A new replacement warranty policy indexed by the product’s correlated failure and usage time

We study a new warranty policy for non-repairable products which is indexed by two correlated random variables, age and usage and covers all failures in (0, t]. Two different warranty costs for the replacement of the failed product are considered, according to its usage being greater or less than a pre-specified level s > 0. A bivariate probability distribution function is applied to incorporate the correlation effect of the two variables. Analytical expressions of the probability density function of the total warranty cost and its expected value, the probability distribution functions of the number of the failed products with usage greater or less than s and their corresponding expected values and costs are derived. Limit results are also obtained. The results obtained are useful measures in establishing the compensation policy and the evaluation of its performance under the proposed warranty. Illustrative numerical examples of the expected cost for Paulson, Pareto and Beta Stacy bivariate distributions are presented and discussed. In particular for Paulson’s bivariate probability distribution, closed form expressions for the expected costs are obtained.     

T-sum of sigmoid-shaped fuzzy intervals

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh’s extension principle based on a t-norm T. A t-norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation, if this arithmetic operation is closed for this class. It is important to know which t-norms are consistent with particular types of fuzzy intervals. Recently, Dombi and Gy?rbíró [J. Dombi, N. Gy?rbíró, Additions of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems, Fuzzy Sets and Systems 157 (2006) 952-963] proved that addition is closed if the Dombi t-norm is used with sigmoid-shaped fuzzy intervals. In this paper, we define a broader class of sigmoid-shaped fuzzy intervals. Then, we study t-norms that are consistent with these particular types of fuzzy intervals. Dombi and Gy?rbíró’s results are special cases of the results described in this paper.     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards

In this paper, we consider a renewal process in which the inter-arrival times and rewards are characterized as fuzzy variables under t-norm-based fuzzy operations. A T-related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables.     

Strong (n, t, n) verifiable secret sharing scheme

A (t, n) secret sharing divides a secret into n shares in such a way that any t or more than t shares can reconstruct the secret; but fewer than t shares cannot reconstruct the secret. In this paper, we extend the idea of a (t, n) secret sharing scheme and give a formal definition on the (n, t, n) secret sharing scheme based on Pedersen’s (t, n) secret sharing scheme. We will show that the (t, n) verifiable secret sharing (VSS) scheme proposed by Benaloh can only ensure that all shares are t-consistent (i.e. any subset of t shares defines the same secret); but shares may not satisfy the security requirements of a (t, n) secret sharing scheme. Then, we introduce new notions of strong t-consistency and strong VSS. A strong VSS can ensure that (a) all shares are t-consistent, and (b) all shares satisfy the security requirements of a secret sharing scheme. We propose a strong (n, t, n) VSS based on Benaloh’s VSS. We also prove that our proposed (n, t, n) VSS satisfies the definition of a strong VSS.     

(?, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring

In this paper, we consider a ring as a universal set and study (?, T)-fuzzy rough approximation operators with respect to a TL-fuzzy ideal of a ring. First, some new properties of generalized (?, T)-fuzzy rough approximation operators are obtained. Then, a new fuzzy algebraic structure - TL-fuzzy rough ideal is defined and its properties investigated. And finally, the homomorphism of (?, T)-fuzzy rough approximation operators is studied.     

Some remarks on the lattice of fuzzy intervals

In this paper we study the connections between three related concepts which have appeared in the fuzzy literature: fuzzy intervals, fuzzy numbers and fuzzy interval numbers (FIN’s). We show that these three concepts are very closely related. We propose a new definition which encompasses the three previous ones and proceeds to study the properties ensuing from this definition. Given a reference lattice (X, ?), we define fuzzy intervals to be the fuzzy sets such that their p-cuts are closed intervals of (X, ?). We show that, given a complete lattice (X, ?), the collection of its fuzzy intervals is a complete lattice. Furthermore we show that, if (X, ?) is completely distributive, then the lattice of its fuzzy intervals is distributive. Finally we introduce a new inclusion measure, which can be used to quantify the degree in which a fuzzy interval is contained in another, an approach which is particularly valuable in engineering applications.     

Atanassov’s intuitionistic (S, T)-fuzzy n-ary sub-hypergroups and their properties

In this paper, we define a new kind of intuitionistic fuzzy n-ary sub-hypergroups of an n-ary hypergroup. This definition, which is based on Atanassov’s intuitionistic fuzzy sets, t-norms and t-conorms, includes earlier definitions of (n-ary) sub-hypergroups, (intuitionistic) fuzzy (n-ary) sub-hypergroups. Then some related properties are investigated. Also, intuitionistic fuzzy relations with respect to t-norms and t-conorms on n-ary hypergroups are discussed.     

The reduction of binary fuzzy relations and its applications

This paper introduces the concept of T-reduction of binary fuzzy relations and establishes its properties. It presents an approach to providing the one-valued restoration of any boundary transitive and T-asymmetric binary fuzzy relation by its T-reduction. A theoretical basis for this approach is established, showing that T-reduction is the smallest fuzzy relation whose T-transitive closure coincides with the given binary fuzzy relation and where an (n × n) fuzzy relation on a finite universe with cardinality n can be represented by n − 1 values. This approach opens up an original way for constructing the membership functions of certain binary fuzzy relations. Important examples for the application of this concept are given.     

Structural relationships among internal locus of control, institutional support, flow, and learner persistence in cyber universities

Although cyber education has become significant due to the development of appropriate information and communication technology, it has a problem as compared to traditional educational methods’ learning outcomes: a low rate of learner persistence. Learning outcomes in the cyber environment are complex phenomena affected by various factors, which need integrative comprehension regarding how the variables affect learning outcomes, for qualitative improvement. The current study aimed to establish a structural equation model explaining the causal relationships among internal locus of control, institutional support, flow, and learner persistence, and to examine the practical direct and indirect effects among them. Therefore, we conducted surveys on these variables, in order to examine the causal relationships among them. We received responses from 568 students at W Cyber University. The results indicate direct effects between learner’s internal locus of control and flow (ß = .273), between institutional support and flow (ß = .487), between internal locus of control and flow (ß = .333), and between flow and learner persistence (ß = .705). In addition, we found flow has an intermediating effect between internal locus of control and learner persistence and between institutional support and learner persistence. We propose strategies to raise learner persistence by improving the learner’s internal locus of control, institutional support, and flow based on these results.     

Fuzzy radicals and prime fuzzy ideals of ordered semigroups

Let S be an ordered semigroup. A fuzzy subset of S is, by definition, an arbitrary mapping f: S → [0, 1], where [0, 1] is the usual interval of real numbers. Motivated by studying prime fuzzy ideals in rings, semigroups and ordered semigroups, and as a continuation of Kehayopulu and Tsingelis’s works in ordered semigroups in terms of fuzzy subsets, in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup S, and give a characterization of prime fuzzy ideals of an ordered semigroup S. We also introduce the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup S, and establish the relationship between the five classes of ideals. Furthermore, we characterize weakly prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of S by their level ideals. Finally, we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points, and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup S can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it. As an application of the results of this paper, the corresponding results of semigroups (without order) are also obtained.     

Measuring incompatibility between Atanassov’s intuitionistic fuzzy sets

The aim of this paper is to establish an axiomatic definition of incompatibility measure in the framework of Atanassov’s intuitionistic fuzzy sets and use geometrical methods to build some families of such incompatibility measures. First, we construct several functions to measure incompatibility for an intuitionistic t-norm that can be represented by an adequate t-norm and t-conorm. Additionally, we establish some relations between some particular cases of these functions. Similarly, we then obtain incompatibility measures for a family of non-representable intuitionistic t-norms.     

On a new class of fuzzy implications: h-Implications and generalizations

A new class of fuzzy implications called the h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yager’s f- and g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called (h, e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function F : [0, 1]² → [0, 1].     

Satellite-based and mesoscale regression modeling of monthly air and soil temperatures over complex terrain in Turkey

Simple regression algorithms were developed to quantify spatio-temporal dynamics of minimum and maximum air temperatures (T_min and T_max, respectively) and soil temperature for a depth of 0-5 cm (T_soil-5cm) across complex terrain in Turkey using Moderate Resolution Imaging Spectroradiometer (MODIS) data at a 500-m resolution. A total of 762 16-day MODIS composites (127 images × 6 bands) between 2000 and 2005 were averaged over a monthly basis to temporally match monthly T_min, T_max, and T_soil-5cm from 83 meteorological stations. A total of 60 (28 temporally averaged plus 32 time series-based) linear regression models of T_min, T_max, and T_soil-5cm were developed using best subsets procedure as a function of a combination of 12 explanatory variables: six MODIS bands of blue, red, near infrared (NIR), middle infrared (MIR), normalized difference vegetation index (NDVI), and enhanced vegetation index (EVI); four geographical variables of latitude, longitude, altitude, and distance to sea (DtS); and two temporal variables of month, and year. The best multiple linear regression models elucidated 65% (RMSE = 5.9 °C), 65% (RMSE = 5.1 °C), and 57% (RMSE = 6.9 °C) of variations in T_min, T_max, and T_soil-5cm, respectively, under a wide range of T_min (−34 to 25 °C), T_max (0.2-47 °C) and T_soil-5cm (−9 to 40 °C) observed at the 83 stations.     

Recovering the initial state of an infinite-dimensional system using observers

Let A be the generator of a strongly continuous semigroup T on the Hilbert space X, and let C be a linear operator from D(A) to another Hilbert space Y (possibly unbounded with respect to X, not necessarily admissible). We consider the problem of estimating the initial state z₀∈D(A) (with respect to the norm of X) from the output function y(t)=CT_tz₀, given for all t in a bounded interval [0,τ]. We introduce the concepts of estimatability and backward estimatability for (A,C) (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating z₀ from y. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell’s principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require T to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.     

Absorption,dissociation, locus of control and presence in virtual reality

The present study aimed to explore the relationship between a number of psychological variables and a reported sense of presence in immersive virtual reality (IVR). It was hypothesised that participants’ scores on measures of absorption, dissociation, and external locus of control would be positively and significantly correlated with a measure of their sense of presence in IVR. A total of 64 people took part. Significant correlations were found between presence and dissociation (r = 0.403, p < 0.01), and presence and locus of control (r = 0.268, p < 0.05). However, the correlation between presence and absorption was not significant (r = −0.037, p = 0.386). The findings reported here suggest a complex interrelationship of psychological variables in relation to presence in IVR that warrants further research.     

Psychometric properties of the Problematic Internet Use Questionnaire

Several instruments have been designed to measure problems associated with excessive, compulsive, or addictive use of the Internet. One such instrument, the 18-item Problematic Internet Use Questionnaire, was recently published with data supporting a three subscale model (Demetrovics et al., 2008). These researches utilized an online format with a sample taken from the general population of Hungary. We utilized an American college student sample and a paper and pencil format to perform a confirmatory factor analysis of the PIUQ. In addition, we examined the reliability and construct validity of the PIUQ by examining the scales’ relationship with several indices of psychological and physical health. CFA results indicate a barely adequate and not completely problem free three factor model for the PIUQ (χ² = 477.40; root mean square error = .097; comparative fit index = .831; Tucker Lewis coefficient = .804). Cronbach’s α for the total scale was .91 while the Cronbach’s α for each subscale were .81, .77, and .79. Construct validity for the model is demonstrated with significant correlations between the subscales and several indices of psychological and physical health. Suggestions for further research are provided.     

Fuzzy measures for a correlation coefficient of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Recently, the sup-min convolution based on Zadeh’s extension principle has been used by Liu and Kao [Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002) 267-275], to calculate a fuzzy correlation coefficient. They used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient. It is well known that T_W (the weakest t-norm)-based addition and multiplication preserve the shape of L-R fuzzy numbers. In this paper, we consider the computational aspect of the T_W-based extension principle when the principle is applied to a correlation coefficient of L-R fuzzy numbers. We give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.     

On fuzzifications of discrete dynamical systems

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277-283] to generalize Zadeh’s original extension principle.In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems.Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.