particle lattices and fuzzy particle lattices abstract this paper introduces several new concepts a particle in a lattice which is a generalization of an atom in a distributive lattice by weakening the minimality property of atom a particle lattice which is a generalization of distributive complete atomic lattices by replacing atoms by particles a fuzzy particle lattice a particle lattice with an order reversing involution a particulate boolean algebra a fuzzy particle lattice whose involution is a complementation operator and other related concepts points fuzzy points boundary points and boundary quasi points the paper establishes representation theorems for fomse of these concepts establishes connections between others and some well established lattice theoretic concepts and provides some elementary properties for some of the new concepts

a classification of types of extensions from top to qftop abstract by making use of the concept of the introduced quasi fuzzy topological space c y zheng fuzzy sets and systems 14 1984 273 280 we introduce the concept of s extensions mainly we generalize and characterize the concepts of good extensions r lowen j math anal appl 56 1976 621 633 weak inducement h w martin j math anal appl 78 1980 634 639 and s extensions by introducing some relations between the category of topological spaces and the category of quasi topological spaces

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

factorizing the factorization a spectral element solver for elliptic equations with linear operation count abstract the paper proposes a novel factorization technique for static condensation of a spectral element discretization matrix that yields a linear operation count of just 13n multiplications for the residual evaluation where n is the total number of unknowns in comparison to previous work it saves a factor larger than 3 and outpaces unfactored variants for all polynomial degrees using the new technique as a building block for a preconditioned conjugate gradient method yields linear scaling of the runtime with n which is demonstrated for polynomial degrees from 2 to 32 this makes the spectral element method cost effective even for low polynomial degrees moreover the dependence of the iterative solution on the element aspect ratio is addressed showing only a slight increase in the number of iterations for aspect ratios up to 128 hence the solver is very robust for practical applications

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

principal superconnected fuzzy topological spaces abstract in this note principal superconnected fuzzy topological spaces are introduced and it is proved that such a space is path connected

event based state estimation of linear dynamic systems with unknown exogenous inputs in this work an event based optimal state estimation problem for linear time varying systems with unknown inputs is investigated by treating the unknown input as a process with a non informative prior the event based minimum mean square error mmse estimator is obtained in a recursive form it is shown that for the general time varying case the closed loop matrix of the optimal event based estimator is exponentially stable and the estimation error covariance matrix is asymptotically bounded for each sample path of the event triggering process the results are also extended to the multiple sensor scenario where each sensor is allowed to have its own event triggering condition the efficiency of the proposed results is illustrated by a numerical example and comparative simulation with the mmse estimators obtained based on time triggered measurements the results are potentially applicable to event based secure state estimation of cyber physical systems

adaptive sampling for linear state estimation when a sensor has continuous measurements but sends occasional messages over a data network to a supervisor which estimates the state the available packet rate fixes the achievable quality of state estimation when such rate limits turn stringent the sensor s messaging policy should be designed anew what are good causal messaging policies what should message packets contain what is the lowest possible distortion in a causal estimate at the supervisor is delta sampling better than periodic sampling we answer these questions for a markov state process under an idealized model of the network and the assumption of perfect state measurements at the sensor if the state is a scalar or a vector of low dimension then we can ignore sample quantization if in addition we can ignore jitter in the transmission delays over the network then our search for efficient messaging policies simplifies first each message packet should contain the value of the state at that time thus a bound on the number of data packets becomes a bound on the number of state samples second the remaining choice in messaging is entirely about the times when samples are taken for a scalar linear diffusion process we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion we stick to finite horizons and impose a hard upper bound n on the number of allowed samples we cast the design as a problem of choosing an optimal sequence of stopping times we reduce this to a nested sequence of problems each asking for a single optimal stopping time under an unproven but natural assumption about the least square estimate at the supervisor each of these single stopping problems are of standard form the optimal stopping times are random times when the estimation error exceeds designed envelopes for the case where the state is a brownian motion we give analytically the shape of the optimal sampling envelopes the shape of the envelopes under optimal delta sampling and their performances surprisingly we find that delta sampling performs badly hence when the rate constraint is a hard limit on the number of samples over a finite horizon we should not use delta sampling

event based state estimation of discrete state hidden markov models the state estimation problem for hidden markov models subject to event based sensor measurement updates is considered in this work using the change of probability approach we assume the measurement updates are transmitted through wired or wireless communication networks for the scenarios with reliable and unreliable communication channels analytical expressions for the probability distributions of the states conditioned on all the past point and set valued measurement information are obtained also we show that the scenario with a lossy channel but without the event trigger can be treated as a special case of the reliable channel results based on these results closed form expressions for the estimated communication rates under the original probability measure are presented which are shown to be the ratio between a weighted 1 norm and the 1 norm of the unnormalized conditional probability distributions of the states under the new probability measures constructed implementation issues are discussed and the effectiveness of the results is illustrated by numerical examples and comparative simulations

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

connectedness on fuzzy topological spaces in this paper properties of connectedness on completely distributive topological lattices definition 2 1 and fuzzy topological spaces 1 are studied some equivalent propositions on the concepts of functions in completely distributive lattices and connected elements in completely distributive lattices and fuzzy topological spaces are proved specially connectedness is characterized both by the r neighborhoods of molecules and c separations for any family li i i i of fuzzy topological spaces the product 1 is connected iff every li i is connected a continuous function maps connected elements onto connected ones the connectedness on the fuzzy real line r l is discussed in the last section as an important example the intervals defined here definition 4 1 must be connected in the fuzzy real line r l and so the intervals defined by rodabaugh in 13 are too any connected element is connected in the fuzzy real line r l if and only if r l r

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

efficient matrix free high order finite element evaluation for simplicial elements with the gap between processor clock speeds and memory bandwidth speeds continuing to increase the use of arithmetically intense schemes such as high order finite element methods continues to be

some other types of fuzzy connectedness abstract in this paper we investigate level connectedness concepts in fuzzy topological spaces fts in short we introduce connectedness concepts e g c3 and c4 in fuzzy topology and establish their appropriateness through several observations

on set valued kalman filtering and its application to event based state estimation motivated by challenges in state estimation with event based measurement updates the properties of the exact and approximate set valued kalman filters with multiple sensor measurements for linear time invariant systems are investigated in this work first we show that the exact and the proposed approximate set valued filters are independent of the fusion sequence at each time instant second the boundedness of the size of the set of estimation means is proved for the exact set valued filter for the approximate set valued filter if the closed loop matrix is contractive then the set of estimation means has a bounded size asymptotically otherwise a nonsingular linear transform is constructed such that the size of the set of estimation means for the transformed states is asymptotically bounded third the effect of set valued measurements on the size of the set of estimation means is analyzed and conditions for performance improvement in terms of smaller size of the set of estimation means are proposed finally the results are applied to event based estimation which allow the event triggering conditions to be designed by considering requirements on performance and communication rates the efficiency of the proposed results are illustrated by simulation examples and comparison with the approximate event based mmse estimator and the kalman filter with intermittent observations

stochastic link activation for distributed filtering under sensor power constraint we consider the problem of link activation for distributed estimation with power constraint to satisfy the requirement of power consumption we propose a stochastic link activation scheme where each sensor equipped with a distributed estimator sends data to its neighboring sensors according to different probabilities first we design the optimal estimator gain of each sensor to minimize the state estimation error covariance then we find an upper bound of the expected state estimation error covariance and provide a sufficient condition to guarantee the stability of the proposed estimator finally we formulate the link activation problem as an optimization problem and convert it to a convex optimization

two new l fuzzy topologies on r l in this paper two new topologies 1 2 on r l are constructed using i we produced two l fuzzy topologies i on r l other than the l fuzzy topology t introduced by hutton we have proved that 1 is finer than 2 and 2 is finer than t 2 is the coarsest among the induced l fuzzy topologies which are finer than t we have shown that lr l i the induced space of r l i possesses many good properties such as stronger separation suitability etc we proved that closed interval a b l 2 a b l is closed connected and n compact the addition and multiplication defined on r l by rodabaugh are still jointly continuous

event triggered maximum likelihood state estimation the event triggered state estimation problem for linear time invariant systems is considered in the framework of maximum likelihood ml estimation in this paper we show that the optimal estimate is parameterized by a special time varying riccati equation and the computational complexity increases exponentially with respect to the time horizon for ease in implementation a one step event based ml estimation problem is further formulated and solved and the solution behaves like a kalman filter with intermittent observations for the one step problem the calculation of upper and lower bounds of the communication rates from the process side is also briefly analyzed an application example to sensorless event based estimation of a dc motor system is presented and the benefits of the obtained one step event based estimator are demonstrated by comparative simulations

evaluation of convergence rate in the central limit theorem for the kalman filter state space models are used for modeling of many physical and economic processes an asymptotic distribution theory for the state estimate from a kalman filter in the absence of the usual gaussian assumption was presented by spall and wall 1984 they proved the central limit theorem for state estimators when the random terms in the model have arbitrary distribution in this study some convergence rates in the central limit theorem are given these convergence rates are used for the development of a nonparametric test of the validity of the model

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

scaling to the stars a linearly scaling elliptic solver for p multigrid abstract high order methods gain increased attention in computational fluid dynamics however due to the time step restrictions arising from the semi implicit time stepping for the incompressible case the potential advantage of these methods depends critically on efficient elliptic solvers due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom n dof the runtime of the best available multigrid solvers scales with o p n dof this scaling with p significantly lowers the applicability of high order methods to high orders while the operators for residual evaluation can be linearized when using static condensation schwarz type smoothers require their inverses on fixed subdomains no explicit inverse is known in the condensed case and matrix matrix multiplications scale with p n dof this paper derives a matrix free explicit inverse for the static condensed operator in a cuboidal cartesian subdomain it scales with p 3 per element i e n dof globally and allows for a linearly scaling additive schwarz smoother yielding a p multigrid cycle with an operation count of o n dof the resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n dof for polynomial degrees at least up to 48 furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a cpu leading to time stepping for the incompressible navier stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers

fuzzy pretopological spaces and their representation today it is obvious that topology and uniformity and the many related theories have and will have a fundamental play in applied sciences every representation of real entities in a mathematical language necessarily implies a topological study of its goodness it is a problem of linguistic translation continuity as defined by hammer 71 this translation must preserved some important features of the original the power of actual digital computers enables us to use new ways in automatic pattern recognition but at the same time it shows us the many difficulties we encounter in searching a proper formulation of the topological concepts that everyone has in mind and uses in a naturally and more or less conscientious fashion one of the first axiomatisations of the concept of topological space was done by hausdorff and it is not so old it is only recently that some authors have thought if necessary to generalize these concepts and have obtained practical applications for instance appert 11 hammer 6 71 brissaud 3 mougeot auray duru 2 this has been especially motivated by the necessity of finding new concepts which well apply to finite sets and approx imations as hammer remarks topologists actually had one model in ind the real continuum and its generated euclidean spaces but not approximation and certainly not finite sets we think that the effort of interpretation and application to many fields such as linguistics system theory and education made by hammer is of great importance 7 and its references an important discovery in system theory which has been inforpally stated as a principle by zadeh 111 is as the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics this principle is very close to limit problems and paradoxes encountered by physicians in particular the similitude with the heisenberg uncertainty prin ciple is clear it is certainly a motivation for the use of fuzzy set theory but these two aspects pretopology or extended topology as named by hammer

a fast spectral element solver combining static condensation and multigrid techniques we propose a spectral element multigrid method for the two dimensional helmholtz equation discretized on regular grids combining p multigrid with static condensation the method achieves nearly linear complexity with an order independent convergence rate for solving the condensed equations for smoothing we consider two groups of edge based relaxation schemes the best of which attains a multigrid convergence rate of 0 014 to 0 028 numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom in comparison with a fast finite difference solver the latter is clearly outperformed already for errors of one percent or lower

on nearly compact and 0 rigid fuzzy sets in fuzzy topological spaces abstract this paper intends to initiate the study of nearly compact and rigid fuzzy sets in a fuzzy topological space these two classes of fuzzy sets are seen to be properly contained in the class of fuzzy compact sets and properly contain the class of fuzzy almost compact sets several characterizations and certain other properties of these two types of fuzzy sets are established by using the concepts of fuzzy closures and closures of fuzzy sets and fuzzy filterbases

an event triggered approach to state estimation with multiple point and set valued measurements abstract in this work we consider state estimation based on the information from multiple sensors that provide their measurement updates according to separate event triggering conditions an optimal sensor fusion problem based on the hybrid measurement information namely point and set valued measurements is formulated and explored we show that under a commonly accepted gaussian assumption the optimal estimator depends on the conditional mean and covariance of the measurement innovations which applies to general event triggering schemes for the case that each channel of the sensors has its own event triggering condition closed form representations are derived for the optimal estimate and the corresponding error covariance matrix and it is proved that the exploration of the set valued information provided by the event triggering sets guarantees the improvement of estimation performance the effectiveness of the proposed event based estimator is demonstrated by extensive monte carlo simulation experiments for different categories of systems and comparative simulation with the classical kalman filter

l topological meet theorems kindler gave a purely topological condition for an intersection of subsets of some set to be nonempty which plays an important role in topology this paper generalizes kindler s theorems to the fuzzy lattice based setting due to a proper choice of compactness and connectedness axioms it shows that almost all basic results of kindler survive in the extended setting

event based state estimation with time synchronous updates to reduce the amount of data transfer in networked systems measurements are usually taken only when an event occurs rather than at each synchronous sample instant however this complicates estimation problems considerably especially in the situation when no measurement is received anymore the goal of this paper is therefore to develop a state estimator that can successfully cope with event based measurements and attains an asymptotically bounded error covariance matrix to that extent a general mathematical description of event sampling is proposed this description is used to set up a state estimator with a hybrid update i e when an event occurs the estimated state is updated using the measurement while at synchronous instants the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space furthermore to minimize computational complexity of the estimator the algorithm is implemented using a sum of gaussians approach the benefits of this implementation are demonstrated by an illustrative example of state estimation with event sampling

event based state estimation of hidden markov models through a gilbert elliott channel in this note the problem of event based state estimation for a finite state hidden markov model under a generic stochastic event triggering condition and an unreliable communication channel is investigated the effect of packet dropout is characterized with a gilbert elliott process utilizing the change of probability measure approach the packet dropout model and the event triggered measurement information available to the estimator analytical expressions for the conditional probability distributions of the states are obtained based on which the optimal event based state estimates can be further calculated together with a closed form expression of the average sensor to estimator communication rate the effectiveness of the proposed results is illustrated by an application to a wireless automated machine health monitoring problem

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

on fuzzy weakly semi continuous functions abstract this paper is devoted to the introduction and study of fuzzy weakly semi continuous functions between fuzzy topological spaces some properties of these functions are characterized in terms of quasi coincidence quasi neighborhoods neighborhoods etc furthermore some relations connecting fuzzy weakly semi continuous functions and fuzzy retractions have been established

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

fuzzy s continuous functions and semilocally connected fuzzy topological spaces abstract the concept of s continuous functions kohli 1978 is translated in fuzzy setting into two concepts fuzzy s continuous functions and fuzzy os continuous functions the concepts of semiconnected functions kohli 1978 and semilocally connected spaces kohli 1978 do the same interactions between those concepts in fuzzy setting and fuzzy continuity chang 1968 are discussed also characterizations of semilocally connected and semilocally o connected fuzzy topological spaces in terms of fuzzy s continuous functions and fuzzy os continuous functions respectively are obtained

connectedness in l fuzzy topological spaces based on the consideration to the layer structures of fuzzy lattices and the level topologies of l fuzzy topological spaces a connectedness is defined for an arbitrary l fuzzy set in this paper this definition reflects the degree of connectivity of an l fuzzy set it is shown that the connectedness of l fuzzy topological spaces is an l good extension multiplicative and preserved under continuous l valued zadeh functions and that the inverse limit of continuums is a continuum general l fuzzy intervals and h intervals are defined and connectedness of them are proved it is also shown that the connectedness of an l fuzzy topological space is equivalent to the connectedness of its induced i l fuzzy topological space

an efficient direct parallel spectral element solver for separable elliptic problems an efficient direct parallel elliptic solver based on the spectral element discretization is developed the direct solver is based on a matrix decomposition approach which reduces multi dimensional separable problems to a sequence of one dimensional problems that can be efficiently handled by a static condensation process thanks to the spectral accuracy and the localized nature of a spectral element discretization this elliptic solver is spectrally accurate and can be efficiently parallelized and it can serve as an essential building block for large scale high performance solvers in computational fluid dynamics and computational materials science

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

fuzzy path and fuzzy connectedness abstract this paper discusses fuzzy connectedness in fuzzy topology zadeh s classical paper 1 has introduced the fundamental concept of a fuzzy set upon which the foundation of fuzzy mathematics rests the theory of fuzzy topology is a branch of such mathematics our purpose is to introduce some basic concepts as fuzzy path fuzzy path connected set locally fuzzy path connected set in the fuzzy topological spaces and to prove some fundamental properties related to fuzzy connectedness

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

approximate optimal periodic scheduling of multiple sensors with constraints a constrained periodic multiple sensor scheduling problem is considered in this paper for each sensor constraints on dwell time and activation times are imposed at each time instant only one sensor can update its measurement with the estimator and the objective is to minimize the average state estimation error an approximation framework is proposed to calculate the objective function which transforms the original scheduling problem into an approximate optimal scheduling problem aosp an upper bound on the approximation error is presented to evaluate the performance of the framework to solve the aosp a necessary condition is first proposed on the optimal schedules when no constraints on activation times exist a dynamic programming based algorithm is devised to identify the optimal schedule with polynomial computational complexity when activation time constraints exist we show that the aosps can be solved by solving traveling salesman problems examples are provided to illustrate the proposed results

fuzzy strongly semiopen sets and fuzzy strong semicontinuity abstract in this paper we first introduce and study the fuzzy strongly semiopen and fuzzy strongly semiclosed sets in fuzzy space then we introduce the strong semi interior and strong semiclosure of a fuzzy set and kuratowski s 14 sets theorem of general topology is generalized finally we introduce the fuzzy strongly semicontinuous fuzzy strongly semiopen and fuzzy strongly semiclosed mappings on fuzzy topological spaces and also establish some of their characteristic properties and discuss relations between fuzzy continuous fuzzy semicontinuous and fuzzy strongly semicontinuous mappings

the cascadic multigrid method for elliptic problems the paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of 10 from the multigrid point of view no correction cycles are needed from the cascade principle view a basic iteration method without any preconditioner is used at successive refinement levels for a prescribed error tolerance on the final level more iterations must be spent on coarser grids in order to allow for less iterations on finer grids a first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of 9 in short ccg method whichused the cg method as basic iteration method on each level in 18 it has been proven that the ccg method is accurate with optimal complexity for elliptic problems in 2d and quasi uniform triangulations the present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers moreover an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non uniform grids is worked out on the basis of a posteriori estimates numerical tests confirm the efficiency and robustness of the cascadic multigrid method