on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

quantitative weights and aggregation based on the strong idempotency of aggregation operators quantitative weights are incorporated into the fusion process our general method is compared with some previous specific cases more details about weighted aggregation based on some penalty function is given further weighted integral based aggregation linked to quantifier based fuzzy measures is investigated especially weighted owa operators several examples are included

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

error analysis of an equation error method for the identification of the diffusion coefficient in a quasi linear parabolic differential equation we consider the inverse problem of reconstructing the diffusion coefficient in a quasi linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain an equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner a detailed error analysis relates the required number of measurements with their accuracy numerical results illustrate the performance of the method

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

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

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

the solution of hutn ik s open problem in this note we give a solution to problem 9 2 which was presented by mesiar and stup v n anov a it open problems from the 12th international conference on fuzzy set theory and its applications fuzzy sets and systems 2014 this http url j fss 2014 07 012 we show that the class of semicopulas solving problem 9 2 contains only the l ukasiewicz t norm

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

a quantitative study on the stability of quadratic delay difference systems this paper estimates the size of the stability region around zero for quadratic delay difference systems when the initial disturbance is in the asymptotic stability region the solution of the initial value problem of the quadratic delay difference system approaches zero examples are given for a three dimensional system and three one dimensional equations to demonstrate both stability and instability examples 2 4 show that when parameters in the systems do not satisfy the stability conditions the zero solutions can be unstable three evolution features of initial disturbances are shown numerically decaying to zero being amplified but bounded and growing to infinity example 3 further shows that the stable zero solution may not be a global attractor numerical results confirm the conclusions of the main theorem in this paper and imply that our estimation of the size of the stability region are of reasonable accuracy 1998 elsevier science ltd all rights reserved

on the finiteness of uniform sinks abstract we study the finiteness of uniform sinks for flows precisely we prove that for 0 and t 0 if a vector field x has only hyperbolic singularities or sectionally dissipative singularities then x can have only finitely many t uniform sinks this is a generalized version of a theorem of liao 4

consistency degrees of finite theories in łukasiewicz propositional fuzzy logic abstract the concept of divergence degrees of theories in mathematical logic can be used to grade the extent of consistency of theories based on the analysis of relationships among the properties of consistent inconsistent and fully divergent theories the present paper proposes a membership function for reflecting the consistency degrees of finite theories in ukasiewicz propositional fuzzy logic it is proved that the consistency degrees of consistent finite theories are in between 1 2 and 1 and that of inconsistent theories are equal to 0 and that of completely consistent finite theories are equal to 1 finally a sufficient and necessary condition for theories being inconsistent is also proposed

a modified method for high order numerical derivatives in this paper we propose a new regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence estimate under an appropriate choice of the regularization parameter is obtained some interesting numerical tests show that the proposed method is effective and stable

generalized uniform fuzzy partition in this note we propose a solution to an open problem which was presented by mesiar and stup anova we give two counterexamples to show that the hypothesis is not sufficient to get the result and we propose that the sufficient condition of the hypothesis is held by modifying the symmetry condition

mutually exchangeable fuzzy implications recently vemuri and jayaram 2012 have proposed a novel generating method of fuzzy implications called the composition further they have also proposed mutual exchangeability me a generalization of the exchange principle ep to a pair of fuzzy implications and have shown that me plays a central role in the preservation of basic properties functional equations and families of fuzzy implications w r t the composition vemuri and jayaram 2014 due to the important role played by me in this work we investigate the pairs i j of fuzzy implications that satisfy me towards this we show first that there exist pairs i j of fuzzy implications that satisfy me and determine some necessary conditions on such fuzzy implications following this for a given i i we find the set j i of all fuzzy implications j such that the pair i j satisfies me keeping in view the variety of fuzzy implications and the complexity of the functional equation me we restrict our investigations to four important families of fuzzy implications namely s n r f and g implications further we discuss a generalization of the cauchy multiplicative equation whose solutions help us in obtaining the set j i for some families of fuzzy implications

the smallest semicopula based universal integrals i properties and characterizations abstract in this article we provide a detailed study of basic properties of the smallest universal integral i s with s being the underlying semicopula and review some of the recent developments in this direction the class of integrals under study is also known under the name seminormed integrals and includes the well known sugeno and shilkret integrals as special cases we present some representations and equivalent formulations for i s further we discuss basic properties such as translatability linearity homogeneity and maxitivity in particular we provide the characterizations of the two prominent integrals of sugeno and shilkret we also raise and discuss some open questions

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

fuzzy partitions with the connectives t s abstract the notion of a fuzzy partition obtained when the set operations are defined using the operators t and s and an appropriate concept of disjointness is according to butnariu 1983 equivalent with ruspini s definition we prove that such an equivalence holds if and only if the connectives for the fuzzy set operations are t and s

lattice operations on fuzzy implications and the preservation of the exchange principle in this work we solve an open problem related to the preservation of the exchange principle ep of fuzzy implications under lattice operations 3 problem 3 1 we show that generalizations of the commutativity of antecedents ca to a pair of fuzzy implications i j viz the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem further we determine conditions under which these become necessary too finally we investigate the pairs of fuzzy implications from different families such that ep is preserved by the join and meet operations

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

perturbation of bivariate copulas new types of constructions of bivariate copulas are introduced discussed and exemplified based on a given copula c we look for its perturbation into another copula c h possibly close to c a special stress is put on the perturbation of the basic copulas m w our results generalize several methods known from the literature such as the farlie gumbel morgenstern copula family for example an illustrative example when fitting copulas to real data is also added

translation invariant semicopula based integrals in this note we give a solution to problem 9 2 which was presented by mesiar and stup anova 2015 6 we show that the class of semicopulas solving problem 9 2 contains only the ukasiewicz t norm

problems on triangular norms and related operators abstract a number of open problems on triangular norms and related operators was posed during the 24th linz seminar on fuzzy set theory triangular norms and related operators in many valued logics held in february 2003 they are collected here together with some other open problems in this context and with some problems which were posed earlier and have been solved in the meantime

expansiveness hyperbolicity and hausdorff dimension we show that there exists a simple upper bound on the dimension of a hyperbolic compact set of a dynamical system in terms of topological entropy and a uniform contraction rate on the stable and unstable manifolds this allows us to give proofs of several apparently unrelated theorems

fourier truncation method for high order numerical derivatives we consider a classical ill posed problem numerical differentiation with a new method we propose fourier truncation method to compute high order numerical derivatives a holder type stability estimate is obtained a numerical implementation is described numerical examples show that the proposed method is effective and stable

on a new construction of 1 lipschitz aggregation functions quasi copulas and copulas a new method of constructing 1 lipschitz aggregation functions is proposed and studied as a by product a new method of constructing quasi copulas from a given quasi copula c is obtained newly constructed copulas are negative quadrant dependent for any original copula c the only invariant copula under this construction is the frechet hoffding lower bound w the method preserves several important properties of copulas such as modularity absolute continuity or symmetry and commutes with some other constructions e g the construction of survival copulas

consistency degrees of theories in some systems of propositional fuzzy logic based on the analysis of relationships among the properties of consistent inconsistent and fully divergent theories and by means of the concept of divergence degrees the present paper proposes a new membership function for reflecting the consistency degrees of general theories in four different types of fuzzy logic systems i e lukasiewicz fuzzy logic system luk godel fuzzy logic system god product fuzzy logic system pro and the r 0 fuzzy logic system l

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

on the redundancy of fuzzy partitions abstract we introduce a concept of redundancy for families of fuzzy subsets based on a left continuous t norm and its corresponding operator and study its basic properties including a characterization of families with redundancy zero in some important special cases examples illustrate that in the case of t norms with zero divisors several definitions for fuzzy partitions from the literature are insensitive to the phenomenon of redundancy finally a geometric interpretation of the redundancy based on the ukasiewicz t norm and an upper bound for its value are given

on the dynamics of dominated splitting let f m m be a c 2 dieomorphism of a compact surface we give a complete description of the dynamics of any compact invariant set having dominated splitting in particular we prove a spectral decomposition theorem for the limit set l f under the assumption of dominated splitting moreover we describe all the bifurcations that these systems can exhibit and the dierent types of dynamics that could follow for small c r perturbations

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

a perturbation method for numerical differentiation we discuss the numerical differentiation from the noisy data we propose a method i e adding a differential term for the differentiation problem an error estimate shows that the method allows for the approximate recovery of the derivative function some interesting numerical tests display the properties of the proposed method

a regularization framework for mildly ill posed problems connected with pseudo differential operator abstract recently filter based regularization methods have been well investigated for ill posed problems when the forward operators are compact there are many ill posed problems connected with pseudo differential operators but there is no uniform method for this kind of problems the work on generalization of filter based regularization methods to pseudo differential operator is necessary in this paper we present a regularization framework for solving the mildly ill posed problems involved pseudo differential operators a general regularization method for this kind of problems is given the order optimal error estimates are derived under the usual source conditions as an example a new fractional tikhonov regularization method could be cast into the general framework numerical experiments are conducted for showing the validity of the new fractional tikhonov method

the truncation method for identifying an unknown source in the poisson equation this paper deals with the problem of determining an unknown source which depends only on one variable in two dimensional poisson equation with the aid of an extra measurement at an internal point the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we obtain the regularization solution by the truncation method for the regularization solution the holder type stability estimate between the regularization solution and the exact solution is given numerical results are presented to illustrate the accuracy and efficiency of this method

t subnorms with strong associated negation some properties in this work we investigate t subnorms m that have strong associated negation firstly we show that such t subnorms are necessarily t norms following this we investigate the inter relationships between different algebraic and analytic properties of such t subnorms viz archimedeanness conditional cancellativity left continuity nilpotent elements etc in particular we show that under this setting many of these properties are equivalent our investigations lead us to two open problems which are also presented

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

the smallest semicopula based universal integrals ii in this paper we continue studying the smallest universal integral i s having s as the underlying semicopula we present convergence theorems for i s integral sequences including monotone almost everywhere almost uniform in measure and in mean converging sequences of measurable functions respectively it emerges that these convergences characterize the underlying measure properties such as null additivity monotone autocontinuity and autocontinuity we provide many examples and counter examples as well as a few interesting open problems

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

a universal integral as common frame for choquet and sugeno integral the choquet and the sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required however their applicability is restricted because of the special operations used in the construction of these integrals therefore we provide a concept of integrals generalizing both the choquet and the sugeno case for functions with values in the nonnegative real numbers universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures for a fixed pseudo multiplication on the nonnegative real numbers the smallest and the greatest universal integrals are given finally another construction method for obtaining universal integrals is introduced and the restriction to the unit interval i e to fuzzy integrals is considered

on s homogeneity property of seminormed fuzzy integral we give an answer to problem 9 3 stated by mesiar and stup anova 8 we show that the class of semicopulas solving this problem contains any associative semicopula s such that for each a 0 1 the function x s a x is continuous and increasing on a countable number of intervals

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

for numerical differentiation dimensionality can be a blessing finite difference methods such as the mid point rule have been applied successfully to the numerical solution of ordinary and partial differential equations if such formulas are applied to observational data in order to determine derivatives the results can be disastrous the reason for this is that measurement errors and even rounding errors in computer approximations are strongly amplified in the differentiation process especially if small step sizes are chosen and higher derivatives are required a number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data the size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h in this paper it is initially shown how first and higher order single variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one dimensional data the result is then extended to the multivariate differentiation of higher dimensional data the nature of the trade off between convergence and stability is explicitly characterized and the complexity of various implementations is examined

a c 2 generic trichotomy for diffeomorphisms hyperbolicity or zero lyapunov exponents or the c 1 creation of homoclinic bifurcations palis conjectured that densely in diff m r 1 diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations we prove a generic trichotomy for c2 diffeomorphisms an axiom a diffeomorphism with no cycles or kupka smale ones admitting zero lyapunov exponents or the c1 creation of homoclinic bifurcations i e the creation of homoclinic tangencies or heterodimensional cycles by some c1 small perturbations

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

on comonotone commuting and weak subadditivity properties of seminormed fuzzy integrals abstract in the paper we characterize the class of all the binary operators commuting with the generalized sugeno integral generated by a strict t norm for the sugeno integral this problem was solved by ouyang and mesiar we also present the necessary and sufficient conditions for the weak subadditivity of the generalized sugeno integral

the truncation method for identifying the heat source dependent on a spatial variable this paper deals with the ill posed problem of determining an unknown source depending only on the spatial variable in the heat equation in this problem the truncation method is employed to get the regularization solution moreover the holder type error estimate is also obtained between this regularization solution and the exact solution various typical examples are given to verify the efficiency and accuracy of the proposed method

meager projections in orthocomplete homogeneous effect algebras abstract we define meager projections on homogeneous effect algebras and discuss their properties as an application we prove that if e is an orthocomplete homogeneous effect algebra such that s e is lattice ordered then e is a lattice effect algebra which gives an affirmative answer to an open problem stated by mesiar and stup anova

discrete mollification and automatic numerical differentiation abstract an automatic method for numerical differentiation based on discrete mollification and the principle of generalized cross validation is presented with data measured at a discrete set of points of a given interval the method allows for the approximate recovery of the derivative function on the entire domain no information about the noise is assumed error estimates are included together with several numerical examples of interest

asymptotic measure expansive diffeomorphisms abstract in this paper the notion of asymptotic measure expansiveness is introduced and its relationship with dominated splitting is considered it is proved that if a diffeomorphism admits a co dimension one dominated splitting then it is asymptotic measure expansive also a diffeomorphism with a homoclinic tangency can be perturbed to a non asymptotic measure expansive diffeomorphism