Erdos Space And Homeomorphism Groups Of Manifolds
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| Author | : Jan Jakobus Dijkstra |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 76 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821846353 |
Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M. Consider the topological group H(M,D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M,D) as follows. If M is a one-dimensional topological manifold, then we proved in an earlier paper that H(M,D) is homeomorphic to Qω, the countable power of the space of rational numbers. In all other cases we find in this paper that H(M,D) is homeomorphic to the famed Erdős space E E, which consists of the vectors in Hilbert space l2 with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space.
| Author | : Steve Hofmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 91 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821852388 |
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.
| Author | : Wilfrid Gangbo |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 90 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821849395 |
Let $\mathcal{M}$ denote the space of probability measures on $\mathbb{R}^D$ endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in $\mathcal{M}$ was introduced by Ambrosio, Gigli, and Savare. In this paper the authors develop a calculus for the corresponding class of differential forms on $\mathcal{M}$. In particular they prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For $D=2d$ the authors then define a symplectic distribution on $\mathcal{M}$ in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper the authors emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of $\mathbb{R}^D$.
| Author | : K.P. Hart |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 898 |
| Release | : 2013-12-11 |
| Genre | : Mathematics |
| ISBN | : 946239024X |
The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland, 1992 and 2002). The book was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in 2002. The following areas experienced significant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems.
| Author | : Michael Handel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 117 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821869272 |
"September 2011, volume 213, number 1004 (end of volume)."
| Author | : Daniel Allcock |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 89 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821847511 |
"Volume 209, number 985 (fourth of 5 numbers)."
| Author | : Dillon Mayhew |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 110 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821848267 |
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.
| Author | : Thomas Lam |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 103 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821846582 |
The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.
| Author | : Alfonso Castro |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 87 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821847260 |
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.
| Author | : Ralph Greenberg |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 198 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 082184931X |
This paper shows that properties of projective modules over a group ring $\mathbf{Z}_p[\Delta]$, where $\Delta$ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve $E$. Modular representation theory for the group $\Delta$ plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a $\mu$-invariant. The author then studies $\lambda$-invariants $\lambda_E(\sigma)$, where $\sigma$ varies over the absolutely irreducible representations of $\Delta$. He shows that there are non-trivial relationships between these invariants under certain hypotheses.
