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Articles 1 - 11 of 11
Full-Text Articles in Physical Sciences and Mathematics
Double Lie Algebroids And Representations Up To Homotopy, A. Gracia-Saz, M. Jotz Lean, K. C. H. Mackenzie, Rajan Amit Mehta
Double Lie Algebroids And Representations Up To Homotopy, A. Gracia-Saz, M. Jotz Lean, K. C. H. Mackenzie, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.
Constant Symplectic 2-Groupoids, Rajan Amit Mehta, Xiang Tang
Constant Symplectic 2-Groupoids, Rajan Amit Mehta, Xiang Tang
Mathematics Sciences: Faculty Publications
We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.
Germs Of Fibrations Of Spheres By Great Circles Always Extend To The Whole Sphere, Patricia Cahn, Herman Gluck, Haggai Nuchi
Germs Of Fibrations Of Spheres By Great Circles Always Extend To The Whole Sphere, Patricia Cahn, Herman Gluck, Haggai Nuchi
Mathematics Sciences: Faculty Publications
We prove that every germ of a smooth fibration of an odd-dimensional round sphere by great circles extends to such a fibration of the entire sphere, a result previously known only in dimension three.
Symplectic Structures On The Integration Of Exact Courant Algebroids, Rajan Amit Mehta, Xiang Tang
Symplectic Structures On The Integration Of Exact Courant Algebroids, Rajan Amit Mehta, Xiang Tang
Mathematics Sciences: Faculty Publications
We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a “Lagrangian” sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrable Dirac structure integrates to a presymplectic groupoid.
A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin
A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin
Mathematics Sciences: Faculty Publications
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
Clustered Networks Protect Cooperation Against Catastrophic Collapse, Gwen Spencer
Clustered Networks Protect Cooperation Against Catastrophic Collapse, Gwen Spencer
Mathematics Sciences: Faculty Publications
Assuming a society of conditional cooperators (or moody conditional cooperators), this computational study proposes a new perspective on the structural advantage of social network clustering. Previous work focused on how clustered structure might encourage initial outbreaks of cooperation or defend against invasion by a few defectors. Instead, we explore the ability of a societal structure to retain cooperative norms in the face of widespread disturbances. Such disturbances may abstractly describe hardships like famine and economic recession, or the random spatial placement of a substantial numbers of pure defectors (or round-1 defectors) among a spatially-structured population of players in a laboratory …
The Fivethirtyeight R Package: ‘Tame Data’ Principles For Introductory Statistics And Data Science Courses, Albert Y. Kim, Chester Ismay, Jennifer Chunn
The Fivethirtyeight R Package: ‘Tame Data’ Principles For Introductory Statistics And Data Science Courses, Albert Y. Kim, Chester Ismay, Jennifer Chunn
Mathematics Sciences: Faculty Publications
As statistics and data science instructors, we often seek to use data in our courses that are rich, real, realistic, and relevant. To this end we created the fivethirtyeight R package of data and code behind the stories and interactives at the data journalism website FiveThirtyEight.com. After a discussion on the conflicting pedagogical goals of "minimizing prerequisites to research" (Cobb 2015) while at the same time presenting students with a realistic view of data as it exists "in the wild," we articulate how a desired balance between these two goals informed the design of the package. The details behind this …
The Dihedral Genus Of A Knot, Patricia Cahn, Alexandra Kjuchukova
The Dihedral Genus Of A Knot, Patricia Cahn, Alexandra Kjuchukova
Mathematics Sciences: Faculty Publications
Let K ⊂ S3 be a Fox p-colored knot and assume K bounds a locally flat surface S ⊂ B4 over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X → S4 . Its branching set is a closed surface embedded in S4 locally flatly away from one singularity whose link is K. When S is homotopy ribbon and X a definite four-manifold, a condition relating the signature of X and the Murasugi signature of K guarantees that S in fact realizes the four-genus of K. We exhibit an infinite …
How Often Does The Best Team Win? A Unified Approach To Understanding Randomness In North American Sport, Michael J. Lopez, Gregory J. Matthews, Benjamin Baumer
How Often Does The Best Team Win? A Unified Approach To Understanding Randomness In North American Sport, Michael J. Lopez, Gregory J. Matthews, Benjamin Baumer
Mathematics Sciences: Faculty Publications
Statistical applications in sports have long centered on how to best separate signal (e.g. team talent) from random noise. However, most of this work has concentrated on a single sport, and the development of meaningful cross-sport comparisons has been impeded by the difficulty of translating luck from one sport to another. In this manuscript, we develop Bayesian state-space models using betting market data that can be uniformly applied across sporting organizations to better understand the role of randomness in game outcomes. These models can be used to extract estimates of team strength, the between-season, within-season, and game-to-game variability of team …
Action Of The Symmetric Group On The Free Lanke: A Catalanke Theorem, Tamar Friedmann, Philip Hanlon, Richard P. Stanley, Michelle L. Wachs
Action Of The Symmetric Group On The Free Lanke: A Catalanke Theorem, Tamar Friedmann, Philip Hanlon, Richard P. Stanley, Michelle L. Wachs
Mathematics Sciences: Faculty Publications
We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S2n−1 on the multilinear component of the free LAnKe with S2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of S3n−2 on the multilinear component the free LAnKe …
Strong Comparison Principle For P-Harmonic Functions In Carnot-Caratheodory Spaces, Luca Capogna, Xiaodan Zhou
Strong Comparison Principle For P-Harmonic Functions In Carnot-Caratheodory Spaces, Luca Capogna, Xiaodan Zhou
Mathematics Sciences: Faculty Publications
We extend Bony’s propagation of support argument to C1 solutions of the nonhomogeneous subelliptic p-Laplacian associated to a system of smooth vector fields satisfying Hörmander’s finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf.