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Articles 241 - 270 of 294

Full-Text Articles in Mathematics

Linear Conditions Imposed On Flag Varieties, Julianna S. Tymoczko Dec 2006

Linear Conditions Imposed On Flag Varieties, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(ℂ) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of …

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Ahlfors Type Estimates For Perimeter Measures In Carnot-Carathéodory Spaces, Luca Capogna, Nicola Garofalo Dec 2006

Ahlfors Type Estimates For Perimeter Measures In Carnot-Carathéodory Spaces, Luca Capogna, Nicola Garofalo

Mathematics Sciences: Faculty Publications

We study the relationship between the geometry of hypersurfaces in a Carnot-Carathéodory (CC) space and the Ahlfors regularity of the corresponding perimeter measure. To this end we establish comparison theorems for perimeter estimates between an hypersurface and its tangent space, and between a CC geometry and its "tangent" Carnot group structure.

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Local Energy Decay For Solutions Of Multi-Dimensional Isotropic Symmetric Hyperbolic Systems, Thomas C. Sideris, Becca Thomases Dec 2006

Local Energy Decay For Solutions Of Multi-Dimensional Isotropic Symmetric Hyperbolic Systems, Thomas C. Sideris, Becca Thomases

Mathematics Sciences: Faculty Publications

The local decay of energy is established for solutions to certain linear, multidimensional symmetric hyperbolic systems, with constraints. The key assumptions are isotropy and nondegeneracy of the associated symbols. Examples are given, including Maxwell's equations and linearized elasticity. Such estimates prove useful in treating nonlinear perturbations.

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A Note On The Engulfing Property And The R 1+Α -Regularity Of Convex Functions In Carnot Groups, Luca Capogna, Diego Maldonado Nov 2006

A Note On The Engulfing Property And The R 1+Α -Regularity Of Convex Functions In Carnot Groups, Luca Capogna, Diego Maldonado

Mathematics Sciences: Faculty Publications

We study the engulfing property for convex functions in Carnot groups. As an application we show that the horizontal gradient of functions with this property is Hölder continuous.

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Exponents For B-Stable Ideals, Eric Sommers, Julianna Tymoczko Aug 2006

Exponents For B-Stable Ideals, Eric Sommers, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1, m 2,. . .,m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n, B n, C n and some other types. When I = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the …

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A Topological Representation Theorem For Oriented Matroids, Jürgen Bokowski, Simon King, Sussane Mock, Ileana Streinu Apr 2005

A Topological Representation Theorem For Oriented Matroids, Jürgen Bokowski, Simon King, Sussane Mock, Ileana Streinu

Computer Science: Faculty Publications

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

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Schwinger Pair Creation Of Kaluza-Klein Particles: Pair Creation Without Tunneling, Tamar Friedmann, Herman Verlinde Mar 2005

Schwinger Pair Creation Of Kaluza-Klein Particles: Pair Creation Without Tunneling, Tamar Friedmann, Herman Verlinde

Mathematics Sciences: Faculty Publications

We study Schwinger pair creation of charged Kaluza-Klein (KK) particles from a static KK electric field. We find that the gravitational backreaction of the electric field on the geometry—which is incorporated via the electric KK-Melvin solution—prevents the electrostatic potential from overcoming the rest mass of the KK particles, thus impeding the tunneling mechanism which is often thought of as responsible for the pair creation. However, we find that pair creation still occurs with a finite rate formally similar to the classic Schwinger result, but via an apparently different mechanism, involving a combination of the Unruh effect and vacuum polarization due …

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The Aronsson-Euler Equation For Absolutely Minimizing Lipschitz Extensions With Respect To Carnot-Carathéodory Metrics, Thomas Bieske, Luca Capogna Feb 2005

The Aronsson-Euler Equation For Absolutely Minimizing Lipschitz Extensions With Respect To Carnot-Carathéodory Metrics, Thomas Bieske, Luca Capogna

Mathematics Sciences: Faculty Publications

We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the L ∞ variational problem inf∥∇ 0u∥L∞(Ω), u = g ε Lip(∂Ω) on ∂Ω, where Ω ⊂ G is an open subset of a Carnot group, ∇ 0u denotes the horizontal gradient of u: Ω ℝ R, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends …

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Partitioning Regular Polygons Into Circular Pieces Ii: Nonconvex Partitions, Mirela Damian, Joseph O'Rourke Dec 2004

Partitioning Regular Polygons Into Circular Pieces Ii: Nonconvex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal circular partition minimizes the maximum ratio over all pieces in the partition. We show that the equilateral triangle has an optimal 4-piece nonconvex partition, the square an optimal 13-piece nonconvex partition, and the pentagon has an optimal nonconvex partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.

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Unfolding Smooth Prismatoids, Nadia Benbernou, Patricia Cahn, Joseph O'Rourke Oct 2004

Unfolding Smooth Prismatoids, Nadia Benbernou, Patricia Cahn, Joseph O'Rourke

Computer Science: Faculty Publications

We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping “volcano unfolding.” These unfoldings keep the base intact, unfold the sides outward, splayed around the base, and attach the top to the tip of some side rib. Our result answers a question for smooth prismatoids whose analog for polyhedral prismatoids remains unsolved.

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Distinguishing Numbers For Graphs And Groups, Julianna Tymoczko Sep 2004

Distinguishing Numbers For Graphs And Groups, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

A graph G is distinguished if its vertices are labelled by a map φ: V(G) → {1,2,..., k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ: X → {1, 2,..., k} is distinguishing if no element of Γ preserves Γ except those which fix each element of X. The distinguishing number of the group action on X is …

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On The Number Of Embeddings Of Minimally Rigid Graphs, Ciprian Borcea, Ileana Streinu Feb 2004

On The Number Of Embeddings Of Minimally Rigid Graphs, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the …

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A 2-Chain Can Interlock With A K-Chain, Julie Glass, Stefan Langerman, Joseph O'Rourke, Jack Snoeyink, Jianyuan K. Zhong Jan 2004

A 2-Chain Can Interlock With A K-Chain, Julie Glass, Stefan Langerman, Joseph O'Rourke, Jack Snoeyink, Jianyuan K. Zhong

Computer Science: Faculty Publications

One of the open problems posed in [3] is: what is the minimal number k such that an open, flexible k-chain can interlock with a flexible 2-chain? In this paper, we establish the assumption behind this problem, that there is indeed some k that achieves interlocking. We prove that a flexible 2-chain can interlock with a flexible, open 16-chain.

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Computational Geometry Column 45, Joseph O'Rourke Jan 2004

Computational Geometry Column 45, Joseph O'Rourke

Computer Science: Faculty Publications

The algorithm of Edelsbrunner for surface reconstruction by "wrapping" a set of points in R3 is described.

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Open Problems From Cccg 2002, Erik D. Demaine, Joseph O'Rourke Jun 2003

Open Problems From Cccg 2002, Erik D. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

No abstract provided.

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Unification Scale, Proton Decay, And Manifolds Of G2 Holonomy, Tamar Friedmann, Edward Witten Jan 2003

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 …

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Planar Minimally Rigid Graphs And Pseudo-Triangulations, Ruth Haas, David Orden, Günter Rote, Francisco Santos, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley Jan 2003

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 …

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A Dynamical System For Plant Pattern Formation: A Rigorous Analysis, Pau Atela, Christophe Golé, S. Hotton Jan 2003

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.

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Long Time Behavior Of Solutions To The 3d Compressible Euler Equations With Damping, Thomas C. Sideris, Becca Thomases, Dehua Wang Jan 2003

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.

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Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke Jan 2003

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 …

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Regularity Of Minimizers Of The Calculus Of Variations In Carnot Groups Via Hypoellipticity Of Systems Of Hörmander Type, Luca Capogna, Nicola Garofalo Jan 2003

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.

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Computational Geometry Column 44, Joseph O'Rourke Jan 2003

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.

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On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke Jan 2003

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.

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On The Quantum Moduli Space Of M-Theory Compactifications, Tamar Friedmann Jul 2002

On The Quantum Moduli Space Of M-Theory Compactifications, Tamar Friedmann

Mathematics Sciences: Faculty Publications

We study the moduli space of M-theories compactified on G2 manifolds which are asymptotic to a cone over quotients of S3 × S3. We show that the moduli space is composed of several components, each of which interpolates smoothly among various classical limits corresponding to low energy gauge theories with a given number of massless U (1) factors. Each component smoothly interpolates among supersymmetric gauge theories with different gauge groups.

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Computational Geometry Column 43, Joseph O'Rourke Jun 2002

Computational Geometry Column 43, Joseph O'Rourke

Computer Science: Faculty Publications

The concept of pointed pseudo-triangulations is defined and a few of its applications described.

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Nonorthogonal Polyhedra Built From Rectangles, Melody Donoso, Joseph O'Rourke May 2002

Nonorthogonal Polyhedra Built From Rectangles, Melody Donoso, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].

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Enumerating Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke Mar 2002

Enumerating Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke

Computer Science: Faculty Publications

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infinite.

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Vertex-Unfoldings Of Simplicial Manifolds, Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, Joseph O'Rourke Jan 2002

Vertex-Unfoldings Of Simplicial Manifolds, Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, Joseph O'Rourke

Computer Science: Faculty Publications

We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.

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Polygonal Chains Cannot Lock In 4d, Roxana Cocan, Joseph O'Rourke Nov 2001

Polygonal Chains Cannot Lock In 4d, Roxana Cocan, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that, in all dimensions d ≥ 4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of “moves.” These results contrast to those known for d = 2, where trees can “lock,” and for d = 3, where open and closed chains can lock.

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Computational Geometry Column 42, Joseph S. B. Mitchell, Joseph O'Rourke Oct 2001

Computational Geometry Column 42, Joseph S. B. Mitchell, Joseph O'Rourke

Computer Science: Faculty Publications

A compendium of thirty previously published open problems in computational geometry is presented.

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