Casimir Effect between Superconducting Plates in the Mixed State

The Casimir effect between type-II superconducting plates in the coexisting phase of a superconducting phase and a normal phase is investigated. The dependence of the optical conductivity of the superconducting plates on the external magnetic field is described in terms of the penetration depth of the incident electromagnetic field, and the permittivity along the imaginary axis is represented by a linear combination of the permittivities for the plasma model and Drude models. The characteristic frequency in each model is determined using the force parameters for the motion of the magnetic field vortices. The Casimir force between parallel YBCO plates in the mixed state is calculated, and the dependence on the applied magnetic field and temperature is considered.

On Convolution and Convex Combination of Harmonic Mappings

In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.

Complex Eigenvalue Splitting for the Dirac Operator

We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.

Regularized Asymptotic Solutions of Integro-Differential Equations with Fast and Slow Variables

The article considers a nonlinear integro-differential system of equations with fast and slow variables. Such systems were not considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. The known studies were mainly devoted to construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the first variation matrix (on the degenerate solution) is located strictly in the open left-half plane of a complex variable. If the spectrum of this matrix falls on the imaginary axis, the S.A. Lomov regularization method is commonly used. However, this method was mainly developed for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this article, the regularization method is generalized for two-dimensional integro-differential equations with fast and slow variables.

A Review of the Role of Intangible Axis toward the Pathok Negoro’s Design Concept, Yogyakarta

The research aimed to review the role of the ‘mystical’ imaginary axis in Javanese cosmology in the Yogyakarta that might affect the inner spaces of the Pathok Negoro mosques (Ploso Kuning, Babadan, Dongkelan, and Mlangi). The research question was how Yogyakarta’s imaginary axis affected the inner spaces of these four Pathok Negoro. Yogyakarta was known to have a unique Javanese cosmology, cultural, and social life character compared to several cities on the island of Java. Yogyakarta kept a harmonious power between human life and the universe and still hold strong Javanese animism and was still tightly coupled with the city’s modern life. Human harmony with the universe was seen in the axis of a city that consisted of two main axes in the direction of the compass, namely north to south and east to west. The research applied a qualitative method by doing a literature study and observation. The research shows that the imaginary axis plays an essential role in Yogyakarta’s urban design and creates a sustainable concept of the four Pathok Negoro. However, it is not directly involved in the order of the inner space of the architecture design of each mosque.