Planar Case Research Articles

Surface plasmon modes in single mode fiber coated with silver films

Surface plasmon (SP) modes can be excited in the single mode optical fiber coated with a thin silver film. To analysis these modes, a full-vector method is employed to solve Maxwell equations in this work. By comparing with the planar case, the resonance coupling between the fiber cladding modes and SP modes can be understood. Moreover, the bound thin-film SP mode can also be solved by this method. In the particular case, both the short range and long range thin-film SP mode can be supported in the fiber. The analysis reveals that the thin-film SP modes on the cylindrical silver film are sensitive to the surrounding medium with high refractive index.

Periodic solutions of symmetric Kepler perturbations and applications

We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations in an independent way, namely, when the unperturbed part corresponds to a Kepler problem in inertial cartesian coordinates and when it corresponds to a Kepler problem in rotating coordinates on ℝ3. Moreover, the characteristic multipliers of the symmetric periodic solutions are characterized. The planar case arises as a particular case. Finally, we apply these results to study the existence and stability of periodic orbits of the Matese-Whitman Hamiltonian and the generalized Størmer model.

Ultrashort Optical Pulses in a Fermi Liquid and Duality of Gauge Gravitation

The problem of the propagation of ultrashort pulses, including both two-dimensional and three-dimensional pulses, in a Fermi liquid is considered with the help of representations of the duality of gauge gravitation. The electromagnetic field is considered classically on the basis of the Maxwell equations. The effective equation so obtained is analyzed numerically and the dynamics of the state of the electromagnetic field are elucidated in the planar case, and also when it is localized in two/three spatial dimensions.

Quasicharacteristic radiation of relativistic electrons at orientation motion in lithium halides crystals along charged planes and axes

The paper deals with the investigation of the orientation motion of relativistic electrons in charged (111) planes and charged [110] axes of lithium halides ionic crystals of LiF, LiCl, LiBr and LiI. On the basis of these investigations the spectra of quasicharacteristic radiation for the electron beams with various Lorentz-factors both in planar and axial cases have been calculated numerically.

Analysis and Optimization of the Efficiency of Induction Heating Applications With Litz-Wire Planar and Solenoidal Coils

Optimization of the efficiency of an induction heating application is essential in order to improve both reliability and performance. For this purpose, multistranded cables with litz structure are often used in induction heating applications. This paper presents an analysis and optimization of the efficiency of induction heating systems focusing on the optimal copper volume of the winding with respect to different constraints. The analysis is based on the concept of a one-strand one-turn coil, which captures the dissipative effects of an induction heating system and reduces the number of variables of the analysis. An expression for the efficiency of the induction heating system is derived. It is found that, with the geometry and the other parameters of the system fixed, efficiency depends on the copper volume of the windings. In order to use this result to optimize the efficiency of an application, volume restrictions, the packing factor and the window utilization factor are also considered. The optimum frequency for an induction heating system is also studied in this study. An experimental verification for both planar and solenoidal cases is also presented.

Non-Collision Singularities in the Planar Two-Center-Two-Body Problem

In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers $Q_1=(-\chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $\mu\ll 1$. They interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlev\'{e} conjecture.

Modified Delaunay empty sphere condition in the problem of approximation of the gradient

The classical Schwarz example shows that piecewise- linear approximation of smooth functions does not necessary yield convergence of the derivatives. However, in the planar case, the required convergence holds if the triangulation of the grid satisfies the empty sphere condition (that is, it is a Delaunay triangulation). These results do not extend to the multidimensional case, as is shown by our published examples. We give a modified empty sphere condition that also guarantees the necessary approximation in the multidimensional case.

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.

Near-Optimal Minimum-Time Guidance Under Spatial Angular Constraint in Atmospheric Flight

A near-optimal minimum-time guidance law to a fixed point under a terminal angular manifold is developed. The planar case and the spatial case are separately considered. The problems are solved in several stages with increased level of complexity: from the simple case of a constant speed with a fixed terminal angular direction (Dubins problem), through the case of a constant speed with a terminal conical manifold, to the more general case of atmospheric flight with a terminal conical manifold.

Curvature-driven foam coarsening on a sphere: A computer simulation.

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state.

Efficient yellow and green emitting InGaN/GaN nanostructured QW materials and LEDs

Efficient green emitting LEDs and monolithic white light emitting LEDs require the extension of the range of efficient light emission in the GaN/InGaN materials system. We demonstrate high efficiency green and yellow light emitting multiple quantum well (MQW) structures grown on GaN nanostripe templates. The structures show promise for realizing high efficiency phosphor – free white LEDs. The nanostripe dimensions range from 100 to 300 nm and have separations that range from 300 nm to 1 μm. The MOCVD growth conditions strongly affect surfaces expressed in the GaN nanostripes whose sidewalls can be controlled to be nearly vertical or inclined and intersecting. Single quantum well (QW) structures are grown on these different stripes. Photoluminescence (PL) measurement shows that QW grown on stripes with the {10−11} surfaces and triangular shape emit the longest peak wavelength and highly efficient PL emission peak wavelengths as long as 570 nm are realized. PL and electroluminescence (EL) spectra show narrow linewidth that is comparable to the planar case and CL studies further demonstrate the uniform emission wavelength along the sidewalls of the structures. Finally, we have grown and fabricated green emitting LEDs on {10−11} faceted nanostripes with promising device characteristics.

Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture

In this note we prove some Rogers-Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.

A model for the quasistatic growth of cracks with fractional dimension

We study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated.

A Limb Compliant Sensing Strategy for Robot Collision Reaction

This paper introduces a compliant limb sensor (CLS) concept for collision detection during robot–human contact. The CLS consists of an external rigid shell compliantly connected to the robot link with collision inferred from measured shell displacements. Measuring displacement of a rigid shell allows customizable compliance and high sampling rates due to the small number of required sensors. The proposed sensor is prototyped for the planar case using LED/light-to-voltage (LTV) sensors for shell pose measurement and foam as the compliant link between the shell and base. A physically motivated model for the output of LED/LTV sensor pairs is formulated for the estimation of the shell pose. Voltage measurements of redundant LTVs and a calibrated shell model are used with an iterative optimization routine to estimate the shell pose at high frequencies. Sensor performance is tested using five trajectories: rest, compression, shear, rotation, and arbitrary motion. Experiments confirmed that the CLS can sense the presence, direction, and intensity of impact. The potential application of the proposed sensor to safety in physical human–robot interaction is discussed. The novel sensing methodology also enables a new method of 3-D human–computer interaction due to the ability to modify the compliance and operating range of the CLS.

Hausdorff dimension, intersections of projections and exceptional plane sections

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our first main result considers the orthogonal projections of two Borel sets A , B ⊂ R 2 A,B \subset \mathbb {R}^{2} into one-dimensional subspaces. Under the assumptions dim ⁡ A ≤ 1 &gt; dim ⁡ B \dim A \leq 1 &gt; \dim B and dim ⁡ A + dim ⁡ B &gt; 2 \dim A + \dim B &gt; 2 , we prove that the intersection of the projections P L ( A ) P_{L}(A) and P L ( B ) P_{L}(B) has dimension at least dim ⁡ A − ϵ \dim A - \epsilon for positively many lines L L , and for any ϵ &gt; 0 \epsilon &gt; 0 . This is quite sharp: given s , t ∈ [ 0 , 2 ] s,t \in [0,2] with s + t = 2 s + t = 2 , we construct compact sets A , B ⊂ R 2 A,B \subset \mathbb {R}^{2} with dim ⁡ A = s \dim A = s and dim ⁡ B = t \dim B = t such that almost all intersections P L ( A ) ∩ P L ( B ) P_{L}(A) \cap P_{L}(B) are empty. In case both dim ⁡ A &gt; 1 \dim A &gt; 1 and dim ⁡ B &gt; 1 \dim B &gt; 1 , we prove that the intersections P L ( A ) ∩ P L ( B ) P_{L}(A) \cap P_{L}(B) have positive length for positively many L L . If A ⊂ R 2 A \subset \mathbb {R}^{2} is a Borel set with 0 &gt; H s ( A ) &gt; ∞ 0 &gt; \mathcal {H}^{s}(A) &gt; \infty for some s &gt; 1 s &gt; 1 , it is known that A A is “visible” from almost all points x ∈ R 2 x \in \mathbb {R}^{2} in the sense that A A intersects a positive fraction of all lines passing through x x . In fact, a result of Marstrand says that such non-empty intersections typically have dimension s − 1 s - 1 . Our second main result strengthens this by showing that the set of exceptional points x ∈ R 2 x \in \mathbb {R}^{2} , for which Marstrand’s assertion fails, has Hausdorff dimension at most one.

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.

Improved convergence theorems for bubble clusters. I. The planar case

Improved convergence theorems for bubble clusters. I. The planar case

Lattice Point Inequalities for Centered Convex Bodies

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the $o$-symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes.

CORRELATION EFFECTS ON THE MIMO CAPACITY FOR CONFORMAL ANTENNAS ON A PARABOLOID

The use of conformal antennas in a MIMO link scenario is investigated. Conformal slot antennas are considered both in the transmitter and the receiver. First, a new modified correlation coefficient is derived that goes beyond the Clarke coefficient and takes into account the element radiation pattern. Secondly, a hybrid formulation that accounts for the impact of the mutual coupling and the pattern dependent correlation on the capacity is presented. The mutual coupling for slots placed circumferentially on a paraboloid substrate is derived using a rigorous approach based on Uniform Theory of Diffraction (UTD). The capacity is evaluated for the case of Rayleigh fading channel considering the new pattern dependent correlation coefficient and the conformal antenna mutual coupling. The planar case is included as a limiting case. It is shown that for conformal antennas on a paraboloid the capacity degradation compared to the planar case is up to 0.5 bps/Hz due to coupling and correlation.

Synthesis of the force and displacement methods of structural mechanics. The planar truss case

The papers delivers a new computational method of statical analysis of frames, well suited for both the 2D and 3D settings and easily generalizable to the case of the presence of big axial forces. The method is based on the compatibility equations which have their roots in the force method, while the unknowns refer to the displacement method. These equations involve also the unknown cross sectional forces which are the force method unknowns. The method proposed can be viewed as the displacement method with an implicit procedure of deriving the stiffness equations.