Planar End Research Articles

Irrational rotations and $2$ -filling rays

Abstract We study a skew product transformation associated to an irrational rotation of the circle $[0,1]/\sim $ . This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of $\{0,1/2\}$ in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of $2$ -filling rays based at that end. These $2$ -filling rays are relevant to Bavard and Walker’s loop graphs.

Regularity of ends of zero mean curvature surfaces in $\mathbf{R}^{2,1}$

In this paper, we analyze ends of zero mean curvature surfaces of mixed (or non-mixed) type in the Lorentzian 3-space $\mathbf{R}^{2,1}$. Among these, we show that spacelike or timelike planar ends are $C^{\infty}$ in the compactification $\hat{L}$ of $\mathbf{R}^{2,1}$ as in the case of minimal surfaces in the Euclidean 3-space $\mathbf{R}^3$. On the other hand, lightlike planar ends are not $C^{\infty}$. Each lightlike planar end of a mixed type surface has the following additional parts: the converging part (a lightlike line in $\mathbf{R}^{2,1}$), the diverging part (the set of the points in $\hat{L} \setminus \mathbf{R}^{2,1}$ corresponding to zero-divisors), and the border of these two parts. We show that such an end is $C^{\infty}$ on the first two parts almost everywhere, while there appears an isolated singularity in the form of $(x^3, x\tau + \mbox{``higher order terms''}, \tau)$ on the border. We also show that conelike singularities of mixed type appear on the lightlike lines in special cases.

Ribaucour surfaces of harmonic type

We introduce the class of Ribaucour surfaces of harmonic type (in short HR-surfaces) that generalizes the Ribaucour surfaces related to a problem posed by Élie Cartan. We obtain a Weierstrass-type representation for these surfaces which depends on three holomorphic functions. As application, we classify the HR-surfaces of rotation, present examples of complete HR-surfaces of rotation with at most two isolated singularities and an example of a complete HR-surface of rotation with one catenoid type end and one planar end. Also, we present a 5-parameter family of cyclic HR-surfaces foliated by circles in non-parallel planes. Moreover, we classify the isothermic HR-surfaces with planar lines of curvature.

Nonorientable minimal surfaces with catenoidal ends

Starting from the pioneering work by Meeks, complete nonorientable minimal surfaces with finite total curvature have been studied by many researchers. However, it seems that there are no known examples all of whose ends are embedded except for Kusner’s flat-ended N-noids. In this paper, we show the existence of a 1-parameter family of complete $${\mathbf{Z}}_N$$ -invariant conformal minimal immersions from finitely punctured real projective planes into $${\mathbf{R}}^3$$ , each of which has $$N+1$$ catenoidal ends, for any odd integer $$N\ge 3$$ . This family gives a deformation from an $$(N+1)$$ -noid with N catenoidal ends and a planar end to Kusner’s flat-ended N-noid. We also give a nonexistence result for such surfaces for any even integer $$N\ge 2$$ .

Rigidity results of free boundary maximal hypersurfaces in the region bounded by a de Sitter space

This paper provides the rigidity results of free boundary maximal hypersurfaces in a region bounded by a de Sitter space in the Lorentz–Minkowski space. First, it is proved that any smooth, compact free boundary maximal hypersurface in a de Sitter ball is the spacelike coordinate planar disk passing through the center of the de Sitter space. Second, a smooth, noncompact, complete free boundary maximal hypersurface in the exterior of a de Sitter ball is considered. For maximal hypersurfaces with one planar end in the exterior of a de Sitter ball, every complete noncompact free boundary maximal hypersurface is shown to be a part of the spacelike coordinate hyperplane.

METHOD OF DESIGNING FUZZY SYSTEM FOR PLANNING ROBOT MANIPULATOR REAL-TIME MOVEMENT IN A STATIC ENVIRONMENT

Under rapid development of science and industry the industrial robots and robotic systems are more widely used in the field of industry. The robots and robotic systems help to manage and master new technological processes efficiently and effectively. In this article there was carried out the analysis of the two degree of freedom (DOF) robot arm, which is widely used in practice in order to reveal its functions and to identify its key dynamic parameters using fuzzy logic system. The mathematical model of robot arm dynamics is composed, which includes the mathematical models of the robot arm links twirling motors and the direct positions task model generating the end point of the robot. After the analysis of the robot model the expanded research was performed – after calculating the missing parameters of experimental 2DOF robot arm motors there was created the robot arm model with „V-REP“ and “MATLAB Simulink” software packages that allowed to set the control voltages and to observe the changing in time processes of link velocity trajectories. There is calculated designed planar robot end point trajectory deviation from the dynamic trajectory.

Finite topology self-translating surfaces for the mean curvature flow in [formula omitted

Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface M⊂R3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S(t)=M+tez evolves by mean curvature flow, or equivalently, that M satisfies the equation HM=ν⋅ez where HM denotes mean curvature, ν is a choice of unit normal to M, and ez is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

Faraday Waves-Based Integrated Ultrasonic Micro-Droplet Generator and Applications.

An in-depth review on a new ultrasonic micro-droplet generator which utilizes megahertz (MHz) Faraday waves excited by silicon-based multiple Fourier horn ultrasonic nozzles (MFHUNs) and its potential applications is presented. The new droplet generator has demonstrated capability for producing micro droplets of controllable size and size distribution and desirable throughput at very low electrical drive power. For comparison, the serious deficiencies of current commercial droplet generators (nebulizers) and the other ultrasonic droplet generators explored in recent years are first discussed. The architecture, working principle, simulation, and design of the multiple Fourier horns (MFH) in resonance aimed at the amplified longitudinal vibration amplitude on the end face of nozzle tip, and the fabrication and characterization of the nozzles are then described in detail. Subsequently, a linear theory on the temporal instability of Faraday waves on a liquid layer resting on the planar end face of the MFHUN and the detailed experimental verifications are presented. The linear theory serves to elucidate the dynamics of droplet ejection from the free liquid surface and predict the vibration amplitude onset threshold for droplet ejection and the droplet diameters. A battery-run pocket-size clogging-free integrated micro droplet generator realized using the MFHUN is then described. The subsequent report on the successful nebulization of a variety of commercial pulmonary medicines against common diseases and on the experimental antidote solutions to cyanide poisoning using the new droplet generator serves to support its imminent application to inhalation drug delivery.

Unsteady Stokes flow induced by a point source of electric current in a non-Darcy hemispherical porous medium

Abstract The time development of the magneto-hydrodynamic Stokes flow in a hemispherical porous medium induced by the symmetric discharge of an electric current from a point source at the center of the planar end of the hemisphere is investigated. Assuming the validity of the Brinkman model, a semi-analytical method is proposed for the determination of the flow field. The streamlines drawn in an axial cross section are found to be in the form of expanding vortices, one on either side of the axis of symmetry. Irrespective of the changes in the evolution of the flow field with time, the polar axis on which the stagnation points of the fluid motion lie from time to time remains stationary without any deviation from the initial position. A comparison of the porous and non-porous media solutions shows that in respect of pure fluids, the steady-state is established much earlier than that in a porous medium. The present analysis plays a significant role in the understanding of the welding processes.

A minimal stable vertical planar end in ℍ2× ℝ has finite total curvature

We prove that a minimal oriented stable annular end in H^2 x R whose asymptotic boundary is contained in two vertical lines has finite total curvature and converges to a vertical plane. Furthermore, if the end is embedded then it is a horizontal graph.

Effective End Group Modification of Poly(3-hexylthiophene) with Functional Electron-Deficient Moieties for Performance Improvement in Polymer Solar Cell.

A series of end-functionalized poly(3-hexylthiophene)s (P3HTs) were synthesized by end-capping with electron-deficient moieties (EDMs, oxadiazole (OXD) and triazole (TAZ)) to prevent the negative influence of bromine chain ends in the common uncapped P3HT in polymer solar cell (PSC) applications. On the basis of the electron-withdrawing capability of the planar OXD end groups, P3HT-end-OXD relative to the uncapped P3HT exhibits a raised absorption coefficient, extended exciton lifetime, and increased crystalline order in the blend with PCBM, leading to an effectual improvement in photovoltaic parameters. However, P3HT-end-TAZ has an opposite result even worse than that of the uncapped P3HT, arising from bulky TAZ end groups. As a consequence, P3HT-end-OXD gives a power conversion efficiency (PCE) of 4.24%, which is higher than those of the uncapped P3HT (3.28%) and P3HT-end-TAZ (0.50%). The result demonstrates that the EDM modification is a valuable method to tailor the structural defect of polymer chain ends. However, the efficacy is dependent on the structure of EDM.

Manipulating the electron distribution through a combination of electron injection and MacKenzie’s Maxwell Demon

Experiments on electron heating are performed in a biased hot filament-produced argon plasma. Electrons are confined by multi-dipole magnetic fields on the radial wall of the cylindrical chamber but not the planar end walls. Electron heating is provided by a combination of cold electron injection (Hershowitz N and Leung K N 1975 Appl. Phys. Lett. 26 607) and a MacKenzie Maxwell Demon (Mackenzie K R et al 1971 Appl. Phys. Lett. 18 529). This approach allows the manipulation of the electrons by introducing a depleted tail into the electron energy distribution function or by removing a depleted tail. It is found that the injected electrons mimic and thermalize with the electron species with the closest average energy or temperature. The effect of the injected electrons is optimal when they mimic the secondary electrons emitted from the wall instead of the degraded primary electrons. Both approaches combine to achieve increases in electron temperature Te from 0.67 to 2.8 eV, which was not significantly higher than using each approach alone.

Fabrication of a chirped artificial compound eye for endoscopic imaging fiber bundle by dose-modulated laser lithography and subsequent thermal reflow

A chirped artificial compound eye on a curved surface was fabricated using an optical resin and then mounted on the end of an endoscopic imaging fiber bundle. The focal length of each lenslet on the curved surface was variable to realize a flat focal plane, which matched the planar end surface of the fiber bundle. The variation of the focal length was obtained by using a photoresist mold formed by dose-modulated laser lithography and subsequent thermal reflow. The imaging performance of the fiber bundle was characterized by coupling with a coaxial light microscope, and the result demonstrated a larger field of view and better imaging quality than that of an artificial compound eye with a uniform focal length. Accordingly, this technology has potential application in stereoscopic endoscopy.

Cylindrical radial superlattice conductors for low loss microwave components

Theory and experimental demonstration of a cylindrical radial superlattice (CRS) conductor composed of alternating nanoscopic non-ferromagnetic/ferromagnetic metal layers are presented with focus on low conductor loss in a K-band microwave spectrum. The dynamic frequency response of the ferromagnetic thin films has been extracted using the Landau-Lifshitz-Gilbert equation which shows a negative magnetic permeability value in the frequencies above its ferromagnetic resonance. The reduction of the conductor loss results from the eddy current canceling (ECC) effect in the CRS conductors, where the negative-permeability ferromagnetic and positive-permeability non-ferromagnetic metal layers produce a zero effective permeability, resulting in virtually infinite skin depth at the targeted frequency. The closed and uniform boundary conditions inherent in the radial shape conductors preclude discontinuity effects occurring at the edges of the planar superlattice conductor and end up with a more effective ECC effect in practice. The design aspects with regards to the CRS materials and structural configuration are discussed. Simulations using a full-wave finite element method high frequency structure simulator are performed to show the ECC effect inside the CRS conductors. An air-lifted inductor made of the CRS conductor has been implemented to prove the effectiveness of the conductor loss reduction with the CRS conductor. The inductor shows an inductance value of 1–2 nH and a Q-factor of 45 at 18 GHz, which is the highest value reported at the frequency by now.

Electron-optical systems with planar field-emission cathode matrices for high-power microwave devices

Reasons for limits on application of field-emission structures in high-power microwave vacuum tubes are studied on the basis of the performed analysis and the methods for their overcoming are discussed. Prospects for application of planar end field-emission structures (PEFESs) as a long-lived current source in high-power vacuum microwave amplifiers are substantiated. The results of design of the proposed electronoptical system with cathode matrices consisting of PEFES cells, which forms electron flows with a power over 100 kW in the contiuous-wave operating conditions, are presented.

Design and construction of an ultrawideband backward wave directional coupler

In this study the design and construction of an ultrawideband backward wave directional coupler is presented. The symmetric, non-uniform coupler covers the frequency range from 2 to 50 GHz with an average coupling of 16.6±2.4 dB while establishing directivity values greater than 10 dB from 10 MHz to 33.8 GHz and 8 dB up to 48.4 GHz. The design is based on coupled microstrip lines suitable for integration into planar front end modules of modern measurement systems. To account for the dispersive behaviour of such waveguides, wiggly-line technique is utilised. Accurate extraction of the waveguide frequency-dependent distributed equivalent circuit parameters is done by 2D finite element electromagnetic field (EM) simulation. To balance out the remaining mistuning appropriate scaling parameters are introduced. The influence of common manufacturing and assembly tolerances on the device performance is analysed. Measurement results of the synthesised coupler with planar matched isolated port are presented.

A Costa-Hoffman-Meeks type surface in $\mathbb H^{2} \times\mathbb R $

We show the existence in the space ℍ2 x ℝ of a family of embedded minimal surfaces of genus 1 ≤ k < +∞ and finite total extrinsic curvature with two catenoidal type ends and one middle planar end. The proof is based on a gluing procedure.

Field-Emission Characteristics of Molded Molybdenum Nanotip Arrays With Stacked Collimation Gate Electrodes

Double-gate field-emission characteristics of metallic field-emitter-array (FEA) cathodes fabricated by molding with stacked collimation gate electrodes with planar end plane are reported. The collimation of field-emission electron beam with minimal reduction of emission current is demonstrated when a negative bias is applied to the collimation gate, whereas when the two electrodes are at the same potential, the emission characteristic of the double-gate device is the same as that of the single-gate device that shows an emission current of ~1 mA from 40 × 40 tip arrays. Results indicate that the device structure of the fabricated double-gate FEAs is promising for high-brilliance cathode applications.

MAXIMAL ANNULI WITH PARALLEL PLANAR BOUNDARIES IN THE THREE-DIMENSIONAL LORENTZ–MINKOWSKI SPACE

AbstractWe prove that maximal annuli in 𝕃3bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann’s example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli have a planar end. Moreover, we extend Shiffman’s convexity result to maximal annuli; but by using Perron’s method we construct a maximal annulus with a planar end where a Shiffman-type result fails.

Calculating vapor attraction for high-coercivity permanent magnets in planar magnetic systems and end clutches

An expression is given for the vapor attraction in high-coercivity constant magnets in the form of rectangular prisms in planar magnetic systems or end magnetic clutches. The expression enables one to correct the attraction of the vapor by the constant magnets as an additional force or load in designing special devices for assembling and disassembling sealed machines and apparatus, in the design of which there are planar magnetic systems or end magnetic clutches, and also for rolling bearings.