Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

We study integrable geodesic flows on Stiefel varieties V _n,r ?=?SO(n)/SO(n?r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on V _n,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T ^* V _n,r )/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian G _n,r and on a sphere S ^n?1 in presence of Yang–Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety W _n,r ?=?U(n)/U(n?r), the matrix analogs of the double and coupled Neumann systems.     

Examples of frames on the Cantor dyadic group

We present two ways to construct frames on the locally compact Cantor dyadic group. The first approach gives a Parseval frame related to the generalized Walsh?CDirichlet kernel, while the second approach includes the Daubechies-type ??admissible condition?? and leads to dyadic compactly supported wavelet frames. The corresponding wavelet constructions on the Cantor and Vilenkin groups (as well as on the half-line $ {\mathbb{R}_{+} } $ ) requires an additional constraint related to the requirement that the masks have no blocking sets.     

Lie Superautomorphisms on Associative Algebras,II

We consider Lie superautomorphisms of prime associative superalgebras. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.     

Controlled synthesis of gold nanostars by using a zwitterionic surfactant

By replacing cetyltrimethylammonium bromide (CTAB) with the zwitterionic lauryl sulfobetaine (LSB) surfactant in the classical seed-growth synthesis, monocrystalline gold nanostars (m-NS) and pentatwinned gold asymmetric nanostars (a-NS) were obtained instead of nanorods. The main product under all synthetic conditions was a-NS, which have branches with high aspect ratios (AR), thus leading to LSPR absorptions in the 750-1150?nm range. The percentage of m-NS versus a-NS, the aspect ratio of the a-NS branches, and consequently the position of their LSPR absorption can be finely tuned simply by regulating the concentration of reductant, the concentration of surfactant, or the concentration of the "catalytic" Ag(+) cation. The m-NS have instead shorter and larger branches, the AR of which is poorly influenced by synthetic conditions and displays an LSPR positioned around 700?nm. A growth mechanism that involves the direct contact of the sulfate moiety of LSB on the surface of the nano-object is proposed, thereby implying preferential coating of the {111} Au faces with weak interactions. Consistent with this, we also observed the straightforward complete displacement of the LSB surfactant from the surface of the nanostars. This was obtained by the simple addition of thiols in aqueous solution to yield extremely stable coated a-NS and m-NS that are resistant to highly acidic, basic, and in similar to in vivo conditions.     

Synthesis and optical properties of silver nanoparticles in ORMOCER

Experimental results on synthesis of metal nanoparticles in ORMOCER by ion implantation are presented. Silver ions were implanted into organic/inorganic matrix at an accelerating energy of 30?keV and doses in the range of 0.25?10¹⁷ to 0.75?10¹⁷?ion/cm². The silver ions form metal nanoparticles, which demonstrate surface plasmon absorption at the wavelength of 425?C580?nm. The nonlinear absorption of new composite materials is measured by Z-scan technique using 150?fs laser pulses at 780?nm wavelength. ORMOCER matrix shows two-photon nonlinear absorption, whereas ORMOCER with silver nanoparticles demonstrates saturated absorption. Some optical applications of these composite materials are discussed.     

Self-assembly of a ternary architecture driven by cooperative Hg2+ ion binding between cucurbit[7]uril and crown ether macrocyclic hosts

A novel supramolecular assembly comprising CB[7], styrylpyridinium dye (1) and Hg(2+) forms in aqueous solution based on the hydrophobic effect and metal-ligand and ion-dipole interactions. The binding of Hg(2+) to 1·CB[7] displays positive cooperativity relative to 1 itself.     

Growth-morphology-luminescence correlation in ZnO-containing nanostructures synthesized in different media

Zinc hydroxide particles were prepared by a two-step process employing zinc nitrate hexahydrate, urea, ethylene glycol, water and p-toluene-sulfonic acid monohydrate (p-TSA). We used different concentrations of the reactants as well as different volume ratios of the solvents. ZnO particles were obtained by thermal treatment of the reaction products at two different temperatures: 350 °C and 500 °C. The samples were characterized by scanning field emission electron microscopy (SEM), X-ray diffraction (XRD) spectroscopy, BET analysis, thermogravimetry (TG) analysis and photoluminescence (PL) spectroscopy. It was found that after the thermal treatment particles become smaller, with the p-TSA concentration strongly affecting the morphology of the particles. Luminescence properties of the samples probed by PL at 8 K and room temperature exhibited a remarkable correlation with specimens′ nanomorphology. Luminescent features at ～2.0–2.2 eV, ～2.4–2.5 eV, ～2.65 eV, ～2.9 eV, ～3.0–3.1 eV and ～3.3 eV were observed in most specimens, although their relative intensity and temperature dependence were specific to an individual group of samples vis-à-vis their growth history and morphology.     

Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays

We study modulational instability in nonlinear arrays of subwavelength metallic nanoparticles and analyze numerically nonlinear scenarios of the instability development. We demonstrate that modulational instability can lead to the formation of regular periodic or quasiperiodic modulations of the polarization. We reveal that such nonlinear nanoparticle arrays can support long-lived standing and moving oscillating nonlinear localized modes--plasmon oscillons.     

Knotted solitons in nonlinear magnetic metamaterials

We demonstrate that nonlinear magnetic metamaterials comprised of a lattice of weakly coupled split-ring resonators driven by an external electromagnetic field may support entirely new classes of spatially localized modes--knotted solitons, which are stable self-localized dissipative structures in the form of closed knotted chains. We demonstrate different topological types of stable knots for the subcritical coupling between resonators and instability-induced breaking of the chains for the supercritical coupling.