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Greedy optimal homotopy and homology generators
 Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms
, 2005
"... Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops t ..."
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Cited by 76 (12 self)
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Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.
Folding and Unfolding in Computational Geometry
"... Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain ..."
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Cited by 53 (4 self)
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Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?
Unfolding convex polyhedra via quasigeodesics: Abstract
 In Proc. 17th Annu. Fall Workshop Comput. Comb. Geom., November 2007. [IOV09] [IV08a] [IV08b] Jinichi Itoh, Joseph O’Rourke, and Costin
"... We show that cutting shortest paths from every vertex of a convex polyhedron to a simple closed quasigeodesic, and cutting all but a short segment of the quasigeodesic, unfolds the surface to a planar simple polygon. 1 ..."
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Cited by 5 (4 self)
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We show that cutting shortest paths from every vertex of a convex polyhedron to a simple closed quasigeodesic, and cutting all but a short segment of the quasigeodesic, unfolds the surface to a planar simple polygon. 1
Inflating polyhedral surfaces
, 2006
"... Abstract. We prove that all polyhedral surfaces in R 3 have volumeincreasing isometric deformations. This resolves the conjecture of Bleecker who proved it for convex simplicial surfaces [B1]. A version of this result is proved for all convex surfaces in R d. We also discuss limits on the volume of ..."
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Cited by 3 (0 self)
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Abstract. We prove that all polyhedral surfaces in R 3 have volumeincreasing isometric deformations. This resolves the conjecture of Bleecker who proved it for convex simplicial surfaces [B1]. A version of this result is proved for all convex surfaces in R d. We also discuss limits on the volume of such deformations, present a number of conjectures and special cases.
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
, 2009
"... We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (nonoverlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but ..."
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Cited by 2 (0 self)
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We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (nonoverlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.
Conjugate Points in Length Spaces
, 709
"... In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting ..."
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In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We next focus on CBA(κ) spaces, proving Rauchtype comparison theorems. In particular, much like the Riemannian setting, we prove that there are no ultimate conjugate points less than π apart in a CBA(1) space. We also prove a relative Rauch comparison theorem to estimate the distance between nearby geodesics. We close with applications and open problems.
Unfolding Convex Polyhedra
, 2008
"... We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q ..."
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We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut. 1
Contents
, 2013
"... This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by G. Craciun, F. Nazarov, and C. Pantea. The main result states that the global attractor conjecture holds for complexbalanced systems that are strongly endotactic: every trajectory wi ..."
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This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by G. Craciun, F. Nazarov, and C. Pantea. The main result states that the global attractor conjecture holds for complexbalanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by D. F. Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes powerlaw systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch’s theorem, a wellknown result about intersections of affine subspaces with manifolds parameterized by monomials.