Projective Plane Research Articles

Semiaffine stable planes

AbstractA locally compact stable plane of positive topological dimension will be called semiaffine if for every line L and every point p not in L there is at most one line passing through p and disjoint from L. We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.

GRAPHICAL AND ANALYTICAL DEFINITION OF PARAMETRIC CURVES IN EXPLICIT FORM

The aim of the work is to establish the relationship between graphical and analytical methods of solving problems on the example of the definition of parametric curves in explicit form. The idea is to represent the system of parametric equations of a plane curve in the space of Otxy coordinate system as a system of two projections of the curve on a complex Monge drawing. Then the desired curve can be obtained explicitly as a discrete set of points on the profile plane of projections in the Otxy coordinate system. If we consider the system of curve projections as a single curve in three-dimensional space Otxy, then this curve can be flat and spatial. For planar curves located in a partial position plane, it is easy to make the transition from parametric setting of the curve to an explicit form due to the linear relationship between one of the variables (x or y) and the parameter t. If the curve in the Otxy coordinate system belongs to the general position plane, we can use the projection plane replacement method to transform it to a projecting position. This allows a linear relationship to be established between one of the variables and a parameter to obtain the equation of the curve in explicit form, followed by an inverse transformation of the planes of projections. Mathematically, replacing the plane of projections is equivalent to replacing the coordinate system and reduces it to a rotation around the origin of the coordinates. Formulas for transforming the coordinate system, taking into account the peculiarities of using Monge's complex drawing and for determining the angle of rotation of the coordinate system are given.

INTEGRATED GIS TECHNIQUES FOR LAND SUITABILITY ASSESSMENT IN SOUTHERN IDAHO FOR SPRING WHEAT

The accurate evaluation of land suitability for special agriculture production plays an important role to decrease the negative environmental impacts. The current study aimed to apply GIS techniques for analyzing land suitability to wheat production in 5474.27 ha of semi-arid regions of Bingham and Bonneville counties of southern Idaho. This research used the Idaho West State Plane projection for the least distorted projection for its investigation. "Convert feature class to feature class" and "copy raster" procedures were used to ensure that all required data layers are stored in the active database. DEM data was gathered in the form of multiple rasters and then turned into a single mosaic dataset so that all of the data are converted to a single mosaic dataset. The results indicated that Bingham County is more suitable for wheat than Bonneville County. A change in irrigation, cropping pattern, rotation, and land use type of Bonneville County might be helpful to increase its wheat production. It was concluded that integrating GIS techniques could be an accurate and effective method for better land use planning and management for agriculture productions.

Differences In Knee Valgus Degrees In Cutting And Drop Landing On The Dominant And Non-Leg Dominant In High School Basketball Players

Background: Basic techniques, physical endurance, and teamwork in basketball are components needed to improve individual quality to be able to play with good performance. Basic techniques and movements often occur quickly and suddenly, including cutting and drop landing. Both of these movements are prone to knee injury. The majority of lower extremity injuries in basketball players are anterior cruciate ligament (ACL) strains. Observation of the knee valgus angle is required to assess the risk of knee injury. One of the risks has the potential to cause a knee injury is the dominant and non-dominant knee. This is because the leg have an important role in controlling posture and stability. Purpose: this study was to determine the differences in the risk factors for the dominant and non-dominant leg in cutting and drop landing. Methods: The research was analytic observational, a cross-sectional study design. The population in this study were high school basketball players in Malang City with a total sample of 22 people the inclusion and exclusion criteria (n=22). Data collection was carried outthe Waterloo Footedness Questionnaire-Revised (WFQ-R) and Frontal Plane Projection Angle (FPPA) as measuring tools using the kinovea software. Results: Based on the analysis Wilcoxon test forcutting and drop landing movements, a value of 0.036 (p <0.05) was obtained for the non-dominant leg, which means that there is a significant difference between the two movements. Conclusion: It canbe concluded that there are differences movements of cutting and drop landing on the non-dominant leg compared to the dominant leg.

New good large (n,r)-arcs in PG(2,29) and PG(2,31)

An $(n, r)$-arc is a set of $n$ points of a projective plane such that some $r$, but no $r+1$ of them, are collinear. The maximumsize of an $(n, r)$-arc in $\PG(2,q)$ is denoted by $m_r(2,q)$. In this article a $(477, 18)$-arc, a $(596,22)$-arc, a $(697,25)$-arc in PG(2,29) and a $(598, 21)$-arc, a $(664, 23)$-arc, a $(699, 24)$-arc, a $(769, 26)$-arc, a $(838,28)$-arc in PG(2,31) are presented. The constructed arcs improve the respective lower bounds on $m_r(2,29)$ and $m_r(2,31)$ in \cite{MB2019}. As a consequence there exist eight new three-dimensional linear codes over the respective finite fields.\\

Exploring the behavior feature of complex trajectories of ships with Fourier transform processing: a case from fishing vessels

The significant uncertainty and complexity of vessels at sea poses challenges for regulatory bodies in the fishing industry. This paper presents a method for identifying fishing vessel trajectory characteristics involving the Fourier series transform. The model utilizes the Fourier series and Gaussian mixture clustering to address the complexity and uncertainty issues in fishing vessel trajectories. First, the vessel trajectories undergo a process of dimensionality expansion and projection along the temporal axis. The relationship between trajectories and complex plane projection was elucidated in this process. Second, a vessel trajectory identification model involving Fourier transformation was constructed. Subsequently, the phase spectrum was assigned binary values using differentiation, and the phase spectrum characteristics of the transformed trajectories through Fourier transformation were analyzed. Finally, six encoding formats for fishing vessel motion trajectories in phase spectrum encoding are introduced, along with the determination of uncertain vessel motion range through mixed Gaussian clustering. This method has been validated using a dataset comprising 7,000 fishing vessel trajectories collected from the Beidou satellite positioning system. The results demonstrate that the range of uncertain vessel motion was able to be obtained with the assistance of Gaussian mixture clustering, with an 80% probability position of approximately 1,000 m and a 50% probability position of around 2,000 m. Effective identification of fishing vessel operating and navigational states was achieved, leading to the determination of a safety distance for fishing vessels in the range of 1,000m–2,000 m. This research holds important reference value for fishery regulatory agencies in terms of supervising fishing vessels and maintaining a safe navigational distance.

Ehrhart theory of paving and panhandle matroids

Abstract We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields

We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27 -dimensional \mathrm{E_6} module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.

Dielectrophoretic Movement of Cell Passing Between Surface Electrodes in Flow Channel

Abstract In this study, cell behavior in a microchannel was tracked for the application of dielectrophoresis to biological cell sorting. A pair of titanium surface electrodes was fabricated inside the microchannel by photolithographic techniques: a triangular electrode with a tip angle of 0.26 rad and a rectangular electrode. A periodic alternating current of square wave with a period of 1 μs was introduced between the electrodes to induce an asymmetric electric field perpendicular to the mainstream direction. The behavior of mouse myoblasts (C2C12: mouse myoblast cell line) was measured in vitro while the suspension was flowing. The relationship between cell shift motion near the electrode and cell shape on the two-dimensional projection plane was investigated. Experimental results showed that cell movement in the direction perpendicular to the mainstream increased with geometries away from the circle in the two-dimensional projection plane. This method can be applied to sort cells according to their degree of shape deviation from a sphere. The dielectrophoretic effect can be applied to sort cells not only by cell size but also by cell deformation.

Sampling based spherical transformer for 360 degree image classification

Using convolutional neural networks for 360° images can induce sub-optimal performance due to distortions entailed by a planar projection. The distortion gets deteriorated when a rotation is applied to the 360° image. Thus, many researches based on convolutions attempt to reduce the distortions to learn accurate representation. In contrast, we leverage the transformer architecture to solve image classification problems for 360° images. Using the proposed transformer for 360° images has two advantages. First, our method does not require the erroneous planar projection process by sampling pixels from the sphere surface. Second, our sampling method based on regular polyhedrons makes low rotation equivariance errors, because specific rotations can be reduced to permutations of faces. In experiments, we validate our network on two aspects, as follows. First, we show that using a transformer with highly uniform sampling methods can help reduce the distortion. Second, we demonstrate that the transformer architecture can achieve rotation equivariance on specific rotations. We compare our method to other state-of-the-art algorithms using the SPH-MNIST, SPH-CIFAR, and SUN360 datasets and show that our method is competitive with other methods.

Fractal interpolation on the real projective plane

Fractal interpolation on the real projective plane

Biadjoint scalars and associahedra from residues of generalized amplitudes

In the Grassmannian formulation of the S-matrix for planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super Yang-Mills, Nk−2MHV scattering amplitudes for k negative and n − k positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of (k − 2)(n − k − 2)-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian G(k, n). In this paper, we replace the Grassmannian G(3, n) with its torus quotient, the moduli space of n points in the projective plane in general position, and planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM with generalized biadjoint scalar amplitudes mn3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {m}_n^{(3)} $$\\end{document} as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such (n − 5)-dimensional residue of mn3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {m}_n^{(3)} $$\\end{document} an entire biadjoint scalar partial amplitude mn2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {m}_n^{(2)} $$\\end{document}, that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for k ≥ 4.

Geometry of knots in real projective 3-space

This paper discusses some geometric ideas associated with knots in real projective 3-space [Formula: see text]. These ideas are borrowed from classical knot theory. Since knots in [Formula: see text] are classified into three disjoint classes: affine, class-[Formula: see text] non-affine and class-[Formula: see text] knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in [Formula: see text]. We then study the notion of companionship of knots in [Formula: see text] and using it we provide geometric criteria for a knot to be affine. We also define a notion of “genus” for knots in [Formula: see text] and study some of its properties. We prove that this genus detects knottedness in [Formula: see text] and gives some criteria for a knot to be affine and of class-[Formula: see text]. We also prove a “non-cancellation” theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-[Formula: see text] non-affine knots with genus [Formula: see text]. Thus we highlight that, [Formula: see text] admits a knot theory with a truly different flavor than that of [Formula: see text] or [Formula: see text].

Femoral neck version in the spinopelvic and lower limb 3D alignment: a full-body EOS® study in 400 healthy subjects.

The goal of this study was to better understand the variation of femoral neck version according to spinopelvic and lower limb 3D alignment using biplanar X-rays in standing position. This multicentric study retrospectively included healthy subjects from previous studies who had free-standing position biplanar radiographs. Subjects were excluded if they presented spinal or any musculo-skeletal deformity, and reported pain in the spine, hip or knee. Age, sex, and the following 3D-reconstructed parameters were collected: spinal curvatures, pelvic parameters, sagittal vertical axis (SVA), T1 pelvic angle (TPA), spino-sacral angle (SSA), femoral torsion angle (FTA), sacro-femoral angle (SFA), knee flexion angle (KA), ankle angle (AA), pelvic shift (PS) and ankle distance. Femoral neck version angle (FVA) was calculated between horizontal plane projection of the bi-coxo-femoral axis and the line passing through the femoral neck barycenter and femoral head center. Analysis according to age subsets was performed. A total of 400 subjects were included (219 females); mean age was 29 ± 18years (range: 4-83). Subjects with high pelvic tilt values presented significantly higher FVA than average and low-PT individuals, respectively, 7.8 ± 7.1°, 2 ± 9° and 2.1 ± 9.5° (p < 0.001). These subjects also presented lower lumbar lordosis values and higher acetabulum anteversion in the horizontal plane than the two other groups. SVA correlation with FVA was weaker (r = 0.1, p = 0.03) than SSA and TPA (r = - 0.3 and r = 0.3, respectively, p < 0.001). A strong correlation was found with femoral torsion (r = 0.5, p < 0.001). SFA (r = - 0.3, p < 0.001), pelvic shift (r = 0.2, p < 0.001) and ankle distance (r = 0.3, p < 0.001) were also significantly correlated. Multivariate analysis confirmed significant association of age, pelvic tilt, lumbar lordosis, pelvic shift, ankle distance and femoral torsion with FVA. Patients with lower lumbar lordosis present pelvic retroversion which induces a higher femoral neck version. This finding may help positioning implants in total hip replacement procedures. Higher pelvic shift, age, male gender and increased femoral torsion were also correlated with higher FVA. II (Diagnostic: individual cross-sectional studies with consistently applied reference standard and blinding).

Fast 3D Object Measurement Based on Point Cloud Modeling

Automated object measurement is becoming increasingly important due to its ability to reduce manual costs, increase production efficiency, and minimize errors in various fields. In this paper, we present a novel approach to three-dimensional (3D) object measurement based on point cloud modeling. Our method introduces a fast point cloud modeling computation framework consisting of five stages: coordinate centralization, rotation and translation, noise filtering, plane projection, and geometric computation. Furthermore, we propose a fast convex hull optimization algorithm to reduce the high complexity problem of traditional convex hull calculation. Our extensive experiments demonstrate that our approach outperforms existing methods in terms of measurement error rate and time savings, with a maximum time saving of 31.03% under certain error conditions.

Influence of self-perceived disability on squatting kinematics in individuals with patellofemoral pain

BackgroundIndividuals with patellofemoral pain have a heterogeneous presentation of symptoms during functional tasks; however, biomechanical data often negates self-reported disability. The single leg squat is common in patellofemoral pain literature but may not be a pain provoking task for all individuals. Therefore, our study examined the influence of self-perceived squatting disability in individuals with patellofemoral pain on lower extremity squatting kinematics. MethodsWe analyzed two-dimensional hip frontal plane projection angle, knee frontal plane projection angle, lateral trunk movement, and pelvic drop in 100 participants, 82 with patellofemoral pain and 18 pain-free controls. Participants with patellofemoral pain were dichotomized based on the level of disability reported during squatting on the anterior knee pain scale. An analysis of variance with post hoc testing was used to compare differences in lower extremity and trunk kinematics between groups, p < 0.05. FindingsParticipants who reported only being able to squat with partial weight bearing had greater hip frontal plane projection angles than those who reported squatting painful each time (p = 0.017). The partial weight bearing group had greater knee frontal plane projection angles compared those who reported pain with repeated squatting and the pain-free group, (p < 0.034). We found no significant differences in lateral trunk motion or pelvic drop between groups. InterpretationIndividuals with patellofemoral pain who self-reported worse squatting disability had greater hip and knee frontal plane motion compared to individuals with less disability and pain-free controls. Clinicians and researchers should consider specific pain provoking tasks when evaluating and treating patients with patellofemoral pain.

On the achromatic number of the Cartesian product of two complete graphs

A vertex colouring f: V(G) → C of a graph G is complete if for any c1, c2 ∈ C with c1 ≠ c2 there are in G adjacent vertices v1, v2 such that f(v1) = c1 and f(v2) = c2. The achromatic number of G is the maximum number achr(G) of colours in a proper complete vertex colouring of G. Let G1▫G2 denote the Cartesian product of graphs G1 and G2. In the paper achr(Kr2 + r + 1▫Kq) is determined for an infinite number of q’s provided that r is a finite projective plane order.

SOME RESULTS ON THE SMALLEST CARTESIAN GROUP PLANE

Let π be the projective plane of order 25 coordinatised by elements of the smallest cartesian group. In this work, in some cases depending on the choice of the regular quadrangle it is shown that there is no any projective subplane of order 3 of π.

Almost complex manifolds with total Betti number three

In this paper, we show the minimal total Betti number of a closed almost complex manifold of dimension [Formula: see text] is four, thus confirming a conjecture of Sullivan except for dimension [Formula: see text]. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.

The crossing number of the generalized Petersen graph P(3k,k) in the projective plane

ABSTRACT The crossing number of a graph G in a surface Σ, denoted by c r Σ ( G ) , is the minimum number of pairwise intersections of edges in a drawing of G in Σ. Let k be an integer satisfying k ≥ 3 , the generalized Petersen graph P ( 3 k , k ) is the graph with vertex set V ( P ( 3 k , k ) ) = { u i , v i ∣ i = 1 , 2 , … , 3 k } and edge set E ( P ( 3 k , k ) ) = { u i u i + 1 , u i v i , v i v k + i ∣ i = 1 , 2 , … , 3 k } , the subscripts are read modulo 3 k . In this paper we investigate the crossing number of P ( 3 k , k ) in the projective plane. We determine the exact value of c r N 1 ( P ( 3 k , k ) ) is k–2 when 3 ≤ k ≤ 7 , moreover, for k ≥ 8 , we get that k − 2 ≤ c r N 1 ( P ( 3 k , k ) ) ≤ k − 1.