基于伪类光超曲面奇点分类的教学研究

在微分几何的教学中,曲线,曲面理论是最主要的基础理论知识.欧氏空间中密切曲线在微分几何学中具有重要的研究价值.主要运用具有类光向量的费雷内标架讨论在四维Minkowski空间中与欧氏空间不同的一类特殊密切曲线(伪类光曲线)的一些几何性质,同时通过横截性原理给出了由伪类光曲线生成的伪类光超曲面的局部几何性质与奇点分类.     

The distance function from the boundary in a Minkowski space

Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the rôle of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

     

On the quantum Lorentz group

The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the SL(2,C) and Lorentz quantum groups. Five matrices of the quantum Lorentz group are constructed in terms of the R matrix of SL(2,C) group. These matrices satisfy Yang–Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.     

资产定价第一基本定理的拓扑描述

资产定价的第一基本定理是数量金融学中核心的定理之一 ,本文证明了在 L∞ 的弱 * 拓扑 σ(L∞ ,L1)中的凸集分离定理 ,并在此定理的基础上给出了没有无风险免费午餐的拓扑描述 ,证明了市场公平性与没有无风险免费午餐条件的等价性 ,从而重新证明了资产定价的第一基本定理 .     

The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

基于扩展多面体的离散单元法及其作用于圆桩的冰载荷计算

对于具有复杂几何形态的多面体单元,线性接触模型不能准确地计算不同接触模式下的作用力,且接触变形和作用力方向也不易判断.基于闵可夫斯基和(Minkowski sum)方法的扩展多面体单元能够准确描述非规则颗粒单元的几何形态,并可精确计算单元间的接触碰撞作用.该方法具有接触判断简单、计算效率高的特点.它将基本多面体和扩展球体相叠加以形成具有光滑棱边和角点的扩展多面体单元.考虑扩展多面体单元相互作用过程中角点、棱边和平面之间的不同接触模式,发展了相应的非线性黏弹性接触模型. 该接触模型将不同接触模型下的法向刚度统一表述为单元接触中接触点处等效曲率半径的函数;黏滞力和切向弹性力接触模型则借鉴球体单元非线性接触模型的处理方法. 为检验扩展多面体的可靠性,对碎冰区冰块对圆桩结构的冰载荷进行了离散元分析. 采用沃洛诺伊(Voronoi)切割算法获得了碎冰的初始随机分布状态,并考虑了海冰在运动过程中的海水浮力和拖曳力.计算表明该扩展多面体单元可描述海冰在海流拖曳下的运动过程以及圆桩结构的动冰力特性.在此基础上进一步分析了冰速和冰块尺寸对圆桩冰力的影响,并确定了冰力在圆桩上的分布规律. 最后,讨论了目前扩展多面体单元在计算冰载荷方面的局限性和改进方法.      

On the Ingredients of the Twin Paradox

If M is a 4-dimensional connected, orientable flat spacetime manifold endowed with a time-arrow, and if the existence of a finite speed limit to energy/information transfers over M can be assessed, then the “twin paradox” necessarily follows (and indeed, the full special relativity theory). Two other implicit ingredients of the paradox are also identified.     

Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently

In this paper, we introduce the notion of a constrained Minkowski sum: for two (finite) point-sets P,Q⊆ℝ² and a set of k inequalities Ax≥b, it is defined as the point-set (P ⊕ Q) _Ax≥b ={x=p+q∣p∈P,q∈Q,Ax≥b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(Nlog N), where N=|P|+|Q| if k is fixed. For the special case where P and Q consist of points with integer x ₁-coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solved. T. Bernholt gratefully acknowledges the Deutsche Forschungsgemeinschaft for the financial support (SFB 475, “Reduction of complexity in multivariate data structures”).     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

Curves in the Minkowski space are very well suited to describe the medial axis transform (MAT) of planar domains. Among them, Minkowski Pythagorean hodograph (MPH) curves correspond to domains where both the boundaries and their offsets admit rational parameterizations (Choi et al., Comput Aided Design 31:59–72, 1999; Moon, Comput Aided Geom Design 16:739–753; 1999). We construct MPH quintics which interpolate two points with associated first derivative vectors and analyze the properties of the system of solutions, including the approximation order of the ‘best’ interpolant.      

Modified regular black holes with time delay and 1-loop quantum correction

We develop the regular black hole solutions by incorporating the 1-loop quantum correction to the Newton potential and a time delay between an observer at the regular center and one at infinity. We define the maximal time delay between the center and the infinity by scanning the mass of black holes such that the sub-Planckian feature of the Kretschmann scalar curvature is preserved during the process of evaporation. We also compare the distinct behavior of the Kretschmann curvature for black holes with asymptotically Minkowski cores and those with asymptotically de-Sitter cores, including Bardeen and Hayward black holes. We expect that such regular black holes may provide more information about the construction of effective metrics for Planck stars.