Projective Plane Research Articles

Strong surjections from two-complexes with odd order top-cohomology onto the projective plane

Strong surjections from two-complexes with odd order top-cohomology onto the projective plane

A new framework for understanding the evolution of early-type galaxies

Context. We have recently suggested that the combination of the scalar virial theorem (Ms ∝ Reσ2) and the L = L0′σβ law, with L0′ and β changing from galaxy to galaxy (and with time), can provide a new set of equations valid for investigating the evolution of early-type galaxies. These equations are able to account for the tilt of the fundamental plane and to explain the observed distributions of early-type galaxies in all its projections. Aims. In this paper we analyze the advantages offered by these equations, derive the β and L0′ parameters for real and simulated galaxies, and demonstrate that depending on the value of β galaxies can move only along some permitted directions in the fundamental plane projections. Then we show that simple galaxy models that grow in mass by infall of gas and form stars with a star formation rate depending on the stellar velocity dispersion nicely reproduce the observed distributions of early-type galaxies in the fundamental plane projections and yield βs that agree with the measured values. Methods. We derive the mutual relationships among the stellar mass, effective radius, velocity dispersion, and luminosity of early-type galaxies as a function of β and calculate the coefficients of the fundamental plane. Then, using the simple infall models, we show that the star formation history of early-type galaxies is compatible with the σ-dependent star formation rate, and that both positive and negative values of β are possible in a standard theory of galaxy evolution. Results. The parameter β(t) offers a new view of the evolution of early-type galaxies. In brief, it gives a coherent interpretation of the fundamental plane and of the motions of galaxies in its projections; it is the fingerprint of their evolution; it measures the degree of virialization of early-type galaxies; and finally it allows us to infer their evolution in the near past.

Comparison of the Frontal Plane Projection Angle and the Dynamic Valgus Index to Identify Movement Dysfunction in Females with Patellofemoral Pain.

Clinicians typically measure the knee frontal plane projection angle (FPPA) during a single-leg squat to identify females with patellofemoral pain (PFP). A limitation of this measure is minimal attention to movement of the pelvis on the femur that can create knee valgus loading. The dynamic valgus index (DVI) may be a better assessment. The purpose of this study was to compare the knee FPPA and DVI between females with and without PFP and determine if the DVI better identified females with PFP than the knee FPPA. Case-control. Sixteen females with and 16 without PFP underwent 2-dimensional motion analysis when performing five trials of a single-leg squat. The average peak knee FPPA and peak DVI were analyzed. Independent t-tests determined between-group peak knee FPPA and peak DVI differences. Receiver operating characteristic (ROC) curves determined the area under the curve (AUC) scores for sensitivity and 1 - specificity of each measure. Paired-sample area difference under the ROC curves was conducted to determine differences in the AUC for the knee FPPA and DVI. Positive likelihood ratios were calculated for each measure. The significance level was p < 0.05. Females with PFP exhibited a higher knee FPPA (p = 0.001) and DVI (p = 0.015) than controls. AUC scores were .85 (p = 0.001) and .76 (p = 0.012) for the knee FPPA and DVI, respectively. Paired-sample area difference under the ROC curves showed a similar (p = 0.10) AUC for the knee FPPA and DVI. The knee FPPA had 87.5% sensitivity and 68.8% specificity; the DVI had 81.3% sensitivity and 81.0% specificity. Positive likelihood ratios for the knee FPPA and DVI were 2.8 and 4.3, respectively. The DVI during a single-leg squat may be another useful tool for discriminating between females with and without PFP. 3a.

On equations of fake projective planes with automorphism group of order $21$

We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective planes with this automorphism group. This includes, in particular, the fake projective plane discovered by J. Keum.

Three-dimensional computed tomographic angular measurements of the canine tibia using a bone-centered coordinate system.

Canine tibial alignment is determined by two-dimensional angular measurements, and tibial torsion is challenging. Aim of the study was the development and evaluation of a CT technique to measure canine tibial varus and torsion angles independent from positioning and truly three-dimensional. A bone-centered 3D cartesian coordinate system was introduced into the CT-scans of canine tibiae and aligned with the anatomical planes of the bone based on osseous reference points. Tibial torsion, and varus (or valgus) angles were calculated based on geometric definition of projection planes with VoXim® medical imaging software using 3D coordinates of the reference points. To test accuracy of the tibial torsion angle measurements, CT scans of a tibial torsion model were performed in 12 different hinge rotation setups ranging from the normal anatomical situation up to +/ 90° and compared to goniometer measurements. Independency of tibial positioning on the CT scanner table was evaluated in 20 normal canine tibiae that were scanned in a position parallel to the z-axis and two additional off-angle double oblique positions having 15° and 45° deviation in direction of the x- and y-axes. Angular measurements in oblique positions were compared with the normal parallel position by subtraction. Precision was tested using clinical CT scans of 34 canine patients with a clinical diagnosis of patellar luxation. Accuracy testing in the tibial torsional deformity model revealed a difference of 0.2° demonstrated by Passing-Bablok analysis and Bland-Altman-Plots. Testing for independency from tibial positioning resulted in mean differences less than 1.3°. Precision testing in clinical patients resulted in coefficients of variation for repeated measurements of 2.35% (intraobserver agreement) and 0.60% (interobserver agreement) for the tibial torsion angle, and 2.70% (intraobserver agreement) and 0.97% (interobserver agreement) for the tibial varus (or valgus) angle. The technique is lacking determination of bone deformities in the sagittal plane, and demonstration of accuracy in severe complex bone deformities in multiple planes.In conclusion, we developed a method to measure canine tibial torsional and varus or valgus deformities, that calculates in 3D space, and we demonstrated its accuracy in a torsional deformity model, and its precision in CT data of clinical patients.

Electronic, elastic and thermal properties of hexagonal TM5Si3N investigated by first-principles calculations

During the current study, structural, electrical, elastic, and thermal properties of TM5Si3N (TM = V, Nb, and Ta) Nowotny phase were researched using first-principles calculations. The calculated formation enthalpies and phonon dispersion spectrums of these TM5Si3N Nowotny phases indicate that they are dynamically and thermodynamically stable. Because of the strong hybridization between TM-d state and Si-p and N-p states, TM5Si3N can form TM-Si and TM-N bonds. The obtained elastic anisotropy indexes, two-dimensional (2D) planar projection, and three-dimensional (3D) surface constructions demonstrate that the elastic modulus of TM5Si3N is anisotropic, and the order of elastic anisotropy is Ta5Si3N > Nb5Si3N > V5Si3N. In addition, the minimum thermal conductivity of TM5Si3N was calculated using the Clarke's and Cahill's models, which was found to be anisotropic and in the following order of Ta5Si3N > Nb5Si3N > V5Si3N.

A Projection-Based Evolutionary Algorithm for Multi-Objective and Many-Objective Optimization

Many-objective optimization problems (MaOPs) are challenging optimization problems in scientific research. Research has tended to focus on algorithms rather than algorithm frameworks. In this paper, we introduce a projection-based evolutionary algorithm, MOEA/PII. Applying the idea of dimension reduction and decomposition, it divides the objective space into projection plane and free dimension(s). The balance between convergence and diversity is maintained using a Bi-Elite queue. The MOEA/PII is not only an algorithm, but also an algorithm framework. We can choose a decomposition-based or dominance-based algorithm to be the free dimension algorithm. When it is an algorithm framework, it exhibits a better performance. We compare the performance of the algorithm and the algorithm with the MOEA/PII framework. The performance is evaluated by benchmark test instances DTLZ1-7 and WFG1-9 on 3, 5, 8, 10, and 15 objectives using IGD-metric and HV-metric. In addition, we investigated its superior performance on the wireless sensor networks deployment problem using C-metric. Moreover, determining objective domain for the objects of the wireless sensor networks deployment problem reduces the time and makes the solution set more responsive to user needs.

Linking number of monotonic cycles in random book embeddings of complete graphs

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining “interior” edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers of disjoint monotonic cycles can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is [Formula: see text], where [Formula: see text] is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of disjoint monotonic [Formula: see text]-cycles in [Formula: see text] grows linearly in [Formula: see text].

Pinwheels as Lagrangian barriers

The complex projective plane [Formula: see text] contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation, see for example [J. Evans and I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143–1180]. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of [Formula: see text], meaning that pinwheels are Lagrangian barriers in the sense of [P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464]. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below [Formula: see text].

Klein’s arrangements of lines and conics

In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of 21 lines in the complex projective plane.

Properties and Construction Method for Symmetric Balanced Incomplete Block Design with λ=1

Symmetric Balanced Incomplete Block Designs with λ=1 is a common class of BIBDs which are mostly used in incomplete experimental block design set up because of their simplicity in set up and also in analysis. Over the years since development of the BIBDs by Yates in the year 1939. A number of research has been done on the design to establish properties of the design and also to determine the construction methods of the design. In terms of properties, the studies have only been able to establish necessary but not sufficient not sufficient conditions for the existence of the design. For the symmetric BIBDs the studies have also determined the non-existence properties for such designs. However, the sufficient existence property for the design have not been established. In terms of construction, the studies have been able to derive several construction methods for BIBDs. However, these methods have been determined not to be adequate in constructing all the BIBDs which still leave the existence of some BIBDs as unknown. For symmetric BIBDs with λ=1 which are also known as projective planes, the studies have not been able to establish the sufficient properties for existence of this class of BIBDs just like the other classes of symmetric BIBDs. Therefore, this give room for investigating other properties of this class of BIBDs. The present study therefore, aimed at deriving the properties of the design from the known properties of BIBDs and also using the properties to determine the construction technique that would be suitable used in constructing this class of BIBDs. The study used the known properties of symmetric BIBDs to derive new properties of symmetric BIBDs, then restricted it to the case of λ=1. Which aided in derivation of new properties of the design and also the construction method. The study was able to derive three new properties for this class of BIBD and it was also able to show that the class of BIBD would be best constructed using PG(2,S).

Adapted in place: A digital geometric analysis of the muqarnas dome of al-Bimaristan al-Nuri in Damascus

The dome of al-Bimaristan al-Nuri is regarded as the first muqarnas dome in Damascus. It is a unique example of a suspended muqarnas dome in the region. The 2D projection plane of the dome displays non-uniform cells, which has led researchers to assume that it was an experimental attempt to replicate a new architectural element. This study aims to deliver an in-depth 3D analysis of construction irregularities and deviations to examine the experimental nature of the design and execution. It provides a 3D workflow to document, analyze, and visualize a muqarnas dome. By exploring the geometrical properties and relations of the dome using terrestrial laser scanning (TLS), the study presents geometrical corroboration of the in-situ adaptation of the dome.

Triangular arrangements on the projective plane

In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a Roots-of-Unity-Arrangement, which is a particular class of triangular arrangements. Among these Roots-of Unity-Arrangements, we provide conditions that ensure their freeness. Finally, we give two triangular arrangements having the same weak combinatorics, such that one is free but the other one is not.

Admissible subcategories of del Pezzo surfaces

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (−1)-curve in a surface is generated by some twist of the structure sheaf of that curve.

Scaling of Networks and their Key Systems Based on Combined Combinatorial Block Designs

A new method of scaling wireless sensor networks, as well as key distribution schemes and key spaces in them based on a combined combinatorial block design formed on the set of all pairs composed of blocks and corresponding to their elements dual blocks of a given combinatorial block design is presented. Estimates are given for the cardinalities of scaled circuits based on the combined projective plane, the combined modified residual block design, and the combined residual combinatorial transversal block design. It is shown that the size of the key memory of the nodes of such networks does not depend on the parameters of the combinatorial block design and is two key units, but in schemes based on the combined projective plane, the length of the shortest route between the vertices of the key matching graph increases to three. The consequences of node capture are evaluated and ways to limit them with a guaranteed security parameter are proposed. For schemes based on a combined cyclic projective plane, as applied to the w-safe WSN of a "smart home", communication schemes between nodes along routes of length from one to three and an example of a protocol are presented. An example of such a WSN containing 910 nodes that are child nodes from 31 to 91 routers is given.

ANALYSIS OF THE IMPACT OF CONVERTING THE COORDINATES OF TURNING POINTS OF LAND PLOTS FROM THE SK-63 SYSTEM TO USK-2000 ON THEIR LINEAR PARAMETERS AND AREA

With the introduction into use of the new state reference system USK-2000, the issue of converting the coordinates of the turning points of the boundaries of land parcels into USK-2000 arises, which is important for geodetic support of the design, construction of engineering structures, etc. The article examines the influence of the transformation parameters on the divergence of the linear parameters of the turning points from the SK-63 coordinate system to the USK-2000 system. The analysis of coordinates and linear parameters of land plots was carried out using conversion coefficients using the affine method according to formulas and compared with those calculated in the Digitals program. Regardless of the area and orientation of the land plot, the sum of increments of its coordinates is equal to zero in the SK-63 and USK- 2000 systems, while the area discrepancy depends on the size of the land plot. To clarify the actual area of a land plot on the physical surface of the Earth, which has a boundary length of more than 500 m, it is necessary to reduce to the reference surface the reference ellipsoid and the Gauss-Kruger projection plane. if the site was connected to the points of the state network. &#x0D; The relevance of the work is to determine the influence of the transformation of the coordinates of the turning points of land plots in the USK-2000 system on linear parameters (coordinate increments, lengths between the turning points of borders) and the area of small land plots. &#x0D; The goal of the work. Analyze the discrepancy of linear parameters and the area of land plots after converting the coordinates of turning points from the SK-63 system to the USK-2000 system in the Digitals program and by the affine method using formulas on specific examples. Research results can be taken into account when considering the issue of determining the area of land plots in the USK-2000 coordinate system up to 25 hectares.

𝒬-conic arrangements in the complex projective plane

We study the geometry of Q \mathcal {Q} -conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such Q \mathcal {Q} -conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.

Optimal lifting for the projective action of SL3(ℤ)

Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon}$, whose projection to $SL_{3}(F_q)$ sends $x$ to $y$. The exponent $1/3$ is optimal and the result is a high rank generalization of Sarnak's optimal strong approximation theorem for $SL_2(Z)$.

Toward Efficient and Complete Line Segment Extraction for Large-Scale Point Clouds via Plane Segmentation and Projection

An efficient method based on plane segmentation and projection is proposed to extract as complete boundary (straight) line segments as possible. To efficiently detect boundary points, 3-D planar point clouds (patches) generated by plane segmentation are converted into 2-D images by graphical projection to avoid point-based nearest neighbors’ searching. To preserve boundary details, the projection resolution for each plane is determined through accurate statistics of average local intervals between points in voxels composing corresponding planar point clouds. To alleviate erroneous line segmentation extraction caused by uneven densities of planar point clouds, improved mathematical morphological operation, whose structural element order is self-adaptive to local point density, is proposed for crack inpainting of projected images. As for extraction of 3-D boundary line segments, the back-projection results of 2-D boundary line segments detected from images are utilized as references for clustering of boundary points, and a 3-D straight line fitting algorithm based on recursive weighted least squares is employed to reduce the effect of boundary noise. A series of refinement strategies are further presented to control the quality of extracted 3-D line segments. Experimental results demonstrate that the proposed line segment extraction method can provide efficient and fine processing results, as well as automatic processing ability, which satisfies the efficient processing demand for large-scale point clouds.

Determination of intramedullary nail based on centerline adaptive registration.

Objective: Internal fixation with intramedullary nails is a gold standard for the treatment of femoral shaft fractures. However, both the mismatch between intramedullary nails and the medullary cavity and inaccurate positioning of entry points will lead to deformation of intramedullary nails after implantation. The study aimed to determine a suitable intramedullary nail with an optimal entry point for a specific patient based on centerline adaptive registration. Method: A homotopic thinning algorithm is employed to extract centerlines of the femoral medullary cavity and the intramedullary nail. The two centerlines are registered to obtain a transformation. The medullary cavity and the intramedullary nail are registered based on the transformation. Next, a plane projection method is employed to calculate the surface points of the intramedullary nail laid outside the medullary cavity. According to the distribution of compenetration points, an iterative adaptive registration strategy is designed to decide an optimal position of the intramedullary nail in medullary cavity. The isthmus centerline is extended to the femur surface, where the entry point of the intramedullary nail is located. The suitability of an intramedullary nail for a specific patient was calculated by measuring the geometric quantities reflecting the interference between the femur and nail, and the suitability values of all nails are compared and the most suitable one is determined. Results: The growth experiment indicated that the bone to nail alignment is indeed affected by the extension of the isthmus centerline, including the extension direction and velocity. The geometrical experiment showed that this method could find the best registration position of intramedullary nails and select the optimal intramedullary nail for a specific patient. In the model experiments, the determined intramedullary nail could be successfully placed into the medullary cavity through the optimal entry point. A pre-screening tool to determine nails which can be successfully used has been given. In addition, the distal hole was accurately located within 14.28s. Conclusion: These results suggest that the proposed method can select a suitable intramedullary nail with an optimal entry point. The position of the intramedullary nail can be determined in the medullary cavity, while deformation is avoided. The proposed method can determine the largest diameter intramedullary nail with as little damage to the intramedullary tissue as possible. The proposed method provides preparation aid for internal fixation with intramedullary nails guided by navigation systems or extracorporeal aimers.