115,859 research outputs found
Intersection theorems with a continuum of intersection points
In all existing intersection theorems conditions are given under which acertain subset of acollection of sets has a non-empty intersection. In this paper conditions are formulated under which the intersection is a continuum of points satisfying some interesting topological properties. In this sense the intersection theorems considered in this paper belong to a new class. The intersection theorems are formulated on the unit cube and it is shown that both the vector of zeroes and the vector of ones lie in the same component of the intersection. This is interesting for some specific applications. The theorems give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz, of Sperner, of Shapley, and of Ichiischi. Moreover the results can be used to sharpen the usual formulation of the Sperner Lemma on the cube.Optimization;operations research
A Tropical Toolkit
We give an introduction to Tropical Geometry and prove some results in
Tropical Intersection Theory. The first part of this paper is an introduction
to tropical geometry aimed at researchers in Algebraic Geometry from the point
of view of degenerations of varieties using projective not-necessarily-normal
toric varieties. The second part is a foundational account of tropical
intersection theory with proofs of some new theorems relating it to classical
intersection theory.
Revised version includes many corrections, more examples, and improved
exposition.Comment: 38 page
The geometry of p-convex intersection bodies
Busemann's theorem states that the intersection body of an origin-symmetric
convex body is also convex. In this paper we provide a version of Busemann's
theorem for p-convex bodies. We show that the intersection body of a p-convex
body is q-convex for certain q. Furthermore, we discuss the sharpness of the
previous result by constructing an appropriate example. This example is also
used to show that IK, the intersection body of K, can be much farther away from
the Euclidean ball than K. Finally, we extend these theorems to some general
measure spaces with log-concave and -concave measure
Graph-Links
The present paper is a review of the current state of Graph-Link Theory
(graph-links are also closely related to homotopy classes of looped
interlacement graphs), dealing with a generalisation of knots obtained by
translating the Reidemeister moves for links into the language of intersection
graphs of chord diagrams. In this paper we show how some methods of classical
and virtual knot theory can be translated into the language of abstract graphs,
and some theorems can be reproved and generalised to this graphical setting. We
construct various invariants, prove certain minimality theorems and construct
functorial mappings for graph-knots and graph-links. In this paper, we first
show non-equivalence of some graph-links to virtual links.Comment: 32 pages, 21 figure
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