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

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Weak topological insulator with protected gapless helical states

A workable model for describing dislocation lines introduced into a three-dimensional topological insulator is proposed. We show how fragile surface Dirac cones of a weak topological insulator evolve into protected gapless helical modes confined to the vicinity of dislocation line. It is demonstrated that surface Dirac cones of a topological insulator (either strong or weak) acquire a finite-size energy gap, when the surface is deformed into a cylinder penetrating the otherwise surface-less system. We show that when a dislocation with a non-trivial Burgers vector is introduced, the finite-size energy gap play the role of stabilizing the one-dimensional gapless states.Comment: 8 pages, 17 figure

Lattice-like operations and isotone projection sets

By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.Comment: Proofs of Theorem 1 and Corollary 4 have been corrected. arXiv admin note: substantial text overlap with arXiv:1210.232

Linear complementarity problems on extended second order cones

In this paper, we study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We state necessary and sufficient conditions for a point to be a solution of the converted problem. We also present solution strategies for this problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally, we present some numerical examples

Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables

We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J. Theor. Phy

The Low Redshift survey at Calar Alto (LoRCA)

The Baryon Acoustic Oscillation (BAO) feature in the power spectrum of galaxies provides a standard ruler to measure the accelerated expansion of the Universe. To extract all available information about dark energy, it is necessary to measure a standard ruler in the local, z<0.2, universe where dark energy dominates most the energy density of the Universe. Though the volume available in the local universe is limited, it is just big enough to measure accurately the long 100 Mpc/h wave-mode of the BAO. Using cosmological N-body simulations and approximate methods based on Lagrangian perturbation theory, we construct a suite of a thousand light-cones to evaluate the precision at which one can measure the BAO standard ruler in the local universe. We find that using the most massive galaxies on the full sky (34,000 sq. deg.), i.e. a K(2MASS)<14 magnitude-limited sample, one can measure the BAO scale up to a precision of 4\% and 1.2\% using reconstruction). We also find that such a survey would help to detect the dynamics of dark energy.Therefore, we propose a 3-year long observational project, named the Low Redshift survey at Calar Alto (LoRCA), to observe spectroscopically about 200,000 galaxies in the northern sky to contribute to the construction of aforementioned galaxy sample. The suite of light-cones is made available to the public.Comment: 15 pages. Accepted in MNRAS. Please visit our website: http://lorca-survey.ft.uam.es