With dichroism (anisotropic absorption), the dielectric tensor of a crystal becomes non-Hermitian, and each optic axis (diabolical point) splits into two singular axes (branch-point degeneracies). For a Gaussian beam incident on a dichroic crystal in a direction near the singular axes, the polarized light beyond the crystal is given by the same diffraction integrals as for a transparent crystal...

In this paper, we discuss the recognition problem for Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors’ previous work “the geometry of fronts” for surfaces. The crucial point to prove our criteria for Ak-singularities is to introduce a suitable parametrization of the singularities ...

Abstract. Mathieu's differential equation, y" + (a — 2q cos 2x)y = 0, admits of solutions of period ir or 2x for four countable sets of characteristic values, aiq), which can be ordered as ariq), r = 0, 1, • • -. The power series expansions for the ariq) converge up to the first double point for that order in the complex plane. [At a double point, ar(g) = ar+2iq).] The present work furnishes th...

Given a multivariate generating function F (z1, . . . , zd) = ∑ ar1,... ,rdz r1 1 · · · z rd d , we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat sin...

A bstract . For Sturm-Liouville operators on the half line, we show that the property of having singular, singular continuous, or pure point spectrum for a set of boundary conditions of positive measure depends only on the behavior of the potential at infinity. We also prove that existence of recurrent spectrum implies that of singular spectrum and that “almost sure” existence of L2-solutions i...

Motivated by Vityuk and Golushkov (2004), using the Schauder Fixed Point Theorem and the Contraction Principle, we consider existence and uniqueness of positive solution of a singular partial fractional differential equation in a Banach space concerning with fractional derivative.

There exists a property “structural stability” for “4-dimensional canards” which is singular-limit solution in slow-fast system with bifurcation parameter. It means that the includes possibility to have some critical values on Corresponding these values, pseudo-singular point, singular point time-scaled-reduced should be changed another one. Then, canards may fly if possible. Can fly? The struc...

In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter λ. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper studies the existence, and multiplicity of positive solutions of a singular boundary value problem for second-order differential systems with impulse effects. By using the upper and lower solutions method and fixed point ind...

Positive singular radial entire solutions of a biharmonic equation with subcritical exponent are obtained via the entire radial solutions of the equation with supercritical exponent and the Kelvin transformation. The expansions of such singular radial solutions at the singular point 0 are presented. Using these singular radial entire solutions, we construct solutions with a prescribed singular ...