Curvature Properties Research Articles

Involute-evolute curves with modified orthogonal frame in Galilean space G 3

UDC 514 We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space G 3 . The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

Adaptive indefinite kernels in hyperbolic spaces

Learning embeddings in hyperbolic space has gained increasing interest in the community, due to its property of negative curvature, as a way of encoding data hierarchy. Recent works investigate the improvement of the representation power of hyperbolic embeddings through kernelization. However, existing developments focus on defining positive definite (pd) kernels, which may affect the intriguing property of hyperbolic spaces. This is due to the structures of hyperbolic spaces being modeled in indefinite spaces (e.g., Kreĭn space). This paper addresses this issue by developing adaptive indefinite kernels, which can better utilize the structures in the Kreĭn space. To this end, we first propose an adaptive embedding function in the Lorentz model and define indefinite Lorentz kernels (iLks) via the embedding function. Due to the isometric relationship between the Lorentz model and the Poincaré ball, these iLks are further extended to the Poincaré ball, resulting in the development of what are termed indefinite Poincaré kernels (iPKs). We evaluate the proposed indefinite kernels on a diversity of learning scenarios, including image classification, few-shot learning, zero-shot learning, person re-identification, knowledge distillation, etc. We show that the proposed indefinite kernels can bring significant performance gains over the baselines and enjoy better representation power from RKKSs than pd kernels.

Deep-Saddle-Shaped Nanographene Induced by Four Heptagons: Efficient Synthesis and Properties.

The construction of multiple heptagonal rings in nanographene is the key step for obtaining exotic carbon nanostructures with a negative curvature and intriguing properties. Herein, a novel saddle-shaped nanographene (1) with four embedded heptagons is synthesized via a highly efficient one-shot Scholl reaction from a predesigned oligophenylene precursor. Notably, a quadruple [6]helicene intermediate was also obtained and isolated by controlling the Scholl reaction conditions. Interestingly, the single crystal structures of 1 display a saddle geometry induced by the four embedded heptagons, resulting in a deep curvature with a width of 16.5 Å and a depth of 8.0 Å. Theoretical calculations at the molecular level suggest a weak antiaromatic character of the heptagons in 1. Remarkably, compound 1 exhibits dual fluorescence from S1 and S2. The deep-saddle-shaped geometry in 1 defines host-guest interactions with fullerenes, which were explored in titration experiments and by theoretical methods. The resulting 1@C60 are stable and are subject to an electron transfer from photoexcited 1 to C60. Our current study underscores the influence of heptagon rings on the photophysical, self-assembly, and electron-donating properties of NGs.

Equivalence problem for first-order and second-order ODEs with a quadratic restriction

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 [Formula: see text]-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.

The impact of diffusion tensor imaging tractography settings on muscle fascicle architecture and diffusion parameter estimates: Tract length, completion, and curvature are most sensitive to tractography settings.

Diffusion-tensor (DT)-MRI tractography provides information about properties relevant to muscle health and function, including estimates of architectural properties such as fascicle length, pennation angle, and curvature and diffusion properties such as mean diffusivity (MD) and fractional anisotropy (FA). Tractography settings, including integration algorithms, thresholds for early tract termination, and tract smoothing approaches, impact the accuracy of the muscle property estimates. However, muscle DT-MRI tractography is performed using a variety of these settings, complicating comparisons between different studies. The effects of different tractography settings on muscle architecture estimates have not been fully explored, and optimized settings for muscle tractography have not yet been determined. We examined the influence of integration algorithm and termination check settings combined with a range of step sizes, termination criteria, and smoothing polynomial orders on tract characteristics, completion/reason for termination, and goodness of fit between fiber tracts and smoothing polynomials using 3-T DT-MR images of the lower leg muscles of seven healthy adults. We found that tract length and completion were highly sensitive to strict FA and intersegment angle thresholds (25%-69% reduction in complete fiber tracts from lowest to highest minimum FA threshold and 11%-36% reduction from highest to lowest intersegment angle threshold). Higher order polynomials (third and fourth order vs. second order) better fit the muscle fiber trajectories, but curvature estimates were highly sensitive to smoothing polynomial order (3.9-6.6 m-1 increase for second- vs. fourth-order fitting polynomials). Step size impacted curvature estimates, albeit to a lesser degree. Integration algorithm had little impact, and mean pennation angle, and tract-based FA and MD, were relatively insensitive to all parameters. The results demonstrate which muscle diffusion measures and architectural estimates are most sensitive to varying tractography settings and support the need for consistent reporting of tractography details to aid interpretation and comparison of results between studies.

Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections

In this study, we investigate the tangent bundle TM of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇˜. Our primary goal is to establish the necessary and sufficient conditions for TM to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for TM to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to TM, alongside specific constraints on the conformal factor λ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve λ and the Hessian of the potential function. Similarly, for TM to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of λ, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry.

Some examples of Finsler metrics with special curvature properties

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.

Impact of Standing and Sitting Postures on Spinal Curvature and Muscle Mechanical Properties in Young Women: A Photogrammetric and MyotonPro Analysis.

BACKGROUND This study aimed to evaluate the effect of standing and sitting positions on spinal curvatures evaluated using projection moire and muscle tone and stiffness using the MyotonPRO hand-held device in young women. MATERIAL AND METHODS Thirty-three healthy women, aged 21 to 23 years, volunteered in the study. We used the projection moire method to examine spinal curvatures in both positions and the MyotonPRO device to measure the tone and stiffness of muscles in 3 regions. We evaluated the effects of positions (standing vs sitting), regions (cervical, thoracic, and lumbar), and side factor (right vs left) using multivariate analysis. RESULTS The sitting position significantly decreased the lumbosacral and thoracolumbar angles (P<0.001), but had no effect on the superior thoracic angle. Muscle tension and stiffness were the highest (P<0.001) in the cervical region and did not differ between positions (P>0.05) in this region. We found significantly higher muscle tone and stiffness in the thoracic and lumbar regions during sitting than during standing (P<0.001). There was symmetry in the muscle tone and the stiffness between the right and left sides of the spine. CONCLUSIONS The sitting posture decreased lumbosacral and thoracolumbar angles but increased muscle tension and stiffness in the lumbar and thoracic regions only. The symmetry of muscle tone and transverse stiffness in both positions was the normative value. This study provides insight into the adaptive physiological changes in spinal curvature and muscle mechanical properties in young women and serves as an important reference point for clinical studies of women.

Geometric analysis of Riemannian doubly warped product submersions

This paper introduces the concept of Riemannian doubly warped product submersions as a natural generalization of Riemannian warped product submersions. We study some fundamental properties of such submersions and derived the conditions under which the Riemannian doubly warped product submersions have totally geodesic, totally umbilical, and minimal fibers. Furthermore, we analyze the curvature properties associated with Riemannian doubly warped product submersions and establish the Einstein conditions. Some non-trivial examples are explored to demonstrate the existence of such submersions.

Mathematical modeling of flexible printed circuit configuration: a study in deformation and optimization

This manuscript offers an exhaustive analysis of Flexible Printed Circuits (FPCs), concentrating on enhancing their design to surmount two primary challenges. Firstly, it seeks to obviate contact with proximate components. Secondly, it aspires to adhere to pre-established curvature constraints. Predicated on the curvature properties of FPCs, we have developed a model adept at accurately forecasting FPC deformation under diverse conditions. Our inquiry entails a thorough examination of various FPC configurations, including bell, 'U', and 'S' shapes. Central to our methodology is the strategic optimization of FPC spatial arrangements, aiming to avert mechanical interference and control curvature, thus mitigating mechanical strain. This dual-faceted strategy is pivotal in enhancing the durability and operational reliability of FPCs, particularly in contexts demanding elevated flexibility and precision. Our research offers essential insights into the refinement of FPC design, skillfully addressing the complexities associated with curvature and physical interaction. Collectively, this study advocates a comprehensive framework for the design and implementation of FPCs, significantly advancing the field of contemporary electronics by ensuring these components meet the evolving demands of the industry.

Analyzing Curvature Properties and Geometric Solitons of the Twisted Sasaki Metric on the Tangent Bundle over a Statistical Manifold

Let (M,∇,g) be a statistical manifold and TM be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of the tangent bundle TM. The second objective is to explore conformal vector fields and Ricci, Yamabe, and gradient Ricci–Yamabe solitons on the tangent bundle TM according to the twisted Sasaki metric G.

Rotational Self-Shrinkers in Euclidean Spaces

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second named authour and at. all recently have considered $3-$dimensional rotational embedded submanifolds in $\mathbb{E}^{5}$. They gave some basic curvature properties of this type of submaifolds. Self-similar flows emerge as a special solution to the mean curvature flow that preserves the shape of the evolving submanifold. In this article we consider self-similar submanifolds in Euclidean spaces. We obtained some results related with self-shrinking rotational submanifolds in Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and sufficient conditions for these type of submanifolds to be homothetic solitons for their mean curvature flows.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U(p+q)-invariant Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×U(1), there exists no U(n+1)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U(p+q)-invariant Kähler metrics on P, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

Geodesics and magnetic curves in the 4-dim almost Kähler model space F4

Abstract We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study K\"{a}hler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

Investigating peristaltic flow of Newtonian fluid in a permeable channel: Effects of nonlinear curvature and wall properties

This investigation delves into the peristaltic flow characteristics of a Newtonian fluid within a permeable channel, emphasizing the influence of nonlinear curvature and wall properties. The study employs wave trains of varying amplitudes and phases to induce an asymmetric configuration along the channel boundaries. The curvature effects are contingent upon the ratio of channel width to wavelength, and the accuracy of results is maintained up to the second order in δ. To facilitate the derivation of closed-form solutions at higher levels, a domain transformation is executed, converting a channel with a variable cross section into one with a uniform cross section. To impose constraints on parameters and achieve a singular flux in the presence of a specified pressure gradient, a distinctiveness criterion is formulated. This study investigates the interplay of inertia and curvature on pumping and trapping, examining both symmetric and asymmetric channels. The pressure difference escalates with wave number in both channel configurations. Moreover, in the absence of inertial forces and flux, the impact of curvature on the pressure difference demonstrates a direct proportionality. Conversely, an augmentation in the permeability of the porous channel correlates with a reduction in the pressure difference. An intriguing observation arises as phase differences increase: the visibility of the bolus diminishes, and the streamlines become more prominent. This establishes a favorable environment for fluid flow, enhancing our understanding of the intricate dynamics at play in peristaltic flow within permeable channels.

Some non-Riemannian curvature properties of the new class of $$(\alpha , \beta )$$-metrics

Some non-Riemannian curvature properties of the new class of $$(\alpha , \beta )$$-metrics

Some curvature properties of spherically symmetric Finsler metrics

In this paper, we study some important Remannian and non-Riemannian curvature properties of spherically symmetric Finsler metrics. Under a condition on the geodesic coefficient, we find the necessary and sufficient conditions under which spherically symmetric metrics are of scalar flag curvature, [Formula: see text]-quadratic or projectively Ricci-flat. For spherically symmetric metrics of relatively isotropic Landsberg curvature, we find the necessary and sufficient conditions under which these metrics are of constant flag curvature or Ricci-quadratic. Finally, we prove a rigidity result that every spherically symmetric metric of relatively isotropic Landsberg curvature is Ricci-quadratic if and only if it is a Berwald metric. Moreover, if the Finsler metric is negatively complete then it reduces to a Riemannian metric.

A SIR Epidemic Model Allowing Recovery

The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, S(t) and R(t), respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of −S′(0), the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if S′(0)&gt;0 and to the SIR model if S′(0)&lt;0. The separating case is completely soluble and S(t) is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms.

Visualizing high-dimensional loss landscapes with Hessian directions

Analyzing the geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural-network structure, implementation attributes, and learning performance. In this paper, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are rarely correctly identified as such in the expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e. the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable for correctly identifying saddle information, we propose to study projections along the dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to approximately 7×106 parameters.