The ring current model of the pentaprismane molecule

Three-dimensional models of the quantum-mechanical current density J(B) , induced in the electron cloud of the C(10)H(10) pentaprismane molecule by a magnetic field B applied along the C(5) (a C(2)) symmetry axis, orthogonal to the pentagonal (a rectangular) face, and denoted by B(‖) (B(⊥)), have been constructed. Predictions of near Hartree-Fock quality are reported for the diagonal components of magnetic tensors, magnetizability (ξ), nuclear shielding of carbon (σ(C)) and hydrogen (σ(H)), and virtual shielding at the center of mass (σ(CM)). The complicated spatial features of the induced electronic current-density field have been rationalized and compactly described via stagnation graphs that elucidate the details of its topological structure. A representation of J(B) is obtained by three-dimensional perspective plots and by planar maps visualizing phase portraits of electron flow in a series of molecular domains. Both streamline J(B) /|J(B) | and modulus |J(B) | are analyzed. These graphic tools illustrate the competition between diatropic and paratropic regimes which determine the magnitude of various components of magnetizability and magnetic shielding of hydrogen and carbon nuclei. Shielding density maps show that the differential Biot-Savart law explains magnetic shielding at hydrogen and carbon nuclei, and virtual shielding at ring and cage centers. Similarities and/or contrasting ring current effects on magnetotropicity are discussed by a comparison with triprismane C(6)H(6) and cubane C(8)H(8) .     

A characterization of duality through section/projection correspondence in the finite dimensional setting

In this paper, we show that the well-known duality operation in the context of convex bodies in Rn is completely characterized by its property of interchanging sections with projections. Our results are compared to results by Böröczky-Schneider and Artstein-Milman, who showed that in many cases, the property of order reversing is sufficient to determine a duality operation, up to obvious linear modifications. In fact, we provide another result that recovers a known characterization of duality by the property of order reversing, and up to a mild condition, also a characterization of duality by interchanging sections by projections.     

Approximation of common fixed points of weakly compatible pairs using the Jungck iteration

We show that the Jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces. The existence of coincidence points are also discussed for those pair of maps. The results are generalizations of well known results of the convergence of Picard iterations for single self maps of Banach spaces. In particular, the results improve, generalize and extend the recent results of Berinde [V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213 (2009) 348-354] and answers the open question posed in the paper.     

Strongly amenable semigroups and nonlinear fixed point properties

A semi-topological semigroup is strongly left amenable if there is a compact left ideal group in the spectrum of its LUC-compactification. In this paper, we want to study those objects, and study some fixed point property related to non-expansive mapping and other similar kind of mapping.     

Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.