Efficiency Analysis and Ranking of DMUs with Fuzzy Data

In this paper, a fuzzy version of CCR model (Charnes, Cooper and Rhodes (1978)) with asymmetrical triangular fuzzy number is presented and a procedure is suggested for its solution. The basic idea is to transform the fuzzy CCR model into a crisp linear programming problem by applying an alternative -cut approach. Thereby, the problem is converted to an interval programming. In this method, instead of comparing the equality (or inequality) of two intervals, a variable is defined in the interval, not only satisfies the set of constraints, but also maximizes the efficiency value. We also propose a ranking method for fuzzy DMUs using presented fuzzy DEA approach. To demonstrate the concept, numerical examples are solved and solutions are compared with Guo and Tanaka (2001).     

On exponents in arithmetical semigroups

Let be the factoring of a natural numbern into primes. In 1969Niven found the average order of the functionsh(n)= min {₁,... _r } andH(n)= max {₁,... _r }. In this note a generalization ofNiven's results is discussed, namely the average order of the corresponding functions in certain arithmetical semigroups.     

A Complete Metric in the Function Space D α[0,∞)

Subspaces D of D[0,1] and of D[0,) are defined. With constructed metrics d , D are shown to be complete separable metric spaces. Conditions for tightness of probability measures in D are given.     

The evolution of a population for a certain viability system

Summary For the Fisher-Wright-Haldane selection model with fitness parametersf _ij =1+ _{i ij} ( _i –1) a complete global analysis is performed.

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Zusammenfassung Für das Fisher-Wright-Haldane-Selektionsmodell mit den Fitneßparameternf _ij =1+ _{i ij} ( _i –1) wird eine vollständige globale Analyse, durchgeführt.

     

Invariants of contact structures and transversally oriented foliations

We exhibit new invariants of the contact structure E(), the contact flow F and the transverse symplectic geometry of a contact manifold (M, ). The invariant of contact structures generalizes to transversally oriented foliations. We focus on the particular cases of orientations of smooth manifolds and transverse orientations of foliations. We define the transverse Calabi invariants and determine their kernels.Supported in part by NSF grants DMS 90-01861 and DMS 94-03196.     

Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let be an absolutely convexly tight Radom semi-stable probability measure onE with index 1<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of -stable probability measures 1<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0<<1 are also discussed.The research of this author was partially supported by AFSOR Grant No. 90-0168The research of this author was supported by KBN Grant, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

On monotone and convex approximation by algebraic polynomials

The following results are obtained: If >0, 2, [3, 4], andf is a nondecreasing (convex) function on [–1, 1] such thatE _n (f) n ^– for any n>, then E _n ⁽¹⁾ (f)Cn ^– (E _n ⁽²⁾ (f)Cn ^–) for n>, where C=C(), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n–1), and E _n ⁽¹⁾ (f) (E _n ⁽²⁾ (f)) are the best monotone and convex approximations, respectively. For =2 ( [3, 4]), this result is not true.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1266–1270, September, 1994.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

A variational approach to some boundary value problems in the half-line

We study the existence of solutions for two kinds of boundary value problem in the interval [0,[. The problems are suggested by models in Mathematical Physics. In the first kind of problem the condition at the left endpoint is u(0) = while in the second kind a homogeneous Neumann condition u = 0 is imposed. In both cases solutions should satisfy u(+) = 0. Our approach is variational, solutions being obtained as minimizers or mountain pass critical points of some functional.Received: October 21, 2003; revised: June 2, 2004Supported by Fundção para a Ciência e a Tecnologia, program POCTI (Portugal/FEDER-EU).