Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Singular solutions of a Hénon equation involving a nonlinear gradient term

<p style='text-indent:20px;'>Here, we consider positive singular solutions of</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p &amp; \text{in}&amp; \Omega \backslash\{0\},\\ u = 0&amp;\text{on}&amp; \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a small smooth perturbation of the unit ball in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> are parameters in a certain range. Using an explicit solution on <inline-formula><tex-math id="M5">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.</p>

Polarimetric Parameters of Scattered Radiation in the Magnetized Plasma

Second order statistical moments of scattered electromagnetic waves in the turbulent magnetized plasma slab with electron density fluctuations are calculated applying the modify stochastic smooth perturbation theory and the boundary conditions. The obtained results are valid for arbitrary correlation function of electron density fluctuations. Stokes parameters are analyzed both analytically and numerically. The theory predicts that depolarization effect caused by second Stokes parameter may be important in scintillation effects. Numerical calculations are carried out for new spectral function of electron density fluctuations containing both anisotropic Gaussian and power-law spectral functions using the experimental data. Polarimetric parameters are calculated for different anisotropy factor and inclination angle of elongated small-scale irregularities with respect to the magnetic lines of forces. The relationship between the scintillations and the polarimetric parameters is important.

On wave propagation in a random micropolar generalized thermoelastic medium

This paper endeavours to study aspects of wave propagation in a random generalized-thermal micropolar elastic medium. The smooth perturbation technique conformable to stochastic differential equations has been employed. Six different types of waves propagate in the random medium. The dispersion equations have been derived. The effects due to random variations of micropolar elastic and generalized thermal parameters have been computed. Randomness causes change of phase speed and attenuation of waves. Attenuation coefficients for high frequency waves have been computed. Second moment properties have been briefly discussed with application to wave propagation in the random micropolar elastic medium. Integrals involving correlation functions have been transformed to radial forms. A special type of generalized thermo-mechanical auto-correlation functions has been used to approximately compute effects of random variations of parameters. Uncoupled problem has been briefly outlined.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

Near-Field Imaging of Interior Cavities

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.