Infinitely many solutions for anisotropic elliptic equations with variable exponent

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.

On the missing anisotropic space inside atoms in quantum mechanics

Schrödinger developed his famous equation from the standard wavelength. However, as demonstrated here, inside the atom, the electron does not move according to de Broglie-Einstein’s postulate λ = h/p, because the wavelength of the electron’s motion varies with the distance to the nucleus. Therefore, Schrödinger’s equation does not quantify the real electron’s motion in atoms. Here, the equation of a variable wavelength for electron motion inside atoms is introduced. The calculation, applied to the hydrogen atom, achieves energy level values very close to the experimental values. This successful result can provide a deeper understanding of the behavior of electrons in atoms and improve the fundamentals of quantum mechanics (QM). However, beyond the question concerning the postulate λ = h/p, two other fundamental principles may be missing in modern QM, and they are: an anisotropic space inside atoms and a motion of the electron through a helical trajectory.

Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

On the Validity of the Cosmic No-hair Conjecture in some Conformal-violating Maxwell Models

We will present main results of our recent investigations on the validity of the cosmic no-hair conjecture proposed by Hawking and his colleagues in some conformal-violating Maxwell models, in which a scalar field or its kinetic term is non-trivially coupled to the electromagnetic field. In particular, we will show that the studied models really admit the Bianchi type I metrics, which are homogeneous but anisotropic space time, as their stable cosmological solutions. Hence, these models turn out to be counterexamples to the cosmic no-hair conjecture. Keywords: Cosmic no-hair conjecture, cosmic inflation, Bianchi type I space time, Maxwell theory.

Adaptations of Relief for continuous domains of bioinformatics

Relief occupies a niche among feature selection methods for data classification. Filters are faster, wrappers are much slower. Relief is feature-set-aware, same as wrappers. However, it is thought being able to deselect only irrelevant, but not redundant features, same as filters. Iterative Reliefs seek to increase the separation margin between classes in the anisotropic space defined by weighted features. Reliefs for continuous domains are much less developed than for categorical domains. The paper discusses a number of adaptations for continuous spaces with Euclidean or Manhattan metric. The ability of Relief to detect redundant features is demonstrated. A dramatic reduction of the feature-set is achieved in a health diagnostics problem.