Projective Plane Research Articles

A geographical study ofM‾2P2,4main $\overline{\mathcal{M}}_2\left(\mathbb{P}^2, 4\right)^{\text {main }}$

Abstract We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation of via logarithmic geometry and Gorenstein singularities.

A density bound for triangle‐free 4‐critical graphs

We prove that every triangle-free 4-critical graph G $G$ satisfies e ( G ) ≥ 5 v ( G ) + 2 3 $e(G)\ge \frac{5v(G)+2}{3}$ . This result gives a unified proof that triangle-free planar graphs are 3-colourable, as well as that graphs of girth at least five which embed in either the projective plane, torus or Klein Bottle are 3-colourable, which are results of Grötzsch, Thomassen, and Thomas and Walls. Our result is nearly the best possible, as Davies has constructed triangle-free 4-critical graphs G $G$ such that e ( G ) = 5 v ( G ) + 4 3 $e(G)=\frac{5v(G)+4}{3}$ . To prove this result, we prove a more general result characterizing sparse 4-critical graphs with few vertex-disjoint triangles.

Reliability and criterion validity of two-dimensional movement assessments in those with patellofemoral pain

ObjectivesDetermine criterion validity and intra/inter-rater reliability of 2-dimensional (2D) knee frontal plane projection angle (kFPPA), hip frontal plane projection angle (hFPPA), and dynamic valgus index (DVI) during forward step-downs in those with patellofemoral pain (PFP). DesignCross-sectional. SettingUniversity research laboratory. Participants39 participants with PFP (34.18 ± 7.41years, 170± .1 cm, 81.03 ± 19.36 kg, duration of pain: 68.67 ± 85.08months, anterior knee pain scale: 80.49 ± 7.87, visual analog scale:2.08 ± 2.02) Main outcome measuresAverage 3D hip and knee sagittal, frontal, and transverse joint angles and 2D kFPPA, hFPPA, and DVI at maximum knee flexion were variables of interest. 3D DVI was calculated as the sum of hip and knee frontal and transverse angles. 2D kFPPA, hFPPA, and DVI were calculated by two raters independently on two occasions. ResultsIntra- and inter-rater reliability of all 2D angles were excellent. kFPPA was moderately correlated to 3D knee transverse angles. hFPPA was moderately correlated to 3D hip frontal and transverse angles and largely correlated to 3D DVI. 2D DVI was moderately correlated to hip transverse angles. ConclusionkFPPA, hFPPA, and DVI are reliable. hFPPA may be reflective of 3D hip and knee frontal and transverse motion during forward step-downs in those with PFP.

Generalized [formula omitted]-Einstein real hypersurfaces with constant coefficient

In this paper, we prove that a real hypersurface in a complex projective plane is generalized D-Einstein with constant coefficient if and only if it is pseudo-Einstein. This, together with the result related to higher dimensions, gives a characterization of pseudo-Einstein real hypersurfaces in a complex projective space. But the corresponding conclusion is much different from the case when the ambient space is a complex hyperbolic plane.

Optimization of Planar Phased Arrays for Vehicles

Radar sensors are used for the detection of objects surrounding a vehicle to enable advanced driver assistance systems. Wide angle azimuth field-of-view (FOV) sensor having high angular resolution is beneficial, but presents challenges for low-cost implementation where element-to-element spacing may be larger than one-half wavelength. Irregular placement of array elements in a two-dimensional (2-D) array can achieve low-cost targets, but may generate large sidelobes if done without care. We introduce an optimization method to determine placement of array elements in a vehicular volumetric beam-scanning radar to minimize sidelobe power for typical automotive beam steering angles. Optimization with array constraints is realized using a gradient-based nonlinear programming technique. Optimization is accelerated by extending <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uv</i> -projection planes beyond the unit circle. For a grating lobe-free scan area based on vehicular applications, the azimuth scanning FOV is increased by 11° bidirectionally to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 54° compared to a reference triangular grid array with 24 elements. The validity of an antenna design generated using the optimization method is established by a three-dimensional (3-D) electromagnetic wave simulation of a representative optimized array configuration.

Semantic Segmentation Guided Coarse-to-Fine Detection of Individual Trees from MLS Point Clouds Based on Treetop Points Extraction and Radius Expansion

Urban trees are vital elements of outdoor scenes via mobile laser scanning (MLS), accurate individual trees detection from disordered, discrete, and high-density MLS is an important basis for the subsequent analysis of city management and planning. However, trees cannot be easily extracted because of the occlusion with other objects in urban scenes. In this work, we propose a coarse-to-fine individual trees detection method from MLS point cloud data (PCD) based on treetop points extraction and radius expansion. Firstly, an improved semantic segmentation deep network based on PointNet is applied to segment tree points from the scanned urban scene, which combining spatial features and dimensional features. Next, through calculating the local maximum, the candidate treetop points are located. In addition, the optimized treetop points are extracted after the tree point projection plane was filtered to locate the candidate treetop points, and a distance rule is used to eliminate the pseudo treetop points then the optimized treetop points are obtained. Finally, after the initial clustering of treetop points and vertical layering of tree points, a top-down layer-by-layer segmentation based on radius expansion to realize the complete individual extraction of trees. The effectiveness of the proposed method is tested and evaluated on five street scenes in point clouds from Oakland outdoor MLS dataset. Furthermore, the proposed method is compared with two existing individual trees segmentation methods. Overall, the precision, recall, and F-score of instance segmentation are 98.33%, 98.33%, and 98.33%, respectively. The results indicate that our method can extract individual trees effectively and robustly in different complex environments.

Planar projections of graphs

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that K n has a representation in ⌈ n / 2 + 2 ⌉ dimensions and K m , n has a representation in ⌈ min ( m , n ) + 1 ⌉ dimensions. In 3 dimensions, we show that there exist graphs with 6 n − 15 edges that can be projected onto two orthogonal planes, and that this is best possible. Finally, we obtain bounds in terms of geometric thickness and caterpillar arboricity, and show connections with unleveled planar graphs. Using such a bound, we show that every graph of maximum degree 5 has a plane-projectable representation in 3 dimensions.

Structure of Projective Planar Subgraphs of the Graph Obstructions for Fixed Surface

Consider the problem of studying the metric properties of a subgraph G\v, where v is an arbitrary vertex of obstruction graphs G of a nonorientable genus, which will determine the sets of points of attachment of one subgraph to another and allow constructing prototypes of graphs-obstruction with number of vertices greater than 10 nonorientable genus greater than 1. This problem is related to Erdosh's hypothesis [3] on the coverage of obstruction graphs of an undirected surface of the genus k, where k &gt; 0, the smallest inclusion of the set of k + 1st graph of the homeomorphic K5, or K3,3, in [4] constructively proved for 35 minors of obstruction graphs of the projective plane, a set of 62 with no more than 10 vertices of obstruction graphs and their splits for the Klein surface, as well as some obstruction graphs for other surfaces. In [5], the existence of a finite set of obstruction graphs for a non-orienting surface was proved. A similar problem was considered in [6], where models or prototypes of obstruction graphs were considered. The prototype of the graph-obstruction of the undirected genus, we will call the graphs that have their own subgraph graph-obstruction of the undirected genus. In [7, 8] the tangent problem of covering the set of vertices with the smallest number of cycles-boundaries of 2-cells was considered, the concept of cell distance is given in [9, 10], where the boundaries of an oriented genus of graphs formed from planar graphs and a simple star glued to some of its peaks. Hypothetically, it is possible to obtain them by recursive φ-transformation of the graph-obstruction of the projective plane and a copy of its planar subgraph given on vertices, edges or parts of edges, or simple chains, i.e. achievable parts of the so-called graph-basis (graph of homeomorphic graph Kuratovsky plane). We assume that instead of one subgraph there can be several copies of subgraphs of graphs-obstructions of the projective plane.The article has an introduction and two parts, in which the structural properties of subgraphs of obstruction graphs for an undirected surface, presented as a φ-image of one of the Kuratovsky graphs and at least one planar graph, are investigated. The metric properties of the minimal embeddings of the subgraphs of the obstruction graphs for undirected surfaces are considered, and the main result is Theorems 1, 2, and Lemma 3 as the basis of the prototype construction algorithm. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.

Projective invariants of images

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

Pedal Sets of Unitals in Projective Planes of Order 16

In this article, we perform computer searches for pedal sets of all known unitals in the known planes of order 16. Special points of unitals having at least one special tangent are studied in detail. It is shown that unitals without special points exist. Open problems regarding the computational results presented in this study are discussed. A conjecture about the numbers of line types of an unital $U$ and its dual unital $U^\perp$ is formulated.

Euler top and freedom in supersymmetrization of one-dimensional mechanics

In this paper we formulate the Euler top as a system on the complex projective plane, playing the role of phase space, i.e., as a one-dimensional mechanical system. Then we suggest the supersymmetrization scheme of the generic one-dimensional systems with positive Hamiltonian which yields á priori integrable family of N=2k supersymmetric Hamiltonians parameterized by N/2 arbitrary real functions.

Waring rank of binary forms, harmonic cross-ratio and golden ratio

We discuss the Waring rank of binary forms of degree 4 and 5, without multiple factors, and point out unexpected relations to the harmonic cross-ratio, j-invariants and the golden ratio. These computations of ranks for binary forms are used to show that the combinatorics of a line arrangement in the complex projective plane does not determine the Waring rank of the defining equation even in very simple situations.

Is spinopelvic sagittal alignment correlated with pain level, functional disability and frontal plane projection angle in women with patellofemoral pain syndrome? A cross-sectional study

There is a growing interest concerning the understanding of measurements of spinal and leg alignment in patients with patellofemoral pain syndrome (PFPS). However, evaluating spinopelvic alignment in the sagittal plane with pain level, functional disability and frontal plane projection angle (FPPA) in PFPS has not been adequately addressed. The aim of the study was to identify the correlation of spinopelvic parameters with pain level, functional disability and frontal plane projection angle in women with PFPS. This was a cross–sectional study involving sixty female patients diagnosed with PEPS, with a mean age of 32±6.47. The measurements used in the study included: radiographic parameters (X-Ray), the numeric pain rating scale (NPRS), the Arabic anterior knee pain scale (AAKP/Kujala) and the 2D-FPPA. Spearman correlation analysis was conducted to evaluate the relationship between these parameters. The results of our study demonstrated a strong positive correlation of lumbar lordosis (LL) with pain level and FPPA (r=0.825, r=0.812, p=0.0001). Also, a strong positive correlation of sacral slope (SS) with pain level (r = 0.0.820) and FPPA (r= 0.783). Pelvic tilt (PT) showed a moderate correlation with pain level (r = 0.0.614) and FPPA (r= 0.605), while a weak negative correlation was found between LL, SS and PT and functional disability score (r=-0.397, r=-0.385 and, r=-0.215 p=0.002). It was concluded that LL, SS and PT were significantly related to the pain level, functional disability and frontal plane projection angle in women with PFPS. These spinal alignments should be considered in clinical evaluation of knee-related disorders associated with PFPS.

Assessment of the Elastoplastic Nonlinear Static and Dynamic Behavior of Guyed Antenna Towers Subjected to Cable Rupture

AbstractCable‐stayed antenna towers are slender structures composed of a central mast and several cable levels and exhibit a highly non‐linear behavior under static and dynamic loads, due to the mast‐cable interaction and the tower height. Many accidents reported in literature are due to cable rupture. Thus, the aim of the present work is to study the effect of cables rupture and plasticity on the load capacity, stability and nonlinear oscillations of a cable stayed tower considering the influence of both geometric and material nonlinearity. For this, finite element models of cable‐stayed towers with two levels of pre‐tensioned cables are developed, considering both fan and parallel cable distributions. Initially, the linear behavior of the structure is evaluated through the natural frequencies and critical loads. Next, the non‐linear equilibrium paths of each model are obtained, applying the concept of modal imperfection. Finally, forced free and damped vibration analyses are performed. For this, time responses, phase plane projections, Fourier transforms and spectrographs are used. The results confirm and quantify the strong influence of the loss of one or more cables on the response of the guyed tower.

Effects of Side Profile on Acoustic Streaming by Oscillating Microstructures in Channel.

In microchannels, microstructure-induced acoustic streaming can be achieved at low frequencies, providing simple platforms for biomedicine and microfluidic manipulation. Nowadays, microstructures are generally fabricated by photolithography or soft photolithography. Existing studies mainly focused on the projection plane, while ignoring the side profile including microstructure’s sidewall and channel’s upper wall. Based on the perturbation theory, the article focuses on the effect of microstructure’s sidewall errors caused by machining and the viscous dissipation of upper wall on the streaming. We discovered that the side profile parameters, particularly the gap (gap g between the top of the structure and the upper wall of the channel), have a significant impact on the maximum velocity, mode, and effective area of the streaming.To broaden the applicability, we investigated boundary layer thickness parameters including frequency and viscosity. Under different thickness parameters, the effects of side profile parameters on the streaming are similar. But the maximum streaming velocity is proportional to the frequency squared and inversely proportional to the viscosity. Besides, the ratio factor of the maximum streaming velocity to the vibration velocity is affected by the side profile parameter gap g and sidewall profile angle .

A stable and quantitative method for dimensionality reduction of aortic centerline.

Aortic dissection (AD) is a fatal aortic disease with high mortality. Assessing the morphology of the aorta is critical for diagnostic and surgical decisions. Aortic centerline projection methods have been used to evaluate the morphology of the aorta. However, there is a big difference between the current model of primary plane projection (PPP) and the actual shape of individuals, which is not conducive to morphological statistical analysis. Finding a method to compress the three-dimensional information of the aorta into two dimensions is helpful to clinical decision-making. In this paper, the evaluation parameters, including contour length (CL), enclosure area, and the sum of absolute residuals (SAR), were introduced to objectively evaluate the optimal projection plane rather than artificial subjective judgment. Our results showed that the optimal projection plane could be objectively characterized by the three evaluation parameters. As the morphological criterion, SAR is optimal among the three parameters. Compared to the optimal projection plane selected by traditional PPP, our method has better AD discrimination in the analysis of aortic tortuosity, and is conducive to the clinical operation of AD. Thus, it has application prospects for the preprocessing techniques for the geometric morphology analysis of AD.

Simplicial complexes from finite projective planes and colored configurations

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as T1⊔T2, such that each Ti represents the lines of a copy of the Fano plane PG(2,F2). We generalize this observation by constructing, for each prime power q, a simplicial complex Xq with q2+q+1 vertices and 2(q2+q+1) facets consisting of two copies of PG(2,Fq). Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring χ:E(G)→[k], such that for all v∈V(G), a,b,c∈[k], following edges of colors a,b,c,a,b,c from v brings us back to v. We give one-to-one correspondences between (1) Sidon sets of order 2 and size k+1 in groups with order n, (2) linear codes with radius 1 and index n in the lattice Ak, and (3) colored (k+1)-configurations with n points and n lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.

Global calibration method for non-overlapping cameras based on rigidly connected stereo targets.

In complex field of view (FOV) environments, a single camera's FOV measurement range is limited and cannot cover the entire object under test for global calibration. Multiple cameras are used mostly for large FOV environment measurements, but the traditional one- and two-dimensional targets used for global calibration in large FOV environments are prone to overlapping FOV. Furthermore, other large-sized targets are difficult to produce and process, and the laser projection method and plane mirror calibration methods are easily affected by the outdoor environment. To solve this problem, a non-common FOV binocular calibration method based on rigidly connected stereo targets is proposed. The calibration process is as follows: First, the rigidly connected target, which is composed of two plane targets with a checkerboard, is placed in front of the two cameras, and the vision sensor captures the corresponding sub-target image; then, the target is moved multiple times, and the transformation relationship between multiple vision sensors is obtained according to the spatial constraint characteristics of the rigidly connected target. Hence, the method overcomes the limitation of the non-overlapping FOV calibration method that relies on large measuring instruments. The experimental results show that the RMS error of the 13 mm distance is 0.16 mm. The proposed method is effective, simpler to operate than other methods, and does not rely on the constraint of complex targets. More importantly, this measurement method solves the difficult problem of measurement in non-public FOV, meeting the requirements of large FOV measurement ranges.

The relationship between ankle dorsiflexion range of motion, frontal plane projection angle, and patellofemoral pain syndrome

BackgroundAnkle range of motion abnormalities have been often linked with alteration in knee kinematics leading to the development of patellofemoral pain syndrome (PFPS). Literature exploring the relationship between ankle dorsiflexion range of motion (DF ROM) and knee kinematics during functional tasks is scanty. This study aims to assess the relation between ankle DF ROM and frontal plane projection angle (FPPA), one of the knee kinematic variables, in individuals with and without PFPS during a step-down test. MethodsThis is a case-control study in which seventy PFPS patients and other 70 asymptomatic control subjects had their ankle DF ROM measured using an inclinometer with the knee flexed and extended. Their FPPA angles were measured using Kinovea software while doing the step-down test. ResultsWhen the two groups were compared, ankle DF ROM measured with the knee flexed was higher in the control group (33.15 ± 4.96) than in the PFPS group (30.20 ± 6.93) (p = 0.03). In both the PFPS group and the control group, the correlation between FPPA and ankle DF ROM with the knee flexed was statistically insignificant (p = 0.075 and 0.323 respectively). ConclusionDecreased ankle DF ROM can be one of the contributing factors to the development of PFPS in the context of greater dynamic knee valgus.

Non-symplectic automorphisms of order 3, 4, and 6 of K3 surfaces with smooth quotient

Let X be a K3 surface, and g be a primitive non-symplectic automorphism of X of order 3, 4, or 6. We write the quotient space as and the quotient morphism as . In this paper, we study curves on X obtained by pulling back curves on by p. As a result, we show that if is smooth, then is the projective plane or Hirzebruch surfaces of degree , or 12. In addition, we show that if is smooth, then is determined by the set of fixed points of g and the action of g on rational curves on X.