generic normal cloud model abstract cloud model is a cognitive model which can realize the bidirectional cognitive transformation between qualitative concept and quantitative data based on probability statistics and fuzzy set theory it uses the forward cloud transformation fct and the backward cloud transformation bct to implement the cognitive transformations between the intension and extension of a concept as one of the most important cloud models the normal cloud models especially the 2 nd order normal cloud model based on normal distribution and gaussian membership function has been extensively researched and successfully applied to many fields in this paper a 2 nd order generic normal cloud model which establishes a relationship between normal cloud and normal distribution is proposed and the 2 nd order generic forward normal cloud transformation algorithm 2 nd gfct is presented whereafter an ideal backward cloud transformation algorithm of the 2 nd order generic normal cloud model 2 nd gibct is designed based on the mutually inverse features of fct and bct in which the distribution of all the cloud drops generated in 2 nd gfct is used meanwhile a 2 nd order generic backward cloud transformation algorithm 2 nd gbct which does not use the distribution of cloud drops is also proposed to solve real life problems since it is impossible to know the distribution of all the cloud drops in advance in real life applications the relationships between the generic backward cloud transformation algorithms are further studied which help reach the finding that the two backward cloud transformation algorithms presented by wang and xu 26 34 are two special cases of the 2 nd gbct in addition the 2 nd order generic normal cloud model is further generalized to p th order generic normal cloud model and the p th order generic forward normal cloud transformation algorithm p th gfct and the backward cloud transformation algorithm p th gbct are presented finally the performances of the 2 nd gibct and the 2 nd gbct are illustrated by simulation experiment the effectiveness of the 2 nd gbct is shown by the results of image segmentation

computing with words via turing machines a formal approach computing with words cw as a methodology means computing and reasoning by the use of words in place of numbers or symbols which may conform more to humans perception when describing real world problems in this paper as a continuation of a previous paper we aim to develop and deepen a formal aspect of cw according to the previous paper the basic point of departure is that cw treats certain formal modes of computation with strings of fuzzy subsets instead of symbols as their inputs specifically 1 we elaborate on cw via turing machine tm models showing the time complexity is at least exponential if the inputs are strings of words 2 a negative result of 6 not holding is verified which indicates that the extension principle for cw via tms needs to be re examined 3 we discuss cw via context free grammars and regular grammars and the extension principles for cw via these formal grammars are set up 4 some equivalences between fuzzy pushdown automata respectively fuzzy finite state automata fuzzy context free grammars respectively fuzzy regular grammars are demonstrated in the sense that the inputs are instead strings of words 5 some instances are described in detail summarily formal aspect of cw is more systematically established more deeply dealt with while some new problems also emerge

h adaptive least squares finite element methods for the 2d stokes equations of any order with optimal convergence rates abstract the adaptive least squares finite element method ls fem for the stokes equations has recently been based on alternative error estimators in bringmann and carstensen 2016 for the lowest order case since the first order div ls fem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation k f without a mesh size factor this enforces separate marking with an overall abstract theory carstensen and rabus 2016 this paper contributes a a discussion of the scaling of the ls fem b optimal rates of the adaptive h version of any order k and c comparing numerical results in 2d with all details on inhomogeneous dirichlet data covered by the analysis

which logic is the real fuzzy logic this paper is a contribution to the discussion of the problem whether there is a fuzzy logic that can be considered as the real fuzzy logic we give reasons for taking imtl bl email protected and ev l fuzzy logic with evaluated syntax as those fuzzy logics that should be indeed taken as the real fuzzy logics

extended fuzzy logic sets and systems the concepts of sets and approximate reasoning within extended fuzzy logic fle provide a systematic procedure for transforming unprecisiated knowledge into a nonlinear mapping over what we define here as f sets an f set differs from a fuzzy set in that it is associated with the restriction of validity in addition to that of possibility therefore by f set we can simultaneously deal with two different types of uncertainties one that is related to ill known objects represented by incomplete information information with its one or more aspects being imprecise vague partial nonspecific undetermined and another that is related to truth values considering gradualness here we define new concepts of vartheta cuts and alpha vartheta cuts introduce the f extension principle and consider arithmetic computations within fle we then address other aspects of the proposed fle system such as fuzzification and validification operations in input processing stage set conversion and defuzzification in output processing stage and inferencing in fact in this paper we intend to develop fle theoretically and practically from the stands of sets and systems to extend the concept of approximate reasoning as a consequence of this development we assert that considering the validity degree of methods and information can lead to more reasonable and trustworthy results through capturing more uncertainty

global dynamics of a state dependent delay model with unimodal feedback abstract we obtain the global attractivity of nonnegative stationary states for a parameterized state dependent delay equation with unimodal feedback under certain mild conditions we show that the equation with unimodal type nonlinearity can generate rich dynamics as the parameter varies to be specific global attractivity of the positive stationary state is obtained in a set of nonnegative bounded continuous functions when the stationary state is less than a value implicitly determined by a condition on the unimodal feedback the general results of global attractivity are illustrated through two examples arising from population dynamics moreover hopf bifurcations are demonstrated in the examples when the positive stationary states lose global attractivity

a qualified description of extended fuzzy logic abstract the sense of approximate reasoning within fuzzy logic has given rise to speculation as to whether precise reasoning within classical logic can even be considered a feasible way of handling most of the world s problems now another question can be raised can fuzzy logic by itself justify handle the sort of uncertainty found in problems that belong to the class of open world problems this family of problems corresponds to the case of ambiguous mathematical theories mainly due to the fact that there is just imperfect incomplete information about them recently zadeh introduced an extension of fuzzy logic claimed to bridge the gaps in conventional approaches to the open world problems in this paper we present a basic background of extended fuzzy logic fle we address several main concepts such as f transformation and f validity enhance the definition of generalized constraints and propose that fle s rules have two layers one based on the implication model and another based on the conjunction model in addition we explore how fle can handle those conditions described in antonym pairs as well as those occurred without considering rationality in the framework both cases challenge most decision making theories we also suggest fle and second order probability are f complementary rather than f competitive the paper asserts that fle is highly capable of constructing mathematical solutions to problems including computational problems that are stated in natural language

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

global stability lobes of turning processes with state dependent delay we obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state dependent delays in the process we transform the models with state dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters we find that the spindle speed control strategy that we investigated in siam j appl math 72 2012 pp 1 24 can provide essential improvement on the stability of turning processes with state dependent delay and furthermore we show the existence of a proper subset of the stability region which is independent of system damping numerical simulations are presented to illustrate the general results

axioms of adaptivity with separate marking for data resolution mixed finite element methods with flux errors in h div norms and div least squares finite element methods require a separate marking strategy in obligatory adaptive mesh refining the refinement indicator sigma 2 mathcal t k eta 2 mathcal t k mu 2 k of a finite element domain k in an admissible triangulation mathcal t consists of some residual based error estimator eta mathcal t k with some reduction property under local mesh refining and some data approximation error mu k separate marking means either dorfler marking if mu 2 mathcal t leq kappa eta 2 mathcal t or otherwise an optimal data approximation algorithm with controlled accuracy the axioms are sufficient conditions on the estimators eta mathcal t k and data approximation errors mu k for optimal asymptotic convergence rates the enfolded set of axioms of this paper simplifies c carstensen m feischl m page and d praetorius comput math appl 67 2014 pp 1195 1253 for collective m

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

global hopf bifurcation of differential equations with threshold type state dependent delay abstract we develop global hopf bifurcation theory of differential equations with state dependent delay using the s 1 equivariant degree and investigate a two degree of freedom mechanical model of turning processes for the model of turning processes we show that the extreme points of each vibration component of the non constant periodic solutions can be embedded into a manifold with explicit algebraic expression this observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point this termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter in the process several sufficient conditions for the non existence of non constant periodic solutions are also obtained

the effects of membership function on fuzzy reasoning abstract fuzzy modus ponens has been considered a generalization of the classic modus ponens logic inference rule it has been expected to provide a more powerful and useful inference capability in the process of fuzzy reasoning the definitions of membership functions of fuzzy sets used in the reasoning process may however adversely affect the reasoning results if the insignificant parts of the membership functions are incompatible this paper explains these undesirable effects and proposes a method to resolve this problem in addition an algorithm for the compositional rule of inference is developed to perform the fuzzy modus ponens syllogism test results shown in this paper indicate that the algorithm and the underlying proposed methods are reasonable approaches to a robust fuzzy logic reasoning system

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

discussion probability theory and fuzzy logic are complementary rather than competitive the relationship between probability theory and fuzzy logic has long been an object of discussion and some controversy the position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real world settings to enhance its effectiveness probability theory needs an infusion of concepts and techniques drawn from fuzzy logic especially the concept of a linguistic variable and the calculus of fuzzy if then rules in the final analysis probability theory and fuzzy logic are complementary rather than competitive

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

an adaptive least squares fem for linear elasticity with optimal convergence rates adaptive mesh refining is of particular importance in computational mechanics and established here for the lowest order locking free least squares finite element scheme which solely employs conforming p 1 approximations for the displacement and lowest order raviart thomas approximations for the stress variables this forms a competitive discretization in particular in three dimensional linear elasticity with traction boundary conditions although the stress approximation does not satisfy the symmetry condition exactly the paper introduces an adaptive mesh refining algorithm based on separate marking and exact solve with some novel explicit a posteriori error estimator and proves optimal convergence rates the point is robustness in the sense that the crucial input parameters theta for the dorfler marking and kappa for the separate marking as well as the equivalence constants in the asymptotic convergence rates do not degenerate as the lame parameter lambda tends to infty

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

what does a probabilistic interpretation of fuzzy sets mean in this paper we propose a statistics model to better understand the probabilistic interpretation for fuzzy sets the model shows the relationship between the two kinds of statistics the set valued statistics for fuzzy sets and the conventional probability statistics for membership functions our model overcomes the shortcomings in previous approaches the results of this paper provide a theoretical interpretation to zadeh s insight that possibility relates to soft constraints restricting the values of language variables

an adaptive least squares fem for the stokes equations with optimal convergence rates this paper introduces the first adaptive least squares finite element method ls fem for the stokes equations with optimal convergence rates based on the newest vertex bisection with lowest order raviart thomas and conforming p 1 p1 discrete spaces for the divergence least squares formulation in 2d although the least squares functional is a reliable and efficient error estimator the novel refinement indicator stems from an alternative explicit residual based a posteriori error control with exact solve particular interest is on the treatment of the data approximation error which requires a separate marking strategy the paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non linear approximation class it extends and generalizes the approach of carstensen and park siam j numer anal 53 43 62 2015 from the poisson model problem to the stokes equations

fuzzy weighted averages revisited the fuzzy weighted average operations are common operations in risk and decision analysis based on zadeh s extension principle dong and wang fuzzy sets and systems 21 1987 183 proposed a fwa method a method based on the cut representation of fuzzy sets and interval analysis the fwa method provides a discrete approximation of the cuts of fuzzy weighted average the fwa method has an exponential complexity subsequently liou and wang fuzzy sets and systems 49 1992 307 proved that indeed the fwa method can be simplified to be a nonconstrainted linear fractional programming problem with lower and upper bounds for each of the variables they proposed an improved algorithm called ifwa which is of o n2 complexity where n is the number of decision variables in ifwa guh et al comput math appl 32 8 1996 115 proposed a max min paired elimination method pfwa they claimed their pfwa was of order n in complexity yet the pfwa is a heuristic method lacking any proof of convergence in 1997 lee and park proposed an algorithm efwa which is of o nlogn complexity most recently kao and liu fuzzy sets and systems 120 2001 435 solved the fwa by transforming the fractional program into a linear program in this note we shall relate the ifwa framework with the nonconstrainted 0 1 integer linear fractional programming problem in the mathematical programming literature several o n time algorithms have been existing for instance robillard naval res logist quart 18 1971 47 and hansen et al math programming 52 1991 255 the algorithm of hansen et al will be presented for easy reference

what are fuzzy rules and how to use them abstract fuzzy rules have been advocated as a key tool for expressing pieces of knowledge in fuzzy logic however there does not exist a unique kind of fuzzy rules nor is there only one type of fuzzy logic this diversity has caused many a misunderstanding in the literature of fuzzy control the paper is a survey of different possible semantics for a fuzzy rule and shows how they can be captured in the framework of fuzzy set and possibility theory it is pointed out that the interpretation of fuzzy rules dictates the way the fuzzy rules should be combined the various kinds of fuzzy rules considered in the paper gradual rules certainty rules possibility rules and others have different inference behaviors and correspond to various intended uses and applications the representation of fuzzy unless rules is briefly investigated on the basis of their intended meaning the problem of defining and checking the coherence of a block of parallel fuzzy rules is also briefly addressed this issue has been neglected in the fuzzy control literature although it looks important for validation purposes

fuzzy weighted average the linear programming approach via charnes and cooper s rule abstract to improve the method of dong and wong fuzzy sets and systems 21 1987 183 199 for obtaining the fuzzy weighted average a steepest descent ascent method was proposed by liou and wang fuzzy sets and systems 49 1992 307 315 in this paper we propose to replace the steepest descent ascent method by linear programming which is a much more powerful approach for large problems the reduction of the original nonlinear programming problem into a linear one is achieved by the use of the charnes and cooper s linear transformation method the example used by dong and wong is again used to illustrate the approach

fast computation in adaptive tree approximation adaptive methods of approximation arise in many settings including numerical methods for pdes and image processing they can usually be described by a tree which records the adaptive decisions this paper is concerned with the fast computation of near optimal trees based on n adaptive decisions the best tree based on n adaptive decisions could be found by examining all such possibilities however this is exponential in n and could be numerically prohibitive the main result of this paper is to show that it is possible to find near optimal trees using computations linear in n

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

aggregation using the fuzzy weighted average as computed by the karnik mendel algorithms by connecting work from two different problems the fuzzy weighted average fwa and the generalized centroid of an interval type 2 fuzzy set a new alpha cut algorithm for solving the fwa problem has been obtained one that is monotonically and superexponentially convergent this new algorithm uses the karnik mendel km algorithms to compute the fwa cut end points it appears that the km cut algorithms approach for computing the fwa requires the fewest iterations to date and may therefore be the fastest available fwa algorithm to date

an efficient algorithm for fuzzy weighted average in multisensor intelligent systems the information fusion plays an important role several algorithms have been proposed for the purpose of aggregating imprecise sensory information represented by fuzzy numbers this paper proposes an efficient algorithm to compute fuzzy weighted average which turned out to be superior to the previous works by reducing the number of comparisons and arithmetic operations to o n log n

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

fuzzy weighted average a max min paired elimination method by the use of max min paired elimination an efficient method for obtaining the fuzzy weighted average fwa was developed this approach is much more efficient than either the approach developed by dong and wong 1 or the improved approach developed by liou and wang 2 the computational requirements for the method of dong and wong are 22n and the largest computational requirements for the improved method of liou and wang are 2 n n 1 the computational requirements for the proposed approach are proportional to the problem size n and the total computational requirements are 2 n 1 the same example used by dong and wong is used to illustrate the approach in addition a more complicated problem is also solved

method for higher order polynomial sugeno fuzzy inference systems a method for higher order polynomial sugeno fuzzy inference systems formation is presented compared to other existing higher order sugeno implementations it uses fewer parameters and compared to zero and 1st order sugeno fuzzy systems it has overall improved model performance while best models are not always obtained via a higher order representation in our proposed method it is possible to choose the polynomial order which best fits the desired data its input is a previously established model found by a clustering algorithm subtractive algorithm in this case afterward parameters of all higher order polynomials are adjusted using recursive least square algorithm for experimental validation multiple benchmark datasets are tested using hold out and k fold validation as well as data forecasting various performance measures are used although akaike information criterion is used as a primary measure to demonstrate that our proposed higher order polynomials have overall better model performance over zero and 1st order polynomials

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

fuzzy transforms theory and applications the technique of direct and inverse fuzzy f transforms of three different types is introduced and approximating properties of the inverse f transforms are described all three types of the direct f transform are transformations from a function space to a finite dimensional vector space the first ordinary f transform is constructed on the basis of the ordinary algebra of reals while the other two types of the f transform are constructed on the basis of residuated lattice the core idea of the technique of f transforms is a fuzzy partition of a universe into fuzzy subsets factors clusters granules etc we claim that for a sufficient representation of a function defined on this universe we may consider its average values over fuzzy subsets from the partition thus a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values a number of theorems establishing best approximation properties of the inverse f transforms are proved in fact three types of the inverse f transform are the best approximations in average from below and from above respectively as one of many possible applications we present a method of image compression and reconstruction on the basis of the f transform

stabilization in a state dependent model of turning processes we consider a two degree of freedom model for turning processes which involves a system of differential equations with state dependent delay depending on process parameters e g spindle speed depth of cut the cutting tool can exhibit unwanted vibrations resulting in a nonsmooth surface of the workpiece in this paper we propose a feedback law to stabilize the turning process for a large range of system parameters the feedback law introduces a generic nonhyperbolic stationary point into the model which generates the main technical challenge of this work we establish the stability equivalence between the differential equations with state dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value then we show the stability of that nonlinear system with constant delay by computing its normal form finally we obtain conditions on system parameters which guarantee the stability of the state dependent delay model at the nonhyperbolic stationary point

a short note on plain convergence of adaptive least squares finite element methods abstract we show that adaptive least squares finite element methods driven by the canonical least squares functional converge under weak conditions on pde operator mesh refinement and marking strategy contrary to prior works our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters finally we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions e g the preconditioned conjugate gradient method with optimal preconditioner the results apply within a fairly abstract framework which covers a variety of model problems