Twofold fuzzy sets and rough sets—Some issues in knowledge representation

This paper deals with the representation of sets where the membership of some elements may be ill-known rather than just a matter of degree as in a fuzzy set. The notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete. A twofold fuzzy set is made of a nested pair of fuzzy sets: the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong. Twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by Pawlak. Set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined. Twofold fuzzy relations are also introduced. Finally, various applications of twofold fuzzy sets in knowledge representation are briefly discussed.     

L—fuzzy Ring and L—fuzzy Module

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Entropy and similarity measure for Atannasov’s interval-valued intuitionistic fuzzy sets and their application

In this study, we first point out the problem of the similarity measure in the literature and then define a new entropy and similarity measure. In order to explore the inter-dependent or interactive characteristics between elements in a set, several Shapley-weighted similarity measures of Atannasov’s interval-valued intuitionistic fuzzy sets are defined by using the well-known Shapley function, which can be seen as an extension of the associated weighted similarity measures. Moreover, if the information about the weights is completely unknown or partially known, models for the optimal fuzzy measures are established, by which the optimal weight vector can be obtained. Finally, an approach to pattern recognition and multi-criteria decision making is developed, and the associated numerical examples are provided to verify the developed methods and demonstrate their practicality and feasibility.     

Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.     

Capturing k-ary existential second order logic with k-ary inclusion–exclusion logic

In this paper we analyze k-ary inclusion–exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k].We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.     

Structure-dependent improved Wilson-θ method with higher order of accuracy and controllable amplitude decay

The Wilson-θ method has been proven to have unconditional stability as a numerical direct time integration method in structural dynamics when θ ≥ 1.37 is adopted. Notwithstanding this great advantage of the method, it has also been proven that the method suffers from two shortcomings: high amount of uncontrollable amplitude decay and period elongation. In other words, the unconditional stability allows time step to be stretched, but as the time step grows longer, amplitude decay and period elongation errors grow higher, resulting in a low level of accuracy. The improved version of the Wilson-θ method negates the disadvantages of the classic method by raising the order of acceleration to vary in quadratic scheme over time step domain and by introducing an accelerator coefficient to the acceleration formula in order to control the amount of amplitude decay and lessen the period elongation error. The stability and accuracy of the proposed method has been analyzed, and the results show that unconditional stability is obtained if θ ≥ 1.38 is adopted. A formula is derived for the accelerator coefficient to make it applicable to various types of structural dynamic problems. Numerical examples are presented to provide a practical assessment of the method, along with the classic Wilson-θ and other methods of similar class.     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

A nonautonomous Beverton–Holt equation of higher order

In this paper, we discuss a certain nonautonomous Beverton–Holt equation of higher order. After a brief introduction to the classical Beverton–Holt equation and recent results, we solve the higher-order Beverton–Holt equation by rewriting the recurrence relation as a difference system of order one. In this process, we examine the existence and uniqueness of a periodic solution and its global attractivity. We continue our analysis by studying the corresponding second Cushing–Henson conjecture, i.e., by relating the average of the unique periodic solution to the average of the carrying capacity.     

Hermite-Fejér interpolation of higher order with varying weights

The authors obtain precise estimations for the coefficients of Hermite-Fejér interpolation of higher order based on Generalized Jacobi zeros.     

A remark on the extended Hermite—Fejér type interpolation of higher order

This paper is devoted to study the interpolation of higher order by the nodes $$\left\{ { \pm 1} \right\} \cup \left\{ {\cos \frac{{(2k - 1)\pi }}{{2n}}} \right\}_{k = 1}^n ,n = 1,2,...\,.$$      

On*-operfect sets and α-*-closed sets

The main purpose of this paper is to introduce *-operfect, τ*-clopen, α-*-closed, strongly α-*-closed and pre-*-closed sets. We compare them and obtain a diagram to show their relationships among these sets and related sets.     

On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price. Discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities. In pricing barrier options, certain aspects are triggered if the asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied. Moreover, these unwanted oscillations become worse when estimating the hedging parameters, e.g., Delta and Gamma. We have an improved smoothing strategy for the Crank–Nicolson method which is unique in achieving optimal order convergence for barrier option problems. Numerical experiments are discussed for one asset and two asset problems. Time evolution graphs are obtained for one asset problems to show how option prices change with respect to time. This smoothing strategy is then extended to higher order methods using diagonal (m,m)$(m,m)$—Padé main schemes under a smoothing strategy of using as damping schemes the (0,2m-1)$(0,2m-1)$ subdiagonal Padé schemes.