A FUNDAMENTAL SOLUTION FOR THE LAPLACE OPERATOR ON THE QUATERNIONIC HEISENBERG GROUP

In this paper, the author studies the Laplace operator on the quaternionic Heisenberg group, construct a fundamental solution for it and use this solution to prove the L-p-boundedness and the weak (1-1) boundedness of certain singular convolution operators on the quaternionic Heisenberg group.

BESOV SPACES B_p^(α,p) ON DOMAINS AND LAPLACE OPERATORS

Let be a bounded smooth domain. In this paper, the authors define the Besov spaces Bpap on , establish the atomic decomposition of these spaces, and obtain the regularity estimate of the Dirichlet problem and the Neumann problem for the Laplace operator on these spaces.

A new method of mesh simplification for 3-Dimension terrain using Laplace operator

This paper proposes a new method to simplify mesh in 3D terrain. The 3D terrain is presented by digital elevation model. First, Laplace operator is introduced to calculate sharp degree of mesh point, which indicates the variation trend of the terrain. Through setting a critical value of sharp degree, feature points are selected. Second, critical mesh points are extracted by an recursive process, and constitute the simplified mesh. Third, the algorithm of linear-square interpolation is employed to restore the characteris- tics of the terrain. Last, the terrain is rendered with color and texture. The experimental results demonstrate that this method can compress data by 16% and the error is lower than 10%.

Interpolation in R^2 with Minimum Value of the Uniform Norm of the Laplace Operator

Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis of their given values at points of a grid. Interpolating functions can be chosen by many various ways. In the paper the authors are interested in interpolating functions, for which the Laplace operator, applied to them, has a minimal norm. The authors interpolate infinite bounded sequences at the knots of the square grid in Euclidian space. The considered problem is formulated as an extremal one. The main result of the paper is the theorem, in which certain estimates for the uniform norm of the Laplace operator applied to smooth interpolating functions of two real variables are established for the class of all bounded （in the corresponding discrete norm） interpolated sequences. Also connections of the considered interpolation problem with other problems and with embeddings of the Sobolev classes into the space of continuous functions are discussed. In the final part of the main section of the paper, the authors formulate some open problems in this area and sketch possible approaches to the search of solutions. In order to prove the main results, the authors use methods of classical mathematical analysis and the theory of polynomial splines of one variable with equidistant knots.

Best constants of harmonic approximation of some convolution classes associated with Laplace operator in d-variables

In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.

TWO-WEIGHT NORM INEQUALITY FOR IMAGINARY POWERS OF A LAPLACE OPERATOR

We study two-weight norm inequality for imaginary powers of a Laplace operator in R^n, n ≥ 1, especially from weighted Lebesgue space Lv^p（R^n） to weighted Lebesgue space Lμ^p（R^n）, where 1 〈 p 〈 ∞. We prove that the two-weighted norm inequality holds whenever for some t 〉 1, （μ^t, v^t） ∈ Ap, or if （μ, v） ∈Ap, where μ and v^-1/（p-1） satisfy the growth condition and reverse doubling property.

ASYMPTOTICS FOR POSITIVE SOLUTIONS OF THE WEIGHTED LAPLACE EQUATIONS INVOLVINGCRITICAL GROWTH

A general weighted second order elliptic equation involving critical growth is considered on bounded smooth. domain in n-dimension space. There is the singular point for the weighted coefficients in the domain. With generalized blow up method, some results are obtained for asymptotic behavior of positive solutions. This problem includes Laplacian operators as special cases.

Interpolation and approximation for data living on manifold surfaces

Meshed surfaces are ubiquitous in digital geometry processing and computer graphics. The set of attributes associated with each vertex such as the vertex locations, curvature, temperature, pressure or saliency, can be recognized as data living on mani- fold surfaces. So interpolation and approximation for these data are of general interest. This paper presents two approaches for mani- fold data interpolation and approximation through the properties of Laplace-Beltrami operator （Laplace operator defined on a mani- fold surface）. The first one is to use Laplace operator minimizing the membrane energy of a scalar function defined on a manifold. The second one is to use bi-Laplace operator minimizing the thin plate energy of a scalar function defined on a manifold. These two approaches can process data living on high genus meshed surfaces. The approach based on Laplace operator is more suitable for manifold data approximation and can be applied manifold data smoothing, while the one based on bi-Laplace operator is more suit- able for manifold data interpolation and can be applied image extremal envelope computation. All the application examples demon- strate that our procedures are robust and efficient.

BOCHNER TECHNIQUE IN REAL FINSLER MANIFOLDS

Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.

Numerical Solution of Klein-Gordon Equation on Manifold Using DEC

In physics,the Klein-Gordon equation describes the motion of a quantum scalar or pseudoscalar field.Itis important to find actual values of its solutions in general timespace manifold.The paper deals with description ofdiscrete exterior calculus method for solving this equation numerically on space manifold and the time.The analysis ofstable condition and error for this method is also accomplished.

Simulation of detection and scattering of sound waves by the lateral line of a fish

A solvable model of lateral line of a fish based on a wave equation with additional boundary conditions on a set of isolated points is proposed.Within the framework of this model it is shown that the ratio of pressures on lateral lines on different fish flanks,as well as the cross section of sound scattering on both the lines,strongly depends on angles of incidence of incoming sound waves.The strong angular dependence of the pressure ratio seems to be sufficient for the fish to determine the directions from which the sound is coming.

NEW SPECTRAL CHARACTERIZATIONS OF EXTREMAL HYPERSURFACES

Let M be a closed extremal hypersurface in S^n＋1 with the same mean curvature of the Willmore torus Wm,n-m.We proved that if Spec^p（M） = Spec^p（Wm,n-m ） for p = 0, 1, 2, then M is Wm,m.

THE SPECTRUM OF COMPACT HYPERSURFACE IN SPHERE

Let M be a compact minimal hypersurface of sphere Sn+1(1). Let M be H(r)-torus of sphere Sn+1(1).Assume they have the same constant mean curvature H, the result in [1] is that if Spec0 (M, g) =Spec0(M, g),then for 3 ≤ n ≤ 6,r2 ≤n-1/n or n > 6,r2 ≥ n-1/n, then M is isometric to M. We improvedthe result and prove that: if Spec0(M, g) =Spec0(M, g), then M is isometric to M. Generally, if Specp(M,g) =Specp(M, g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to M.

On a Right Inverse of a Polynomial of the Laplace in the Weighted Hilbert Space L^(2)(R^(n),e^(−|x|^(2)))

Let P(∆)be a polynomial of the Laplace operator∆=n∑j=1∂^(2)∂x^(2)_(j) on R^(n).We prove the existence of a bounded right inverse of the differential operator P(∆)in the weighted Hilbert space with the Gaussian measure,i.e.,L^(2)(R^(n),e^(−|x|^(2))).

高温作业专用服装设计研究

服装作为人与环境的中间体,充当着人体第二皮肤的作用,热防护功能就是一项重要且被持续关注的功能,因此,在高温环境 下建立高温专业服装的设计模型就十分重要。本文通过建立一维非稳态偏微分方程模型研究了在不同温度环境,不同防护层厚度的条件下, 温度与导热时间以及四层防护服厚度的关系。 对问题一,在给出条件下的温度分布,首先根据 Fourier's Law 建立四个热传导方程,通过寻找一个初值条件及两个边界条件组成方程 组建立一维偏微分模型,利用差分运算等算法进行编程,最终取整体的矩阵进行编程运算得到精确解,提取可视化图的数据固定 x 的值得 到温度随时间的关系图,与实际较为吻合。 对问题二的研究,基于问题一建立的模型求解一个带有约束条件的偏微分方程优化问题,在题目给定的约束条件下,寻找第 II 层的最 优厚度以使得高温衣服的质量达到最小,采用二分算法等算法,并使用 Matlab 等数学软件进行编程,得到第 II 层的最优厚度为 18.6915mm。 对问题三的研究基于问题一建立的一维非稳态偏微分模型,根据题目给出的条件,采用搜索算法遍历所有可能取值从而寻找符合约束 条件的第 II、IV 层的厚度,即找到目标函数 { } 1 1 2 2min r d r d + 使得高温服装的质量最轻,防护性能最好。Matlab 编程求得第 II 层的最优厚 度 20.1mm,第 IV 层的最优厚度 6.2mm。

Sharp lower bound of spectral gap for Schrodinger operator and related results

We give an easy proof of Andrews and Clutterbuck＇s main results [J. Amer. Math. Soc., 2011, 24（3）： 899-916], which gives both a sharp lower bound for the spectral gap of a Schr5dinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the ＇double coordinate＇ approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of SchSdinger operator.

Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative

In this paper,we aim to explore the properties and applications on the operators consisting of the Hodge star operator together with the exterior derivative,whose action on an arbitrary p-form field in n-dimensional spacetimes makes its form degree remain invariant.Such operations are able to generate a variety of p-forms with the even-order derivatives of the p-form.To do this,we first investigate the properties of the operators,such as the Laplace–de Rham operator,the codifferential and their combinations,as well as the applications of the operators in the construction of conserved currents.On the basis of two general p-forms,then we construct a general n-form with higher-order derivatives.Finally,we propose that such an n-form could be applied to define a generalized Lagrangian with respect to a p-form field according to the fact that it includes the ordinary Lagrangians for the p-form and scalar fields as special cases.

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper, oscillation of solutions to a class of impulsive delay parabolic partial differential equations system with higher order Laplace operator is studied. Under two different boundary value conditions, we establish some sufficient criteria with respect to the oscillations of such systems, employing first-order impulsive delay differential inequalities. The results fully reflect the influence action of impulsive and delay in oscillation.

Existence of Solutions to a Semilinear Elliptic System Through Generalized Orlicz-Sobolev Spaces

This paper is concerned with the existence theory of a semilinear elliptic system. In particular, we will prove that the system has a nontrivial positive solution in some appropriate solution spaces.

A Fast Direct Solver for a Class of 3-D Elliptic Partial Differential Equation with Variable Coefficient

We propose a direct solver for the three-dimensional Poisson equation with a variable coefficient,and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix.Introducing some appropriate finite difference operators,we derive a second-order scheme for the solver,and then two suitable high-order compact schemes are also discussed.For a cube containing N nodes,the solver requires O(N^(3/2)log^(2)N)arithmetic operations and O(NlogN)memory to store the necessary information.Its efficiency is illustrated with examples,and the numerical results are analysed.