Using the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space R ⊕ V , where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyrosubgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors.
In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ∆ (α,β,γ) + := D 1+α x + 0 + D 1+β y + 0 + D 1+γ z + 0 , where (α, β, γ) ∈ ]0, 1] 3 , and the fractional derivatives D 1+α x + 0 , D 1+β y + 0 , D 1+γ z + 0 are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator ∆ (α,β,γ) + in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.
In this paper we consider a Möbius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example we will present the factorization of the Möbius gyrogroup of the unit ball in R n linked to the proper Lorentz group Spin + (1, n). (2000). Primary: 20N05; Secondary: 30G35.
Mathematics Subject Classification
We consider the conformal group of the unit sphere S n−1 , the so-called proper Lorentz group Spin + (1, n), for the study of spherical continuous wavelet transforms. Our approach is based on the method for construction of general coherent states associated to square integrable group representations over homogeneous spaces. The underlying homogeneous space is an extension to the whole of the group Spin + (1, n) of the factorization of the gyrogroup of the unit ball by an appropriate gyro-subgroup. Sections on it are constituted by rotations of the subgroup Spin(n) and Möbius transformations of the type ϕ a (x) = (x − a)(1 + ax) −1 , where a belongs to a given section on a quotient space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations on the unit sphere.
In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation with time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS expressed in terms of a multivariate MittagLeffler function in the Fourier domain. 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 n and of the fractional parameters α and β.
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The growth of the higher education population and different school paths to access an academic degree has increased the heterogeneity of students inside the classroom. Consequently, the effectiveness of traditional teaching methods has reduced. This paper describes the design, development, implementation and evaluation of a tutoring system (TS) to improve student's engagement in higher mathematics. The TS design was based on the Personalized System of Instruction of the Mastery Learning pedagogical approach and can be implemented in any higher education course with mathematics needs. The TS consists on small self‐paced modularized units of educational contents, including tutorial videos, notes and formative e‐assessment with personalized feedback. The TS ensures that the student is only allowed to proceed to the next unit after he or she achieves the required mastery criterion of the current unit. The TS was implemented in the Quantitative Methods course of an undergraduate degree and received good acceptance from students. It was also recognized that TS contributed to learning and engagement with the discipline. Through an experimental research experience, it has been shown that the imposition of restrictions on the advance to the next level by a mastery criterion leads to a significant improvement in student's engagement and performance.
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