Non-standard cut classification of fuzzy sets

Several important non-standard cut sets of lattice-valued fuzzy sets are investigated. These are strong cuts, “not less” and “neither less nor equal” cuts. In each case it is proved that collection of all cuts of any lattice-valued fuzzy set form a complete lattice under inclusion. Decomposition theorem (representation by cuts) is proved for “neither less nor equal” cuts. Necessary and sufficient conditions under which two lattice-valued fuzzy sets with the same domain have equal families of corresponding cut sets are given.     

Regular ordered semigroups in terms of fuzzy subsets

Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S → [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids-semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128-132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231-236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13-28] where S has been endowed with the structure of an ordered semigroup and defined “fuzzy” analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of “fuzzy” ordered semigroups. Some of our results are true for ordered groupoids in general.     

The pasting constructions of lattice ordered effect algebras

The aim of this paper is to present some ways of constructing a lattice ordered effect algebra from the given family of MV-algebras. The loop lemma of lattice ordered effect algebras is proved. Then the definitions of Greechie diagrams of effect algebras are given. All these results have generalized those of orthomodular lattices. As applications of loop lemma and Greechie diagrams, some lattice ordered effect algebras without states are presented.     

Lattice-valued fuzzy frames

In this paper we aim to introduce the concept of lattice-valued fuzzy frame or L-fuzzy frame, related to traditional frames analogously to how L-fuzzy (resp. L-fuzzifying) topological spaces related to L-topological spaces (resp. traditional topological spaces). Some properties of L-fuzzy frames are introduced.     

Dynamics of fisheries with prey reserve and harvesting

A model for two fish species and one predator in a patchy environment is formulated using a deterministic model to study the dynamics of fishery in two homogeneous patches, a free fishing zone and a refuge for prey reserve in which fishing is prohibited. The system is analysed around steady states; the criteria for local and global stabilities are established. The existence of bionomic equilibrium of the system is determined and the conditions for their existence are derived. The optimal harvesting policy is studied by using Pontryagin's maximal principle. Sensitivity analysis is carried out and it is observed that the populations are more sensitive to growth, dispersal and predation rates, least sensitive to the catchability coefficient. Statistical analysis is employed to estimate the parameters and to assess both the uncertainty in the model parameters and in the model-based predictions. Graphical representations of the model are provided.     

On the ideal convergence of sequences of fuzzy numbers

In this paper we introduce and study the concepts of I-convergence, I^∗-convergence and I-Cauchy sequence for sequences of fuzzy numbers where I denotes the ideal of subsets of N, the set of positive integers.     

Computing Interval Enclosures for Definite Integrals by Application of Triple Adaptive Strategies

In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered. Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation. In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons of different methods are given.     

Rates of convergence of extremes for mixed exponential distributions

The limiting distributions of the extremes of mixed exponential distributions and the associated rates of convergence are derived. The practical values of the results are illustrated by a numerical study.     

Compositions of invariant fuzzy implications

We examine sup-min compositions in a finite family of fuzzy implications. Since the composition of invariant fuzzy implications is an invariant function, then we get a kind of `multiplication table' for such implications. A multistage proof of such table is presented. As a result we obtain examples of finite semigroups of fuzzy implications.     

On a family of copulas constructed from the diagonal section

We characterize the class of copulas that can be constructed from the diagonal section by means of the functional equation C(x,y)+|x−y|=C(x∨y,x∨y), for all (x,y) in the unit square such that C(x,y)>0. Some statistical properties of this class are given.     

Notes on redefined fuzzy implicative filters of lattice implication algebras

The concept of -fuzzy implicative filters of lattice implication algebras is introduced and some of its related properties are investigated. In particular, the relationships among the ordinary fuzzy implicative filters, the (∈,∈∨q)-fuzzy implicative filters and the -fuzzy implicative filters of lattice implication algebras will be studied.     

Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

A sequence ω is Martin-Löf random if and only if it appears early in every Lebesgue measure one set of computably enumerable intervals.     

λ-Statistical limit points of the sequences of fuzzy numbers

Aytar has introduced the concepts of statistical limit and cluster points of a sequence of fuzzy numbers based on the definitions given in Fridy’s study for sequences of real numbers. In this paper, we define λ-statistical limit and λ-statistical cluster points of sequences of fuzzy numbers and discuss the relations among the sets of ordinary limit points, λ-statistical limit points and λ-statistical cluster points of sequences of fuzzy numbers.     

Smolyak's Construction of Cubature Formulas of Arbitrary Trigonometric Degree

We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree in dimension d, we need about function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree. Received: April 8, 1998     

Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations

Using a fixed point technique, the sequence of successive approximations and a recent quadrature formula for fuzzy-number-valued functions, it is constructed a numerical method for the solution of nonlinear fuzzy Fredholm integral equations. Moreover, the error estimate of the method and a criterion to stop the corresponding algorithm are given.     

On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

On the Existence of Totally Anti-Symmetric Quasigroups of Order 4<Emphasis Type="Italic">k</Emphasis>+2

Totally anti-symmetric quasigroups are employed in check digit systems. We give counterexamples to a conjecture of Ecker and Poch and show that there are totally anti-symmetric quasigroups for all orders n0,1,34, n=4(5k+2)+2, and n=4(7k+3)+2. We prove that finite totally anti-symmetric quasigroups possess a transversal, and we give a useful condition for quasigroups not to have a transversal.     

On the Crossing Number of Complete Graphs

Let (G) denote the rectilinear crossing number of a graph G. We determine (K₁₁)=102 and (K₁₂)=153. Despite the remarkable hunt for crossing numbers of the complete graph K_n – initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11. Based on these findings, we establish a new upper bound on (K_n) for general n. The bound stems from a novel construction of drawings of K_n with few crossings.