Three-dimensional Hypersurface Research Articles

Complete Contraction of Tensor and Its Application

This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any λ type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple.

Modeling the workspace of a three-link planar manipulator

A study of the working space of a three-link planar manipulator was carried out. The basis is taken from analytical dependencies that allow solving the direct problem of kinematics, i.e. determine the coordinates of the gripper center point using three generalized coordinates of the manipulator. The analysis performed made it possible to give a geometric interpretation of the dependencies. It has been established that the workspace of the manipulator is a three-parameter set of points. On a plane, this set of points is represented in the form of two disks consisting of ring cells, for which the corresponding analytical dependencies are obtained. The geometric image of this set is a three-dimensional torus. The resulting models are visualized, which facilitates the solution of this problem. To determine the values of the generalized coordinates of the gripper center points, a mapping was carried out by orthogonal projection of the families of circles obtained in the work into four-dimensional space. As a result, three-dimensional hypersurfaces in four-dimensional space were obtained. It is proposed to study them by constructing hypersurface sections models by hyperplanes. Such models in visualization mode allow solving direct and inverse kinematics problems of the manipulator under study.

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds, which in turn yields the first initial data characterization of Ricci-flat pp-waves. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of -Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows in Riemannian signature.

A Curvature Inequality Characterizing Totally Geodesic Null Hypersurfaces

A well-known application of the Raychaudhuri equation shows that, under geodesic completeness, totally geodesic null hypersurfaces are unique which satisfy that the Ricci curvature is nonnegative in the null direction. The proof of this fact is based on a direct analysis of a differential inequality. In this paper, we show, without assuming the geodesic completeness, that an inequality involving the squared null mean curvature and the Ricci curvature in a compact three-dimensional null hypersurface also implies that it is totally geodesic. The proof is completely different from the above, since Riemannanian tools are used in the null hypersurface thanks to the rigging technique.

Universal and flexible design for high-sensitivity and wide-ranging surface plasmon resonance sensors based on a three-dimensional tuning hypersurface

Surface plasmon resonance (SPR), generally including propagating SPR (PSPR) and localized SPR (LSPR), has attracted wide attention due to the uniqueness of the strong evanescent field. The PSPR has the merit of high sensitivity over the LSPR, but lacks flexibility to tune resonant wavelength and serves limited application. Here we revisit the fundamental PSPR wavevector matching theory and construct a three-dimensional (3D) tuning hypersurface for the parameters of resonant wavelength, incident angle, and surrounding refractive index, which provides the performance tuning guide map. With assistance of the hypersurface, a hypersurface-tuning PSPR (HT-PSPR) sensor is designed with high sensitivity and ultra-wide detection range. Experiments demonstrate that the sensor can selectively tune the resonance and render the sensitivity to a theoretical upper limit of ∼105 nm/RIU, or an ultra-wide detection range of 0.518 RIU, which satisfies the requirement for most gas and liquid sensing applications. Using D-biotin as the detection sample of small molecule (MW=244 Da), the HT-PSPR sensor achieves an ultra-low detection limit down to 27.5 fM. The hypersurface-tuning has created an intrinsic, universal and flexible method for tuning PSPR at the desired resonant wavelength, sensitivity and detection range, which is expected to find great significance in SPR design and applications.

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

Hodge numbers of hypersurfaces in ℙ4 with ordinary triple points

Abstract We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities.

Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues
 are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.

Эффективность регенерации отработанного моторного масла в зависимости от параметров процесса

<b>Object and purpose of research. </b>The purpose of this study is to construct regulatory diagram of water-fuel fraction stripping from a thin slick of spent motor oil in a cyclonic vacuum-thermal evaporator and estimate the conditions in which this regeneration process is the most efficient <br><b> Subject matter and methods.</b>Regulatory diagram are based on the plots of plane sections with the isolines of water-fuel fraction stripping efficiency, three-dimensional hypersurfaces of stripping functions in natural values characterizing the efficiency of water-fuel fraction stripping from spent motor oil slick with reference to the temperature and vacuum pressure in cyclonic vacuum-thermal evaporator. <br><b>Main results. </b>Regulatory diagrams are a part of integrated mathematical model describing “soft” regeneration of spent motor oil as per the procedure suggested by the author for both separate and joint stripping of water and light fuels from spent motor oil of a marine diesel. These regulatory diagrams indicate the temperature of spent motor oil and vacuum pressures offering the greatest efficiency of fuel and water stripping for vacuum-thermal evaporators of various capacity. <br><b> Conclusion. </b>Regulatory diagrams are necessary to predict the results of regeneration process bringing viscosity and flash point of regenerated motor oil well above the minimum acceptable levels stipulated in regulatory documents, so that the oil thus regenerated becomes again suitable for use in internal combustion engines.

𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms

Abstract Kaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H 2(c) with tanh ⁡ ( | c | 2 r ) = 1 2 $\begin{array}{} \displaystyle \tanh(\frac{\sqrt{|c|}}{2}r) = \frac{1}{2} \end{array}$ or a ruled real hypersurface.

Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Photon region (PR) in the strong gravitational field is defined as a compact region where photons can travel endlessly without going to infinity or disappearing at the event horizon. In Schwarzschild metric PR degenerates to the two-dimensional photon sphere r=3r_g/2 where closed circular photon orbits are located. The photon sphere as a three-dimensional hypersurface in spacetime is umbilic (its second quadratic form is pure trace). In Kerr metric the equatorial circular orbits have different radii for prograde, r_p, and retrograde, r_r, motion (where r is Boyer–Lindquist radial variable), while for r_p<r<r_r the spherical orbits with constant r exist which are no more planar, but filling some spheres. These spheres, however, do not correspond to umbilic hypersurfaces. In more general stationary axisymmetric spacetimes not allowing for complete integration of geodesic equations, the numerical integration show the existence of PR as well, but the underlying geometric structure was not fully identified so far. Here we suggest geometric description of PR in generic stationary axisymmetric spacetimes, showing that PR can be foliated by partially umbilic hypersurfaces, such that the umbilic condition holds for classes of orbits defined by the foliation parameter. New formalism opens a way of analytic description of PR in stationary axisymmetric spacetimes with non-separable geodesic equations.

Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

Let [Formula: see text] be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that [Formula: see text] is [Formula: see text]-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in [Formula: see text] or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in [Formula: see text]. Moreover, [Formula: see text]-parallel real hypersurfaces in [Formula: see text] and [Formula: see text] under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with [Formula: see text]-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

Generating functional for gravitational null initial data

A field theory on a three-dimensional manifold is introduced, whose field equations are the constraint equations for general relativity on a three-dimensional null hypersurface. The underlying boundary action consists of two copies of the dressed Chern–Simons term for self-dual Ashtekar variables, a kinetic term for the null flag at the boundary plus additional junction conditions for the spin coefficients across the interface. In fact, there is a doubling of the field content, because the null hypersurface is treated as an internal boundary between two adjacent slabs of spacetime. The paper concludes with a proposal for the construction of the gravitational transition amplitudes in the bulk via a gluing prescription for edge states across two-dimensional corners, thus providing a proposal for a quasi-local realisation of the holographic principle at the light front.

Hamiltonian analysis of 4-dimensional spacetime in Bondi-like coordinates**Supported by the National Natural Science Foundation of China (11690022)

We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,xa,a = 2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T±(2), whose Lie algebra (1,3) is decomposed into and t±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in BF theory. General relativity can be reformulated as the four-dimensional coframe () and connection () dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e−, e+ and two spacelike 1-forms e2, e3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about . The consistency conditions of the primary constraints can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints and 40 second-class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n0, l0, and are Ricci identities. The equations of motion of the canonical variables have also been shown.

Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = ϕ A - A ϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified.

Automorphisms of singular three-dimensional cubic hypersurfaces

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group G, which are not G-rational and have no birational structure of G-Mori fiber space with the base of positive dimension. Also we prove the $$\mathfrak {A}_{5}$$ -birational superrigidity of the Segre cubic.

The complete classification of a class of conformally flat Lorentzian hypersurfaces in ℝ14

A three-dimensional Lorentzian hypersurface [Formula: see text] is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of [Formula: see text]. In this paper, using the projective light-cone model, we give a complete classification of those ones whose shape operators have two distinct real eigenvalues and cannot be diagonalizable. These hypersurfaces are conformal equivalent to cones, cylinders, or rotational hypersurfaces generated by B-scrolls (over null Frenet curves) in three-dimensional Lorentzian space forms.

Derivatives on Real Hypersurfaces of Two-Dimensional Non-flat Complex Space Forms

The aim of this paper is to answer the question if there are three-dimensional real hypersurfaces in non-flat complex space forms whose covariant derivative with respect to the kth generalized Tanaka–Webster connection of a tensor field of type (1, 1) coincides with the Lie derivative of it either in direction of any vector field orthogonal to \(\xi \) or in direction of \(\xi \). The answer is given, in case the tensor field is the shape operator of a real hypersurface or the structure Jacobi operator of it.

Uniqueness of static photon surfaces: Perturbative approach

A photon surface $S$ is defined as a three-dimensional timelike hypersurface such that any null geodesic initially tangent to $S$ continues to be included in $S$, like $r=3M$ of the Schwarzschild spacetime. Using analytic solutions to static perturbations of a Schwarzschild spacetime, we examine whether a nonspherical spacetime can possess a distorted static photon surface. It is shown that if the region outside of $r=3M$ is vacuum, no distorted photon surface can be present. Therefore, we establish the perturbative uniqueness for an asymptotically flat vacuum spacetime with a static photon surface. It is also pointed out that if matter is present in the outside region, there is a possibility that a distorted photon surface could form.

A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of their Generalized Tanaka–Webster Lie Derivative

AbstractOn a real hypersurface M in a non-flat complex space form there exist the Levi–Civita and the k-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.