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- Symbolic computation applied to the study of the kernel of special classes of paired singular integral operatorsPublication . Conceição, Ana C.Operator theory has many applications in several main scientific research areas (structural mechan-ics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and theimportance of its study is globally acknowledged. On the study of the operator’s kernel some progress has beenachieved for some specific classes of singular integral operators whose properties allow the use of particular strate-gies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes ofsingular integral operators but are not designed to be implemented on a computer. The main goal of this paper is toshow how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel ofspecial classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle.It is described how some factorization algorithms can be used to compute the dimension of the kernel of specialclasses of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar],and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility ofimplementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Forthe essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The papercontains some interesting and nontrivial examples obtained with the use of a computer algebra system.
- Dynamic and interactive tools to support teaching and learningPublication . Conceição, Ana C.The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the theoretical results and often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. The main goal of this work is to show how some new dynamic and interactive tools, created with Mathematica and available in the Computable Document Format (CDF), can be used as active learning tools to promote better student activity and engagement in the learning process. The CDF format allows anyone with a computer to use them, at no cost, even without an active Wolfram Mathematica license. Besides that, the presented tools are very intuitive to use which makes it suitable for less experienced users. Some tools applicable to several mathematics concepts taught in higher education will be presented. This kind of tools can be used either in a remote or classroom learning environment. The corresponding CDF files are made available as supplement of the online edition of this article.
- Symbolic computation applied to cauchy type singular integralsPublication . Conceição, Ana C.; Pires, Jéssica C.The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.