embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

some questions about zadeh s extension on metric spaces abstract for a continuous function f on a hausdorff space x we prove that f u f u for each u f x and 0 1 where f is the zadeh s extension of f by means of this result some results on locally compact spaces and the zadeh s extension are generalized given a metric space x d we introduce skorokhod s metric d 0 on the set f x of the family of all upper semicontinuous fuzzy sets u x 0 1 with compact support and such that u 1 1 is non empty we show that if f x d x d is a continuous function then its zadeh s extension f to f x d 0 is also continuous and that x d is separable if and only if f x d 0 is separable we also present a fuzzy version of the so called hutchinson operator a valuable tool in fractal theory

weak laws of large numbers for fuzzy random variables abstract in this paper we establish weak laws of large numbers for sums of convex compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in rp as a result weak laws of large numbers for sums of convexly tight fuzzy random variables are given

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

topological properties on the space of fuzzy sets abstract in this paper we introduce a new metric on the space f rp of fuzzy sets and prove that f rp is separable and complete also a characterization of compact subsets of f rp is presented and some examples are given

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

structure selection for bounded parameter models consistency conditions and selection criterion strong consistency conditions for structure selection in bounded parameter models are studied a certain robust selection criterion based on the volume of the exact parameter bounding polytope is proposed for linear regression models the effectiveness of the polytope volume criterion is demonstrated on a model nonlinear in its variables but linear in its parameters its strong consistency is proved for a large class of noise distributions the usual assumptions on the noise namely independence constant variance or martingale difference properties need not be made but asymptotic independence is assumed

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

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

a characterization of compact subsets of fuzzy number space abstract in this paper a characterization of compact subsets on the space of fuzzy numbers endowed with the skorokhod topology is presented this characterization is an analogue of the arzela ascoli theorem which characterizes compact subsets of the space c 0 1

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

feasibility of integer knapsacks given a matrix a in mathbb z m times n satisfying certain regularity assumptions we consider the set mathcal f a of all vectors boldsymbol b in mathbb z m such that the associated knapsack polytope p a boldsymbol b boldsymbol x in mathbb r n geq0 a boldsymbol x boldsymbol b contains an integer point when m 1 the set mathcal f a is known to contain all consecutive integers greater than the frobenius number associated with a in this paper we introduce the diagonal frobenius number mathrm g a which reflects in an analogous way feasibility properties of the problem and the structure of mathcal f a in the general case we give an optimal upper bound for mathrm g a and also estimate the asymptotic growth of the diagonal frobenius number on average

compactness and convexity on the space of fuzzy sets abstract in this paper a criterion for which the convex hull of k is relatively compact is given when k is a relatively compact subset of the space f r p of fuzzy sets endowed with the skorokhod topology also some examples are given to illustrate the criterion

strong convergence for weighted sums of fuzzy random sets in this paper we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper semicontinuous fuzzy sets in r p

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

on the structure of reduced kernel lattice bases lattice based reformulation techniques have been used successfully both theoretically and computationally one such reformulation is obtained from the kernel lattice associated with an input matrix some of the hard instances in the literature that have been successfully tackled by lattice based techniques have randomly generated input since the considered instances are very hard even in low dimension less experience is available for larger instances recently we have studied larger instances and observed that the lll reduced basis of the kernel lattice has a specific sparse structure in particular this translates into a map in which some of the original variables get a rich translation into a new variable space whereas some variables are only substituted in the new space if an original variable is important in the sense of branching or cutting planes this variable should be translated in a nontrivial way in this paper we partially explain through a probabilistic analysis the obtained struct

the skorokhod topology on space of fuzzy numbers in this paper we introduce the skorokhod metric on the space f r of fuzzy numbers and prove that f r is separable and complete

continuity of functions defined on the space of fuzzy sets in this paper we investigate the continuity of many useful functions defined on the space f rp of normal and upper semicontinuous fuzzy sets in rp with compact support

set membership binormalized data reusing lms algorithms this paper presents and analyzes novel data selective normalized adaptive filtering algorithms with two data reuses the algorithms the set membership binormalized lms sm bn drlms algorithms are derived using the concept of set membership filtering smf these algorithms can be regarded as generalizations of the previously proposed set membership nlms sm nlms algorithm they include two constraint sets in order to construct a space of feasible solutions for the coefficient updates the algorithms include data dependent step sizes that provide fast convergence and low excess mean squared error mse convergence analyzes in the mean squared sense are presented and closed form expressions are given for both white and colored input signals simulation results show good performance of the algorithms in terms of convergence speed final misadjustment and reduced computational complexity

compactness and convexity on the space of fuzzy sets ii abstract in this paper we establish some characterizations of convex and relatively compact subsets of the space f r p of normal upper semicontinuous and compactly supported fuzzy sets in rp equipped with the skorokhod metric

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

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

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

integer knapsacks average behavior of the frobenius numbers the largest integer that cannot be represented as a nonnegative integral combination of given set of positive integers is called the frobenius number of these integers we show that the asymptotic growth of the frobenius number on average is significantly slower than the growth of the maximum frobenius number

l metric on the space of l fuzzy numbers abstract in this article a natural l topology is constructed on the set of l fuzzy numbers which is called the standard l topology it is proved that the standard l topology can be induced by an l metric the space of trapezoidal fuzzy numbers can be regarded as a subspace of the space of l fuzzy numbers an l metric on the set of trapezoidal fuzzy numbers is presented

multiweight optimization in optimal bounding ellipsoid algorithms optimal bounding ellipsoid obe algorithms offer an attractive alternative to traditional least squares methods for identification and filtering problems involving affine in parameters signal and system models the benefits including low computational efficiency superior tracking ability and selective updating that permits processor multi tasking are enhanced by multiweight mw optimization in which the data history is considered in determining update times and optimal weights on the observations mw optimization for obe algorithms is introduced and an example mw obe algorithm implementation is developed around the recent quasi obe algorithm optimality of the solution is discussed and simulation studies are used to illustrate performance benefits

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

an embedding theorem for convex fuzzy sets in this paper we embed the space of upper semicontinuous convex fuzzy sets on a banach space into a space of continuous functions on a compact space the following structures are preserved by the embedding convex cone metric sup semilattice the indicator function of the unit ball is mapped to the constant function 1 two applications are presented strong laws of large numbers for fuzzy random variables and korovkin type approximation theorems

some properties of a new metric on the space of fuzzy numbers abstract this paper deals with some properties of a new metric here called the hausdorff skorokhod metric previously introduced by joo and kim on the space of fuzzy numbers in the euclidean space r k we first prove that convergence in the hausdorff skorokhod metric implies convergence in the sendograph metric also we show that the hausdorff skorokhod metric is not absolutely homogeneous and not translation invariant and investigate the continuity of operations and convexity on the space of fuzzy numbers equipped with the hausdorff skorokhod metric

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

hard equality constrained integer knapsacks we consider the following integer feasibility problem given positive integer numbersa0 a1 a n with gcd a1 a n 1 anda a1 a n does there exist a vectorx z n 0satisfyinga x a0 we prove that if the coefficientsa1 a nhave a certain decomposable structure then the frobenius number associated witha1 a n i e the largest value ofa0 for whicha x a0 does not have a nonnegative integer solution is close to a known upper bound in the instances we consider we takea0 to be the frobenius number furthermore we show that the decomposable structure ofa1 a nmakes the solution of a lattice reformulation of our problem almost trivial since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small for branch and bound such instances are difficult to solve since they are infeasible and have large values ofa0 a i 1 i n we illustrate our results by some computational examples

supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings this paper generalizes a classical result about the space of bounded closed sets with the hausdorff metric and establishes the completeness of cb x with respect to the completeness of the metric space x where cb x is the class of fuzzy sets with nonempty bounded closed a cut sets equipped with the supremum metric d which takes the supremum on the hausdorff distances between the corresponding a cut sets in addition some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory

on equivalence of major relaxation methods for minimum ellipsoid covering intersection of ellipsoids abstract this paper investigates the problem on the minimum trace or volume ellipsoid covering intersection of ellipsoids this problem can be solved by three major relaxation methods involving semi definite programming relaxation s procedure relaxation and parameterized bounding ellipsoid relaxation they are proposed from the different ideas or viewpoints so that it is difficult to judge which relaxation method is the most suitable for the different practical implementations to the best of our knowledge the problem of proving the convexity of the optimization problem derived by the third relaxation method remains open by rigorous analysis we reveal the equivalence among the three relaxation methods and address the open problem i e the optimization problem derived by the third relaxation method can be equivalently transformed to a convex optimization problem the computation complexity of these relaxation methods are also discussed and the equivalent result is helpful to set membership filter and multisensor fusion obtaining a tighter ellipsoid to contain the state with lower computation complexity finally the performance of each method is evaluated by several typical numerical examples in information fusion and filtering

the asymptotic distribution of frobenius numbers the frobenius number f a of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a we show that if a is taken to be random in an expanding d dimensional domain then f a has a limit distribution which is given by the probability distribution for the covering radius of a certain simplex with respect to a d 1 dimensional random lattice this result extends recent studies for d 3 by arnold bourgain sinai and shur sinai ustinov the key features of our approach are a a novel interpretation of the frobenius number in terms of the dynamics of a certain group action on the space of d dimensional lattices and b an equidistribution theorem for a multidimensional farey sequence on closed horospheres

testing integer knapsacks for feasibility abstract we present a new approach for determining whether there exist nonnegative integers x1 x2 xn satisfying a1x1 a2x2 anxn b where a1

asymptotically convergent modified recursive least squares with data dependent updating and forgetting factor for systems with bounded noise continual updating of estimates required by most recursive estimation schemes often involves redundant usage of information and may result in system instabilities in the presence of bounded output disturbances an algorithm which eliminates these difficulties is investigated based on a set theoretic assumption the algorithm yields modified least squares estimates with a forgetting factor it updates the estimates selectively depending on whether the observed data contain sufficient information the information evaluation required at each step involves very simple computations in addition the parameter estimates are shown to converge asymptotically at an exponential rate to a region around the true parameter

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space