probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

transmutations l bases and complete families of solutions of the stationary schrodinger equation in the plane an l basis associated to a linear second order ordinary differential operator l is an infinite sequence of functions k k 0 such that l k 0 for k 0 1 l k k k 1 k 2 for k 2 3 and all k satisfy certain prescribed initial conditions we study the transmutation operators related to l in terms of the transformation of powers of the independent variable x x0 k k 0 to the elements of the l basis and establish a precise form of the transmutation operator realizing this transformation we use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary schrodinger equation from a certain class the system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification the obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory

spectral parameter power series for perturbed bessel equations a spectral parameter power series spps representation for regular solutions of singular bessel type sturm liouville equations with complex coefficients is obtained as well as an spps representation for the entire characteristic function of the corresponding spectral problem on a finite interval it is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the sturm liouville problem based on the spps representation a new mapping property of the transmutation operator for the considered perturbed bessel operator is obtained and a new numerical method for solving corresponding spectral problems is developed the range of applicability of the method includes complex coefficients complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator on a set of known test problems we show that the developed numerical method based on the spps representation is highly competitive in comparison to the best available solvers such as sleign2 matslise and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the spps method delivers excellent numerical results

fuzzy plane geometry i points and lines abstract we introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry we consider the fuzzy distance between fuzzy points and show it is a weak fuzzy metric we study various definitions of a fuzzy line develop their basic properties and investigate parallel fuzzy lines

the categorical topology approach to fuzzy topology and fuzzy convergence it is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology we first look at interesting subcategories of fts and determine whether they are bireflective and or coreflective we also describe the set of all subcategories of fts which are at the same time bireflective and coreflective due to the fact that top is one of these subcategories of fts ffs itself can be neither cartesian closed nor extensional we therefore secondly construct a supercategory of bts which is a topological quasitopos the construction of this quasitopos is based on the theory of convergence in fts

progressive hedging as a meta heuristic applied to stochastic lot sizing abstract in a great many situations the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved this gives rise to a need for stochastic programming sp methods that create solutions that are hedged against future uncertainty the progressive hedging algorithm pha of rockafellar and wets is a general method for sp we cast the pha in a meta heuristic framework with the sub problems generated for each scenario solved heuristically rather than using an approximate search algorithm for the exact problem as is typically the case in the meta heuristic literature we use an algorithm for sub problems that is exact in its usual context but serves as a heuristic for our meta heuristic computational results reported for stochastic lot sizing problems demonstrate that the method is effective

liouville transformation analytic approximation of transmutation operators and solution of spectral problems a method for solving spectral problems for the sturm liouville equation p v q v r v 0 based on the approximation of the delsarte transmutation operators combined with the liouville transformation is presented the problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients p q and r by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced by khmelnytskaya et al 2013 and kravchenko and torba 2015 the method allows one to compute both lower and higher eigendata with an extreme accuracy several necessary results concerning the action of the liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the sturm liouville operator 1 r d d x p d d x q

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

a theory of fuzzy uniformities with applications to the fuzzy real lines abstract for each completely distributive lattice l with order reversing involution the fuzzy real line r l is uniformizable by a uniformity which both generates the canonical fuzzy topology and induces a pseudometric generating the canonical topology if l is also a chain the usual addition and multiplication defined on r r 0 1 extend jointly fuzzy continuously to and on r l three fundamental questions in fuzzy sets until now are question a if l 1 l 2 is r l 1 uniformly isomorphic to r l 2 in some sense question b for each chain l is jointly uniformly continuous in a sense which guarantees its joint continuity on r l question c is r l a complete pseudometric space in some sense we construct categories qu and u using the quasi uniformities of b hutton which enable us to answer these questions in the affirmative these results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative question d does fuzzy topology have deep specific canonical examples

topological properties of the spaces of regular fuzzy sets abstract by identifying the space f 0 x of regular fuzzy sets with a subspace t 0 x of some compact set valued function space we define a uniform topology on f 0 x and discuss several topological properties such as first countability separability connectedness etc when x is a reflexive banach space f 0 x is imbedded into a locally convex topological vector space and a definition of integration and differentiation for f 0 x valued functions is given

dynamic topologies and their applications to crisp topologies fuzzifications of crisp topologies and fuzzy topologies on the crisp real line abstract we introduce and develop topologies whose members are dynamic sets a la zeleny this development spawns the notions of phase and co phase spaces fuzzy dual dynamic dual and co fuzzy dual with the following results being obtained result a the notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of hutton gantner steinlage warren klein and lowen result b the dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic result c the fuzzy real lines re interpreted as fuzzy duals of r with the usual euclidean topology t induce on r via the notion of co fuzzy dual a class of canonical stratified fuzzy topologies co r l e g if l is a demorgan algebra and lc 1 then the compactness of a r is equivalent to the ordinary compactness of a w r t t and if l is a hutton algebra r l co r l is metrizable in the sense of hutton and erceg with erceg metric d l such that d l p r 1 p s 1 r s where pr is the crisp point or impulse with support r i e d l extends the euclidean metric on r

a fuzzy lagrange interpolation theorem abstract the following problem was posed by l a zadeh suppose we are given n 1 points x0 xn in r and for each of these points a fuzzy value in r rather than a crisp one is it then possible to construct some function on r with range also a collection of fuzzy values which coincides on the given n a points with the given fuzzy values and which fulfills some natural smoothness condition in this paper we shall present a solution to this problem based on the fundamental and well known polynomial interpolation theorm of lagrange

the fuzzy hausdorff bourbaki structures and their relations with the fuzzy vietoris structures abstract in 7 lowen introduced starting from a uniform space the hausdorff bourbaki fuzzy uniformity on the hyperspace of fuzzy sets in this paper we shall introduce two coarser structures the supremum of which is the hausdorff bourbaki fuzzy uniformity we also study the relations of these so called semi hausdorff bourbaki structures with the semi vietoris fuzzy neighbourhood structures introduced in 11

fixed point theorems for fuzzy numbers abstract in this paper we study the problem of the existence of fixed points for fuzzy maps we first define uniform structures on sets of fuzzy subsets and describe the closed convergence and myope convergence structures which have properties close to the corresponding traditional ones we then give two ways in order to obtain fixed point results we finally give applications of these results particularly to fuzzy markov processes

errata complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines abstract we construct a fuzzy multiplication for the case l a chain in the l fuzzy real line r l which is jointly fuzzy continuous and which is the unique piecewise consistent extension of ordinary multiplication to r l distributing over the fuzzy addition constructed by the author in 47 we also specify the fuzzy integers and give an extensive analysis of fuzzy contractions and expansions these results the constructions of this paper and the results of 47 are used to prove r l and the stratified r c l are nontopologically isomorphic complete fuzzy topological hyperfields if l is a chain with l 3 we can now completely answer for l a chain the following question of 27 in the affirmative can a set of fuzzy numbers its fuzzy topology and its fuzzy addition and multiplication be defined such that these operations are jointly fuzzy continuous extensions of the usual addition and multiplication of r and can we specify the fuzzy integers these results in combination with recent papers of several authors would seem to answer in the affirmative the following philosophical question cf 34 does fuzzy topology have deep specific canonical examples

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

representation of solutions to the one dimensional schrodinger equation in terms of neumann series of bessel functions abstract a new representation of solutions to the equation y q x y 2 y is obtained for every x the solution is represented as a neumann series of bessel functions depending on the spectral parameter due to the fact that the representation is obtained using the corresponding transmutation operator a partial sum of the series approximates the solution uniformly with respect to which makes it especially convenient for the approximate solution of spectral problems the numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

spectral parameter power series for fourth order sturm liouville problems a general solution of the fourth order sturm liouville equation is presented in the form of a spectral parameter power series spps the uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure based on the spps representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their taylor series expansions this leads to a simple and efficient numerical method for solving the spectral problems for fourth order sturm liouville equations several examples of application are discussed

conditions on a semigroup to be a fuzzy neighborhood group abstract our major purpose is to determine conditions under which a fuzzy neighborhood semigroup is a fuzzy neighborhood group in doing so we produce several characterization theorems using a level topological criterion

modified spectral parameter power series representations for solutions of sturm liouville equations and their applications abstract spectral parameter power series spps representations for solutions of sturm liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems they are based on a computation of recursive integrals sometimes called formal powers in this paper new relations between the formal powers are presented which considerably improve and extend the application of the spps method for example originally the spps method at a first step required to construct a nonvanishing in general a complex valued particular solution corresponding to the zero value of the spectral parameter the obtained relations remove this limitation additionally equations with nasty sturm liouville coefficients 1 p or r can be solved by the spps method we develop the spps representations for solutions of sturm liouville equations of the form p x u q x u k 1 n k r k u x a b where r k u r k x u s k x u k 1 n the complex valued functions p q r k s k are continuous on the finite segment a b several numerical examples illustrate the efficiency of the method and its wide applicability

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

fuzzy preuniform structures and the structures they induce 2 applications in this second article we use the methodology and results we have introduced in the first part in this text we principally develop applications of what we have called the stabilization mechanism in the case of uniform spaces in fact we identify the set of the fuzzy subsets of a set e with the quotient set for an equivalence relation of a larger family we endow this family with uniform and topological structures and we study the quotient structures induced on the set of the fuzzy subsets of e we especially detail the case where e is a classic uniform space when e is a locally compact metric and separable space we can generate the classic closed convergence structure on the family of the closed subsets of e and the myope convergence structure on the family of the compact subsets of e we extend these structures to the case of fuzzy subsets and we obtain results which are generally close to the corresponding classic ones we think that a fuzzy mapping is a concept which is close to the concept of correspondence and then the continuity of fuzzy mappings must be defined in a way similar to those used for correspondences so we define the upperand lowerhemi continuity of fuzzy mappings finally we apply these results to the case where e is a linear space and more especially where e r we consider the fuzzy vectors on r and the corresponding extended operations we give continuity results for these operations and applications to fuzzy real analysis we have quoted some linked works for instance 5 71 in this article we use the notations and conventions of the first article

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

an experimental study on fuzzy distances for skull face overlay in craniofacial superimposition abstract skull face overlay is the most time consuming and error prone stage in craniofacial superimposition an important skeleton based forensic identification technique this task focuses on achieving the best possible overlay of an unknown skull found and a single ante mortem image of a candidate missing person the process is influenced by some sources of uncertainty since two objects of different nature are involved i e a skull and a face in previous works we have developed a computer aided craniofacial superimposition system aimed to assist forensic anthropologists in obtaining the best possible skull and face overlay the system has successfully allowed us to reduce the processing time simplify the forensic anthropologists work and make the process more objective and reproducible our approach is based on automatically overlaying a skull three dimensional model onto a facial photograph by minimizing the distance between two subsets of corresponding cranial and facial landmarks the proposed method properly deals with the inherent uncertainty sources to the skull face overlay process by considering fuzzy sets to model imprecise landmark location and imprecise cranial and facial landmarks spatial correspondence resulting from the presence of soft tissues in the face accordingly our methodology requires computing two kinds of distance metrics between a point and a fuzzy set and between two fuzzy sets this contribution is devoted to study the performance and influence of the most significant and suitable fuzzy distances proposed in the specialized literature as well as other new ones proposed on our skull face overlay system in particular we have tested the behavior of our automatic method when considering eight different distance measurements the system performance has been objectively evaluated considering 18 case studies resulting from a ground truth dataset following a rigorous statistical experimental setup the fact that the choice of a good distance metric is crucial to our method has been demonstrated since it significantly affects the quality of the final solutions it has been shown that our skull face overlay approach presents the best performance using the weighted mean distance in most of the cases and that the results are both more accurate and robust than the other studied metrics

fuzzy plane geometry ii circles and polygons abstract this paper continues our research in fuzzy plane geometry previously we studied fuzzy points and fuzzy lines now we investigate fuzzy circles and fuzzy polygons we show that the fuzzy area of a fuzzy circle or a fuzzy polygon is a fuzzy number we also argue that the fuzzy perimeter of a fuzzy circle or a fuzzy polygon is a fuzzy number

some questions in fuzzy topology the purpose of this paper is to discuss some basic questions related mainly to hutton s l fuzzy unit interval some of those questions are stated here for the first time while some of them are long standing related results are surveyed in some cases

dispersion equation and eigenvalues for the zakharov shabat system using spectral parameter power series a general solution of the zakharov shabat system with a real valued potential is obtained in terms of spectral parameter power series it is used for deriving a dispersion characteristic equation for the zakharov shabat eigenvalue problem with a compactly supported potential due to its convenient form the dispersion equation lends itself to numerical solution providing a simple and accurate numerical method for solving the zakharov shabat eigenvalue problem

a hybrid scenario cluster decomposition algorithm for supply chain tactical planning under uncertainty abstract we propose a hybrid scenario cluster decomposition hscd heuristic for solving a large scale multi stage stochastic mixed integer programming ms mip model corresponding to a supply chain tactical planning problem the hscd algorithm decomposes the original scenario tree into smaller sub trees that share a certain number of predecessor nodes then the ms mip model is decomposed into smaller scenario cluster multi stage stochastic sub models coordinated by lagrangian terms in their objective functions in order to compensate the lack of non anticipativity corresponding to common ancestor nodes of sub trees the sub gradient algorithm is then implemented in order to guide the scenario cluster sub models into an implementable solution moreover a variable fixing heuristic is embedded into the sub gradient algorithm in order to accelerate its convergence rate along with the possibility of parallelization the hscd algorithm provides the possibility of embedding various heuristics for solving scenario cluster sub models the algorithm is specialized to lumber supply chain tactical planning under demand and supply uncertainty an ad hoc heuristic based on lagrangian relaxation is proposed to solve each scenario cluster sub model our experimental results on a set of realistic scale test cases reveal the efficiency of hscd in terms of solution quality and computation time

analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems a method for approximate solution of spectral problems for sturm liouville equations based on the construction of the delsarte transmutation operators is presented in fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in khmelnytskaya et al 2013 and kravchenko and torba 2014 the method allows one to compute both lower and higher eigendata with an extreme accuracy

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

wave polynomials transmutations and cauchy s problem for the klein gordon equation abstract we prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials solutions of the klein gordon equation with a variable coefficient using the transmutation transformation operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the cauchy problem for the klein gordon equation based on this result we develop a numerical method for solving the cauchy problem and test its performance

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

the problem of completeness for p mean symmetric difference metric in this paper by applying lebesgue s measure on the symmetric difference of sets the p mean symmetric difference metric d p is established to measure the difference between fuzzy numbers the basic properties of cauchy sequences of e1 d p are discussed and we show e1 k d p is complete where e1 k a a e1anda0 k k is a closed interval a0 is the closure of support set of a further an example where the space e1 d p is not complete is given

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

concerning the constants in fuzzy topology on etudie les cas mathematiquement importants qui justifient l etude des topologies quasi floues

a branch and cut algorithm for the stochastic uncapacitated lot sizing problem this paper addresses a multi stage stochastic integer programming formulation of the uncapacitated lot sizing problem under uncertainty we show that the classical l s inequalities for the deterministic lot sizing polytope are also valid for the stochastic lot sizing polytope we then extend the l s inequalities to a general class of valid inequalities called the inequalities and we establish necessary and sufficient conditions which guarantee that the inequalities are facet defining a separation heuristic for inequalities is developed and incorporated into a branch and cut algorithm a computational study verifies the usefulness of the inequalities as cuts

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

a branch and price algorithm for multistage stochastic integer programming with application to stochastic batch sizing problems in this paper we present a branch and price method to solve special structured multistage stochastic integer programming problems we validate our method on two different versions of a multistage stochastic batch sizing problem sbsp one version adopts a recourse formulation and the other is based on probabilistic constraints our algorithmic approach is applicable to both formulations our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch and price tree the success of our approach calls for extensions in methodology as well as applications

extending lattice structures on essaie d etendre des operations d une structure reticulee et de definir une sous classe d ensembles flous sur lesquels les extensions induisent un treillis

fuzzy neighborhood convergence spaces we introduce a supercategory of the category nfs of fuzzy neighborhood spaces this supercategory is defined by means of local axioms similar to those of fns but the objects can also nicely be characterized internally via a notion of convergence in future work it shall be shown that this new category allows for a good treatment of compactifications

the hyperspace of fuzzy sets a natural nontopological fuzzy topological space let x be a uniform topological space then on the family ix resp d x of all nonzero functions resp nonzero uppersemicontinuous functions from x to the unit interval i a fuzzy uniform topology is constructed such that 2x resp 61 x the family of all nonvoid resp nonvoid closed subsets of x equipped with the hausdorff bourbaki structure is isomorphically injected in ix resp i x the main result of this paper is a complete description of convergence in ix by means of a notion of degree of incidence of members of ix immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications and that second it follows from its construction and properties that for each ordinary uniform topological space x there exists a natural nontopological fuzzy uniform topology on ix

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