The systems approach and fuzzy set theory bridging the gap between mathematical and language-oriented economists

Ideas are presented to show how fuzzy mathematics can be applied in macro-economics in combination with the systems approach in order to bridge the gap between mathematical and language-oriented economists.Two reasons are given.From a mathematical point of view, fuzzy sets, fuzzy relations and fuzzy logic are not fuzzy at all. They are all well defined, but tend to be more complicated than traditional, mathematical concepts used in economics.From a language-oriented economist's point of view, fuzzy sets, etc. are used to express mathematically the type of concepts which are typical in language and most valuable in dealing with complex systems like an economy.The paper deals with economics in general terms, but examples are provided to illustrate the ideas.     

The personalization of security selection: An application of fuzzy set theory

We apply the theory of fuzzy subsets to the multiple objective decision problem of stock selection. We allow our objectives to have varying degrees of importance. We discuss various criteria used in selecting stocks. We indicate some procedures for subjectively evaluating the membership functions associated with these criteria.     

Successive Proportional Additive Numeration Using Fuzzy Linguistic Labels (Fuzzy Linguistic SPAN)

The SPAN (Successive Proportional Additive Numeration or Social Participatory Allocation Network) is a procedure that converts individual judgments into a group decision. The procedure is based on a voting design by which individual experts allocate their votes iteratively between their preferred options and other experts. The process ends when all the votes are allocated to options, and the one with the highest number of votes is selected. The method requires the experts to specify an exact allocation of votes to both options and other experts. The Fuzzy Linguistic SPAN allows experts to allocate their votes using linguistic labels such as “most of” or “a few”, and determine the preferred option. This method is demonstrated using the Max–Min aggregation function used to develop a proportional representation of the option and member voting schemes. The method is also demonstrated using the LOWA aggregation function. The Fuzzy Linguistic SPAN method is beneficial since the linguistic voting process is easier for the experts and significantly reduces the computational process compared to the traditional SPAN. The paper presents the method and two examples with comparisons to the numerical SPAN method.     

Even-tempered Slater-type orbitals revisited: from hydrogen to krypton

Even-tempered Slater-type orbital basis sets were developed in 1973, based on total atomic energy optimization. Here, we revisit ET STOs and propose new sets based on past experience and recent computational studies. From preliminary atomic and molecular tests, these sets are shown to be very well balanced and to perform, at lower cost, almost as well as a very large (close to complete) basis set.     

Remarks on the formula for γ-interior

The paper presents a formula for the γ-interior of a set under special conditions for , more general than those in the previous paper [Acta Math. Hungar. 80 (1998) 89-93]. There are also some applications.     

On the Green Function for a Complement of a Finite Number of Real Intervals

We give a new proof (and a generalization) of recent remarkable result by V. Totik concerning the smoothness properties of the equilibrium measure for compact subsets of the real line and the Green function for complement of these subsets taken with respect to the complex plane.     

Schopenhauer's Prolegomenon to Fuzziness

Prolegomenon means something said in advance of something else. In this study, we posit that part of the work by Arthur Schopenhauer (1788–1860) can be thought of as a prolegomenon to the existing concept of fuzziness. His epistemic framework offers a comprehensive and surprisingly modern framework to study individual decision making and suggests a bridgeway from the Kantian program into the concept of fuzziness, which may have had its second prolegomenon in the work by Frege, Russell, Wittgenstein, Peirce and Black. In this context, Zadeh's seminal contribution can be regarded as the logical consequence of the Kant-Schopenhauer representation framework.     

Immersions with fractal set of points of zero Gauss-Kronecker curvature

We construct, for any good Cantor set F of S ^n-1, an immersion of the sphere S ⁿ with set of points of zero Gauss-Kronecker curvature equal to F × D ¹, where D ¹ is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do Carmo-Elbert and Barbosa-Fukuoka-Mercuri.To professor João Lucas Barbosa, in occasion of his 60th birthday. Supported by CNPq/Brazil. Supported by Faperj/Brazil.     

Looking for Difference Sets in Groups with Dihedral Images

We prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p with p an odd prime, contains a nontrivial (v, k, ) difference set D with order n = k – prime to p and self-conjugate modulo p. If G has an image of order p, then 0 2a + 2 for a unique choice of = ±1, and for a = (k – )/2p. If G has an image of order 2p, then and ( – 1)/( – 1). There are further constraints on n, a and . We give examples in which these theorems imply no difference set can exist in a group of a specified order, including filling in some entries in Smith's extension to nonabelian groups of Lander's tables. A similar theorem covers the case when p|n. Finally, we show that if G contains a nontrivial (v, k, ) difference set D and has a dihedral image D _2m with either (n, m) = 1 or m = p ^t for p an odd prime dividing n, then one of the C ₂ intersection numbers of D is divisible by m. Again, this gives some non-existence results.