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1.
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.  相似文献   

2.
《Set-Valued Analysis》2008,16(2-3):307-318
In this paper we study a class of closed convex sets introduced recently by Ernst et al. (J Funct Anal 223:179–203, 2005) and called by these authors slice-continuous sets. This class, which plays an important role in the strong separation of convex sets, coincides in ℝ n with the well known class of continuous sets defined by Gale and Klee in the 1960s. In this article we achieve, in the setting of reflexive Banach spaces, two new characterizations of slice-continuous sets, similar to those provided for continuous sets in ℝ n by Gale and Klee. Thus, we prove that a slice-continuous set is precisely a closed and convex set which does not possess neither boundary rays, nor flat asymptotes of any dimension. Moreover, a slice-continuous set may also be characterized as being a closed and convex set of non-void interior for which the support function is continuous except at the origin. Dedicated to Boris Mordukhovich in honour of his 60th birthday.  相似文献   

3.
Algorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree   to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3R3, and other forms of geometrical convolution, are briefly discussed.  相似文献   

4.
The paper deals with the properties of the exterior algebra ℝ(Λn) related to the Euclidean structure on ℝ(Λn) induced by the scalar product in ℝ(Λn). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λn), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.  相似文献   

5.
Minkowski Geometric Algebra of Complex Sets   总被引:2,自引:0,他引:2  
A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connections arise from the geometric algebra based on these operations. Minkowski products with lines and circles are intimately related to problems of wavefront reflection or refraction in geometrical optics. The Minkowski algebra is also the natural extension, to complex numbers, of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations. The Minkowski sums and products offer basic 'shape operators' for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables play a fundamental role – Fourier analysis, conformal mapping, stability of control systems, etc.  相似文献   

6.
Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations of Wiener-Hopf-typeK±u ±=f on ℝ n , whereKS′ andu ± are boundary values of monogenic functions in ℝ+ n+1 and ℝ_ n+1 respectivly.  相似文献   

7.
It has been recently conjectured that, in the context of the Heisenberg group ℍn endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of ℍn, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn–Minkowski inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F⊂ℍn of class C2 can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed. Mathematics Subject Classification (2000) 28A75, 22E25, 49Q20  相似文献   

8.
We find necessary and sufficient conditions for a curve in ℝ m×n to be the gradient range of a C 1-smooth function υ: Ω ⊂ ℝ n → ℝ m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ m×n are hyperplanes.  相似文献   

9.
The Ruelle Sullivan map for an ℝn-action on a compact metric space with invariant probability measure is a graded homomorphism between the integer Cech cohomology of the space and the exterior algebra of the dual of ℝn. We investigate flows on tori to illuminate that it detects geometrical structure of the system. For actions arising from Delone sets of finite local complexity, the existence of canonical transversals and a formulation in terms of pattern equivariant functions lead to the result that the Ruelle Sullivan map is even a ring homomorphism provided the measure is ergodic.  相似文献   

10.
We obtain new necessary conditions for ann-dimensional semialgebraic subset of ℝ n to be a polynomial image of ℝ n . Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers. Supported by Spanish GAAR BFM2002-04797 and European RAAG HPRN-CT-2001-00271.  相似文献   

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