双曲型Lagrangian函数*

双曲复数与Minkowski几何相对应,由四维时空间隔不变量和双曲型Lorentz变换可导出双曲型Lagrangian方程和Hamilton-Jacobi方程.     

Minkowski平面中的角度

讨论了Minkowski平面中定义角度的新方法,并在Minkowski平面中给出了接近欧式空间的新角度,同时解决了关于类光向量的角度问题.最后,依据此角度给出了Minkowski平面中的伪勾股定理.     

双曲复空间的拓扑结构与应用

双曲复空间与Minkowski空间相对应,具有时空方向异性的特点。以双曲复空间为原空间,可以抽象出一类双曲拟、虚度量空间和多拓扑结构。     

狭义相对论数学基础的探讨

狭义相对论的变革点就是相对时空观,而相对论时空与非欧几何学有着密切的联系.在介绍了传统的Minkowski空间后,引入双曲虚单位,其所构造的双曲复数对应双曲Minkowski复空间.利用双曲Minkowski空间复数运算规则,可以使高速运动客体的物理规律与复数的性质结合起来,为解决狭义相对论的普遍形式提供新的数学工具.     

有类间距离因素聚类结果的比较分析

本文对于有类间距离因素聚类结果的比较，提出了类结构的空间描述方法和比较相似度的度量指标──夹角余弦，并推导出它的一些性质．最后，用蒙特卡洛模拟的结果阐明用夹角余弦作为聚类结果的相似性度量指标是合理的．     

基于距离相关系数的分层聚类法

随着大数据时代的到来，各个领域涌现出海量数据且结构复杂.如变量的维数不同、尺度不同等.而现实中变量之间往往存在着不确定关系，经典的Pearson相关系数仅能反映两个同维变量间的线性相关关系，不足以完全刻画变量间的相关关系.2007年Szekely等提出的距离相关系数则能描述不同维数变量间的非线性关系.为了探索变量之间的内在信息，本文基于距离相关系数提出了最大距离相关系数法对变量聚类，且有超度量性和空间收缩性.为充分发挥距离相关系数的优势，对上述方法改进得到类整体距离相关系数法.该方法在刻画两类间相似性时，将每类中的所有变量合并成一个整体，再计算这两个不同维数的整体间的距离相关系数.最后，将类整体距离相关系数法应用到几个实际问题中，验证了算法的有效性.     

三维Minkowski空间中基于Killing向量场的磁力曲线

本文在三维Minkowski空间中,研究了基于Killing向量场的磁力曲线.首先给出了三维Minkowski空间中对应于平移和旋转的所有Killing向量场,并对不同的Killing向量所对应的磁力曲线进行了分类,最后给出了一些磁力曲线的图像.     

度量测度空间中Sobolev类Banach空间值函数的刻画（英文）

本文研究了在指标是无穷大时欧式空间情形下Sobolev函数类理论和指标是有限常数时度量空间下Sobolev类Banach空间值函数理论.利用Banach空间理论和位势理论的方法,得到了在指标是无穷大时度量测度空间中Sobolev类Banach空间值函数的各种刻画,进而比较了该Sobolev类与对应的Lipschitz类和Hajlasz-Sobolev类.所获结果推广了欧式空间和度量测度空间中Sobolev函数类相应的结论.     

类拟距离空间中含有Jachymski函数的不动点定理

首先证明拟距离空间中的w-距离或mw-距离可以生成一个类拟距离.然后,在类拟距离空间中建立一些含有Jachymski函数的不动点定理.这些结果推广了Alegre等人最近的两个结果.最后,给出一些例子支持我们结果.     

一类量子态的迹距离相干度量的最优化证明方法

迹距离相干度量是基于迹范数提出的量化相干的一种度量.然而,很难给出一般量子态迹距离相干度量的表达式并且找到对应的最近非相干态.通过最优化方法给出了一类d×d量子态的迹距离相干度量,并且证明了这类量子态的最近非相干态就是由该量子态去掉所有非对角元素得到的对角矩阵.     

A new theory of complex rays

A new approach to the theory of complex rays is presented. Itis shown that the three-dimensional Minkowski space, the variantof the well known four-dimensional space–time Minkowskispace of the special theory of relativity, is more appropriatefor describing both real and complex rays than the usual Euclideanspace. It turns out that in this space complex rays, as realones, may have quite definite directions and magnitudes. Thisallows us to understand the geometrical meaning of the complexmagnitudes such as complex distances and complex angles, intensivelydiscussed over the last several decades. From this point ofview a new interpretation of the Gaussian beams and reflectionlaws is presented.     

Hyperbolic Harmonic Function in Minkowski Space

The paper discusses the properties of four-dimensional Minkowski space by using Clifford algebra, then gives the concept of hyperbolic harmonic function by constructing a system (P₄) in the four-dimensional Minkowski space, and obtains several properties and a sufficient and necessary condition for the solvability of the system (P₄) .     

Hyperbolic Hilbert space

The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex space as original spaces. We can abstract a class of the hyperbolic inner product space and the hyperbolic Hilbert space.     

The geometrical interpretation of uncertainty relation in Minkowski space

Clifford algebraic geometry corresponds to Minkowski space. Using the discrete structure of Minkowski space, we can abstract a class n-dimensional hyperbolic Hilbert phase space. To discussing the causality of physical event in Minkowski space, we can obtain the geometrical interpretation of uncertainty relation.     

Hyperbolic Schrödinger equation

Clifford algebra corresponds to Minkwoski space. The coupling between real object particles and light quantums can be discussed by Minkowski space’s directional strangeness. We introduce Galilei transfomation and Schr?dinger equation into Minkowski space and give a geometrical explanation for classical quantum theory.     

Can Galileons support Lorentzian wormholes?

We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in general relativity coupled to a generalized Galileon field π. Assuming that the Minkowski space–time is obtained at ?_π = 0, we find that there is tension between the properties of the energy–momentum tensor required to support a wormhole (violation of the average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In three-dimensional space–time, this tension is strong enough to rule out wormholes with the above properties. In higher dimensions, including the most physically interesting case of four-dimensional space–time, wormholes, if any, must have fairly contrived shapes.     

An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.     

Three-component WKI equation and curve motion flow in Euclidean and Minkowski space

Motion of curves in the four-dimensional Euclidean and Minkowski space are discussed. It is shown that the three-component WKI equation and its hyperbolic type arise from certain curve motion flows. They are obtained by using the relation between curvatures of the curves and their graph. Group-invariant solutions to the three-component WKI equation and its hyperbolic type are also derived.     

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.