In magnetically confined fusion plasmas, the role played by zonal $E \times B$ flow shear layers in the suppression of turbulent transport is relatively well understood. However, less is understood about the role played by the weak shear regions that arise in the non-monotonic radial electric field profiles often associated with these shear layers. In electrostatic simulations from the global total- $f$ gyrokinetic particle-in-cell code XGC, we demonstrate how shearless regions with non-zero flow curvature form zonal ‘jets’ that, in conjunction with neighbouring regions of shear, can act as robust barriers to particle transport and turbulence spreading. By isolating quasi-coherent fluctuations radially localised to the zonal jets, we construct a map model for the Lagrangian dynamics of gyrokinetic test particles in the presence of drift waves. We identify the presence of shearless invariant tori in this model and verify that these tori act as partial phase-space transport barriers in the simulations. We also demonstrate how avalanches impinging on these shearless tori cause eddy detachment events that form ‘cold/warm core ring’ structures analogous to those found in oceanic jets, facilitating transport across the barriers without destroying them completely. We discuss how shearless tori may generically arise from tertiary instabilities or other types of discrete eigenmodes, suggesting their potential relevance to broader classes of turbulent fluctuations.

Abstract Using metric techniques introduced by Berndtsson, we show a result on the constancy of families dominated by a constant variety and, on the opposite side, results on the strong non-isotriviality of certain families of surfaces with positive index and, in arbitrary dimension, in terms of the complex conjugate of a suitable representative of the Kodaira–Spencer class. We also give a metric interpretation of the liftability of relative volume forms.

Abstract Introduction Penile fracture (PF) is a urological emergency characterized by rupture of the tunica albuginea of the corpora cavernosa, occurring either in isolation or alongside a urethral tear in up to 15% of cases. Early surgical intervention is crucial to restore normal function. However, long-term outcomes after PF repair-including erectile function, penile deformity, and plaque formation-remain poorly defined. Additionally, for patients with concurrent urethral injury, the risk of urethral stricture disease after successful repair remains uncertain. This study aims to evaluate long-term urinary and sexual functional outcomes following PF, with or without urethral injury, using validated questionnaires. Objective This study aims to evaluate long-term urinary and sexual functional outcomes following PF, with or without urethral injury, using validated questionnaires. Methods All patients who have had a penile fracture repair done at a tertiary referral center were sent out a questionnaire including the International Index of Erectile Function- Erectile Function Domain (IIEF-EFD) as well as the International Prostate Symptoms Score. There were additional questions related to penile curvatures. This study was registered as an audit. Results To date, 52/220 men responded. The mean age at PF was 41 (12) years. Imaging to diagnose PF was utilized in 90% of patients with predominantly ultrasounds in 43 and MRI in 6 cases. 50% had a ventral longitudinal incision, 43% penoscrotal incision and 6% degloving incision. 9 cases had bilateral corporal injury whilst remaining had unilateral corporal fracture. A urethral injury was identified and repaired in 12 cases using 4-0 vicryl or 4-0 monocryl. Corporal tunical fracture was repaired with 0/0 PDS in 85% of cases and 2/0 PDS in the remainder. Mean duration post PF for questionnaire completion was 6.3 years SD=(4.9). The mean IIEF-EFD, and the IPSS score for patients were 25 (6) and 4 (6) respectively. In terms of the IIEF-EFD, there was no statistically significant difference in patients with urethral injury 23 (9) and those without 25 (6) p=44. IPSS mean score in patients without urethral injury was 4.2 (5.9) and those with urethral injury 3 (4) p=0.5. 7 of patients were taking PDE5i at time of filling questionnaires. 27% of patients complained of penile curvature (n=14). Of those a third of them stated they had only minimal curvatures and 1 patient is currently awaiting penile curvature correction surgery. No cases proceeded to have any form of urethral stricture surgery or penile implant surgery. Conclusions Timely PF repair with a urethral injury restores erectile function and does not predispose to urethral stricture disease. Approximately a quarter of patients after PF repair develop a form of penile curvature. Disclosure No

The forward and reverse knife technique is the core method used in black pottery carving, yet traditional carving tools are mostly handcrafted by artisans using improvised materials, often resulting in poor ergonomics and low efficiency. To address these issues, this study develops an improved carving tool aimed at enhancing both comfort and productivity. A pressure-based reverse engineering experiment was conducted to analyze the distribution of hand pressure during carving, providing quantitative data for ergonomic optimization. Based on the operational requirements of the forward and reverse knife techniques and the habitual postures of artisans, the handle’s curvature and grip form were refined to better fit the natural contours of the hand. The optimized handle effectively distributes and alleviates pressure on the palm and fingers during prolonged use, minimizing fatigue and reducing the risk of hand strain. Overall, the improved tool design not only enhances carving efficiency and operational stability but also contributes to the sustainable protection of artisans’ hand health, offering a practical reference for ergonomic tool innovation in traditional handicrafts.

From bees to blue whales, it has long been assumed that brain size scales with body size according to a simple log-linear relationship – with differences in the slope and intercept observed amongst different groups of animals. However, recent analyses in mammals contradict this view, revealing size dependency in the form of curvature in the brain and body mass relationship. Here, we use data from 4679 species across seven animal classes and spanning nearly 12 orders of magnitude to uncover near universal curvilinearity. We demonstrate that this body size dependence is a metaphenomenon emerging from a pattern of diminishing allometry within species with increasing body mass. This has fundamental implications for how we interpret macroevolutionary patterns – which can arise as a consequence of within-lineage dynamics. Our integration of inter- and intra-specific allometries reshapes perspectives on morphological evolution by providing a broader framework for understanding how microevolutionary within-species dynamics shape macroevolutionary phenomena.

Hodge decomposition of the Sobolev space H1 on a space form of nonpositive curvature

The paper considers the problem of dynamic longitudinal–transverse bending of an elastic rod with initial curvatures under short-term axial action. The motivation of the research is the current problems related to the analysis of the stability and strength of structural elements under shock loads, applicable, in particular, to the problems of launching aircraft, pile foundations and vehicles. The main attention is paid to two special cases: the effect of a short-term longitudinal impact of constant force and the application of an axial force that increases linearly over time. The problem statement specifically takes into account the fact that the time range of the load application is half the time of wave propagation along the length of the rod, which makes it possible to correctly simulate wave effects. Unlike classical approximate models, in which the propagation of a longitudinal wave is assumed to be instantaneous and the voltage is assumed to be constant along the rod, equations are used in this work taking into account the finite velocity of propagation of longitudinal waves. To numerically study the dynamics of small transverse deviations, eigenfunction decomposition is applied and a system of ordinary differential equations for Fourier amplitudes is solved. The influence of various shapes of the initial bend is analyzed in detail, as well as changes in the maximum oscillation amplitudes for various parameters of the model. A comparative analysis of the behavior of the rod for simple and complex initial forms of curvature is carried out, and the occurrence of irregularities close to those observed in physical experiments is shown. The results obtained confirm the importance of taking into account wave effects in short-term dynamic tasks and expand the scope of the developed models.

Abstract This work investigates the geometric and dynamical structure of a spacetime governed by the straight spinning string (SSS) metric. The analysis begins with a comprehensive study of the spacetime geometry, including computation of Christoffel symbols, Ricci tensor, and curvature forms in both coordinate and Cartan formalisms. Despite nontrivial global features such as frame-dragging and conical singularities, the spacetime is shown to be locally flat and Ricci-flat. The symmetries of geodesic motion are then explored via the Noether symmetry approach, yielding conserved quantities associated with the metric's isometries. A Hamiltonian formulation is subsequently developed on a four-dimensional configuration space, incorporating the effects of frame-dragging through canonical momenta and enabling a symplectic reduction. The resulting reduced Hamiltonian reveals hidden integrals linked to dynamical and geometric symmetries. The study proceeds to derive an effective potential for radial geodesic motion, elucidating the interplay between geometry, conserved charges, and singularities. Finally, symbolic and numerical analyses of the potential's behavior uncover a smooth profile near critical radii and offer insights into geodesic confinement and causal structure. Altogether, the results contribute to a deeper understanding of relativistic systems with angular-temporal couplings and topological defects.

 All calculations, numerical evaluations, and plots of the effective potential were performed using \texttt{Maple 2020} to ensure accuracy and clarity in visualization.

Understanding how different forms of supramolecular curvature arise during assembly is crucial to designing and tuning the microstructure of hierarchically self-assembled materials. Here, we show that in crystalline phases of mesogenic oligomers, the oligomer length is a critical parameter that determines the type of curvature (Gaussian or cylindrical) exhibited by the self-assembled structures. We use iterative exponential growth to synthesize monodisperse mesogenic oligomers ranging from dimer to octamer. By analyzing their phase behavior and microstructure, we elucidate how length-dependent thermodynamic and kinetic effects tune their hierarchical degree of ordering. The oligomers’ length-dependent crystalline order drives the formation of scrolled sheets in shorter oligomers and twisted ribbons in longer oligomers. These studies highlight how oligomer length interplays with mesogen geometry and crystalline packing to drive self-assembly, introducing oligomer length as a powerful design parameter toward tailored applications of mesogenic systems.

Abstract The article is devoted to studying steel I-beams under lateral-torsional buckling with operational geometric bow imperfections revealed during inspections of damaged elements of reconstructed buildings. The influence of imperfection in the form of initial curvature in the horizontal plane on the stability of structures is analyzed. An unresolved before part of the general problem of analyzing the condition of steel elements which is the subject of this article is to determine the permissible design values of steel I-beam curvature in the horizontal plane when restoring damaged parts of buildings. Then attention is focused on ensuring the stability and efficiency of further operation of the beams. The possibility of using bearing capacity resources, namely lateral bracing restraint with the help of structures attached to a steel beam, is considered. The proposed approaches are aimed at optimizing the use of materials and increasing savings by detailing the verification calculations of damaged steel beams.

Introduction. Idiopathic scoliosis (IS) is a common form of spinal curvature that occurs in children and adolescents. The analysis of the quality of life in such patients is an urgent topic. The PedsQL 4.0 questionnaire is widely used to determine changes in quality of life (QOL). The aim of the study was to determine changes in the quality of life in children with idiopathic scoliosis, depending on the used tactics of surgical treatment. Materials and methods. There were examined 88 patients aged 13–18 years, the average age was 15.1 ± 0.61 years, including 20 boys and 68 girls who underwent surgery from 2017 to 2024 for idiopathic scoliosis types I–IV according to L. Lenke score. The patients were divided into 3 groups. The 1st (control) group consisted of 29 children who underwent a single-stage dorsal correction of scoliosis, supplemented by G1 osteotomy according to F. Schwab. Group 2 consisted of 29 patients who underwent two-stage surgical treatment: at the first stage, preliminary mobilization of the anterior support column of the spine was performed, followed by pelvic traction; at the second stage, dorsal correction of spinal deformity was performed. Group 3 consisted of 30 patients who underwent surgery using a single-stage dorsal correction supplemented by an apical multilevel G2 osteotomy according to F. Schwab. The minimum follow-up period after surgery was 12 months. Results. The patients of group 1 initially showed high QOL scores on all scales of the questionnaire, while the children of group 2 had low QOL scores in the field of physical and role functioning. The parameters of social and emotional functioning in patients of groups 2 and 3 were similar. Changes in QOL in patients 12 months after surgical treatment turned out to be almost the same. At the same time, a significant improvement in QOL was revealed in patients of groups 2 and 3 compared with children of the control group. An analysis of patients based on baseline values of thoracic kyphosis revealed an improvement in QOL in patients of group 2 with hypokyphosis, which is associated with improved physical health and overall psychosocial functioning (p < 0.05). Patients in group 2 with normokyphosis showed marked improvement in the physical and psychosocial spheres, in contrast to children in group 3. In hyperkyphosis children, the improvement in QOL was more significant in patients of group 3 compared with patients of group 2, especially in the field of general psychosocial functioning (p < 0.02). Conclusion. The choice of surgical intervention tactics based on the severity of thoracic kyphosis has significantly improved the quality of life in patients after surgical treatment of idiopathic scoliosis. There was a significant increase in scores on the scales of emotional, social, and role functioning. The established patterns can be applied by orthopedists to optimize the choice of tactics for surgical treatment of idiopathic scoliosis in children.

Abstract Recent advances in cell biology and experimental techniques using reconstituted cell extracts have generated significant interest in understanding how geometry and topology influence active fluid dynamics. In this work, we present a comprehensive continuum theory and computational method to explore the dynamics of active nematic fluids on arbitrary surfaces without topological constraints. The fluid velocity and nematic order parameter are represented as the sections of the complex line bundle of a two-manifold. We introduce the Levi–Civita connection and surface curvature form within the framework of complex line bundles. By adopting this geometric approach, we introduce a gauge-invariant discretization method that preserves the continuous local-to-global theorems in differential geometry. We establish a nematic Laplacian on complex functions that can accommodate fractional topological charges through the covariant derivative on the complex nematic representation. We formulate advection of the nematic field based on a unifying definition of the Lie derivative, resulting in a stable geometric semi-Lagrangian (sL) discretization scheme for transport by the flow. In general, the proposed surface-based method offers an efficient and stable means to investigate the influence of local curvature and global topology on the two-dimensional hydrodynamics of active nematic systems.

Problem statement. Current methods for diagnosing musculoskeletal disorders, especially when assessing posture in the frontal and sagittal planes, are limited in analyzing dynamic conditions, such as walking or other forms of natural movement. Static methods do not reflect the actual behavior of the spine during movement, which reduces the accuracy of diagnosing pathologies such as scoliosis, pelvic asymmetry, and hyperlordosis. Additional difficulties arise when analyzing video due to camera shake, changes in viewing angles, and frame quality. Optical tracking systems are also subject to perspective distortion, which complicates the quantitative assessment of posture parameters. Objective. Development of an improved algorithm for measuring and classifying spinal deformities during movement. The main goal is to provide a stable coordinate system adaptive to patient movement, automatic and accurate determination of spinal markers, as well as quantitative analysis of curvature and asymmetry of the spinal column. Particular attention is paid to the correct processing of video data, extraction of key frames, compensation of jitter, and the use of neural network methods for classification of the deformation type. Results. A complex algorithm was developed that includes several interrelated stages. At the pre-processing stage, video material is standardized (up to MP4 format, 1080p), jitter compensation is performed using stabilization, and key frames are selected using computer vision methods based on convolutional and recurrent neural networks (CNN and RNN). To ensure the accuracy of the analysis, reduced compression is used, which allows maintaining the high image quality necessary for accurate assessment of textures and anatomical landmarks of the spine. Next, the camera is initialized and the coordinate axes are synchronized; if necessary, the video is matched with biomechanical motion signals. Deformation measurements are performed based on the YOLOv6 algorithm, with subsequent refinement of the results using CNN and LSTM models, which ensures accurate extraction of spine coordinate data under motion conditions. The post-processing stage includes noise filtration, smoothing and refinement of the spinal curvature boundaries. Then, the detected deformities are automatically classified, including the determination of the type and severity of such pathologies as scoliosis, kyphosis and lordosis. The final stage is the visualization of the results in the form of curvature and deviation graphs, as well as the generation of a structured report with a quantitative assessment and recommendations for clinical use. Practical significance. The proposed algorithm opens up new prospects in the diagnosis of posture disorders, providing the ability to accurately measure in conditions of natural patient movement without the need for expensive equipment or strictly controlled conditions. Its use allows for an objective assessment of the effectiveness of various orthopedic devices, such as corsets, insoles and exoskeletons, as well as monitoring the progression of spinal pathologies at early stages. In addition, the algorithm can be adapted to individualize the rehabilitation process depending on the dynamics of changes in spinal curvature, which makes it a valuable tool in clinical practice. Due to its versatility and efficiency, the development is easily integrated into mobile and telemedicine solutions, which significantly expands the scope of its application. In total, this approach represents an important step towards the creation of intelligent diagnostic systems of the new generation, combining high accuracy, adaptability and accessibility.to biological tissues associated with the specifics of MIS interventions and allow to improve the quality and safety of surgical care.

In this paper, a type of Riemannian manifold, namely generalized pseudo conformally symmetric manifold is studied. Several geometric properties of such spaces are studied. By imposing different restrictions on the conformal curvature tensor, we have obtained several properties. If the conformal curvature tensor is harmonic, then the form of the scalar curvature is obtained. Also, the relations among the 1-forms under various conditions are obtained.

In the language of L ∞ L^\infty -modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space ( M , m ) (M,\mathfrak {m}) carrying a quasi-regular, strongly local Dirichlet form \usefont {U}{BOONDOX-cal}{m}{n}E {\text {\usefont {U}{BOONDOX-cal}{m}{n}E}} . Furthermore, we develop a second order calculus if ( M , \usefont {U}{BOONDOX-cal}{m}{n}E , m ) (M,{\text {\usefont {U}{BOONDOX-cal}{m}{n}E}},\mathfrak {m}) is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented.

We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan’s structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü–Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed.

Curvature form of Raychaudhuri equation and its consequences: A geometric approach

This article deals with a new roulette of special curves formed by a circle rolling along a line which are given the name of Laithoid curves. The new curve is a new special form of cycloid produced by rolling a circle along a horizontal line of 4 times the rolling circle's radius. It is the locus traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle), as it rolls along a straight line. In this paper, a set of 6 forms of new curvatures within two groups are produced depending on a rolling circle on the Laithoid's curve, and their geometrical and algebra proportions are graphically formed. Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enable comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.

In the first part we extend the construction of the smooth normal-crossing divisors compactification of projectivized strata of abelian differentials given by Bainbridge, Chen, Gendron, Grushevsky and Möller to the case of k-differentials. Since the generalized construction is closely related to the original one, we mainly survey their results and justify the details that need to be adapted in the more general context.In the second part we show that the flat area provides a canonical hermitian metric on the tautological bundle over the projectivized strata of finite area k-differentials whose curvature form represents the first Chern class. This result is useful in order to apply Chern–Weil theory tools. It has already been used as an assumption in the work of Sauvaget for abelian differentials and is also used in a paper of Chen, Möller and Sauvaget for quadratic differentials.