Set Of Planes Research Articles

Closure and connected component of a planar global semianalytic set defined by analytic functions definable in o-minimal structure

We consider a global semianalytic set defined by real analytic functions definable in an o-minimal structure. When the o-minimal structure is polynomially bounded, we show that the closure of this set is a global semianalytic set defined by definable real analytic functions. We also demonstrate that a connected component of a planar global semianalytic set defined by real analytic functions definable in a substructure of the restricted analytic field is a global semianalytic set defined by definable real analytic functions.

Column planarity and partially-simultaneous geometric embedding

We introduce the notion of column planarity of a subset $R$ of the vertices of a graph $G$. Informally, we say that $R$ is column planar in $G$ if we can assign $x$-coordinates to the vertices in $R$ such that any assignment of $y$-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of $G$. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on $n$ vertices contains a column planar set of size at least $14n/17$ and for any $\epsilon > 0$ and any sufficiently large $n$, there exists an $n$-vertex tree in which every column planar subset has size at most $(5/6 + \epsilon)n$. In addition, we show that every outerplanar graph has a column planar set of size at least $n/2$. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs $G_1$ and $G_2$ allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct $k$-PSGEs, which are PSGEs in which at least $k$ vertices are mapped to the same point for both graphs. In particular, we show that every two trees on $n$ vertices admit an $11n/17$-PSGE and every two outerplanar graphs admit an $n/4$-PSGE.

On Edge’s correspondence associated with $$\cdot 222$$ · 222

We describe explicitly the correspondence of Edge between the set of planes contained in the Fermat cubic fourfold in characteristic 2, and the set of lattice points T of the Leech lattice such that OABT is a regular tetrahedron, where O is the origin of , and A and B are fixed points of such that OAB is a regular triangle of edge length 2. Using this description, we present Conway’s isomorphism from to in terms of matrices.

An axiomatic approach to decoupling indicators for green growth

The concept of decoupling was introduced to measure and analyze the controversial trade-off between economic development and environmental sustainability; in particular, several empirical studies concern the construction and use of decoupling indicators. We elaborate on a descriptive comparison by Conte Grand (2016) of the three main ones, respectively DO by OECD (2002), Dϵ by Tapio (2005), and Dt by Lu et al. (2011), and introduce an axiomatic approach into the subject that articulates in the identification of some properties that appear indispensable or at least desirable for any decoupling indicator and in the assessment of their validity for the indices under scrutiny and/or in the construction of new indices that satisfy them. A graphical examination of the aggregation function level sets in the Cartesian plane is a relevant part of the method. Under such analysis, the index DO turns out to show milder defects than Dϵ and Dt. We then propose a suitable modification DN in order to remove the defects and fulfill all the given compatible axioms; in particular, DN is cumulative over sub-periods.It may also be opportune for a decoupling index to differentiate the treatment of the case when, during economic growth, environmental stress (e.g., polluting emissions) decreases from the case when it increases, although less than economy (the so-called absolute and relative decoupling, respectively), as well as to capture the rebound effect phenomenon, whereby the efforts to reduce environmental intensity may eventually result in a smaller overall environmental improvement than predicted or intended. To this end we build another index DP by applying to DN a correction (that can be calibrated via a global parameter) for the distance from what we define symmetric decoupling, the case when the variations of economy and of environmental pressure are inversely proportional.We conclude by testing the novel indices DN and DP against DO on data from OECD (2017) of 103 world countries for the most recent completely available decade 2003–2013.

Bicovariograms and Euler characteristic of regular sets

AbstractWe establish an expression of the Euler characteristic of ar‐regular planar set in function of some variographic quantities. The usualframework is relaxed to aregularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set ofin terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non‐connected polyrectangular grains in. Applications to excursions of smooth bivariate random fields are derived in the companion paper, and applied for instance toGaussian fields, generalising standard results.

AUTOMATED COARSE REGISTRATION OF POINT CLOUDS IN 3D URBAN SCENES USING VOXEL BASED PLANE CONSTRAINT

Abstract. For obtaining a full coverage of 3D scans in a large-scale urban area, the registration between point clouds acquired via terrestrial laser scanning (TLS) is normally mandatory. However, due to the complex urban environment, the automatic registration of different scans is still a challenging problem. In this work, we propose an automatic marker free method for fast and coarse registration between point clouds using the geometric constrains of planar patches under a voxel structure. Our proposed method consists of four major steps: the voxelization of the point cloud, the approximation of planar patches, the matching of corresponding patches, and the estimation of transformation parameters. In the voxelization step, the point cloud of each scan is organized with a 3D voxel structure, by which the entire point cloud is partitioned into small individual patches. In the following step, we represent points of each voxel with the approximated plane function, and select those patches resembling planar surfaces. Afterwards, for matching the corresponding patches, a RANSAC-based strategy is applied. Among all the planar patches of a scan, we randomly select a planar patches set of three planar surfaces, in order to build a coordinate frame via their normal vectors and their intersection points. The transformation parameters between scans are calculated from these two coordinate frames. The planar patches set with its transformation parameters owning the largest number of coplanar patches are identified as the optimal candidate set for estimating the correct transformation parameters. The experimental results using TLS datasets of different scenes reveal that our proposed method can be both effective and efficient for the coarse registration task. Especially, for the fast orientation between scans, our proposed method can achieve a registration error of less than around 2 degrees using the testing datasets, and much more efficient than the classical baseline methods.

MORE ABOUT CANTOR LIKE SETS IN HYPERBOLIC NUMBERS

In this paper, we discuss the construction of new Cantor like sets in the hyperbolic plane. Also, we study the arithmetic sum of two of these Cantor like sets, as well as of those previously introduced in the literature. An hyperbolization, in the sense of Gromov, of the commutative ring of hyperbolic numbers is also given. Finally, we present the construction of a Cantor-type set as hyperbolic boundary.

Improved scatter correction with factor analysis for planar and SPECT imaging.

Quantitative nuclear medicine imaging is an increasingly important frontier. In order to achieve quantitative imaging, various interactions of photons with matter have to be modeled and compensated. Although correction for photon attenuation has been addressed by including x-ray CT scans (accurate), correction for Compton scatter remains an open issue. The inclusion of scattered photons within the energy window used for planar or SPECT data acquisition decreases the contrast of the image. While a number of methods for scatter correction have been proposed in the past, in this work, we propose and assess a novel, user-independent framework applying factor analysis (FA). Extensive Monte Carlo simulations for planar and tomographic imaging were performed using the SIMIND software. Furthermore, planar acquisition of two Petri dishes filled with 99mTc solutions and a Jaszczak phantom study (Data Spectrum Corporation, Durham, NC, USA) using a dual head gamma camera were performed. In order to use FA for scatter correction, we subdivided the applied energy window into a number of sub-windows, serving as input data. FA results in two factor images (photo-peak, scatter) and two corresponding factor curves (energy spectra). Planar and tomographic Jaszczak phantom gamma camera measurements were recorded. The tomographic data (simulations and measurements) were processed for each angular position resulting in a photo-peak and a scatter data set. The reconstructed transaxial slices of the Jaszczak phantom were quantified using an ImageJ plugin. The data obtained by FA showed good agreement with the energy spectra, photo-peak, and scatter images obtained in all Monte Carlo simulated data sets. For comparison, the standard dual-energy window (DEW) approach was additionally applied for scatter correction. FA in comparison with the DEW method results in significant improvements in image accuracy for both planar and tomographic data sets. FA can be used as a user-independent approach for scatter correction in nuclear medicine.

X-ray pole figure analysis of catalyst free InAs nanowires on Si substrate

We present crystallographic analysis of InAs nanowires by X-ray pole figure measurements and reciprocal space mapping in this report. The InAs nanowires have been grown on Si (111) substrates by a catalyst free approach using molecular beam epitaxy technique. The pole figure measurement of InAs nanowires for (111) and (220) reflections reveals important details such as the nanowire crystal structure, orientation, presence of twin defects and the epitaxial relation of the nanowires with the substrate. Angular relationship between several set of twin planes in each pole figure reflection is analyzed and the twin direction in each case is determined using a quantitative approach. Possibility of presence of mixed zinc blende /wurtzite phases in the nanowires is explored from an analysis of pole figure measurement for the (311) and (10−15) reflections and an asymmetric reciprocal space map (RSM) measurement.

Room temperature green synthesis of anisotropic gold nanoparticles using novel biological fruit extract

ABSTRACTHerein, we reported the room temperature green synthesis of anisotropic gold nanoparticles from the aqueous Cordia myxa fruit extract, which can concurrently serve as the source for both the reducing and stabilizing agents. In the X-ray diffraction pattern, the identified Bragg reflections corresponded to the (111), (200), and (220) sets of lattice planes, which can be indexed based on the face center cubic structures of gold. They are in different morphologies such as multiple twinned pentagonal shaped octahedron and hexagonal shaped icosahedron with their size ranging from 20 to 50 nm as evident from the transmission electron microscopy images.

Geometric design and continuity conditions of developable λ-Bézier surfaces

In this paper, two explicit methods are presented for the computer-aided design of developable λ-Bézier surfaces associated with shape parameter. Based on the duality between points and planes in 3D projective space, a developable λ-Bézier surface associated with a shape parameter is designed by using a set of control planes with λ-Bézier basis functions. The shape of developable λ-Bézier surface can be easily adjusted by modifying the value of the shape parameter. When the shape parameter takes on different values, a family of developable λ-Bézier surfaces can be constructed, which keeps most of beneficial properties of traditional Bézier surfaces. In order to tackle the problem that an engineering complex developable surface is usually hard to be constructed by using a single developable surface, we also derive the necessary and sufficient conditions for G1 continuity, Farin-Boehm G2 continuity and G2 Beta continuity between two adjacent developable λ-Bézier surfaces. Finally, the properties and applications of developable λ-Bézier surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which greatly improve the problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.

A β-accurate linearization method of Euclidean distance for the facility layout problem with heterogeneous distance metrics

Most existing research on facility layout problems (FLPs) considers a single distance metric, mainly Rectilinear distance, in the calculation of the material handling cost between departments. However, there are many industrial cases in which heterogeneous distance metrics may need to be used simultaneously to cater for different styles of material handling, such as the Euclidean distance metric for conveyor belts and the Tchebychev distance metric for overhead cranes. In this paper, we study the unequal area facility layout problem with heterogeneous distance metrics (UA-FLP-HDM), considering a hybrid use of three metrics, i.e., Rectilinear, Euclidean, and Tchebychev, as distance measures of different styles of material handling in the production system. We propose a β-accurate linearization method that uses a set of tangent planes to convert the non-linear Euclidean distance constraint into a set of linear constraints that guarantee the approximation error within a given percentage β, e.g., as small as −0.01% in our experiments, and develop linear constraints for the Tchebychev distance metric as well. Based on these contributions, we present a mixed-integer linear programming (MILP) model for the UA-FLP-HDM. Computational experiments are carried out to test the performance of the MILP model with five benchmark problems in the literature and compare the layout designs using different distance metrics. Numerical results indicate that different distance metrics may lead to significantly different solutions and a hybrid use of heterogeneous distance metrics fits better for real industrial applications.

Order on Order Types

Given P and $$P'$$Pź, equally sized planar point sets in general position, we call a bijection from P to $$P'$$Pźcrossing-preserving if crossings of connecting segments in P are preserved in $$P'$$...

On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems

This paper introduces a new coordinate formulation for the kinematic and dynamic analysis of planar multibody systems composed of rigid bodies. The methodology presented in this work is called planar reference point coordinate formulation (RPCF) with Euler parameters. In the planar RPCF with Euler parameters, the rotational coordinates used for describing the body orientation are the redundant components of a two-dimensional unit quaternion that identify a planar set of Euler parameters. It is shown in the paper that the planar RPCF with Euler parameters allows for obtaining consistent kinematic and dynamic descriptions of two-dimensional rigid bodies. In the numerical solution of the equations of motion, the well-known generalized coordinate partitioning method can be effectively utilized to stabilize the violation of the algebraic constraints at the position and velocity levels leading to physically correct and numerically stable dynamic simulations. Furthermore, a standard numerical integration procedure can be employed for calculating an approximate solution of the equations of motion resulting from the planar RPCF with Euler parameters. In the paper, the computer implementation of the proposed formulation approach is demonstrated considering four rigid multibody systems which serve as simple benchmark problems.

Dimension maximizing measures for self-affine systems

In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and the strong separation condition. The main result of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space.

A distance-limited continuous location-allocation problem for spatial planning of decentralized systems

We introduce a new continuous location-allocation problem where the facilities have both a fixed opening cost and a coverage distance limitation. The problem has wide applications especially in the spatial planning of water and/or energy access networks where the coverage distance might be associated with the physical loss constraints. We formulate a mixed integer quadratically constrained problem (MIQCP) under the Euclidean distance setting and present a three-stage heuristic algorithm for its solution: In the first stage, we solve a planar set covering problem (PSCP) under the distance limitation. In the second stage, we solve a discrete version of the proposed problem where the set of candidate locations for the facilities is formed by the union of the set of demand points and the set of locations in the PSCP solution. Finally, in the third stage, we apply a modified Weiszfeld’s algorithm with projections that we propose to incorporate the coverage distance component of our problem for fine-tuning the discrete space solutions in the continuous space. We perform numerical experiments on three example data sets from the literature to demonstrate the performance of the suggested heuristic method.

The Assouad dimension of self-affine carpets with no grid structure

Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a ‘grid structure’, thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urbański sets to the lower local dimensions of Bernoulli convolutions.

Whole-body Magnetic Resonance Imaging in Inflammatory Arthritis: Systematic Literature Review and First Steps Toward Standardization and an OMERACT Scoring System.

Whole-body magnetic resonance imaging (WB-MRI) is a relatively new technique that can enable assessment of the overall inflammatory status of people with arthritis, but standards for image acquisition, definitions of key pathologies, and a quantification system are required. Our aim was to perform a systematic literature review (SLR) and to develop consensus definitions of key pathologies, anatomical locations for assessment, a set of MRI sequences and imaging planes for the different body regions, and a preliminary scoring system for WB-MRI in inflammatory arthritis. An SLR was initially performed, searching for WB-MRI studies in arthritis, osteoarthritis, spondyloarthritis, or enthesitis. These results were presented to a meeting of the MRI in Arthritis Working Group together with an MR image review. Following this, preliminary standards for WB-MRI in inflammatory arthritides were developed with further iteration at the Working Group meetings at the Outcome Measures in Rheumatology (OMERACT) 2016. The SLR identified 10 relevant original articles (7 cross-sectional and 3 longitudinal, mostly focusing on synovitis and/or enthesitis in spondyloarthritis, 4 with reproducibility data). The Working Group decided on inflammation in peripheral joints and entheses as primary focus areas, and then developed consensus MRI definitions for these pathologies, selected anatomical locations for assessment, agreed on a core set of MRI sequences and imaging planes for the different regions, and proposed a preliminary scoring system. It was decided to test and further develop the system by iterative multireader exercises. These first steps in developing an OMERACT WB-MRI scoring system for use in inflammatory arthritides offer a framework for further testing and refinement.

On the [formula omitted] of a planar point set

We study the Oβ-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle β. Given a set P of n points in the plane, we show how to maintain the Oβ-hull of P while β runs from 0 to π in Θ(nlog⁡n) time and O(n) space. With the same complexity, we also find the values of β that maximize the area and the perimeter of the Oβ-hull and, furthermore, we find the value of β achieving the best fitting of the point set P with a two-joint chain of alternate interior angle β.

The uniqueness of meromorphic functions in k-punctured complex plane

Abstract The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k-punctured complex plane. It is proved that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if EΩ(Sj, f) = EΩ(Sj, g)(j = 1,2).