# The compactness theorem

Xem 1-11 trên 11 kết quả The compactness theorem
• ### The Invariance of the Index of Elliptic Operators, elliptic operators

Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodiﬀerential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under suﬃciently small perturbations.

• ### Logic For Everyone

The discipline known as Mathematical Logic will not speciﬁcally be deﬁned within this text. Instead, you will study some of the concepts in this signiﬁcant discipline by actually doing mathematical logic. Thus, you will be able to surmise for yourself what the mathematical logician is attempting to accomplish. Consider the following three arguments taken from the disciplines of military science, biology, and set-theory, where the symbols (a), (b), (c), (d), (e) are used only to locate speciﬁc sentences....

• ### Báo cáo " On the matheron theorem for topological spaces"

In this paper we study the extending of the Matheron theorem for general topological spaces. We also show some examples about the spaces F such that the miss-and-hit topology on those spaces are unseparated or non-Hausdorff. 1. Introduction The Choquet theorem (see [1, 2]) plays very importance role in theory of random sets. The proof of this theorem is based on the Matheron theorem and especially, the locally compact property of the space F , where F is a space of all close subsets of a given space E and F is equipped with the miss-and-hit topology (see [1]). ...

• ### Đề tài " A quantitative version of the idempotent theorem in harmonic analysis "

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ ∗ µ = µ, or alternatively if µ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ G : µ(γ) = 1} belongs to the coset ring of G, 1. Introduction Let

• ### Đề tài " Geometrization of 3dimensional orbifolds "

This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramiﬁcation locus, then O is geometric (i.e. has a metric of constant curvature or is Seifert ﬁbred). As a corollary, any smooth orientationpreserving nonfree ﬁnite group action on S 3 is conjugate to an orthogonal action. Contents 1. Introduction 2. 3-dimensional orbifolds 2.1. Basic deﬁnitions 2.2. Spherical and toric decompositions 2.3. Finite group actions on spheres with ﬁxed points 2.4.

• ### Đề tài " A Paley-Wiener theorem for reductive symmetric spaces "

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-ﬁnite compactly supported smooth functions on X is characterized. Contents 1. Introduction 2. Notation 3. The Paley-Wiener space. Main theorem 4. Pseudo wave packets 5. Generalized Eisenstein integrals 6. Induction of Arthur-Campoli relations 7. A property of the Arthur-Campoli relations 8. Proof of Theorem 4.4 9.

• ### Đề tài " Density of hyperbolicity in dimension one "

Annals of Mathematics In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of inﬁnity.

• ### One-Parameter Semigroups for Linear Evolution Equations

The theory of one-parameter semigroups of linear operators on Banach spaces started in the first half of this century, acquired its core in 1948 with the Hille–Yosida generation theorem, and attained its first apex with the 1957 edition of Semigroups and Functional Analysis by E. Hille and R.S. Phillips. In the 1970s and 80s, thanks to the efforts of many different schools, the theory reached a certain state of perfection, which is well represented in the monographs by E.B. Davies [Dav80], J.A. Goldstein [Gol85], A. Pazy [Paz83], and others.

• ### Đề tài " Invariant measures and arithmetic quantum unique ergodicity "

We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classiﬁcation can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the ﬁnite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result. ...

• ### Đề tài " Quasi-actions on trees I. Bounded valence "

Given a bounded valence, bushy tree T , we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T . This theorem has many applications: quasi-isometric rigidity for fundamental groups of ﬁnite, bushy graphs of coarse PD(n) groups for each ﬁxed n; a generalization to actions on Cantor sets of Sullivan’s theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. ...