Curvature Conditions Research Articles

Evaluation of Secondary Buckling and Ultimate Strength Behaviour According to Boundary Conditions of Curved Plate Subjected to Compressive Load

This paper analyzed the buckling and ultimate strength characteristics of plate members with curvature, which are essentially used in ships and marine structures, through numerical analysis. Major structural members are affected by rotational velocity depending on the rigidity of structures placed around them, and it is very important to check how this affects buckling and ultimate strength. The aspect ratio of the curved plate was adopted as 2.5, 3.0, and 3.5 to reflect the plate thickness range and high-strength steel characteristics actually used. In the case of no curvature, and in the case of curvature, 5 degrees, 10 degrees, 15 degrees, 20 degrees, 30 degrees, and 45 degrees were considered. Under certain thickness and curvature conditions, secondary buckling occurs and the buckling state of the plate changes rapidly. Therefore, the final strength is evaluated lower than in the case where there is no curvature. Buckling and ultimate strength were high in a clamped condition around the curved plate, and there was no specificity in behavior depending on the aspect ratio change. In the four-side simply supported condition, the in-plane stiffness gradient changes depending on the aspect ratio due to the secondary buckling effect, and this characteristic is a phenomenon that is not considered at all in the existing classification method.

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry

n Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is given in this review article, which also summarizes their historical evolution, core ideas, important findings, and applications. We investigate the complex interactions between curvature conditions and geometric inequalities as well as the several kinds of Ricci solitons, such as expanding, steady, and shrinking solitons. We also go over current developments, unresolved issues, and possible paths for further study in this fascinating area.

Temperature-insensitive and cost-effective distributed NP-Doped optical fiber sensors

This paper presents the development of a cost-effective distributed optical fiber sensor for temperature-insensitive assessment of mechanical disturbances along an optical fiber cable. The proposed sensor system uses a nanoparticle (NP)-doped optical fiber with enhanced Rayleigh backscattering to provide higher sensitivity and spatial resolution using the transmission and reflection analysis (TRA) approach, where the transmitted and backscattered optical powers are analyzed as a function of the mechanical disturbance. In addition, Fiber Bragg Gratings (FBGs) are used as wavelength filters to provide the wavelength division multiplexing of the proposed device, which enable the use of 3 different NP-doped optical fiber sections for simultaneous detection of multiple curvature conditions in a cost-effective distributed sensing approach. The sensor characterization tests are performed by means of applying curvature angles from 360° to 1080° at different positions along NP-doped fibers (namely 25 mm, 100 mm and 175 mm) at 4 different temperatures of 25 °C, 30 °C, 40 °C and 50 °C. The results indicate the feasibility of the proposed approach, where the temperature variations lead only to a wavelength shift of the Bragg wavelength, whereas the mechanical disturbances (the curvatures) lead only to variations in the transmitted and reflected optical powers. Thus, by analyzing the transmitted and reflected optical powers in conjunction with the Bragg wavelength shift, it is possible to estimate both the mechanical disturbance amplitude (i.e., curvature angle) and the position along each NP-doped optical fiber section. Results indicate a relative error of around 3 % and 4 % for the mechanical disturbance location and absolute value, respectively. Moreover, the temperature cross-sensitivity in this case is below 2 % considering both amplitude and location of the mechanical disturbance. The proposed approach can be applied in structural health monitoring of different types of structures by integrating the fibers in the structures themselves with the possibility of measuring the strain distribution along the fibers (instead of in different points along the fiber) using a lower cost hardware when compared with similar distributed optical fiber sensing approaches.

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g) is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric dg induced on the boundary ∂M? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in R3 can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

Visualization study on the dynamic behavior of shear thinning droplets impacting superhydrophobic spheres

The surface and droplet characteristics significantly affect the dynamic of droplet impact on solid surfaces. In present work, a platform is utilized to experimentally study the dynamic behavior of shear thinning fluid droplets impacting superhydrophobic spheres, where the effects of different concentrations, Weber numbers (We), particle size ratios on droplets impact are studied. The results showed that on the superhydrophobic sphere, as the shear thinning droplet concentration increases, the diffusion diameter increases, and the contact time decreases. For the same curvature, the contact time between the droplet and the sphere decreases with the increase of the We. When the We reaches 75 or above, the contact time remains constant with an increase in the We. For various curvature conditions, the contact time increases as the curvature increasing. When the diameter ratio of the sphere to the droplet is D∗≥6, the maximum diffusion coefficient remains almost constant. However, when D∗<6, the maximum diffusion dimensionless diameter significantly increases as the particle size ratio decreases. More importantly, a theoretical model considering gravity effect is presented to predict the maximum dimensionless diameter of shear thinning droplets on superhydrophobic spheres according to energy conservation. The predicted results are rationally agreement with experiments with the deviation within ±10 %.

η-Einstein soliton on α-Sasakian manifold with respect to Zamkovoy connection

The purpose of this paper is to study some properties of [Formula: see text]-Sasakian manifolds with respect to Zamkovoy connection. We study [Formula: see text]-Einstein soliton on pseudo-projectively flat [Formula: see text]-Sasakian with respect to Zamkovoy connection. Further, we discuss [Formula: see text]-Einstein soliton on this manifold under various curvature conditions.

The Projectivity of Compact Kähler Manifolds with Mixed Curvature Condition

The Projectivity of Compact Kähler Manifolds with Mixed Curvature Condition

Gradient conformal stationarity and the CMC condition in LRS spacetimes

Abstract We study the existence of gradient conformal Killing vectors (CKVs) in the class of locally rotationally symmetric (LRS) spacetimes which generalizes spherically symmetric spacetimes, and investigate some implications for the evolutionary character of marginally outer trapped surfaces. We first study existence of gradient CKVs via the obtention of a relationship between the Ricci curvature and the gradient of the divergence of the CKV. This provides an alternative set of equations, for which the integrability condition is obtained, to analyze the existence of gradient CKVs. A uniqueness result is obtained in the case of perfect fluids, where it is demonstrated that the Robertson-Walker solution is the unique perfect fluid solution with a nonvanishing pressure, admitting a timelike gradient CKV. The constant mean curvature condition for LRS spacetimes is also obtained, characterized by three distinct conditions which are specified by a set of three scalars. Linear combinations of these scalars, whose vanishing define the constant mean curvature condition, turn out to be related to the evolutions of null expansions of 2-spheres along their null normal directions. As such, some implications for the existence of black holes and the character of the associated horizons are obtained. It is further shown that dynamical black holes of increasing area, with a non-vanishing heat flux across the horizon, will be in equilibrium, with respect to the frame of the conformal observers.

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Gradient estimates for Yamabe type equations under different curvature conditions and applications

The aim of this paper is to give various aspects of gradient estimates for positive solutions to the Yamabe type equation ut=Δfu+au+buα. We study the problem in many different settings. As result, we obtain several types of gradient estimates for the solution. When the manifold is complete without boundary, we study gradient estimates of Shi type, and of Hamilton type. Moreover, we also derive gradient estimates if the manifold is evolved under the (k,m)-super Perelman-Ricci flows. Besides, we prove gradient estimates under the integral Ricci curvature bounds. Finally, when the manifold has compact boundary, we investigate gradient estimates with Dirichlet boundary conditions. Our theorems generalize and improve many previous works.

Understanding the impact of sky diffuse irradiance on curved photovoltaic modules

Modules utilized in vehicle-integrated photovoltaics (VIPV) have particularities that make them differ from standard PV modules. These particularities are related to their curvature and operating conditions (changing locations, orientations, etc.) and they need to be addressed in the characterization. A vital characterization test is the angular response test, which allows obtaining the electrical response of the module under different angles of incidence and orientations. Results from these tests on real commercial modules presented some dispersion, which is related to diffuse irradiance. Two outdoors experiments involving a programmable dual-axis tracker were conducted to verify its impact: the first one was made with a 1D highly curved module under cloudy sky conditions to identify the effect of diffuse irradiance on such modules and the second experiment assessed the angular response of a 2D commercial curved module under clear sky conditions. Based on the findings, recommendations are given to minimize dispersion in outdoors angular response measurements.

$L^{p}$ improving properties and maximal estimates for certain multilinear averaging operators

In this article we focus on L^{p} estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the L^{1} -improving estimates of multilinear averaging operators. We also obtain L^{p} -improving estimates for certain multilinear averages by means of the nonlinear Brascamp–Lieb inequality.

Complete stable minimal hypersurfaces in positively curved 4-manifolds

We show that the combination of non-negative sectional curvature (or 2 -intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a 4 -manifold with bounded curvature. Our work leads to new comparison results. We also construct various examples showing that rigidity of stable minimal hypersurfaces can fail under other curvature conditions.

Lattice eigenfunction equations of KdV-type *

We develop lattice eigenfunction equations of the lattice KdV equation, which are equations obeyed by auxiliary functions, or eigenfunctions, of the Lax pair of the lattice KdV equation. These equations are three-dimensionally consistent quad-equations, that are closely related to lattice equations in the Adler-Bobenko-Suris (ABS) classification. The connection between the H3(δ), Q1(δ), Q2 and Q3(δ) equations in the ABS classification and the lattice eigenfunction equations is explicitly showed. In particular, we provide a natural interpretation of the δ term in those equations. This can be understood as ‘interactions’ between the eigenfunctions. Other integrable properties of the eigenfunction equations, such as exact solutions, discrete zero curvature conditions are also provided. We believe that the approach presented in this paper can be used as a means to search for integrable lattice equations.>

Geometric Classifications of Perfect Fluid Space-Time Admit Conformal Ricci-Bourguignon Solitons

This paper is dedicated to the study of the geometric composition of a perfect fluid space-time with a conformal Ricci-Bourguignon soliton, which is the extended version of the soliton to the Ricci-Bourguignon flow. Here, we have delineated the conditions for conformal Ricci-Bourguignon soliton to be expanding, steady, or shrinking. We have studied certain curvature conditions on the spacetime that admit conformal Ricci-Bourguignon soliton. We have also discussed conformal Ricci-Bourguignon soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid, and radiation era.

Almost η-Ricci solitons on two classes of almost Kenmotsu manifolds

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting almost η-Ricci solitons. In this context, we have shown that in a (k, µ) and (k, µ)' -almost Kenmotsu manifold admitting an almost η-Ricci soliton the curvature conditions (i) the manifold is Einstein, (ii) the manifold is Ricci symmetric (∇S = 0), (iii) the manifold is Ricci semisymmetric (R · S = 0) and (iv) the manifold is projective Ricci semisymmetric (P · S = 0) are equivalent. Also, we have shown that the curvature condition Q · P = 0 in a (k, µ)-almost Kenmotsu manifold admitting an almost η-Ricci soliton holds if and only if the manifold is locally isometric to the hyperbolic space H2n+1(−1) and if a (k, µ)' -almost Kenmotsu manifold admitting an almost η-Ricci soliton satisfies the curvature condition Q · R = 0, then it is locally isometric to the Riemannian product H n+1(−4) × ℝn. n.

Compact gradient shrinking ρ-Einstein solitons with pinching conditions

We prove that an n-dimensional, n≥4, compact gradient shrinking ρ-Einstein soliton satisfying suitable pinching conditions and curvature conditions is isometric to a quotient of the round sphere Sn. Our results extend the rigidity theorems given by Huang (Integral pinched gradient shrinking ρ-Einstein solitons, 2017) in dimension 4≤n≤6.

On Wintgen Ideal Submanifolds Satisfying Some Pseudo-symmetry Type Curvature Conditions

Let M be a Wintgen ideal submanifold of dimension n in a real space form Rn+m(k) of dimension (n + m) and of constant curvaturek, n ≥ 4, m ≥ 1. Let g, R, Ricc, g ∧ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms Rn+m(k), n ≥ 4, m ≥ 1, satisfying the following pseudo-symmetry type curvature conditions: (i) the tensors R · C and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (ii) the tensors C · R and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (iii) the tensors R·C -C ·R and Q(g, R) (resp., Q(g, C), Q(g, g∧Ricc), Q(Ricc, R) or Q(Ricc, g∧Ricc)) are linearly dependent.

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

Abstract 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.