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Laboratory experiments and numerical simulations were performed to explore the influence of intersection geometry on fluid flow and solute transport in fractures. Fractures were engraved and sealed into an acrylic plate and two orthogonal intersections with different geometry were constructed. The effects of curvature and relative shear displacement at intersections on preferential flow and solute transport were investigated. By solving the Navier–Stokes (NS) equation, the fluid mixing and solute distribution were predicted. The results showed that the geometric characteristics at the intersection have a significant effect on the preferential flow and solute distribution. The results agreed well with the experimental results, in terms of flow direction, preferential flow rate, and heterogeneous solute distribution. With an increase in curvature, the flow difference between the two outlets increases gradually. Increasing curvature can reduce the preferential flow and weaken the inhomogeneity of solute distribution. An increase of relative shear displacement decreases the pressure gradient and flow rate at the entrance of the two branch fractures, and thereby increases preferential flow and inhomogeneity of solute distribution. The results provide a basis and reference for further exploring the relationship between the geometric characteristics of fracture intersections and flow behaviors. [ABSTRACT FROM AUTHOR]

The renewed focus on eye growth in preterm‐born children was primarily triggered by Danish cohort studies, including the Copenhagen Project, which focused on children born from 1959–1961. The retinotoxic effects of excessive oxygen on premature neonates had long been clarified and therapeutically adjusted for. Later, ultrasound oculometry and keratometry established that ocular size deficits, linked to development, also occurred in normally developing children, not just frail outliers. This indicated that general catch‐up had not been achieved. This paper discusses whether one early segment of eye development does not occur in preterm, and here even in more robust neonates, without later compensation. [ABSTRACT FROM AUTHOR]

In this work, we prove a compactness theorem on the space of all Hamiltonian stationary Lagrangian submanifolds in a compact symplectic manifold with uniform bounds on area and total extrinsic curvature. This generalizes the compactness theorems in Chen and Warren [J. Differential Geom. 126 (2024), pp. 65–97] and Chen and Ma [Var. Partial Differential Equations 60 (2021), Paper No. 75, 23]. [ABSTRACT FROM AUTHOR]

We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows. [ABSTRACT FROM AUTHOR]

L ν operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of L ν 2 operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of L ν 2 operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the L I I operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate. [ABSTRACT FROM AUTHOR]

In the realm of five-axis Computer Numerical Control (CNC) machining, the challenge of minimizing velocity, acceleration, and jerk fluctuations has prompted the development of various local corner smoothing methods. However, existing techniques often rely on symmetrical spline curves or impose pre-set transition length constraints, limiting their effectiveness in reducing curvature and maintaining velocity at critical corners. To comprehensively address these limitations comprehensively, a novel approach for local smoothing of five-axis linear toolpaths is presented in this paper. The proposed method introduces two asymmetrical B-splines at corners, effectively smoothing both the tool-tip position in the workpiece coordinate system and the tool orientation in the machine coordinate system. To fine-tune transition curve scales and minimize velocity disparities between adjacent corners, an overlap elimination scheme is employed. Furthermore, the two-step strategy emphasizes the synchronization of tool-tip position and tool orientation while considering maximal approximation error. The outcome is a blended five-axis toolpath that significantly reduces curvature extremes in the smoothed path, achieving reductions ranging from 36.28 % to 45.51 %. Additionally, the proposed method streamlines the interpolation process, resulting in time savings ranging from 5.68 % to 8.78 %, all the while adhering to the same geometric and kinematic constraints. [ABSTRACT FROM AUTHOR]

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in ${\mathbb R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows one, in particular, to determine their expectations and covariance structure. We also show the convergence of the rescaled expectations and variances of the intrinsic volumes of the construction steps to the expectations and variances of the limit variables, and we give rates for this convergence in some cases. These results significantly extend our previous work, which addressed only limits of expectations of intrinsic volumes. [ABSTRACT FROM AUTHOR]

Projectively flat Finlser metrics on a convex domain U in R n are regular solutions to Hilbert's Fourth Problem. In this paper, we study projectively flat Finlser metrics on U. We find equations that characterize these metrics with weakly orthogonal invariance, refining a theorem due to Sol o ´ rzano-Le o ´ n. As its application, we obtain infinitely many new projectively flat Finlser metrics on S n + 1 and determine their scalar flag curvature. These metrics contain Bryant's projective spherically symmetric Finsler metric of constant flag curvature 1. [ABSTRACT FROM AUTHOR]

This paper presents the details of the analyses which were conducted to study the effects of steel reinforcing bars with unintended high strength on the behaviors of reinforced concrete (RC) columns. The influence of these bars on the column strength and strength-reduction factors were investigated. The former was studied with the help of column axial load-moment interaction diagrams, while a reliability analysis was carried out for the latter. Four different column cross sections reinforced with reinforcement ratios varying from 1 to 4% were included in the analysis. Other variables included concrete compressive and reinforcing bar yield strengths. The effects of the aforementioned variables were also considered on the development length of the reinforcing bars in tension and compression. It was found that the use of reinforcing bars with unintended high strength could change column behavior to compression-controlled at a lesser axial load level, which is accompanied by a reduction in the curvature capacity. Modifications have been suggested to control the negative effects of unintended high strength of bars on the column behavior and bar development length. Strength-reduction factors for RC sections ranging from compression-controlled to tension-controlled regions have also been proposed, which differ from those suggested by the prevalent code of practice. [ABSTRACT FROM AUTHOR]

In contrast to the conventional etching that makes nanoparticles rounder and our previous sharpening etching mode that causes serrated edges, here, we developed a new boring etching mode that targets the faces of Au nanoplates to make holes. The critical factors are the pre-incubation step with the ligand 2-mercapto-5-benzimidazolecarboxylic acid (MBIA) and the subsequent removal of excess ligands in the solution. Thus, etching is focused onto the few sites with initial loss of ligands, which cannot be quickly replaced. The choice of ligand MBIA is also of importance, as it carries negative charge and repels each other. Its inability of forming a dense layer probably plays a critical role in the site-selectivity for faces, because ligands at the higher curvature edges and corners are expected to have less repulsion. The etching results from the comproportionation reaction between Au3+ and Au0 in the nanoplates, where Br− coordination to Au and the extra stabilization from cetyltrimethylammonium bromide (CTAB) are essential. We believe that the ability of boring holes is an important tool for future synthetic designs. [ABSTRACT FROM AUTHOR]

We prove an analogue statement to an estimate by De Lellis–Müller in R 3 on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is W 2 , 2 -close to a round surface provided the trace-free part of a scalar second fundamental form A is sufficiently small in L 2 . To determine the correct intrinsically round cross section of reference, we define an associated 4-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis–Topping. [ABSTRACT FROM AUTHOR]

We obtain a complete description of divergent Palais–Smale sequences for the prescribed Ricci curvature functional on compact homogeneous spaces. As an application, we prove the existence of saddle points on generalized Wallach spaces and several types of generalized flag manifolds. We also describe the image of the Ricci map in some of our examples. [ABSTRACT FROM AUTHOR]

In this paper, we classify the hypersurfaces of S 2 × S 2 with constant sectional curvature. We prove that the constant sectional curvature can only be 1 2 . We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in S 2 × S 2 , and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous "sinh-Gordon equation" ∂ 2 ∂ u 2 + ∂ 2 ∂ v 2 h = - 1 2 sinh 2 h . [ABSTRACT FROM AUTHOR]

We show that if the universal cover of a closed smooth manifold admitting a metric with non-negative Ricci curvature is formal, then the manifold itself is formal. We reprove a result of Fiorenza–Kawai–Lê–Schwachhöfer, that closed orientable manifolds with a non-negative Ricci curvature metric and sufficiently large first Betti number are formal. Our method allows us to remove the orientability hypothesis; we further address some cases of non-closed manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative proof for the longtime behavior of classical curve shortening flow by using support function to estimate the curvature of evolving curves after an area-preserving rescaling, and then showing the convergence of the rescaled curvature via analyzing the structure of its $ \omega $-limit set. [ABSTRACT FROM AUTHOR]

A directionally‐split volume‐of‐fluid (VOF) methodology for evolving interfaces under curvature‐dependent speed is devised. The interface is reconstructed geometrically and the volume fraction is advected with a technique to incorporate a topological volume conservation constraint. The proposed approach uses the idea that the role of curvature in a speed function V$$ \mathbf{V} $$ is analogous to the role of viscosity in the corresponding hyperbolic conservation law to propagate complex interfaces where singularities may exist. The approach has the advantage of simple implementation and straightforward extension to more complex multiphase systems by formulating the interface evolution problem using energy functionals to derive an expression for the interface‐advecting velocity. The numerical details of the volume‐of‐fluid based formulation are discussed with emphasis on the importance of curvature estimation. Finally, canonical curves and surfaces traditionally investigated by the level set (LS) method are tested with the devised approach and the results are compared with existing work in LS. [ABSTRACT FROM AUTHOR]

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Robertson–Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the cosmological expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow. [ABSTRACT FROM AUTHOR]

In this paper, we thoroughly study ϱ-Ricci–Bourguignon almost soliton and gradient ϱ-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on K-paracontact and para-Sasakian manifolds. Here, we prove that if the metric g of the K-paracontact manifold endows a ϱ-Ricci–Bourguignon almost soliton with the nonzero potential vector field V parallel to ξ, then the manifold is an Einstein with Einstein constant − 2n. Next, we show that if a para-Sasakian manifold represents a gradient ϱ-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature − 2n(2n + 1). We also discuss ϱ-Ricci–Bourguignon almost soliton on (κ,μ)-paracontact manifold. [ABSTRACT FROM AUTHOR]

In this study, we explore a topologically charged higher-dimensional traversable defect wormhole, with a specific emphasis on five dimensions. Particularly noteworthy is the fact that the matter–energy distribution attributed to this charged wormhole configuration adheres to the weak energy condition, thus presenting an instance of a five-dimensional wormhole supported by non-exotic matter. Furthermore, our analysis shows that scalar quantities related to space-time curvature and parameters associated with the matter–energy distribution remain finite at the wormhole throat. Moreover, they diminish as distance extends toward infinity, indicating the asymptotic flatness inherent in our model. [ABSTRACT FROM AUTHOR]

Given a random map (T1,T2,T3,T4,p1,p2,p3,p4), we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards. [ABSTRACT FROM AUTHOR]

New invariants for two-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems about these invariants, computes examples and lays out a programme of conjectures. [ABSTRACT FROM AUTHOR]

A higher-order spectral problem is introduced, based on a special Lie subalgebra of the general linear algebra. An associated matrix Liouville integrable hierarchy, each of which consists of four submatrix equations, is generated by means of the zero curvature formulation. The corresponding Hamiltonian structure is established by the trace variational identity and two integrable reductions, of which one is real and the other is complex, are constructed under similarity transformations. [ABSTRACT FROM AUTHOR]

Cavitation, a confusing yet essential natural process, remains the target of intensive scientific research. Discovered through propeller cavitation nearly a century ago, initial theories suggest that bubble collapse initiates cavitation. The mechanism of bubble collapse in cavitation remains unclear, especially in modern cavitation scenarios like sonodynamic tumor therapy or aerodynamic noise without natural bubbles. Therefore, a comprehensive visualization-based cavitation mechanism is vital to comprehend and investigate various cavitation phenomena, from historical propeller cavitation to modern in vivo tumor treatment. This study introduces and discusses the direct conversion between two fundamental motions, sound and vortex, and its application as a universal cavitation mechanism to initiate and sustain tailored long-life cavitation from droplets. The results demonstrate a special acoustic phenomenon and its potential technological applications for harnessing sound with customized acoustics cavitation devoid of bubbles. [ABSTRACT FROM AUTHOR]

We consider the equivalence problem for first-order ordinary differential equations (ODEs) and recover the well-known result that any two first-order ODEs are equivalent under a point transformation by Cartan’s equivalence method. Curvature properties of some first-order and second-order ODEs motivate us to define the equivalence problem for such equations as the metric equivalence problem which defines the {e}-structure on the corresponding manifolds. Accordingly, we show that there are first-order ODEs and linear second-order ODEs which may not be equivalent under a class of fiber-preserving diffeomorphisms. [ABSTRACT FROM AUTHOR]

Heusler compounds belong to a large family of materials and exhibit numerous physical phenomena with promising applications, particularly ferromagnetic Weyl semimetals for their use in spintronics and memory devices. Here, anomalous Hall transport is reported in the room‐temperature ferromagnets NiMnSb (half‐metal with a Curie temperature (TC) of 660 K) and PtMnSb (pseudo half‐metal with a TC of 560 K). They exhibit 4 µB/f.u. magnetic moments and non‐trivial topological states. Moreover, NiMnSb and PtMnSb are the first half‐Heusler ferromagnets to be reported as Weyl semimetals, and they exhibit anomalous Hall conductivity (AHC) due to the extended tail of the Berry curvature in these systems. The experimentally measured AHC values at 2 K are 1.8 × 102 Ω−1 cm−1 for NiMnSb and 2.2 × 103 Ω−1 cm−1 for PtMnSb. The comparatively large value between them can be explained in terms of the spin‐orbit coupling strength. The combined approach of using ab initio calculations and a simple model shows that the Weyl nodes located far from the Fermi energy act as the driving mechanism for the intrinsic AHC. This contribution of topological features at higher energies can be generalized. [ABSTRACT FROM AUTHOR]

In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott–Sturm–Villani [Sturm, On the geometry of metric measure spaces. I, Acta Math. 196(1) (2006) 65–131; On the geometry of metric measure spaces. II, Acta Math. 196(1) (2006) 133–177; Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169(3) (2009) 903–991], and a newly introduced notion that we call strong Brunn–Minkowski inequality SBM(K,N). This condition is a reinforcement of the generalized Brunn–Minkowski inequality BM(K,N), which is known to hold in CD(K,N) spaces. Our result is a first step toward providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which has been recently proved in [M. Magnabosco, L. Portinale and T. Rossi, The Brunn–Minkowski inequality implies the CD condition in weighted Riemannian manifolds, Nonlinear Anal. 242 (2024) 113502] in the framework of weighted Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Highlights Application of the global Gauss–Bonnet theorem for the prediction of changes in Gaussian curvature for double curvature laminate geometries due to shrinkage is developed for non‐isometric deformation states where in surface area change accompanies changes in curvature. The transition between isometric and non‐isometric deformation is strongly influenced by the ratio of the principal radii of curvature of the double curved surface and a relationship is proposed by the authors to accommodate the transition. Examples for the radii of curvature ratios of 2, 3, and 4 are compared to results from finite‐element simulations and the developed relationship is shown to be accurate within 0.55%. Furthermore, these results demonstrate that isometric deformation state can be taken as an adequate representation for double curvature toroidal laminate geometries of ρ2/ρ1$$ {\rho}_2/{\rho}_1 $$ >10 and the class of materials studied in this work. Shrinkage induced deformation in multi‐curvature composite laminates is a recurring issue in polymer composite manufacturing. Thermal, crystallization and cure shrinkage are the dominant sources of shrinkage deformation in polymer composite systems. Isometric deformation of multi‐curvature, symmetric laminates composite is defined by the absence of membrane strain. The Gauss–Bonnet theorem provides a platform for determining radii of curvature changes in multi‐curvature symmetric laminates, Non‐isometric deformation of symmetric laminates can be treated by the Gauss–Bonnet theorem for radii of curvature ratios that correspond to geometric self‐constraint. The above highlights yield insight into the anticipated geometric distortions due to shrinkage in polymer composites. [ABSTRACT FROM AUTHOR]

We consider the short-wave diffraction of a cylindrical wave by a contour whose curvature has a Hölder type discontinuity at a point. The incidence is nontangential at the point of singularity. In the framework of the Kirchhoff method, we find an asymptotic description for the outgoing wavefield inside the transition zone at both small and moderate distances. Analysis of ray formulas allows us to characterize applicability areas of the expressions obtained. [ABSTRACT FROM AUTHOR]

We define |$0$| -shifted and |$+1$| -shifted contact structures on differentiable stacks, thus laying the foundations of shifted Contact Geometry. As a side result we show that the kernel of a multiplicative |$1$| -form on a Lie groupoid (might not exist as a Lie groupoid but it) always exists as a differentiable stack, and it is naturally equipped with a stacky version of the curvature of a distribution. Contact structures on orbifolds provide examples of |$0$| -shifted contact structures, while prequantum bundles over |$+1$| -shifted symplectic groupoids provide examples of |$+1$| -shifted contact structures. Our shifted contact structures are related to shifted symplectic structures via a Symplectic-to-Contact Dictionary. [ABSTRACT FROM AUTHOR]

SLAM (simultaneous localization and mapping) is essential for accurate positioning and reasonable path planning in outdoor mobile robots. LiDAR SLAM is currently the dominant method for creating outdoor environment maps. However, the mainstream LiDAR SLAM algorithms have a single point cloud feature extraction process at the front end, and most of the loop closure detection at the back end is based on RNN (radius nearest neighbor). This results in low mapping accuracy and poor real-time performance. To solve this problem, we integrated the functions of point cloud segmentation and Scan Context loop closure detection based on the advanced LiDAR-inertial SLAM algorithm (LIO-SAM). First, we employed range images to extract ground points from raw LiDAR data, followed by the BFS (breadth-first search) algorithm to cluster non-ground points and downsample outliers. Then, we calculated the curvature to extract planar points from ground points and corner points from clustered segmented non-ground points. Finally, we used the Scan Context method for loop closure detection to improve back-end mapping speed and reduce odometry drift. Experimental validation with the KITTI dataset verified the advantages of the proposed method, and combined with Walking, Park, and other datasets comprehensively verified that the proposed method had good accuracy and real-time performance. [ABSTRACT FROM AUTHOR]

A high Fe content easily produces Fe-rich phases with a harmful morphology, resulting in a huge detrimental effect on the properties and recycling ability of Al-Si alloys. Therefore, finding ways to effectively transform Fe-rich phases to form a beneficial phase or shape is of great significance. Accordingly, Al-Si-based alloys with Fe contents ranging from 0.1 wt.% to 2.0 wt.% were modified by different Mn additions. Moreover, experiments combined with simulations were utilized to comprehensively analyze the mechanism of Mn on the morphology and microstructural evolution of Fe-rich phases from different perspectives. The current findings determine that adding different Fe contents changes the phase-transition reactions in alloys. Without Mn, and by increasing the Fe content from 0.1 wt.% to 2.0 wt.%, the Fe-rich phases gradually convert from a skeleton-shaped α-Al8Fe2Si (<0.25 wt.%) to β-Al9Fe2Si2 with a fibrous (0.5 wt.%), needle-like (1.0 wt.%) and plate-like shape without curvatures (2.0 wt.%). The maximum length and mean aspect ratio increase from 12.01 μm to 655.66 μm and from 1.96 to 84.05, while the mean curvature decreases from 8.66 × 10−2 μm−1 to 8.25 × 10−4 μm−1. The addition of 0.35 wt.% Mn promotes a new Chinese-character and petal-shaped α-Al15(FeMn)3Si2, with an atomic ratio of Fe and Mn of 1:1 when the Fe content is lower than 0.5 wt.%, while it transforms to β-Al15(FeMn)3Si2 with an atomic ratio of 5:1, presenting as a refined plate-like shape with a certain curvature, as the Fe content increases to 2.0 wt.%. Mn alters the phase reactions and increases the threshold of the Fe content required for β-Al15(FeMn)3Si2, limiting the formation and growth of them simultaneously in time and space. The enrichment of Mn atoms and solute diffusion at the growth front of β-Al15(FeMn)3Si2, as well as the strong atomic-binding ability, can deflect the growth direction of β-Al15(FeMn)3Si2 for it to have a certain curvature. Additionally, the enriched Mn atoms easily form α-Al15(FeMn)3Si2 and cause the long β-Al15(FeMn)3Si2 to be broken and refined to further reduce the damages caused to the alloy's performance. Ultimately, the maximum length and mean aspect ratio can be effectively reduced to 46.2% and 42.0%, respectively, while the mean curvature can be noticeably increased by 3.27 times with the addition of Mn. [ABSTRACT FROM AUTHOR]

A novel stress suppression design for flexible RF MEMS switches has been presented and demonstrated through theoretical and experimental research to isolate the stress caused by substrate bending. An RF MEMS switch with an S-shaped microspring structure was fabricated by the two-step etching process as a developmental step toward miniaturization and high reliability. The RF MEMS switches with an S-shaped microspring exhibited superior microwave performance and stable driving voltage under different substrate curvatures compared to the conventional non-microspring switches, demonstrating that the bending stress is successfully suppressed by the S-shaped microspring and the island structure. Furthermore, this innovative design could be easily extended to other flexible devices. [ABSTRACT FROM AUTHOR]

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with F -statistical connections are examined. The equations of Gauss and Codazzi of F -statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to F -statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared F -mean curvature of a statistical submanifold of statistical manifolds. [ABSTRACT FROM AUTHOR]

Motivated by Guo–Luo’s generalized circle packings on surfaces with boundary [Guo and Luo, Geom. Topol. 13(3) (2009) 1265–1312], we introduce the generalized Thurston’s sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized Thurston’s sphere packings. We prove that the generalized Thurston’s sphere packings are locally determined by the combinatorial scalar curvatures. We further prove the infinitesimal rigidity that the generalized Thurston’s sphere packings cannot be deformed while keeping the combinatorial Ricci curvatures fixed. [ABSTRACT FROM AUTHOR]

Translation abstract A mechanically consistent approach for stability verification of timber beams in compression and bending around the strong axis – part 2: experimental tests and numerical simulationSlender timber beams in bending can be designed using different methods in view of out‐of‐plane stability. Lateral torsional buckling may be verified either using the so called equivalent member method or by means of 2nd order calculations, considering equivalent imperfections. Applying current codes, however, determining the lateral torsional buckling resistance using either of these both methods, leads to different results. In this paper, the background of both verifications is investigated in deep. Moreover, as an alternative approach, a non‐linear calculation with an imperfection shape in affinity to the critical shape was presented in part 1 of this paper. By scaling the curvature, the same results as by the equivalent member method can be achieved. It was shown that plasticity effects can be considered as well. In this part 2, results of lateral torsional buckling tests on glued laminated timber beams are presented, with nearly ideal boundary conditions and load direction remaining vertical through‐out the test. Based on these test first tentative values can be obtained for the imperfection compatible with the alternative lateral torsional buckling approach presented. [ABSTRACT FROM AUTHOR]

AbstractThe primary objective of the current work is to establish the impacts of planar boundary and non-planar boundary on the displacement components (DCs) of Rayleigh-type wave propagation in an elastic structure. The elastic structure, precisely, incorporates a pre-stressed nearly incompressible elastic material in the half-space along with the non-planar boundary at the free surface. The methodology applied features variable separable technique and Taylor’s series expansion along with substitution method to establish the expressions for DCs and frequency relation. The frequency relation of Rayleigh-type wave is obtained for the situation considering planar free surface, which is further reduced as a special case of the study and validated with the classical result present in the literature. The DCs of Rayleigh-type wave have been numerically computed, and their reliance on wave number have been exhibited graphically. In addition to this, the influences of affecting parameters viz., hydrostatic stress, principal Cauchy stress and curvature parameter on DCs have been traced out. The computational result also manifests the domain of existence of Rayleigh-type wave propagating in the considered model in the presence of planar free surface. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of differential equations describing a mathematical model of a phase-locked loop with a delay. We introduce the concept of the curvature of a cycle, which is applied to analyzing the proximity of cycles of nonphase and phase systems. [ABSTRACT FROM AUTHOR]

An equilibrium thermoelastic state of two plates connected by a thin layer is examined. The problem is reduced to a system of six second-order differential equations. Temperature deformations of a bimetallic plate clamped at one edge and free at the other are considered. In the case of zero thickness of the interaction layer, the law of change of the interface line and the distribution of stresses along this line are obtained. In a uniform state (pure bend), the curvature of the interface line becomes constant. For zero Poisson ratio, the temperature dependence of the curvature is obtained. [ABSTRACT FROM AUTHOR]

Autonomous driving is a complex problem that requires intelligent decision making, and it has recently garnered significant interest due to its potential advantages in convenience and safety. In autonomous driving, conventional path planning to reach a destination is a time-consuming challenge. Therefore, learning-based approaches have been successfully applied to the controller or decision-making aspects of autonomous driving. However, these methods often lack explainability, as passengers cannot discern where the vehicle is headed. Additionally, most experiments primarily focus on highway scenarios, which do not effectively represent road curvature. To address these issues, we propose a reinforcement-learning-based trajectory learning in the Frenet frame (RLTF), which involves learning trajectories in the Frenet frame. Learning trajectories enable the consideration of future states and enhance explainability. We demonstrate that RLTF achieves a 100% success rate in the simulation environment, considering future states on curvy roads with continuous obstacles while overcoming issues associated with the Frenet frame. [ABSTRACT FROM AUTHOR]

We show that, for any given q ≥ 0 {q\geq 0} , any Sasakian structure on a closed manifold M is approximated in the C q {C^{{q}}} -norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C 0 {C^{0}} -norm to a C q {C^{{q}}} -approximation. [ABSTRACT FROM AUTHOR]

A surface is minimal if the mean curvature ℋ mean {\mathcal{H}_{\rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} . In particular, we completely classify minimal surfaces. [ABSTRACT FROM AUTHOR]

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damping part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove the well-posedness of the corresponding system in energy space and establish the existence of a regular global attractor under certain conditions on the nonlinearities and coefficients of the damped part only. Besides, we study the singular limits of the problem under consideration when curvature tends to zero, or curvature tends to zero, and simultaneously shear moduli tend to infinity and perform numerical modeling for these processes. [ABSTRACT FROM AUTHOR]

Purpose: To improve the setup reproducibility of neck curvature using real-time optical surface imaging (OSI) guidance on 2 regions of interest (ROIs) to infer cervical spine (c-spine) curvature for surface-guided radiotherapy (SGRT) of head-and-neck (HN) and c-spine cancer. Methods: A novel SGRT setup approach was designed to reproduce neck curvature with 2 ROIs: upper-chest ROI and open-face ROI. It was hypothesized that the neck curvature could be reproduced if both ROIs were aligned within ±3 mm/2˚ tolerance. This was tested prospectively in 7 volunteers using real-time 3D-OSI guidance and lateral 2D-photography verification after the 3D and 2D references were captured from the initial conventional setup. Real-time SGRT was performed to align chest-ROI and face-ROI, and the longitudinal distance between them was adjustable using a head-support slider. Verification of neck curvature anteriorly and posteriorly was achieved by overlaying edge-extracted lateral pictures. Retrospectively, the relationship between anterior surface and spinal canal alignment was checked in 11 patients using their simulation CT (simCT) and setup cone-beam CT (CBCT). After the anterior surface was rigidly aligned, the spinal canal alignment was checked and quantified using the mean-distance-to-agreement (MDA) and DICE similarity index, and surface-to-spine correlation was calculated. Results: The reproducibility of neck curvatures using the 2xROI SGRT setup is verified and the mean neck-outline-matching difference is within ±2 mm in lateral photographic overlays. The chest-ROI alignment takes 110 ± 58 s and the face-ROI takes 60 ± 35 s. When the anterior body surface is aligned (MDA = 1.1 ± 0.6 mm, DICE = 0.96 ± 0.02,) the internal spinal canal is also aligned (MDA = 1.0 ± 0.3 mm, DICE = 0.84 ± 0.04) in 11 patients. The surface-to-spine correlation is c = 0.90 (MDA) and c = 0.85 (DICE). Conclusion: This study demonstrates the feasibility of the novel 2-ROI SGRT setup technique to achieve reproducible neck and c-spine curvature regardless of neck visibility and availability as ROI. Staff training is needed to adopt this unconventional SGRT technique to improve patient setup. [ABSTRACT FROM AUTHOR]

The Frenet frame is not suitable for describing the behavior of the curve in the Galilean space since it is not defined everywhere. In this study, an alternative frame, the so-called quasi-frame, is investigated in Galilean 4-space. Furthermore, the quasi-formulas in Galilean 4-space are deduced and quasi- curvatures are obtained in terms of the quasi-frame and its derivatives. Quasi- rectifying, quasi-normal, and quasi-osculating curves are studied in Galilean 4-space. We prove that there is no quasi-normal and accordingly normal curve in Galilean 4-space. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of a negative corona discharge in air at atmospheric pressure. To ignite it, the edge (cathode) –plane (anode) system has been used with interelectrode gap of 10–20 mm long and edge curvature radius of ≈20 μm. For the first time, it has been established that in a certain voltage range, the Trichel pulse mode after ignition becomes unstable and is periodically replaced by the glow discharge mode with a current ≲ 1 μA. It is shown that at the stage preceding the first Trichel pulse, the discharge current is not recorded within measurement error of ±0.2 nA. [ABSTRACT FROM AUTHOR]

We study the entanglement entropy associated with a holographic RG flow from AdS7 to AdS4 × ℍ3, where ℍ3 is a 3-dimensional hyperbolic manifold with curvature κ. The dual six-dimensional RG flow is disconnected from Lorentz-invariant flows. In this context we address various notions of central charges and identify a monotonic candidate c-function that captures IR aspects of the flow. The UV behavior of the holographic entanglement entropy and, in particular its universal term, display an interesting dependence on the curvature, κ. We then contrast our holographic results with existing field theory computations in six dimensions and find a series of new corrections in curvature to the universal term in the entanglement entropy. [ABSTRACT FROM AUTHOR]

In the present paper, we have studied the curvature tensors of PSasakian manifold. For a P-Sasakian manifold, W1 ·S = 0, W1 · e Z = 0 and W9 · W1 = 0 cases are considered. According these cases, PSasakian manifolds have been characterized such as η-Einstein and Einstein. In addition, we research W1-flat and W9-flat for a PSasakian manifold. The results are interesting and give an idea about the geometry of P-Sasakian manifold. [ABSTRACT FROM AUTHOR]

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold: first, we obtain various curvature inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Kenmotsu space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion. The equality cases of all the inequalities are studied. Moreover, these curvature inequalities are studied under two different cases: the structure vector field ξ being vertical or horizontal. [ABSTRACT FROM AUTHOR]

The volume of fluid (VOF) method is a popular technique for the direct numerical simulations of flows involving immiscible fluids. A discrete volume fraction field evolving in time represents the interface, in particular, to compute its geometric properties. The height function method (HF) is based on the volume fraction field, and its estimate of the interface curvature converges with second-order accuracy with grid refinement. Data-driven methods have been recently proposed as an alternative to computing the curvature, with particular consideration for a well-balanced input data set generation and symmetry preservation. In the present work, a two-layer feed-forward neural network is trained on an input data set generated from the height function data instead of the volume fraction field. The symmetries for rotations and reflections and the anti-symmetry for phase swapping have been considered to reduce the number of input parameters. The neural network can efficiently predict the local interface curvature by establishing a correlation between curvature and height function values. We compare the trained neural network to the standard height function method to assess its performance and robustness. However, it is worth noting that while the height function method scales perfectly with a quadratic slope, the machine learning prediction does not. [ABSTRACT FROM AUTHOR]