How to project onto extended second order cones

The extended second order cones were introduced by S. Z. N\'emeth and G. Zhang in [S. Z. N\'emeth and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. N\'emeth and G. Zhang in [S.Z. N\'emeth and G. Zhang. Positive operators of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods

On the spherical convexity of quadratic functions

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cones are given

Instabilities in Nuclei

The evolution of dynamical perturbations is examined in nuclear multifragmentation in the frame of Vlasov equation. Both plane wave and bubble type of perturbations are investigated in the presence of surface (Yukawa) forces. An energy condition is given for the allowed type of instabilities and the time scale of the exponential growth of the instabilities is calculated. The results are compared to the mechanical spinodal region predictions. PACS: 25.70 MnComment: 22 pages, latex, with 5 PS figures, available at http://www.gsi.de/~papp

Effects of Photosensitization and Low-Power Helium-Neon Laser Irradiation on Liposomes and Cell Membranes

Low-power He-Ne laser irradiation causes a well-defined and energy dependent cell destruction of in vitro cultured cell lines sensitized by hematoporphyrin derivative (HPD). The mechanism of this photosensitization was studied by measuring with polarization microscopic, scanning electron microscopic, and electron-spin-resonance (ESR) spectroscopic parameters. The cell damage caused by photosensitization and laser irradiation seems to be a complex process, however the biological membranes seem to be one of the primary targets. The energy of laser light causes rotation and resonance changes of macromolecules and the water molecules, resulting in an increased structural order of the submembraneous components in the living cells, detectable microscopically. During the photosensitization process, the red (630 nm) He-Ne laser light, during a one-photon energy activation, causes excitation of hematoporphyrin molecules to their triplet state. The excitation of HPD molecules results in a multi-step, free-radical generating effect, measured by ESR spectroscopy and studied by the ultrastructural changes of membrane organization and cell shape. Similar effects could be observed on in vitro lipid-water liposome membranes