On Operads, Bimodules and Analytic Functors

On Operads, Bimodules and Analytic Functors

Author: Nicola Gambino

Publisher: American Mathematical Soc.

Published: 2017-09-25

Total Pages: 122

ISBN-13: 1470425769

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The authors develop further the theory of operads and analytic functors. In particular, they introduce the bicategory of operad bimodules, that has operads as -cells, operad bimodules as -cells and operad bimodule maps as 2-cells, and prove that it is cartesian closed. In order to obtain this result, the authors extend the theory of distributors and the formal theory of monads.


Colored Operads

Colored Operads

Author: Donald Yau

Publisher: American Mathematical Soc.

Published: 2016-02-29

Total Pages: 458

ISBN-13: 1470427230

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The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality. The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.


(Co)end Calculus

(Co)end Calculus

Author: Fosco Loregian

Publisher: Cambridge University Press

Published: 2021-07-22

Total Pages: 331

ISBN-13: 1108746128

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This easy-to-cite handbook gives the first systematic treatment of the (co)end calculus in category theory and its applications.


The Stability of Cylindrical Pendant Drops

The Stability of Cylindrical Pendant Drops

Author: John McCuan

Publisher: American Mathematical Soc.

Published: 2018-01-16

Total Pages: 122

ISBN-13: 1470409380

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The author considers the stability of certain liquid drops in a gravity field satisfying a mixed boundary condition. He also considers as special cases portions of cylinders that model either the zero gravity case or soap films with the same kind of boundary behavior.


On Sudakov's Type Decomposition of Transference Plans with Norm Costs

On Sudakov's Type Decomposition of Transference Plans with Norm Costs

Author: Stefano Bianchini

Publisher: American Mathematical Soc.

Published: 2018-02-23

Total Pages: 124

ISBN-13: 1470427664

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The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost with , probability measures in and absolutely continuous w.r.t. . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in , where are disjoint regions such that the construction of an optimal map is simpler than in the original problem, and then to obtain by piecing together the maps . When the norm is strictly convex, the sets are a family of -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map is straightforward provided one can show that the disintegration of (and thus of ) on such segments is absolutely continuous w.r.t. the -dimensional Hausdorff measure. When the norm is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set and then in . The strategy is sufficiently powerful to be applied to other optimal transportation problems.


Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb {R}^4$

Systems of Transversal Sections Near Critical Energy Levels of Hamiltonian Systems in $\mathbb {R}^4$

Author: Naiara V. de Paulo

Publisher: American Mathematical Soc.

Published: 2018-03-19

Total Pages: 118

ISBN-13: 1470428016

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In this article the authors study Hamiltonian flows associated to smooth functions R R restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point in the zero energy level . The Hamiltonian function near is assumed to satisfy Moser's normal form and is assumed to lie in a strictly convex singular subset of . Then for all small, the energy level contains a subset near , diffeomorphic to the closed -ball, which admits a system of transversal sections , called a foliation. is a singular foliation of and contains two periodic orbits and as binding orbits. is the Lyapunoff orbit lying in the center manifold of , has Conley-Zehnder index and spans two rigid planes in . has Conley-Zehnder index and spans a one parameter family of planes in . A rigid cylinder connecting to completes . All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to in follows from this foliation.


Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries

Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries

Author: Francis Nier

Publisher: American Mathematical Soc.

Published: 2018-03-19

Total Pages: 156

ISBN-13: 1470428024

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This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.


Entire Solutions for Bistable Lattice Differential Equations with Obstacles

Entire Solutions for Bistable Lattice Differential Equations with Obstacles

Author: Aaron Hoffman

Publisher: American Mathematical Soc.

Published: 2018-01-16

Total Pages: 132

ISBN-13: 1470422018

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The authors consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions they show that wave-like solutions exist when obstacles (characterized by “holes”) are present in the lattice. Their work generalizes to the discrete spatial setting the results obtained in Berestycki, Hamel, and Matuno (2009) for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.


Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Author: Xiao Xiong

Publisher: American Mathematical Soc.

Published: 2018-03-19

Total Pages: 130

ISBN-13: 1470428067

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This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative -torus (with a skew symmetric real -matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces.