positive decreasing solutions of quasi linear difference equations the second order nonlinear difference equation is considered a full characterization of limit behavior of all positive decreasing solutions is established the obtained results answer some open problems formulated for sturm liouville discrete operator a comparison with the continuous case jointly with similarities and discrepancies is given as well

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

notes on solution sets of inf t fuzzy relational equations on complete brouwerian lattices and fuzzy relational equations on complete brouwerian lattices this paper clarifies several hypotheses and results given in q q xiong and x p wang solution sets of fuzzy relational equations on complete brouwerian lattices information sciences 177 21 47574767 2007 and in q q xiong and x p wang fuzzy relational equations on complete brouwerian lattices information sciences 193 0 141152 2012 which in their current form are confusing and need to be improved

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

design and analysis of fuzzy morphological algorithms for image processing a general paradigm for lifting binary morphological algorithms to fuzzy algorithms is employed to construct fuzzy versions of classical binary morphological operations the lifting procedure is based upon an epistemological interpretation of both image and filter fuzzification algorithms are designed via the paradigm for various fuzzifications and their performances are analyzed to provide insight into the kind of liftings that produce suitable results algorithms are discussed for three image processing tasks shape detection edge detection and clutter removal detailed analyses are given for the effect of noise and its mitigation owing to fuzzy approaches it is demonstrated how the fuzzy hit or miss transform can be used in conjunction with a decision procedure to achieve word recognition

bounded solutions and wavefronts for discrete dynamics this paper deals with the second order nonlinear difference equation dd r k dd u k q kg u k 1 0 where r k and q k are positive real sequences defined on n cup 0 and the nonlinearity g r to r is nonnegative and nontrivial sufficient and necessary conditions are given for the existence of bounded solutions starting from a fixed initial condition u 0 the same dynamic with f instead of g such that uf u 0 for u not 0 was recently extensively investigated on the contrary our nonlinearity g is of a small appearance in the discrete case its introduction is motivated by the analysis of wavefront profiles in biological and chemical models the paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f some applications in the study of wavefronts complete this work

forty years of hutton fuzzy unit interval this note is a very short history of a specific fuzzy topological space the hutton s fuzzy unit interval 0 1 l which occupies one of the central positions in fuzzy topology on that occasion the superiority of l valued topology over 0 1 valued topology is being discussed we then show how the hutton s concept evolved after obvious identifications we can today say that 0 1 l is the tensor product 0 1 l of the lattices 0 1 and l an account of the consequences of this new view point is hoped to appear thereafter

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

oscillatory and asymptotic behavior of second order nonlinear difference equations in this paper we are dealing with the oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form rn xn f n xn 0 n n n0 1 we obtain the following results a if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following four types k k k 0 k 0 b if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following three types k0 k k0 c necessary and sufficient conditions for 1 to have a nonoscillatory solution which belongs to k k k 0 k0 or k are given depending on whether f is a superlinear or sublinear function all these results include and improve b szmanda s results in bull polish acad sci math 34 nos 3 4 1986 133 141

a radon nikodym theorem for fuzzy valued measures abstract we study measures mapping into the set of fuzzy real numbers and in particular prove a radon nikodym theorem where the radon nikodym derivatives are fuzzy valued functions

on fuzzy associative memory with multiple rule storage capacity kosko s fuzzy associative memory fam is the very first neural network model for implementing fuzzy systems despite its success in various applications the model suffers from very low storage capacity i e one rule per fam matrix a lot of hardware and computations are usually required to implement the model and hence it is limited to applications with small fuzzy rule base in this paper the inherent property for storing multiple rules in a fam matrix is identified a theorem for perfect recalls of all the stored rules is established and based upon which the hardware and computation requirements of the fam model can be reduced significantly furthermore we have shown that when the fam model is generalized to the one with max bounded product composition single matrix implementation is possible if the rule base is a set of semi overlapped fuzzy rules rule modification schemes are also developed and the inference performance of the established high capacity models is reported through a numerical example

on initial and final l topological groups in this paper a notion of l topological group introduced by ahsanullah in 1984 is studied some basic properties related to these l topological groups are proved in 1992 kubiak had generalized the functors w and i defined by lowen in 1976 for any complete lattice l to the functors w l and i l we show here that for this notion of l topological group w l and i l are functors moreover to justify this notion of l topological group we show in this paper that all initial and final lifts exist uniquely in the concrete category l topgrp of l topological groups and hence all initial and final l topological groups exist and can be characterized as consequences the l topological subgroups l topological product groups and l topological quotient groups are exist some examples of l topological groups are given

on unbounded positive solutions of second order difference equations with a singular nonlinear term abstract in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form r n x n f n x n 0 n n n 0 where n n0 n0 n0 1 n0 is a fixed nonnegative integer rn 0 for n n0 such that n n0 1 rn and f n n0 r r is a real valued continuous function such that f n x 0 for all n x n n0 0

adaptive mathematical morphology for edge linking in this paper adaptive mathematical morphology and its application to edge linking are presented to fill in the gaps between edge segments broken edges are extended along their slope directions by using the adaptive dilation operation with the suitable sized elliptical structuring elements the size and orientation of the structuring element are adjusted according to the local properties such as slope and curvature postprocesses of thinning and pruning are also applied the edge linking operation is performed in an iterative manner so that the broken edges can be linked up gradually and smoothly while the details of the object shape are preserved

gray scale morphological associative memories neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns thus far the emphasis in research on morphological associative memory systems has been on binary models although a number of notable features of autoassociative morphological memories amms such as optimal absolute storage capacity and one step convergence have been shown to hold in the general gray scale setting in this paper we make extensive use of minimax algebra to analyze gray scale autoassociative morphological memories specifically we provide a complete characterization of the fixed points and basins of attractions which allows us to describe the storage and recall mechanisms of gray scale amms computer simulations using gray scale images illustrate our rigorous mathematical results on the storage capacity and the noise tolerance of gray scale morphological associative memories mams finally we introduce a modified gray scale amm model that yields a fixed point which is closest to the input pattern with respect to the chebyshev distance and show how gray scale amms can be used as classifiers

oscillations and nonoscillations of half linear difference equations generated by deviating arguments for the half linear difference equation y k 1 y k i 1npi k y gi k 1y gi k k a where 0 we shall offer sufficient conditions for the oscillation of all solutions as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions several examples which dwell upon the importance of our results are also included

nonoscillatory solutions for the one dimensional p laplacian abstract in this paper we study the existence of nonoscillatory solutions of the equation xk 1 f k xk 0 k 1 2 where r r is defined by s s p 2 s with p 1 and xk 1 is a nonnegative sequence with infinitely many positive terms

invariant probabilistic metrizability of fuzzy neighbourhood groups we introduce the notions of n local compactness and n openess in fuzzy neighbourhood spaces and in fuzzy neighbourhood groups together with some characterizations for them in particular we prove that a fuzzy neighbourhood space is n locally compact if and only if its level spaces for all 0 1 are locally compact a similar criterion is established for n open functions we also introduce the notion of fuzzy absolute value functions on groups we show that there exists a 1 1 correspondence between invariant fuzzy probabilistic pseudo metrics on groups and fuzzy absolute value functions we establish theorems on fuzzy probabilistic pseudo metrizability of fuzzy neighbourhood groups finally fuzzy probabilistic metrizability of quotient fuzzy neighbourhood groups is also taken into account

suitability in fuzzy topological spaces abstract an l fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset basic properties of and sufficient conditions for suitable spaces are derived characterizations of the suitable subspaces of the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are obtained suitability is l fuzzy productive nondegenerate 1 hausdorff spaces are suitable the fuzzy unit interval the fuzzy open unit interval and the fuzzy real line are not suitable and no suitable subspace of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line is a fuzzy retract of the fuzzy unit interval the fuzzy open unit interval or the fuzzy real line respectively without restrictions there cannot be a fuzzy extension theorem

a general framework for fuzzy morphological associative memories fuzzy associative memories fams can be used as a powerful tool for implementing fuzzy rule based systems the insight that fams are closely related to mathematical morphology mm has recently led to the development of new fuzzy morphological associative memories fmams in particular implicative fuzzy associative memories ifams as the name fmam indicates these models belong to the class of fuzzy morphological neural networks fmnns thus each node of an fmam performs an elementary operation of fuzzy mm clarifying several misconceptions about fmams that have recently appeared in the literature we provide a general framework for fmams within the class of fmnn we show that many well known fam models fit within this framework and can therefore be classified as fmams moreover we employ certain concepts of duality that are defined in the general theory of mm in order to derive a large class of strategies for learning and recall in fmams

bounded linear transformations between probabilistic normed vector spaces we denote by r i the set of all nonnegative fuzzy real numbers that is functions n from r 0 to the unit interval i which are left continuous and nonascending and satisfy 0 1 and 0 this set r i canonically includes r in the obvious way it is ordered by the usual pointwise order of real functions

a lattice of continuities for fuzzy topological spaces abstract several continuities are defined for fuzzy topological spaces and characterizations of these continuities are obtained it is shown that these continuities form a lattice ordered by implication whose supremum is fuzzy continuity and whose infimum is the continuity of general topology

some results on i l and r l abstract this paper gives a partial solution to problem 1 in wang and lu 1992 and a suggestion about how to construct a better fuzzy unit interval it defines a new kind of fuzzy real line and investigates a series of its properties

on central algorithms of approximation under fuzzy information we consider the problem of approximation of an operator by information described by n real characteristics in the case when this information is fuzzy we develop the well known idea of an optimal error method of approximation for this case it is a method whose error is the infimum of the errors of all methods for a given problem characterized by fuzzy numbers in this case we generalize the concept of central algorithms which are always optimal error algorithms and in the crisp case are useful both in practice and in theory in order to do this we define the centre of an l fuzzy subset of a normed space the introduced concepts allow us to describe optimal methods of approximation for linear problems using balanced fuzzy information

convergence in the i fuzzy real line infinite sums and sequences in the fuzzy real line have been investigated by wang in this paper a necessary and sufficient condition for a sequence in to be convergent is given in terms of cut set it is proved that fuzzy convergence of sequences in is equivalent to weak convergence as an application convergence of sequences of fuzzy numbers in the sense of chang and zadeh is discussed

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

fuzzy subcontinuity inverse fuzzy subcontinuity and a new category of fuzzy topological spaces the notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting adopting fuzzy net theoretic approach various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied a category of fuzzy topological spaces is constructed canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity in the process n compactness is characterized and fuzzy unit interval is shown to be n compact also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous

fuzzy semi inner product of fuzzy points abstract in an earlier paper using the set of non negative fuzzy real numbers r i we constructed a fuzzy normed algebra c i in this paper the concept of fuzzy semi inner product of fuzzy points is introduced on a c i module x also some properties of fuzzy linear functionals on the fuzzy semi inner product space of fuzzy points are studied

implicative fuzzy associative memories associative neural memories are models of biological phenomena that allow for the storage of pattern associations and the retrieval of the desired output pattern upon presentation of a possibly noisy or incomplete version of an input pattern in this paper we introduce implicative fuzzy associative memories ifams a class of associative neural memories based on fuzzy set theory an ifam consists of a network of completely interconnected pedrycz logic neurons with threshold whose connection weights are determined by the minimum of implications of presynaptic and postsynaptic activations we present a series of results for autoassociative models including one pass convergence unlimited storage capacity and tolerance with respect to eroded patterns finally we present some results on fixed points and discuss the relationship between implicative fuzzy associative memories and morphological associative memories

summation inequalities and half linear difference equations abstract the aim of this paper is to present new summation inequalities and to show that the half linear difference equation a n x n sgn x n b n x n 1 sgn x n 1 exhibits not only similarities but also some discrepancies with the linear case especially as regards the possible coexistence of the so called extremal solutions some of our results are new also for the linear difference equation

morphological perceptrons with competitive learning lattice theoretical framework and constructive learning algorithm a morphological neural network is generally defined as a type of artificial neural network that performs an elementary operation of mathematical morphology at every node possibly followed by the application of an activation function the underlying framework of mathematical morphology can be found in lattice theory with the advent of granular computing lattice based neurocomputing models such as morphological neural networks and fuzzy lattice neurocomputing models are becoming increasingly important since many information granules such as fuzzy sets and their extensions intervals and rough sets are lattice ordered in this paper we present the lattice theoretical background and the learning algorithms for morphological perceptrons with competitive learning which arise by incorporating a winner take all output layer into the original morphological perceptron model several well known classification problems that are available on the internet are used to compare our new model with a range of classifiers such as conventional multi layer perceptrons fuzzy lattice neurocomputing models k nearest neighbors and decision trees

unbounded solutions of quasi linear difference equations we study positive increasing solutions of the nonlinear difference equation an p n bnf n 1 p u u p 2u p 1 where an bn are positive real sequences for n 1 fr r is continuous with uf u 0 for u 0 a full characterization of limit behavior of all these solutions in terms of an bn is established examples showing the essential role of used hypotheses are also included the tools used are the schauder fixed point theorem and a comparison method based on the reciprocity principle

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

swarm based translation invariant morphological prediction method for financial time series forecasting in this paper we present a method to overcome the random walk rw dilemma for financial time series forecasting called swarm based translation invariant morphological prediction stmp method it consists of a hybrid model composed of a modular morphological neural network mmnn combined with a particle swarm optimizer pso which searches for the best time lags to optimally describe the time series phenomenon as well as estimates the initial sub optimal parameters of the mmnn weights architecture and number of modules an additional optimization is performed with each particle of the pso population a distinct mmnn using the back propagation bp algorithm after the mmnn parameters adjustment we use a behavioral statistical test and a phase fix procedure to adjust time phase distortions observed in financial time series finally we conduct an experimental analysis with the proposed method using four real world stock market time series where five well known performance metrics and a fitness function are used to assess the prediction performance the obtained results are compared with those generated by classical models presented in the literature

bounded and monotone properties of solutions of second order quasilinear forced difference equations abstract the authors consider the quasilinear difference equation p n 1 y n 1 q n y n r n and obtain results on the asymptotic behavior of solutions of including sufficient conditions for all solutions to be bounded or unbounded some results on the existence and behavior of nonincreasing solutions of are also obtained examples are inserted to illustrate the results

global l neighborhood groups this paper introduces and studies the notion of global l neighborhood group which is defined as a group equipped with a global l neighborhood structure in sense of gahler et al such that both the binary operation and the unary operation of the inverse are continuous with respect to this global l neighborhood structure some examples of global l neighborhood groups are given it is shown that the l topological groups given by ahsanullah in 1984 and later by bayoumi in 2003 are special global l neighborhood groups we also show that all initial and final lifts and hence all initial and final global l neighborhood groups uniquely exist in the category l gnghgrp of global l neighborhood groups these initial and final global l neighborhood groups are defined using the initial and final global l neighborhood structures moreover we show that the l neighborhood groups defined by ahsanullah using the l neighborhood structures in sense of lowen are special global l neighborhood groups for l i is the closed unit interval

morphological regularization neural networks abstract in this paper we establish a relationship between regularization theory and morphological shared weight neural networks msnn we show that a certain class of morphological shared weight neural networks with no hidden units can be viewed as regularization neural networks this relationship is established by showing that this class of msnns are solutions of regularization problems this requires deriving the fourier transforms of the min and max operators the fourier transforms of min and max operators are derived using generalized functions because they are only defined in that sense

bounded and zero convergent solutions of second order difference equations abstract in this paper we study boundedness and monotonicity properties of a homogeneous second order linear difference equation under certain conditions it is shown that every solution must eventually be monotonic the question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated a necessary and sufficient condition for all solutions to be bounded is presented comparison theorems are also discussed

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

nonoscillation criteria for half linear second order difference equations abstract nonoscillation criteria for the half linear second order difference equation are established these criteria are derived using the riccati and variational technique