MMI bakalaureusetööd – Bachelor's theses. Kuni 2015

Selle kollektsiooni püsiv URIhttps://hdl.handle.net/10062/30415

Sirvi

Nüüd näidatakse 1 - 20 27

Aasia optsioonide hindamine
(Tartu Ülikool, 2013) Kask, Rain; Raus, Toomas, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
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).
Absoluutselt summeerivad operaatorid ruumil B (F)
(Tartu Ülikool, 2013) Izotova, Jekaterina; Põldvere, Märt,juhendaja; Tamme, Tõnu, juhendaja; Tartu Ülikool.Matemaatika-informaatikateaduskond; Tartu Ülikool.Matemaatika instituut
Agrawali, Kayali ja Saxena teoreem algarvulisuse kohta
(Tartu Ülikool, 2013) Räni, Ave; Laan, Valdis, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
This bachelor’s thesis gives an overview about prime numbers and different methods how to determine if an integer is prime or composite. It is based on Andrew Granville’s article "It is easy to determine if a given integer is prime", where he introduces and gives a proof of the primality theorem of Agrawal, Kayal and Saxena. The theorem was first published in 2002 in a paper "PRIMES is in P" by Manindra Agrawal, Neeraj Kayal and Nitin Saxena. This thesis consists of 3 parts. In the first part, the main definitions are given that are used throughout the whole thesis. The second part gives some examples about different theorems which have been used to determine primality before the theorem of Agrawal, Kayal and Saxena. In the third part we give a detailed proof of the main theorem of the thesis. It is formulated as follows. For a given integer n 2, let r be a positive integer < n, for which n has order > (log2 n)2 (mod r). Then n is prime if and only if 1) n is not a perfect power, 2) n does not have any prime factor r, 3) (x + a)n xn + a mod (n; xr =< 1) for each integer a, 1 a p r log n. Based on this theorem M. Agrawal, N. Kayal and N. Saxena created a deterministic primality-proving algorithm. This algorithm determines whether a positive integer n is prime or composite within polynomial time with respect to the number of digits of n.
B-splainid ja nende rakendused
(Tartu Ülikool, 2007) Kuljus, Elina; Fischer, Malle, juhendaja; Oja, Peeter, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
DEA meetodi rakendamine Kagu-Eesti gümnaasiumide efektiivsuse hindamisel
(Tartu Ülikool, 2013) Muru, Liina; Miidla, Peep, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
The aim of this Bachelor Thesis is to introduce Data Envelopment Analysis method and to apply it to the secondary schools of Southeastern Estonia. DEA method is a mathematical method that calculates the efficiency by solving linear programming problem. This method is special because it can calculate efficiency of non-profit organizations like hospitals, libraries and administrative units. This paper consists of four parts. The first part is the introduction of the DEA method. The second part is about choosing the inputs and outputs. DEA method needs different inputs and outputs and to find the best parameters, we contacted the County Governments of the three Southeastern Estonian counties, Põlva, Valga and Võru, which schools we analyzed in the paper. Every county government proposed a different set of inputs and outputs and so we got three sets to use in the DEA method. The fourth set was conducted using the information found in the earlier studies from all over the world. The third part is application of the method to the secondary schools of Southeastern Estonia and the fourth part is results. We analyzed 20 schools. To apply the method on the sets we used Data Envelopment Analysis (Computer) Program (DEAP) which is a DOS-program from the nineties, but inspite of the age is a very good program. It makes applying the DEA method easy. It only needs two .txt files, which consist of instructions and data. The program has many choices when it comes to the type of DEA. User can pick from output or input oriented method, one-, two- or multistage method, Malmquist or cost DEA. Results of the analyze depend on the inputs and outputs. With one set there were only three efficient schools and in another there were ten. The mean value of efficiency varied from 0,732 to 0,950. This study can be used by the profesionals to analyze the educational system. The most important part of the analyze is to find the right inputs and outputs to describe the educational system.
Dünaamiliste süsteemide ja geneetika mudelid
(Tartu Ülikool, 2013) Lillo, Karin; Puman, Ella, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
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.
Elastsete plaatide painde ülesanded
(Tartu Ülikool, 2013) Kasepuu, Kaile; Lellep, Jaan, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
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.
Hulkade ja funktsioonide r-kumerus
(Tartu Ülikool, 2005) Kivi, Karin; Soomer, Virge, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Hüpoteeside kontrollimine ja võimsuse arvutamine Analyst Application abil
(Tartu Ülikool, 2002) Bileva, Anna; Parring, Anne-Mai, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Integraali keskväärtusteoreemid: keskväärtust määravate punktide asümptootiline käitumine
(Tartu Ülikool, 2013) Maadik, Inger-Helen; Leiger, Toivo, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
The purpose of this thesis is to study the asymptotic behaviour of intermediate points in mean value theorems for integrals. The most simple mean value theorem states that if f : [a; b] ! R is a continuous function then there exists a number c 2 (a; b) such that Z b a f(t) dt = f(c)(b - a): In the case of the simpler mean value theorems for integrals the intermediate point c(x) asymptotically approaches the midpoint of the interval [a; x] and in addition lim sup x!a c(x) - a x - a 1 e : The simplest weighted mean value theorem for integrals states that if f : [a; b] ! R is a continuous function and g : [a; b] ! [0;1) is an integrable function then there exsists a number c 2 [a; b] such that Z b a f(t)g(t) dt = f(c) Z b a g(t) dt: In the case of the weighted mean value theorems for integrals the intermediate point asymptotically approaches the value a + (x - a) k q 1 k+1, where k 2 N is the number of times the function f di erentiable at the point a. Also when f and g are di erentiable then the intermediate point satis es the equation lim x!a R c(x) a g(t) dt R x a g(t) dt = 1 2 : 38 If the conditions set on the functions in the weighted mean value theorems for integrals are expanded to functions that aren't di erentiable at the point a then the approximate value c(x) a + (x - a) r r s + 1 r + s + 1 ; where r 2 (-1; 0) [ (0;1), s 2 (-1;1) and r +s > -1, can be used to provide close approximiations to certain physics' problems.
Interpoleerimine kuupsplainidega
(Tartu Ülikool, 2013) Koovit, Annika; Pedas, Arvet, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Joonteparvede mähisjooned
(Tartu Ülikool, 2013) Org, Riina; Rahula, Maido, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
The point of this bachelor’s thesis is to explore the envelopes of family of curves and curves characteristic points on the plane. In this thesis it has been studied enveloping surface of family of spheres (cone of sound in the tail of aircraft), caustic of lighting the inside of container, curved lines formed by characteristic points of spokes of wheel (for example Pascal’s lima¸con in the spokes of bicycle’s wheel). The Result is miscellaneous possible applications in interpretation of wave formation and shock waves.
Kvaasi-aritmeetilised keskmised ja Lagrange'i keskmised
(Tartu Ülikool, 2013) Päll, Kärt; Leiger, Toivo, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
In this thesis, a continuous function is called a proper mean if it is symmetric, reflexive and monotonic. We mainly observe quasi-arithmetic and Lagrangian means. Lagrangian means are de ned as means obtained from applying the classical mean value formula to a continuous and strictly monotonic function. Quasi-arithmetic means are transformations of the common arithmetic mean. We de ne the quasiarithmetic mean Qf associated with f : I ! I as Qf (x; y) := f-1 f (x) + f (y) 2 x; y 2 I and Lagrangian mean Lf as Lf (x; y) := 8< : f1 1 yx Ry x f(t)dt ; if x 6= y; x; if x = y: Many well-known means like arithmetic and geometric mean are both Lagrangian and quasi-arithmetic. The harmonic mean however is not Lagrangian but it is quasi-arithmetic. It is shown that there is a close relationship between Lagrangian and quasi-arithmetic means. This relationship can be made explicit through a homeomorphism. The purpose of this thesis is to give answers for two questions: 1. Which means can be represented according to some function f as quasiarithmetic and Lagrangian mean? 2. How to describe function classes with the same quasi-arithmetic and Lagrangian mean? The answers are presented in the second and third chapter. 41 In the rst chapter, some of the well-known means are discussed. The de nition of a proper mean is given and illustrated with some examples. The second chapter of the thesis, is dedicated to quasi-arithmetic means. It is shown that Qf = Qg if and only if g = f + where ; 2 R and 6= 0: A proof of the famous Acz el theorem is given. In the last chapter of the thesis, Lagrangian means are observed. It is shown that Lf = Lg if and only if g = f + where ; 2 R and 6= 0.
M(a,B,c)-ideaalid Banachi ruumides
(Tartu Ülikool, 2013) Rozhinskaya, Ksenia; Zolk, Indrek, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
In their 1973 paper “Structure in real Banach spaces” [AE], E. Alfsen and E. Effros introduced the notion of an M-ideal. It turned out that a closed subspace of a Banach space that is an M-ideal enjoys some properties (e.g. uniqueness of a norm-preserving extension) which do not necessarily occur in arbitrary subspaces. In [GKS], G. Godefroy, N. Kalton and P. Saphar introduced the notion of an ideal. It allowed them to make a connection between M-ideals and u-ideals, which were first introduced in [CK]. They also presented a natural strengthening of the definition of a u-ideal, an h-ideal. Another important step towards the generalization of previously studied ideals was made by J. Cabello and E. Nieto in [CN], where they defined an M(r; s)-ideal. The idea of studying M(a;B; c)-ideals dates back to a paper [O1] by E. Oja from 2000. In [OZ], one finds the following definition. Let a, c > 0 and let B K be a compact set of scalars. We shall say that a Banach space X satisfies the M(a;B; c)-inequality if kax + b Xx k + ck Xx k 6 kx k 8b 2 B; 8x 2 X ; where X : X ! X is the canonical projection. Based on this definition, we define an M(a;B; c)-ideal for an arbitrary closed subspace and aim to study some properties of M(a;B; c)-ideals. The pursuit to studyM(a;B; c)-ideals is motivated by the fact that the definition of anM(a;B; c)- ideal encompasses all previously studied special cases of ideals and makes it possible to handle them with a more unified approach. This bachelor thesis consists of four chapters. In the first chapter, we give a brief overview of some basic definitions and results required for further work. In the second chapter, we introduce the notion of an M(a;B; c)-ideal and study some basic properties of M(a;B; c)-ideals. We also take a closer look at M-, u- and h-ideals. The aim of the third chapter is to study M(a;B; c)-ideals in particular Banach spaces. First we give necessary and sufficient conditions for a one-dimensional subspace of a Banach space `2 1 to be an M(a;B; c)-ideal in `2 1. We also provide a theoretical result which can be used to derive examples of M(a;B; c)-ideals in L(X). In the fourth chapter, we study the transitivity of M(a;B; c)-inequality. First we show that ideal projections are closely connected to Hahn-Banach extension operators. Using this knowledge, we show as a first main result of this bachelor thesis that if X is an M(a;B; c)-ideal in Y and Y is an M(d;E; f)-ideal in Z, then X is an ideal satisfying a certain type of inequality in Z. Relying on this result, we show as a second main result of this thesis that if X is an M(a;B; c)-ideal in its second bidual X , then X is an ideal satisfying a certain type of inequality in X(2n) for every n 2 N.
Matemaatika võimalikest rakendustest muusikas
(Tartu Ülikool, 2013) Simson, Mai; Abel, Mati, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
The aim of this bachelor’s thesis is to give an overview of some mathematical applications which can be used in music. The thesis is mainly based on three books: “Simple Gamma” by Georgi Evgenevitš Šilov, “Occidents’ History of Music” by Igor Garšnek and “Music: A Mathematical Offering” by Dave Benson. The thesis consists of four chapters. In the first chapter we give definitions to different terms that are used in the thesis. The second chapter briefly describes the common history in music and mathematics. The aim of the third chapter is to explain the construction of the musical scale using logarithms and chain fractures. The final chapter focuses on different mathematical applications used in music. We will discuss Iannis Xenakis’s theory about the algebraic qualities of the intervals, the theory that Pierre Boulez used to compose his “Structures I”, Urmas Sisask using mathematics and astronomy to compose his choir piece “Gloria Patri” and also some mathematical methods that could be used to describe music.
Matemaatilisest mõtlemisest
(Tartu Ülikool, 2013-05-31) Murruste, Markus; Palm, Reimo, juhendaja; Tartu Ülikool. Matemaatika-informaatika teaduskond; Tartu Ülikool. Matemaatika instituut
Minkowski aegruumi geomeetriast
(Tartu Ülikool, 2013) Lätt, Priit; Abramov, Viktor, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
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.
Murrulised tuletised
(Tartu Ülikool, 2013) Heinsalu, Märten; Pedas, Arvet, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
This thesis consists of five parts. The first section contains definitions and theorems that are needed in following sections. These mostly regard Euler’s gamma and beta functions. This includes the domain of Euler’s gamma function, the relationship between the two functions and the gamma function’s limit representation. The second section consists of three chapters and provides Grünwald-Letnikov’s, Riemann-Liouville’s and Caputo’s approaches to fractional derivatives. Each chapter includes the definitions, the reasoning behind those definitions and some of the more important properties of those definitions. The third section concerns the relationships between the three previously stated definitions of fractional derivatives. Firstly, it explores the connection between Riemann-Liouville’s and Caputo’s fractional differential operators, and secondly, the relationship between Riemann-Liouville’s and Grünwald-Letnikov’s definitions of fractional derivatives. The fourth section gives examples on how to find fractional derivatives of some power functions based on the three different definitions explored in this paper. The example functions are (x − a)b, √x, x2, x3 and the constant function f(x) = c. The final section gives examples of the applications of fractional derivatives. It primarily focuses on how to solve Abel’s integral equation and equations that are equivalent to Abel’s integral equation using fractional order derivatives and how to model some special cases of diffusion.
Osakeste parvega optimeerimise rakendamine elektribörsil
(Tartu Ülikool, 2013) Rakaselg, Gerda; Miidla, Peep, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
This bachelor thesis consists of giving an overview of the electircity market in Estonia, introducing the method of particle swarm optimization (PSO) and implementing PSO on a problem associated with electricity market. Electricity market was opened only in the beginning of year 2013 in Estonia. Thus this subject is topical at the moment. We concentrate on power producers in the day-ahead market, specifically on submitting an offer curve to the day-ahead market. In order to maximize profit, power companies need appropriate bidding strategies. In this thesis one strategy is described and a method for calculating necessary prices and quantities for electricity is introduced. Particle swarm optimization is a computation technique, which is inspired by swarm behaviour such as fish schooling or bird flocking. PSO is initialized with a population of random particles and it searches for solution by updating generations. The particles fly through the n-dimensional problem space moving towards both current individual and current global optimum position. In last chapter of this thesis a theoretical example of finding an offer curve for two electricity generators is solved implementing particle swarm optimization. It is modelled using MATLAB.
Otsesuunatud tehisnärvivõrgud paketis R
(Tartu Ülikool, 2013) Liivoja, Merili; Miidla, Peep, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Human brain is a complex and powerful system that is able to solve a wide variety of tasks. The aim of many scientists is to develop a computer simulation that mimics the brain functions and solves problems the way our brains do. Very simplified models of biological neural networks are artificial neural networks. There are two different types of artificial neural networks – feed forward neural networks and recurrent neural networks. This thesis gives an overview of feed-forward neural networks and their working principles. The thesis is divided into two main parts. The first part is the theory of feed-forward neural networks and the second part is a practical example of neural network with software R. The first part gives an overview of the artificial neuron and its history. Also different types of artificial neurons are introduced. The first part includes instructions of how feed-forward neural networks are composed and explains how they calculate the results. Separate chapter is devoted to training artificial neural networks. The chapter gives an overview of two main training algorithms – perceptron training algorithm and back-propagation algorithm. The first is designed to train perceptrons and the second is often used in training multi-layer feed-forward neural networks. The last topic explains how to construct feed-forward neural networks with software R. It includes a tutorial of how to build a neural network that calculates the square root. The tutorial will produce a neural network which takes a single input and produces a single output. Input is the number that we want square rooting and the output is the square root of the input.

Sirvi

Sirvi MMI bakalaureusetööd – Bachelor's theses. Kuni 2015 Märksõna "bakalaureusetööd" järgi