Set Of Planes Research Articles

A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure

For every beta in (0,infty ), beta ne 1, we prove that a positive measure subset A of the unit square contains a point (x_0,y_0) such that A nontrivially intersects curves y-y_0 = a (x-x_0)^beta for a whole interval Isubseteq (0,infty ) of parameters ain I. A classical Nikodym set counterexample prevents one to take beta =1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.

Chains, Koch Chains, and Point Sets with Many Triangulations

We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x -coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations. We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω (9.08 n ) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω (8.65 n ) for the maximum number of triangulations of planar point sets.

Ultrafast cadmium-zinc-telluride-based renal single-photon emission computed tomography: clinical validation

BackgroundOne of the main limitations of 99mtechnetium-dimercaptosuccinic acid (DMSA) scan is the long acquisition time.ObjectiveTo evaluate the feasibility of short DMSA scan acquisition times using a cadmium-zinc-telluride-based single-photon emission computed tomography (SPECT) system in children.Materials and methodsThe data of 27 children (median age: 4 years; 16 girls) who underwent DMSA SPECT were retrospectively analyzed. Both planar and SPECT DMSA were performed. SPECT images were analyzed using coronal-simulated planar two-dimensional images. A reduction in SPECT acquisition time was simulated to provide 4 series (SPECT-15 min, SPECT-10 min, SPECT-5 min and SPECT-2.5 min). A direct comparison of the planar and SPECT series was performed, including semi-quantification reproducibility, image quality (mean quality score on a scale of 0 to 2) and inter- and intra-observer reproducibility of the scintigraphic patterns.ResultsThe overall image quality score (± standard deviation) was 1.3 (± 0.6) for the planar data set, 1.6 (± 0.5) for the SPECT-15 min data set, 1.4 (± 0.5) for the SPECT-10 min data set, 1.0 (± 0.5) for the SPECT-5 min data set and 0.6 (± 0.6) for the SPECT-2.5 min data set. Median Kappa coefficients for inter-observer agreement between planar and SPECT images were greater than 0.83 for all series and all readers except one reader for the SPECT-2.5 min series (median Kappa coefficient = 0.77).ConclusionShortening SPECT acquisitions to 5 min is feasible with minimal impact on images in terms of quality and reproducibility.Graphical

Algorithms for constructing suboptimal coverings of plane figures with disks in the class of regular lattices

The problem of covering a compact planar set $M$ with a set of congruent disks is considered. It is assumed that the centers of the circles belong to some lattice. The criterion of optimality in one case is the minimum of the number of elements of the covering, and in the other case — the minimum of the Hausdorff deviation of the union of elements of the covering from the set $M$. To solve the problems, transformations of parallel transfer and rotation with the center at the origin can be applied to the lattice. Statements concerning sufficient conditions for sets of circles that provide solutions to the problems are proved. Numerical algorithms based on minimizing the Hausdorff deviation between two flat compacts are proposed. Solutions of a number of examples are given for various figures of $M$.

On the crack inverse problem for pressure waves in half-space

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks Γ1 and Γ2 under the assumption that R3∖Γ1∪Γ2‾ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if Γ1 and Γ2 are smooth and “almost” flat. Finally, we show in the case where R3∖Γ1∪Γ2‾ is not necessarily connected that after excluding a discrete set of frequencies, Γ1 and Γ2 can again be differentiated from overdetermined boundary data.

Plücker-Type Inequalities for Mixed Areas and Intersection Numbers of Curve Arrangements

Abstract Any collection of $n$ compact convex planar sets $K_1,\dots , K_n$ defines a vector of ${n\choose 2}$ mixed areas ${\operatorname {V}}(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for $n=4$, these inequalities completely describe the space of all mixed area vectors $({\operatorname {V}}(K_i,K_j)\,:\,1\leq i<j\leq 4)$. For arbitrary $n\geq 4$, we show that this space has a semialgebraic closure of full dimension. As an application, we show that the pairwise intersection numbers of any collection of $n$ tropical curves satisfy the Plücker-type inequalities. Moreover, in the case of four tropical curves, any homogeneous polynomial relation between their six intersection numbers follows from the corresponding Plücker-type inequalities.

Disjoint sets in projective planes of small order

In this paper, results of a computer search for disjoint sets associated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4, 8) and 156-sets of type (8, 12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.

On the limits of Particle Image Velocimetry with continuous wave lasers

The different contributions to the loss of correlation between image pairs in Particle Image Velocimetry (PIV) have been largely studied when a pulsed laser is used to illuminate the flow. In this case, the flow is well illuminated and the particles appears in the image without blurring. Nevertheless, if a continuous laser (CW) is used to illuminate the flow, light deficiency and particle blurring may appear and contribute to increase the loss of correlation introducing limits to the spatial resolution achievable of velocity field. Here, we develop mathematical expressions for the added loss of correlation which allows to estimate a relationship between the main influence variables of planar PIV set up with CW lasers and the spatial resolution achievable at the cross-correlation step. The mathematical model also allows to find the optimal time of exposure.

Isoperimetric Sets and p-Cheeger Sets are in Bijection

Given an open, bounded, planar set Omega , we consider its p-Cheeger sets and its isoperimetric sets. We study the set-valued map mathfrak {V}:[1/2,+infty )rightarrow mathcal {P}((0,|Omega |]) associating to each p the set of volumes of p-Cheeger sets. We show that whenever Omega satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of Gamma -convergence. Moreover, when restricted to (1/2, 1) such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regularity result.

A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L^2-eigenfunctions of Schrödinger-type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is , where is the usual Euclidean distance to the union of the set of collision planes {mathcal {F}}. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification {overline{X}} of the underlying space X and we first blow up the spheres {mathbb {S}}_Y subset {mathbb {S}}_X at infinity of the collision planes Y in {mathcal {F}} to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes {mathcal {F}}. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher-order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

Order of uniform approximation by polynomial interpolation in the complex plane and beyond

Let X denote a compact set in the complex plane ℂ and zn≔{zn,j}j=0n be a family of points that lie on X. Then the norm ‖Azn‖ of the operator Azn that maps the Banach space C(X) to the space Πn of polynomials of degree ≤n, defined by Aznf=pn, where pn∈Πn is the polynomial that interpolates f∈C(X) at zn, is called the Lebesgue constant of the family zn. While the bulk of this paper is on the study of Lagrange polynomial interpolation at the Fejér points zn∗≔{zn,j∗}j=0n, and their proper adjustment, that lie on an open arc X=γ in ℂ which does not cross itself, the general spirit of our presentation carries over to other settings of polynomial representations of functions on a simple closed piece-wise smooth curve X=Γ, which is the boundary of a Jordan domain D⊂ℂ; and instead of polynomial interpolation, Azn is replaced by the linear operator AFn, that maps the (generalized) Hardy space Hp(D), for p≥1, of analytic functions f in D with non-tangential limit f∗∈Lp(Γ), to the nth partial sum Sn(⋅;f) of the Faber series representation of f in D. The importance of the Lebesgue constant ‖AFn‖ is that if Pn∗∈Πn is the best polynomial approximant of f in Hp(D), with approximation error ϵn(f;Γ)≔‖f−Pn∗(⋅;f)‖Hp, then Sn(⋅;f) can be used to replace Pn∗ with error of approximation ‖f−Sn(⋅;f)‖Hp≤(1+‖AFn‖)ϵn(f;Γ). This consideration, along with the discussion of related problems and results inspired by the work of Prof. J. Korevaar, will be presented in the final section of the present paper.For Lagrange polynomial interpolation on open arcs X=γ in ℂ, it is well-known that the Lebesgue constant for the family of Chebyshev points xn≔{xn,j}j=0n on [−1,1]⊂R has growth order of O(log(n)). The same growth order was shown in Zhong and Zhu (1995) for the Lebesgue constant of the family zn∗∗≔{zn,j∗∗}j=0n of some properly adjusted Fejér points on a rectifiable smooth open arc γ⊂ℂ. On the other hand, in our recent work Chui and Zhong (2021), it was observed that if the smooth open arc γ is replaced by an L-shape arc γ0⊂ℂ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz–Zygmund inequalities are no longer valid for the family of Fejér points zn∗≔{zn,j∗}j=0n on γ, and that the rate of growth for the corresponding Lebesgue constant Lzn∗ is as fast as clog2(n) for some constant c>0.The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the L-shape arc γ0 consisting of two line segments of the same length that meet at the angle of π/2, the growth rate of the Lebesgue constant Lzn∗ is at least as fast as O(log2(n)), with limsupLzn∗log2(n)=∞; secondly, the corresponding (modified) Marcinkiewicz–Zygmund inequalities fail to hold; and thirdly, a proper adjustment zn∗∗≔{zn,j∗∗}j=0n of the Fejér points on γ will be described to assure the growth rate of Lzn∗∗ to be exactly O(log2(n)).

Analysis on fracture mechanics theory of roof cutting instability mechanism with large mining height face in shallow coal seam

The roof in a fully-mechanized face of a shallow coal seam with large mining height is prone to form a combined cantilever—articulated rock beam structure. When the support resistance is insufficient, the articulated rock beam will sink. This will make the combined cantilever beam rotate and fracture. It is then easy to induce the sliding and instability of the articulated rock beam, which results in large-scale roof cutting and the support crushing. Taking the combined cantilever beam structure as the main research object, and considering the mining damaged characteristics of cantilever beam rock stratum, the rock beam was regarded as a finite plate model with an edge crack of arbitrary dip angle. In addition, a fracture mechanics model controlled by a set of structural planes was established, the instability conditions of rock beam and the main control factors were analyzed, and the method of determining the support resistance were discussed. The results show that the cantilever beam rotates and fractures. This causes a chain reaction of the rock beam that leads to fracture. The combined cantilever beam then loses stability with the increase of the length of the crack length and the crack dip angle, and therefore it is easier to penetrate the cantilever beam and cause roof instability. The necessary condition for rock beam instability was crack activation, and the sufficient condition was the airfoil branch crack propagate through the rock beam. The influence degree of each parameter on the support resistance was thus determined: crack length a > crack dip angle β > rock thickness h > weighting interval l. The theoretical analysis results were proven to be reasonable by an in situ monitoring example of no. 22,310 working face in the Daliuta coal mine, China. On this basis, the reasonable value of support resistance was obtained. The conclusions of this research provide a new method for researching the roof instability mechanism. They are also conducive to the green and sustainable development of mines.

Estimation of soil overburden thickness/depth of rock strata using geo-physical survey at Himalayan region


 
 
 Estimation of soil overburden and depth of competent strata for construction of infrastructures in hilly terrain is a challenging job for engineers. Modern technologies like Electrical Resistivity Survey (ERS) techniques with site geological mapping have been carried out at the ridge area near Palace, Gangtok. Two independent 2D-electrical resistivity surveys profile sections using hybrid Schlumberger-Wenner method were conducted along the ridge line trending in the North-South direction. The ERS profile sections were taken on both the slopes of the ridge along its eastern and western flanks. With the collective information from field and geo-technical data, a comparatively stable slope has been identified with respect to geological conditions in the present work, which focuses on the vulnerable slope failure with respect to slope direction, local geological condition, depth of competent strata, thickness of soil overburden, water saturation zones and resistivity of the materials. Diamond core drilling of 15meters each was carried out at both the flanks of the slope to understand the sub-surface strata and correlate it with resistivity data generated by ERS survey. The result indicates that Sandy/silty soil with flakes of mica having resistivity of 107 ohm-m and weathered mica schist having resistivity more than 300 ohm-m in the present study area. Geological mapping in 1:3000 scale was carried in the area demarcation with various litho-units and rock type. The area is characterized by medium grade metamorphic rock sequence represented in the area by mica schist having dip of foliations towards NE direction and three sets of joint planes. The geometry of the rock orientation and slope direction plays a vital role for determining the overall stability condition of the area. The present study will provide technical input for structural engineers to design the structures in such geological conditions. Further, the thickness of overburden estimated from ERS has been validated by drilling data.
 
 

Efficient Algorithms and Edge Crossing Properties of Euclidean Minimum Weight Laman Graphs

We investigate the Euclidean minimum weight Laman graph on a planar point set P, for short. Bereg et al. (2016) studied geometric properties of and showed that the upper and lower bounds for the total number of edge crossings in are and , respectively. In this paper, we improve these upper and lower bounds to and for any , respectively. For improving the upper bound, we introduce a novel counting scheme based on some geometric observations. We also propose an time algorithm for computing , which was regarded as one of interesting future works by Bereg et al. (2016).

Optimal Area Polygonization by Triangulation and Visibility Search

The aim of the “CG:SHOP Challenge 2019” was to generate optimal area polygonizations of a planar point set. We describe here the algorithm that won the challenge. It is a two-phase algorithm based on the node-insertion move technique, which comes from the TSP. In the first phase, we use constrained triangulations to check efficiently the simplicity of the generated polygonizations. In the second phase, we perform visibility searches to be able to generate a wider variety of polygonizations. In both phases, the simulated annealing metaheuristic is implemented to approach the optimum.

Context-Consistent Generation of Indoor Virtual Environments Based on Geometry Constraints.

In this article, we propose a system that can automatically generate immersive and interactive virtual reality (VR) scenes by taking real-world geometric constraints into account. Our system can not only help users avoid real-world obstacles in virtual reality experiences, but also provide context-consistent contents to preserve their sense of presence. To do so, our system first identifies the positions and bounding boxes of scene objects as well as a set of interactive planes from 3D scans. Then context-consistent virtual objects that have similar geometric properties to the real ones can be automatically selected and placed into the virtual scene, based on learned object association relations and layout patterns from large amounts of indoor scene configurations. We regard virtual object replacement as a combinatorial optimization problem, considering both geometric and contextual consistency constraints. Quantitative and qualitative results show that our system can generate plausible interactive virtual scenes that highly resemble real environments, and have the ability to keep the sense of presence for users in their VR experiences.

Compression of a Beam of Fast Argon Atoms for Surface Polishing

The results of studying a beam source of fast argon atoms obtained by neutralizing charges of accelerated ions during their flight through an accelerating grid in the form of a set of plane–parallel plates are presented. Ions are accelerated from the plasma emitter by a voltage of negative polarity at the grid, fly into the gaps between its plates, and, when approaching their surfaces, cause the emission of electrons that neutralize the ion charge. The equivalent beam current of fast atoms emitted from the grid is up to 90% of the current in its circuit, and their energy is equal to the ion energy that corresponds to the accelerating voltage between the plasma emitter and the grid. To increase the beam current density of fast atoms on the treated surface, the sections of the grid plates facing the plasma emitter are made in the form of a segment of a circle. The movement of ions and fast atoms that are formed inside the grid to the center of this circle makes it possible to reduce the beam width by an order of magnitude and increase the flow density of fast atoms without reducing the angle of their incidence on the treated surface. The beam compression significantly improves the efficiency of polishing a surface with a beam of fast neutral argon atoms with an angle of incidence on the surface of 80°.

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C2-smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold.

What Can You Draw?

We address the problem of which planar sets can be drawn with a pencil and eraser. The pencil draws any union of black open unit disks in the plane . The eraser produces any union of white open unit disks. You may switch tools as many times as desired. Our main result is that drawability cannot be characterized by local obstructions: a bounded set can be locally drawable, while not being (globally) drawable. We also show that if drawable sets are defined using closed unit disks, then the cardinality of the collection of drawable sets is strictly larger compared with the definition involving open unit disks.

Equilateral triangulations and the postcritical dynamics of meromorphic functions

We show that any dynamics on any planar set S discrete in some domain D can be realized by the postcritical dynamics of a function holomorphic in D, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain D can be equilaterally triangulated with triangles whose diameters \(\rightarrow 0\) (at any prescribed rate) near \(\partial D\).