A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

Analytical Solution of the Dynamics Equations for a Wave Solid-State Gyroscope Using the Angular Rate Linear Approximation

The exact solution is provided of the dynamics equation for an elastic inextensible ring being the basic model of a wave solid-state gyroscope with the linear law of the base angular rotation rate alteration. This solution is presented in terms of the parabolic cylinder functions (Weber function). Asymptotic approximations are used in the device certain operating modes. On the basis of the solution obtained, the analytical solution to the equation of the ring dynamics in case of piecewise linear approximation of the angular rate arbitrary profile on a time grid is derived. This significantly expands the class of angular rate dependences, for which the solution could be written down analytically. Earlier, in addition to the simplest case of constant angular rate, solutions were obtained for angular rate varying according to the square root law with time (Airy function), as well as according to the harmonic law (Mathieu function). Error dependence of such approximation on the discretization step in time is estimated numerically. Results obtained make it possible to reduce the number of operations, when it is necessary to study long-term evolutions of the dynamic system oscillations, as well as to quantitatively and qualitatively control convergence of finite-difference schemes in solving dynamics equations for a wave solid-state gyroscope with the ring resonator

Study of a parabolic cylinder elastic-plastic contact model

Based on Hertz contact theory, two parabolic cylinder normal contact models are established. The effect of contact angle on normal approach, actual contact area, and normal contact stiffness are investigated, and the effect of the distance from the focus to the directrix (focus distance) on the mechanical characteristics of the models is further analyzed. The parabolic cylinder contact model was verified by simulation analysis and comparison with cylinder contact model. The results demonstrated that the contact angle, focal distance, and load have significant effects on the mechanical properties of the model. The simulation data are basically consistent with the contact model data, and the parabolic cylinder contact model and cylinder contact model have the same change trend. The results verify the correctness of the parabolic cylinder contact model and reveal the variation of the mechanical properties of the contact model.

Radiation collector systems comparison in Contaminants of Emerging Concern degradation by solar heterogeneous photocatalysis

Wastewater with Contaminants of Emerging Concern (CEC) can be generated from different sources as industry, agriculture and urban and hospital wastes. Heterogeneous Photocatalysis (HP) with TiO2 is one of the Advanced Oxidation Processes (AOPs) most suitable for water treatment with CEC. In this research, three CEC: Safranin T (SF), 2,4-dichlorophenoxyacetic acid (2,4-D) and Sulfacetamide (SAM) degradation was evaluated by solar-HP in a quartz wall reactor. First, 365 nm wavelength radiation was used and the best operating conditions was determined under the high flow and aeration configuration, obtaining a removal rate of 48.05% for SF, 11.64% for 2,4-D and 6.98 for SAM. Then, under these conditions, SF, SAM and 2,4-D degradation with solar lighting was made on 4 radiation collector systems configurations, Flat Plate Collector (FPC), V Collector (VC), Parabolic Collector (PC) and Compound Parabolic Cylinder Collector (CPC) until reaching the same value of accumulated energy (122.77 kJ m− 2) finding that the PC had the best performance in the treatment for the three pollutants. Finally, the Collector Impact Ratio Factor (CIRF) for the pollutants was calculated, achieving until 12 times degradation for SAM.

Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes

Bidirectional Grid Constrained (BGC) stochastic processes (BGCSPs) are constrained Itô diffusions with the property that the further they drift away from the origin, the more the resistance to movement in that direction they undergo. The underlying characteristics of the BGC parameter Ψ X t , t are investigated by examining its geometric properties. The most appropriate convex form for Ψ , that is, the parabolic cylinder is identified after extensive simulation of various possible forms. The formula for the resulting hidden reflective barrier(s) is determined by comparing it with the simpler Ornstein–Uhlenbeck process (OUP). Applications of BGCSP arise when a series of semipermeable barriers are present, such as regulating interest rates and chemical reactions under concentration gradients, which gives rise to two hidden reflective barriers.

Controlling Film Thickness Distribution by Magnetron Sputtering with Rotation and Revolution

The laterally graded multilayer collimator is a vital part of a high-precision diffractometer. It is applied as condensing reflectors to convert divergent X-rays from laboratory X-ray sources into a parallel beam. The thickness of the multilayer film varies with the angle of incidence to guarantee every position on the mirror satisfies the Bragg reflection. In principle, the accuracy of the parameters of the sputtering conditions is essential for achieving a reliable result. In this paper, we proposed a precise method for the fabrication of the laterally graded multilayer based on a planetary motion magnetron sputtering system for film thickness control. This method uses the fast and slow particle model to obtain the particle transport process, and then combines it with the planetary motion magnetron sputtering system to establish the film thickness distribution model. Moreover, the parameters of the sputtering conditions in the model are derived from experimental inversion to improve accuracy. The revolution and rotation of the substrate holder during the final deposition process are achieved by the speed curve calculated according to the model. Measurement results from the X-ray reflection test (XRR) show that the thickness error of the laterally graded multilayer film, coated on a parabolic cylinder Si substrate, is less than 1%, demonstrating the effectiveness of the optimized method for obtaining accurate film thickness distribution.

Simulative and Experimental Characterization of an Adaptive Astigmatic Membrane Mirror

Adaptive optical (AO) components play an important role in numerous optical applications, from astronomical telescopes to microscope imaging systems. For most of these AO components, the induced wavefront correction, respectively added optical power, is based on a rotationally symmetric or segmented design of the AO component. In this work, we report on the design, fabrication, and characterization of a micro-electronic-mechanical system (MEMS) adaptive membrane mirror in the shape of a parabolic cylinder. In order to interpret the experimental characterization results correctly and provide a tool for future application development, this is accompanied by the setup of an optical simulation model. The characterization results showed a parabolically deformable membrane mirror with an aperture of 8 × 2 mm2 and an adaptive range for the optical power from 0.3 to 6.1 m−1 (dpt). The optical simulation model, using the Gaussian beamlet propagation method, was successfully validated by laser beam profile measurements taken in the optical characterization setup. This MEMS-based adaptive astigmatic membrane mirror, together with the accompanying simulation model, could be a key component for the rapid development of new optical systems, e.g., adaptive laser line generators.