Maximal rank for ΩPn

International Mathematical Forum, Vol. 6, 2011, no. 8, 389 - 398Full Tex

Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs

Manifestly non-Hermitian quantum graphs with real spectra are introduced and shown tractable as a new class of phenomenological models with several appealing descriptive properties. For illustrative purposes, just equilateral star-graphs are considered here in detail, with non-Hermiticities introduced by interactions attached to the vertices. The facilitated feasibility of the analysis of their spectra is achieved via their systematic approximative Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting bound-state spectra are found real in a discretization-independent interval of couplings. This conclusion is reinterpreted as the existence of a hidden Hermiticity of our models, i.e., as the standard and manifest Hermiticity of the underlying Hamiltonian in one of less usual, {\em ad hoc} representations ${\cal H}_j$ of the Hilbert space of states in which the inner product is local (at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples of these (of course, Hamiltonian-dependent) hermitizing inner products are offered in closed form. In this way each initial quantum graph is assigned a menu of optional, non-equivalent standard probabilistic interpretations exhibiting a controlled, tunable nonlocality.Comment: 33 pp., 6 figure

PT-symmetric deformations of Calogero models

We demonstrate that Coxeter groups allow for complex PT-symmetric deformations across the boundaries of all Weyl chambers. We compute the explicit deformations for the A2 and G2-Coxeter group and apply these constructions to Calogero–Moser–Sutherland models invariant under the extended Coxeter groups. The eigenspectra for the deformed models are real and contain the spectra of the undeformed case as subsystems

q-breathers in Discrete Nonlinear Schroedinger lattices

$q$-breathers are exact time-periodic solutions of extended nonlinear systems continued from the normal modes of the corresponding linearized system. They are localized in the space of normal modes. The existence of these solutions in a weakly anharmonic atomic chain explained essential features of the Fermi-Pasta-Ulam (FPU) paradox. We study $q$-breathers in one- two- and three-dimensional discrete nonlinear Sch\"{o}dinger (DNLS) lattices -- theoretical playgrounds for light propagation in nonlinear optical waveguide networks, and the dynamics of cold atoms in optical lattices. We prove the existence of these solutions for weak nonlinearity. We find that the localization of $q$-breathers is controlled by a single parameter which depends on the norm density, nonlinearity strength and seed wave vector. At a critical value of that parameter $q$-breathers delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into strong mode-mode interaction regime. In particular this breakdown takes place at one of the edges of the normal mode spectrum, and in a singular way also in the center of that spectrum. A stability analysis of $q$-breathers supplements these findings. For three-dimensional lattices, we find $q$-breather vortices, which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.Comment: 19 pages, 9 figure

Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be published in J. Phys. A: Mathematical and Genera

Detectable Viral Load in Late Pregnancy among Women in the Rwanda Option B+ PMTCT Program: Enrollment Results from the Kabeho Study.

There are limited viral load (VL) data available from programs implementing Option B+, lifelong antiretroviral treatment (ART) to all HIV-positive pregnant and postpartum women, in resource-limited settings. Extent of viral suppression from a prevention of mother-to-child transmission of HIV program in Rwanda was assessed among women enrolled in the Kigali Antiretroviral and Breastfeeding Assessment for the Elimination of HIV (Kabeho) Study. ARV drug resistance testing was conducted on women with VL\u3e2000 copies/ml. In April 2013-January 2014, 608 pregnant or early postpartum HIV-positive women were enrolled in 14 facilities. Factors associated with detectable enrollment VL (\u3e20 copies/ml) were examined using generalized estimating equations. The most common antiretroviral regimen (56.7%, 344/607) was tenofovir/lamivudine/efavirenz. Median ART duration was 13.5 months (IQR 3.0-48.8); 76.1% of women were on ART at first antenatal visit. Half of women (315/603) had undetectable RNA-PCR VL and 84.6% (510) had36 months compared to those on ART 4-36 months (72/191, 37.7% versus 56/187, 29.9%), though the difference was not significant. The odds of having detectable enrollment VL decreased significantly as duration on ART at enrollment increased (AOR = 0.99, 95% CI: 0.9857, 0.9998, p = 0.043). There was a higher likelihood of detectable VL for women with lower gravidity (AOR = 0.90, 95% CI: 0.84, 0.97, p = 0.0039), no education (AOR = 2.25, (95% CI: 1.37, 3.70, p = 0.0004), nondisclosure to partner (AOR = 1.97, 95% CI: 1.21, 3.21, p = 0.0063) and side effects (AOR = 2.63, 95% CI: 1.72, 4.03, p36 months with genotyping available. Most women were receiving ART at first antenatal visit, with relatively high viral suppression rates. Shorter ART duration was associated with higher VL, with a concerning increasing trend for higher viremia and drug resistance among women on ART for \u3e3 years

An exactly solvable quantum-lattice model with a tunable degree of nonlocality

An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the "smearing of the lattice coordinates" in the model.Comment: 35 p

Molecular dynamics of a short range ordered smectic phase nanoconfined into porous silicon

4-n-octyl-4-cyanobiphenyl (8CB) has been recently shown to display an unusual sequence of phases when confined into porous silicon (PSi). The gradual increase of oriented short-range smectic (SRS) correlations in place of a phase transition has been interpreted as a consequence of the anisotropic quenched disorder induced by confinement in PSi. Combining two quasielastic neutron scattering experiments with complementary energy resolutions, we present the first investigation of the individual molecular dynamics of this system. A large reduction of the molecular dynamics is observed in the confined liquid phase, as a direct consequence of the dynamical boundary conditions imposed by the confinement. Temperature fixed window scans (FWS) reveal a continuous 'glass-like' reduction of the molecular dynamics of the confined liquid and SRS phases on cooling down to 250 K, where a solid-like behavior is finally reached by a two steps crystallization process