Projective Plane Research Articles

Acute effects of artificially increased frontal plane projection angle on gait biomechanics in healthy subjects: A pilot study

Acute effects of artificially increased frontal plane projection angle on gait biomechanics in healthy subjects: A pilot study

Nonisomorphic two‐dimensional algebraically defined graphs over R ${\mathbb{R}}$

AbstractFor , let be a two‐dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of and two vertices and are adjacent if and only if . It is known that has girth 6 and can be extended to the point‐line incidence graph of the classical real projective plane. However, it was unknown whether there exists such that has girth 6 and is nonisomorphic to . This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of for families of bivariate functions .

In Myotonic Dystrophy Type 1 Head Repositioning Errors Suggest Impaired Cervical Proprioception.

Background: Myotonic dystrophy type 1 (DM1) is a rare multisystemic genetic disorder with motor hallmarks of myotonia, muscle weakness and wasting. DM1 patients have an increased risk of falling of multifactorial origin, and proprioceptive and vestibular deficits can contribute to this risk. Abnormalities of muscle spindles in DM1 have been known for years. This observational cross-sectional study was based on the hypothesis of impaired cervical proprioception caused by alterations in the neck spindles. Methods: Head position sense was measured in 16 DM1 patients and 16 age- and gender-matched controls. A head-to-target repositioning test was requested from blindfolded participants. Their head was passively rotated approximately 30° leftward or rightward and flexed or extended approximately 25°. Participants had to replicate the imposed positions. An optoelectronic system was adopted to measure the angular differences between the reproduced and the imposed positions (joint position error, JPE, °) concerning the intended (sagittal, horizontal) and unintended (including the frontal) planar projections. In DM1 patients, JPEs were correlated with clinical and balance measures. Static balance in DM1 patients was assessed through dynamic posturography. Results: The accuracy and precision of head repositioning in the intended sagittal and horizontal error components did not differ between DM1 and controls. On the contrary, DM1 patients showed unintended side-bending to the left and the right: the mean [95%CI] of frontal JPE was -1.29° [-1.99°, -0.60°] for left rotation and 0.98° [0.28°, 1.67°] for right rotation. The frontal JPE of controls did not differ significantly from 0° (left rotation: 0.17° [-0.53°, 0.87°]; right rotation: -0.22° [-0.91°, 0.48°]). Frontal JPE differed between left and right rotation trials (p < 0.001) only in DM1 patients. No correlation was found between JPEs and measures from dynamic posturography and clinical scales. Conclusions: Lateral head bending associated with head rotation may reflect a latent impairment of neck proprioception in DM1 patients.

The effect of different subtalar joint pronation amounts on postural stability, function and lower extremity alignment in healthy individuals

IntroductionThe prone foot posture has a negative effect on postural stability, function, and knee valgus, but to our knowledge, the contribution of the degree of pronation has not been examined. Methods39 participants aged 18–40, with Foot Posture Index (FPI) scores between 6–12 and without any pain complaints were included. Participants with 6–9 points were included in the pronation group (PG) (n = 19), and participants with 10–12 points were included in the hyperpronation group (HPG) (n = 20). Static and dynamic postural stability, Foot and Ankle Ability Measure (FAAM), and frontal plane projection angles (FPPA) were measured for all participants. ResultsThe initial data of the participants are distributed homogeneously. In the intergroup evaluation only FPI-1 (p = 0.001; p < 0.05), FPI-4 (p = 0.00; p < 0.05), FPI-5 (p = 0.00; p < 0.05) and FPI-T (p = 0.000; p < 0.05) scores were found significantly different. ConclusionPronation and hyperpronation of the subtalar joint did not lead to a difference in postural stability, function, and knee valgus in healthy individuals. It may be more beneficial to focus on the prone posture rather than the degree of pronation.

Optimal control of a solar sail

SummaryOptimal control of a solar sail orbiting around a fixed center of mass is considered. The sail is modelled as a plane surface with two sides having similar optical properties. The control is assumed to be the attitude of the sail and is represented as an element of the projective plane. Mapping this plane into the three dimensional ambient space to compute the actual force generated by a given attitude results in a generally non‐convex set of admissible control values. A suitable convex relaxation is introduced to study the optimality system associated with maximising the change of the sail orbital parameters in a given direction along one orbit. Existence for both the relaxed and the original problems are deduced. Building on previous works, a refined analysis of the control structure is given, proving rigorously that it contains only switchings, and a global bound of the number of switchings is obtained. In order to compute effective solutions, a multiple shooting approach is retained that is well suited for systems with a sequence of switchings between various control subarcs. Three issues are addressed: initialization of shooting by means a tailored semi‐definite relaxation that takes advantage of good convergence properties of modern convex optimization algorithms; changes in the control structure that are accommodated by coupling shooting with differential continuation; implicit character of the Hamiltonian maximization when using Pontrjagin maximum principle that is taken care of by incorporating the associated equation for the dynamical feedback into the shooting procedure. The method is illustrated on an example from NASA for which the sail inclination is optimally changed over one orbital period.

Mutations of numerically exceptional collections on surfaces

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

Effect of Menstrual Cycle and Hormonal Contraception on Musculoskeletal Health and Performance: Protocol for a Prospective Cohort Design and Cross-Sectional Comparison.

Women of reproductive age experience cyclical variation in the female sex steroid hormones 17β-estradiol and progesterone during the menstrual cycle that is attenuated by some hormonal contraceptives. Estrogens perform a primary function in sexual development and reproduction but have nonreproductive effects on bone, muscle, and sinew tissues (ie, ligaments and tendons), which may influence injury risk and physical performance. The purpose of the study is to understand the effect of the menstrual cycle and hormonal contraceptive use on bone and calcium metabolism, and musculoskeletal health and performance. A total of 5 cohorts of physically active women (aged 18-40 years) will be recruited to participate: eumenorrheic, nonhormonal contraceptive users (n=20); combined oral contraceptive pill (COCP) users (n=20); hormonal implant users (n=20); hormonal intrauterine system users (n=20); and hormonal injection users (n=20). Participants must have been using the COCP and implant for at least 1 year and the intrauterine system and injection for at least 2 years. First-void urine samples and fasted blood samples will be collected for biochemical analysis of calcium and bone metabolism, hormones, and metabolic markers. Knee extensor and flexor strength will be measured using an isometric dynamometer, and lower limb tendon and stiffness, tone, and elasticity will be measured using a Myoton device. Functional movement will be assessed using a single-leg drop to assess the frontal plane projection angle and the qualitative assessment of single leg loading. Bone density and macro- and microstructure will be measured using ultrasound, dual-energy x-ray absorptiometry, and high-resolution peripheral quantitative computed tomography. Skeletal material properties will be estimated from reference point indentation, performed on the flat surface of the medial tibia diaphysis. Body composition will be assessed by dual-energy x-ray absorptiometry. The differences in outcome measures between the hormonal contraceptive groups will be analyzed in a one-way between-group analysis of covariance. Within the eumenorrheic group, the influence of the menstrual cycle on outcome measures will be assessed using a linear mixed effects model. Within the COCP group, differences across 2 time points will be analyzed using the paired-samples 2-tailed t test. The research was funded in January 2020, and data collection started in January 2022, with a projected data collection completion date of August 2024. The number of participants who have consented at the point of manuscript submission is 66. It is expected that all data analysis will be completed and results published by the end of 2024. Understanding the effects of the menstrual cycle and hormonal contraception on musculoskeletal health and performance will inform contraceptive choices for physically active women to manage injury risk. ClinicalTrials.gov NCT05587920; https://classic.clinicaltrials.gov/ct2/show/NCT05587920. DERR1-10.2196/50542.

Comparison of hip abductors’ strength and 2D frontal plane projection angle during two functional tests, in females with and without patellofemoral pain

IntroductionHip abductors eccentrically maintain leg alignment during weight-bearing exercises. Hip abductors' weakness may be a significant aspect to take into account in the conservative management of Patellofemoral Pain (PFP). The aim of the study was to compare hip abductors’ eccentric strength and hip kinematics during two functional tests, in female subjects with PFP. MethodIn a cross-sectional study using convenience sampling, isokinetic Isokinetic unilateral eccentric hip abductors’ mean peak moment (MPM) at 60°/sec and Frontal Plane Projection Angle (FPPA) during Single-Leg Squat (SLS) and Single-Leg Landing (SLL) tests, were assessed in 17 asymptomatic individuals and 19 PFP patients. ResultsThe FPPA during SLL (p = 0.771) and SLS (p = 0.587), as well as the isokinetic values of the eccentric adductors' MPM at 60°/sec (p = 0.769), did not differ between the groups. The isokinetic values and the SLL (r = −0.009 to p = 0.970) and SLS (r = −0.002 to p = 0.993) in the PFP group and the control group (SLS r = 0.160 to p = 0.540) did not show any statistically significant relationships. ConclusionPFP and asymmetry in hip abduction strength are not substantially correlated. It is necessary to conduct more research to comprehend how these impairments interact and to find out if there are patient subgroups with PFP.

Intrinsic Geometries of the Simple Group of Order 168

It is well known that, up to isomorphism, there is a unique simple group of order 168. The projective special linear groups PSL(2, 7) and PSL(3, 2) are concrete examples of such groups, and are therefore isomorphic. This isomorphism is not obvious, but it has been clarified from several viewpoints. Here we use purely group-theoretic methods—the Sylow theorems, Poincaré’s theorem, and the orbit-stabilizer theorem—to show that any simple group of order 168 has a natural action on the projective line over F 7 , so it is isomorphic to PSL(2, 7). We discuss an analogous action on the projective plane over F 2 described by Smith and Tabachnikova. Both geometries are constructed using conjugacy classes of maximal subgroups. This gives another proof of the uniqueness theorem and may dispel some of the air of mystery that often surrounds the isomorphism between PSL(2, 7) and PSL(3, 2).

Adaptive projection plane and reference point strategy for multi-objective particle swarm optimization

Achieving a balance between convergence and diversity and their mutual enhancement is a complex task in the process of algorithm improvement. This is crucial because it is directly related to the effectiveness of the algorithm in obtaining accurate and uniformly distributed Pareto frontiers. Although significant progress has been made in particle swarm algorithms, exploring new approaches is necessary. In this paper, we construct a projection plane (projection line in 2D) based on the extreme values of the non-dominated solutions, select a set of uniform reference points on the projection plane, and then project the non-dominated solutions onto the constructed projection plane to form projection points. The reference points and projection points on the projection plane are thus utilized to guide the updating of the population as well as the maintenance of the external archive, a strategy that enhances the algorithm's global exploration and local exploitation capabilities. Secondly, we aggregate the target values of particles into a single scalar value and combine the idea of particle fusion to design a scheme for the particle selection of individual optimal particles. This paper further improves the algorithm's overall performance by using the information between populations to select individual optimal particles. Lastly, it is evaluated against a number of multi-objective algorithms that are currently in use and perform well on 22 test problems. The findings demonstrate that the algorithm this paper proposes performs better when solving multi-objective problems.

Staircase patterns in Hirzebruch surfaces

The ellipsoidal capacity function of a symplectic four manifold X measures how much the form on X must be dilated in order for it to admit an embedded ellipsoid of eccentricity z . In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, X is said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfaces H_{b} formed by blowing up the projective plane with weight b . We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parameters b that are blocked, i.e. have no staircase, and an uncountable number of other values of b that do admit staircases. The remaining b -values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long as b is not one of these special rational values, any staircase in H_{b} has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations on H_{b} is echoed in the structure of the set of blocked b -intervals.

Algebraic Invariants of Codes on Weighted Projective Planes

Algebraic Invariants of Codes on Weighted Projective Planes

Sets of mutually orthogoval projective and affine planes

Sets of mutually orthogoval projective and affine planes

Examples of harmonic foliated currents and singular Levi-flats on the projective plane

We provide examples of foliations on the complex projective plane CP2 carrying positive foliated harmonic currents whose supports coincide with singular Levi-flats which, in turn, can be chosen real-analytic (but non-algebraic) or merely continuous with fractal transverse nature. Furthermore, non-trivial examples as above can already be found among foliations of degree 2 and 3. In addition, the space of positive foliated harmonic currents for these foliations is fully characterized and it contains a unique harmonic (non-closed) current supported on the Levi-flat in question. Finally, we also provide examples of foliations carrying diffuse positive foliated closed currents related to a theorem due to Brunella as well as a general criterion for the existence of singular real analytic Levi-flats for Riccati foliations.

An engineering applicable method for multiaxial fatigue under proportional and non-proportional loads based on the octahedral plane projection

Present multiaxial fatigue models have achieved satisfactory performance in predicting multiaxial fatigue life of different loading paths; however, many of them require critical plane calculations and additional material constants. In this study, a novel multiaxial fatigue life prediction model based on octahedral shear strain energy is proposed. In the proposed model, the maximum octahedral shear strain energy serves as the damage parameter without requiring any additional material parameters. All material constants in this model can be derived from material strength constants and uniaxial fatigue constants, greatly simplifying the life prediction process. Moreover, based on the actual physical phenomenon of non-proportional additional hardening, the multiaxial non-proportional correction factor is incorporated into the plastic stage of the life equation, aligning more closely with fundamental principles in physics. Then, by utilizing the experimental data from the literature for four materials, a comparison is made between the prediction results of the novel model and those of four commonly employed models. The findings demonstrate that the new model exhibits superior simplicity and accuracy. Subsequently, through life prediction outcomes for DZ411 material across various paths, the stability and precision of the octahedral energy model are substantiated, rendering it suitable for practical engineering applications.

Correlation between frontal plane projection angle and pain, dynamic balance and function in patients with patellofemoral pain syndrome

Correlation between frontal plane projection angle and pain, dynamic balance and function in patients with patellofemoral pain syndrome

Sinusoidal oscillator parametrically forced to robust hyperchaotic states: the lumpkin case

The objective of this paper is to showcase the capability of the conventional circuit structure known as the Lumpkin oscillator, widely employed in practical applications, to operate in robust chaotic or hyperchaotic steady states. Through numerical analysis, we demonstrate that the generated signals exhibit a significant level of unpredictability and randomness, as evidenced by positive Lyapunov exponents, approximate entropy, recurrence plots, and other indicators of complex dynamics. We establish the structural stability of strange attractors through design and practical construction of a flow-equivalent fourth-order chaotic oscillator, followed by experimental measurements. The oscilloscope screenshots captured align well with the plane projections of the approximate solutions derived from the underlying mathematical models.

Precyzyjne wyznaczanie geodezyjne w budowie obiektów inżynierskich z orientacją w określonym kierunku

In the construction practice, there are engineering structures, where some complex requirements in the process of their building have to be meet, e.g., their positioning, geographical orientation and geometric determination. The current publication examines the specificity of geodetic activities, regarding the orientating of the main construction axes of the foundation and the supporting structures of the terrestrial telecommunication antennas in a predetermined direction, for example, in the parallel direction to the geographical meridian. The focus is on the assessment of the accuracy and the choice of appropriate methods and equipment in order to create the required geodetic network, as well as to perform accurate setting out works. The combination of demands, namely for a specific geographic orientation and precise relative accuracy of the setting elements, implies a suitable transition from the ellipsoidal geodetic coordinates to the projection plane of the setting out project. In this connection, the influence of the meridian convergence on the accuracy of the planning setting out works is examined. A mathematical approach is proposed to assess the need of elimination of this factor, depending on the value of the constructional tolerance and the geographic location of the site.The necessary precision for determining the geodetic ellipsoidal coordinates by GNSS (Global Navigation Satellite System) measurements as well as the required accuracy of the planning and elevation setting out works are shown.Using different geodetic methods and tools, a preliminary accuracy assessment approach and the choice of the appropriate coordinate projection and height reference system is also proposed. The real data, which were obtained in different Earth’s locations by the explained methodology, are presented. The established geodetic networks provide orientation of the main construction axes of the bases of the terrestrial communication antennas in the parallel direction to the geodetic meridian with an azimuth standard error - МА = ± 2÷5' (minutеs) and allowable error in distances and eleventions of the setting out elements, respectively - MdS ≤ ±3 mm, Mdh ≤ ± 2 mm.

SC-AOF: A Sliding Camera and Asymmetric Optical-Flow-Based Blending Method for Image Stitching.

Parallax processing and structure preservation have long been important and challenging tasks in image stitching. In this paper, an image stitching method based on sliding camera to eliminate perspective deformation and asymmetric optical flow to solve parallax is proposed. By maintaining the viewpoint of two input images in the mosaic non-overlapping area and creating a virtual camera by interpolation in the overlapping area, the viewpoint is gradually transformed from one to another so as to complete the smooth transition of the two image viewpoints and reduce perspective deformation. Two coarsely aligned warped images are generated with the help of a global projection plane. After that, the optical flow propagation and gradient descent method are used to quickly calculate the bidirectional asymmetric optical flow between the two warped images, and the optical-flow-based method is used to further align the two warped images to reduce parallax. In the image blending, the softmax function and registration error are used to adjust the width of the blending area, further eliminating ghosting and reducing parallax. Finally, by comparing our method with APAP, AANAP, SPHP, SPW, TFT, and REW, it has been proven that our method can not only effectively solve perspective deformation, but also gives more natural transitions between images. At the same time, our method can robustly reduce local misalignment in various scenarios, with higher structural similarity index. A scoring method combining subjective and objective evaluations of perspective deformation, local alignment and runtime is defined and used to rate all methods, where our method ranks first.

A 3D local feature description algorithm based on point distribution

It is well known that the local feature descriptor is playing an important role in 3D image recognition. But the current local feature descriptors have poor ability to describe target features in situations such as noise interference, be obscured, and changes in data resolution. On the basis of point distribution, we propose a new 3D local feature descriptor, the Point Distribution Signature Histogram(PDSH), which is relatively simple and easy to implement and has a fast identification speed by simplifying Local Reference Frame(LRF). We also rotate the local surface of the point cloud from multiple angles and project them onto three coordinate planes of the LRF. The projection planes are segmented at the same angle, and the distance information of the edge points and the count of projected points in each segment are statistically accumulated. Finally, the sub-feature vectors of the three coordinate planes are concatenated to obtain the feature descriptor of the key points. Experimental results show that our PDSH descriptor achieves a descriptiveness of 98 % on the pollution-free point cloud data set and better performance than the classical descriptors.