Noncommutative deformation of instantons

We study instanton solutions on noncommutative Euclidean 4-space which are deformations of instanton solutions on commutative Euclidean 4-space. We show that the instanton numbers of these noncommutative instanton solutions coincide with the commutative solutions and conjecture that the instanton number in R⁴${\mathbb{R}}^4$ is preserved for general noncommutative deformations. We also study noncommutative deformation of instanton solutions on a T⁴$T^4$ with twisted boundary conditions.     

Phonon energy inversion in graphene during transient thermal transport

This work reports on the phonon energy inversion in graphene nanoribbons: after initial localized thermal excitation, the energy of initial cold phonons (flexural mode: FM) becomes higher than that of local hot phonons (longitudinal and transverse modes: LM/TM). Such energy inversion holds for about 50 picoseconds. Two physical factors combine together to give rise of this phenomenon: one is the much faster heat conduction by FM phonons than that by LM/TM phonons, and the other factor is the strongly temperature-dependent energy exchange rate between FM and LM/TM phonons: 3.7×10¹⁰ s⁻¹$3.7\times {10}^{10}{\text{s}}^{-1}$ at 84 K to 20.3×10¹⁰ s⁻¹$20.3\times {10}^{10}{\text{s}}^{-1}$ at around 510 K.     

Abelian vortices from sinh-Gordon and Tzitzeica equations

It is shown that both the sinh-Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in R^2,1${\mathbb{R}}^{2,1}$, or on a surface conformally related to a hyperbolic affine sphere in R³${\mathbb{R}}^3$. In both cases the Higgs field and the U(1)$U(1)$ vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh-Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlevé transcendents are used to compute explicit values for the strength of the vortices.     

Integrable boundary interaction in 3D target space: The “pillow-brane” model

We propose a model of boundary interaction, with three-dimensional target space, and the boundary values of the field X∈R³$\mathbf{X}\in {\mathbb{R}}^3$ constrained to lay on a two-dimensional surface of the “pillow” shape. We argue that the model is integrable, and suggest that its exact solution is described in terms of certain linear ordinary differential equation.     

Formulae for null curves deriving from elliptic curves

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C³${\mathbb{C}}^3$ and thus an algebraic minimal surface in R³${\mathbb{R}}^3$. We derive simple formulae for the coordinate functions of such a null curve.     

Numerical solution of an inverse medium scattering problem for Maxwell’s Equations at fixed frequency

Consider a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in R³${\mathbb{R}}^3$. In this paper, well-posedness of the variational problem for the direct scattering is examined. An energy estimate for the scattered field is obtained on which the Born approximation is based. A regularized recursive linearization method for the inverse medium scattering, which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field, is developed. The algorithm requires only single-frequency data. Using an initial guess from the Born approximation, each update is obtained via continuation on the spatial frequency of a two-parameter family of plane waves by solving one direct problem and one adjoint problem of the Maxwell equation.     

Generalized Weyl quantization on the cylinder and the quantum phase

Generalized Weyl quantization formalism for the cylindrical phase space S¹×R¹$S^1\times {\mathbb{R}}^1$ is developed. It is shown that the quantum observables relevant to the phase of the linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space L²(S¹)$L^2\left(S^1\right)$.     

Curvatures,volumes and norms of derivatives for curves in Riemannian manifolds

The well-known formulas express the curvature and the torsion of a curve in R³${\mathbb{R}}^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in arbitrary Riemannian manifolds. Our motivation comes from physics. It follows that regular curves in Rⁿ${\mathbb{R}}^n$ are determined up to isometry by the norms of their n$n$ consecutive derivatives. We extend this fact to two-point homogeneous spaces.     

On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data

We examine the scaling regime for the detrended fluctuation analysis (DFA)—the most popular method used to detect the presence of long-term memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of time series length L$L$ and the correlation coefficient of the linear regression R²$R^2$ at various confidence levels. Next, a similar analysis for artificial short series of data with long-term memory is performed. In both cases the scaling range λ$\lambda$ is found to change linearly—both with L$L$ and R²$R^2$. We show how this dependence can be generalized to a simple unified model describing the relation λ=λ(L,R²,H)$\lambda =\lambda (L,R^2,H)$ where H$H$ (1/2≤H≤1$1/2\le H\le 1$) stands for the Hurst exponent of the long range autocorrelated signal. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where a huge number of short time series has to be analyzed at the same time, without possibility of checking the scaling range in each of them separately.     

Singular self-dual Zollfrei metrics and twistor correspondence

We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean’s self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP³$\mathbb{R}{\mathbb{P}}^3$ to CP³$\mathbb{C}{\mathbb{P}}^3$ which has one singular point. This embedding corresponds to an odd function on R$\mathbb{R}$ that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.     

Ribaucour transformations for isometric immersions of pseudo-Riemannian space forms

In this paper, we give the explicit construction for isometric immersions of pseudo-Riemannian space forms via loop group actions. As a consequence, we obtain a Ribaucour transformation and its permutability theorem for isometric immersions of pseudo-Riemannian space forms, which is a generalization of the permutability theorem for surfaces in R³$R^3$, and a family of new isometric immersions of the pseudo-Riemannian space forms from a given one via a purely algebraic algorithm.     

Distribution of xp in some molecular rotational states

Developing the analysis of the distribution of the so-called posmom xp to some molecular rotational states for diatomic molecules and spherical cage molecules, we obtain posmometry (introduced recently by Bernard and Gill, 2010 [5]) of the spherical harmonics and demonstrate that it is similar to the momentum distributions of the stationary states for a one-dimensional simple harmonic oscillator. The results are not only potentially experimentally testable but also reflect a fact that the embedding of the two-dimensional spherical surface S²$S^2$ in three-dimensional flat space R³$R^3$ is physically self-consistent and appealing.     

Tamm plasmon-polaritons: First experimental observation

We report on the first experimental observation of Tamm plasmon-polaritons (TPP) formed at the interface between a metal and a dielectric Bragg reflector (DBR). Contrary to conventional surface plasmons, TPPs have an in-plane wave vector less than the wave vector of light in vacuum, which allows for their direct optical excitation, and can be formed in both the TE and TM polarizations. The angular resolved reflectivity and transmission spectra of a GaAs/AlAs DBR covered by Au films of various thicknesses show the resonances associated with the TPP at low temperatures and at room temperature. The in-plane dispersion of TTPs is parabolic with an effective mass 4×10⁻⁵$4\times 10^{-5}$ of the free electron mass.     

A Jenkins–Serrin problem on the strip

We describe the family of minimal graphs on strips with boundary values ±∞$\pm \infty$ disposed alternately on edges of length 1, and whose conjugate graphs are contained in horizontal slabs of width 1 in R³${\mathbb{R}}^3$. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfaces and the singly periodic Scherk minimal surface of angle π/2$\pi /2$.     

Geometry of projective plane and Poisson structure

V.I. Arnold [V. I. Arnold, Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane, Journal of Geometry and Physics, 53 (4) (2005), 421–427] gave an alternative proof to the Lobachevsky triangle altitudes theorem by using a Poisson bracket for quadratic forms and its Jacobi identity, and showed that the orthocenter theorem can be extended on RP²$\mathbb{R}{P}^2$. In this paper, we find a new identity in the Poisson algebra of quadratic forms. Following Arnold’s idea, the goal of this article is to give alternative proofs to theorems, of Desargues, Pascal, and Brianchon, in RP²$\mathbb{R}{P}^2$, by using the Poisson bracket and the identity.     

Bertrand curves in the three-dimensional sphere

A curve α$\alpha$ immersed in the three-dimensional sphere S³${\mathbb{S}}^3$ is said to be a Bertrand curve if there exists another curve β$\beta$ and a one-to-one correspondence between α$\alpha$ and β$\beta$ such that both curves have common principal normal geodesics at corresponding points. The curves α$\alpha$ and β$\beta$ are said to be a pair of Bertrand curves in S³${\mathbb{S}}^3$. One of our main results is a sort of theorem for Bertrand curves in S³${\mathbb{S}}^3$ which formally agrees with the classical one: “Bertrand curves in S³${\mathbb{S}}^3$ correspond to curves for which there exist two constants λ≠0$\lambda \ne 0$ and μ$\mu$ such that λκ+μτ=1$\lambda \kappa +\mu \tau =1$”, where κ$\kappa$ and τ$\tau$ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S³${\mathbb{S}}^3$ as the only twisted curves in S³${\mathbb{S}}^3$ having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S³${\mathbb{S}}^3$ and (1,3)-Bertrand curves in R⁴${\mathbb{R}}^4$.