inequalities and bounds for generalized complete elliptic integrals computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained also bounds for some combinations and products for integrals under discussion are established it has been proven that both families of integrals are logarithmically convex as functions of the first parameter this property has been employed to obtain several inequalities involving integrals in question

nonlinearity of the fuzzy integral we prove that the fuzzy integral has some linearity properties only for small classes of fuzzy measures we also show that an extension of this concept to bounded functions as proposed by kandel 2 5 is not well defined in general

monotonicity properties of generalized elliptic integrals with respect to the parameter abstract for a 0 1 2 and r 0 1 let k a r and e a r k r and e r denote the generalized elliptic integrals the complete elliptic integrals respectively of the first and second kinds respectively in this paper the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a 0 1 2 by these results sharp bounds expressed in terms of k r e r and elementary functions are obtained for k a r and e a r thus showing the dependence on the parameter a of k a r and e a r and substantially improving the known related results

inequalities for the generalized elliptic integrals and modular functions abstract in this paper the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions such as m a r and a r making use of these results the authors obtain some sharp inequalities for the so called ramanujan s generalized modular functions

generalized elliptic integrals and the legendre m function abstract we study monotonicity and convexity properties of functions arising in the theory of elliptic integrals and in particular in the case of a schwarz christoffel conformal mapping from a half plane to a trapezoid we obtain sharp monotonicity and convexity results for combinations of these functions as well as functional inequalities and a linearization property

analytic bethe ansatz and functional equations associated with any simple root systems of the lie superalgebra sl r 1 s 1 abstract the lie superalgebra sl r 1 s 1 admits several inequivalent choices of simple root systems we have carried out analytic bethe ansatz for any simple root systems of sl r 1 s 1 we present transfer matrix eigenvalue formulae in dressed vacuum form which are expressed as the young supertableaux with some semistandard like conditions these formulae have determinant expressions which can be viewed as quantum analogue of jacobi trudi and giambelli formulae for sl r 1 s 1 we also propose a class of transfer matrix functional relations which is specialization of hirota bilinear difference equation using the particle hole transformation relations among the bethe ansatz equations for various kinds of simple root systems are discussed

precise and fast computation of the general complete elliptic integral of the second kind we developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind b m and d m by using their taylor series expansions the definition of jacobi s nome and legendre s relation the developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1 3 machine epsilons and it runs drastically faster around 5 times faster than bulirsch s cel2 and 16 times faster than carlson s rf and rd

generating fuzzy measures fuzzy measure and integration theory was introduced by sugeno 131 in order to deal with events affected by nonstatistical uncertainty further developments were concerned both with the analytical treatment of the concept of fuzzy measure and its links with the classical measure theory see e g 3 5 8 9 11 14 and the determination of a unique framework into which fuzziness and randomness can be set see 7 the main feature of the fuzzy measure in comparison with the classical one is the loss of additivity conditions while continuity and monotonicity axioms are assumed our present aim is to exhibit some constructive examples of fuzzy measures in terms of compositions of o additive measures and suitable real valued functions furthermore we show some properties that are peculiar to the related class of the fuzzy measures above

toward a general theory of fuzzy variables abstract a general framework for a theory is presented that encompasses both statistical uncertainty which falls within the province of probability theory and nonstatistical uncertainty which relates to the concept of a fuzzy set and possibility theory l a zadeh j fuzzy sets 1 1978 3 28 the concept of a fuzzy integral is used to define the expected value of a random variable properties of the fuzzy expectation are stated and a mean value theorem for the fuzzy integral is proved comparisons between the fuzzy and the lebesgue integral are presented after a new concept of dependence is formulated various convergence concepts are defined and their relationships are studied by using a chebyshev like inequality for the fuzzy integral the possibility of using this theory in bayesian estimation with fuzzy prior information is explored

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

asymptotical bounds for complete elliptic integrals of the second kind abstract in this paper we establish several asymptotical bounds for the complete elliptic integrals of the second kind e r and improve the well known conjecture e r 1 1 r 2 3 4 2 2 3 2 for all r 0 1 proposed by m vuorinen

lattice valued fuzzy measure and lattice valued fuzzy integral abstract in this paper 1 the concepts of lattice valued fuzzy measure with no valuation property and lower resp upper lattice valued fuzzy integral are proposed which give the unified discription to the fuzzy measures and fuzzy integrals studied by delgado and moral qiao ralescu adams sugeno wang and zhang 2 some asymptotic structural characteristics of lattice valued fuzzy measures are introduced and some relations between them are given 3 some concepts of convergences for lattice valued functions are defined and riesz theorem egoroff s theorems and lebesgue s theorem for lattice valued measurable function are proved 4 the monotone increasing resp decreasing convergence theorem and almost resp pseudo almost everywhere convergence theorrem for lower resp upper lattice valued fuzzy integral are shown under some weak conditions

generalized hersch pfluger distortion function and complete elliptic integrals abstract in this paper we study the relationship between the generalized hersch pfluger distortion function k a r and complete elliptic integrals of the second kind e r and establish several inequalities for k a r and e r

generalized convexity and inequalities abstract let r 0 and let m be the family of all mean values of two numbers in r some examples are the arithmetic geometric and harmonic means given m 1 m 2 m we say that a function f r r is m 1 m 2 convex if f m 1 x y m 2 f x f y for all x y r the usual convexity is the special case when both mean values are arithmetic means we study the dependence of m 1 m 2 convexity on m 1 and m 2 and give sufficient conditions for m 1 m 2 convexity of functions defined by maclaurin series the criteria involve the maclaurin coefficients our results yield a class of new inequalities for several special functions such as the gaussian hypergeometric function and a generalized bessel function

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

lacunarity of self similar and stochastically self similar sets let k be a self similar set in rdd of hausdorff dimension d and denote by ik c l the d dimensional lebesgue measure of its 6 neighborhood we study the limiting behavior of the quantity e d d ik e i as 6 e 0 it turns out that this quantity does not have a limit in many interesting cases including the usual ternary cantor set and the sierpinski carpet we also study the above asymptotics for stochastically self similar sets the latter results then apply to zero sets of stable bridges which are stochastically self similar in the sense of the present paper and then more generally to level sets of stable processes specifically it follows that if kt is the zero set of a realvalued stable process of index a c 1 2 run up to time t then e siit e i converges to a constant multiple of the local time at 0 simultaneously for all t 0 on a set of probability one the asymptotics for deterministic sets are obtained via the renewal theorem the renewal theorem also yields asymptotics for the mean 1e ik c in the random case while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks

precise estimates for differences of the gaussian hypergeometric function abstract by showing certain combinations of the gaussian hypergeometric functions f a b a b 1 x c and f a b a b 1 x d to be monotone on 0 1 for given a b c d 0 a b and c d the authors study the problem of comparing these two functions they find sup 0 a f a b a b 1 x c f a b a b 1 x d for x 0 1 thus solving an open problem and a recent conjecture

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

representations of the fuzzy integral abstract several representation formulas of the fuzzy integral have been found both for general integrand 2 3 and for integrand having suitable continuity properties 2 in this paper some representations of the fuzzy integral are considered the case of a topological structure added is treated furthermore an expression of the fuzzy integral of a fuzzy variable in terms of a decreasing rearrangement of f is given

fuzzy measures and convergence abstract several theorems have been proved e g by ralescu ralescu and adams puri and ralescu and klement in order to unify the approach to uncertainty of both statistical and fuzzy origin the present paper is motivated by the same instances and deals with some extensions of the measure theoretical bases of fuzzy measures and convergence properties e g a b levi like theorem