A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

The Persistence of Finnish Equity Fund Performance

Tutkimuksen tavoitteena on selvittää, esiintyykö suomeen sijoittavilla osakerahastoilla menestyksen pysyvyyttä. Tutkimusaineisto koostuu kaikista suomalaisista osakerahastoista, jotka toimivat ajanjaksolla 15.1.1998-13.1.2005. Aineisto on vapaa selviytymisvinoumasta. Suorituskyvyn mittareina käytetään CAPM-alfaa sekä kolmi- ja nelifaktori-alfaa. Empiirisessä osassa osakerahastojen menestyksen pysyvyyttä testataan Spearmanin järjestyskorrelaatiotestillä. Evidenssi menestyksen pysyvyydestä jäi vähäiseksi, vaikkakin sitä esiintyi satunnaisesti kaikilla menestysmittareilla joillakin ranking- ja sijoitusperiodin yhdistelmillä. CAPM-alfalla tarkasteltuna tilastollisesti merkitsevää menestyksen pysyvyyttä esiintyi selvästi useammin kuin muilla menestysmittareilla. Tulokset tukevat viimeaikaisia kansainvälisiä tutkimuksia, joiden mukaan menestyksen pysyvyys riippuu usein mittaustavasta. Menestysmittareina käytettyjen regressiomallien merkitsevyystestit osoittavat multifaktorimallien selittävän osakerahastojen tuottoja CAPM:a paremmin. Lisätyt muuttujat parantavat merkittävästi CAPM:n selitysvoimaa.

Distribution development in global business environment

The purpose of this master’s thesis was to analyze a case company’s distribution and its elements. With different distribution structure options a customer can be served with different ways. However, cost elements and service elements create different trade-offs. In this work the case company’s distribution’s present state was analyzed and conclusions were made. Based on these conclusions different alternatives for distribution were created. These different alternatives were analyzed through simulations and with more traditional calculations to compose the total operative costs of each alternative. Also other aspects were considered, such as required organizational changes and the possible better customer service. With the suggested 3-stock distribution structure shorter delivery times and possibly more punctual deliveries for products kept in stock could be achieved with the same operative costs as in current direct delivery model. Also, similar benefits could be achieved with other products with the 3-stock model.