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.

Energiatehokkuuden edistäminen rakentamisen ja asumisen alalla

Suomi haluaa energiatehokkuuden edelläkävijäksi maailmassa, mutta se ei onnistu ilman rakennusalan osallistumista energiatehokkuustalkoisiin. Toistaiseksi energiatehokkuuden kehitys asumisen ja rakentamisen osalta ei ole ollut niin nopeaa kuin muualla Euroopassa. Energiatehokkuustalkoiden takana on huoli ilmastomuutoksen vaikutuksista. Ilmastomuutoksen torjuminen aiheuttaa yhteiskunnalle kustannuksia, mutta pidemmällä aikavälillä se on halvempaa kuin puuttumatta jättäminen. Omakotitalojen suosio lisääntyy koko ajan ja energian kulutus niissä kasvaa. Erityisesti sähkönkäyttö lisääntyy koko ajan erityisesti viihde-elektroniikan suosion myötä. Energiatehokkuuteen yritetään vaikuttaa monenlaisilla ohjauskeinoilla, joita ovat muun muassa taloudelliset ja lainsäädännölliset ohjauskeinot. Lainsäädännön lisäksi yksi tärkeimmistä kannustimista energiatehokkuuteen on öljyn hinta. Rakennusten energiatehokkuudessa kokonaisuus ratkaisee ja erityisesti tiiveys. Julkisen sektorin rooli energiatehokkuuden edistämisessä on merkittävä sekä säädösten laatijana että esimerkin näyttäjänä. Vastuuta ilmastotalkoista halutaan jakaa myös kunnille.

Vesiteiden pato- ja sulkurakenteiden korjaustarpeen määrittäminen Suomen merkittävimmissä vesivoimalaitoksissa

Tämän diplomityön tavoitteena oli kartoittaa suomalaisten vesivoimalaitosten vesiteiden pato – ja sulkurakenteiden tulevia korjaustarpeita seuraavien viiden vuoden aikana. Tutkimuskohteiksi rajattiin yli 1 MW:n ja ennen vuotta 1975 valmistuneet sekä vielä peruskorjaamattomat kohteet. Näillä rajauksilla muodostui 73 vesivoimalaitoksen joukko. Näissä vesivoimalaitoksissa oli yhteensä 133 koneistoa ja 394 erilaista sulkurakennetta. Tutkimusaineisto hankittiin haastattelemalla 30 henkilöä yhteensä 24:stä eri yhtiöstä. Haastateltavat olivat pääasiassa laitosten käyttö- tai kunnossapitopäälliköitä, mutta myös omistajayhtiöiden ylintä johtoa. Strukturoidun haastattelulomakkeen avulla toteutettuna haastattelu oli siten lähes puhdas kyselytutkimus, jonka vastausprosentiksi muodostui 100 %. Haastattelu mahdollisti epäselvien yksityiskohtien selvittämisen ja siten tutkimuksen luotettavuus muodostui suuremmaksi kuin pelkällä kyselylomake - mittauksella. Tulokset osoittivat, että tulevia korjaustarpeita havaittiin noin kolmasosassa tutkimuskohteista. Tyypillisimmät korjaustarpeet ovat vesiteiden betonirakenteisiin tulleissa vaurioissa sekä sulkurakenteiden teknisen ikääntymisen myötä tulevissa tiiviysongelmissa sekä toiminnallisissa vioissa.

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.

Liquid cooling solutions for rotating permanent magnet synchronous machines

In the design of electrical machines, efficiency improvements have become very important. However, there are at least two significant cases in which the compactness of electrical machines is critical and the tolerance of extremely high losses is valued: vehicle traction, where very high torque density is desired at least temporarily; and direct-drive wind turbine generators, whose mass should be acceptably low. As ever higher torque density and ever more compact electrical machines are developed for these purposes, thermal issues, i.e. avoidance of over-temperatures and damage in conditions of high heat losses, are becoming of utmost importance. The excessive temperatures of critical machine components, such as insulation and permanent magnets, easily cause failures of the whole electrical equipment. In electrical machines with excitation systems based on permanent magnets, special attention must be paid to the rotor temperature because of the temperature-sensitive properties of permanent magnets. The allowable temperature of NdFeB magnets is usually significantly less than 150 ˚C. The practical problem is that the part of the machine where the permanent magnets are located should stay cooler than the copper windings, which can easily tolerate temperatures of 155 ˚C or 180 ˚C. Therefore, new cooling solutions should be developed in order to cool permanent magnet electrical machines with high torque density and because of it with high concentrated losses in stators. In this doctoral dissertation, direct and indirect liquid cooling techniques for permanent magnet synchronous electrical machines (PMSM) with high torque density are presented and discussed. The aim of this research is to analyse thermal behaviours of the machines using the most applicable and accurate thermal analysis methods and to propose new, practical machine designs based on these analyses. The Computational Fluid Dynamics (CFD) thermal simulations of the heat transfer inside the machines and lumped parameter thermal network (LPTN) simulations both presented herein are used for the analyses. Detailed descriptions of the simulated thermal models are also presented. Most of the theoretical considerations and simulations have been verified via experimental measurements on a copper tooth-coil (motorette) and on various prototypes of electrical machines. The indirect liquid cooling systems of a 100 kW axial flux (AF) PMSM and a 110 kW radial flux (RF) PMSM are analysed here by means of simplified 3D CFD conjugate thermal models of the parts of both machines. In terms of results, a significant temperature drop of 40 ̊C in the stator winding and 28 ̊C in the rotor of the AF PMSM was achieved with the addition of highly thermally conductive materials into the machine: copper bars inserted in the teeth, and potting material around the end windings. In the RF PMSM, the potting material resulted in a temperature decrease of 6 ̊C in the stator winding, and in a decrease of 10 ̊C in the rotor embedded-permanentmagnets. Two types of unique direct liquid cooling systems for low power machines are analysed herein to demonstrate the effectiveness of the cooling systems in conditions of highly concentrated heat losses. LPTN analysis and CFD thermal analysis (the latter being particularly useful for unique design) were applied to simulate the temperature distribution within the machine models. Oil-immersion cooling provided good cooling capability for a 26.6 kW PMSM of a hybrid vehicle. A direct liquid cooling system for the copper winding with inner stainless steel tubes was designed for an 8 MW directdrive PM synchronous generator. The design principles of this cooling solution are described in detail in this thesis. The thermal analyses demonstrate that the stator winding and the rotor magnet temperatures are kept significantly below their critical temperatures with demineralized water flow. A comparison study of the coolant agents indicates that propylene glycol is more effective than ethylene glycol in arctic conditions.