Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

By the application of the coinduction method as well as the Magri method to the ideal of real Hilbert–Schmidt operators we construct the hierarchies of integrable Hamiltonian systems on Banach Lie–Poisson spaces which consist of such types of operators. We also discuss their algebraic and analytic properties and solve them in dimensions, N=2,3,4$N=2,3,4$.     

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m$m$th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski (n+1)$(n+1)$-spaces (n≥3$n\ge 3$) of nonzero constant m$m$th mean curvature (m≤n−1$m\le n-1$) with two distinct principal curvatures λ$\lambda$ and μ$\mu$ satisfying inf(λ−μ)²>0$inf{(\lambda -\mu )}^2>0$ are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder Hⁿ⁻¹(c)×R${\mathbb{H}}^{n-1}\left(c\right)\times \mathbb{R}$ in terms of square length of the second fundamental form.     

Curve flows in Lagrange–Finsler geometry,bi-Hamiltonian structures and solitons

Methods in Riemann–Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N$N$-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G/SO(n)$G/SO\left(n\right)$ provided with an N$N$-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.     

Abelian functions for cyclic trigonal curves of genus 4

We discuss the theory of generalized Weierstrass σ$\sigma$ and ℘$\wp$-functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve y³=x⁵+_λ4x⁴+_λ3x³+_λ2x²+_λ1x+_λ0$y^3=x^5+{\lambda }_4{x}^4+{\lambda }_3{x}^3+{\lambda }_2{x}^2+{\lambda }_{1}x+{\lambda }_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘$\wp$-functions, and the derivation of addition formulae.     

On the geometry of isomonodromic deformations

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇)$(V,\nabla )$ of vector bundles and connections as being obtained by “twists” supported over points of a fixed vector bundle _V0$V_0$ with a fixed connection _∇0${\nabla }_0$; this gives two deformations, one, isomonodromic, of (V,∇)$(V,\nabla )$, and another induced from the isomonodromic deformation of (_V0,_∇0)$(V_0,{\nabla }_0)$. The difference between the two will be Hamiltonian.     

Rotating Einstein–Maxwell–dilaton black holes in D dimensions

We construct exact charged rotating black holes in Einstein–Maxwell–dilaton theory in D   spacetime dimensions, D?5$D?5$, by embedding the D  -dimensional Myers–Perry solutions in D+1$D+1$ dimensions, and performing a boost with a subsequent Kaluza–Klein reduction. Like the Myers–Perry solutions, these black holes generically possess N=[(D−1)/2]$N=[(D-1)/2]$ independent angular momenta. We present the global and horizon properties of these black holes, and discuss their domains of existence.     

Template-free sonochemical synthesis of flower-like ZnO nanostructures

Flower-like ZnO nanostructures have been successfully synthesized via a facile and template-free sonochemical method, using zinc acetate and potassium hydroxide as reactants only. The as-synthesized flower-like ZnO nanostructures were composed of nanorods with the width of ∼300–400 nm$\sim 300\text{–}400\text{nm}$ and the length of ∼2–3 μm$\sim 2\text{–}3\text{μm}$. The structures, morphologies and optical properties of the as-prepared products were characterized by X-ray diffraction, scanning electron microscope, transmission electron microscopy, UV-Vis spectrophotometry and Raman-scattering spectroscopy. A plausible formation mechanism of flower-like ZnO nanostructures was studied by SEM which monitors an intermediate morphology transformation of the product at the different ultrasonic time (t=80,90,95,105, and 120 min$t=80,90,95,105\text{, and}120\text{min}$).     

Flow-induced noise of a wall-mounted finite airfoil at low-to-moderate Reynolds number

This paper presents an experimental investigation of the flow-induced noise created by a wall-mounted finite airfoil at low-to-moderate Reynolds number and zero angle of attack. Far-field noise measurements have been taken at a single observer location and with two perpendicular microphone arrays in an anechoic wind tunnel at Reynolds numbers of _Rec=9.2×10⁴–1.6×10⁵${\mathit{Re}}_c=9.2\times {10}^4–1.6\times {10}^5$, based on chord, and for a variety of airfoil aspect ratios (length to chord ratio of L/C=0.2–2$L/C=0.2–2$, corresponding to length to thickness ratio of L/T=1.7–16.7$L/T=1.7–16.7$). Additionally, surface oil-film visualisation images and unsteady velocity measurements taken in the near trailing edge wake are related to far-field noise measurements to determine the flow mechanisms responsible for noise generation. The results show that the wall-mounted finite airfoil radiates noise similar to a two-dimensional airfoil when L/T>8.3$L/T>8.3$. Despite the incoming boundary layer height at the junction being 1.30≤δ/T≤1.46$1.30\le \delta /T\le 1.46$, junction and tip flow suppresses tonal noise production for airfoil?s up to L/T=8.3$L/T=8.3$ at _Rec=9.2×10⁴–1.2×10⁵${\mathit{Re}}_c=9.2\times {10}^4–1.2\times {10}^5$. Trailing edge noise is found to be the dominant airfoil noise generation mechanism at frequencies above 1 kHz with the position of the noise source along the trailing edge determined by the proportion of the airfoil span influenced by flow at the airfoil–wall junction.     

Loop group decompositions in almost split real forms and applications to soliton theory and geometry

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action–by the whole subgroup of loops which extend holomorphically to the exterior disc–on the U$U$-hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1$1+1$ wave maps) from surfaces into compact symmetric spaces. An “Iwasawa-type” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given.     

Magnetic properties of smooth terminating dipole bands in Te

Three strongly coupled sequences have been established in ^110,112Te up to high spins. They are interpreted in terms of deformed structures built on proton 1-particle–1-hole excitations that reach termination at I∼40?$I\sim 40?$. This is the first observation of smooth terminating dipole structures in this mass region. Lifetime measurements have allowed the extraction of experimental B(M1;I→I−1)$B(\mathrm{M}1;I\to I-1)$ and B(E2;I→I−2)$B(\mathrm{E}2;I\to I-2)$ reduced transition rates for one of the dipole bands in ¹¹⁰Te. The results support the deformed interpretation.     

Flame wrinkles from the Zhdanov–Trubnikov equation

The Zhdanov–Trubnikov equation describing wrinkled premixed flames is studied, using pole decompositions as starting points. Its one-parameter (−1?c?+1$-1?c?+1$) nonlinearity generalises the Michelson–Sivashinsky equation (c=0$c=0$) to a stronger Darrieus–Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c≈−1$c\approx -1$, which numerical resolutions confirm. Large wrinkles are analysed via   a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner–Pollaczek polynomials, which numerical results confirm for −1<c?0$-1<c?0$ (reduced stabilisation). Although locally ill-behaved if c>0$c>0$ (over-stabilisation) such analytical solutions can yield accurate flame shapes for 0?c?0.6$0?c?0.6$. Open problems are invoked.