Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

A Fractional Perspective to the Bond Graph Modelling of World Economies

Inspired in dynamic systems theory and Brewer’s contributions to apply it to economics, this paper establishes a bond graph model. Two main variables, a set of inter-connectivities based on nodes and links (bonds) and a fractional order dynamical perspective, prove to be a good macro-economic representation of countries’ potential performance in nowadays globalization. The estimations based on time series for 50 countries throughout the last 50 decades confirm the accuracy of the model and the importance of scale for economic performance.

Modeling Vegetable Fractals by means of Fractional-order Equations

This paper characterizes four ‘fractal vegetables’: (i) cauliflower (brassica oleracea var. Botrytis); (ii) broccoli (brassica oleracea var. italica); (iii) round cabbage (brassica oleracea var. capitata) and (iv) Brussels sprout (brassica oleracea var. gemmifera), by means of electrical impedance spectroscopy and fractional calculus tools. Experimental data is approximated using fractional-order models and the corresponding parameters are determined with a genetic algorithm. The Havriliak-Negami five-parameter model fits well into the data, demonstrating that classical formulae can constitute simple and reliable models to characterize biological structures.

Fractional Dynamics in the Rayleigh's Piston

This paper studies the dynamics of the Rayleigh piston using the modeling tools of Fractional Calculus. Several numerical experiments examine the effect of distinct values of the parameters. The time responses are transformed into the Fourier domain and approximated by means of power law approximations. The description reveals characteristics usual in Fractional Brownian phenomena.

Fractional State Space Analysis of Economy Systems

This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners’ modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.

Challenges in the creation and development of a mathematics MOOC

The fast development of distance learning tools such as Open Educational Resources (OER) and Massive Open Online Courses (MOOC or MOOCs) are indicators of a shift in the way in which digital teaching and learning are understood. MOOC are a new style of online classes that allow any person with web access, anywhere, usually free of charge, to participate through video lectures, computer graded tests and discussion forums. They have been capturing the attention of many higher education institutions around the world. This paper will give us an overview of the “Introduction to Differential Calculus” a MOOC Project, created by an engaged volunteer team of Mathematics lecturers from four schools of the Polytechnic Institute of Oporto (IPP). The MOOC theories and their popularity are presented and complemented by a discussion of some MOOC definitions and their inherent advantages and disadvantages. It will also explore what MOOC mean for Mathematics education. The Project development is revealed by focusing on used MOOC structure, as well as the quite a lot of types of course materials produced. It ends with a presentation of a short discussion about problems and challenges met throughout the development of the project. It is also our goal to contribute for a change in the way teaching and learning Mathematics is seen and practiced nowadays, trying to make education more accessible to as many people as possible and increase our institution (IPP) recognition.

Visualizing control systems performance: A fractional perspective

This article presents a novel method for visualizing the control systems behavior. The proposed scheme uses the tools of fractional calculus and computes the signals propagating within the system structure as a time/frequency-space wave. Linear and nonlinear closed-loop control systems are analyzed, for both the time and frequency responses, under the action of a reference step input signal. Several nonlinearities, namely, Coulomb friction and backlash, are also tested. The numerical experiments demonstrate the feasibility of the proposed methodology as a visualization tool and motivate its extension for other systems and classes of nonlinearities.

Aplicação das FPAA na realização de sistemas de ordem fraccionária

Esta tese de dissertação tem como principal objetivo a implementação de controladores fracionários utilizando diapositivos analógicos FPAA (Field Programable Analog Array). Embora estes dispositivos já não sejam um tecnologia recente, não tiveram grande aceitação comercial, daí não ter sido grande a sua evolução nesta última década. Mas para a elaboração de alguns circuitos analógicos, nomeadamente filtros, amplificadores e mesmo controladores PID (Proporcional-Integrativo-Derivativo) analógicos torna-se numa ferramenta que pode facilitar o projeto e implementação. Para a realização deste estudo, utilizou-se a placa de desenvolvimento da Anadigm AN231K04-DVLP3 juntamente com o software disponibilizado pela mesma empresa, o AnadigmDesigner2. Para a simulação e observação dos resultados foi utilizada a DAQ (Data Acquisition) Hilink da Zelton juntamente com o software Matlab. De forma a testar a implementação dos controladores fracionários nas FPAA foram realizados alguns circuitos no software e enviados para a FPAA comparando os resultados obtidos na simulação com os visualizados no osciloscópio. Por último foi projetado um controlador PIlDm recorrendo aos métodos de aproximação inteira descritos neste documento implementados na FPAA recorrendo ao uso de filtros de primeira e segunda ordem.

Projeto de estabilidade de internato masculino

O presente documento corresponde à realização de um projeto de estruturas em betão armado e estrutura metálica. A escolha deste tema teve como propósito aplicar os conhecimentos adquiridos ao longo do percurso académico, sobretudo em disciplinas do ramo de estruturas. O trabalho desenvolvido é um Projeto de Estabilidade de Betão Armado e Estrutura Metálica de um Internato Masculino para Padres em Onameva, Cunene – Angola. A elaboração deste projeto decorre da realização de um estágio formal de 8 meses na empresa CALCULUS, Miguel Barros – Engenharia, LDA, com o objetivo principal de reunir os requisitos necessários à admissão na Ordem dos Engenheiros. No contexto laboral de um gabinete de projetos de engenharia civil, foi feita a integração de forma gradual e sustentada das competências adquiridas ao longo da formação académica e vivenciadas situações reais de trabalho profissional. De entre as atividades desenvolvidas no estágio, foi escolhido o projeto de estabilidade acima referido pela oportunidade de realizar todas as etapas do seu desenvolvimento, desde a análise da arquitetura até à elaboração das peças escritas e desenhadas, dando assim resposta às exigências necessárias para a conclusão do Mestrado em Engenharia Civil, no ramo de Estruturas, pelo Instituto Superior de Engenharia do Porto.

On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

A Playful Glance at Hierarchical Questions for Two-Way Alternating Automata

Two-way alternating automata were introduced by Vardi in order to study the satisfiability problem for the modal μ-calculus extended with backwards modalities. In this paper, we present a very simple proof by way of Wadge games of the strictness of the hierarchy of Motowski indices of two-way alternating automata over trees.

Large amplitude motions in the S1 state of formic acid /

A detailed theoretical investigation of the large amplitude motions in the S, excited electronic state of formic acid (HCOOH) was done. This study focussed on the the S, «- So electronic band system of formic acid (HCOOH). The torsion and wagging large amplitude motions of the S, were considered in detail. The potential surfaces were simulated using RHF/UHF ab-initio calculations for the two electronic states. The energy levels were evaluated by the variational method using free rotor basis functions for the torsional coordinates and harmonic oscillator basis functions for the wagging coordinates. The simulated spectrum was compared to the slit-jet-cooled fluorescence excitation spectrum allowing for the assignment of several vibronic bands. A rotational analysis of certain bands predicted that the individual bands are a mixture of rotational a, b and c-type components.The electronically allowed transition results in the c-type or Franck-Condon band and the electronically forbidden, but vibronically allowed transition creates the a/b-type or Herzberg-Teller components. The inversion splitting between these two band types differs for each band. The analysis was able to predict the ratio of the a, b and c-type components of each band.

Infrared spectroscopy of Mg-doped SrRuO3 thin films /

The reflectance of thin films of magnesium doped SrRu03(Mg-SR0) produced by pulsed laser deposition on SrTiOa (100) substrates has been measured at room temperature between 100 and 7500 cm~^. The films were chosen to have wide range of thickness, stoichiometry and electrical properties. As the films were very thin (less than 300 nm), and some were insulating the reflectance data shows structures due to both the film and the substrate. Hence, the data was analyzed using Kramers-Kronig constrained variational fitting (VDF) method to extract the real optical conductivity of the Mg-SRO films. Although the VDF technique is flexible enough to fit all features of the reflectance spectra, it seems that VDF could not eliminate the substrate's contribution from fllm conductivity results. Also the comparison of the two different programs implementing VDF fltting shows that this technique has a uniqueness problem. The optical properties are discussed in light of the measured structural and transport properties of the fllms which vary with preparation conditions and can be correlated with differences in stoichiometry. This investigation was aimed at checking the VDF technique and also getting answer to the question whether Mg^"*" substitutes in to Ru or Sr site. Analysis of our data suggests that Mg^+ goes to Ru site.

Elliptical orbital studies of H21 and H2 molecules /|nS. K. Gupta. -- 260 St. Catharines, Ont. : [s. n.],

Calculations are performed on the \S <:Jd ground states of d ' + the H and HC) molecules using a basis set of non-integral ~ ~ I elliptical orbitals. Different variational wavefunctions constructed i- for H~ involved one parameter to three par~~eter variation. In order to l"'educe the ntunber of parameters in most commonly 0- used basis orbitals set, the importance of the term (,+~) Y\ over the term ;u 'Where n is a variational pararneter and the value of cr may be given by boundary condition or cusp condition is outlined in Chapters II and III. It is found that the two parameter -+ variational energies than any wavefunction published to date for small and moderate internuclear separations. c;. In order to find out the importance of the term (I +~ ) Y\ over ;U for the two electron problem, the variational energy is computed for the H~ molecule from unrestricted two parameter closed shell wavefunctions including the term U+ft)