OAR@UM Collection:

The non-linear Van der Pol equation

2021-07-23T10:03:29Z

Title: The non-linear Van der Pol equation Abstract: A two-dimensional dynamical system is defined by two coupled first order differential equations of the form : x = P(x,y), y = Q(x,y), d (.):= dt, (II) where P and Q are two functions of the variables x and y parametrised by the time independent variable t. The dynamical system (Tl) appears very often within several branches of science, such as biology, chemistry, astrophysics, mechanics, electronics, fluid mechanics, etc. The most important problem connected with the study of system (II) is the limit ·cycle. Stable limit-cycles are very important in science. They model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing. Of the countless examples that could be given, we mention only a few : The beating of a heart, chemical reactions that oscillate spontaneously, self-excited vibrations in bridges and aeroplane wings, etc. In each case, there is a standard oscillation of some preferred period, and amplitude. If the system is slightly perturbated, it always returns to the standard cycle. Limit-cycles are an inherently non-linear phenomena; they cannot occur in linear systems. Description: M.SC.MATHS

Two-fold orbital digraphs and other constructions

2023-05-15T05:04:27Z

Title: Two-fold orbital digraphs and other constructions Abstract: The developments of group theory, such as the classification of finite simple groups, stimulated recent developments in algebraic graph theory. One of the remarkable tasks accomplished is the determination of all vertex-transitive graphs or order equal to the product of two primes ( cf, [6], [13] and [12] ). The construction of orbital digraphs is one of the basic tools in the study of vertex-transitive digraphs. In principle, the group-theoretical method used to construct orbital digraphs may not only be used to generate all vertex-transitive digraphs (cf. [3] and [7] ), but also makes it clear whether these vertex-transitive digraphs are arc-transitive or not. Description: M.SC.MATHS

qm-Sequence characterizations of certain measure-theoretic and topological properties

2021-07-23T05:49:14Z

Title: qm-Sequence characterizations of certain measure-theoretic and topological properties Abstract: In this thesis we shall study some relations between the concepts of Measure and Topology. The spaces which are considered are assumed to be at least Tychonoff, that is a T1 space X on which every point x and every closed set F disjoint from x are functionally seperated. Let us denote by M(X), Mo(X), MT(X) and Mt(X) the sets of all regular measures, o-additive measures, t-additive measures and tight measures on a Tychnonoff space X respectively, and by T(X), To(X), Tt(X) and D(X) the sets of all two-valued measures, two-valued o-additive measures, two-values T-additive measures and Dirac measures on a Tychonoff space X respectively. Description: M.SC.MATHS

Lexicographic products on linearly ordered topological spaces

2021-07-23T05:47:02Z

Title: Lexicographic products on linearly ordered topological spaces Abstract: A linear order < defined on a set X is a relation such that: • For all x EX, x < x does not hold. (Non-reflexivity) • x < y and y < z --} x < z (Transitivity) • For 011 x, y C X, we have x < y or y < x or x < y (Totality) The linear order topology defined on X is the topology generated by the family of all open intervals in X, and the obtained topological space is called a linearly ordered topological space (LOTS). A subspace of a LOTS is called a generalized ordered space (or GO-space). Clearly, a LOTS is also a GO-space. In this thesis, we study several well-known properties like compactness, paracompactness, etc in GO-spaces. In addition, we will study lexicographic products of LOTS. In Chapter 1, we provide some utility results for the study of properties of GO-spaces. The most important results are those about gaps and pseudo gaps in GO-spaces. As we will see, in the following chapters, gaps and pseudo gaps are intensively used to characterize different properties of GO-spaces. In addition, gaps also play a critical role in our study of lexicographic products of LOTS. Since gaps can be considered to be points in the Dedekind compactification of a LOTS or GO-space, our study about gaps are mainly in the context of Dedekind compactification. In Theorem 1.5.5, we show that a gap in the lexicographic product must be generated from a gap in one of its factor spaces. This result is quite intuitive, but the proof is not so simple. Then in Lemma 1 5 7 and Lemma 1 5 10, we discuss circumstances, in which a gap in a factor space will give a gap in the lexicographic product. Description: M.SC.MATHS