Lower Approximations by Fuzzy Consequence Operators

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Three aspects of bosonized supersymmetry and linear differential field equation with reflection

Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N=1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to `up' or `down' field component gives rise to a linear differential equation with reflection whose `square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration.Comment: 18 pages, LaTeX, minor typos corrected, ref updated, to appear in Nucl. Phys.

On the construction of a finite Siegel space

In this note we construct a finite analogue of classical Siegel's Space. Our approach is to look at it as a non commutative Poincare's half plane. The finite Siegel Space is described as the space of Lagrangians of a $2n$ dimensional space over a quadratic extension $E$ of a finite base field $F$. The orbits of the action of the symplectic group $Sp(n,F)$ on Lagrangians are described as homogeneous spaces. Also, Siegel's Space is described as the set of anti-involutions of the symplectic group.2

Thermal Decomposition of Diphenyl Tetroxane in Chlorobenzene Solution

The thermal decomposition of Cyclic Diperoxide of Benzaldehyde 3,6-diphenyl-1,2,4,5-tetroxane, (DFT) in chlorobenzene solution in the studied temperature range (130°C - 166°C) satisfactorily satisfies a first order law up to 60% conversions of diperoxide. DFT would decompose through a mechanism in stages and initiated by the homolytic breakdown of one of the peroxidic bonds of the molecule, with the formation of the corresponding intermediate biradical. The concentration studied was very low, so that the effects of secondary reactions of decomposition induced by free radicals originated in the reaction medium can be considered minimal or negligible. The activation parameters for the unimolecular thermal decomposition reaction of the DFT are ΔH# = 30.52 ± 0.3 kcal·mol-1 and ΔS# = -6.38 ± 0.6 cal·mol-1 K-1. The support for a step-by-step mechanism instead of a process concerted is made by comparison with the theoretically calculated activation energy for the thermal decomposition of 1,2,4,5-tetroxane.Fil: Bordón, Alexander Germán. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; ArgentinaFil: Pila, Andrea Natalia. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Modelado e Innovación Tecnológica. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Instituto de Modelado e Innovación Tecnológica; ArgentinaFil: Profeta, Mariela Inés. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; ArgentinaFil: Jorge, María J.. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; ArgentinaFil: Jorge, Lilian Cristina. Universidad Nacional del Nordeste. Facultad de Ciencias Veterinarias; ArgentinaFil: Romero, Jorge Marcelo. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; ArgentinaFil: Jorge, Nelly Lidia. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura; Argentin