On the periodic orbits of the third-order differential equation x ′′′ − µx ′′ + x ′ − µx = εF (x, x ′ , x ′′)

In this paper we study the periodic orbits of the third-order differential equation x ′′′−µx ′′+ x ′ − µx = εF (x, x ′ , x ′′), where ε is a small parameter and the function F is of class C 2 .

Geometrical coupling fields of a hypercomplex type

This paper presents an application of Laplace's equation obtained from a quaternionic function that satisfies the Cauchy-Riemann conditions determined earlier by Borges and Machado [#!BorgesZeMarcio!#]. Therefore, we show that it is possible to express in a single equation gravity, electric and magnetic potential fields, and this expression can only be provided due to a function that will be called here the coupling function.

Geometrical hypercomplex coupling between electric and gravitational fields

The present work shows a coupling of electrical and gravitational fields through Cauchy-Riemann conditions for quaternions present in a previous paper [1]. It is also obtained an extended version of the Laplace-like equations for quaternions, now written in terms of both electric and gravitational fields.

A note on the hypercomplex riemann-cauchy like relations for quaternions and laplace equations

In this Note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier [1]. As in the theory of functions of a complex variable, it is expected that this new set of Laplace-Like equations might be applied to a large number of Physical problems, providing new insights in the Classical Fields Theory.

Fourier series for quaternions and the square of the error theorem

In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

A note on pairs of regular foliations on the solid torus

We discuss the geometry of the pair of foliations on a solid torus given by the Reeb foliation together with discs transverse to the boundary of the torus.

A decoding procedure which improves code rate and error corrections

Corresponding to $C_{0}[n,n-r]$, a binary cyclic code generated by a primitive irreducible polynomial $p(X)\in \mathbb{F}_{2}[X]$ of degree $r=2b$, where $b\in \mathbb{Z}^{+}$, we can constitute a binary cyclic code $C[(n+1)^{3^{k}}-1,(n+1)^{3^{k}}-1-3^{k}r]$, which is generated by primitive irreducible generalized polynomial $p(X^{\frac{1}{3^{k}}})\in \mathbb{F}_{2}[X;\frac{1}{3^{k}}\mathbb{Z}_{0}]$ with degree $3^{k}r$, where $k\in \mathbb{Z}^{+}$. This new code $C$ improves the code rate and has error corrections capability higher than $C_{0}$. The purpose of this study is to establish a decoding procedure for $C_{0}$ by using $C$ in such a way that one can obtain an improved code rate and error-correcting capabilities for $C_{0}$.

On the zeros of a class of generalized hypergeometric polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The spin-orbit resonant problem including core-mantle gravitational coupling

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Processing optimized for symmetry in the problem of evasive maneuvers

The increasing number of space debris in operating regions around the earth constitutes a real threat to space missions. The goal of the research is to establish appropriate scientific-technological conditions to prevent the destruction and/or impracticability of spacecraft in imminent collision in these regions. A definitive solution to this problem has not yet been reached with the degree of precision that the dynamics of spatial objects (vehicle and debris) requires mainly due to the fact that collisions occur in chains and fragmentation of these objects in the space environment. This fact threatens the space missions on time and with no prospects for a solution in the near future. We present an optimization process in finding the initial conditions (CIC) to collisions, considering the symmetry of the distributions of maximum relative positions between spatial objects with respect to the spherical angles. For this, we used the equations of the dynamics on the Clohessy-Witshire, representing a limit of validation that is highly computationally costly. We simulate different maximum relative positions values of the corresponding initial conditions given in terms of spherical angles. Our results showed that there are symmetries that significantly reduce operating costs, such that the search of the CIC is advantageously carried out up to 4 times the initial processing routine. Knowledge of CIC allows the propulsion system operating vehicle implement evasive maneuvers before impending collisions with space debris.

Searching for capture and escape trajectories around Jupiter using its Galilean satellites

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A study of the errors of the averaged models in the restricted three-body problem in a short time scale

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Celestial mechanics: from the errant stars to guidance of spacecrafts

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Pareto Frontier for the time-energy cost vector to an Earth-Moon transfer orbit using the patched-conic approximation

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Práticas algébricas no contexto da modelagem compreendida como proposta pedagógica

Pós-graduação em Educação Matemática - IGCE