normal forms in bl algebra and their contribution to universal approximation of functions this paper continues the investigation of approximating properties of generalized normal forms in fuzzy logic the problem is formalized and solved algebraically normal forms are considered in two variants infinite and finite it is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any l valued function where l is a support set of any complete bl algebra the estimation of the quality of approximation is suggested moreover functions which can be precisely represented by the discrete normal forms are considered

towards a higher degree f transform the aim of this study is to show how the f transform technique can be generalized from the case of constant components to the case of polynomial components after a general presentation of an f m transform m 0 a detailed characterization of the f 1 transform is given we apply a numeric integration technique in order to simplify the computation of f 1 transform components the inverse f m transform m 0 is defined similarly to the ordinary inverse f transform the quality of approximation using the inverse f m transform increases with an increase in m

renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

fuzzy transform and least squares approximation analogies differences and generalizations investigating the relations between the least squares approximation techniques and the fuzzy transform in this paper we show that the discrete fuzzy transform is invariant with respect to the interpolating and least squares approximation additionally the fuzzy transform is evaluated at any point by simply resampling the continuous approximation underlying the input data using numerical linear algebra we also derive new properties e g stability to noise additivity with respect to the input data and characterizations e g radial and dual membership maps of the discrete fuzzy transform finally we define the geometry and confidence driven discrete fuzzy transforms which take into account the intrinsic geometry and the confidence weights associated to the data

a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

fuzzy projection versus inverse fuzzy transform as sampling interpolation schemes in this study the inverse fuzzy transform recently introduced by perfilieva is compared to a new representation of signals that we name the fuzzy projection both representations are sampling interpolation schemes in a given fuzzy context fuzzy projection is closely related to fuzzy and inverse fuzzy transforms and is presented in this framework we deal with these two operators in the space of finite energy signals while proposing theoretical and numerical comparisons after presenting convergence results for fuzzy projection some numerical tests are performed on signals they confirm the theoretical studies and show the superiority of fuzzy projection for handling in the l 2 sense signals without too high frequencies since both representations can be regarded as low pass filtering processes they also show the robustness of fuzzy projection with respect to noise suggesting future studies in simultaneous compression and denoising

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

global convergence of a class of non interior point algorithms using chen harker kanzow smale functions for nonlinear complementarity problems we propose a class of non interior point algorithms for solving the complementarity problems cp find a nonnegative pair x y 2n satisfying y f x and xiyi 0 for every i 1 2 n where f is a continuous mapping from n to n the algorithms are based on the chen harker kanzow smale smoothing functions for the cp and have the following features a it traces a trajectory in 3n which consists of solutions of a family of systems of equations with a parameter b it can be started from an arbitrary not necessarily positive point in 2n in contrast to most of interior point methods and c its global convergence is ensured for a class of problems including not strongly monotone complementarity problems having a feasible interior point to construct the algorithms we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition and propose a class of algorithms involving suitable neighborhoods of the trajectory we also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

hukuhara difference of z numbers abstract real world problems in decision analysis economics optimization and other fields are characterized by fuzzy and partially reliable information in this regard prof zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers a and b the first of which describes a linguistic value and the second one is a fuzzy value of probability measure of the first one playing a role of reliability of information construction of mathematical models to handle z number based information is a new challenge which requires development of z number valued analysis including z number valued equations z number valued derivative and integral and other concepts one of the basic concepts in this realm is an inverse of addition operation in the existing literature devoted to set valued arithmetic and fuzzy arithmetic this operation is referred to as hukuhara difference for interval numbers and fuzzy numbers the existence conditions of hukuhara difference are derived unfortunately the concept of hukuhara difference for z numbers is not developed in this work for the first time we suggest a fundamental approach for development of the concept of hukuhara difference of z numbers examples are provided in the paper to show validity of the suggested approach

relational closure and partition powerset theories abstract for general powerset theories in categories new notions of relational closure or partition powerset theories in these categories are introduced these new types of powerset theories are defined as categories whose objects are the original powerset objects with relational closure or partition structures defined on these objects this construction generalizes classical constructions realized on powerset objects of all subsets of a given set or all fuzzy sets on a given set and it enables to define these structures on more general powerset objects examples of these new powerset objects are also shown

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

lattice valued f transforms and similarity relations abstract the relationships between lattice valued f transforms and similarity relations are investigated several similarity relations are naturally associated with f transforms three types of such similarity relations related to lattice valued f transforms are investigated namely similarity relationships that are related to spaces with fuzzy partitions as ground structures of the f transform relationships between extensional hulls with respect to similarity relations and f transforms and relationships between similarities of pairs of functions on the one hand and similarities of the f transforms of these functions on the other some examples of possible applications are also presented

a segmentation method for images compressed by fuzzy transforms in this paper we describe a segmentation method applied to images which are compressed by using fuzzy transforms the segmentation of the images is realized via the fgfcm fast generalized fuzzy c means clustering algorithm which is robust to noise and outliers the optimal number of clusters is determined via the pcaes partition coefficient and exponential separation validity index we use a similarity measure defined via lukasiewicz t norm for comparison between the original image and the reconstructed images the best results are obtained if this similarity measure overcomes a threshold value experimentally determined from the analysis of the trend of it with respect to the psnr peak signal to noise ratio

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

a smoothing filter based on fuzzy transform this paper is devoted to the smoothing of discrete functions using the fuzzy transform introduced by perfilieva we generalize a smoothing filter based on the fuzzy transform recently proposed by us to obtain a better control on the smoothed functions for this purpose a generalization of the concept of fuzzy partition is suggested and the smoothing filter is defined as a combination of the direct discrete fuzzy transform and a slightly modified inverse continuous fuzzy transform an approximation behavior total variation of smoothed functions and statistical properties including the description of the white noise reduction and the asymptotic expression of bias and variance are investigated and discussed the proposed filter is compared with the nadaraya watson estimator and the results are illustrated assuming financial data

correction and linearization of resonant vector fields and diffeomorphisms we extend the classical siegel brjuno russmann linearization theorem to the resonant case by showing that under a d brjuno s diophantine condition any resonant local analytic vector field resp diffeomorphism possesses a well defined correction which 1 depends on the chart but in any given chart is unique 2 consists solely of resonant terms and 3 has the property that when substracted from the vector field resp when factored out of the diffeomorphism the vector field or diffeomorphism thus corrected becomes analytically linearizable with a privileged or canonical linearizing map moreover in spite of the small denominators and contrary to a hitherto prevalent opinion the correction s analyticity can be established by pure combinatorics without any analysis

a neural network approach to the fuzzy transform the paper deals with the f transform technique which was introduced as a method for an approximate representation of continuous functions the same task is solved by many methods from different areas neural networks also belong to techniques which have been proved to be powerful for approximation objectives they provide us with many advantages especially incremental way of learning parameters the paper inherits neural approaches to the f transform method and presents experiments justifying the proposed approach incremental way of determination of certain parameters of the f transform method which were up to now given by batch formula enriches possible areas of application of the method by fast on line processes moreover the neural approach is demonstrated to be an appropriate one for finding such fuzzy partition of a domain which respects better a given set of measured samples which are to be approximated by a continuous function with no predetermined shape

a color image reduction based on fuzzy transforms we present a new method for color image reduction based on the concept of fuzzy transform any image in a single band can be considered as a fuzzy matrix which is subdivided into submatrices called blocks each block is compressed with various compression rates by means of a fuzzy transform in two variables we compare our method with recent three algorithms due to g beliakov h bustince and d paternain based on the minimizing penalty functions defined over a discrete lattice the quality of the reduced image is measured by the mean square error mse and penalty function pen obtained by comparing both magnified and original images we also point out a threshold of the compression rate beyond which the mse follows a linear trend and the corresponding loss of information is still acceptable

approximation of extensional fuzzy relations over a residuated lattice we investigate a relationship between extensionality of fuzzy relations and their lipschitz continuity on generalized metric spaces the duality of these notions is shown and moreover two particular applications of the extensionality property in the field of approximation are given

on the structural properties of f m transform with applications abstract the aim of this study is to obtain some direct formulas for f m transforms of derivative integral and multiplication of functions in order to extend the application and flexibility of f m transform method using the derived explicit formula one can use f m transform method straightforwardly to solve various linear and nonlinear problems including differential and integral equations in this study the general form of the higher order fuzzy transforms are constructed using any arbitrary basis functions instead of the classical polynomial ones

regularization of p 0 functions in box variational inequality problems two recent papers f facchinei math oper res 23 1998 pp 735 745 and f facchinei and c kanzow siam j control optim 37 1999 pp 1150 1161 have shown that for a continuously differentiable p0 function f the nonlinear complementarity problem ncp f varepsilon corresponding to the regularization f varepsilon x f x varepsilon x has a unique solution for every varepsilon 0 that dist x varepsilon mathop rm sol f rightarrow 0 as varepsilon rightarrow 0 when the solution set sol f of ncp f is nonempty and bounded and ncp f is stable if and only if the solution set is nonempty and bounded these results are proved via the fischer function and the mountain pass theorem in this paper we generalize these nonlinear complementarity results to a box variational inequality problem corresponding to a continuous p0 function where the regularization is described by an integral we also describe an upper semicontinuity property of the inverse of a weakly univalent function and study its consequences

construction of image reduction operators using averaging aggregation functions in this work we present an image reduction algorithm based on averaging aggregation functions we axiomatically define the concepts of image reduction operator and local reduction operator we study the construction of the latter by means of averaging functions and we propose an image reduction algorithm image reduction operator we analyze the properties of several averaging functions and their effect on the image reduction algorithm finally we present experimental results where we apply our algorithm in two different applications analyzing the best operators for each concrete application

supporting trading strategies by inverse fuzzy transform trading in finance requires to define strategies able to identify early trading buy sell signals i e conditions which suggest to enter or exit a position in order to exploit early mover advantage trading signals are generally identified by looking at the time series of prices and volumes several technical analysis indicators and strategies have been proposed and are commonly in use in this paper we propose inverse fuzzy transform as a means for building a new class of technical indicators experimental results show that this approach outperforms simple and exponential moving average when embedded in common strategies

limiting behavior of trajectories generated by a continuation method for monotone complementarity problems defining the mapping f from the 2n dimensional euclidean space r2n into itself by fx y x1y1 xn yn y1 f1x yn fnx for every x y r2n we write the cp the complementarity problem with a mapping f rn rn as the system of equations fx y 0 and x y r 2n where r 2n denotes the nonnegative orthant of r2n under the assumption that f is a monotone function on r n we show that f maps the positive orthant r 2n of r2n homeomorphically this result then ensures the existence of a trajectory consisting of the solutions x y t of the system of equations fx y wt and x y t r 2n 0 1 where w is a continuous mapping from the unit interval 0 1 into fr 2n such that w0 0 if t 0 the system coincides with the cp we study limiting behavior of the trajectory as t 0 and give some sufficient conditions for the trajectory to lead to a solution of the cp this approach is an extension of the one used in the polynomially bounded algorithm recently given by kojima mizuno and yoshise for solving positive semidefinite linear complementarity problems

fuzzy transforms for compression and decompression of color videos we use fuzzy transforms for coding decoding color frames of videos and we compare the results with the same frames reconstructed with the standard jpeg method we classify all the frames in intra frames and predictive frames by adopting a similarity measure based on the lukasiewicz t norm and a pre processing phase which determines the best similarity threshold value the compression is made on particular frames called d frames obtained from a suitable difference defined on the values of the pixels of an intra frame and a predictive frame under high compression rates we see that the peak signal to noise ratio of the frames obtained with the fuzzy transforms is averagely close to the psnr obtained with the jpeg method we use the videos at url sampl eng ohiostate edu sampl database htm

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

non additive interval valued f transform this article proposes a new interval valued fuzzy transform its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built the main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward this interpretation relates to a traditional sampling reconstruction framework where little is known about the sampling and or reconstructing kernels numerous properties of the proposed approach are proved that could be useful for function analysis and comparison in the experimental section we illustrate some properties of the proposed transform while highlighting interesting features of the obtained framework

f transform with parametric generalized fuzzy partitions the fuzzy transform f transform has some interesting properties that allow its use in curve fitting and smoothing we suggest the following a a generalization of fuzzy partitions for the f transform to control and improve the smoothing effect and b a parametrization of the fuzzy numbers constituting the basic functions related to the fuzzy partitions an optimization criterion is then suggested to obtain the f transform with best approximation properties by estimating the basic functions from a family of parametric fuzzy numbers the generalized fuzzy partitions are characterized by the number of subintervals the bandwidth of the partition forming the core of each basic function and we suggest two procedures for bandwidth selection one based on taut string variation and a second based on generalized cross validation some illustrative examples are included

multi dimensional fuzzy transform and projection fuzzy and inverse fuzzy transforms introduced by i perfilieva is an important tool for signal and image fuzzy representation it led to several variants as the natural least squares minimization named fuzzy projection we deal with this transform proposing first to show its simplicity and its easy numerical implementation in any dimension of space secondly we point out that a key parameter permits to control its numerical robustness thirdly we show that this parameter also governs the stability of the representation by fuzzy projection when choosing different partitions of the space we conclude in discussing linear representation especially triangular representation for signals and pyramidal representation for images

interval valued probability density estimation based on quasi continuous histograms proof of the conjecture the sensitivity of histogram computation to the choice of a reference interval and number of bins can be attenuated by replacing the crisp partition on which the histogram is built by a fuzzy partition this involves replacing the crisp counting process by a distributed weighted voting process the counterpart to this low sensitivity is some confusion in the count values a value of 10 in the accumulator associated with a bin can mean 10 observations in the bin or 40 observations near the bin this confusion can bias the statistical decision process based on such a histogram in a recent paper we proposed a method that links the probability measure associated with any subset of the reference interval with the accumulator values of a fuzzy partition based histogram the method consists of transferring counts associated with each bin proportionally to its interaction with the considered subset two methods have been proposed which are called precise and imprecise pignistic transfer imprecise pignistic transfer accounts for the interactivity of two consecutive cells in order to propagate in the estimated probability measure counting confusion due to fuzzy granulation imprecise pignistic transfer has been conjectured to include precise pignistic transfer the present article proposes a proof of this conjecture

general approximation of fuzzy numbers by f transform in this paper we will prove that in most of the cases the extended inverse fuzzy transform preserves the quasi concavity of a fuzzy number and hence it can be used to generate fuzzy numbers by approximating the restriction of the membership function to its support in the case of continuous fuzzy numbers with cores containing more than one element the rate of uniform convergence is of linear type and the same holds when we approximate the important characteristics of a fuzzy number such as the value or the ambiguity moreover we have the preservation of the support and the convergence of the core which in addition can be determined precisely in the case of continuous fuzzy numbers with one element core it is in general necessary to normalize the approximation but the support is preserved again and the core can be determined exactly in this case too moreover the approximations have again linear rate of uniform convergence

differentiation by the f transform and application to edge detection in this contribution we consider integrable functions of many in particular two variables and extend for them the technique of f transform to f1 transform we prove that linear coefficients in the representation of the f1 transform component approximate the first partial derivatives we give estimates for this approximation finally we discuss the application to the edge detection problem we propose a new algorithm where the technique of f1 transform is combined with the canny detector a theoretical justification and advantages of the proposed approach are emphasized and illustrative examples are presented

axiomatic of lattice valued f transform abstract axioms for upper and lower fuzzy transformation systems are introduced and it is proved that a transformation function satisfies axioms for upper or lower respectively transformation systems if and only if it is an upper or lower respectively f transform categories of upper lower respectively transformation systems are introduced and it is proved that both categories are isomorphic to the category of spaces with fuzzy partitions we prove that lower and upper f transforms preserve extensional fuzzy sets associated with spaces with fuzzy partitions

a new reconstruction from the f transform components a new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its f transform components is analyzed an analogy with the nyquistshannonkotelnikov sampling theorem is discussed special attention is paid to the case where sample values of a function come with noise we show that the double reconstruction through the f transform components significantly reduces the influence of noise

development of autofocusing algorithm based on fuzzy transforms in this paper we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera the focus measure operations play a key role in a passive auto focusing method the proposed focus measure operation is based on a localized version of image variance method the localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and fuzzy transform in order to validate our approach experimental comparative studies are conducted on test images with the use of different focus measure operators the selected operators have been chosen from an extensive review of the state of the art approaches a comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods

fuzzy transforms of monotone functions with application to image compression this paper is focused on the special properties of functions monotonicity lipschitz continuity that are invariant with fuzzy transform on the basis of the monotonicity invariance we propose an efficient algorithm for improved image compression and reconstruction based on fuzzy transform

łukasiewicz transform and its application to compression and reconstruction of digital images we define the lukasiewicz transform as a residuated map and a homomorphism between semimodules over the semiring reducts of an mv algebra then we describe the lukasiewicz transform based ltb algorithm for image processing demonstrating its applicability

structural and stability properties of p 0 nonlinear complementarity problems we consider p0 nonlinear complementarity problems and study the connectedness and stability of the solutions by applying degree theory and the mountain pass theorem to a smooth reformulation of the complementarity problem we show that the solution set is connected and bounded if a bounded isolated component of the solution set exists and that a solution is locally unique if and only if it is globally unique furthermore we prove that a solution is stable in ha s sense if and only if it is globally unique while the complementarity problem is stable if and only if the solution set is bounded

a new fuzzy approximation method to cauchy problems by fuzzy transform we present new numeric methods based on the first and second degree f transform for solving the cauchy problem we show that they outperform the second order rungekutta method especially when a right hand function is oscillating and or a solution is requested on a long interval moreover a new numeric method to solve fuzzy initial value problem using the f transform is presented the suitability of the presented new methods is theoretically justified and illustrated on various examples

the logical content of triangular bases of fuzzy sets in łukasiewicz infinite valued logic abstract continuing to pursue a research direction that we already explored in connection with godel dummett logic and ruspini partitions we show here that ukasiewicz logic is able to express the notion of pseudo triangular basis of fuzzy sets a mild weakening of the standard notion of triangular basis en route to our main result we obtain an elementary logic independent characterisation of triangular bases of fuzzy sets

forecasting seasonal time series based on fuzzy techniques abstract this paper is devoted to a method for the forecasting of seasonal time series the core of our approach is based on the fuzzy transform and fuzzy natural logic fnl techniques under the assumption that a time series can be additively decomposed into a trend cycle a seasonal component and an irregular fluctuation the forecasting is a combination of individual forecasting of each of these constituents more precisely the trend cycle and the seasonal component are predicted with the help of fuzzy transform pattern recognition and fuzzy natural logic techniques to model the irregular component we apply the box jenkins approach in the paper we compare the suggested method with two other well known methods namely stl see 14 and arima ones

approximation properties of fuzzy transforms we enlarge the class of fuzzy transforms f transforms by considering different types of fuzzy partitions new types of f transforms are constructed based on b splines shepard kernels bernstein basis polynomials and favard szasz mirakjan type operators we study approximation properties of f transforms particular cases of the classical korovkin s theorems are presented together with new error estimates for different types of fuzzy transforms it turns out that f transforms can be studied in the framework of classical approximation theory

locating the least 2 norm solution of linear programs via a path following method a linear program has a unique least 2 norm solution provided that the linear program has a solution to locate this solution most of the existing methods were devised to solve certain equivalent perturbed quadratic programs or unconstrained minimization problems we provide in this paper a new theory which is different from these traditional methods and is an effective numerical method for seeking the least 2 norm solution of a linear program the essence of this method is a interior point like path following algorithm that traces a newly introduced regularized central path that is fairly different from the central path used in interior point methods one distinguishing feature of our method is that it imposes no assumption on the problem the iterates generated by this algorithm converge to the least 2 norm solution whenever the linear program is solvable otherwise the iterates converge to a point which gives a minimal kkt residual when the linear program is unsolvable

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

filtering out high frequencies in time series using f transform abstract in this paper we focus on application of fuzzy transform f transform to analysis of time series under the assumption that the latter can be additively decomposed into trend cycle seasonal component and noise we prove that when setting properly width of the basic functions the inverse f transform of the time series closely approximates its trend cycle this means that the f transform almost completely removes the seasonal component and noise the obtained theoretical results are demonstrated on two artificial time series whose trend cycle is precisely known and on three real time series at the same time comparison with three classical methods namely stl method ssa method and low pass butterworth filter is also provided

isochronous centers of polynomial hamiltonian systems and a conjecture of jarque and villadelprat abstract we study the conjecture of jarque and villadelprat stating that every center of a planar polynomial hamiltonian system of even degree is nonisochronous this conjecture has already been proved for quadratic and quartic systems using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields we are able to describe a very large class of nonisochronous hamiltonian systems of even arbitrarily large degree

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

fuzzy relation equations for coding decoding processes of images and videos we adopt fuzzy relation equations with continuous triangular norms for compression decompression processes of grey images colour images in the rgb space and frames of videos by comparing the results of the reconstructed images with standard methods like jpeg and mpeg 4 any image is subdivided in blocks and each block is coded decoded using arbitrary fuzzy sets as coders in the fuzzy equations we evaluate the peak signal to noise ratio psnr on the decompressed images for several values of the compression rates as measure of the quality of images and frames reconstructed the original frames of a video are classified in intra frames and predictive frames by using a similarity measure based on the well known lukasiewicz t norm

properties of fuzzy transform obtained from l p minimization and a connection with zadeh s extension principle abstract in 2006 perfilieva proved that the components fk of the fuzzy transform are actually minimizers of certain weighted l2 type distances obtained from a fuzzy partition inspired by this property we consider and analyze the properties of a fuzzy transform when the components fk are obtained as minimizers with respect to weighted lp type distances where p 1 in this way we can generate new approximation operators based on the fuzzy transform method for these operators we obtain good rates of uniform convergence and what is more they satisfy all the properties required for an effective approximation operator such as additivity and homogeneity if we consider the extended fuzzy transform and inverse fuzzy transform respectively in this setting of lp type distances then we get some nice supplementary properties like monotonicity of the operator and preservation of monotone functions finally we consider fuzzy valued versions of the l1 and l2 type fuzzy transforms proposed recently as quantile and expectile smoothing tools and we prove that the l1 type fuzzy valued transform of a continuous monotonic function f coincides with the zadeh s fuzzy extension fep evaluated at suitable fuzzy numbers obtained from the basic functions of the used fuzzy partition

existence and limiting behavior of a non interior point trajectory for nonlinear complementarity problems without strict feasibility condition for p0 complementarity problems most existing non interior point path following methods require the existence of a strictly feasible point for a p complementarity problem the existence of a strictly feasible point is equivalent to the nonemptyness and the boundedness of the solution set in this paper we propose a new homotopy formulation for complementarity problems by which a new non interior point continuation trajectory is generated the existence and the boundedness of this non interior point trajectory for p0 complementarity problems are proved under a very mild condition that is weaker than most conditions used in the literature one prominent feature of this condition is that it may hold even when the often assumed strict feasibility condition fails to hold in particular for a p problem it turns out that the new non interior point trajectory exists and is bounded if and only if the problem has a solution we also study the convergence of this trajectory and characterize its limiting point as the parameter approaches zero

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

fuzzy transform as an additive normal form in this paper we recall a class of approximating formulas since they arose as a generalization of classical normal forms we denote them by the same term moreover we will show that the fuzzy transform f transform introduced in perfilieva i fuzzy approach to solution of differential equations with imprecise data application to reef growth problem in r v demicco g j klir eds fuzzy logic in geology academic press amsterdam 2003 pp 275 300 chapter 9 perfilieva i fuzzy transforms in j f peters a skowron eds transactions on rough sets ii rough sets and fuzzy sets lecture notes in computer science vol 3135 2004 pp 63 81 as an approximation method for continuous functions can be viewed as a particular case of normal form therefore the results valid for normal forms may be applied to f transforms as well

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

compression and decompression of images with discrete fuzzy transforms in foregoing papers we have used the compression decompression method of images based on the concept of discrete fuzzy transform and its inverse of a function f defined on a real interval with respect to the fuzzy sets a 1 a n forming a fuzzy partition of such interval here we make a detailed experimental comparison with the similar method based on the fuzzy transforms f 6 and f 7 of f defined via a continuous triangular norm and its corresponding residuum respectively we consider some images of sizes 256x256 pixels extracted from the well known database corel galery arizona directory by using the same compression rate in both methods we have that the psnr peak signal to noise ratio obtained with the discrete fuzzy transform and its inverse of f is more higher than the psnr determined with the operators f 6 and f 7 defined via the usual lukasiewicz product and minimum triangular norms moreover we compare our results with the classical jpeg method for values of compression rate approximately equal to those used in the previous methods

arithmetic fuzzy models it is well known that a fuzzy rule base can be interpreted in different ways from a logical point of view the conjunctive interpretation is preferred while from a practical point of view the disjunctive interpretation has been dominantly present each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base basic interpolation requirements naturally suggest a corresponding inference mechanism the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation interpolation then corresponds to solvability of some system of fuzzy relational equations in this paper we show that other types of fuzzy relations which are closely related to takagi sugeno t s models are of major interest as well these fuzzy relations are based on addition and multiplication only from which we get the name arithmetic fuzzy models under some mild requirements these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations the impact of these results is both theoretical and practical there exist simple solutions to systems of fuzzy relational equations other than the extremal solutions that have received all the attention so far which are moreover easy to implement

a penalized fischer burmeister ncp function we introduce a new ncp function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations this new ncp function turns out to have stronger theoretical properties than the widely used fischer burmeister function and other ncp functions suggested previously moreover numerical experience indicates that a semismooth newton method based on this new ncp function performs considerably better than the corresponding method based on the fischer burmeister function

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

a measure of market volatility based on f transform abstract volatility is a tool of fundamental importance for capturing price movements in financial markets in general terms volatility is the tendency of prices to vary along a period of time measuring volatility is a quantitative proxy of uncertainty in prices there are different definitions of volatility measure one of the most common is based on the standard deviation of daily returns in this paper we propose an alternative definition based on f transform and its inverse the method is compatible with the theory of risk originally proposed by duncan luce as experimental setting we considered the nifty 50 stock market index within the period september 2000 and february 2017 results show that the proposed measure is smoother and better correlated between stocks

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers

fuzzy transforms method in prediction data analysis we present a prediction method based on fuzzy transforms for forecasting problems and we compare it with the well known wang mendel s one another comparison is made on the local linear wavelet neural network method via forecasting time series we apply these concepts to a mackey glass chaotic time series and to the monthly nao climatic index time series

an interval valued inversion of the non additive interval valued f transform as shown in a recent paper the fuzzy transform can be reinterpreted as a sampling process with an ill known sampling kernel or point spread function this reinterpretation leads to an interval valued direct fuzzy transform the non additive fuzzy transform it provides the convex set of all sampled values that should have been obtained by sampling a continuous signal with a convex set of sampling kernels its dual transform the non additive inverse fuzzy transform allows computation of an interval valued reconstruction of a sampled signal which is the set of all values that would have been obtained by interpolating or approximating this sampled signal when using a convex set of reconstruction kernels in this paper we consider using a new interval valued inversion process within this new framework to upsample a signal i e reconstruct a high resolution signal with a low resolution signal in a semi blind context as illustrated in the experimental part of the paper the main advantage of using this method compared to previous techniques is its inherent robustness with respect to modeling the sampling process