Narrow Results

Following the publication online, it has been brought to the attention of the Editorial Board of Communications in Contemporary Mathematics (CCM) that the result of this paper was known since 1992 and appeared in the article by Luis Alías, Angel Ferrández and Pascual Lucas, Surfaces in the 3-dimensional Lorentz–Minkowski space satisfying $\Delta x=\mathrm{A}x+B$, Pacific J. Math.156(2) (1992) 201–208.

The information has been verified, and therefore the Editors-in-Chief of CCM have decided to retract the acceptance of this paper.

Let $\mathrm{\Omega }={\mathrm{\Omega }}_0\setminus \overline{\mathrm{\Theta }}\subset {ℝ}^n$, $n\ge 2$, where ${\mathrm{\Omega }}_0$ and $\mathrm{\Theta }$ are two open, bounded and convex sets such that $\overline{\mathrm{\Theta }}\subset {\mathrm{\Omega }}_0$ and let $\beta <0$ be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to $\mathrm{\Omega }$, with Robin boundary condition on $\partial {\mathrm{\Omega }}_0$ and Neumann boundary condition on $\partial \mathrm{\Theta }$. In [47] it is proved that the spherical shell is the only maximizer for the first Robin–Neumann eigenvalue in the class of domains $\mathrm{\Omega }$ with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed.

We show that if the Ricci curvature of a left-invariant metric g on S³ is greater than that of the standard metric g₀, then the eigenvalues of Δ_g are greater than the corresponding eigenvalues of Δ_go.

The eigentime identity for random walks on networks is the expected time for a walker going from a node to another node. In this paper, our purpose is to calculate the eigentime identities of flower networks by using the characteristic polynomials of normalized Laplacian and recurrent structure of Markov spectrum.

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green’s function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green’s function explicitly.

Three-dimensional (3D) human mesh sequences obtained by 3D scanning equipment are often used in film and television, games, the internet, and other industries. However, due to the dense point cloud data obtained by 3D scanning equipment, the data of a single frame of a 3D human model is always large. Considering the different topologies of models between different frames, and even the interaction between the human body and other objects, the content of 3D models between different frames is also complex. Therefore, the traditional 3D model compression method always cannot handle the compression of the 3D human mesh sequence. To address this problem, we propose a sequence compression method of 3D human mesh sequence based on the Laplace operator, and test it on the complex interactive behavior of a soccer player bouncing the ball. This method first detects the mesh separation degree of the interactive object and human body, and then divides the sequence into a series of fragments based on the consistency of separation degrees. In each fragment, we employ a deformation algorithm to map keyframe topology to other frames, to improve the compression ratio of the sequence. Our work can be used for the storage of mesh sequences and mobile applications by providing an approach for data compression.

This paper deals with an eigenvalue problem for the Laplace operator on a bounded domain with smooth boundary in ℝ ^N (N ≥ 3). We establish that there exist two positive constants λ_* and λ* with λ_* ≤ λ* such that any λ ∈ (0, λ_*) is not an eigenvalue of the problem while any λ ∈ [λ*, ∞) is an eigenvalue of the problem.

In this paper, we classify translation surfaces in the three-dimensional simply isotropic space ${𝕀}_3^1$ under the condition ${\mathrm{\Delta }}^i{x}_i={\lambda }_i{x}_i$ where $\mathrm{\Delta }$ is the Laplace operator with respect to the first and second fundamental forms and $\lambda$ is a real number. We also give explicit forms of these surfaces.

In [M. K. Karacan, D. W. Yoon and B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space ${𝕀}_3^1$, Int. J. Geom. Methods Mod. Phys.13(7) (2016) 1650088], there is a mistake in Theorem 5 that appeared in the paper. We here provide a correct theorem.

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space ${𝕀}^3$ under the condition

In this paper, power options pricing is driven via time-fractional PDE when the dynamic of underlying asset price follows a regime switching model in which the risky underlying asset depends on a continuous-time hidden Markov chain process. An exact solution for power options pricing is driven under our considered model.

By using a measure transformation method, the essential spectrum of the Laplacian in a noncompact Riemannian manifold is studied. The principle result given includes the corresponding main theorems in both Kumura [16] and Donnelly [9]. The essential spectrum of the Laplacian on one-forms is also studied.

The following sections are included:

• Introduction

• A concrete case: the case of the nonlinear Robin boundary conditions

• Acknowledgments

• References

The following sections are included:

• Introduction

• Existence of a family of solutions and corresponding functional representation around ∊ = 0

• Functional analytic representation for energy integrals

• Acknowledgement

• References