Current Issue : April - June Volume : 2019 Issue Number : 2 Articles : 5 Articles
In this paper, we consider the global existence and decay rates of strong solutions\nto the three-dimensional compressible quantum Hall-magneto-hydrodynamics\nequations. By combing the Lp-Lq estimates for the linearized equations and a\nstandard energy method, the global existence and its convergence rates are\nobtained in various norms for the solution to the equilibrium state in the\nwhole space when the initial perturbation of the stationary solution is small in\nsome Sobolev norms. More precisely, the decay rates in time of the solution\nand its first order derivatives in L2-norm are obtained when the L1-norm of\nthe perturbation is bounded....
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In this paper, we derive Fourier series expansions for functions related to sums of finite\nproducts of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier\nseries expansions, we are able to express those two kinds of sums of finite products of polynomials as\nlinear combinations of Bernoulli polynomials....
The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular\ncase of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves.The solutions of\nthe generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and\ndegree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the\nwhole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation.\nThe regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions;\nthese reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree...
By Invoking symmetry principle, we present a self-consistent interpretation\nof the existing quantum theory which explains why our world is fundamentally\nindeterministic and that why non-local quantum jumps occur. Symmetry\nprinciple dictates that the concept of probability is more fundamental\nthan the notion of the wave function in that the former can be derived directly\nfrom symmetries rather than have to be assumed as an additional axiom. It\nis argued that the notion of quantum probability and that of the wavefunction\nare intimately connected....
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