The Analysis of Rotated Vector Field for the Pendulum

The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $\beta$ and the periodic solution, and a conclusion that the critical value of $\beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics.Comment: 11 pages With 8 figures. A mathematical focus version separated from cond-mat/0702061. cond-mat/0702061 is replaced with a physics focus version. In Ver.2, some sentences are revised on rhetoric, and the bibliography is renewe

PT{\cal PT} symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling

By introducing the next-nearest-neighboring (NNN) intersite coupling, we investigate the eigenenergies of the $\cal PT$-symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials at the end sites. It is found that with the strengthening of NNN coupling, the particle-hole symmetry is destroyed. As a result, the bonding band is first narrowed and then undergoes the top-bottom reversal followed by the its width's increase, whereas the antibonding band is widened monotonously. In this process, the topological state extends into the topologically-trivial region, and its energy departs from the energy zero point, accompanied by the emergence of one new topological state in this region. All these results give rise to the complication of the topological properties and the manner of $\cal PT$-symmetry breaking. It can be concluded that the NNN coupling takes important effects to the change of the topological properties of the non-Hermitian SSH system.Comment: 8 pages and 8 figure

Skew-spectra and skew energy of various products of graphs

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ consisting of all the eigenvalues of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$. In this paper, we give orientations of the Kronecker product $H\otimes G$ and the strong product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an arbitrary graph. Then we determine the skew-spectra of the resultant oriented graphs. As applications, we construct new families of oriented graphs with maximum skew energy. Moreover, we consider the skew energy of the orientation of the lexicographic product $H[G]$ of a bipartite graph $H$ and a graph $G$.Comment: 11 page

Chi-Square Test Neural Network: A New Binary Classifier based on Backpropagation Neural Network

We introduce the chi-square test neural network: a single hidden layer backpropagation neural network using chi-square test theorem to redefine the cost function and the error function. The weights and thresholds are modified using standard backpropagation algorithm. The proposed approach has the advantage of making consistent data distribution over training and testing sets. It can be used for binary classification. The experimental results on real world data sets indicate that the proposed algorithm can significantly improve the classification accuracy comparing to related approaches

Interior Regularity for a generalized Abreu Equation

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.Comment: 22 pages,1 figure. Any comments are welcome. arXiv admin note: text overlap with arXiv:1305.087

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations

Some Estimates for a Generalized Abreu's Equation

We study a generalized Abreu equation and derive some estimates.Comment: 18 pages. Any comments are welcom

Fast and Accurate Computation of Exact Nonreflecting Boundary Condition for Maxwell's Equations

We report in this paper a fast and accurate algorithm for computing the exact spherical nonreflecting boundary condition (NRBC) for time-dependent Maxwell's equations. It is essentially based on a new formulation of the NRBC, which allows for the use of an analytic method for computing the involved inverse Laplace transform. This tool can be generically integrated with the interior solvers for challenging simulations of electromagnetic scattering problems. We provide some numerical examples to show that the algorithm leads to very accurate results.Comment: 6 pages, 2 figure

An accurate spectral method for Maxwell equations in Cole-Cole dispersive media

In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields and the induced electric polarisation (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then we solve to IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show such an enriched collocation method is convergent and accurate. % analysis of the scheme is carried out.Comment: 22 page

Accurate Simulation of Ideal Circular and Elliptic Cylindrical Invisibility Cloaks

The coordinate transformation offers a remarkable way to design cloaks that can steer electromagnetic fields so as to prevent waves from penetrating into the {\em cloaked region} (denoted by $\Omega_0$, where the objects inside are invisible to observers outside). The ideal circular and elliptic cylindrical cloaked regions are blown up from a point and a line segment, respectively, so the transformed material parameters and the corresponding coefficients of the resulted equations are highly singular at the cloaking boundary $\partial \Omega_0$. The electric field or magnetic field is not continuous across $\partial\Omega_0.$ The imposition of appropriate {\em cloaking boundary conditions} (CBCs) to achieve perfect concealment is a crucial but challenging issue. Based upon the principle that finite electromagnetic fields in the original space must be finite in the transformed space as well, we obtain CBCs that intrinsically relate to the essential "pole" conditions of a singular transformation. We also find that for the elliptic cylindrical cloak, the CBCs should be imposed differently for the cosine-elliptic and sine-elliptic components of the decomposed fields. With these at our disposal, we can rigorously show that the governing equation in $\Omega_0$ can be decoupled from the exterior region $\Omega_0^c$, and the total fields in the cloaked region vanish. We emphasize that our proposal of CBCs is different from any existing ones