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To improve the water solubility of hydroxypropyl-β-cyclodextrin(HP-β-CD, 500–800 mg/mL), a novel hydroxypropyl-β-cyclodextrin polymer (HP-β-CDP) was firstly synthesized and characterized by Fourier Transform Infrared spectroscopy (FT-IR), Nuclear magnetic resonance (1H-NMR), and X-ray diffraction spectroscopy (XRD) comprehensively. The solubility of HP-β-CDP was determined as 40.2 g/mL, which is about 46.6 times more than HP-β-CD. The inclusion complexes were prepared with HP-β-CD and HP-β-CDP by saturated solution method to improve the solubility and in vitro release behavior of insoluble drug AZM. The formulation was optimized by orthogonal design with three factors and three levels L9 (34). Solubility, FT-IR, XRD and scanning electron microscope investigations demonstrated that both HP-β-CD and HP-β-CDP effectively formed the inclusion complex. The resultant HP-β-CDP complex presented an increased AZM solubility (628.5 ± 0.25 μg/mL) than HP-β-CD (311.3 ± 0.26 μg/mL) compared with AZM (34.2 ± 0.30 μg/mL) in water. The in vitro accumulative release studies revealed that more than 90 % of AZM could be released from HP-β-CDP inclusion complex within 1 h, which is far more than free AZM (27.5 %) and HP-β-CD inclusion complex (46.5 %). The studies indicated that HP-β-CDP with better water solubility could improve remarkably the solubility and release properties of AZM than HP-β-CD.

The α-migrative uninorms over a fixed uninorm with different neutral elements are presented. All cases when both uninorms lay in any one of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for some possible combinations. The solutions obtained generalize the results where both uninorms have the same neutral elements.

This paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms. In particular, the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent. Moreover, we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict.

Recently the distributivity equation was discussed in families of certain operations (e.g. triangular norms, conorms, uninorms and nullnorms (or called t-operators)). In this paper we describe the solutions of distributivity for semi t-operators over uninorms. Previous results about distributivity for nullnorms over uninorms can be obtained as a corollary.

Distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives. For this reason, the distributivity property has been extensively studied for several families of operations like triangular norms and conorms, some kinds of uninorms and nullnorms (also called t-operators) and even for some generalizations of them. In this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms, leading to many new solutions.

In this paper, quasi-homogeneous t-norms are generalized and then pseudo-homogeneous t-norms are introduced and characterized. It is shown that only some particular strict t-norms and the minimum t-norm T M are pseudo-homogeneous. The concepts of pseudo-homogeneous t-conorms and proper uninorms are given as well. Different from t-norms, only the maximum t-conorm S M is pseudo-homogeneous and any proper uninorm is not pseudo-homogeneous.

In this paper, the notions of α-migrative uninorms over semi t-operators and α-migrative semi t-operators over uninorms are introduced and investigated. All solutions of the migrativity equations for all possible combinations of semi t-operators and uninorms are characterized.

In this paper, we firstly introduce two new classes of fuzzy implications generated from one-variable functions, called [Formula: see text]- and [Formula: see text]-implications, respectively. Then we give a series of necessary and sufficient conditions that these implications satisfy: left neutrality property, identity principle, ordering principle, law of contraposition, modus ponens and modus tollens, respectively. We also discuss the relations between [Formula: see text]- implication ([Formula: see text]-implications, respectively) and other known classes of fuzzy implications.

An extension of fuzzy implications and coimplications, called fuzzy boundary weak implications (shortly, fuzzy bw-implications), is introduced and discussed in this paper. Firstly, by weakening the boundary conditions of fuzzy implications and coimplications, we introduce the concept of fuzzy bw-implications. And then, we investigate some of their basic properties. Next, the concept of fuzzy pseudo-negations is introduced and the natural pseudo-negations of fuzzy bw-implications are investigated. Finally, the fuzzy bw-implications generated, respectively, by aggregation operators and generator functions are discussed in details. This work is motivated by the fact that in real applications there are used some operators which are not fuzzy implications. We hope that such an extension of fuzzy (co)implications can provide a certain theoretical foundation for the real applications.

Uninorms, as binary operations on the unit interval, have been widely applied in information aggregation. The class of almost equitable uninorms appears when the contradictory information is aggregated. It is proved that among various uninorms of which either underlying t-norm or t-conorm is continuous, only the representable uninorms belong to the class of almost equitable uninorms. As a byproduct, a characterization for the class of representable uninorms is obtained.

Conditional distributivity of nullnorm over uninorm with continuous operators is discussed.The structure of uninorms with idempotent operators is discussed. The problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory. This paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t-norm and t-conorm.

In this paper, two classes of generalized (U, N )−implications derived from commutative semi-uninorms and pseudo- uninorms are investigated, respectively. Then some properties of UI,N operations derived from implications are explored. Last the necessary and sufficient conditions such that a (UCS ,N )−operation (or (UP ,N )−operation) is an implication are given out.

This study achieved a nanocomposite structure of nickel oxide (NiO)/titanium dioxide (TiO2) heterojunction on a TiO2 film surface. The photocatalytic activity of this structure evaluated by decomposing methylene blue (MB) solution was strongly correlated to the conductive behavior of the NiO film. A p-type NiO film of high concentration in contact with the native n-type TiO2 film, which resulted in a strong inner electrical field to effectively separate the photogenerated electron-hole pairs, exhibited a much better photocatalytic activity than the controlled TiO2 film. In addition, the photocatalytic activity of the NiO/TiO2 nanocomposite structure was enhanced as the thickness of the p-NiO film decreased, which was beneficial for the migration of the photogenerated carriers to the structural surface.

The distributivity and conditional distributivity of a uninorm and a continuous t-conorm present an open problem recalled by Klement in the Linz2000 closing session. In 2006, Ruiz and Torrens 14 solved this problem for the most usual known classes of uninorms. Recently, Rak 13 solved the same problem for several classes of semi-uninorms, i.e., a special class of conjunctive semi-uninorms, a special class of disjunctive semi-uninorms and the class of idempotent semi-uninorms. In this work we continue to investigate the same topic as the above by focusing on two other classes of semi-uninorms, i.e., representable semi-uninorms and continuous semi-uninorms. The obtained results are different from the case of representable uninorms and also different from the cases of other known classes of semi-uninorms. Moreover, the dual case of distributivity and conditional distributivity of the above two classes of semi-uninorms over continuous t-norms is also investigated and similar results are obtained.

In this paper we generalize (U,N)-implications. First we introduce (US, N)-operation by replacing the uninorm U with a semi-uninorm US in the definition of a (U, N)-operation. Then the necessary and sufficient conditions that a (US, N)-operation is a implication are given out. Furthermore, we characterize the (US, N)-implications. Last we generalize the definition of rotation invariant and investigate the intersection between the residual implications and (US,N)-implications derived from some semi-uninorms with the generalized rotation invariant.

Recently, ( α , U 0 ) -migrative uninorms U were introduced and discussed, where U and U 0 are two uninorms with neutral element e. In this paper, we will further study ( α , U 2 ) -migrative uninorms U 1 , where U 1 and U 2 are two uninorms with different neutral elements e 1 and e 2 , respectively. All cases when the uninorm U 1 and U 2 are in different families out of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for some possible combinations of U 1 and U 2 .

A new class of fuzzy implications, called ordinal sum implications, is introduced by means of the ordinal sum of a family of given implications, which is similar to the ordinal sum of t -norms (or t -conorms). Basic properties of ordinal sum implications are discussed. It is shown that the ordinal sum implication is really a new class, which is different from the known ( S , N ) -, R -, QL - and Yager’s f - and g -implications.

Pseudo-uninorms are a generalization of uninorms by removing the commutativity from the axioms of the uninorms. In this paper, we further study coimplications generated from pseudo-uninorms on a complete lattice. Firstly, we further discuss the properties of residual coimplications generated from pseudo-uninorms on a complete lattice. Then, we recall the induced operators by coimplications on a complete lattice and give conditions such that they are pseudo-uninorms. Finally, we give out some characterizations of the residual coimplications generated from (right) infinitely ?-distributive disjunctive pseudo-uninorms on a complete lattice.

The aim of this paper is mainly to solve the functional equations of distributivity for 2-uninorms over semi­t-operators. Our investigations are motivated by the couple of distributive logical connectives and their generalizations, one of which covering both uninorms and nullnorms are 2-uninorms. In this work, we discuss all possible cases of the distributivity equations for 2-uninorms over semi-t-operators, and give the sufficient and necessary conditions that 2-uninorms are distributive over semi-t-operators.

Since the introduction of uninorm in 1996, it has been widely used in many fields. In this paper, the class of uninorms of which the underlying operators are involutive and left-continuous is discussed. The structure of these classes of uninorms is described.

Bodjanova, Kalina and Kral’ recently introduced a construction method, called paving, which enables to define a new associative, commutative and increasing operation from a given one and a discrete representable partial operation. As a matter of fact, not every discrete t-norm is representable, i.e. it can not always be generated by some additive generator, and this also holds for t-conorms and uninorms. Inspired by this fact and the method of paving, we construct some new associative, commutative and increasing operations on the unit interval from a t-norm on the unit interval and a discrete t-norm, t-superconorm, t-conorm or uninorm. Because of the duality between t-norms and t-conorms, we also define some operations from a t-conorm and a discrete t-norm, t-subnorm, t-conorm or uninorm.

In this paper, we characterize all weakly smooth uninorms i.e., uninorms with smooth underlying operators defined on a finite chain. It is proved that any such uninorm is determined by three unary functions and vice versa. As a by-product, we obtain the characterizations of smooth t-norms, smooth t-conorms through additive generators and show that on a finite chain, there exists no counterpart of the class of uninorms continuous in ]0, 1[2.

The α-migrativity is an important property of a binary operation, and many authors discussed this property for various operators. In this paper, the migrative property for nullnorms is investigated. All solutions of the migrativity equation for all possible combinations of nullnorms are analyzed and characterized.

This paper shows by an example that Proposition 15(iv)(c) in Mas et al. (2013) about migrative property for uninorms in U"c"o"s","m"i"n is false, and corrects these propositions.

The aim of this paper is to correct and improve some results obtained in the paper “Quantitative logic” [Information Sciences 179 (2009) 226–247].

Uninorms, as binary operations on the unit interval, have been widely applied in fuzzy set theory. In this paper, we study uninorms with nilpotent underlying t-norm and t-conorm. We prove that such a uninorm belongs to [Formula: see text] or [Formula: see text]. Moreover, some construction methods of uninorms from given t-norm and t-conorm are discussed.

In this paper, Yager’s g -implications are generalized, and a new class of implications, called ( g , u ) -implications, is introduced. It is shown that ( g , u ) -implications are not only the generalization of Yager’s g -implications, but also the generalization of ( S , N ) -, R -, QL -implications. Basic properties and characterization of these implications are discussed. Furthermore, the distributive equations of these implications are investigated.

First, we show that the class of left-continuous triangular norms whose associated negations are discontinuous only at one point is not generally continuous except for the trivial case. Second, we find the strongest left-continuous triangular norm with a given associated negation and shows that there is no weakest left-continuous triangular norm with a given associated negation.

A new class of fuzzy implications, called (h,min)-implications, is introduced by means of generalized h-generators. Basic properties of these implications are discussed. It is shown that the (h,min)-implications are really a new class different from the known (S,N)-, R-, QL- and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz., law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are established, under which the corresponding functional equations are satisfied.

By means of the generalized additive generators of representable uninorms, we introduce a kind of fuzzy implications, which are called h-1-implications. These implications are shown to be different from the known (S, N)-, R-, QL-, generalized-h and h-generated, and Yager's f- and g -implications. Generalizations of three classical logic tautologies with implications are investigated. A series of necessary and sufficient conditions are established, under which the corresponding functional equations are satisfied.

Recently, the distributive equations of fuzzy implications based on t -norms or t-conorms have become a focus of research. The solutions to these equations can help people design the structures of fuzzy systems in such a way that the number of rules is largely reduced. This paper studies the distributive functional equation I(x,S1(y,z))=S2(I(x,y),I(x,z)), where S1 and S2 are two continuous t -conorms given as ordinal sums, and I:[0,1]2→ [0,1] is a binary function which is increasing with respect to the second place. If there is no summand of S2 in the interval [I(1,0),I(1,1)], we get its continuous solutions directly. If there are summands of S2 in the interval [I(1,0),I(1,1)], by defining a new concept called feasible correspondence and using this concept, we describe the solvability of the distributive equation above and characterize its general continuous solutions. When I is restricted to fuzzy implications, it is showed that there is no continuous solution to this equation. We characterize its fuzzy implication solutions, which are continuous on (0,1] × [0,1].

This paper investigates the functional equation of distributivity of uninorms over nullnorms. We consider the case where the uninorm is continuous in (0,1)^2 or is representable. It has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds. And then combining the structures of the uninorm and the nullnorm, we give the characterization of the pair of the functions of the uninorm and the nullnorm. Moreover, when the nullnorm is continuous, we obtain the sufficient and necessary conditions under which the distributive equation holds.

Since Yager introduced f-generated implications in 2004, this class of fuzzy implications has been extensively investigated. In this paper, we generalize f-generated implications and get a new class of fuzzy implications called (f,g)-implications, which is different from the usual known classes of fuzzy implications. We discuss the basic algebraic properties of (f,g)-implications and study some classical logic tautologies (i.e., law of importation, contrapositive symmetry and distributivity over t-norms or t-conorms) for (f,g)-implications. Characterization of solutions to the corresponding fuzzy functional equations is obtained.

Uninorms are an important class of aggregation functions in information aggregation. It is well known that there exist many different classes of uninorms in references. In this paper, the relationships among several classes of uninorms are discussed. Moreover, a complete characterization of the class of almost equitable uninorms is presented. As a byproduct, a characterization of the class of representable uninorms is obtained.

In this paper, we investigate the distributive functional equation I(x,S"1(y,z))=S"2(I(x,y),I(x,z)), where I:[0,1]^2->[0,1] is an unknown function, S"2 a continuous Archimedean t-conorm and S"1 a continuous t-conorm given as an ordinal sum. First, based on the special case with one summand in the ordinal sum of S"1, all the sufficient and necessary conditions of solutions to the distributive equation above are given and the characterization of its continuous solutions is derived. It is shown that the distributive equation does not have continuous fuzzy implication solutions. Subsequently, we characterize its non-continuous fuzzy implication solutions. Finally, it is pointed out that the case with finite summands in the ordinal sum of S"1 is equivalent to the one with one summand.

One class of fuzzy implications is introduced by means of the additive generators of continuous Archimedean t-norms. Basic properties of these implications are discussed. These implications are shown to be different from the known (S,N)-, R-, QL-, and Yager's f- and g-implications. Several functional equations with fuzzy implications are investigated. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.

An extension of a uninorm called a semi-uninorm is introduced and discussed in this paper. First, we introduce the concept of semi-uninorms on a complete lattice. Then, we discuss two kinds of residual operators of semi-uninorms and give conditions such that the operators are implications. We also give equivalent conditions for infinitely @?-distributive left- and right-conjunctive semi-uninorms. Furthermore, we define two classes of induced operators by implications on a complete lattice and give conditions such that they are semi-uninorms. We also provide the equivalent conditions for the infinitely @?-distributive implications in their second variables.

This paper investigates an iterative Boolean-like law with fuzzy implications derived from uninorms. More precisely, we characterize the solutions to the functional equation I(x,y)=I(x,I(x,y)) that involve RU-, (U,N)- and QLU-implications generated by the most usual classes of uninorms.

A new class of fuzzy implications, called (g, min )-implications, is introduced by means of the additive generators of continuous Archimedean t-conorms, called g-generators. Basic properties of these implications are discussed. It is shown that the (g, min )-implications are really a new class different from the known ( S , N )-, R -, QL - and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz. law of importation, contraction law and distributivity over triangular norms ( t -norms) and triangular conorms ( t -conorms) are investigated. A series of necessary and sufficient conditions are proposed, under which the corresponding functional equations are satisfied.

Anycast routing can raise the QoS of networks and offers load-balance, it is the leading issue in the domain of computer, electric and communication. With the study of the anycast routing algorithm based on genetic algorithm, we put forth an improved anycast routing algorithm in the thesis. Through a lot simulation experiment on simulating platform, we compare and evaluate the performance discrepancy of the improved algorithm and the old one. The improved anycast-routing algorithm befits large-scale network, can satisfy the demand of the development of the net, It has more competent application merit.

The reverse and α-reverse triple I reasoning methods based on Łukasiewi -cz implication IL are established in new manners which correct the mistakes in the existing literature. Furthermore, the α-reverse triple I reasoning method is extended to a new form, called α(u, v)-reverse triple I reasoning method, which can contain the reverse triple I reasoning method as its particular case. This is another improved point to the existing results.

Two kinds of extensions of triangular norms (t-norms) are proposed, and the relations between these extensions and fuzzy implications are discussed in this paper. First, two classes of pseudo-t-norms (ps-t-norms), called type-A and type-B ps-t-norms, and their respective residual operators are defined. Then, we prove that these residual operators are fuzzy implications and satisfy the left neutral property. For these two classes of pseudo-t-norms, we give a series of equivalent conditions of left-continuity with respect to their first or second variable. By combining the axioms of the two classes of pseudo-t-norms, we simply get the definition of the triangular seminorms. Furthermore, we define two classes of induced operators from fuzzy implications and give the conditions under which they are type-A ps-t-norms, type-B ps-t-norms and t-seminorms. For a fuzzy implication, a series of equivalent conditions of right-continuity with respect to its second variable are established. Finally, another method inducing type-A ps-t-norms, type-B ps-t-norms and t-seminorms by implications is proposed.

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.

The goal of this paper is to extend the scope of previously proposed ''fully implicational restriction methods'' for fuzzy reasoning, and provide a unified form of these methods which can employ diverse implications. By means of the residual implications generated by left-continuous t-norms, we obtain the supremum of triple I (the abbreviation of triple implications) restriction solutions for fuzzy modus ponens and the supremum of multiple I restriction solutions for general fuzzy reasoning respectively. Furthermore, we obtain the infimum of triple I restriction solutions and the supremum of triple I^* restriction solutions for fuzzy modus tolens. Finally, the optimal (i.e., the largest or smallest, depending on context) restriction solutions of both basic and general reasoning problems are derived for four specific residual implications.

The aim of this paper is to correct a result about a class of implications in interval-valued fuzzy set theory introduced in paper-Fuzzy Sets and Systems 153 (2005) 229-248. The properties of the modified implications are also discussed.

Wang and Fu [1] provided the unified forms of the triple I method for fuzzy modus ponens (FMP for short) and fuzzy modus tollens (FMT for short) and found an interesting duality between them. In this paper, we first answer why there exists the duality, and then establish another unified form of the triple I method for FMT under a weaker condition.