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In this paper, we review a recently offered notion of quasi-static evolution of the axial self-gravitating structures at large scales and the criterium to characterize the corresponding evolutionary aspects under the influence of strong curvature regimes. In doing so, we examine the axial source's dynamic and quasi-static behavior within the parameters of various modified gravity theories. We address the formalism of these notions and their possible implications in studying the dissipative and anisotropic configuration. We initiate by considering higher-order curvature gravity. The Palatini formalism of f (R) gravity is also taken into consideration to analyze the behavior of the kinematical as well as the dynamical variables of the proposed problem. The set of invariant velocities is defined to comprehend the concept of quasi-static approximation that enhances the stability of the system in contrast to the dynamic mode. It is identified that vorticity and distinct versions of the structure scalars Y I , Y II and Y KL play an important role in revealing the significant effects of a fluid's anisotropy. As another example of evolution, we check the influence of Palatini-based factors on the shearing motion of the object. A comparison-based study of the physical nature of distinct curvature factors on the propagation of the axial source is exhibited. This provides the intriguing platform to grasp the notion of quasi-static evolution together with the distinct curvature factors at current time scenario. The importance of slowly evolving axially symmetric regimes will be addressed through the distinct modified gravitational context. Finally, we share a list of queries that, we believe, deserve to be addressed in the near future. [ABSTRACT FROM AUTHOR]

A Friedmann–Lemaitre–Robertson–Walker space–time model with all curvatures k = 0 ,   ± 1 is explored in f (R , T) gravity, where R is the Ricci scalar, and T is the trace of the energy–momentum tensor. The solutions are obtained via the parametrization of the scale factor that leads to a model transiting from a decelerated universe to an accelerating one. The physical features of the model are discussed and analyzed in detail. The study shows that f (R , T) gravity can be a good alternative to the hypothetical candidates of dark energy to describe the present accelerating expansion of the universe. [ABSTRACT FROM AUTHOR]

This study aims to elucidate the longitudinal refractive and ocular biometric alterations in preschool children with high hyperopia who underwent early interventions. We conducted a retrospective analysis of preschool children diagnosed with high hyperopia at Tianjin Medical University Eye Hospital between 2011 and 2023. Inclusion criteria required an initial examination with cycloplegic refraction, bilateral spherical equivalent power (SE) ≥ +5.00D with a difference <1.00D, a minimum two-year follow-up, and at least three ocular biometric measurements. The annual axial growth rate evaluated emmetropization in highly hyperopic children. We applied Restricted Cubic Spline (RCS) models to explore potential nonlinear relationships between age and spherical equivalent, axial length, corneal curvature, and axial length-to-corneal curvature ratio. Additionally, Mixed-effects models were employed to investigate factors associated with changes in refractive error and axial length. The study enrolled 60 eligible subjects, with a median initial diagnosis age of 3.5 years (IQR, 2.8-4.9 years) and a median last visit age of 9.3 years (IQR, 8.1-10.8 years). The average follow-up duration was 5.7 years. RCS analysis revealed notable nonlinear changes in spherical equivalent power, axial length, and axial length-to-corneal curvature ratio, although corneal curvature displayed no statistically significant nonlinear trend. Factors affecting SE changes included the presence of strabismus, the use of cycloplegia, baseline SE, and age. Conversely, changes in axial length solely correlated with baseline axial length and age. Highly hyperopic preschool children undergoing early intervention display a marked emmetropization tendency, though most still remain moderately to highly hyperopic, with the progression of refractive changes showing non-uniform patterns with respect to age. [ABSTRACT FROM AUTHOR]

This study focuses on the lateral safety of autonomous vehicles. The aim is to provide greater safety margins for the motion controller under hazardous conditions by improving the planning methods. The proposed lateral planning method employs a hierarchical structure. First, a graph search method that considers road curvature is presented. This method benefits from the design of jump nodes and can generate a safe corridor in real-time by backtracking. Next, a concise and effective optimization problem is constructed in the Frenet frame. It is designed to optimize the path by incorporating the peak, integral, and differential of curvature into the objective function. By solving this problem using CasADi, the path within the safe corridor is obtained. Finally, the proposed planning method is combined with a model predictive motion controller and tested through simulation and experimentation on a real vehicle platform. This planning method has good scalability, adaptability, and real-time performance. Simulation and experimental results show that this method can help avoid exceeding lateral limits of road surface friction, improve path-following accuracy, and enhance the safety of autonomous vehicles in dangerous conditions. [ABSTRACT FROM AUTHOR]

When representing high-level stimuli, such as faces and animals, we tend to emphasize salient features—such as a face's prominent cheekbones or a bird's pointed beak. Such mental caricaturing leaves traces in memory, which exaggerates these distinctive qualities. How broadly does this phenomenon extend? Here, in six experiments (N = 700 adults), we explored how memory automatically caricatures basic units of visual processing—simple geometric shapes—even without task-related demands to do so. Participants saw a novel shape and then immediately adjusted a copy of that shape to match what they had seen. Surprisingly, participants reconstructed shapes in exaggerated form, amplifying curvature, enlarging salient parts, and so on. Follow-up experiments generalized this bias to new parameters, ruled out strategic responding, and amplified the effects in serial transmission. Thus, even the most basic stimuli we encounter are remembered as caricatures of themselves. [ABSTRACT FROM AUTHOR]

In this paper, we give a lot of new non-projectively flat Finsler warped product metrics with zero flag curvature. We also manufacture new generalized Douglas–Weyl (GDW) metrics which are nontrivial in the sense that these metrics are neither Douglas metrics nor Weyl metrics via hyperbolic spaces. [ABSTRACT FROM AUTHOR]

If the small spheres of a Finslerian projective metric are quadrics, then it is a Riemannian projective metric of constant curvature. [ABSTRACT FROM AUTHOR]

Research suggests structural and connectivity abnormalities in patients with treatment-resistant schizophrenia (TRS) compared to first-line responders and healthy-controls. However, measures of these abnormalities are often influenced by external factors like nicotine and antipsychotics, limiting their clinical utility. Intrinsic-cortical-curvature (ICC) presents a millimetre-scale measure of brain gyrification, highly sensitive to schizophrenia differences, and associated with TRS-like traits in early stages of the disorder. Despite this evidence, ICC in TRS remains unexplored. This study investigates ICC as a marker for treatment resistance in TRS, alongside structural indices for comparison. We assessed ICC in anterior cingulate, dorsolateral prefrontal, temporal, and parietal cortices of 38 first-line responders, 30 clozapine-resistant TRS, 37 clozapine-responsive TRS, and 52 healthy-controls. For comparative purposes, Fold and Curvature indices were also analyzed. Adjusting for age, sex, nicotine-use, and chlorpromazine equivalence, principal findings indicate ICC elevations in the left hemisphere dorsolateral prefrontal (p < 0.001, η2partial = 0.142) and temporal cortices (LH p = 0.007, η2partial = 0.060; RH p = 0.011, η2partial = 0.076) of both TRS groups, and left anterior cingulate cortex of clozapine-resistant TRS (p = 0.026, η2partial = 0.065), compared to healthy-controls. Elevations that correlated with reduced cognition (p = 0.001) and negative symptomology (p < 0.034) in clozapine-resistant TRS. Fold and Curvature indices only detected group differences in the right parietal cortex, showing interactions with age, sex, and nicotine use. ICC showed interactions with age. ICC elevations were found among patients with TRS, and correlated with symptom severity. ICCs relative independence from sex, nicotine-use, and antipsychotics, may support ICC's potential as a viable marker for TRS, though age interactions should be considered. [ABSTRACT FROM AUTHOR]

The unique morphology of 2D van der Waals materials enables them to withstand large deformations and significant nonuniform strain, potentially inducing a strong flexoelectric effect. Despite the size‐dependent flexoelectric effect showing potential for modulating the optoelectronic performance of 2D van der Waals materials, it is far from being fully exploited owing to various challenges. Herein, the use of nanowires with different diameters as the bending media to fabricate 2D α‐In2Se3/β‐InSe heterojunctions with large curvatures of 0.1–1 µm−1 is proposed. The significant band alignment modulation in α‐In2Se3 resulting from the bending‐induced flexoelectric effect is verified through Kelvin probe force microscopy. The strain‐induced piezoelectric effect can be negated because of the weak vdW forces at the interface. The flexoelectric polarization in β‐InSe is screened via the accumulated electrons in the unilateral depleted heterojunction. Compared to the flat heterojunction, the curved heterojunction with an average curvature of 0.9 µm−1 shows 2.48‐fold and 7.62‐fold increases in open‐circuit voltage and zero‐biased responsivity, respectively. This study demonstrates the first successful modulation of photodetection in 2D heterojunctions by exploiting the flexoelectric effect, providing a new perspective for high‐performance 2D vdW optoelectronic devices. [ABSTRACT FROM AUTHOR]

Many factors contribute to bacterial membrane stabilization, including steric effects between lipids, membrane spontaneous curvature, and the difference in the number of neighboring molecules. This forum provides an overview of the physicochemical properties associated with membrane curvature and how this parameter can be tuned to design more effective antimicrobial peptides. [ABSTRACT FROM AUTHOR]

Separating overlapped nuclei is a significant challenge in histopathology image analysis. Recently published approaches have achieved promising overall performance on nuclei segmentation; however, their performance on separating overlapped nuclei is limited. To address this issue, we propose a novel multitask learning network with a bending loss regularizer to separate overlapped nuclei accurately. The newly proposed multitask learning architecture enhances generalization by learning shared representation from the following three tasks: instance segmentation, nuclei distance map prediction, and overlapped nuclei distance map prediction. The proposed bending loss defines high penalties to concave contour points with large curvatures, and small penalties are applied to convex contour points with small curvatures. Minimizing the bending loss avoids generating contours that encompass multiple nuclei. In addition, two new quantitative metrics, the Aggregated Jaccard Index of overlapped nuclei (AJIO) and the accuracy of overlapped nuclei (ACCO), have been designed to evaluate overlapped nuclei segmentation. We validate the proposed approach on the CoNSeP and MoNuSegv1 data sets using the following seven quantitative metrics: Aggregate Jaccard Index, Dice, Segmentation Quality, Recognition Quality, Panoptic Quality, AJIO, and ACCO. Extensive experiments demonstrate that the proposed Bend-Net outperforms eight state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

We show that the mean curvature flow of generic closed surfaces in R 3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R 4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. [ABSTRACT FROM AUTHOR]

We extend the equivariant classification results of Escher and Searle for closed, simply connected, Riemannian n-manifolds with non-negative sectional curvature admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove that any such manifold is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to three. [ABSTRACT FROM AUTHOR]

Introduction: To evaluate the intraocular differences in optical coherence tomography (OCT)-based macular curvature index (MCI) among children with anisomyopia and to investigate the relationship between MCI and the macular microvasculature. Methods: Fifty-two schoolchildren with anisometropia > 2.00 D were enrolled and underwent comprehensive examinations including cycloplegic refraction, axial length (AL), and swept source OCT/OCT angiography. OCT-based MCIs were determined from horizontal and vertical B-scans by a customized curve fitting model in MATLAB R2022 at 1-mm-, 3-mm-, and 6-mm-diameter circles at fovea. Characteristics and topographic variation of MCI was analyzed, and the relationships with microvascularity and its associated factors were investigated. Results: MCI achieved high reliability and repeatability. There were overall larger MCIs in the more myopic eyes than the less myopic eyes in 1-mm-, 3-mm-, and 6-mm-diameter circles at fovea (all p < 0.001). For the topographic variation, horizontal MCI was significantly greater than vertical MCI (all p < 0.001), and was the largest in 6-mm circle, followed by 3-mm and 1-mm circles. Stronger correlation of horizontal MCI with myopic severity than vertical MCI was found. Partial Pearson's correlation found MCI was negatively associated with deep capillary plexus (DCP) vessel density (p = 0.016). Eyes with a higher MCI in a 6-mm circle were more likely to have longer AL (p < 0.001), lower DCP vessel density (p = 0.037), and thinner choroidal thickness (ChT) (p = 0.045). Conclusion: Larger MCI was found in the more myopic eyes of children with anisomyopia and was significantly associated with smaller DCP density, suggesting that MCI was an important indicator of myopia-related retinal microvascularity change, and it could be a valuable metric for myopia assessment in children. [ABSTRACT FROM AUTHOR]

To investigate the system pressure drop distribution when conveying particle using different curvature radius pipes for the pneumatic conveying system, this paper measured the particle velocity distribution, particle-particle collision characteristics, collision energy loss, minimum pressure drop gas velocity, system pressure drop distribution, and power dissipation for R/D = 3.75, R/D = 5, and R/D = 6.25 pipes. Subsequently, the artificial neural network technique is used to predict the pressure drop of the pneumatic conveying system. It is found that the pressure drop of the system is lower when using the pipe with R/D = 6.25 for conveying particles. Compared to the pipe with R/D = 3.75, the reduction in power dissipation is 3.18 and 5.27% for conveying pellets when using R/D = 5 and R/D = 6.25 pipes, respectively. In addition, the energy loss of the system can be effectively reduced when using the pipe with R/D = 6.25 for conveying particles, which is more beneficial for the particles move in the pipe. The pressure drop model built with artificial neural network can predict the pressure drop value of the system more accurately within ±1.5%. [ABSTRACT FROM AUTHOR]

A theorem of Llarull says that if a smooth metric g$g$ on the n$n$‐sphere Sn$\mathbb {S}^n$ is bounded below by the standard round metric and the scalar curvature Rg$R_g$ of g$g$ is bounded below by n(n−1)$n (n - 1)$, then the metric g$g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound Rg⩾n(n−1)$R_g \geqslant n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature Rg$R_g$. We utilize two methods: spinor and spacetime harmonic function. [ABSTRACT FROM AUTHOR]

The aesthetics of abstract shapes — shapes devoid of meaning or familiarity — offer an intriguing subject for study, as it can offer insights into how we perceive and appreciate visual stimuli, shedding light on the underlying mechanisms of visual cognition and the nature of artistic experience. This research investigates the impact of contour type (angular versus rounded edges) and complexity (number of vertices) on aesthetic preferences, including their potential interaction. Additionally, we explored the influence of movement as an aesthetic variable, given its potential to enhance complexity, though the relationship between movement and complexity remains unexplored. Our findings indicate that both contour type and complexity significantly influence preferences, with shapes featuring curved contours and fewer vertices being favoured. This highlights the aesthetic appeal of curvature and simplicity. Contrary to expectations, movement did not have a noticeable effect on aesthetic judgements. While no overall interaction between contour type and complexity was found, this lack of interaction was obscured by significant individual differences. Specifically, within individuals, strong interactions between contour type and complexity were observed. It appears that these individual differences are due more to the varying emphasis (dominance) placed on each variable rather than a difference in the preference for specific characteristics. Future research should further analyse these individual differences to understand the nuanced dynamics of aesthetic preferences. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid inexact regularized Newton and negative curvature method for solving unconstrained nonconvex problems. The descent direction is chosen based on different conditions, either the negative curvature or the inexact regularized direction. In addition, to minimize computational costs while obtaining the negative curvature, we employ a dimensionality reduction strategy to verify if the Hessian matrix exhibits negative curvatures within a three-dimensional subspace. We show that the proposed method can achieve the best-known global iteration complexity if the Hessian of the objective function is Lipschitz continuous on a certain compact set. Two simplified methods for nonconvex and strongly convex problems are analyzed as specific instances of the proposed method. We show that under the local error bound assumption with respect to the gradient, the distance between iterations generated by our proposed method and the local solution set converges to 0 at a superlinear rate. Additionally, for strongly convex problems, the quadratic convergence rate can be achieved. Extensive numerical experiments show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Determining the edge and horizontal position of geologic structures is one of the fundamental steps in interpreting potential field data. Several filters have been introduced that use the concept of curvature to determine the edge of potential field data. However, these filters have advantages and disadvantages in detecting causative sources. Therefore, it seems necessary to introduce more efficient approaches. In this work, the most positive and most negative curvatures of gravity field data were analyzed, and a more efficient filter was introduced and applied that uses the concept of curvature and its combination to delineate the edges of geological structures and buried sources. The proposed method, called the hybrid positive and negative curvature (PNH) approach, combines the most positive and most negative curvatures into one curvature by fitting the formula and weighted summation. The proposed strategy takes advantage of both positive and negative curvatures to improve the edge detection of gravity field data. To this end, the performance of the PNH procedure was investigated considering different density assumptions (positive, negative, and positive-negative) for the relatively imposed synthetic gravity model resulting from buried prisms. The results obtained on synthetic models with and without noise show that the PNH procedure can detect the horizontal boundaries of buried structures relatively well. Of course, due to the use of directional derivatives in the filter of the hybrid positive and negative curvature approach, it seems very necessary to use noise-reducing filters before applying edge detection methods. Moreover, conventional filters such as the second vertical derivative (SVD) and the tilt angle (TDR) were used to compare the performance of the hybrid positive and negative curvature filter on the synthetic model. However, the obtained results show that the second vertical derivative and the tilt angle do not have the required capability to determine the edge of the synthetic model. In the following, the quality of the most positive and most negative curvatures filter and the hybrid positive and negative curvature were investigated using real data from a gold mine in the Witwatersrand area (South Africa) and also gravity data from the Aji-chai salt dome, East Azerbaijan province (Iran) and then using WGM-2012 derived gravity data belonging to the Marian trench area. Due to the sensitivity of the filters to noise, the upward continuation filter was applied before determining the edge of the buried structures. The edge maps from the Witwatersrand area and the data from the Aji-chai salt dome obtained using the hybrid positive and negative curvature determination method, demonstrate acceptable accuracy of this filter in determining the edge and representing the horizontal position of various geological structures. By using the PNH filter, the lateral boundaries of the main structures and other subsurface sources are well detected. Of course, due to the noise sensitivity of this filter, which is due to the use of secondorder gravity derivatives, good quality data without noise must be used. Therefore, it is suggested that noise attenuate filters, such as upward continuation method, must be used prior to creating the maps to determine the edge. Therefore, the PNH edge detection method can be reliably used for qualitative interpretation of gravity field data. [ABSTRACT FROM AUTHOR]

For fixed positive integer n, p ∈ [ 0 , 1) , a ∈ (0 , 1) , we prove that if a function g : S n - 1 → R is sufficiently close to 1, in the C a sense, then there exists a unique convex body K whose L p curvature function equals g. This was previously established for n = 3 , p = 0 by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if p = 0 and n = 4 or n ≤ 3 and p ∈ [ 0 , 1) , and the L p curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies λ - 1 ≤ g ≤ λ , for some λ > 1 , then max x ∈ S n - 1 h K (x) ≤ C (p , λ) , for some constant C (p , λ) > 0 that depends only on p and λ . This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the L p surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the L p -Minkowksi problem, for - n < p < 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of fully nonlinear equations on Riemannian manifolds with negative curvature which naturally arise in conformal geometry. Moreover, we prove the a priori estimates for solutions to these equations and establish the existence results. Our results can be viewed as an extension of previous results given by Gursky–Viaclovsky and Li–Sheng. [ABSTRACT FROM AUTHOR]

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory. [ABSTRACT FROM AUTHOR]

We derive two types of sharp Moser-Trudinger inequalities on complete, simply connected, two-dimensional Riemannian manifolds whose sectional curvatures K satisfy the bounds −b² ≤ K ≤ −a² < 0. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to define the concept of metallic deformation through a relation between the metallic structure and paracontact structure on an almost paracontact para-Norden manifold. A Riemannian connection is obtained on a metallically deformed para-Sasaki-like para-Norden manifold. A φ-connection is obtained via the Riemannian connection on a metallically deformed para-Sasaki-like para-Norden manifold. The curvature tensors, Ricci tensors, scalar curvatures, and ∗-scalar curvatures are investigated with respect to the Riemannian connection and the φ-connection. Finally, an example is given of a metallically deformed 3-dimensional para-Sasaki-like para-Norden manifold. [ABSTRACT FROM AUTHOR]

In the realm of the four-dimensional Minkowski space L4, the focus is on hypersurfaces classified as right conoids and defined by light-like axes. Matrices associated with the fundamental form, Gauss map, and shape operator, all specifically tailored for these hypersurfaces, are currently undergoing computation. The intrinsic curvatures of these hypersurfaces are determined using the Cayley-Hamilton theorem. The conditions of minimality are addressed by the analysis. The Laplace-Beltrami operator for such hypersurfaces is computed, accompanied by illustrative examples aimed at fostering a more profound understanding of the involved mathematical principles. Additionally, scrutiny is applied to the umbilical condition, and the introduction of the Willmore functional for these hypersurfaces is presented. [ABSTRACT FROM AUTHOR]

Retroreflectors are an important optical component, but current retroreflector structures and manufacturing processes are relatively complex. This paper proposes a rapid, low-cost, large-area method for fabricating retroreflectors based on microlens arrays. Tunable microlens arrays with adjustable curvature, fill factor, and sizes were prepared using photolithography and thermal reflow techniques. Subsequently, a two-step nanoimprinting process was used to create a flexible reverse mold and transfer the structure onto the desired substrate. The microlens arrays, with a diameter of 30 μm, a period of 33 μm, a curvature radius ranging from 15.5 to 18.8 μm, and a fill factor ranging from 75.1% to 88.8%, were fabricated this way. In addition, the method also fabricated microlens arrays with diameters ranging from 10 to 80 μm. Retroreflectors were made by sputtering a layer of silver on the MLAs as a reflecting layer, and tests showed that the microlens-based retroreflector exhibited superior retroreflective performance with a wide-angle response of ±75°. Microlens-based retroreflectors have the advantages of simple operation and controllable profiles. The fabrication method in this paper is suitable for large-scale production, providing a new approach to retroreflector design. [ABSTRACT FROM AUTHOR]

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovský and Polácik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity. [ABSTRACT FROM AUTHOR]

In the present study, we consider a generalized Weyl conformal curvature tensor on para-Kenmotsu manifolds (briefly, PK-manifolds). First we describe certain vanishing properties of generalized Weyl conformal curvature tensor (brifly, GWC-curvature tensor) on a PK-manifold. Later, we study generalized Weyl conformally semi-symmetric PK-manifold that turns out to an Einstein manifold. Among others, it has been shown that the generalized Weyl conformally Φ-symmetric PK-manifold is of constant curvature or dr(Ψ) = 0. [ABSTRACT FROM AUTHOR]

The hexachordal theorem is an intriguing combinatorial property of the sets in Z=12Z, discovered and popularized by the musicologist Milton Babbitt (1916-2011). It has been given several explanations and partial generalizations. Here we enhance how this phenomenon can be understood by giving both a geometrical and a probabilistic perspective. [ABSTRACT FROM AUTHOR]

Accurate arbor extraction is an important element of forest surveys. However, the presence of shrubs can interfere with the extraction of arbors. Addressing the issues of low accuracy and weak generalizability in existing Terrestrial Laser Scanning (TLS) arbor point clouds extraction methods, this study proposes a trunk axis fitting (TAF) method for arbor extraction. After separating the point cloud data by upper and lower, slicing, clustering, fitting circles, obtaining the main central axis, filtering by distance, etc. The canopy point clouds are merged with the extracted trunk point clouds to precisely separate arbors and shrubs. The advantage of the TAF method proposed in this study is that it is not affected by point cloud density or the degree of trunk curvature. This study focuses on a natural forest plot in Shangri-La City, Yunnan Province, and a plantation plot in Kunming City, using manually extracted data from a standardized dataset of samples to test the accuracy of the TAF method and validate the feasibility of the proposed method. The results showed that the TAF method proposed in this study has high extraction accuracy. It can effectively avoid the problem of trunk point cloud loss caused by tree growth curvature. The experimental accuracy for both plots reached over 99%. This study can provide certain technical support for arbor parameter extraction and scientific guidance for forest resource investigation and forest management decision-making. [ABSTRACT FROM AUTHOR]

Purpose: Laparoscopic sleeve gastrectomy (LSG) may present many technical modifications depending on the surgeon. However, no significant variation was noted in the chronological order of the surgical steps in the current technique. We present the "Window to Window," a variant LSG technique. Material and Methods: The technique, introduced in our institution in 2007, requires three trocars. Surgical steps, presented chronologically in a descriptive video, start by creating the "paracardial window" then the "antral window." Subsequently, the gastric transection is performed in two phases after progressive retro-gastric liberation. Lastly, the mobilization of greater curvature is carried out followed by specimen extraction and fascial closure. Results: The procedure was performed for approximately 3200 patients and was comparable in terms of medium-term percent of excess weight loss compared with the standard technique. Conclusion: This variant technique of LSG presents many technical advantages for surgeons in terms of safety and exposure compared with the standard technique, while presenting a longer learning curve. [ABSTRACT FROM AUTHOR]

In this paper, we give an estimate of the upper bound of the inner radius for a complete Finsler measure space (M , F , d μ) of weighted Ricci curvature bounded from below, in terms of the d μ -mean curvature bound of the boundary. We also prove the rigidity result that when the boundary is compact, the upper bound is achieved if and only if M is isometric to a forward Finsler geodesic ball of vanishing radical flag curvature. [ABSTRACT FROM AUTHOR]

The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half‐planes when the total curvature of the boundary is positive. [ABSTRACT FROM AUTHOR]

In this paper, inspired by Brendle (Comm. Pure Appl. Math. 76 (2023), 2192) and Johne (arXiv:2103.08496, 2021), we prove a Sobolev inequality on manifolds with density and asymptotically nonnegative Bakry–Émery Ricci curvature. [ABSTRACT FROM AUTHOR]

This paper introduces a two-tier feedback control law for the path tracking of a mobile robot equipped with N on-axle trailers. Initially, through a recursive design process, the curvature-tracking challenge is converted into stabilizing the joint angles at predefined reference values. This design approach is straightforward and can be easily extended to configurations with multiple trailers. Using input-to-state stability analysis, we demonstrate the asymptotic stability of the closed-loop system, which is structured in cascade form. Furthermore, we reformulate the path-tracking problem as a curvature-planning challenge and propose an algorithm to determine the desired curvature for the tail trailer. The simulation results validate the effectiveness of this novel algorithm in truck-trailer systems. [ABSTRACT FROM AUTHOR]

Given a smooth positive function $ K $ defined on the standard half sphere endowed with its standard metric $ g $, we consider the problem of finding a metric $ \tilde{g} $ conformally equivalent to $ g $ and whose scalar curvature is equal to $ K $ and the boundary mean curvature is equal to zero. Using careful study of Hopf-Poincaré counting index formulae, we prove the existence of many of such metrics $ \tilde{g} $ provided that $ K $ is close to the scalar curvature of $ g $. Such Counting index formulae are related to the topological contribution of solutions at the level sets of the associated approximate variational functional. The location of the levels of these solutions are given very precisely which leads to a new type of multiplicity results on half spheres. Such multiplicity results are proved without any assumptions of symmetry or periodicity on the function $ K $. [ABSTRACT FROM AUTHOR]

We study the non-submodular maximization problem, in which the objective function is characterized by parameters, subject to a cardinality or p -system constraint. By adapting the Threshold-Greedy algorithm for the submodular maximization, we present two deterministic algorithms for approximately solving the non-submodular maximization problem. Our analysis shows that the algorithms we propose requires much less function evaluations than existing algorithms, while providing comparable approximation guarantees. Moreover, numerical experiment results are presented to validate the theoretical analysis. Our results not only fill a gap in the (non-)submodular maximization, but also generalize and improve several existing results on closely related optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the k-Hessian problem Sk(D2u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C∞-smooth bounded strictly (k − 1)-convex domain in R N with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C1(0, ∞) (or f ∈ C 1 (R) ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions. [ABSTRACT FROM AUTHOR]

In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed. [ABSTRACT FROM AUTHOR]

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in ("A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 by virtue of the Brendle-Guan-Li's flow ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 (Ω)) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 provided that M is h-convex and Ω is the domain enclosed by M, pk(r) = ωn(λ′)k−1, W 1 (Ω) = 1 n | M | , λ′(r) = coshr, q 1 (r) = W 1 S r n + 1 , the area for a geodesic sphere of radius r, and q 1 − 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in ("Locally constrained curvature flows and geometric inequalities in hyperbolic space," Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022). [ABSTRACT FROM AUTHOR]

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3$\hskip.001pt 3$–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3$\hskip.001pt 3$. [ABSTRACT FROM AUTHOR]

Summary: The use of curved forms in tall buildings has increased recently due to their structural and esthetic advantages. Diagrid structures are typically used for these buildings because of their ability to express curved forms. There are, however, few studies about the curvature of tall buildings. This paper examines the effect of curvature on the wind force on tall curved buildings with diagrid system. So, to achieve this, the form of tall curved buildings is investigated, and the commonly used equations that constitute their form are determined. Then, applying computational fluid dynamics simulation, wind force was calculated on the developed alternatives which were created by the equations. Besides, the structural mass of each model is calculated by optimizing the cross‐section of the members. It is concluded that considering the conditions of the applied model, by increasing the concavity of the form, the suction on the building's facade increases. Further, with the increase of the concavity in the plan and elevation, the along wind force on the structure and overturning moment increases. Also, it is observed that the structural efficiency of curvilinear buildings is only appropriate to a certain extent of curvature, and with its increase, the structural efficiency decreases. This article offers useful information for the preliminary design of tall buildings. [ABSTRACT FROM AUTHOR]

ABSTRACTExtracellular vesicles (EVs), transporting diverse cellular components, play a crucial role in intercellular communication in numerous physiological and pathological processes. EVs have also been recognized as a drug delivery platform for therapeutic purposes and cell-free regenerative medicine. While various approaches have focused on increasing EV production for efficient use therapeutic use of EVs, enhancing the quality of EVs, such as ensuring efficient uptake by their target cells, has not been widely explored. In this study, we linked a negative membrane curvature-forming inverse BAR (IBAR) domain with an integrin β tail-binding talin F3 domain to create the IBAR-F3 fusion protein. We observed that IBAR-F3 can trigger filopodia-like membrane protrusions and attract integrins to those protrusion-rich regions, when expressed in Chinese hamster ovary cells expressing integrin αIIbβ3. Surprisingly, the expression of IBAR-F3 also induced a robust production of EVs, which were then efficiently taken up by nearby cells in an integrin-dependent manner. Moreover, IBAR triggered integrin activation, presumably by inducing negative membrane curvature that likely disrupts the interaction between the integrin α and β transmembrane domain. Therefore, we suggest that IBAR-F3 should be utilized to promote both EV production and efficient uptake mediated by integrins. Furthermore, the negative curvature-inducing integrin activation suggests that integrins on EVs can be activated by the nanoscale change in the curvature of the EV without the need for conventional machinery to activate integrin inside the EVs.