Geometrical wave equation and the cauchy-like theorem for octonions

Riemann surfaces, cohomology and homology groups, Cartan's spinors and triality, octonionic projective geometry, are all well supported by Complex Structures [1], [2], [3], [4]. Furthermore, in Theoretical Physics, mainly in General Relativity, Supersymmetry and Particle Physics, Complex Theory Plays a Key Role [5], [6], [7], [8]. In this context it is expected that generalizations of concepts and main results from the Classical Complex Theory, like conformal and quasiconformal mappings [9], [10] in both quaternionic and octonionic algebra, may be useful for other fields of research, as for graphical computing enviromment [11]. In this Note, following recent works by the autors [12], [13], the Cauchy Theorem will be extended for Octonions in an analogous way that it has recentely been made for quaternions [14]. Finally, will be given an octonionic treatment of the wave equation, which means a wave produced by a hyper-string with initial conditions similar to the one-dimensional case.

Stable singularities of co-rank one quasi homogeneous map germs from (ℂn+1, 0) to (ℂn, 0), n = 2, 3

In this article, we investigate the geometry of quasi homogeneous corank one finitely determined map germs from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. We give a complete description, in terms of the weights and degrees, of the invariants that are associated to all stable singularities which appear in the discriminant of such map germs. The first class of invariants which we study are the isolated singularities, called 0-stable singularities because they are the 0-dimensional singularities. First, we give a formula to compute the number of An points which appear in any stable deformation of a quasi homogeneous co-rank one map germ from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. To get such a formula, we apply the Hilbert's syzygy theorem to determine the graded free resolution given by the syzygy modules of the associated iterated Jacobian ideal. Then we show how to obtain the other 0-stable singularities, these isolated singularities are formed by multiple points and here we use the relation among them and the Fitting ideals of the discriminant. For n = 2, there exists only the germ of double points set and for n = 3 there are the triple points, named points A1,1,1 and the normal crossing between a germ of a cuspidal edge and a germ of a plane, named A2,1. For n = 3, there appear also the one-dimensional singularities, which are of two types: germs of cuspidal edges or germs of double points curves. For these singularities, we show how to compute the polar multiplicities and also the local Euler obstruction at the origin in terms of the weights and degrees. © 2013 Pushpa Publishing House.

On the control of vibrations using synchrophasing

This paper describes the application of a technique, known as synchrophasing, to the control of machinery vibration. It is applicable to machinery installations, in which several synchronous machines, such as those driven by electrical motors, are fitted to an isolated common structure known as a machinery raft. To reduce the vibration transmitted to the host structure to which the machinery raft is attached, the phase of the electrical supply to the motors is adjusted so that the net transmitted force to the host structure is minimised. It is shown that while this is relatively simple for an installation consisting of two machines, it is more complicated for installations in which there are more than two machines, because of the interaction between the forces generated by each machine. The development of a synchrophasing scheme, which has been applied to propeller aircraft, and is known as Propeller Signature Theory (PST) is discussed. It is shown both theoretically and experimentally, that this is an efficient way of controlling the phase of multiple machines. It is also shown that synchrophasing is a worthwhile vibration control technique, which has the potential to suppress vibration transmitted to the host structure by up to 20 dB at certain frequencies. Although the principle of synchronisation has been demonstrated on a one-dimensional structure, it is believed that this captures the key features of the approach. However, it should be realised that the mode-shapes of a machinery raft may be more complex than that of a one-dimensional structure and this may need to be taken into account in a real application. © 2013 Elsevier Ltd.

Cálculo de funções de Wannier eletrônicas para aplicações em ciência dos materiais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Funções de Wannier para cristais fotônicos unidimensionais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

O oscilador harmônico singular revisitado

The one-dimensional Schrödinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive singular oscillator exhibits an infinite number of acceptable solutions provided the parameter responsible for the singularity is greater than a certain critical value, in disagreement with the literature. The problem for the whole line exhibits a two-fold degeneracy in the case of the singular oscillator, and the intrusion of additional solutions in the case of a nonsingular oscillator. Additionally, it is shown that the solution of the singular oscillator can not be obtained from the nonsingular oscillator via perturbation theory.

Visualization of Protein Folding Funnels in Lattice Models

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

Confinement effects and why carbon nanotube bundles can work as gas sensors

Carbon nanotubes have been at the forefront of nanotechnology, leading not only to a better understanding of the basic properties of charge transport in one dimensional materials, but also to the perspective of a variety of possible applications, including highly sensitive sensors. Practical issues, however, have led to the use of bundles of nanotubes in devices, instead of isolated single nanotubes. From a theoretical perspective, the understanding of charge transport in such bundles, and how it is affected by the adsorption of molecules, has been very limited, one of the reasons being the sheer size of the calculations. A frequent option has been the extrapolation of knowledge gained from single tubes to the properties of bundles. In the present work we show that such procedure is not correct, and that there are qualitative differences in the effects caused by molecules on the charge transport in bundles versus isolated nanotubes. Using a combination of density functional theory and recursive Green's function techniques we show that the adsorption of molecules randomly distributed onto the walls of carbon nanotube bundles leads to changes in the charge density and consequently to significant alterations in the conductance even in pristine tubes. We show that this effect is driven by confinement which is not present in isolated nanotubes. Furthermore, a low concentration of dopants randomly adsorbed along a two-hundred nm long bundle drives a change in the transport regime; from ballistic to diffusive, which can account for the high sensitivity to different molecules.

Criação de um banco de itens para avaliar o conhecimento do câncer de cabeça e pescoço

Pós-graduação em Odontologia Preventiva e Social - FOA

Construção de uma escala para avaliação da pro ciência em conteúdos matemáticos básicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Estudo de sistemas não-lineares em mecânica quântica

The main goal of this work is to investigate the effects of a nonlinear cubic term inserted in the Schrödinger equation for one-dimensional potentials studied in Quantum Mechanics textbooks. Being the main tool the numerical analysis in a large number of works, the analysis of this effect by this term in the potential itself, in order to work with an analytical solution, can be considered something new. For the harmonic oscillator potential, the analysis was made from a numerical method, comparing the result with the known results in the literature. In the case of the infinite well potential and the step potential, hoping to work with an analytical solution, by construction we started with the known wavefunction for the linear case noting the effects in the other physical quantities. The coupling of the physical quantities involved in this work has yielded, besides many complications in the calculations, a series of conditions on the existence and validity of the solutions in regard to the system possible configurations