Topological Classification of Families of Diffeomorphisms Without Small Divisors
Topological Classification of Families of Diffeomorphisms Without Small Divisors

Author: Javier Ribón

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 183

ISBN-13: 0821847481

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The author gives a complete topological classification for germs of one-parameter families of one-dimensional complex analytic diffeomorphisms without small divisors. In the non-trivial cases the topological invariants are given by some functions attached to the fixed points set plus the analytic class of the element of the family corresponding to the special parameter. The proof is based on the structure of the limits of orbits when we approach the special parameter.

Classification of Radial Solutions Arising in the Study of Thermal Structures with Thermal Equilibrium or No Flux at the Boundary
Classification of Radial Solutions Arising in the Study of Thermal Structures with Thermal Equilibrium or No Flux at the Boundary

Author: Alfonso Castro

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 87

ISBN-13: 0821847260

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The authors provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, their study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. They describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. They also provide information on the stability-unstability of the radial steady states.

The Internally 4-Connected Binary Matroids with No $M(K_{3,3})$-Minor
The Internally 4-Connected Binary Matroids with No $M(K_{3,3})$-Minor

Author: Dillon Mayhew

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 110

ISBN-13: 0821848267

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The authors give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates
Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Author: Steve Hofmann

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 91

ISBN-13: 0821852388

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Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$
Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

Author: Nicola Gigli

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 173

ISBN-13: 0821853090

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The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

On $L$-Packets for Inner Forms of $SL_n$
On $L$-Packets for Inner Forms of $SL_n$

Author: Kaoru Hiraga

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 110

ISBN-13: 0821853643

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The theory of $L$-indistinguishability for inner forms of $SL_2$ has been established in the well-known paper of Labesse and Langlands (L-indistinguishability forSL$(2)$. Canad. J. Math. 31 (1979), no. 4, 726-785). In this memoir, the authors study $L$-indistinguishability for inner forms of $SL_n$ for general $n$. Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305-379, Contemp. Math. 145 (1993)), they modify the $S$-group and show that such an $S$-group fits well in the theory of endoscopy for inner forms of $SL_n$.

Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates
Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

Author: Jun Kigami

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 145

ISBN-13: 082185299X

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Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.

Chevalley Supergroups
Chevalley Supergroups

Author: Rita Fioresi

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 77

ISBN-13: 0821853007

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In the framework of algebraic supergeometry, the authors give a construction of the scheme-theoretic supergeometric analogue of split reductive algebraic group-schemes, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular, all Lie superalgebras of both basic and strange types are considered. This provides a unified approach to most of the algebraic supergroups considered so far in the literature, and an effective method to construct new ones. The authors' method follows the pattern of a suitable scheme-theoretic revisitation of Chevalley's construction of semisimple algebraic groups, adapted to the reductive case. As an intermediate step, they prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.