Virtual polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility

Virtual Polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

On the Conley-Zehnder index and Sasaki-Einstein manifolds

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. Koert, Otto van .제 2장에서는 저자가 서울대학교 수리과학부에서 학위를 하는 동안 출판한 논문에 대해 소개하였습니다. 구체적으로 Reeb 벡터장을 사교공간상의 경로로 간주하고 그 Conley-Zehnder 지표와 몫공간으로서 생성된 기저 공간의 orbifold 천(Chern) 특성류 사이의 관계를 규명하였습니다. 이렇게 얻어진 관계를 우리에게 매우 익숙한 기본적인 예제들에 적용시켜 구체적인 값을 구하였습니다. 제 3장은 저자가 학위기간 동안 주로 연구한 분야인 사사키-아인슈타인 기하(Sasaki-Einstein geometry)에 대한 조사 보고서입니다. 기본적인 정의, 정리부터 흥미로운 예제, 존재성에 대한 걸림돌 이론(obstruction theory)등에 대해서 살펴보았습니다.In the second chapter, we prove a useful relation between the Conley-Zehnder indices of the Reeb vector flow action along periodic orbits in prequantization bundles and the orbifold Chern class of the base symplectic orbifolds motivated by the well-known case of manifolds. We also apply this method to primary examples. In the third chapter, we survey interesting properties on Sasaki-Einstein geometry from the elementary definitions and theorems to well-known examples and simple obstructions.Abstract i 1 Introduction 1 2 The Conley-Zehnder indices of the Reeb flow action along S1-fibers over certain orbifolds 4 2.1 The Conley-Zehnder index . . . . . . . . . . . . . . . . . . . . 4 2.1.1 The Maslov index . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 The Conley-Zehnder index . . . . . . . . . . . . . . . . 6 2.1.3 The Robbin-Salamon index . . . . . . . . . . . . . . . 7 2.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . 12 2.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 The Boothby-Wang fibration . . . . . . . . . . . . . . . 15 2.3.2 The main theorem . . . . . . . . . . . . . . . . . . . . 16 2.3.3 The weighted projective spaces and their complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Some computations for non-principal orbits . . . . . . 30 2.3.5 Inertia orbifolds . . . . . . . . . . . . . . . . . . . . . . 32 3 A survey on Sasaki-Einstein manifolds 35 3.1 Sasakian structures and Einstein metrics . . . . . . . . . . . . 35 3.1.1 Symplectic manifolds and contact structures . . . . . . 35 3.1.2 Almost contact structures and Sasakian structures . . . 40 3.1.3 General relativity, Einstein manifolds . . . . . . . . . . 45 3.2 Kahler-Einstein metrics . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Einstein conditions in Kahler metrics . . . . . . . . . . 51 3.2.2 Calabi conjecture and Calabi-Yau manifolds . . . . . . 54 3.2.3 Kahler-Einstein metrics on del Pezzo surfaces . . . . . 57 3.3 Sasaki-Einstein manifolds . . . . . . . . . . . . . . . . . . . . 62 3.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . 62 3.3.2 Toric Sasaki-Einstein manifolds . . . . . . . . . . . . . 66 3.3.3 Sasaki-Einstein metrics on Y pq . . . . . . . . . . . . . 75 3.3.4 Simple obstructions . . . . . . . . . . . . . . . . . . . . 80 Abstract (in Korean) 88Docto

Evaluation of automated decisionmaking methodologies and development of an integrated robotic system simulation

A generic computer simulation for manipulator systems (ROBSIM) was implemented and the specific technologies necessary to increase the role of automation in various missions were developed. The specific items developed are: (1) capability for definition of a manipulator system consisting of multiple arms, load objects, and an environment; (2) capability for kinematic analysis, requirements analysis, and response simulation of manipulator motion; (3) postprocessing options such as graphic replay of simulated motion and manipulator parameter plotting; (4) investigation and simulation of various control methods including manual force/torque and active compliances control; (5) evaluation and implementation of three obstacle avoidance methods; (6) video simulation and edge detection; and (7) software simulation validation

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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