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Articles 1 - 9 of 9
Full-Text Articles in Physical Sciences and Mathematics
Open Problems From Cccg 2002, Erik D. Demaine, Joseph O'Rourke
Open Problems From Cccg 2002, Erik D. Demaine, Joseph O'Rourke
Computer Science: Faculty Publications
No abstract provided.
Planar Minimally Rigid Graphs And Pseudo-Triangulations, Ruth Haas, David Orden, Günter Rote, Francisco Santos, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley
Planar Minimally Rigid Graphs And Pseudo-Triangulations, Ruth Haas, David Orden, Günter Rote, Francisco Santos, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley
Mathematics Sciences: Faculty Publications
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide - to the best of our knowledge - the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also …
Unification Scale, Proton Decay, And Manifolds Of G2 Holonomy, Tamar Friedmann, Edward Witten
Unification Scale, Proton Decay, And Manifolds Of G2 Holonomy, Tamar Friedmann, Edward Witten
Mathematics Sciences: Faculty Publications
Models of particle physics based on manifolds of G2 holonomy are in most respects much more complicated than other string-derived models, but as we show here they do have one simplification: threshold corrections to grand unification are particularly simple. We compute these corrections, getting completely explicit results in some simple cases. We estimate the relation between Newton’s constant, the GUT scale, and the value of αGUT , and explore the implications for proton decay. In the case of proton decay, there is an interesting mechanism which (relative to four-dimensional SUSY GUT’s) enhances the gauge boson contribution to p → π …
A Dynamical System For Plant Pattern Formation: A Rigorous Analysis, Pau Atela, Christophe Golé, S. Hotton
A Dynamical System For Plant Pattern Formation: A Rigorous Analysis, Pau Atela, Christophe Golé, S. Hotton
Mathematics Sciences: Faculty Publications
We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram.
Regularity Of Minimizers Of The Calculus Of Variations In Carnot Groups Via Hypoellipticity Of Systems Of Hörmander Type, Luca Capogna, Nicola Garofalo
Regularity Of Minimizers Of The Calculus Of Variations In Carnot Groups Via Hypoellipticity Of Systems Of Hörmander Type, Luca Capogna, Nicola Garofalo
Mathematics Sciences: Faculty Publications
We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.
Long Time Behavior Of Solutions To The 3d Compressible Euler Equations With Damping, Thomas C. Sideris, Becca Thomases, Dehua Wang
Long Time Behavior Of Solutions To The 3d Compressible Euler Equations With Damping, Thomas C. Sideris, Becca Thomases, Dehua Wang
Mathematics Sciences: Faculty Publications
The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t-3/2. While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.
Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke
Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke
Computer Science: Faculty Publications
We explore an instance of the question of partitioning a polygon into pieces, each of which is as “circular” as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a …
On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke
On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke
Computer Science: Faculty Publications
Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex “openings” of a planar convex chain.
Computational Geometry Column 44, Joseph O'Rourke
Computational Geometry Column 44, Joseph O'Rourke
Computer Science: Faculty Publications
The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.