Este trabalho tem como objetivo apresentar uma introdução ao grupo S O ( 4 ) e duas aplicações na Física: uma na Mecânica Clássica e outra na Mecânica Quântica. Os geradores do grupo S O ( 4 ) serão determinados, assim como sua álgebra de Lie. A aplicação na Mecânica Clássica será na obtenção da transformação de Galilei homogênea e na Mecânica Quântica será na obtenção do espectro de energia do átomo de hidrogênio, no regime não-relativístico. A versão quântica do vetor de Laplace-Runge-Lenz será fundamental para a construção da fórmula de Bohr para os níveis de energia do respectivo átomo. Física determinados Lie hidrogênio nãorelativístico. nãorelativístico não relativístico. relativístico não-relativístico LaplaceRungeLenz Laplace Runge Lenz

This work aims to present an introduction to the group S O ( 4 ) and two applications in physics: one in Classical Mechanics and other in Quantum Mechanics. The generators of the S O ( 4 ) group will be determined, as well as their Lie algebra. The application in Classical Mechanics will be in obtaining the homogeneous Galilei transformation and in Quantum Mechanics will be in obtaining the energy spectrum of the hydrogen atom, in the non-relativistic regime. The quantum version of the Laplace-Runge-Lenz vector will be fundamental for the construction of Bohr’s formula for the energy levels of the respective atom. physics determined algebra atom nonrelativistic non relativistic regime LaplaceRungeLenz Laplace Runge Lenz Bohrs Bohr s

Abstract This paper presents the application of recent ansatz for estimation of stability of the Laplace resonance for Galilean moons (Io, Europa, Ganymede). We estimate over time the eccentricity + semi–major axis in a binary system experiencing the net tidal friction, including the additional tidal heating which comes from the transformation of net transfer of angular momentum between the Galilean moons of Jupiter (due to dynamical features of the Laplace resonance). Presumably, there should be a net transfer of angular momentum between Io and Europa (for the reason that tidal heating on Ganymede seems to be negligible with respect to Io and Europa). We established the fact that Laplace resonance should be valid and stable on a timescale of centuries in the future, but there might be chaotic perturbations less than 0.1\% for the accuracy of such phenomenon. Moreover, the presented ansatz can be used to predict a scheme for optimizing the maneuvers of spacecrafts in the vicinity of Ganymede (due to absence of net transfer of angular momentum between Ganymede and other Galilean moons). The main conclusion stems from previously suggested approach (\cite{ershkov2017tidal

En este artículo se presenta una solución alternativa basada en series infinitas para la evaluación precisa y eficiente de la impedancia de retorno por tierra en cables subterráneos. En este método se utiliza la transformación de Wedepohl y Wilcox para descomponer la integral impropia de Pollaczek en un conjunto de funciones de Bessel más una integral definida. La característica principal de las funciones de Bessel es que son fáciles de calcular con herramientas modernas de software matemático como Matlab. Las principales contribuciones de este artículo son la aproximación de la integral definida por una serie infinita, dado que no tiene solución analítica, y su evaluación numérica por medio de una fórmula recursiva. La precisión y eficiencia de la fórmula recursiva se compara contra el método de integración numérica para un amplio rango de frecuencias y configuraciones de cables subterráneos. Finalmente, se utiliza el método propuesto como una subrutina de cálculo de parámetros de cables en la Transformada Numérica de Laplace (NLT) para obtener respuestas transitorias precisas en el dominio del tiempo.

This paper presents an alternative solution based on infinite series for the accurate and efficient evaluation of cable earth return impedances. This method uses Wedepohl and Wilcox's transformation to decompose Pollaczek's integral in a set of Bessel functions and a definite integral. The main feature of Bessel functions is that they are easy to compute in modern mathematical software tools such as Matlab. The main contributions of this paper are the approximation of the definite integral by an infinite series, since it does not have analytical solution; and its numerical solution by means of a recursive formula. The accuracy and efficiency of this recursive formula is compared against the numerical integration method for a broad range of frequencies and cable configurations. Finally, the proposed method is used as a subroutine for cable parameter calculation in the inverse Numerical Laplace Transform (NLT) to obtain accurate transient responses in the time domain.

AbstractIn this paper, modelling and simulation of a multi-body system for its vibration control using sensors and actuators is presented. The model developed on MSC ADAMS consists of six legs connecting moving and base plates using spherical joints with axial rotation of legs constrained. It contains linear actuator collocated with force sensor. Mathematical model of the mechanism representing the Kinematic and Dynamic analysis of model is presented. The Leg Length variation obtained using MATLAB is obtained in order to validate the model developed in MSC ADAMS. Natural frequencies for six different mode shapes are obtained from vibration excitation analysis. A decentralized force feedback controller which uses PID control law with single gain is used for actively attenuating the vibration, which is coming from base platform. The transfer function, defined as the ratio of Laplace transformation of the acceleration of moving plate and base plate, is considered as a measure of isolation. Effect of proportional, derivative and integral feedback is studied separately and together as well. A significant isolation in the resonance transmissibility and natural frequency of the mechanism is demonstrated.

In this article, Perturbation Method (PM) is employed to obtain an approximate solution for Troesch equation. In addition, we will describe the use of Laplace transform and Padé transformation to deal with the truncated series obtained by the PM method, in order to obtain handy approximations. Finally a table of comparison, between proposed solution, and other solutions reported in the literature: Adomian's Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM) and exact numerical solution, shows that our solution is the most accurate one (Average Absolute Relative Error 1.705648354x10-8).

En este artículo el Método de Perturbación (PM) es empleado para obtener una solución aproximada para el problema de Troesch. Además describiremos el uso de la Transformada de Laplace y la Aproximación de Padé para trabajar con las series truncadas obtenidas por el Método de Perturbación, y así obtener soluciones aproximadas compactas. Finalmente se propone una tabla comparativa entre la solución propuesta y otras soluciones reportadas en la literatura: Método de Descomposición de Adomian, Método de Perturbación Homotópica, Método de Análisis Homotópico y la solución numérica exacta. Los resultados muestran que nuestra solución es la más exacta (Error Relativo Absoluto Promedio1.705648354x10-8).

In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, a lower triangular matrix M, in which all entries are determinants, and an upper triangular matrix U whose entries are also in determinant form. From a well-known theorem on the pivot elements of the Doolittle-Gauss elimination process, we deduce a corollary to obtain a diagonal matrix D. With it, we scale the elementary lower triangular matrix of the Doolittle-Gauss elimination process and deduce a new elementary lower triangular matrix. Applying this linear transformation to A by means of both minimum and complete pivoting strategies, we obtain the determinant of A as if it had been calculated by means of a Laplace expansion. If we apply this new linear transformation and the above pivot strategy to an augmented matrix (A|b), we obtain a Cramer's solution of the linear system of equations. These algorithms present an O(n³) computational complexity when (A,b)⊂Rn on hybrid GPU-accelerated multicore systems.

En este trabajo se postula un Nuevo Teorema de Descomposición de una Matriz A en dos matrices: una Matriz triangular inferior M, cuyas entradas son todas expresadas en forma de determinantes, y una matriz triangular superior U cuyas entradas están también expresadas en forma de determinantes. A partir de un muy conocido Teorema sobre los elementos pivotales del proceso de eliminación de Doolittle-Gauss, deducimos un corolario para obtener una Matriz Diagonal D. Usando esta matriz, escalamos la Matriz Elemental Triangular Inferior obtenida durante el proceso de eliminación de Doolittle-Gauss y deducimos una Nueva Matriz Elemental Triangular Inferior. Aplicando esta transformación lineal a la matriz A, por medio de una estrategia de pivoteo total, se obtiene el determinante de A como si hubiera sido calculado a través de la Expansión de Laplace. Si aplicamos esta nueva transformación lineal y la estrategia de pivoteo anteriormente mencionada a la matriz aumentada (A|b) obtenemos la solución de la Regla de Cramer aplicada a un Sistema de Ecuaciones Lineales. Estos algoritmos presentan una complejidad computacional O(n³) cuando (A,b)⊂Rn se calcula en Sistemas Multi-Core Acelerados con GPU.

The Fokker-Planck equation with a sink term is resolved exactly, using the Fujita diffusivity and Parlange relationship between conductivity and diffusivity. In order to obtain the solution, the Kirchhoff potential and the Fujita-Storm mobile coordinate are introduced. The differential equation takes the form of the Burgers equation, which is linear in the diffusive term. The convective coefficient of the latter is replaced by the Hopf-Cole transformation for the purpose of deriving the classical linear heat equation. During the transformation, the sink term is defined functionally, so that the end result is precisely the heat equation without sink term. The exact solution of the Hopf-Cole potential is deduced by using the classical Laplace transform for certain initial and boundary conditions of interest. The solution of the Fokker-Planck equation in the physical space is obtained through the inverse transformations. The solution includes as particular cases both the Sanders et al. and Broadbridge and White solutions. The exact solution can be used to validate numerical solutions of the Fokker-Planck equation and in studies on water extraction by plant roots.

Se resuelve de manera exacta la ecuación Fokker-Planck con un término de sumidero, utilizando la difusividad de Fujita y la relación de Parlange entre la conductividad y difusividad. Para obtener la solución, se introduce primero el potencial de Kirchhoff y enseguida la coordenada móvil de Fujita-Storm; la ecuación diferencial toma la forma de la ecuación de Burgers, que es lineal en el término difusivo. El coeficiente convectivo de esta última se sustituye por la transformación de Hopf-Cole, con la finalidad de deducir la ecuación lineal clásica del calor. Durante las transformaciones, el término de sumidero se define funcionalmente, de modo que el resultado final sea precisamente la ecuación de calor sin término de sumidero. La solución exacta del potencial de Hopf-Cole se obtiene con la transformada clásica de Laplace para algunas condiciones iniciales y de frontera de interés. La solución de la ecuación Fokker-Planck en el espacio físico se obtiene a través de la inversión de las transformaciones utilizadas. La solución incluye como casos particulares las soluciones de Sanders et al., y de Broadbridge y White. La solución exacta puede ser utilizada para validar soluciones numéricas de la ecuación Fokker-Planck y en estudios sobre la extracción de agua por las raíces de las plantas.

Neste trabalho foram consideradas questões associadas à solução de problemas dinâmicos diretos e inversos para os modelos elásticos delgados no domínio espectral. A solução de problemas diretos obtém-se usando a transformação do sistema de Lamé completo no domínio espectral bidimensional utilizando a transformada de Laplace com respeito à variável temporal e a transformada de Fourier-Bessel com respeito às variáveis espaciais. Tais transformadas utilizam sismogramas multicomponentes para cálculo dos espectros correspondentes. Uma questão fundamental é a possibilidade de combinação dos espectros, calculados teoricamente e por sismogramas reais. A dificuldade é que o problema direto, resolvido no domínio espectral, define o sismograma com limites infinitos com respeito às variáveis temporal e espaciais. Ao mesmo tempo as observações reais são registradas com abertura limitada e num intervalo de tempo finito. A transformação de sismogramas limitados e discretos dá um espectro deformado, que pode ser bem diferente da solução teórica. Neste trabalho estuda-se a influência do parâmetro de Laplace e dos filtros de suavização no grau de compatibilidade entre estes dois tipos de espectros. O resultado principal desta investigação foi a obtenção de um nível aceitável de compatibilidade dos espectros para sismogramas sintéticos, o que permite diferenciar vários tipos de modelos e distinguir a mudança dos parâmetros deles.

The questions connected with the solution of direct and inverse dynamic problems for thin layer elastic models in spectral domain are considered in this work. The solution of the corresponding direct problem is obtained using the complete Lamé system represented in two-dimensional spectral domain. Such transition is based on the Laplace transform with respect to the time variable and the Fourier-Bessel transform with respect to the spatial variables. Similar transformations are used for calculation of corresponding two-dimensional spectra by multicomponent seismograms. The possibility of compatibility of these two types of the spectra calculated theoretically and using real seismograms is studied. The matter is that the direct problem, solved in the spectral domain, defines the seismograms in infinite limits on the spatial and time variables. At the same time we have real observation on the limited aperture and in a finite time interval. Transformation of the limited discrete seismogram gives the deformed spectrum which can essentially differ from the theoretical solution. We study the influence of the Laplace parameter and smoothing filters on degree of compatibility of two specified types of spectra. There is shown the possibility to obtain comprehensible degree of compatibility of these spectra for synthetic seismograms, allowing to differentiate types of the models and to distinguish change of their parameters.

Performance evaluation and economic analysis is carried out to predict the steady state operational availability and net present value index for a thermal power plant. Mathematical formulation of the plant's reliability is carried out using probability theory following the Chapman-Kolmogorov Birth-Death process. The system of differential equations obtained is solved using the Laplace transformation technique. The behaviour analysis of performance modules reveals that availability decreases with increasing failure rates, while operational availability improves with an initial increase in repair rates for different sub-systems. From the economic evaluation module, the plant availability and power generation capacity are found to be sensitive, while increasing the redundancies of feed pump or condensate extraction pump sub-systems has less influence on the profitability index. A high-capacity plant reflects better trends. Model predictions indicate a critical condition for repair rates; beyond these rates the plant availability, net present value, and annual revenue earned are found to be insensitive.

Prestasiebepaling en teenswoordige ontleding word uitgevoer om gestadigde operasionele beskikbaarheid en 'n netto teenswoordige indekswaarde vir 'n fossielbrandstofkragstasie te bepaal. 'n Wiskundige model vir aanlegbetroubaarheid word probabilisties geskep met gebruik van die Chapman-Kolmogorov metode. Die model se differensiaalvergelykings word gemanipuleer met gebruik van Laplace-transformasie. Die resulterende sisteemprestasie toon dat beskikbaarheid afneem met toenemende mislukkingstempo's en vice versa. Die ekonomiese ontleding van die model toon sensitiwiteit by lewering van drywing en beskikbaarheid, en dat oortolligheid van sekere hulpsubsisteme nie betekenisvol tot winsgewendheid bydra nie. Die model benadeel die belangrikheid van hersteltempo's.

The mathematical model for two-dimensional unsteady sonic flow, based on the classical diffusion equation with imaginary coefficient, is presented and discussed. The main purpose is to develop a rigorous formulation in order to bring into light the correspondence between the sonic, supersonic and subsonic panel method theory. Source and doublet integrals are obtained and Laplace transformation demonstrates that, in fact, the source integral is the solution of the doublet integral equation. It is shown that the doublet-only formulation reduces to a Volterra integral equation of the first kind and a numerical method is proposed in order to solve it. To the authors' knowledge this is the first reported solution to the unsteady sonic thin airfoil problem through the use of doublet singularities. Comparisons with the source-only formulation are shown for the problem of a flat plate in combined harmonic heaving and pitching motion.