Set Of Planes Research Articles

Integral geometry of pairs of hyperplanes or lines

Crofton’s formula of integral geometry evaluates the total motion invariant measure of the set of k-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.

Atomic-level calculations and experimental study of dislocations in InSb

Plastic deformation in InSb single crystals is governed by the motion of dislocations. Since InSb has a diamond cubic lattice, it possesses two sets of slip planes: a shuffle set and a glide set. Transmission electron microscopy analysis of deformed bulk single crystals shows that, at low temperatures (<20 °C), dislocations have narrow cores, while at higher temperatures, they have extended cores. However, it remains unclear to which slip plane set these dislocations belong. In this paper, by combining experiments and atomic-level calculations, we show that dislocations with narrow and extended cores, respectively, belong to the shuffle and glide sets. The conclusion is reached by calculating the generalized stacking fault energy curves and ideal shear stresses using density functional theory calculations and the intrinsic stacking fault width associated with dislocations using atomistic simulations. It is also found that while the shuffle set dislocations are easier to activate at lower temperatures, dislocations on the glide set become dominant at higher temperatures.

SMALLNESS OF THE SET OF CRITICAL VALUES OF DISTANCE FUNCTIONS IN TWO‐DIMENSIONAL EUCLIDEAN AND RIEMANNIAN SPACES

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

Crystalline evolutions with rapidly oscillating forcing terms

We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is very simple, the limit evolution problem is quite rich, and we discuss some properties such as uniqueness, comparison principle and pinning/depinning phenomena.

Dispersion points and rational curves

We construct two connected plane sets which can be embedded into rational curves. The first is a biconnected set with a dispersion point. It answers a question of Joachim Grispolakis. The second is indecomposable. Both examples are completely metrizable.

A Hypersurface Containing the Support of a Radon Transform must be an Ellipsoid. I: The Symmetric Case

If the Radon transform of a compactly supported distribution f ne 0 in mathbb {R}^n is supported on the set of tangent planes to the boundary partial D of a bounded convex domain D, then partial D must be an ellipsoid. As a corollary of this result we get a new proof of a recent theorem of Koldobsky, Merkurjev, and Yaskin, which settled a special case of a conjecture of Arnold that was motivated by a famous lemma of Newton.

The Mirror: Mother of All Symmetries in Crystals

Mirror symmetry is found to be the only fundamental symmetry in crystalline solids because all other symmetries, such as rotation, inversion, rotoreflection, rotoinversion and translational periodicity can be easily derived from suitable combinations of mirrors. Similarly, the point group symmetries can also be derived from the same. The mirror combination scheme is found to work in accordance with the principle of Wigner-Seitz cells and Brillouin Zones (and not with the conventional unit cells as proposed by Bravais), where the zone boundaries of a Brillouin zone represent different sets of Bragg planes obtained from diffraction pattern of the given crystal, while the diffraction of given crystal takes place in terms of decreasing interplanar spacing in reciprocal space. Because the Wigner Seitz cells, the Brillouin zones and the diffraction patterns possess defined origin and exhibit spherical symmetry, they cannot have translational symmetry of any kind (microscopic or macroscopic). Results obtained on the basis of this concept help us to remove the existing ambiguities in crystallography and make the crystal structure determination simple. Further, prima facie the diffraction patterns are found to take care of the proposed 'systematic absences' arising due to the so called lattice centering, glide planes and screw axes without actually taking them into consideration. This newly and first discovered concept is expected to explain all other complicated or less understood issues related to crystallography.

Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

The sub-additive pressure function for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution s0 to P(s0) = 1, were introduced by Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affinities which define the IFS are matrices which strictly preserve a common cone, the sub-additive pressure is locally real analytic as a function of the matrix coefficients of the linear parts of the affinities. In this setting we also show that the sub-additive pressure is piecewise real analytic in s, implying that the affinity dimension is locally analytic in the matrix coefficients. Combining this with a recent result of Bárány et al we obtain results concerning the analyticity of the Hausdorff dimension for certain families of planar self-affine sets.

A New Technique to Calculate Earthquake Stress Transfer and to Probe the Physics of Aftershocks

ABSTRACTCoseismic stress changes have been the primary physical principle used to explain aftershocks and triggered earthquakes. However, this method does not adequately forecast earthquake rates and diverse rupture populations when subjected to formal testing. We show that earthquake forecasts can be impaired by assumptions made in physics-based models such as the existence of hypothetical optimal faults and regional scale invariability of the stress field. We compare calculations made under these assumptions along with different realizations of a new conceptual triggering model that features a complete assay of all possible ruptures. In this concept, there always exists a set of theoretical planes that has positive failure stress conditions under a combination of background and coseismic static stress change. In the Earth, all of these theoretical planes may not exist, and if they do, they may not be ready to fail. Thus, the actual aftershock plane may not correspond to the plane with the maximum stress change value. This is consistent with observations that mainshocks commonly activate faults with exotic orientations and rakes. Our testing ground is the M 7.2, 2010 El Mayor–Cucapah earthquake sequence that activated multiple diverse fault populations across the United States–Mexico border in California and Baja California. We carry out a retrospective test involving 748 M≥3.0 triggered earthquakes that occurred during a 3 yr period after the mainshock. We find that a probabilistic expression of possible aftershock planes constrained by premainshock rupture patterns is strongly favored (89% of aftershocks consistent with static stress triggering) versus an optimal fault implementation (35% consistent). Results show that coseismic stress change magnitudes do not necessarily control earthquake triggering, instead we find that the summed background stress and coseismic stress change promotes diverse ruptures. Our model can thus explain earthquake triggering in regions where optimal plane mapping shows coseismic stress reduction.

CO Chemisorption on Vicinal Rh(111) Surfaces Studied with a Curved Crystal

Curved crystal surfaces enable the systematic and accurate comparison of physical and chemical processes for a full set of vicinal crystal planes, which are probed in the very same environment. Her...

Cubic crystals in an x-ray polarization-splitting geometry.

Hexagonal and cubic crystals contain paired sets of internal planes that reflect the linearly polarized components of certain x rays into two separate, perpendicular directions. For the cubic crystals, two distinct crystal orientations provide the same polarization-splitting geometry. One of the orientations may have advantages for plasma spectroscopy by suppressing unwanted reflections. This paper demonstrates the two orientations with a germanium crystal and K characteristic lines from copper and zirconium.

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Abstract Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.

Reconstruction of three-dimensional flow fields from two-dimensional data

The objective of this work is to develop a procedure that allows for reconstructing three-dimensional flow fields from two-dimensional information contained in some representative planes, conveniently distributed throughout the domain. The reconstructing tool is based on two recent methods developed by two of the authors, namely the higher order dynamic mode decomposition and the spatio-temporal Koopman decomposition (STKD). The latter method decomposes a given spatio-temporal flow field as a series expansion in Fourier-like modes (including both wavenumbers/frequencies and spatial/temporal growth rates) in time and some distinguished longitudinal spatial coordinates, and spatial modes depending on the remaining transverse spatial coordinates. To obtain the (unknown) three-dimensional reconstruction, the STKD method is first applied to the (known) two-dimensional data in the considered planes. This application of STKD yields both the wavenumbers/frequencies and the spatial/temporal growth rates appearing in the three-dimensional reconstruction. Imposing that the three-dimensional reconstruction coincides with the two-dimensional data in the given set of planes, a system of linear equations results that permits computing the various ingredients appearing in the three-dimensional STKD expansion.The performance of the method is tested in the three-dimensional wake of a circular cylinder at Reynolds number (based on the cylinder diameter and the incoming free-stream velocity) equal to 280. The resulting flow is highly non-linear and quasi-periodic, with two fundamental temporal frequencies, associated with the well-known modes A and B. The method can be applied using experimental or numerical data, allowing to identify the full three-dimensional flow field and its main characteristics from limited information.

Dynamic intersection of multiple implicit Dantzig–Wolfe decompositions applied to the adjacent only quadratic minimum spanning tree problem

In this paper, we introduce a dynamic Dantzig–Wolfe (DW) reformulation framework for the Adjacent Only Quadratic Minimum Spanning Tree Problem (AQMSTP). The approach is dynamic in the sense that the structures over which the DW reformulation takes place are defined on the fly and not beforehand. The idea is to dynamically convexify multiple promising regions of the domain, without explicitly formulating DW master programs over extended variable spaces and applying column generation. Instead, we use the halfspace representation of polytopes as an alternative to mathematically represent the convexified region in the original space of variables. Thus, the numerical machinery we devise for computing AQMSTP lower bounds operates in a cutting plane setting, separating projection cuts associated to the projection of the variables used in the extended DW reformulations. Our numerical experience indicates that the bounds are quite strong and the computational times are mostly spent by linear programming reoptimization and not by the separation procedures. Thus, we introduce a Lagrangian Relax-and-cut algorithm for approximating these bounds. The method is embedded in a Branch-and-Bound algorithm for the AQMSTP. Although it does not strictly dominate the previous state-of-the-art exact method, it is able to solve more instances to proven optimality and is significantly faster for the hardest AQMSTP instances in the literature.

Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

We introduce a novel Conceptual Framework for finding approximations to both Minimum Weight Triangulation (MWT) and optimal Traveling Salesman Problem (TSP) of planar point sets. MWT is a classical problem of Computational Geometry with various applications, whereas TSP is perhaps the most researched problem in Combinatorial Optimization. We provide motivation for our research and introduce the fields of triangulation and polygonization of planar point sets as theoretical bases of our approach, namely, we present the Isoperimetric Inequality principle, measured via Compactness Index, as a key link between our two stated problems. Our experiments show that the proposed framework yields tight approximations for both problems.

Coherence properties of the projectile’s beam: The missing piece of the C6+ + He ionization puzzle

SynopsisWe analyze the influence of the coherence of the projectile’s beam in scattering phenomena. We focus our study in the ionization of He by C6+ projectiles at 100 MeV/amu. Our calculations, which had been previously performed for only the scattering and perpendicular collision planes and within the First Born approximation, were repeated for an ampler set of collision planes and with a more refined model. We obtain a very good agreement with the experimental results.

Using SDN for Controlling the Carbon Footprint of The Internet

Several works have been done in order to balance the energy consumption of the network with traffic which aims to have a positive impact on the CO2 emission. However, CO2 and energy consumption cannot be considered proportionate if the means of electricity production differs. In this paper, we have proposed two different metrics namely Carbon Emission Factor and Non-Renewable Energy usage percentage for achieving green network. We have designed an algorithm considering these metrics as objective functions. We have considered a software defined network approach and provided a set of data and control plane for each metric. Their performances are then analyzed and compared with respect to green policy enabled Shortest Path First algorithm. All the experiments are conducted on GÉANT network with realistic demand size. A comprehensive analysis of the quality of service parameters like the end to end delay and packet loss is also done for each metric of the algorithm.

Improved Pose Graph Optimization for Planar Motions Using Riemannian Geometry on the Manifold of Dual Quaternions

Abstract We present a novel Riemannian approach for planar pose graph optimization problems. By formulating the cost function based on the Riemannian metric on the manifold of dual quaternions representing planar motions, the nonlinear structure of the SE(2) group is inherently considered. To solve the on-manifold least squares problem, a Riemannian Gauss– Newton method using the exponential retraction is applied. The proposed Riemannian pose graph optimizer (RPG-Opt) is further evaluated based on public planar pose graph data sets. Compared with state-of-the-art frameworks, the proposed method gives equivalent accuracy and better convergence robustness under large uncertainties of odometry measurements.

Apple Carving Algorithm to Approximate Traveling Salesman Problem from Compact Triangulation of Planar Point Sets

We propose a modified version of the Convex Hull algorithm for approximating minimum-length Hamiltonian cycle (TSP) in planar point sets. Starting from a full compact triangulation of a point set, our heuristic “carves out” candidate triangles with the minimal Triangle Inequality Measure until all points lie on the outer perimeter of the remaining partial triangulation. The initial candidate list consists of triangles on the convex hull of a given planar point set; the list is updated as triangles are eliminated and new triangles are thereby exposed. We show that the time and space complexity of the “apple carving” algorithm are O(n2) and O(n), respectively. We test our algorithm using a well-known problem subset and demonstrate that our proposed algorithm outperforms nearly all other TSP tour construction heuristics.

Coloring Hypergraphs Defined by Stabbed Pseudo-Disks and ABAB-Free Hypergraphs

What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.