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Mathematics

Smith College

2012

Articles 1 - 10 of 10

Full-Text Articles in Physical Sciences and Mathematics

L ∞-Algebra Actions, Rajan Amit Mehta, Marco Zambon Dec 2012

L ∞-Algebra Actions, Rajan Amit Mehta, Marco Zambon

Mathematics Sciences: Faculty Publications

We define the notion of action of an L -algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of g on a second L -algebra, leading to a notion of "semidirect product" of L -algebras more general than those we found in the literature.

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Group-Theoretical Derivation Of Angular Momentum Eigenvalues In Spaces Of Arbitrary Dimensions, Tamar Friedmann, C. R. Hagen Sep 2012

Group-Theoretical Derivation Of Angular Momentum Eigenvalues In Spaces Of Arbitrary Dimensions, Tamar Friedmann, C. R. Hagen

Mathematics Sciences: Faculty Publications

The spectrum of the square of the angular momentum in arbitrary dimensions is derived using only group theoretical techniques. This is accomplished by application of the Lie algebra of the noncompact group O(2, 1). Illuminating discussions with Jonathan Pakianathan and his comments on a draft are gratefully acknowledged. This work was supported in part by (U.S.) Department of Energy (DOE), Grant No. DE-FG02-91ER40685.

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An Aronsson Type Approach To Extremal Quasiconformal Mappings, Luca Capogna, Andrew Raich Aug 2012

An Aronsson Type Approach To Extremal Quasiconformal Mappings, Luca Capogna, Andrew Raich

Mathematics Sciences: Faculty Publications

We study C 2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a 'localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short-time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

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From Pop-Up Cards To Coffee-Cup Caustics: The Knight's Visor, Stephanie Jakus, Joseph O'Rourke Jun 2012

From Pop-Up Cards To Coffee-Cup Caustics: The Knight's Visor, Stephanie Jakus, Joseph O'Rourke

Computer Science: Faculty Publications

As a pedagogical exercise, we derive the shape of a particularly elegant pop-up card design, and show that it connects to a classically studied plane curve that is (among other interpretations) a caustic of a circle

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A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For 𝔰𝔩3, Julianna Tymoczko Jun 2012

A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For 𝔰𝔩3, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and 𝔰𝔩2. Building on a result of Kuperberg’s, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3 × n and irreducible webs for 𝔰𝔩3whose boundary vertices are all sources. In this paper, we give a simple and explicit map from standard Young tableaux of shape 3 …

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Unfolding Prismatoids As Convex Patches: Counterexamples And Positive Results, Joseph O'Rourke May 2012

Unfolding Prismatoids As Convex Patches: Counterexamples And Positive Results, Joseph O'Rourke

Computer Science: Faculty Publications

We address the unsolved problem of unfolding prismatoids in a new context, viewing a “topless prismatoid” as a convex patch—a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for “petal unfolding” topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads …

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Source Unfoldings Of Convex Polyhedra Via Certain Closed Curves, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu May 2012

Source Unfoldings Of Convex Polyhedra Via Certain Closed Curves, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

Abstract. We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q “lives on a cone” to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon

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Permutation Notations For The Exceptional Weyl Group F4, Patricia Cahn, Ruth Haas, Aloysius G. Helminck, Juan Li, Jeremy Schwartz Jan 2012

Permutation Notations For The Exceptional Weyl Group F4, Patricia Cahn, Ruth Haas, Aloysius G. Helminck, Juan Li, Jeremy Schwartz

Mathematics Sciences: Faculty Publications

This paper describes a permutation notation for the Weyl groups of type F4 and G2. The image in the permutation group is presented as well as an analysis of the structure of the group. This description enables faster computations in these Weyl groups which will prove useful for a variety of applications.

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A New Qcd Effect: The Shrinking Radius Of Hadrons, Tamar Friedmann Jan 2012

A New Qcd Effect: The Shrinking Radius Of Hadrons, Tamar Friedmann

Mathematics Sciences: Faculty Publications

We propose an extended schematic model for hadrons in which quarks as well as diquarks serve as building blocks. The outcome is a reclassification of the hadron spectrum in which there are no radially excited hadrons: all mesons and baryons previously believed to be radial excitations are orbitally excited states involving diquarks. Also, there are no exotic hadrons: all hadrons previously believed to be exotic are states involving diquarks and are an integral part of the model. We discuss the implications of this result for a new understanding of confinement and its relation to asymptotic freedom, as well as its …

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Π/2-Angle Yao Graphs Are Spanners, Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel Smid, Stefanie Wuhrer Jan 2012

Π/2-Angle Yao Graphs Are Spanners, Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel Smid, Stefanie Wuhrer

Computer Science: Faculty Publications

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23√ 2). Enroute to this, we also show that the Yao graph Y∞4 in the L∞ metric is a planar spanner with stretch factor 8.

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