Degree Theory of Immersed Hypersurfaces

Degree Theory of Immersed Hypersurfaces
Title Degree Theory of Immersed Hypersurfaces PDF eBook
Author Harold Rosenberg
Publisher American Mathematical Soc.
Pages 62
Release 2020-09-28
Genre Mathematics
ISBN 1470441853

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The authors develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function.

Degree Theory of Immersed Hypersurfaces

Degree Theory of Immersed Hypersurfaces
Title Degree Theory of Immersed Hypersurfaces PDF eBook
Author Harold Rosenberg
Publisher
Pages
Release 2011
Genre
ISBN

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Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples

Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples
Title Linear Dynamical Systems on Hilbert Spaces: Typical Properties and Explicit Examples PDF eBook
Author S. Grivaux
Publisher American Mathematical Soc.
Pages 147
Release 2021-06-21
Genre Education
ISBN 1470446634

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We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigen-values and admits no non-trivial invariant measure, but is densely distri-butionally chaotic. (ii) A typical upper-triangular operator with coefficients of modulus 1 on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form “diagonal with coefficients of modulus 1 on the diagonal plus backward unilateral weighted shift” is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. (iii) There exist Hilbert space operators which are chaotic and U-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not U-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.

Weakly Modular Graphs and Nonpositive Curvature

Weakly Modular Graphs and Nonpositive Curvature
Title Weakly Modular Graphs and Nonpositive Curvature PDF eBook
Author Jérémie Chalopin
Publisher American Mathematical Soc.
Pages 85
Release 2021-06-18
Genre Education
ISBN 1470443627

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This article investigates structural, geometrical, and topological characteri-zations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive cur-vature” and “local-to-global” properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of mod-ular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises (1–skeletons, collinearity graphs, covering graphs, domains, etc.) in several seemingly-unrelated fields of mathematics: * Metric graph theory * Geometric group theory * Incidence geometries and buildings * Theoretical computer science and combinatorial optimization We give a local-to-global characterization of weakly modular graphs and their sub-classes in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With l1–embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even 􀀁–matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme.

The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners

The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners
Title The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners PDF eBook
Author Paul Godin
Publisher American Mathematical Soc.
Pages 72
Release 2021-06-21
Genre Education
ISBN 1470444216

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We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the L2 Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of L2 Sobolev regularity with respect to the Cauchy data and the external forces.

Łojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals

Łojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals
Title Łojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals PDF eBook
Author Paul M Feehan
Publisher American Mathematical Society
Pages 138
Release 2021-02-10
Genre Mathematics
ISBN 1470443023

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The authors' primary goal in this monograph is to prove Łojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces that impose minimal regularity requirements on pairs of connections and sections.

Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms

Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms
Title Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms PDF eBook
Author Kazuyuki Hatada
Publisher American Mathematical Soc.
Pages 165
Release 2021-06-18
Genre Education
ISBN 1470443341

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