Parallelepipeds and Decomposition of Ranges of Vector Measures in Banach Spaces

We extend a result of A. Neyman about zonoids in ℝⁿ to zonoids in Banach spaces. A zonoid in a Banach space is the closed convex hull of the range of a vector measure. We show that the following conditions on a zonoid Z are equivalent: (1) Z determines univocally the associated conical measure; (2) There exists a vector measure defined on (Ω, Σ) such that every decomposition of Z into sum of zonoids can be obtained by decomposing (Ω, Σ); (3) Z ∩ (—αZ) is a parallelepiped for every α > 0. We also prove other results about decomposition of zonoids; for instance, we prove that if Z is a zonoid and Z ∩ (—Z) is a zonotope, there exists a zonoid L such that Z = Z ∩ (—Z) + L.     

Combined Modelling of Triply Paired Electronic States of HO2+ ion with 3 Symmetry using Coupled Eigen Model

The multi state triply paired coupled adiabatic electronic states of , symmetries for the title ion have been modelled using the triply paired coupled eigen valued functions with the principal aim of getting a set of well optimized triply paired coupled functions, capable of explaining all the microscopic topograhical features present in the chosen coupled electronic states. The high quality size truncated dynamically correlated points at the limit of complete basis set (CBS) were utilized for this purpose. The proposed model function are presented in an “easy to understand” linear form with respect to the pre-defined variables which are directly dependent on the diabatic potentials and coupling strengths found in the diabatic matrix. Its most important feature (an essential criterion to be selected for modelling the triply degenerate seam pathway) is the ability to predict the existence of an uniquely defined triply degenerate iso-energetic geometry among the chosen coupled states in addition to the prediction of the existence of many doubly degenerate iso-energetic geometries in those states and hence can be adaptable to any general molecular system where such triply or doubly degenerate geometries are believed to occur. The title ion is one among them where many doubly iso-energetic geometries were located but not the triply degenerate one as it is a three atomic system unlike the polyatomic case where the triply degenerate geometry will most likely to be found. To do this work, one requires the knowledge of four asymptotic diatomic curves upon atom+diatom dissociation whose correct correlated spectroscopic states are O₂⁺ , O₂ , O₂⁺ , OH⁺ . The quality and the characteristics of analytical surface of the ion is discussed well in detail.