Algorithms for Minkowski products and implicitly‐defined complex sets

Minkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers selected from given complex‐set operands. Whereas Minkowski sums (under vector addition in Rⁿ have been extensively studied, from both the theoretical and computational perspective, Minkowski products in R² (induced by the multiplication of complex numbers) have remained relatively unexplored. The complex logarithm reveals a close relation between Minkowski sums and products, thereby allowing algorithms for the latter to be derived through natural adaptations of those for the former. A novel concept, the logarithmic Gauss maps of plane curves, plays a key role in this process, furnishing geometrical insights that parallel those associated with the “ordinary” Gauss map. As a natural generalization of Minkowski sums and products, the computation of “implicitly‐defined” complex sets (populated by general functions of values drawn from given sets) is also considered. By interpreting them as one‐parameter families of curves, whose envelopes contain the set boundaries, algorithms for evaluating such sets are sketched. This revised version was published online in June 2006 with corrections to the Cover Date.     

Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝⁿ whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝⁿ are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.     

Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distances

In this paper, we extend the concept of the perturbation of fuzzy sets based on normalized Minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets. These operations are induced by triangular norms and conorms. Furthermore, we discuss the perturbation of fuzzy reasoning.     

Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups

In this paper, we introduce the notion of a Minkowski Lie algebra, which is the natural generalization of the notion of a real quadratic Lie algebra (metric Lie algebra). We then study the positive definite Minkowski Lie algebras and obtain a complete classification of the simple ones. Finally, we present some applications of our results to Finsler geometry and give a classification of bi-invariant Finsler metrics on Lie groups. This work was supported by NSFC (No.10671096) and NCET of China.     

关于随机紧凸集的更新定理

该文在讨论了多维更新定理的基础上,重点研究了随机紧凸集的Minkowski和的更新定理,得到了一系列重要结论.     

Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23× speedup for Clifford products and a 33× speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor.     

A refined Brunn–Minkowski inequality for convex sets

Starting from a mass transportation proof of the Brunn–Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.     

Lower S-dimension of fractal sets

The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in Rd (cf. Rataj and Winter (in press) [6]). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in Rataj and Winter (in press) [6] suggest that there is much more flexibility for the lower contents and dimensions. We show that this is indeed the case. There are sets whose lower S-dimension is strictly smaller than their lower Minkowski dimension. More precisely, given two numbers s, m with 0<s<m<1, we construct sets F in Rd with lower S-dimension s+d−1 and lower Minkowski dimension m+d−1. In particular, these sets are used to demonstrate that the inequalities obtained in Rataj and Winter (in press) [6] regarding the general relation of these two dimensions are best possible.     

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.     

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.     

An introduction to Minkowski geometries

The fundamental ideas of Minkowski geometries are presented. Learning about Minkowski geometries can sharpen our students’ understanding of concepts such as distance measurement. Many of its ideas are important and accessible to undergraduate students. Following a brief overview, distance and orthogonality in Minkowski geometries are thoroughly discussed and many illustrative examples and applications are supplied. Suggestions for further study of these geometries are given. Indeed, Minkowski geometries are an excellent source of topics for undergraduate research and independent study.     

An Arithmetic for Matrix Pencils: Theory and New Algorithms

This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case. These operations give an unusual perspective on a variety of pencil related computations. We derive generalizations of monodromy matrices and the matrix exponential. A new algorithm for computing a pencil-arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces.Some of this work was completed at the University of Kansas. Partial support received by Deutsche Forschungsgemeinschaft, grant BE 2174/4-1.This material is based upon work partially supported by the DFG Research Center “Mathematics for Key Technologies” (MATHEON) in Berlin, the University of Kansas General Research Fund allocation 2301062-003 and by the National Science Foundation under awards 0098150, 0112375 and 9977352.     

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) .     

Random sets. A survey of results and applications

A survey of current directions in the theory of random closed sets is presented; these include: the central limit theorem, the law of large numbers for Minkowski sums and unions of random sets, semi-Markov random closed sets, Boolean models and statistical estimation of their parameters, specification of distributions and associated problems of capacity theory. Weak convergence of random closed sets is defined and its application to limit theorems for graphs and epi-graphs of random processes and problems of stochastic optimization is described. Other connections with the theory of random processes (level sets, multivalued and controllable random processes) are also discussed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1587–1599, December, 1991.     

Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

Computation of Minkowski Values of Polynomials over Complex Sets

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X. Given any polynomial P(z) with prescribed coefficients in the complex variable z, the Minkowski value P(X) is defined to be the set of all complex values generated by evaluating P, through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X. The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield different Minkowski value sets P(X). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial value P _m (X), for which the monomial terms are evaluated separately (incurring n(n+1) independent values of z) and summed; the Minkowski factor value P _f (X), where P is represented as the product (z–r ₁)(z–r _n ) of n linear factors – each incurring an independent choice zX – and r ₁,...,r _n are the roots of P(z); and the Minkowski Horner value P _h (X), where the evaluation is performed by nested multiplication and incurs n independent values zX. A new algorithm for the evaluation of P _h (X), when 0X, is presented.     

Brunn–Minkowski and Zhang inequalities for convolution bodies

A quantitative version of Minkowski sum, extending the definition of θ$\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn–Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.     

Characterizing digital microstructures by the Minkowski-based quadratic normal tensor

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures.     

Section and projection means of convex bodies

This paper is concerned with various geometric averages of sections or projections of convex bodies. In particular, we consider Minkowski and Blaschke sums of sections as well as Minkowski sums of projections. The main result is a Crofton-type formula for Blaschke sums of sections. This is used to establish connections between the different averages mentioned above. As a consequence, we obtain results which show that, in some circumstances, a convex body is determined by the averages of its sections or projections.The research of the first author was supported in part by NSF grants DMS-9504249 and INT-9123373