Electromagnetic radiation in a semi-compact space

In this note, we investigate the electromagnetic radiation emitted from a revolving point charge in a compact space. If the point charge is circulating with an angular frequency ${\omega }_0$ on the $(x,y)$-plane at $z=0$ with boundary conditions, $x～x+2\pi R$ and $y～y+2\pi R$, it emits radiation into the z-direction of $z\in [?\infty ,+\infty ]$. We find that the radiation shows discontinuities as a function of ${\omega }_{0}R$ at which a new propagating mode with a different Fourier component appears. For a small radius limit ${\omega }_{0}R?1$, all the Fourier modes except the zero mode on $(x,y)$-plane are killed, but an effect of squeezing the electric field totally enhances the radiation. In the large volume limit ${\omega }_{0}R\to \infty$, the energy flux of the radiation reduces to the expected Larmor formula.     

Exact PsTd invariant and PsTd symmetric breaking solutions,symmetry reductions and Bäcklund transformations for an AB–KdV system

In natural and social science, many events happened at different space–times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, $B=\stackrel{?}{f}A$ for suitable $\stackrel{?}{f}$ operators. A nonlocal AB–KdV system with shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) symmetry where $B=\stackrel{?}{P_s}\stackrel{?}{T_d}A$ is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB–KdV system, including $P_s{T}_d$ invariant and $P_s{T}_d$ symmetric breaking solutions are shown by different methods. The $P_s{T}_d$ invariant solution show that the event happened at A will happen also at B. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton–cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB–KdV system possesses rich structures. Also, a special Bäcklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB–KdV system.     

Percolation on infinitely ramified fractal networks

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff ($D_H$), topological Hausdorff ($D_{tH}$), and spectral ($d_s$) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers ($D_H$, $D_{tH}$, $d_s$), rather than solely by the spatial dimension (d). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension $D_{tH}$, whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension $D_H$. The effect of the network connectivity on the site percolation threshold is revealed.     

Revisiting scaling laws for the diffusion coefficients in simple melts based on the structural deviation from hard-sphere-like case

Molecular simulations of a series of simple melts, including Al, Cu, Ni, Pt, Ti, Si and Ge, are used to study the scaling laws of diffusion coefficients proposed by Rosenfeld and Dzugutov. Our simulation results give strong support to the result that the scaling laws of diffusion coefficients hold true for simple liquids with isotropic many-body interactions but fail for systems with anisotropic interatomic interaction. The failure of the scaling laws in application liquid Si (l-Si) and liquid Ge (l-Ge) is connected to the fact that the excess entropy of them cannot be calculated approximately in terms of the two-body contribution and the original reduction parameters are no longer appropriate. In particular, since the ratio between the positions of the second and first peaks in structure factor ($Q_2/Q_1$) is a direct measure of the structural deviation from the hard-sphere-like case, the temperature dependence of $Q_2/Q_1$ was analyzed. Moreover, in comparison with 1.86, the value of $Q_2/Q_1$ for the hard-sphere-like case, we modified the scaling laws of diffusion coefficients proposed by Rosenfeld and Dzugutov. The modified scaling relations are appropriate not only for simple liquid metals, but also for l-Si and l-Ge with anisotropic interatomic interaction.