To be or not to be, the importance of Digital Identity in the networked society

The emergence of the web has had a deep impact at different levels of our society, changing the way people connect, interact, share information, learn and work. In the current knowledge economy, participatory media seems to play an important part in everyday interactions. The term “digital identity” is becoming part of both our lexicon and our lives. This paper explores some of the aspect s regarding approaches and practices of educators, using web technologies to foster their digital identity within their networks and, at the same time, developing a social presence to complement their professional and academic profiles. In fact, we think it is imperative to discuss the relationship between our social presence and our professional life, as online the two are often intertwined. We present the issues the web poses through dichotomies: open or closed, genuine or fake, single or multiple. We also comment on different approaches to these dichotomies through examples extracted from recent projects, drawing from user’s experiences in building their digital identities. This paper looks at the importance of digital identity in the current networked society, by reviewing the contemporaneous scenario of the participatory web, raising a set of questions about the advantages and implication of consciously developing one’s digital identity, thus opening the discussion regarding openness, uniqueness and integrity in connection with one’s digital identity. This paper is also a reflection of thinking and practice in progress, drawing from examples and real-life situations observed in a diversity of projects. The issue could be reduced, perhaps, to whether one consciously becomes a part of the digital world or not, and how that participation is managed. It is up to us to manage it wisely, and guide knowledge workers in their journey to create theirs

A survey on fractional variational calculus

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in 'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201

Calculus of variations with fractional derivatives and fractional integrals

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.Comment: Accepted (July 6, 2009) for publication in Applied Mathematics Letter

The Variable-Order Fractional Calculus of Variations

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online as a SpringerBrief in Applied Sciences and Technology at [https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos, detected by the authors while reading the galley proofs, were corrected, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating the fractional problem either by discretizing the fractional term or expanding the fractional derivatives as a series involving integer order derivatives. The former method, as a subclass of direct methods in the theory of calculus of variations, uses finite differences, Grunwald-Letnikov definition in this case, to discretize the fractional term. Any quadrature rule for integration, regarding the desired accuracy, is then used to discretize the whole problem including constraints. The final task in this method is to solve a static optimization problem to reach approximated values of the unknown functions on some mesh points. The latter method, however, approximates fractional problems by classical ones in which only derivatives of integer order are present. Precisely, two continuous approximations for fractional derivatives by series involving ordinary derivatives are introduced. Local upper bounds for truncation errors are provided and, through some test functions, the accuracy of the approximations are justified. Then we substitute the fractional term in the original problem with these series and transform the fractional problem to an ordinary one. Hereafter, we use indirect methods of classical theory, e.g. Euler-Lagrange equations, to solve the approximated problem. The methods are mainly developed through some concrete examples which either have obvious solutions or the solution is computed using the fractional Euler-Lagrange equation.Comment: This is a preprint of a paper whose final and definite form appeared in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control (Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova Science Publishers, New York, 2014. (See http://www.novapublishers.com/catalog/product_info.php?products_id=46851). Consists of 39 page