Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems

A semianalytical method is derived for finding the existence and stability of single-impact periodicorbits born in a boundary equilibrium bifurcation in a generaln-dimensional impacting hybridsystem. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3Dexamples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how themethod can easily be embedded within numerical continuation, and some remarks are made aboutnecessary and sufficient conditions in arbitrary dimensional system

On Derivative-Free Optimisation Methods

As part of the field of mathematical optimisation, derivative-free optimisation is the study of optimisation methods that are not granted full access to the derivative of the objective function. In this master's thesis, three derivative-free optimisation methods known from the literature that do not use the derivatives have been studied, namely the Nelder–Mead method, the conditional trust-region method and the discrete gradient method. For each of these methods, besides recalling a description and a convergence statement, focus was given on providing motivation and background for increased understanding of the method without requiring specific prior knowledge in derivative-free optimisation. Different types of differentiability as total differentiability or subdifferentiability have been recalled for general usage in the understanding of those methods. As part of the description of the discrete gradient method, Wolfe's method for finding a minimum norm vector in a convex set is recalled, with a modified statement for proven convergence. Each of the methods are accompanied with an implementation for use with the MATLAB programming and numeric computing platform, or a reference to one such existing implementation is given. Numerical experiments were performed to compare the quality of variants of the methods.Master's Thesis in InformaticsINF399MAMN-INFMAMN-PRO

Bifurcation Analysis of Large Networks of Neurons

The human brain contains on the order of a hundred billion neurons, each with several thousand synaptic connections. Computational neuroscience has successfully modeled both the individual neurons as various types of oscillators, in addition to the synaptic coupling between the neurons. However, employing the individual neuronal models as a large coupled network on the scale of the human brain would require massive computational and financial resources, and yet is the current undertaking of several research groups. Even if one were to successfully model such a complicated system of coupled differential equations, aside from brute force numerical simulations, little insight may be gained into how the human brain solves problems or performs tasks. Here, we introduce a tool that reduces large networks of coupled neurons to a much smaller set of differential equations that governs key statistics for the network as a whole, as opposed to tracking the individual dynamics of neurons and their connections. This approach is typically referred to as a mean-field system. As the mean-field system is derived from the original network of neurons, it is predictive for the behavior of the network as a whole and the parameters or distributions of parameters that appear in the mean-field system are identical to those of the original network. As such, bifurcation analysis is predictive for the behavior of the original network and predicts where in the parameter space the network transitions from one behavior to another. Additionally, here we show how networks of neurons can be constructed with a mean-field or macroscopic behavior that is prescribed. This occurs through an analytic extension of the Neural Engineering Framework (NEF). This can be thought of as an inverse mean-field approach, where the networks are constructed to obey prescribed dynamics as opposed to deriving the macroscopic dynamics from an underlying network. Thus, the work done here analyzes neuronal networks through both top-down and bottom-up approaches

The Libor market model and its calibration to the South African market

The South African interest rate market has mainly been focused on vanilla interest rate products and hence can be seen as underdeveloped in this regard when compared, for instance, to the associated equity market. Market participants subscribe this aspect to a lack of demand and sophistication of investors within the market. This is, however, expected to change given the influx of international banks into the South African market over the past couple of years. The current market methodology, for the pricing of vanilla interest rate options in the South African market, is the standard Black model with some mechanism to incorporate interest rate smiles. This mechanism is typically in the form of the SABR model. The most signi cant drawback of this approach is the fact that it models each forward rate in isolation. Hence, there is no way to incorporate the joint dynamics between different forward rates and consequently cannot be used for the pricing of exotic interest rate options. In anticipation of these new market developments, we explore the possibility of calibrating the LIBOR market model to the South African market. This dissertation follows a bottom up approach and hence considers all aspects associated with such an implementation. The work mainly focuses on the calibration to at-the-money interest rate options. A possible extension to the SABR model, while remaining within the LMM framework, is considered in the final chapter. CopyrightDissertation (MSc)--University of Pretoria, 2012.Mathematics and Applied Mathematicsunrestricte

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Tools to analyze cell signaling models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (v. 2, leaves 345-369).Diseases such as diabetes, some forms of cancer, hyper-tension, auto-immune diseases, and some viral diseases are characterized by complex interactions within the human body. Efforts to understand and treat these diseases have only been partially successful. There is currently a huge commercial and academic effort devoted to computational biology to address the shortfalls of qualitative biology. This research has become relevant due to the vast amounts of data now available from high-throughput techniques such as gene-chips, combinatorial chemistry, and fast gene sequencing. The goal of computational biology is to use quantitative models to test complex scientific hypotheses or predict desirable interventions. Consequently, it is important that the model is built to the minimum fidelity required to meet a specific goal, otherwise valuable effort is wasted. Unlike traditional chemical engineering, computational biology does not solely depend on deterministic models of chemical behavior. There is also widespread use of many types of statistical models, stochastic models, electro-static models, and mechanical models. All of these models are inferred from noisy data. It is therefore important to develop techniques to aide the model builder in their task of verifying and using these models to make quantitative predictions. The goal of this thesis is to develop tools for analysing the qualitative and quantitative characteristics of cell-signaling models. The qualitative behavior of deterministic models is studied in the first part of this thesis and the quantitative behavior of stochastic models is studied in the second part. A kinetic model of cell signaling is a common example of a deterministic model used in computational biology.(cont.) Usually such a model is derived from first-principles. The differential equations represent species conservation and the algebraic equations represent rate equations and equations to estimate rate constants. The researcher faces two key challenges once the model has been formulated: it is desirable to summarize a complex model by the phenomena it exhibits, and it is necessary to check whether the qualitative behavior of the model is verified by experimental observation. The key result of this research is a method to rearrange an implicit index one DAE into state-space form efficiently, amenable to standard control engineering analysis. Control engineering techniques can then be used to determine the time constants, poles, and zeros of the system, thus summarizing all the qualitative behavior of the system. The second part of the thesis focuses on the quantitative analysis of cell migration. It is hypothesized that mammalian cell migration is driven by responses to external chemical, electrical and mechanical stimulus. It is desirable to be able to quantify cell migration (speed, frequency of turning) to correlate output to experimental conditions (ligand concentration, cell type, cell medium, etc). However, the local concentration of signaling molecules and receptors is sufficiently low that a continuum model of cell migration is inadequate, i.e., it is only possible to describe cell motion in a probabilistic fashion ...by David Michael Collins.Ph.D

Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory

This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different artificial neural network (ANN) architectures (such as fully-connected feedforward ANNs, convolutional ANNs, recurrent ANNs, residual ANNs, and ANNs with batch normalization) and different optimization algorithms (such as the basic stochastic gradient descent (SGD) method, accelerated methods, and adaptive methods). We also cover several theoretical aspects of deep learning algorithms such as approximation capacities of ANNs (including a calculus for ANNs), optimization theory (including Kurdyka-{\L}ojasiewicz inequalities), and generalization errors. In the last part of the book some deep learning approximation methods for PDEs are reviewed including physics-informed neural networks (PINNs) and deep Galerkin methods. We hope that this book will be useful for students and scientists who do not yet have any background in deep learning at all and would like to gain a solid foundation as well as for practitioners who would like to obtain a firmer mathematical understanding of the objects and methods considered in deep learning.Comment: 601 pages, 36 figures, 45 source code

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

Essays in financial asset pricing

Three essays in financial asset pricing are given; one concerning the partial differential equation (PDE) pricing and hedging of a class of continuous/generalized power mean Asian options, via their (optimal) Lie point symmetry groups, leading to practical pricing formulas. The second presents high-frequency predictions of S&P 500 returns via several machine learning models, statistically significantly demonstrating short-horizon market predictability and economically significantly profitable (beyond transaction costs) trading strategies. The third compares profitability between these [(mean) ensemble] strategies and Asian option Δ-hedging, using results of the first. Interpreting bounds on arithmetic Asian option prices as ask and bid values, hedging profitability depends largely on securing prices closer to the bid, and settling midway between the bid and ask, significant profits are consistently accumulated during the years 2004-2016. Ensemble predictive trading the S&P 500 yields comparatively very small returns, despite trading much more frequently. The pricing and hedging of (arithmetic) Asian options are difficult and have spurred several solution approaches, differing in theoretical insight and practicality. Multiple families of exact solutions to relaxed power mean Asian option pricing boundary-value problems are explicitly established, which approximately satisfy the full pricing problem, and in one case, converge to exact solutions under certain parametric restrictions. Corresponding hedging parameters/ Greeks are derived. This family consists of (optimal) invariant solutions, constructed for the corresponding pricing PDEs. Numerical experiments explore this family behaviorally, achieving reliably accurate pricing. The second chapter studies intraday market return predictability. Regularized linear and nonlinear tree-based models enjoy significant predictability. Ensemble models perform best across time and their return predictability realizes economically significant profits with Sharpe ratios after transaction costs of 0.98. These results strongly evidence that intraday market returns are predictable during short time horizons, beyond that explainable by transaction costs. The lagged constituent returns are shown to hold significant predictive information not contained in lagged market returns or price trend and liquidity characteristics. Consistent with the hypothesis that predictability is driven by slow-moving trader capital, predictability decreased post-decimalization, and market returns are more predictable midday, on days with high volatility or illiquidity, and during financial crises