weighted norm inequalities gaussian bounds and sharp spectral multipliers abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l whose kernels p t x y have gaussian upper bounds but there is no assumption on the regularity in variables x and y in this article we study weighted l p norm inequalities for spectral multipliers of l we show that sharp weighted hormander type spectral multiplier theorems follow from gaussian heat kernel bounds and appropriate l 2 estimates of the kernels of the spectral multipliers these results are applicable to spectral multipliers for large classes of operators including laplace operators acting on lie groups of polynomial growth or irregular non doubling domains of euclidean spaces elliptic operators on compact manifolds and schrodinger operators with non negative potentials

proofs of the kochen specker theorem based on a system of three qubits a number of new proofs of the kochen specker theorem are given based on the observables of the three qubit pauli group each proof is presented in the form of a diagram from which it is obvious by inspection each of our observable based proofs leads to a system of projectors and bases that generally yields a large number of parity proofs of the kochen specker theorem some examples of such proofs are given and some of their applications are discussed

convergence of sequences of fuzzy numbers and fixed point theorems for increasing fuzzy mappings and application in this paper we discuss some properties of the convergence of sequences of fuzzy numbers and obtain some fixed point theorems for increasing fuzzy mappings and as applications we give two examples one is about fishery production which discusses the balance problems between the quantity of catching fish and the increase in the quantity of fish that is produced by the reproduction and growth of fish another is about the solvability of the equation of fuzzy numbers

two simple proofs of the kochen specker theorem a new proof of the kochen specker theorem uses 33 rays instead of 117 in the original proof if the number of dimensions is increased from 3 to 4 only 24 rays are needed

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

some properties of limit inferior and limit superior for sequences of fuzzy real numbers abstract the limit inferior and limit superior of a bounded sequence of fuzzy real numbers have been introduced by aytar et al 2008 1 in this paper we give a simplified expressions for limit inferior and limit superior the expressions is concise and convenient for use as a straightforward corollary of this expressions we can easily prove some properties of the limit inferior and limit superior

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

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

the core of a sequence of fuzzy numbers in this paper based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties we extend the concept of the core of a sequence of complex numbers first introduced by knopp in 1930 to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval n m where n and m are extreme limit points of the sequence

simple unified form for the major no hidden variables theorems two examples are given that substantially simplify the no hidden variables theorem of kochen and specker greatly reducing the number of observables considered and requiring no intricate geometric argument while one of the examples also obeys a more powerful version of bell s theorem the other does not the examples provide a new perspective on both of these fundamental theorems and on the relation between them

greenberger horne zeilinger paradoxes from qudit graph states one fascinating way of revealing quantum nonlocality is the all versus nothing test due to greenberger horne and zeilinger ghz known as the ghz paradox so far genuine multipartite and multilevel ghz paradoxes are known to exist only in systems containing an odd number of particles here we shall construct ghz paradoxes for an arbitrary number greater than 3 of particles with the help of qudit graph states on a special kind of graphs called ghz graphs furthermore based on the ghz paradox arising from a ghz graph we derive a bell inequality with two d outcome observables for each observer whose maximal violation attained by the corresponding graph state and a kochen specker inequality testing the quantum contextuality in a state independent fashion

the supremum and infimum of the set of fuzzy numbers and its application abstract in this paper we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum as an application we obtain that the continuous fuzzy valued function on a closed interval exists supremum and infimum and give the precise representation we also show that the bounded fuzzy valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of riemann and riemann stieltjes by the usual way

hidden variable theorems for real experiments it has recently been questioned whether the kochen specker theorem is relevant to real experiments which by necessity only have finite precision we give an affirmative answer to this question by showing how to derive hidden variable theorems that apply to real experiments so that noncontextual hidden variables can indeed be experimentally disproved the essential point is that for the derivation of hidden variable theorems one does not have to know which observables are really measured by the apparatus predictions can be derived for observables that are defined in an entirely operational way

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

the extension of the knopp core theorem to the sequences of fuzzy numbers the concept of the core of a sequence of fuzzy numbers has been introduced by aytar et al in the core of a sequence of fuzzy numbers fuzzy sets and systems 159 24 2008 3369 3379 quite recently some matrix transformations on the classical sets of sequences of fuzzy numbers have been characterized by talo and basar in determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations comput math appl 58 2009 717 733 in this paper we have proved a core theorem for sequences of fuzzy numbers which is analogous to the famous knoop core theorem for real sequences to achieve this goal we have studied some properties of the lim sup and lim inf of sequences of fuzzy numbers also we have characterized a class of positive regular matrices of fuzzy numbers such that these matrices leave the core of any bounded sequence of fuzzy numbers invariant

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

on the level convergence of a sequence of fuzzy numbers abstract in this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space e 1 established by wu and wu j math anal appl 210 1997 499 and improve the expressions of the supremum and infimum as a straightforward corollary of this result we obtain a necessary and sufficient condition under which for a bounded sequence of fuzzy numbers u n the pair of functions sup n u n and sup n u n can determine a fuzzy number secondly we give a necessary and sufficient condition for a sequence of fuzzy numbers u n to be levelwise convergent in e 1 and generalize some important theorems in real number spaces to fuzzy number spaces finally we prove the existence of supremum and infimum for the level continuous fuzzy valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained

necessary conditions for extended noncontextuality in general sets of random variables in this paper we explore the graph approach to contextuality to restate the extended definition of noncontextuality as given by kujala et al phys rev lett 115 150401 2015 using graph theoretical terms this extended definition avoids the assumption of the pre sheaf or non disturbance condition which states that if two contexts overlap then the marginal distribution obtained for the intersection must be the same a restriction that will never be perfectly satisfied in real experiments with this we are able to derive necessary conditions for extended noncontextuality for any set of random variables based on the geometrical aspects of the graph approach which can be tested directly with experimental data in any contextuality experiment and which reduce to traditional necessary conditions for noncontextuality if the non disturbance condition is satisfied

some properties of the space of fuzzy valued continuous functions on a compact set in this paper a generalized form of the bolzano theorem in classical analysis to fuzzy number space and a characterization of compact subsets in fuzzy number space are given some properties of the fuzzy valued continuous functions defined on a compact set k are studied completeness of the space c k e 1 of fuzzy valued continuous functions on k endowed with the supremum metric d is proved a characterization of compact subsets in the space c k e 1 d is presented which is a generalization of the arzela ascoli theorem in classical analysis

entropic test of quantum contextuality we study the contextuality of a three level quantum system using classical conditional entropy of measurement outcomes first we analytically construct the minimal configuration of measurements required to reveal contextuality next an entropic contextual inequality is formulated analogous to the entropic bell inequalities derived by braunstein and caves phys rev lett 61 662 1988 that must be satisfied by all noncontextual theories we find optimal measurements for violation of this inequality the approach is easily extendable to higher dimensional quantum systems and more measurements our theoretical findings can be verified in the laboratory with current technology

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

state independent proof of kochen specker theorem with 13 rays quantum contextuality as proved by kochen and specker and also by bell should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified however for the simplest system capable of exhibiting contextuality a qutrit the quantum contextuality is verified only state dependently in experiment because too many at least 31 observables are involved in all the known state independent tests here we report an experimentally testable inequality involving only 13 observables that is satisfied by all noncontextual realistic models while being violated by all qutrit states thus our inequality facilitates a state independent test of the quantum contextuality for an indivisible quantum system we also provide a record breaking state independent proof of the kochen specker theorem with 13 directions determined by 26 points on the surface of a magic cube

general conditions for maximal violation of non contextuality in discrete and continuous variables the contextuality of quantum mechanics can be shown by the violation of inequalities based on measurements of well chosen observables an important property of such observables is that their expectation value can be expressed in terms of probabilities for obtaining two exclusive outcomes examples of such inequalities have been constructed using either observables with a dichotomic spectrum or using periodic functions obtained from displacement operators in phase space here we identify the general conditions on the spectral decomposition of observables demonstrating state independent contextuality of quantum mechanics our results not only unify existing strategies for maximal violation of state independent non contextuality inequalities but also lead to new scenarios enabling such violations among the consequences of our results is the impossibility of having a state independent maximal violation of non contextuality in the peres mermin scenario with discrete observables of odd dimensions

parity proofs of the kochen specker theorem based on the lie algebra e8 the 240 root vectors of the lie algebra e8 lead to a system of 120 rays in a real eight dimensional hilbert space that contains a large number of parity proofs of the kochen specker ks theorem after introducing the rays in a triacontagonal representation due to coxeter we present their ks diagram in the form of a basis table showing all 2025 bases i e sets of eight mutually orthogonal rays formed by the rays only a few of the bases are actually listed but simple rules are given based on the symmetries of e8 for obtaining all the other bases from the ones shown the basis table is an object of great interest because all the parity proofs of e8 can be exhibited as subsets of it we show how the triacontagonal representation of e8 facilitates the identification of substructures that are more easily searched for their parity proofs we have found hundreds of different types of parity proofs ranging from 9 bases or contexts at the low end to 35 bases at the high end and involving projectors of various ranks and multiplicities after giving an overview of the proofs we found we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties in particular we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the ks theorem found to date in eight dimensions

proof of the peres conjecture for contextuality a central result in the foundations of quantum mechanics is the kochen specker theorem in short it states that quantum mechanics cannot be reconciled with classical models that are noncontextual for ideal measurements the first explicit derivation by kochen and specker was rather complex but considerable simplifications have been achieved thereafter we propose a systematic approach to find minimal hardy type and greenberger horne zeilinger type ghz type proofs of the kochen specker theorem these are characterized by the fact that the predictions of classical models are opposite to the predictions of quantum mechanics based on our results we show that the kochen specker set with 18 vectors from cabello et al phys lett a 212 183 1996 pylaag0375 960110 1016 0375 9601 96 00134 x is the minimal set for any dimension verifying a longstanding conjecture by peres our results allow to identify minimal contextuality scenarios and to study their usefulness for information processing

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

statistical limit inferior and limit superior for sequences of fuzzy numbers in this paper we extend the concepts of statistical limit superior and limit inferior as introduced by fridy and orhan statistical limit superior and limit inferior proc amer math soc 125 12 1997 3625 3631 12 to statistically bounded sequences of fuzzy numbers and give some fuzzy analogues of properties of statistical limit superior and limit inferior for sequences of real numbers

necessary and sufficient condition for quantum state independent contextuality we solve the problem of whether a set of quantum tests reveals state independent contextuality and use this result to identify the simplest set of the minimal dimension we also show that identifying state independent contextuality graphs r ramanathan and p horodecki phys rev lett 112 040404 2014 is not sufficient for revealing state independent contextuality

fuzzy continuous function and its properties abstract in this paper we introduce the concepts of fuzzy distance of fuzzy numbers fuzzy limit of fuzzy numbers fuzzy function and fuzzy limit of a fuzzy function and fuzzy continuous function on the set of fuzzy numbers and give some of their elementary properties we also discuss some important theorems of fuzzy numbers and the fuzzy continuous function on the m closed interval

optimal classical simulation of state independent quantum contextuality simulating quantum contextuality with classical systems requires memory a fundamental yet open question is what is the minimum memory needed and therefore the precise sense in which quantum systems outperform classical ones here we make rigorous the notion of classically simulating quantum state independent contextuality qsic in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite qsic set we obtain the minimum memory needed to simulate arbitrary qsic sets via classical systems under the assumption that the simulation should not contain any oracular information in particular we show that while classically simulating two qubits tested with the peres mermin set requires log 2 24 4 585 bits simulating a single qutrit tested with the yu oh set requires at least 5 740 bits