Memory is an important component of effective learning systems and is crucial in non-Markovian as well as partially observable environments. In recent years, Long Short-Term Memory (LSTM) networks have been the dominant mechanism for providing memory in reinforcement learning, however, the success of transformers in natural language processing tasks has highlighted a promising and viable alternative. Memory in reinforcement learning is particularly difficult as rewards are often sparse and distributed over many time steps. Early research into transformers as memory mechanisms for reinforcement learning indicated that the canonical model is not suitable, and that additional gated recurrent units and architectural modifications are necessary to stabilize these models. Several additional improvements to the canonical model have further extended its capabilities, such as increasing the attention span, dynamically selecting the number of per-symbol processing steps and accelerating convergence. It remains unclear, however, whether combining these improvements could provide meaningful performance gains overall. This dissertation examines several extensions to the canonical Transformer as memory mechanisms in reinforcement learning and empirically studies their combination, which we term the Integrated Transformer. Our findings support prior work that suggests gating variants of the Transformer architecture may outperform LSTMs as memory networks in reinforcement learning. However, our results indicate that while gated variants of the Transformer architecture may be able to model dependencies over a longer temporal horizon, these models do not necessarily outperform LSTMs when tasked with retaining increasing quantities of information.

The present MSc thesis discusses some notions of abelian group theory in connection with recent topics of topological entropy of locally compact abelian groups. It has been used the reference of [D. J. S. Robinson, A Course in the Theory of Groups, Springer, 1996, New York], which is a classical textbook in group theory. A list of exercises, relevant to our purposes, has been selected, in order to introduce some recent aspects of topological entropy of locally compact abelian groups. It is worth to mention that many of the exercises, which have been solved in the present thesis, are subject to technicalities which require the application of theorems of decomposition for abelian groups. Therefore the logic of the solutions allows us to describe the topological entropy in presence of an appropriate factorization.

Includes bibliographical references.

Einstein theory of General Relativity was well adapted and accepted until limitations in the form of an unexplained form of energy, referred today as Dark Energy, were observed. For this reason, modifications to the standard Theory of General Relativity were proposed: the so-called f(R) theories. In this dissertation, after a passage on the generalities of cosmology, we use the metric formalism technique to derive the field equations for the general f(R) function. Thereafter we analyse and check the solutions proposed in [85] for the quadratic model in f(R) gravity, for spherically symmetric and static neutron stars, using two different viable equations of state. We also check the accuracy of our code through a forward-backward integration technique, to show that in both directions, we obtain the same results. We then perform a thorough analysis in the case of f(R) = R1+ models. Results will show that for a negative value, we have non-Schwarzschild, but asymptotically flat solutions, for which we can use the backward integration technique to retrieve the solutions from the forward integration. However, for the case of positive values, we will show the existence of horizons, which deny us the possibility of using the backward integration technique. One of the aims of this thesis is to check, through the backward integration technique that we developed, whether the exact exterior solutions proposed in [86], are indeed realistic solutions for neutron stars. We will see that for some cases, we do have realistic profiles, while for some others, although solutions exist, they are rejected due to their disagreement with the equation of state used therein.

Summary in English. / Includes bibliographical references. / The modern approach to modelling of welding leads to the phenomenological description of the complex thermo-mechano-metallurgical (TMM) process and finite element (FE) solution of this problem. The TMM process requires four groups of data that define material properties, welding control variables, boundary and initial conditions and finite element approximation. To assure the better control of this simulation with the very large number of data an interactive database was developed. The development of the database has been one of the major tasks of this M.Sc. thesis.

Includes abstract. / Includes bibliographical references (leaves 112-115). / The aim of this dissertation was to build a basic understanding of hybrid securities with a focus on convertible bonds. We look at various methods to price these complex instruments and learn of the many subtleties they exhibit when traded in the market.

Since the discovery of a method to obtain graphene in 2004, there has been intensive research on this material by several researchers ranging from investigations of its physical and chemical properties to some novel applications of graphene. The discovery of graphene came about despite the fact that some leading researchers such as Landau and Peierls had predicted that twodimensional (2D) crystals were thermodynamically unstable. Andre Geim and Konstantin Novoselov managed to obtain graphene using a rather surprising technique called the scotch tape method. The scotch tape method involves carefully peeling off layer after layer in graphite which is a three-dimensional (3D) material without making any distortions to the subsequent layers. Due to the many applications of graphene, understanding the dynamical behavior of the material is a very important problem. In order to investigate the chaoticity in graphene, empirical force fields appropriate for modeling its dynamics are required. In this study we use such models which have been established to accurately describe bond stretching and bond deformation in graphene. Based on the corresponding Hamiltonian formalism we derive the system's Hamilton equations of motion, whose numerical solution determine the dynamical behavior of graphene, as well as the so-called variational equations needed for the numerical computation of several chaos indicators like the maximum Lyapunov Characteristic Exponent.

In this thesis, we introduce a novel class of many-body quantum system, which we term ‘quantum small worlds'. These are strongly-interacting systems that interpolate between completely ordered (nearest-neighbour, next-to-nearest-neighbour etc.) and completely random interactions. They are systems of quantum spin particles in which the network topology is given by the Watts-Strogatz model of network theory. As such, they furnish a novel laboratory for studying quantum systems transitioning between integrable and non-integrable behaviour. Our motivation is to understand how the dynamics of the system are affected by this transition, particularly with regards to the ability of the system to scramble (quantum) information, and potential emergence of chaotic behaviour. Our work begins with a review of the relevant literature regarding algebraic graph theory and quantum chaos. Next, we introduce the model by starting from a well understood integrable system, a spin- 1 2 Heisenberg, or Ising, chain. We then inject a small number of long-range interactions and study its ability to scramble quantum information using two primary devices: the out-of-time-order correlator (OTOC) and the spectral form factor (SFF). We find that the system shows increasingly rapid scrambling as its interactions become progressively more random, with no evidence of quantum chaos as diagnosed by either of these devices.

Includes bibliographical references. / The main aim of the study was to test the applicability of published EVT-based VaR calculation methods to the South African market. Two methods were tested on a hypothetical portolio of South African stocks, using the standard backtesting technique.

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