in the semi classical limit these intervals approach the range of the potential

loss of stability at turning points in nonlinear elliptic variational inequalities a considerable amount of work has been devoted in recent years to stationary free boundary problems in particular obstacle problems have been the subject of extensive work concerning existence estimates and regularity of the solution or of the coincidence set and occurrence of multiple solutions in 6 8 161 the authors consider obstacle problems with nonlinear forcing terms leading to multiple solutions hence to turning points when a real parameter appears in the forcing term a local study of the branches of solutions is set up the results obtained are the extension in a weaker form of those well known for equations ll 12 however some important classical questions remain open for obstacle problems

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

convex normal functions revisited the lattice l u of upper semicontinuous convex normal functions with convolution ordering arises in studies of type 2 fuzzy sets in 2002 kawaguchi and miyakoshi extended t norms as logical connectives of fuzzy truth values multiple valued logic 8 1 2002 53 69 showed that this lattice is a complete heyting algebra later harding et al lattices of convex normal functions fuzzy sets and systems 159 2008 1061 1071 gave an improved description of this lattice and showed it was a continuous lattice in the sense of gierz et al a compendium of continuous lattices springer berlin 1980 in this note we show the lattice l u is isomorphic to the lattice of decreasing functions from the real unit interval 0 1 to the interval 0 2 under pointwise ordering modulo equivalence almost everywhere this allows development of further properties of l u it is shown that l u is completely distributive is a compact hausdorff topological lattice whose topology is induced by a metric and is self dual via a period two antiautomorphism we also show the lattice l u has another realization of natural interest in studies of type 2 fuzzy sets it is isomorphic to a quotient of the lattice l of all convex normal functions under the convolution ordering this quotient identifies two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere

on the extension of nullnorms and uninorms to fuzzy truth values abstract in this work by applying zadeh s extension principle we extend type 1 proper nullnorms and proper uninorms to fuzzy truth values the definitions of type 2 nullnorms and uninorms are introduced we mainly discuss in which subalgebras of fuzzy truth values extended proper nullnorms and proper uninorms are type 2 nullnorms and uninorms respectively

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

type 2 operations on finite chains the algebra of truth values for fuzzy sets of type 2 consists of all mappings from the unit interval into itself with operations certain convolutions of these mappings with respect to pointwise max and min this algebra generalizes the truth value algebras of both type 1 and of interval valued fuzzy sets and has been studied rather extensively both from a theoretical and applied point of view this paper addresses the situation when the unit interval is replaced by two finite chains most of the basic theory goes through but there are several special circumstances of interest these algebras are of interest on two counts both as special cases of bases for fuzzy theories and as mathematical entities per se

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

localization of the continuous anderson hamiltonian in 1 d we study the bottom of the spectrum of the anderson hamiltonian mathcal h l partial x 2 xi on 0 l driven by a white noise xi and endowed with either dirichlet or neumann boundary conditions we show that as l rightarrow infty the point process of the appropriately shifted and rescaled eigenvalues converges to a poisson point process on mathbf r with intensity e x dx and that the appropriately rescaled eigenfunctions converge to dirac masses located at independent and uniformly distributed points furthermore we show that the shape of each eigenfunction recentered around its maximum and properly rescaled is given by the inverse of a hyperbolic cosine we also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate and we obtain very precise information on the zeros and local maxima of these eigenfunctions finally we show that the eigenvalues eigenfunctions in the dirichlet and neumann cases are very close to each other and converge to the same limits

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

on the largest singular values of random matrices with independent cauchy entries we apply the method of determinants to study the distribution of the largest singular values of large m n real rectangular random matrices with independent cauchy entries we show that for a special one parametric class of statistics the properties of the largest singular values rescaled by a factor 1 m2n2 agree in the limit with the statistical properties of the poisson random point process with the intensity 1 x 3 2 and therefore are different from the tracy widom law among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m n standard wishart ensemble and the real rectangular 2m 2n standard wishart ensemble

concentration of eigenvalues for skew shift schrodinger operators i analyze the microscopic behavior of the eigenvalues of skew shift schrodinger operators and show that their statistics must resemble the one of the anderson model rather than the one of quasi periodic schrodinger operators

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

the algebra of fuzzy truth values the purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type 2 fuzzy sets

analytic solutions of nonlinear elliptic equations on rectangular tori abstract in this paper we consider the nonlinear elliptic equations on rectangular tori using methods in the study of kam theory and anderson localization we prove that these equations admit many analytic solutions

generic continuous spectrum for ergodic schrodinger operators we consider families of discrete schrodinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function we introduce the concepts of topological and metric repetition property assuming that the underlying dynamical system satisfies one of these repetition properties we show using gordon s lemma that for a generic continuous sampling function the set of elements in the associated family of schrodinger operators that have no eigenvalues is large in a topological or metric sense respectively we present a number of applications particularly to shifts and skew shifts on the torus

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

quantum harmonic oscillator systems with disorder we study many body properties of quantum harmonic oscillator lattices with disorder a sufficient condition for dynamical localization expressed as a zero velocity lieb robinson bound is formulated in terms of the decay of the eigenfunction correlators for an effective one particle hamiltonian we show how state of the art techniques for proving anderson localization can be used to prove that these properties hold in a number of standard models we also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one particle eigenfunction correlators in particular we show that static correlations decay exponentially fast if the corresponding effective one particle hamiltonian exhibits localization at low energies regardless of whether there is a gap in the spectrum above the ground state or not our results apply to finite as well as to infinite oscillator systems the eigenfunction correlators that appear are more general than those previously studied in the literature in particular we must allow for functions of the hamiltonian that have a singularity at the bottom of the spectrum we prove exponential bounds for such correlators for some of the standard models

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

on the existence of sobolev quasi periodic solutions of multidimensional nonlinear beam equation in this paper we prove the existence of quasi periodic solutions of the multidimensional nonlinear beam equation with finitely differentiable nonlinearities and quasi periodic forcing in time

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

generalized eigenvalue counting estimates for the anderson model we generalize minami s estimate for the anderson model and its extensions to n eigenvalues allowing for n arbitrary intervals and arbitrary single site probability measures with no atoms as an application we derive new results about the multiplicity of eigenvalues and mott s formula for the ac conductivity when the single site probability distribution is holder continuous

on the integrated density of states for schrodinger operators on ℤ2 with quasi periodic potential in this paper we consider discrete schrodinger operators on the lattice 2 with quasi periodic potential we establish new regularity results for the integrated density of states as well as a quantitative version of a thouless formula as previously considered by craig and simon for real energies and with rates of convergence the main ingredient is a large deviation theorem for the green s function that was recently established by bourgain goldstein and the author for the integrated density of states an argument of bourgain is used finally we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest

eigenvalue statistics for schrodinger operators with random point interactions on rd d 1 2 3 we prove that the local eigenvalue statistics at energy e in the localization regime for schrodinger operators with random point interactions on rd for d 1 2 3 is a poisson point process with the intensity measure given by the density of states at e times the lebesgue measure this is one of the first examples of poisson eigenvalue statistics for the localization regime of multi dimensional random schrodinger operators in the continuum the special structure of resolvent of schrodinger operators with point interactions facilitates the proof of the minami estimate for these models

construction of quasi periodic solutions for delayed perturbation differential equations abstract we employ the craig wayne bourgain method to construct quasi periodic solutions for delayed perturbation differential equations our results not only implement the existing literatures on constructing quasi periodic solutions for dde by the kam method and space splitting technique but also provide an example of application of multi scale analysis method to non selfadjoint problem

answering an open problem on t norms for type 2 fuzzy sets abstract this paper proves that a binary operation on 0 1 ensuring that the binary operation is a t norm or is a t conorm is a t norm where and are special convolution operations defined by f g x sup f y g z y z x f g x sup f y g z y z x for any f g map 0 1 0 1 where and are a continuous t norm and a continuous t conorm on 0 1 answering negatively an open problem posed in 8 besides some characteristics of t norm and t conorm are obtained in terms of the binary operations and

eigenvalue statistics for random schrodinger operators with non rank one perturbations we prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice anderson models constructed with finite rank perturbations are compound poisson distributed this distribution is characterized by the fact that the levy measure is supported on at most a finite set of positive integers determined by the rank the proof relies on a minami type estimate for finite rank perturbations for anderson type continuum models on mathbb r d we prove a similar result for certain natural random variables associated with the local eigenvalue statistics we prove that the compound poisson distribution associated with these random variables has a levy measure whose support is at most the set of positive integers