Set Of Planes Research Articles

Influence of Bifurcation Structures Revealed by Refinement of a Nonlinear Conductance in JosephsonJunction Element

We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.

Steklov eigenvalues of reflection-symmetric nearly circular planar domains.

We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

A New Lower Bound for the Size of an Affine Blocking Set

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geqslant 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

A core decomposition of compact sets in the plane

A Peano continuum means a locally connected continuum. A compact metric space is called a Peano compactum if all its components are Peano continua and if for any constant C>0 all but finitely many of its components are of diameter less than C. Given a compact set K⊂C, there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the core decomposition of K with Peano quotient. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into C. We also construct a concrete continuum K⊂R3 such that the core decomposition of K with Peano quotient does not exist. For specific choices of K⊂C, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map f:Cˆ→Cˆ restricted to its Julia set.

Identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors

This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.

Holomorphic surjectivity near the boundary: topological and algebraic genericity

ABSTRACT Let Ω be an open set in the complex plane. Given a subdomain V that is not relatively compact in Ω, we prove the existence of infinitely generated free subalgebras as well as of dense vector subspaces and of infinite dimensional Banach spaces consisting of holomorphic functions on Ω all of whose non-zero members are surjective on V. Some of these results are extended to several collections of subdomains and to the action of differential operators on this kind of functions.

Updating Dynamic Random Hyperbolic Graphs in Sublinear Time

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points with a probability depending on their hyperbolic distance. We present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities. To process the dynamic changes efficiently, we formalize the concept of a probabilistic neighborhood : Let P be a set of n points in Euclidean or hyperbolic space, q a query point, dist a distance metric, and f : R + → [0, 1] a monotonically decreasing function. Then, the probabilistic neighborhood N ( q , f ) of q with respect to f is a random subset of P and each point p ∈ P belongs to N ( q , f ) with probability f (dist( p , q )). We present a fast, sublinear-time query algorithm to sample probabilistic neighborhoods from planar point sets. For certain distributions of planar P , we prove that our algorithm answers a query in O ((| N ( q , f )| + √ n ) log n ) time with high probability. This enables us to process a node movement in random hyperbolic graphs in sublinear time, resulting in a speedup of about one order of magnitude in practice compared to the fastest previous approach. Apart from that, our query algorithm is also applicable to Euclidean geometry, making it of independent interest for other sampling or probabilistic spreading scenarios.

Crystallographic orientation dependence of nanoindentation hardness in austenitic phase of stainless steel

ABSTRACTThis paper presents the indentation hardness evolution in different in-plane directions of austenite grains whose {111} planes are parallel to the sample surface determined by nanoindentation tests and electron back scattering diffraction (EBSD) analysis. A scanning electron microscopy (SEM) image of the indentation surface around one of the indents indicated the activation of two sets of slip planes with respect to each of the three indenter surfaces for a Berkovich tip. The identification of the slip traces by EBSD data analysis is in accord with Schmid`s law. We therefore proposed a new approach for defining the orientation parameter to interpret the indentation hardness. The orientation parameter was shown to be the minimum value of the three maximum Schmid factors on the secondarily activated slip planes in three directions for a Berkovich tip. Indentation hardness increased with the decrease in the orientation parameter and was dependent on in-plane orientation.

A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm (with O(n^8) being a crude bound on the run-time) to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.

Extensions of planar GC sets and syzygy matrices

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.

Interior of sums of planar sets and curves

AbstractRecently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

Specified holes with pairwise disjoint interiors in planar point sets

A k-hole of a planar point set in general position is a convex k-gon whose vertices are elements of the set and whose interior contains no elements of the set. We discuss the minimum size of a point set that contains specified holes with disjoint interiors.

Topology optimization of modulated and oriented periodic microstructures by the homogenization method

This paper is concerned with the topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstructure at a desired lengthscale, which is a delicate issue, albeit computationally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. The main application of our work is the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Several numerical examples are presented for compliance minimization in 2-d.

Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with Lp data and applications to nonlinear elasticity

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.

Plane sets invisible in finitely many directions

We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with π, for any there exists a set invisible in the chosen directions that contains the circle and is contained in its -neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary.

Angular Radon spectrum for rotation estimation

This paper presents a robust method for rotation estimation of planar point sets using the Angular Radon Spectrum (ARS). Given a Gaussian Mixture Model (GMM) representing the point distribution, the ARS is a continuous function derived from the Radon Transform of such distribution. The ARS characterizes the orientation of a point distribution by measuring its alignment w.r.t. a pencil of parallel lines. By exploting its translation and angular-shift invariance, the rotation angle between two point sets can be estimated through the correlation of the corresponding spectra. Beside its definition, the novel contributions of this paper include the efficient computation of the ARS and of the correlation function through their Fourier expansion, and a new algorithm for assessing the rotation between two point sets. Moreover, experiments with standard benchmark datasets assess the performance of the proposed algorithm and other state-of-the-art methods in presence of noisy and incomplete data.

Accuracy of two midsagittal planes in three-dimensional analysis and their measurement in patients with skeletal mandibular deviation: a comparative study

Our aim was to evaluate the accuracy of two midsagittal planes (MSP) to provide a better reference plane for studying the 3-dimensional structural symmetry in patients with skeletal mandibular deviation. Thirty adult patients with facial asymmetry were admitted to the Department of Orthodontics, Hospital of Stomatology, between January 2015 and 2017. The DICOM data were collected and reconstructed using ProPlan CMF® 2.0 (Materialise). Two sets of reference planes were set up. In the orbital margin plane group, the plane crossing the nasion (N) point and perpendicular to the frontozygomatic (FZ) suture line was used as the MSP. In the skull base plane group, the MSP was established using the sella turcica (S), N, and basion (Ba). The distances from the craniofacial skeletal and soft tissue midline landmarks to the two MSP were separately measured, and the significance of differences between measurements corresponding to two reference planes were assessed using the paired t test. Except for the posterior nasal spine (PNS)-MSP, the distances from other soft and hard tissue landmarks to the MSP in the orbital margin plane group were significantly shorter than those in the skull base plane group. In the 3-dimensional measurement analysis, the skeletal and soft tissue anatomical midline landmarks were closer to the MSP in the orbital margin plane group. The MSP crossing point N, perpendicular to the FZ suture line, was more accurate and suitable.

Validation of fracture data recognition in rock masses by automated plane detection in 3D point clouds

This paper presents (1) an automated method to extract planes and their spatial orientation directly from 3D point clouds, followed by (2) extensive validation tests accompanied by thorough statistical analysis, and (3) a fracture intensity calculation on automatically segmented planes. For the plane extraction, a region growing segmentation algorithm controlled by several input parameters is applied to a point cloud of a granite outcrop. Within its complex surface shape, more than 1000 compass measurements were conducted for validation. In addition, digitally handpicked planes in the software Virtual Reality Geological Studio (VRGS) were used for single plane comparison. In a second test site, we performed fracture intensity calculation in Petrel based on results of the segmentation algorithm on mechanical layers of a clastic sedimentary succession. The comparison of automated segmentation results and compass measurements of three different plane sets shows a deviation of 0.70–2.00°, while the mean single plane divergence amounts to 4.97°. Hence, this study presents a fast, precise, and highly adaptable automated plane detection method, which is reproducible, transparent, objective, and provides increased accuracy in outcrops with rough and complex surfaces. Moreover, output formats of spatial orientation and location of planes are designed for simple handling in other workflows and software.

Changes in nasotracheal tube depth in response to head and neck movement in children.

A tracheal tube is often inserted via the nasal cavity for dental surgery. The position of the tube tip is important, given that the head position sometimes changes during surgery. Head movement induces changes in the length of the trachea (t-length) and/or the distance between the nare and the vocal cords (n-v-distance). In this study, we investigated the changes in t-length and n-v-distance in children undergoing nasotracheal intubation. Eighty patients aged 2-8year undergoing dental surgery were enrolled. After nasotracheal intubation with an uncuffed nasotracheal tube (4.5-6.0mm), the tube was fixed at the patient's nares. The distance between the tube tip and the first carina was measured using a fibrescope with the angle between the Frankfort plane and horizontal plane set at 110°. The location of the tube in relation to the vocal cords was then checked. These measurements were repeated at angles of 80° (flexion) and 130° (extension). The t-length and n-v-distance were then calculated using these measurements. On flexion, the t-length shortened significantly from 87.5±10.4mm to 82.9±10.7mm (P=0.017) and the n-v-distance decreased from 128.1±10.7mm to 125.6±10.4mm (P=0.294). On extension, the t-length increased significantly from 87.5±10.4mm to 92.7±10.1mm (P=0.007) and the n-v-distance increased from 128.1±10.7mm to 129.4±10.7mm (P=0.729). The change in t-length was significantly greater than that in the n-v-distance. A change in the position of the tracheal tube tip in the trachea depends mainly on changes in t-length during paediatric dental surgery.

An improvement of the lower bound on the maximum number of halving lines in planar sets with 32 points

In this paper we give a recursive lower bound on the maximum number of halving lines for sets in the plane and as a consequence we improve the current best lower bound on the maximum number of halving lines for sets in the plane with 32 points.