Fractional calculus analysis of the cosmic microwave background

Cosmic microwave background (CMB) radiation is the imprint from an early stage of the Universe and investigation of its properties is crucial for understanding the fundamental laws governing the structure and evolution of the Universe. Measurements of the CMB anisotropies are decisive to cosmology, since any cosmological model must explain it. The brightness, strongest at the microwave frequencies, is almost uniform in all directions, but tiny variations reveal a spatial pattern of small anisotropies. Active research is being developed seeking better interpretations of the phenomenon. This paper analyses the recent data in the perspective of fractional calculus. By taking advantage of the inherent memory of fractional operators some hidden properties are captured and described.

A satisfação dos trabalhadores com as práticas de gestão de recursos humanos: o caso de uma Instituição do Ensino Superior Público em Portugal

Este estudo tem como objetivos: (1) conhecer as práticas desenvolvidas numa organização do Ensino Superior Público Português; (2) conhecer a tipologia das práticas de GRH de cariz tradicional e de cariz estratégico; (3) perceber em que medida as práticas de GRH estão relacionadas com a área de qualificação dos responsáveis do departamento de RH; (4) averiguar o grau de satisfação que os trabalhadores sentem com as Práticas de Gestão de Recursos Humanos desenvolvidas e a sua relação com a área de qualificação dos responsáveis do departamento de RH. Foi utilizada uma metodologia mista, que possibilita ampliar a obtenção de resultados em abordagens investigativas, proporcionando ganhos relevantes para a pesquisa. É realizado um primeiro estudo exploratório, que utiliza uma metodologia mista quantitativa e qualitativa, com recurso a uma entrevista semiestruturada e inquérito realizados aos responsáveis de RH, e que tem como objetivos identificar e caracterizar as Práticas de GRH vigentes na Organização e, consequentemente, averiguar se se aproximam das designadas na literatura, assim como averiguar o grau de intervenção do DRH no desenvolvimento das PGRH e caraterizar o perfil do responsável de RH na Organização, averiguando se a área de formação de RH influencia as Práticas de GRH desenvolvidas. No segundo estudo, recorremos a uma metodologia quantitativa com recurso ao inquérito por questionário, aplicado aos trabalhadores que exercem funções a tempo integral, para averiguar o grau de satisfação dos trabalhadores em relação às Práticas de Gestão de Recursos Humanos. Na compilação dos dois estudos foi nosso objetivo obter respostas às questões que orientaram a nossa investigação. Na parte final da dissertação são discutidos os principais resultados obtidos e apresentadas as conclusões do estudo aqui levado a cabo. Os resultados sugerem que: 1) as PGRH existentes são essencialmente de cariz tradicional, em especial a gestão administrativa; 2) as PGRH predominantes são: o Planeamento de Recursos Humanos, a Análise e Descrição de Funções, o Recrutamento e Seleção, a Formação e Desenvolvimento, a Gestão Administrativa, a Comunicação e a Partilha de Informação, Ética e Deontologia e o Estatuto Disciplinar; 3) existe pouco recurso ao outsourcing para as PGRH; 4) o grau de intervenção DRH baseia-se em atividades de cariz mais administrativo; 5) as práticas tradicionais de RH são aquelas que requerem mais tempo ao DRH; 6) não existe relação entre o tipo de PGRH e a área de qualificação do responsável do DRH; 7) as PGRH são realizadas seguindo essencialmente as normas legais e regras rígidas da GRH na AP; 8) algumas PGRH não são entendidas em contexto da AP, como importantes pelos gestores, embora já sejam desenvolvidos alguns procedimentos dessas práticas; 9) a PGRH da formação e desenvolvimento não é corretamente desenvolvida e não dá cumprimento ao estipulado na lei; 10) a gestão de carreiras e o sistema de compensação e recompensas são entendidas como inexistentes, porque não existem promoções e progressões desde 2005; 11) a avaliação do desempenho é um sistema burocrático e ritualista com fins de promoção e compensação, sem efeitos práticos no momento atual, e que causa insatisfação e o sentimento de injustiça; 12) existem problemas de comunicação quanto a partilha e uniformização de procedimentos entre UO; 13) a satisfação dos trabalhadores é maior com as PGRH de tipo tradicional, nomeadamente na gestão administrativa, recrutamento e seleção, análise e descrição de funções, acolhimento, integração e socialização 14) a satisfação é menor na gestão de carreiras, no sistema de compensação e recompensas e na avaliação do desempenho; 15) quanto a relação entre o grau de satisfação e as características sócio demográficas e profissionais dos inquiridos, os casos com significância mostram que os trabalhadores com 10 ou mais anos de antiguidade tendem a sentir mais satisfação com as práticas em GRH; 16) existe mais satisfação dos trabalhadores das UO onde o responsável de DRH possui formação na área de RH.

Using interval graphs in an order processing optimization problem

In this paper we address an order processing optimization problem known as minimization of open stacks (MOSP). We present an integer pro gramming model, based on the existence of a perfect elimination scheme in interval graphs, which finds an optimal sequence for the costumers orders.

A mathematical model for supermarket order picking

Order picking consists in retrieving products from storage locations to satisfy independent orders from multiple customers. It is generally recognized as one of the most significant activities in a warehouse (Koster et al, 2007). In fact, order picking accounts up to 50% (Frazelle, 2001) or even 80% (Van den Berg, 1999) of the total warehouse operating costs. The critical issue in today’s business environment is to simultaneously reduce the cost and increase the speed of order picking. In this paper, we address the order picking process in one of the Portuguese largest companies in the grocery business. This problem was proposed at the 92nd European Study Group with Industry (ESGI92). In this setting, each operator steers a trolley on the shop floor in order to select items for multiple customers. The objective is to improve their grocery e-commerce and bring it up to the level of the best international practices. In particular, the company wants to improve the routing tasks in order to decrease distances. For this purpose, a mathematical model for a faster open shop picking was developed. In this paper, we describe the problem, our proposed solution as well as some preliminary results and conclusions.

Resolução do problema de atribuição de salas a exames

Este trabalho pretende resolver o problema das alocações de salas a exames no Departamento de Engenharia Mecânica do Instituto Superior de Engenharia do Porto. A solução desenvolvida atribui salas a exames respeitando as restrições de capacidade de salas e a restrição de realização dum único exame por sala num determinado período, por forma a minimizar a atribuição de salas e, consequentemente, docentes a exames. Foi criado um modelo matemático, que representa as variáveis relevantes do problema, e realiza a sua implementação numa plataforma informática amigável para o utilizador. O modelo matemático foi validado comparando as suas soluções com as obtidas através do processo manual. Os resultados do novo método demonstram a sua supremacia relativamente ao modelo atual. No futuro, poderá ser estudada a possibilidade de usar esta ferramenta na resolução do mesmo problema em realidades diferentes da do Departamento de Engenharia Mecânica do ISEP.

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.

Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

Rhapsody in Fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Pseudo Phase Plane and Fractional Calculus Modelling of Western Global Economic Downturn

This paper applies Pseudo Phase Plane (PPP) and Fractional Calculus (FC) mathematical tools for modeling world economies. A challenging global rivalry among the largest international economies began in the early 1970s, when the post-war prosperity declined. It went on, up to now. If some worrying threatens may exist actually in terms of possible ambitious military aggression, invasion, or hegemony, countries’ PPP relative positions can tell something on the current global peaceful equilibrium. A global political downturn of the USA on global hegemony in favor of Asian partners is possible, but can still be not accomplished in the next decades. If the 1973 oil chock has represented the beginning of a long-run recession, the PPP analysis of the last four decades (1972–2012) does not conclude for other partners’ global dominance (Russian, Brazil, Japan, and Germany) in reaching high degrees of similarity with the most developed world countries. The synergies of the proposed mathematical tools lead to a better understanding of the dynamics underlying world economies and point towards the estimation of future states based on the memory of each time series.

Numerical calculation of the left and right fractional derivatives

The calculation of fractional derivatives is an important topic in scientific research. While formal definitions are clear from the mathematical point of view, they pose limitations in applied sciences that have not been yet tackled. This paper addresses the problem of obtaining left and right side derivatives when adopting numerical approximations. The results reveal the relationship between the resulting distinct values for different fractional orders and types of signals.

What is a fractional derivative?

This paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. Two criteria for required by a fractional operator are formulated. The Grünwald–Letnikov, Riemann–Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria. A Leibniz rule is also obtained for the Riesz potential.

Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

Nonlinear Dynamics for Local Fractional Burgers Equation Arising in Fractal Flow

The local fractional Burgers’ equation (LFBE) is investigated from the point of view of local fractional conservation laws envisaging a nonlinear local fractional transport equation with a linear non-differentiable diffusion term. The local fractional derivative transformations and the LFBE conversion to a linear local fractional diffusion equation are analyzed.

A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.