# MMI bakalaureusetööd – Bachelor's theses. Kuni 2015

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Item Применение принципа равномерной органиченности к теории суммируемости(Tartu Ülikool, 1999) Удальцова, Елена; Soomer, Virge, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Optimaalsete veomarsruutide leidmine kahe lao korral(Tartu Ülikool, 2002) Kuresson, Aire; Kangro, Raul, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Autonoomsete diferentsiaalvõrrandite süsteemide perioodilised lahendid(Tartu Ülikool, 2002) Korobova, Evelin; Miidla, Peep, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Hüpoteeside kontrollimine ja võimsuse arvutamine Analyst Application abil(Tartu Ülikool, 2002) Bileva, Anna; Parring, Anne-Mai, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Hulkade ja funktsioonide r-kumerus(Tartu Ülikool, 2005) Kivi, Karin; Soomer, Virge, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Asümmeetria ja järsakuse karakteristikud(Tartu Ülikool, 2005) Kilgi, Helle; Kollo, Tõnu, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Rajaülesande lahendamine ratsionaalsplainidega kollokatsioonimeetodil(Tartu Ülikool, 2006) Mehide, Inga; Fischer, Malle, juhendaja; Oja, Peeter, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Riccati diferentsiaalvõrrand finantsmatemaatikas(Tartu Ülikool, 2007) Kuld, Anu; Miidla, Peep, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Rajaülesande lahendamine ratsionaalsplainidega kollokatsioonimeetodil(Tartu Ülikool, 2007) Nurk, Anni; Fischer, Malle, juhendaja; Oja, Peeter, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item B-splainid ja nende rakendused(Tartu Ülikool, 2007) Kuljus, Elina; Fischer, Malle, juhendaja; Oja, Peeter, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Ruutsplainidega interpoleerimine ühtlasel võrgul(Tartu Ülikool, 2008) Frolova, Svetlana; Pedas, Arvet, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Ajaskaalal määratud funktsioonide diferentseerimine ja integreerimine(Tartu Ülikool, 2009) Karm, Märten; Leiger, Toivo, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Otsese ja kaudse meetodi võrdlemine aktsiahinna kasvamise tõenäosuse leidmisel päevasisese kauplemise tingimustes(Tartu Ülikool, 2009) Olvik, Ander; Kangro, Raul, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Item Absoluutselt summeerivad operaatorid ruumil B (F)(Tartu Ülikool, 2013) Izotova, Jekaterina; Põldvere, Märt,juhendaja; Tamme, Tõnu, juhendaja; Tartu Ülikool.Matemaatika-informaatikateaduskond; Tartu Ülikool.Matemaatika instituutShow more Item Minkowski aegruumi geomeetriast(Tartu Ülikool, 2013) Lätt, Priit; Abramov, Viktor, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more We consider a four-dimensional real vector space in which three coordinates describe the space, and the fourth coordinate describes time. Such a vector space, equipped with Minkowski metric, is said to be a Minkowski spacetime, denotedM and sometimes referred just as Minkowski space. In the beginning of 20th century it emerged from the works of Hermann Minkowski and Henry Poincar e that this mathematical tool can be really useful in areas like the theory of special relativity. The rst objective of this thesis is to introduce the classical properties of Minkowski spacetime in a way that doesn't get into the details about the physical meaning of the topic. The main focus is on the elements of the spacetime and the orthogonal maps that describe the transformations of these elements. As this topic, the Minkowski spacetime, has many applications in physics, most of the literature on it originates from physics, rather than mathematics. In the books that are written by physicists, strict proofs are not given properly or they are avoided entirely. Sometimes physical discussion is considered as a proof, but as shown in proposition 3.1 for example, things are not so obvious after all in a mathematical point of view. In contrary, the aim of this thesis is to ll this gap by giving rigorous proof to all the statements that are made. The last part of the thesis takes a new course and tries to introduce tools from branch of mathematics called di erential geometry. Apparatus described there can be useful not just describing the structure of Minkowski spacetime, but also wide areas of theoretical physics. At last, we generalize classical spacetime and present the Minkowski superspace { a space with anticommuting coordinates that has become a useful tool in the eld theory. This is all done in a bit less formal way. This thesis consists of four sections. In the rst section we recall such topics that are not strictly connected to the thesis but have great impact on further understanding of the paper. For example Gram{Schmidt orthogonalization process is reminded and we prove one of the fundamental results related to dot product, the Cauchy{Schwartz{Bunjakowski inequality. We also describe the rules of the multipication of block matrices as 42 well as nding the inverse of block matrix. Map called matrix exponential is introduced as it plays a key role in the nal part of the thesis. Considered the geometrical meaning of the paper we nd ourselves stopping for a bit longer to study the idea of topological manifold. Second section sets an objective to describe the elements or events of the Minkowski spacetime. For that we give a strict de nition of dot product and move on to the concept of pseudoeucleidic space. After that we describe the dot product in Minkowski spacetime, which gives us the metric. It appears that by the metric we can factorize the elements of spacetime into three distinct classes, these are spacelike, timelike and nullvectors. The last subsection is there to describe the orthogonal transformations of Minkowski space, referred as Lorentz transformations. We show how these transformations can be represented in both coordinates and matrix form and what is the connection between the transformations and the metric of M. Finally we prove that the set of all Lorentz transformatioins has a very natural group structure, the Lorentz group. In the third section we continue with the observation of Lorentz transformations. In the rts subsection we take a closer look at the properties of such transformations and concentrate on the most important of them. As a result we nd a rotation subgroup of Lorentz group. We proceed with noting that Lorentz transformations remain the distances invariant. That encourages us to look for other such transformations that share the same property. As it happens translations are of this kind and by observing translations with Lorentz transformations together, we get the symmetry group of Minkowski spacetime, the Poincar e group. The fourth section takes another course. In this section the objective is to introduce, via more popular approach, the connections between Minkowski spacetime and di erential geometry, and to uncover the term superspace with the surrounding mathematics. In the part of di erential geometry we de ne Lie algebra with its representation and give examples to clear the topic. As expected we then nd ourselves de ning Lie group which we later connect with Lie algebra by giving an informal description. The section, as well as the paper is then nished with the introduction of Minkowski superspace. For that we de ne terms such as superalgebra and give a well known example, the Grassmann algebra. By avoiding strict mathematical formalism, we show what are functions on superspace and how to nd integral or derivative of such functions.Show more Item Täisarvuliste maatriksite Smithi normaalkuju(Tartu Ülikool, 2013) Loit, Kätlin; Tart, Lauri, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more This bachelor’s thesis gives an overview of the Smith normal form for integral matrices, i.e. matrices whose entries are integers. This normal form is a diagonalization that exists for any integral matrix and, moreover, is uniquely determined. It was first used in the 1861 paper by H.J.S. Smith which considered solving linear diophantine equations and congruences. The Smith normal form has seen an extensive number of applications since then, including diophantine analysis, integer programming, linear systems theory and module theory over principal ideal domains. The thesis itself is a review of fundamentals and contains no original research, but it does strive to be as elementary and self-contained as possible. There are altogether three chapters. The first one introduces a number of definitions and results from elementary matrix algebra and number theory that will be needed later on. The second chapter introduces the notion of equivalent matrices. It also contains the main result of the thesis, the proof that every integral matrix has a Smith normal form, i.e. is equivalent to a specific kind of diagonal matrix. There is also a subchapter on certain invariants of equivalent matrices, namely determinantal divisors and invariant factors, which are used to prove that the Smith normal form is unique. Finally, the process of finding the Smith normal form of an integral matrix is illustrated by a simple numerical example. The last chapter contains an overview of three applications: solving linear diophantine equations, a method for analysing a certain class of combinatorial problems and the fundamental theorem of finitely generated Abelian groups.Show more Item Aasia optsioonide hindamine(Tartu Ülikool, 2013) Kask, Rain; Raus, Toomas, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Determining the correct value of an option is the main problem in option theory. There are several factors which determine the price of an option. In addition to these factors, Asian options also depend on the history of the underlying asset which complicates the correct pricing of an Asian option. Although the structure of an Asian option is more complex than for example European option's, many of the typical numerical methods can still be used to nd the price of an Asian option when modi ed correctly. In the rst part of this thesis some of these typical numerical methods are introduced. The basic idea of lattice method, di erential method and Monte-Carlo method are described by showing how to nd a value of a usual European option step by step. The last and main part of this thesis is dedicated to Asian options and lattice method. It is shown how to modify the lattice method so that it could be used for pricing an Asian option. The modi ed lattice method for pricing an Asian option is called forward shooting grid method (FSG) and was rst used in 1993 by Hull and White for nding the value of Asian and lookback options. The method is described thoroughly and 2 di erent approaches (Barraquand-Pudet method and modi ed Hull-White method) for choosing the average price of an underlying asset are introduced. To ensure that the FSG method can truly be used for pricing an Asian option, some results obtained by using the FSG method with di erent parameters N, and are brought out in the last section of the thesis. The prices found by Barraquand and Pudet method and modi ed Hull and White method are compared with a price of an unrealistic Asian option, which analytical value can be found. For one more realistical case of an Asian option the prices found by FSG method are compared with a price found by Monte-Carlo method. Source codes (written in Python) for FSG method and Monte-Carlo method are brought out in Appendixes (Lisad).Show more Item Elastsete plaatide painde ülesanded(Tartu Ülikool, 2013) Kasepuu, Kaile; Lellep, Jaan, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more The aim of this work is to introduce the Basic equations of the Kirchhoff’s bending theory. The current study consists on two parts. In the first part we derive the governing differential equation for the deflection for thin plate bending. In the second part we study the bending of thin rectangular plates subjected to the transverse pressure. We find the middle surface of the plate in the case of cylindrical bending. It appears that in this case the differential equation of equilibrium can be solved by direct integration. Also we present Navier’s method for rectangular plates with simply supported boundary conditions on all four edges. We look for the solution of the equilibrium equation in the form of double seires. Each term of this series is a product of trigonometrical functions which meet the boundary conditions spontaneously.Show more Item Tõkestamata intervallis ühtlaselt pidevad funktsioonid(Tartu Ülikool, 2013) Bogdanova, Galina; Leiger, Toivo, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Uniform continuity of real functions on unbounded intervals is not a compulsory topic in the basic analysis courses. In the course «Calculus III», the study of the uniform continuity is limited to Cantor’s theorem. The theorem states that if f : [a; b] ! R is a continuous function, then it is uniformly continuous. This result is essential for proving one of the central results of the integral calculus, which states that if a function f : [a; b] ! R is continuous, then it is integrable on [a; b]. We also know that if f : [a;1) ! R is continuous and limx!1 f(x) exists (as a real number), then f is uniformly continuous on the interval [a;1). The aim of this bachelor thesis is to characterize functions f : [a;1) ! R that are uniformly continuous on [a;1). This thesis consists of three chapters and is based on the articles by R. L. Pouso [4] and S. Djebali [1]. In the first chapter of this bachelor thesis, we discuss the definition of a uniform continuity and bring some examples of continuous functions that are not uniformly continuous. In this introductory chapter we use sequences to investigate uniform continuity.We provide an important proposition on the relation between uniformly continuous function and Cauchy sequences, which we will use for further proofs. The second chapter is devoted to examining uniformly continuous functions on unbounded intervals. We present conditions for a function f : [a;1) ! R to be uniformly continuious on an interval [a;1). Then we give an example that shows that the condition limx!1 jf(x)j x = 0 does not guarantee uniform continuity of the function f. At the end of the chapter, we show that if the graph of a function f has a horizontal or oblique asymptote then the function is uniformly continuous. In the third chapter, the relation between uniform continuity and convergence of improper integrals is studied.We examine unifom continuity of a function f : [a;1) ! R, for which improper integral R 1 a f(x)dx convergences. We also provide example of cases when uniform continuity of a function does not guarantee integral convergence.Show more Item Dünaamiliste süsteemide ja geneetika mudelid(Tartu Ülikool, 2013) Lillo, Karin; Puman, Ella, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituutShow more Dynamic systems are systems, which develop or change in time. With every model in this thesis there is given a story about the origin of given system or equation and necessary information. In addition to models there are also different charts to simplify the understanding of the models. Systems in this thesis are based on differential equations, because differential equations describe dynamic systems, which are continuous in time. Attention is paid to systems, in which the differential equations are nonlinear, because these could lead to chaotic solutions. All the models are given with initial conditions and some models also have charts, which show the change of the solution if a parameter has been changed a little. It is clear, that all systems may not lead to chaos with every value, but if the parameters are changed and solutions are analysed, it is seen that even a slightest change may lead to a chaos. With models it is easy to experiment with the real world. Without such models we were often left to manipulate real systems in order to understand the relationships of cause and effect. Models help to realize the consequences of the actions of mankind without harming the nature or the real world. In this thesis, two basic genetic problems were discussed, the mating of two alleles and natural selection and mutation. These models are quite simple, yet they provide enough infor-mation to understand the base of these problems in the real world. It is also possible to make these models more precise, which will lead to more accurate results and more precise predictions to real life cause and effect situations. In conclusion, models just make a lot of things easier to understand.Show more

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