Mathematical Study of Degenerate Boundary Layers: A Large Scale Ocean Circulation Problem
Author: Anne-Laure Dalibard
Publisher: American Mathematical Soc.
Published: 2018-05-29
Total Pages: 111
ISBN-13: 1470428350
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Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem
Author: Gabriella Pinzari
Publisher: American Mathematical Soc.
Published: 2018-10-03
Total Pages: 92
ISBN-13: 1470441020
DOWNLOAD EBOOK →The author proves the existence of an almost full measure set of -dimensional quasi-periodic motions in the planetary problem with masses, with eccentricities arbitrarily close to the Levi–Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V. I. Arnold in the 1960s, for the particular case of the planar three-body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group. The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, a common tool of previous literature.
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
Author: T. Alazard
Publisher: American Mathematical Soc.
Published: 2019-01-08
Total Pages: 108
ISBN-13: 147043203X
DOWNLOAD EBOOK →This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, the authors consider solutions such that the curvature of the initial free surface does not belong to L2. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.
On the Asymptotic Analysis of Large-scale Ocean Circulation
Bellman Function for Extremal Problems in BMO II: Evolution
Author: Paata Ivanisvili
Publisher: American Mathematical Soc.
Published: 2018-10-03
Total Pages: 136
ISBN-13: 1470429543
DOWNLOAD EBOOK →In a previous study, the authors built the Bellman function for integral functionals on the space. The present paper provides a development of the subject. They abandon the majority of unwanted restrictions on the function that generates the functional. It is the new evolutional approach that allows the authors to treat the problem in its natural setting. What is more, these new considerations lighten dynamical aspects of the Bellman function, in particular, the evolution of its picture.
Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems
Author: Laurent Lazzarini
Publisher: American Mathematical Soc.
Published: 2019-02-21
Total Pages: 106
ISBN-13: 147043492X
DOWNLOAD EBOOK →A wandering domain for a diffeomorphism of is an open connected set such that for all . The authors endow with its usual exact symplectic structure. An integrable diffeomorphism, i.e., the time-one map of a Hamiltonian which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory; lower estimates are related to examples of Arnold diffusion. This is a contribution to the “quantitative Hamiltonian perturbation theory” initiated in previous works on the optimality of long term stability estimates and diffusion times; the emphasis here is on discrete systems because this is the natural setting to study wandering domains.
On Fusion Systems of Component Type
Author: Michael Aschbacher
Publisher: American Mathematical Soc.
Published: 2019-02-21
Total Pages: 182
ISBN-13: 1470435209
DOWNLOAD EBOOK →This memoir begins a program to classify a large subclass of the class of simple saturated 2-fusion systems of component type. Such a classification would be of great interest in its own right, but in addition it should lead to a significant simplification of the proof of the theorem classifying the finite simple groups. Why should such a simplification be possible? Part of the answer lies in the fact that there are advantages to be gained by working with fusion systems rather than groups. In particular one can hope to avoid a proof of the B-Conjecture, a important but difficult result in finite group theory, established only with great effort.
Fusion of Defects
Author: Arthur Bartels
Publisher: American Mathematical Soc.
Published: 2019-04-10
Total Pages: 102
ISBN-13: 1470435233
DOWNLOAD EBOOK →Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. They introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. The authors' most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
Crossed Products of Operator Algebras
Author: Elias G. Katsoulis
Publisher: American Mathematical Soc.
Published: 2019-04-10
Total Pages: 85
ISBN-13: 1470435454
DOWNLOAD EBOOK →The authors study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. They develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. They complement their generic results with the detailed study of many important special cases. In particular they study crossed products of tensor algebras, triangular AF algebras and various associated C -algebras. They make contributions to the study of C -envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. They also answer questions from the pertinent literature.
Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance
Author: Jun Kigami
Publisher: American Mathematical Soc.
Published: 2019-06-10
Total Pages: 118
ISBN-13: 1470436205
DOWNLOAD EBOOK →In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0,1]n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0,1]n, density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0,1]2 and self-similar measures. The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.
