Fuzzy subgroups, algebraic and topological points of view and complex analysis

Fuzzy subsets and fuzzy subgroups are basic concepts in fuzzy mathematics. We shall concentrate on fuzzy subgroups dealing with some of their algebraic, topological and complex analytical properties. Explorations are theoretical belonging to pure mathematics. One of our ideas is to show how widely fuzzy subgroups can be used in mathematics, which brings out the wealth of this concept. In complex analysis we focus on Möbius transformations, combining them with fuzzy subgroups in the algebraic and topological sense. We also survey MV spaces with or without a link to fuzzy subgroups. Spectral space is known in MV algebra. We are interested in its topological properties in MV-semilinear space. Later on, we shall study MV algebras in connection with Riemann surfaces. In fact, the Riemann surface as a concept belongs to complex analysis. On the other hand, Möbius transformations form a part of the theory of Riemann surfaces. In general, this work gives a good understanding how it is possible to fit together different fields of mathematics.

Monivaiheisen höyryturbiinin optimointi

Tässä diplomityössä optimoitiin nelivaiheinen 1 MWe höyryturbiinin prototyyppimalli evoluutioalgoritmien avulla sekä tutkittiin optimoinnista saatuja kustannushyötyjä. Optimoinnissa käytettiin DE – algoritmia. Optimointi saatiin toimimaan, mutta optimoinnissa käytetyn laskentasovelluksen (semiempiirisiin yhtälöihin perustuvat mallit) luonteesta johtuen optimoinnin tarkkuus CFD – laskennalla suoritettuun tarkastusmallinnukseen verrattuna oli jonkin verran toivottua pienempi. Tulosten em. epätarkkuus olisi tuskin ollut vältettävissä, sillä ongelma johtui puoliempiirisiin laskentamalleihin liittyvistä lähtöoletusongelmista sekä epävarmuudesta sovitteiden absoluuttisista pätevyysalueista. Optimoinnin onnistumisen kannalta tällainen algebrallinen mallinnus oli kuitenkin välttämätöntä, koska esim. CFD-laskentaa ei olisi mitenkään voitu tehdä jokaisella optimointiaskeleella. Optimoinnin aikana ongelmia esiintyi silti konetehojen riittävyydessä sekä sellaisen sopivan rankaisumallin löytämisessä, joka pitäisi algoritmin matemaattisesti sallitulla alueella, muttei rajoittaisi liikaa optimoinnin edistymistä. Loput ongelmat johtuivat sovelluksen uutuudesta sekä täsmällisyysongelmista sovitteiden pätevyysalueiden käsittelyssä. Vaikka optimoinnista saatujen tulosten tarkkuus ei ollut aivan tavoitteen mukainen, oli niillä kuitenkin koneensuunnittelua edullisesti ohjaava vaikutus. DE – algoritmin avulla suoritetulla optimoinnilla saatiin turbiinista noin 2,2 % enemmän tehoja, joka tarkoittaa noin 15 000 € konekohtaista kustannushyötyä. Tämä on yritykselle erittäin merkittävä konekohtainen kustannushyöty. Loppujen lopuksi voitaneen sanoa, etteivät evoluutioalgoritmit olleet parhaimmillaan prototyyppituotteen optimoinnissa. Evoluutioalgoritmeilla teknisten laitteiden optimoinnissa piilee valtavasti mahdollisuuksia, mutta se vaatii kypsän sovelluskohteen, joka tunnetaan jo entuudestaan erinomaisesti tai on yksinkertainen ja aukottomasti laskettavissa.

On Topological Solitons in The Faddeev-Skyrme Model and its Extensions

The topological solitons of two classical field theories, the Faddeev-Skyrme model and the Ginzburg-Landau model are studied numerically and analytically in this work. The aim is to gain information on the existence and properties of these topological solitons, their structure and behaviour under relaxation. First, the conditions and mechanisms leading to the possibility of topological solitons are explored from the field theoretical point of view. This leads one to consider continuous deformations of the solutions of the equations of motion. The results of algebraic topology necessary for the systematic treatment of such deformations are reviewed and methods of determining the homotopy classes of topological solitons are presented. The Faddeev-Skyrme and Ginzburg-Landau models are presented, some earlier results reviewed and the numerical methods used in this work are described. The topological solitons of the Faddeev-Skyrme model, Hopfions, are found to follow the same mechanisms of relaxation in three different domains with three different topological classifications. For two of the domains, the necessary but unusual topological classification is presented. Finite size topological solitons are not found in the Ginzburg-Landau model and a scaling argument is used to suggest that there are indeed none unless a certain modification to the model, due to R. S. Ward, is made. In that case, the Hopfions of the Faddeev-Skyrme model are seen to be present for some parameter values. A boundary in the parameter space separating the region where the Hopfions exist and the area where they do not exist is found and the behaviour of the Hopfion energy on this boundary is studied.

Development of Color Difference Equations Matching to Human Vision System

Research on color difference evaluation has been active in recent thirty years. Several color difference formulas were developed for industrial applications. The aims of this thesis are to develop the color density which is denoted by comb g and to propose the color density based chromaticity difference formulas. Color density is derived from the discrimination ellipse parameters and color positions in the xy , xyY and CIELAB color spaces, and the color based chromaticity difference formulas are compared with the line element formulas and CIE 2000 color difference formulas. As a result of the thesis, color density represents the perceived color difference accurately, and it could be used to characterize a color by the attribute of perceived color difference from this color.

Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

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.

Flexible multibody dynamics and intelligent control of a hydraulically driven hybrid redundant robot machine

The assembly and maintenance of the International Thermonuclear Experimental Reactor (ITER) vacuum vessel (VV) is highly challenging since the tasks performed by the robot involve welding, material handling, and machine cutting from inside the VV. The VV is made of stainless steel, which has poor machinability and tends to work harden very rapidly, and all the machining operations need to be carried out from inside of the ITER VV. A general industrial robot cannot be used due to its poor stiffness in the heavy duty machining process, and this will cause many problems, such as poor surface quality, tool damage, low accuracy. Therefore, one of the most suitable options should be a light weight mobile robot which is able to move around inside of the VV and perform different machining tasks by replacing different cutting tools. Reducing the mass of the robot manipulators offers many advantages: reduced material costs, reduced power consumption, the possibility of using smaller actuators, and a higher payload-to-robot weight ratio. Offsetting these advantages, the lighter weight robot is more flexible, which makes it more difficult to control. To achieve good machining surface quality, the tracking of the end effector must be accurate, and an accurate model for a more flexible robot must be constructed. This thesis studies the dynamics and control of a 10 degree-of-freedom (DOF) redundant hybrid robot (4-DOF serial mechanism and 6-DOF 6-UPS hexapod parallel mechanisms) hydraulically driven with flexible rods under the influence of machining forces. Firstly, the flexibility of the bodies is described using the floating frame of reference method (FFRF). A finite element model (FEM) provided the Craig-Bampton (CB) modes needed for the FFRF. A dynamic model of the system of six closed loop mechanisms was assembled using the constrained Lagrange equations and the Lagrange multiplier method. Subsequently, the reaction forces between the parallel and serial parts were used to study the dynamics of the serial robot. A PID control based on position predictions was implemented independently to control the hydraulic cylinders of the robot. Secondly, in machining, to achieve greater end effector trajectory tracking accuracy for surface quality, a robust control of the actuators for the flexible link has to be deduced. This thesis investigates the intelligent control of a hydraulically driven parallel robot part based on the dynamic model and two schemes of intelligent control for a hydraulically driven parallel mechanism based on the dynamic model: (1) a fuzzy-PID self-tuning controller composed of the conventional PID control and with fuzzy logic, and (2) adaptive neuro-fuzzy inference system-PID (ANFIS-PID) self-tuning of the gains of the PID controller, which are implemented independently to control each hydraulic cylinder of the parallel mechanism based on rod length predictions. The serial component of the hybrid robot can be analyzed using the equilibrium of reaction forces at the universal joint connections of the hexa-element. To achieve precise positional control of the end effector for maximum precision machining, the hydraulic cylinder should be controlled to hold the hexa-element. Thirdly, a finite element approach of multibody systems using the Special Euclidean group SE(3) framework is presented for a parallel mechanism with flexible piston rods under the influence of machining forces. The flexibility of the bodies is described using the nonlinear interpolation method with an exponential map. The equations of motion take the form of a differential algebraic equation on a Lie group, which is solved using a Lie group time integration scheme. The method relies on the local description of motions, so that it provides a singularity-free formulation, and no parameterization of the nodal variables needs to be introduced. The flexible slider constraint is formulated using a Lie group and used for modeling a flexible rod sliding inside a cylinder. The dynamic model of the system of six closed loop mechanisms was assembled using Hamilton’s principle and the Lagrange multiplier method. A linearized hydraulic control system based on rod length predictions was implemented independently to control the hydraulic cylinders. Consequently, the results of the simulations demonstrating the behavior of the robot machine are presented for each case study. In conclusion, this thesis studies the dynamic analysis of a special hybrid (serialparallel) robot for the above-mentioned special task involving the ITER and investigates different control algorithms that can significantly improve machining performance. These analyses and results provide valuable insight into the design and control of the parallel robot with flexible rods.