Matemaatika instituut. Kuni 2015

Selle valdkonna püsiv URIhttps://hdl.handle.net/10062/14971

2016 moodustas koos Matemaatilise Statistika Instituudiga Matemaatika ja Statistika Instituudi.

Sirvi

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Diferentsiaalgeomeetria meetodite rakendused dünaamiliste süsteemide uurimisel
(Tartu Ülikool, 2015-08-11) Ojaots, Margit; Abramov, Viktor, juhendaja; Liivapuu, Olga, kaasjuhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Käesolev magistritöö põhineb J-M. Ginoux monograafias Differential Geometry Applied to Dynamical Systems [4] kirjeldatud ja artiklites [Diferential geometry and mechanics: Applications to Chaotic Dynamical Systems [5], Slow invariant manifold of heartbeat model [6], Flow curvature method applied to canard explosion [3]] uuritud meetoditel. Selles lähenemises me vaatleme n-dimensionaalse dünaamilise süsteemi trajektoori kõverat kui kõverat Eukleidilises ruumis. Seda meetodit nimetatakse kõveruse muutkonna meetodiks. Punktides, kus voo kõverus on null, saame defineerida muutkonna, mida nimetatakse kõveruse voo muutkonnaks. Konkreetsel juhul rakendame seda meetodit Van der Pol’i ostsillaatorile.
Lie algebroidid
(Tartu Ülikool, 2015-08-11) Org, Riina; Abramov, Viktor, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Käesolevas magistritöös defineeritakse Lie algebroid ja tuuakse ka mõned näited. Enne Lie algebroidide mõiste andmist defineerime muutkonna ja räägime mitmemuutuja funktsiooni diferentseeruvusest. Edasises defineeritakse diferentseeruv muutkond ja antakse ka seosed Lie rühmade ja regulaarsete alammuutkondade vahel. Poissoni muutkonna juures näitame, et Jacobi samasus on võrdne Shouteni tingimusega.
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.
n-Lie superalgebrad
(Tartu Ülikool, 2015-08-11) Lätt, Priit; Abramov, Viktor, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatika instituut
Käesolevas magistritöös tuletame meelde mõned Lie algebrate teooria põhitõed ja vaatame selle klassikalise struktuuri üldistusi. Filippov konstrueeris artiklis [7] n-Lie algebra, kus binaarne kommutaator on asendatud n-aarse analoogiga. Meie kombineerime viimase Lie superalgebra struktuuriga, mis üldistab Lie algebraid kasutades Z2 -gradueeritud vektorruumi ning gradueeringu iseärasusi kommutaatoril tavalise vektorruumi asemel, et saada n-Lie superalgebra, nagu seda on tehtud artiklis [1]. Me uurime n-Lie superalgebra, ehk n-aarse gradueeritud Filippovi samasust rahuldava tehtega superalgebra omadusi, ning rakendades ideid artiklitest [1, 3] indutseerime n-Lie superalgebratest (n +1)-Lie superalgebrad. Viimaks uurime me ternaarseid Lie superalgebraid üle C, kus algebra aluseks oleva supervektorruumi dimensioon on m|n, m + n ≤ 4. Me teeme kindlaks kui palju on erinevaid võimalikke kommutatsioonieeskirju, mida neile tingimustele vastavad 3-Lie superalgebrad omada võivad.

Sirvi

Sirvi Matemaatika instituut. Kuni 2015 Autor "Abramov, Viktor, juhendaja" järgi