Milton Ferreira

Interactive and Multimedia Contents Associated with a System for Computer-Aided Assessment.

This article presents a research study addressing the development, implementation, evaluation, and use of Interactive Modules for Online Training (MITO) of mathematics in higher education. This work was carried out in the context of the MITO project, which combined several features of the learning and management system Moodle, the computer-aided assessment for mathematics STACK, the mathematical software GeoGebra, several packages from the type-setting program LaTeX, and tutorial videos. A total of 1,962 students participated in this study. Two groups of students taking a calculus course were selected for a deeper analysis. With regard to usability and functionality, the results indicate that MITO scored well in almost all aspects, which is fundamental for their introduction into formal university courses. The analysis of the data reveals that the use of MITO educational contents by students mainly occurs about 1 week and a half prior to the evaluations. Moreover, there is a strong correlation between the results of online assessments on MITO in a continuous assessment model and the final grade on the course.

Gyrogroups in Projective Hyperbolic Clifford Analysis

Using the projective hyperbolic model in Clifford analysis we show that different velocities in special relativity theory (non-standard velocities, coordinate velocities and proper velocities) corresponds to taking different realizations of the hyperbolic geometry in the projective model. A full description of the changes is given, together with a proof of the isomorphism of the related gyrogroup structures (Mobius, Einstein, and proper velocity gyrogroups).

Eigenfunctions and Fundamental Solutions of the Caputo Fractional Laplace and Dirac Operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Clifford Analysis and the Continuous Spherical Wavelet Transform

We present a group-theoretical approach for the continuous wavelet transform on the sphere S n−1, based on the Lorentz group Spin(1, n) (the conformal group of the unit sphere). We introduce transformations on the sphere based on the decomposition of the group Spin(1, n) into the maximal compact subgroup of rotations (Spin(n)) and the set of Mobius transformations in ℝn of the form ϕ a (x) = (x − a)(1 + ax)−1, |a| < 1. This approach presents an advantage of allowing the full use of the whole of the conformal group Spin(1, n), and in such way, it is a generalization of the continuous wavelet transform defined by J. P. Antoine and P. Vandergheynst (see [1], [2]). We will give an account of the influence of the parameter a arising in the definition of dilatations / contractions on the sphere. Finally we give different representations (with different properties) for the Hilbert space L 2 (S n−1) and the Hardy space H 2.

Monogenic Wavelets Over the Unit Ball

In this article we de ne a monogenic wavelet transform for quaternion valued functions on the unit ball B in R3 based on representations of the group of Mobius transformations which maps the unit ball onto itself.

Fundamental solution of the multi-dimensional time fractional telegraph equation

Abstract In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives

Abstract In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator where and the fractional derivatives , , are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.

Harmonic analysis on the proper velocity gyrogroup

In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter z and on the radius t of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherels Theorem. In the limit of large t, t → +∞, the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on R, thus unifying hyperbolic and Euclidean harmonic analysis.

Complex dilations, shearings, and rotations

In this short paper we show how to construct some complex affine transformations such as complex dilations, complex shearings, and complex rotations (Euclidean or hyperbolic), in the pseudo-Hermitian space Hn,m. These transformations appear as spin-1 actions of appropriate elements of the complex Lie algebra of complex blades in Hn,m.

Monogenic frames for an integral transform on the unit sphere

In this article we construct Banach frames of monogenic functions and prove Jackson-type theorems for the best n-point approximation.