Solução da equação advectivo-difusiva utilizando regras para manipulação de exponenciais de operadores diferenciais e simetrias de Lie : aplicações em engenharia ambiental

Vários métodos analíticos, numéricos e híbridos podem ser utilizados na solução de problemas de difusão e difusão-advecção. O objetivo deste trabalho é apresentar dois métodos analíticos para obtenção de soluções em forma fechada da equação advectivo-difusiva em coordenadas cartesianas que descreve problemas de dispersão de poluentes na água e na atmosfera. Um deles é baseado em regras de manipulação de exponenciais de operadores diferenciais, e o outro consiste na aplicação de simetrias de Lie admitidas por uma equação diferencial parcial linear. Desenvolvem-se regras para manipulação de exponenciais de operadores diferenciais de segunda ordem com coeficientes constantes e para operadores advectivo-difusivos. Nos casos em que essas regras não podem ser aplicadas utiliza-se uma formulação para a obtenção de simetrias de Lie, admitidas por uma equação diferencial, via mapeamento. Define-se um operador diferencial com a propriedade de transformar soluções analíticas de uma dada equação diferencial em novas soluções analíticas da mesma equação. Nas aplicações referentes à dispersão de poluentes na água, resolve-se a equação advectivo-difusiva bidimensional com coeficientes variáveis, realizando uma mudança de variáveis de modo a reescrevê-la em termos do potencial velocidade e da função corrente correspondentes ao respectivo escoamento potencial, estendendo a solução para domínios de contornos arbitrários Na aplicação referente ao problema de dispersão de poluentes na atmosfera, realiza-se uma mudança de variáveis de modo a obter uma equação diferencial parcial com coeficientes constantes na qual se possam aplicar as regras de manipulação de exponenciais de operadores diferenciais. Os resultados numéricos obtidos são comparados com dados disponíveis na literatura. Diversas vantagens da aplicação das formulações apresentadas podem ser citadas, a saber, o aumento da velocidade de processamento, permitindo a obtenção de solução em tempo real; a redução da quantidade de memória requerida na realização de operações necessárias para a obtenção da solução analítica; a possibilidade de dispensar a discretização do domínio em algumas situações.

The Interval Constructor on classes of ML-algebras

Monoidal logic, ML for short, which formalized the fuzzy logics of continuous t-norms and their residua, has arisen great interest, since it has been applied to fuzzy mathematics, artificial intelligence, and other areas. It is clear that fuzzy logics basically try to represent imperfect or fuzzy information aiming to model the natural human reasoning. On the other hand, in order to deal with imprecision in the computational representation of real numbers, the use of intervals have been proposed, as it can guarantee that the results of numerical computation are in a bounded interval, controlling, in this way, the numerical errors produced by successive roundings. There are several ways to connect both areas; the most usual one is to consider interval membership degrees. The algebraic counterpart of ML is ML-algebra, an interesting structure due to the fact that by adding some properties it is possible to reach different classes of residuated lattices. We propose to apply an interval constructor to ML-algebras and some of their subclasses, to verify some properties within these algebras, in addition to the analysis of the algebraic aspects of them

Irreducibility and compositeness in q-deformed harmonic oscillator algebras

A study of the reducibility of the Fock space representation of the q-deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is carried out by using the properties of the Gauss polynomials. When the deformation parameter is a root of unity, an interesting result comes out in the form of a reducibility scheme for the space representation which is based on the classification of the primitive or nonprimitive character of the deformation parameter. An application is carried out for a q-deformed harmonic oscillator Hamiltonian, to which the reducibility scheme is explicitly applied.

Mass generation from Lie algebra extensions

Applied to the electroweak interactions, the theory of Lie algebra extensions suggests a mechanism by which the boson masses are generated without resource to spontaneous symmetry breaking. It starts from a gauge theory without any additional scalar field. All the couplings predicted by the Weinberg-Salam theory are present, and a few others which are nevertheless consistent within the model.

Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa-Holm equation

In this Letter we investigate Lie symmetries of a (2 + 1)-dimensional integrable generalization of the Camassa-Holm (CH) equation. Through the similarity reductions we obtain four different (1 + 1)-dimensional systems of partial differential equations in which one of them turns out to be a (1 + 1)-dimensional CH equation. We establish their integrability by providing the Lax pair for all of them. Further, we present a brief analysis for some types of particular solutions which include the cuspon, peakon and soliton solutions for the two-dimensional generalization of the CH equation. (C) 2000 Published by Elsevier B.V. B.V.

Solitons from dressing in an algebraic approach to the constrained KP hierarchy

The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1 + 1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy.The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time Bows which distinguishes them From the conventional structure of the Darboux-Backlund-Wronskian solutions of the constrained KP hierarchy.

On Clifford subalgebras, spacetime splittings and applications

Z(2)-gradings of Clifford algebras are reviewed and we shall be concerned with an alpha-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced alpha-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.

Quantum V-algebras

The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all H-reduced G(r)-current algebras belong to one of the following basic algebraic structures: local quadratic W-algebras, rational U-algebras, nonlocal W-algebras, nonlocal quadratic WV-algebras and rational nonlocal UV-algebras. The main new features of the quantum Ir-algebras and their heighest weight representations are demonstrated on the example of the quantum V-3((1,1))-algebra.

MANY-BODY PROBLEMS WITH COMPOSITE-PARTICLES AND Q-HEISENBERG ALGEBRAS

We propose to employ deformed commutation relations to treat many-body problems of composite particles. The deformation parameter is interpreted as a measure of the effects of the statistics of the internal degrees of freedom of the composite particles. A simple application of the method is made for the case of a gas of composite bosons.

THE ALGEBRA OF NONLOCAL CHARGES IN NONLINEAR SIGMA-MODELS

We obtain the exact classical algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry group O(N). As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, now containing a calculable correction of order one unit lower. The relation with Yangians and the role of the results in the context of Lie-Poisson algebras are also discussed.

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

So(2,1) Lie algebra and the Green's functions for the conditionally exactly solvable potentials

The Green's functions of the recently discovered conditionally exactly solvable potentials are computed. This is done through the use of a second-order differential realization of the so(2,1) Lie algebra. So we present the dynamical symmetry underlying the solvability of such potentials and show that they belong to a general class of solvable and partially solvable potentials. © 1994 The American Physical Society.

Affine Lie algebraic origin of constrained KP hierarchies

An affine sl(n + 1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non-linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Bäcklund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy. © 1995 American Institute of Physics.