a fuzzy and bipolar approach to preference modeling with application to need and desire fuzziness and bipolarity allow representing human knowledge taking into account the gradual and the dialectic properties of language focusing on the meaning of concepts under this cognitive and linguistic approach we explore preference relations examining their semantic decomposition through fuzzy preference structures and the specification of meaningful opposites in particular we introduce the preference aversion p a model which allows analyzing under an independent aggregation methodology the possible gains and losses like pros and cons towards a given set of alternatives as an attractive feature of this proposal we show that the p a model allows distinguishing between need and desire contrary to common preference models where both notions are indistinguishable

an axiomatic approach to fuzzy preference modelling abstract we propose an axiomatic approach to the definition of fuzzy strict preference p indifference i and incomparability j associated with a fuzzy preference relation r expression for p i and j are given via solving two systems of functional equations

coherent continuous systems and the generalized functional equation of associativity a theory of coherent continuous systems is outlined systems are to be decomposed hierarchically into modules for this purpose some results on the structure of continuous functions are adapted to generalize the classic 3 modules theorem of binary systems consideration of the modular structure of coherent continuous systems leads naturally to the generalized functional equation of associativity ggu v w hu hv w involving four unknown 2 place continuous functions the functional equation is solved under some regularity conditions motivated by the properties of coherent continuous systems the solutions obtained give rise to generalized notions of parallel and series system structures

another paraconsistent algebraic semantics for lukasiewicz pavelka logic abstract as recently proved in a previous work of turunen tsoukias and ozturk starting from an evidence pair a b on the real unit square and associated with a propositional statement we can construct evidence matrices expressed in terms of four values t f k u that respectively represent the logical valuations true false contradiction both true and false and unknown neither true nor false regarding the statement the components of the evidence pair a b are to be understood as evidence for and against respectively moreover the set of all evidence matrices can be equipped with an injective mv algebra structure thus the set of evidence matrices can play the role of truth values of a lukasiewicz pavelka fuzzy logic a rich and applicable mathematical foundation for fuzzy reasoning and in such a way that the obtained new logic is paraconsistent in this paper we show that a similar result can be also obtained when the evidence pair a b is given on the real unit triangle since the real unit triangle does not admit a natural mv structure we introduce some mathematical results to show how this shortcoming can be overcome and another injective mv algebra structure in the corresponding set of evidence matrices is obtained also we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives

on numerical improvement of gauss legendre quadrature rules among all integration rules with n points it is well known that n point gauss legendre quadrature email protected 1 1f x email protected i 1 nw if x i has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n 1 where nodes x i are zeros of legendre polynomial p n x and w i s are corresponding weights in this paper we are going to estimate numerical values of nodes x i and weights w i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance e 0 say e 0 10 8 for monomial functionsf x x j j 0 1 2n 1 two monomials more than precision degree of gauss legendre quadrature rules we also consider some conditions under which the new rules act numerically more accurate than the corresponding gauss legendre rules some examples are given to show the numerical superiority of presented rules

on numerical improvement of gauss lobatto quadrature rules it is well known that gauss lobatto quadrature email protected 11f x email protected i 1nw if x i pf 1 qf 1 is exact for polynomials of degree at most 2n 1 in this paper we are going to find a formula which is approximately exact for monomials x j j 0 1 2n 3 instead of being analytically exact for monomials x j j 0 1 2n 1 we also consider a class of functions for which the new formula produces better results

valued preference structures abstract an approach to the axiomatics of valued preference modelling is suggested in the framework of this approach general definitions of strict preference indifference and incomparability relations associated with a valued preference relation are established via solving a system of functional equations any solution of this system satisfies most of the classical properties characterizations of important particular solutions are also proved

symmetries of a 2 1 dimensional breaking soliton equation infinitely many symmetries for a 2 1 dimensional breaking soliton equation are constructed via the infinitesimal version of the dressing method these symmetries are proved to constitute an infinite dimensional lie algebra which contains some abelian and virasoro subalgebras the hierarchies of equations generated by these symmetries are also considered these hierarchies of equations are proved to be associated with the isospectral and non isospectral deformations of the akns spectral problem

new families of exact solutions to the integrable dispersive long wave equations in 2 1 dimensional spaces the integrable dispersive long wave equations especially the higher dimensional ones are of current interest in both physics and mathematics obtained in this paper via a symbolic computation based method are new families of exact solutions to the 2 1 dimensional integrable dispersive long wave equations sample solutions from those families are presented solitary waves are merely a special case in one family

exact solutions of some systems of nonlinear partial differential equations using symbolic computation abstract in this paper we present a solution method that utilizes symbolic computation to obtain exact traveling wave solutions of some systems of nonlinear partial differential equations the solution method is demonstrated by obtaining solutions to the variant shallow water wave equations

on numerical improvement of closed newton cotes quadrature rules abstract in this paper we discuss about numerical improvement of the open newton cotes integration rules that are in forms of a x 1 b x n 1 x 1 n 2 h f x d x k 0 n b k n f x 1 k 1 h it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if the integral bounds are considered as two additional variables i e a and h in fact we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n 2 in this way some numerical tests are given to show the numerical superiority of our idea with respect to the usual open newton cotes integration rules

some properties of choice functions based on valued binary relations abstract choice functions based on t norms of valued binary relations are introduced strict preference is also specified with the use of a t norm properties of the choice functions are investigated and rationality conditions are studied some classical particular cases are presented

the semi open newton cotes quadrature rule and its numerical improvement in this paper we discuss about numerical improvement of the semi open newton cotes integration rules that are in forms of a x 1b x n 1 x 1 n 1 hf x email protected k 0nb k n f x 1 k 1 h it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if the integral bounds are considered as two additional variables i e a and h in fact we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n 2 in this way some computational tests are given to show the numerical superiority of our approach with respect to the usual semi open newton cotes integration rules

new exact solutions of the double sine gordon equation using symbolic computations the double sine gordon equation is studied some complex hyperbolic functions are proposed to derive travelling wave solutions based on the symbolic computation program maple many new exact solutions are obtained

fuzzy rationality measures fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives it is therefore natural to extend to fuzzy preferences the notion of rationality or consistency with this goal in mind we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality specifically we establish a collection of conditions that any fuzzy rationality measure must satisfy some examples of fuzzy rationality measures are then given and analyzed

on numerical improvement of the second kind of gauss chebyshev quadrature rules one of the integration methods is the second kind of gauss chebyshev quadrature rule denoted by 11f x 1 x 2dx pn 1 k 1nsin 2k pn 1fcosk pn 1 p2 2 n 1 2n f 2 n h 1 h 1 according to gauss quadrature rules the precision degree of above formula is the highest i e 2n 1 hence it is not possible to increase the precision degree of second kind of gauss chebyshev integration formulas anymore but on the other hand we claim that we can improve the above formula numerically to do this we consider the integral bounds as two unknown variables this causes to numerically be extended the monomial space f x x j from j 0 1 2n 1 to j 0 1 2n 1 this means that we have two monomials more than second kind gauss chebyshev integration method in other words we give an approximate formula as abf x 1 x 2dx i 1nw if x i in which a b and w 1 w 2 w n and x 1 x 2 x n are all unknowns and the formula is almost exact for the monomial basis f x x j j 0 1 2n 1 some important examples are finally given to show the excellent superiority of the proposed nodes and coefficients with respect to the second kind gauss chebyshev nodes and coefficients let us add that in this part we give also some wonderful 2 point 3 point and 4 point formulas that are respectively comparable with 103 point 261 point and 108 point formulas of second kind gauss chebyshev quadrature rules in average

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

strict preference relations based on weak t norms abstract in this paper we introduce and investigate weak t norms and related concepts in order to give a general representation of valued strict preference relations associated with a given valued preference relation all the well known forms of valued strict preference relations are particular cases of our results moreover when some types of rationality assumptions are made our unified approach seems to be the only possible one

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

derivative based closed newton cotes numerical quadrature abstract a new family of numerical integration formula of closed newton cotes type is presented that uses both the function value and the derivative value on uniformly spaced intervals since there are more unknowns when using including derivative values in addition to function values the order of accuracy of these numerical integration formula are higher than the standard closed newton cotes formula these new formula are derived via the method of undetermined coefficients based on the concept of the precision of the quadrature formula the error terms are found in three different ways using the concept of precision using taylor series expansions about the interval midpoint and using polynomial approximating functions which is how the error terms for the standard closed newton cotes formula were obtained the concept of precision and the taylor series methods yield the same error terms as the polynomial based method but there are certain unverifiable assumptions in their use quadrature formula using first derivatives at all points throughout the interval increase the order of accuracy to 2 n 2 quadrature formula using the first derivatives at the endpoints of the interval obtain an increase of two orders of accuracy over the closed newton cotes formula while quadrature formula involving higher order derivatives result in substantially higher orders of accuracy being d 1 n 1 where d is the number of derivatives involved in the formula the computational cost for these methods are analyzed for two different examples showing the number of function and derivative evaluations necessary to reduce the error below a certain level these two numerical examples are used to demonstrate that the theoretical order of accuracy is achieved by the numerical implementation of these formula additionally a generalized rolle s theorem with derivatives and a theorem for the error in the general interpolating polynomial including derivatives are provided in the appendix which leads to the proof of some of the error terms in the derivative based closed newton cotes type quadrature formula

exact solutions of some nonlinear evolution equations using symbolic computations abstract in this paper we present a solution methodology that utilizes symbolic computations to obtain analytic solutions of some nonlinear evolution equations by balancing the nonlinear and the dispersive effects the solution method is demonstrated by obtaining solutions to burgers equation the nonlinear heat equation the modified kdv equation and the kuramoto sivashinsky equation

using symbolic computation to exactly solve for the bogoyavlenskii s generalized breaking soliton equation abstract in this paper the symbolic computation systems help us find some exact solutions for the bogoyavlenskii s generalized breaking soliton equation which are the sums of the arbitrary functions of time and parameterized solitary waves

on fuzzy strict preference indifference and incomparability relations abstract the existence of fuzzy models for strict preference indifference and incomparability satisfying all classical properties is established a particular class of such models is completely described

the second kind chebyshev quadrature rules of semi open type and its numerical improvement abstract one of the less known integration methods is the weighted newton cotes quadrature rule of semi open type which is shown by a x 0 b x n 1 x 0 n 1 h f x w x d x k 0 n w k f x 0 kh where w x is a weight function on a b and h b a n 1 is the step size there are various cases for the weight function w x that one can use here we consider the weight function w x 1 x 2 which is important in numerical analysis therefore in this paper we would face with the following formula 1 1 f x 1 x 2 d x k 0 n w k f 1 2 k n 1 it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s but we consider the upper and lower bounds as two additional variables to reach a nonlinear system that numerically improves the precision degree of above formula up to degree n 2 in this way we also give some examples to show the numerical superiority of our idea

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

the first kind chebyshev newton cotes quadrature rules closed type and its numerical improvement one of the less known integration methods is the weighted newton cotes of closed type quadrature rule which is denoted by a x 0b x n x 0 nhf x w x email protected k 0nw kf x 0 kh where w x is a positive function and h b an is a positive value there are various cases for the weight function w x that one can use because of special importance of the weight function of gauss chebyshev quadrature rules i e w x 11 x 2 in numerical analysis we consider this function as the main weight hence in this paper we face with the following formula in fact 1 1f x 1 x email protected k 0nw kf 1 2kn it is known that the precision degree of above formula is n 1 for even n s and is n for odd n s however if we consider its bounds as two additional variables we reach a nonlinear system that numerically improves the precision degree of above formula up to degree n 2 in this way we give several examples which show the numerical superiority of our approach

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

a general model for deriving preference structures from data in this paper we comment upon the integrated model for valued preferences introduced by fodor ovchinnikov and roubens in particular while on one hand we revise basic assumptions and point out their intuitive meaning on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results but also a functional characterization of the whole family of solutions

a 2 1 dimensional extension for the sine gordon equation starting from the breaking soliton equation the author obtains a new integrable equation in 2 1 dimensions though the equation has no exchange symmetry of the space variables x and y the model reduces back to the known 1 1 dimensional sine gordon or liouville equation

on numerical improvement of gauss radau quadrature rules it is known that gauss radau quadrature rule 11f x dx i 1na if b i pf 1 orqf 1 is exact for polynomials of degree at most 2n in this paper we intend to find a formula which is nearly exact for monomial functions x j j 0 1 2n 2 instead of being analytically exact for the basis space x j j 0 1 2n in this way several examples are also given to show the numerical superiority of the presented rules with respect to usual gauss radau quadrature rules

on numerical improvement of the first kind gauss chebyshev quadrature rules one of the integration methods of the equality type is gauss chebyshev quadrature rule which is in the following form 11f x 1 x 2dx pn k 1nfcos 2k 1 p2n 2 p2 2 n 2n f 2 n h 1 h 1 according to gauss quadrature rules the precision degree of above formula is the highest i e 2n 1 hence it is not possible to increase the precision degree of gauss chebyshev integration formulas anymore in this way we present a matrix proof for this matter but on the other hand we claim that we can improve the above formula numerically to do this we consider the integral bounds as two unknown variables this causes to numerically be extended the monomial space f x x j from j 0 1 2n 1 to j 0 1 2n 1 this means that we have two monomials more than gauss chebyshev integration method in other words we give an approximate formula as abf x 1 x 2dx i 1nw if x i in which a b and w 1 w 2 w n and x 1 x 2 x n are all unknowns and the formula is almost exact for the monomial basis f x x j j 0 1 2n 1 some important examples are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual gauss chebyshev nodes and weights let us add that in this part we have also some wonderful 2 point formulas that are comparable with 71 point formulas of gauss chebyshev quadrature rules in average