Inversion transformation

Abstract: We extend the results presented by Acena et al in the afore mentioned paper, [1], to the case of axisymmetric, maximal initial data which are invariant under an inversion transformation.

Abstract: We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

Abstract: The electrostatic problems of the configurations of two dielectric spheres are solved with the aid of the inversion transformation. The potentials due to a unit point charge located on the symmetry axis or that excited by a uniform external field are obtained. In the case of conducting spheres, our results are in agreement with the previous results obtained by the method of images.

Abstract: Based on our previous study on the elementary characterization of the fuzzy symmetry, we inquired the static state fuzzy symmetry of some molecules and their molecular orbitals (MO). Now we will analyze the fuzzy symmetry of some simple linear tri-atomic dynamic systems in connection with the reaction. Three related transformations will mainly be studied in detail. These three transformations are (1) the space inversion transformation about the mid-atom as the center, (2) the reaction reversal transformation in relation to the reaction B + AC→BA + C and (3) the joint transformation of the above two. We examined the variation for the internal configuration of these systems owing to the operation of above three transformations, and then establish methods to calculate the fuzzy symmetry characterization, such as the membership functions for the MOs of such linear tri-atomic dynamic systems in relation to these transformations. We examined the variation regularity in relation to the fuzzy symmetry characterization for the MOs of these systems along the intrinsic reaction coordinate (IRC) and dividing line. The variation regularity and the distribution for the fuzzy symmetry characterization in related internal configuration coordinate space are also analyzed. An IRC-scale is suggested for internal configuration coordinate space in this paper.