Quasihomeomorphisms and Skula spaces

Let [Formula: see text] be a topological space. By the Skula topology (or the [Formula: see text]-topology) on [Formula: see text], we mean the topology [Formula: see text] on [Formula: see text] with basis the collection of all [Formula: see text]-locally closed sets of [Formula: see text], the resulting space [Formula: see text] will be denoted by [Formula: see text]. We show that the following results hold: (1) [Formula: see text] is an Alexandroff space if and only if the [Formula: see text]-reflection [Formula: see text] of [Formula: see text] is a [Formula: see text]-space. (2) [Formula: see text] is a Noetherian space if and only if [Formula: see text] is finite. (3) If we denote by [Formula: see text] the Alexandroff extension of [Formula: see text], then [Formula: see text] if and only if [Formula: see text] is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, [Formula: see text]. A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology [Formula: see text] of [Formula: see text]: The indiscrete components of [Formula: see text] are of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is the intersection of all open sets of [Formula: see text] containing [Formula: see text] (equivalently, [Formula: see text]). We show that [Formula: see text]

Desain Pembelajaran Refleksi dan Translasi Berkonteks Klenteng Sam Poo Kong Semarang

Reflection and translation material are critical to be mastered by students in learning transformation material. However, reflection and translation material are still difficult for students to understand. Therefore, this study aims to develop a learning trajectory that will assist ninth-grade students of Junior High School in grasping the notion of reflection and translation in the context of the Sam Poo Kong Temple in Semarang. The research used the design research method, which consisted of three stages: the preliminary design, the design experiment (pilot experiment and teaching experiment), and retrospective analysis. The PMRI approach was used to develop learning activities in this research. This study involved 32 ninth-grade students at a junior high school in Semarang city. The results of this study is a learning trajectory that includes a series of learning processes in four activities. These are observing the Sam Poo Kong Semarang video and analyzing the properties of reflection, finding the reflection formula, analyzing the properties of translation, discovering the translation formula, and solving contextual problems related to reflection and translation. The activities conducted can enable students to develop a better understanding of material reflection and translation. The research's findings indicate that using the Sam Poo Kong Semarang temple context can equip ninth-grade students to comprehend the concept of material reflection and translation. Additionally, the outcome of this study offers additional local wisdom options that can be used as a context in mathematics.

Application of the general matching law on the study of multi-coated radar absorbing materials

In this paper, the Generalized Matching Law of electromagnetic parameters was given for in the application of radar absorbing materials, and applied in the research of absorbing properties of multi -coated absorbing materials. We gave the power reflection formula, defined the constant of general matching law M, and discussed the relationship of general matching constant M with power reflectivity RP, thickness d, relative permittivity εr, relative permeability μr. We found that the enhancement of RP was not obvious at the ε″r was optimized, but the μ″r was primary. In short, this is consistent with panorama analysis method.

Equivalence of zeta function technique and Abel–Plana formula in regularizing the Casimir energy of hyper-rectangular cavities

Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. By contrast, Abel–Plana formula method permits explicit separation of divergent terms. In regularizing the Casimir energy for a massless scalar field in a D-dimensional rectangular box, we give the rigorous proof of the equivalence of the two methods by deriving the reflection formula of Epstein zeta function from repeatedly application of Abel–Plana formula and giving the physical interpretation of the infinite integrals. Our study may help with the confidence of choosing any regularization method at convenience among the frequently used ones, especially the zeta function method, without the doubts of physical meanings or mathematical consistency.

Elastic impedance in weakly anisotropic media

The original formulation for the P-wave elastic impedance (EI) equation ignores seismic anisotropy. Incorporation of anisotropy effects into the EI formula requires a suitable approximation for reflection coefficients. In order to derive an anisotropic EI equation, this paper uses an approximation for PP-wave reflection [Formula: see text] coefficients which holds for weak-contrast interfaces separating weakly anisotropic media of arbitrary symmetry. Inserting the chosen [Formula: see text] coefficient approximation into the original formalism provides an anisotropic EI formula, which is written as a product of two terms: a modified version for the isotropic EI equation and a correction because of weak anisotropy. The latter term shows dependence of the anisotropic EI formula on the so-called weak anisotropy (WA) parameters, on a reference isotropic medium, and on the azimuthal and incident phase angles. Numerical tests show the performance of the EI formula in calculating anisotropic [Formula: see text] coefficients and in constructing azimuthal far-offset EI logs. Since EI allows applying poststack algorithms without modification, an inversion methodology can be designed for investigating anisotropy in sedimentary formations.

MICROSTRUCTURES AND STRAIN RELAXATION IN MODULATION-DOPED AlxGa1-xN/GaN HETEROSTRUCTURES

Modulation-doped Al 0.22 Ga 0.78 N / GaN heterostructures with various thickness of Si-doped Al 0.22 Ga 0.78 N barrier (n- AlGaN ) were deposited on (0001)-oriented sapphire (α- Al 2 O 3) by atmosphere-pressure metal-organic chemical vapor deposition (MOCVD). The reciprocal space mappings (RSMs) of symmetric reflection (0002) and asymmetric reflection [Formula: see text] were measured by means of the high resolution X-ray diffraction (HRXRD). The results indicate that the microstructures and the strain status of the n- AlGaN barrier correlate to those of the underlying i- GaN layer. The strained n- AlGaN barrier starts to relax when its thickness is 75 nm. It is found that there exists an "abnormal" relaxation state (the strain parameter γ>1) in modulation-doped Al 0.22 Ga 0.78 N / GaN heterostructures, which maybe results from the internal defects in Al 0.22 Ga 0.78 N barrier and the strain relaxation status at the i- GaN /α- Al 2 O 3 interfaces.