homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

quotient mi groups a many identities group mi group for short is a special algebraic structure in which identity like elements called pseudoidentities are specified and collected into a monoidal substructure in this way many algebraic structures such as monoids of fuzzy intervals numbers or convex bodies possessing behavior very similar to that of a group structure may be well described and investigated using a new approach which seems to be superfluous for the classical structures the concept of mi groups was recently introduced by holcapek and st pnicka in the paper mi algebras a new framework for arithmetics of extensional fuzzy numbers to demonstrate how a standard structure can be generalized in terms of mi algebras this paper is a continuation of the development of mi group theory and is focused on the construction of quotient mi groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for mi groups

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

dithered a d conversion of smooth non bandlimited signals the classical method for sampling a smooth non bandlimited signal requires a lowpass anti aliasing filter in applications like distributed sampling where sampling and quantization operations precede filtering aliasing error is inevitable motivated by such applications the sampling of smooth and bounded non bandlimited signals whose spectra have a finite absolute first moment without the use of an analog anti alias lowpass filter is studied in a centralized setup upper bounds for the distortion rate function are derived by first upper bounding the distortion with a linear combination of errors due to aliasing and quantization and then balancing their contributions by selecting an appropriate reconstruction bandwidth for a class of dithered sampling methods it is shown that a lower quantizer precision can be traded for a higher sampling rate without affecting the realizable high rate asymptotic distortion rate characteristics these results are applied to signals with exponentially and polynomially decaying spectral characteristics and truncated bandlimited signals to uncover the realizable distortion rate characteristics for these signal classes

on generalized quotient mi groups abstract many identities group mi group for short is an algebraic structure generalizing the group structure where an involutive anti automorphism satisfying certain properties is used instead of the standard group inversion the concept of mi group in a more general form than in this article has been introduced by holcapek and st pnicka in the paper mi algebras a new frame work for arithmetics of extensional fuzzy numbers to describe properties of different approaches to arithmetics of vaguely specified quantities e g stochastic or fuzzy quantities in a unified way this article is a continuation of the effort to develop the theory of mi groups and is focused on a generalization of the construction of quotient mi groups induced by so called normal full mi subgroups which has been introduced by holcapek et al recently in the paper quotient mi groups besides a more general definition of quotient mi groups we prove three isomorphism theorems for mi groups in this new framework

exponential weighting algorithms for reconstruction of bandlimited signals the problem of reconstruction of bandlimited signals from discrete noisy observations is considered whittaker shannon interpolation series based estimates involving the weighting factors are proposed it is shown that they converge as the sampling rate increases to infinity the rate of convergence of the double exponential weighting algorithm is established these results are corroborated in computer simulations

dyadic sampling approximations for non sequency limited signals dyadic sampling approximations as well as error estimates are derived for non random signals which are walsh stieltjes transforms and for dyadic stationary and walsh harmonizable random signals also derived are inversion formulae for walsh stieltjes transforms which are used in this paper

fuzzy decision networks and deconvolution abstract three kinds of networks namely fuzzy minimal spanning tree fuzzy pert and fuzzy shortest path are analyzed by the use of a recently developed fuzzy ranking method to handle the various fuzzy quantities to overcome the problem of double inclusion of the amounts of uncertainties involved in the fuzzy quantities when extended subtraction is used for problems such as fuzzy pert fuzzy deconvolution is used since we are only interested in the ranking and aggregation of the results the existence or nonexistence of the results from deconvolution does not influence the resulting analysis a technique based on the ranking method is developed to handle negative spreads in the resulting fuzzy quantities with the use of this fuzzy ranking method the structure of the network is maintained and conventional algorithms can be applied with appropriate modifications emphasis is placed on the use of the subjective decision maker s opinion in the proposed fuzzy network analysis approach numerical examples are given to illustrate the approach

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

from fuzzy set theory to fuzzy set theory some critical remarks on existing concepts abstract our aim in this paper is to give some foundational remarks on the nature of fuzziness and their modeling by fuzzy sets we will reveal some fundamental misunderstandings which have pervaded fuzzy set theory just from its very beginnings at the end of the paper we give three examples demonstrating the differences and advantages of our new conception our approach in heavily based on an operational and set theoretical background contrasting hisdal s tee model

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

notes on fuzzy complex analysis several fuzzy complex analysis problems are discussed pointing out some errors in literature and some weaknesses of introduced concepts a modified approach to fuzzy complex analysis is proposed and its impact is discussed

moving average restoration of bandlimited signals from noisy observations the purpose of this paper is to describe the extension of the whittaker shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean uncorrelated noise it is shown that the classical whittaker shannon sampling scheme is not consistent in the case of noisy measurements and new reconstruction algorithms based on the moving average smoothing are proposed the weak and strong consistency of the algorithms is established and the rate of convergence is investigated the theory is verified in the computer simulations

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

a representation of fuzzy numbers abstract each fuzzy number can be written in a unique way as the sum of a symmetric and a skew fuzzy number these two kinds of fuzzy numbers correspond bijectively to certain functions on the unit interval that are of bounded variation these bijections give rise to a representation of fuzzy numbers by pairs of functions of bounded variation as application mathematical structures on the set of functions of bounded variation are transported to that of fuzzy numbers making the set of fuzzy numbers into a complete metric space a commutative semiring and a commutative residuated lattice

linear problems in linear problems in pth order and stable processes this work extends to processes with finite moments of order p 1 p 2 and to symmetric alpha stable processes 1 alpha 2 some of the basic linear theory known for processes with finite second moments p 2 and for gaussian processes alpha 2 here the covariation plays a role analogous to the covariance specifically stochastic integrals of two types are introduced and studied for pth order processes and in particular for symmetric stable processes regression estimates and linear estimates on certain symmetric stable processes are evaluated including regression and linear filtering of signal in noise also for certain symmetric stable inputs the identification of a linear system from the input covariation and the input output cross covariation is considered and the way the distribution of the output depends on the linear system is studied

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

restoring causal signals by analytic continuation a generalized sampling theorem for causal signals a band limited part x 0 t of a non band limited causal signal x t is used to completely restore x t by using analytic continuation as a consequence a generalized sampling theorem for causal signals is derived the total causal signal x t can be completely restored by sampling the band limited part x 0 t with finite sampling frequency

additive decomposition of fuzzy quantities abstract the object of this paper is to define a more refined equivalent relation than mares 3 and to improve mares result about additive decomposition of fuzzy quantities concerning an equivalent relation this result proves one of his open questions

optimal reconstruction of bandlimited bounded signals a formula is presented for restoring bounded bandlimited signals from a finite number of samples the restoration is linear in the samples used and the error bound of the method is the theoretical minimum

finite sampling approximations for non band limited signals finite sampling approximations along with bounds on the approximation error are derived for certain deterministic and random signals which are not band limited

high frequency propagation for the schrodinger equation on the torus the main objective of this paper is understanding the propagation laws obeyed by high frequency limits of wigner distributions associated to solutions to the schrodinger equation on the standard d dimensional torus td from the point of view of semiclassical analysis our setting corresponds to performing the semiclassical limit at times of order 1 h as the characteristic wave length h of the initial data tends to zero it turns out that in spite that for fixed h every wigner distribution satisfies a liouville equation their limits are no longer uniquely determined by those of the wigner distributions of the initial data we characterize them in terms of a new object the resonant wigner distribution which describes high frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase space t td this construction is related to that of the so called two microlocal semiclassical measures we prove that any limit of the wigner distributions corresponding to solutions to the schrodinger equation on the torus is completely determined by the limits of both the wigner distribution and the resonant wigner distribution of the initial data moreover follows a propagation law described by a family of density matrix schrodinger equations on the periodic geodesics of td finally we present some connections with the study of the dispersive behavior of the schrodinger flow in particular with strichartz estimates among these we show that the limits of sequences of position densities of solutions to the schrodinger equation on t2 are absolutely continuous with respect to the lebesgue measure

semiclassical completely integrable systems long time dynamics and observability via two microlocal wigner measures we look at the long time behavior of solutions to a semi classical schr odinger equation on the torus we consider time scales which go to infinity when the semi classical parameter goes to zero and we associate with each time scale the set of semi classical measures associated with all possible choices of initial data on each classical invariant torus the structure of semi classical measures is described in terms of two microlocal measures obeying explicit propagation laws we apply this construction in two directions we first analyze the regularity of semi classical measures and we emphasize the existence of a threshold for time scales below this threshold the set of semi classical measures contains measures which are singular with respect to lebesgue measure in the position variable while at and beyond the threshold all the semi classical measures are absolutely continuous in the position variable reflecting the dispersive properties of the equation second the techniques of two microlocal analysis introduced in the paper are used to prove semiclassical observability estimates the results apply as well to general quantum completely integrable systems

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

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

bilinear eigenfunction estimates and the nonlinear schrodinger equation on surfaces we study the cubic non linear schrodinger equation nls on compact surfaces on the sphere mathbb s 2 and more generally on zoll surfaces we prove that for s 1 4 nls is uniformly well posed in hs which is sharp on the sphere the main ingredient in our proof is a sharp bilinear estimate for laplace spectral projectors on compact surfaces

additive decomposition of fuzzy quantities with finite supports abstract the results on group properties of the addition operation over fuzzy quantities presented in a previous paper by the author 5 imply some interesting consequences concerning the equivalence relation between fuzzy quantities the equivalence is necessary for the validity of the group axioms for the class of fuzzy quantities more detailed analysis of the equivalence properties implies some conclusions regarding the interpretation of different sources of fuzziness entering the fuzzy quantities model the method used to prove the main theorem of this paper demands to limit the investigation to fuzzy quantities with finite support this limitation however unpleasant does not influence the applicability of the presented results in a wide class of practical situations

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

on restoring band limited signals the problem of reconstruction of band limited signals from discrete and noisy data is studied the reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined in particular the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated the main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

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