Shape Operator Research Articles

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

Determination of in-plane anisotropy of rolled sheets, taking into account the effect of strain intensity

The results of experimental studies of the mechanical properties anisotropy of rolled sheets made of NKh260YD steel are presented. Tension was applied to flat specimens cut in directions of 0, 45, and 90 degrees with respect to the rolling direction. A dividing grid was applied to the samples with a diamond indenter in the area of the estimated length on an instrumental microscope. Based on the results of measurements of the samples linear dimensions before and after deformation in the places where the dividing grid was applied, the logarithmic deformations were calculated over the width and thickness of the sample in each of the considered sections. The results of measurements and calculations showed the uneven intensity of deformations in its various sections. For further processing, data in sections under conditions of uniform deformation were used. The anisotropy coefficients were determined as the ratio of logarithmic strains over the width and thickness of the sample. The tension was carried out in three stages, which made it possible to establish the dependences of the anisotropy coefficients on the strain intensity, which were used to assess the degree of in-plane anisotropy of the material under study. Graphic dependences are presented illustrating the nature of the change in the anisotropy coeffi cients and the degree of anisotropy of the material under study depending on the intensity of deformation. A significant dependence of the HX260YD steel planar anisotropy degree on the magnitude of the strain intensity is shown, which must be taken into account when developing technological processes for production of critical articles from anisotropic sheet material by plastic shaping operations.

Chen inequalities on spacelike hypersurfaces of a GRW spacetime

In this paper, we establish Chen-Ricci inequality and Chen inequalities for spacelike hypersurfaces in GRW spacetimes. Considering these inequalities, we give some characterizations. Equality cases are also considered. We show that M is a totally geodesic or a totally umbilical in case of equalities. Moreover, we find some basic inequalities for shape operator of such hypersurfaces and obtain some inequalities for spacelike slices in GRW spacetimes.

Scenario Expression Method for Regional Geological Structures

Knowing the GIS expression of geological phenomena is an important basis for the combination of geology and GIS. Regional geological structures include folds, faults, strata, rocks, and other typical geological phenomena and are the focus of geological GIS research. However, existing research on the GIS expression of regional geological structure focuses on the expression of the spatial and attribute characteristics of geological structures, and our knowledge of the expression of the semantic, relationship, and evolution processes of geological structures is not comprehensive. In this paper, a regional geological structure scene expression model with the semantic terms positional accuracy, geometric shape, relationship type, attribute type, and time-type attributes and operations is proposed. A regional geological structure scenario markup language (RGSSML) and a method for mapping it with graphics are designed to store and graphically express regional geological structure information. According to the geological time scale, a temporal reference coordinate system is defined to dynamically express the evolution of regional geological structures. Based on the dynamic division of the time dimension of regional geological structures, the expression method of “time dimension + space structure” for the regional geological structure evolution process is designed based on the temporal model. Finally, the feasibility and effectiveness of the regional geological structure scene expression method proposed in this paper is verified using the Ningzhen Mountain (Nanjing section) as an example. The research results show that the regional geological structure scene expression method designed in this paper has the following characteristics: (1) It can comprehensively express the spatial characteristics, attribute characteristics, semantics, relationships, and evolution processes of regional geological structures; (2) it can be used to realize formalized expression and unified storage of regional geological information; and (3) it can be used to realize dynamic expression of the regional geological structure evolution process. Moreover, it has significant advantages for the expression of regional geological structure semantics, relationships, and evolution processes. This study improves our knowledge of the GIS expression of regional geological structures and is expected to further promote the combination and development of geology and GIS.

Ricci solitons on pseudo–Riemannian hypersurfaces of 4–dimensional Minkowski space

In this paper, we get classification theorems for Ricci solitons on pseudo–Riemannian hypersurfaces of a Minkowski space E14 by taking the potential vector field as the tangent component of the position vector of the pseudo–Riemannian hypersurfaces. First, we obtain the necessary and sufficient condition for a pseudo–Riemannian hypersurface in E14 which admits such a Ricci soliton. According to the forms of the shape operator of a pseudo–Riemannian hypersurface, we obtain characterization results about Ricci solitons on the pseudo–Riemannian hypersurfaces in E14. More precisely, we show that totally umbilical isoparametric hypersurfaces, hyperbolic and a pseudo–spherical cylinder in E14 admit shrinking Ricci solitons whose potential vector fields are the tangent part of the position vectors in E14. Furthermore, we also get that a generalized umbilical Lorentzian hypersurface in E14 admits a shrinking Ricci soliton. Finally, we obtain that there do not exist Ricci solitons on the Lorentzian hypersurfaces in E14 whose minimal polynomials have complex roots or real roots with multiplicity three.

Seismic attenuation compensation with spectral-shaping regularization

Because of the viscoelasticity of the subsurface medium, seismic waves will inherently attenuate during propagation, which lowers the resolution of the acquired seismic records. Inverse-Q filtering, as a typical approach to compensating for seismic attenuation, can efficiently recover high-resolution seismic data from attenuation. Whereas most efforts are focused on compensating for high-frequency energy and improving the stability of amplitude compensation by inverse-Q filtering, low-frequency leakage may occur as the high-frequency component is boosted. In this article, we propose a compensation scheme that promotes the preservation of low-frequency energy in the seismic data. We constructed an adaptive shaping operator based on spectral-shaping regularization by tailoring the frequency spectra of the seismic data. We then performed inverse-Q filtering in an inversion scheme. This data-driven shaping operator can regularize and balance the spectral-energy distribution for the compensated records and can maintain the low-frequency ratio by constraining the overcompensation for high-frequency energy. Synthetic tests and applications on prestack common-reflection-point gathers indicated that the proposed method can preserve the relative energy of low-frequency components while fulfilling stable high-frequency compensation.

Semisymmetric hypersurfaces in complex hyperbolic two-plane Grassmannians

In this paper, we introduce new notions of symmetric operators such as semisymmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with such notions.

Linear Interpolation of Shape Operators for Umbilical Points Through Local Parametrization

Linear Interpolation of Shape Operators for Umbilical Points Through Local Parametrization

Seismic Resolution Enhancement by Spectral Shaping Using Shaping-Regularized Inversion

Seismic deconvolution aims to improve the vertical seismic resolution by compressing the seismic wavelet and extending the seismic frequency bandwidth. Deconvolution in the frequency domain is normally implemented in three steps: wavelet estimation, deconvolution operator construction, and data filtering. We propose a seismic resolution enhancement approach by spectral shaping using shaping-regularized inversion. Instead of designing a deconvolution operator based on the extracted wavelet, we formulate an inverse problem using the seismic spectrum as the diagonal kernel matrix and a predefined filter as the expected spectrum. Assuming the randomness in the reflectivity and smoothness of the spectral shaping operator, we estimate the spectral shaping operator by inversion via a shaping regularization scheme, which imposes constraints by shaping the estimated spectral shaping operator model to the admissible model space. The role of the shaping regularization in inversion is to ensure the continuity and smoothness of the spectral shaping operator. We use a synthetic model and real land seismic data to demonstrate the effectiveness of the proposed approach in seismic resolution enhancement.

Classification of f-biharmonic submanifolds in Lorentz space forms

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

Hypersurfaces of the nearly Kähler twistor spaces $\mathbb{C}\mathrm{P}^3$ and ${\mathbb{F}_{1,2}}$

In this article, we show that a hypersurface of the nearly Kähler ${\mathbb{C}\mathrm{P}}^3$ or ${\mathbb{F}_{1,2}}$ cannot have its shape operator and induced almost contact structure commute together. This settles the question for six-dimensional homogeneous nearly Kähler manifolds, as the cases of ${\mathbb{S}}^6$ and ${\mathbb{S}}^3 \times {\mathbb{S}}^3$ were previously solved, and provides a counterpart to the more classical question for the complex space forms ${\mathbb{C}\mathrm{P}}^n$ and ${\mathbb{C}\mathrm{H}}^n$. The proof relies heavily on the construction of ${\mathbb{C}\mathrm{P}}^3$ and ${\mathbb{F}_{1,2}}$ as twistor spaces of ${\mathbb{S}^{4}}$ and ${\mathbb{C}\mathrm{P}}^2$.

GEOIDS AS TWO DIMENSIONAL HYPERSURFACES

<jats:p>We propose a research of geoids as real hypersufaces. Moreover, we give adjustment of some classical results on hyperusrfaces to CR submanifolds. Our main focus is to study the properties of second fundamental form which give us information on the shape of a hypersurface. Keywords: geoids, hypersurfaces, shape operator, CR submanifolds.</jats:p>

Parallel spinors on globally hyperbolic Lorentzian four-manifolds

We investigate the differential geometry and topology of globally hyperbolic four-manifolds (M, g) admitting a parallel real spinor $$\varepsilon $$ . Using the theory of parabolic pairs recently introduced in [22], we first formulate the parallelicity condition of $$\varepsilon $$ on M as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on an appropriate Cauchy surface $$\Sigma \hookrightarrow M$$ . The existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on $$\Sigma $$ , which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding $$\Sigma \hookrightarrow M$$ . Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair $$({\mathfrak {e}},\Theta )$$ , where $${\mathfrak {e}}$$ is a coframe and $$\Theta $$ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of $${\mathbb {R}}^2$$ and are isomorphic to certain torus bundles over $$S^1$$ , whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.

Elastic-hereditary errors of fixation of non-rigid box blanks

The literature review has shown that so far no studies have been carried out on the elastic deformations of open box blanks during installation and fixation in machine tools. In the analysis of the installation of non-rigid axisymmetric blanks, it is indicated that the existing shape error of the base surface elastically changes its shape during fastening, and the processed surface gets the correct shape, which, after removing the clamping force, is distorted, and the base surface is not restored to its original shape due to the removal of the allowance layer. Typical operations of preliminary shaping of an open box-shaped workpiece obtained from medium-alloy steel 30HGSA with increased strength are described. The workpiece has errors in the relative position of the elements up to ± 0.3 mm. In production conditions the elastic-hereditary errors of fixing non-rigid box blanks were determined when compensating for the initial position of the deformed wall of the blank, by monitoring with the built-in sensor of the machine. It has been determined that such compensation is insufficient. For theoretical justification of the new proposals, the calculation of the stress-strain state of the construction using finite element analysis was made, which showed the deviation values of the side walls of the blank. Errors that most affect the deviations of the side walls when processing blanks of reduced rigidity are noted. A method of fixing and processing of thin-walled body blanks is proposed, in which the base side of the blank is measured using an RMP40 sensor before clamping force and after it, followed by adding value difference compensations in the control program for processing on a CNC milling machine. A comparative analysis of the dimensional deviation under the action of clamping force and after release from the clamps was made. A trial batch of blanks was processed, confirming the results of theoretical studies.

Operations Decisions and the Influence of Host-Country Institutions: A Study under the Paraguayan “Maquila Regime”

This study analyzes how formal institutions shape the firm’s decisions while exploring two polar cases of Brazilian textile-apparel firms that expanded their operations to Paraguay under the “Maquila regime.” It is guided by the institution-based view of international business. Different aspects of the institutional environment were analyzed to explain how the uncertainty can hamper firms from making quick operations decisions. Mechanisms of control and compliance imposed by institutions can negatively impact product-related decisions and innovation. Our research unveils the role of formal institutions in shaping operations decisions, helping to bridge the research gap about interactions between institutions and firm-level strategies.

Lie Derivatives of the Shape Operator of a Real Hypersurface in a Complex Projective Space

We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, B^{(k)}_T, related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor A^{(k)}_T associated with the shape operator A of M.

Ultrasensitive detection of endocrine disruptors via superfine plasmonic spectral combs

The apparent increase in hormone-induced cancers and disorders of the reproductive tract has led to a growing demand for new technologies capable of detecting endocrine disruptors. However, a long-lasting challenge unaddressed is how to achieve ultrahigh sensitive, continuous, and in situ measurement with a portable device for in-field and remote environmental monitoring. Here we demonstrate a simple-to-implement plasmonic optical fiber biosensing platform to achieve an improved light–matter interaction and advanced surface chemistry for ultrasensitive detection of endocrine disruptors. Our platform is based on a gold-coated highly tilted fiber Bragg grating that excites high-density narrow cladding mode spectral combs that overlap with the broad absorption of the surface plasmon for high accuracy interrogation, hence enabling the ultrasensitive monitoring of refractive index changes at the fiber surface. Through the use of estrogen receptors as the model, we design an estradiol–streptavidin conjugate with the assistance of molecular dynamics, converting the specific recognition of environmental estrogens (EEs) by estrogen receptor into surface-based affinity bioassay for protein. The ultrasensitive platform with conjugate-induced amplification biosensing approach enables the subsequent detection for EEs down to 1.5 × 10−3 ng ml−1 estradiol equivalent concentration level, which is one order lower than the defined maximal E2 level in drinking water set by the Japanese government. The capability to detect EEs down to nanogram per liter level is the lowest limit of detection for any estrogen receptor-based detection reported thus far. Its compact size, flexible shape, and remote operation capability open the way for detecting other endocrine disruptors with ultrahigh sensitivity and in various hard-to-reach spaces, thereby having the potential to revolutionize environment and health monitoring.

Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space

In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed $H$, the strategy consists in rewriting the equation for $H$, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for surfaces of revolution with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for $S$, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of $H$ and $S$.

The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects.