A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization

Chunmei Wang

Displaying similar documents to “A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization”

On $B_{2 k}$ -sequences

Martin Helm (1993)

Acta Arithmetica

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Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_{r}$ -sequence A satisfies $l i m i n f_{n \to \infty} (A (n) / (n^{1 / r})) = 0$ . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof...

Inessentiality with respect to subspaces

Michael Levin (1995)

Fundamenta Mathematicae

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Let X be a compactum and let $A = (A_{i}, B_{i}) : i = 1, 2, . . .$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_{i}$ separating $A_{i}$ and $B_{i}$ the intersection $(\cap F_{i}) \cap Y$ is not empty. So A is inessential on Y if there exist closed $F_{i}$ separating $A_{i}$ and $B_{i}$ such that $\cap F_{i}$ does not intersect Y. Properties of inessentiality are studied and applied to prove: Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

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We study ergodicity of cylinder flows of the form $T_{f} : T \times ℝ \to T \times ℝ$ , $T_{f} (x, y) = (x + α, y + f (x))$ , where $f : T \to ℝ$ is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that $D^{k} f$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of $D^{k} f$ have some good properties, then $T_{f}$ is ergodic. Moreover, there exists $ε_{f} > 0$ such that if $v : T \to ℝ$ is a function with zero integral such that $D^{k} v$ is of bounded...

Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski, Ireneusz Recław (1995)

Fundamenta Mathematicae

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We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^{w} \times 2^{w}$ and continuous functions $e, f : w^{w} \to w^{w}$ such that • N is $G_{δ}$ and $N_{x} : x \in w^{w}$ , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^{w}$ ; • M is $F_{σ}$ and $M_{x} : x \in w^{w}$ is a basis for the ideal of meager subsets of $2^{w}$ ; • $\forall x, y N_{e (x)} \subseteq N_{y} \Rightarrow M_{x} \subseteq M_{f (y)}$ . From this we derive that for a separable metric space X, •if for all Borel (resp. $G_{δ}$ ) sets...

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 1. We improve this result of Sternfeld...

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and $1 \leq α < ω_{1}^{C K}$ , if for some $η < ℵ_{1}$ and all u ∈ WO of length η, a is $Σ_{α}^{0} (u)$ , then a is $Σ_{α}^{0}$ . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: $Σ_{1}^{1}$ -Turing-determinacy implies the existence of $0$ .

A note on Tsirelson type ideals

Boban Veličković (1999)

Fundamenta Mathematicae

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Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_{0}$ or $l_{p}$ , for p ≥ 1, we construct a simple Borel ideal $I_{T}$ such that the Borel cardinalities of the quotient spaces $P (ℕ) / I_{T}$ and $P (ℕ) / I_{0}$ are incomparable, where $I_{0}$ is the summable ideal of all sets A ⊆ ℕ such that $\sum_{n \in A} 1 / (n + 1) < \infty$ . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

On the positivity of the number of t-core partitions

Ken Ono (1994)

Acta Arithmetica

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A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n .$ Here we define a special class of partitions. 1. Let $t \geq 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n .$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,4,6]. If $t \geq 1$ and $n \geq 0$ ,...

Co-H-structures on equivariant Moore spaces

Martin Arkowitz, Marek Golasiński (1994)

Fundamenta Mathematicae

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Let G be a finite group, $𝕆_{G}$ the category of canonical orbits of G and $A : 𝕆_{G} \to 𝔸$ b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $E x t^{n - 1} (A, A \otimes A)$ . Then the case $G = ℤ_{p^{k}}$ leads to an example of...

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

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We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in $A (ℰ \otimes ℬ (X))$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential...

Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones (1995)

Fundamenta Mathematicae

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Given two topologies, $T_{1}$ and $T_{2}$ , on the same set X, the intersection topologywith respect to $T_{1}$ and $T_{2}$ is the topology with basis $U_{1} \cap U_{2} : U_{1} \in T_{1}, U_{2} \in T_{2}$ . Equivalently, T is the join of $T_{1}$ and $T_{2}$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and $ω_{1}$ -compactness in this class of topologies. We demonstrate that the majority of his results...

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x = 1 / d ₁ (x) + 1 / (d ₁ (x) d ₂ (x)) + . . . + 1 / (d ₁ (x) d ₂ (x) . . . d_{n} (x)) + . . .$ , where $d_{j} (x), j \geq 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j + 1} (x) \geq d_{j} (x)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $l i m_{n \to \infty} d_{n}^{1 / n} (x) = e .$ He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $d i m_{H} x \in (0, 1] : (2) f a i l s = 1$ . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $d i m_{H}$ to denote...

A strongly non-Ramsey uncountable graph

Péter Komjáth (1997)

Fundamenta Mathematicae

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It is consistent that there exists a graph X of cardinality $ℵ_{1}$ such that every graph has an edge coloring with $ℵ_{1}$ colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors).

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_{j}$ is $j^{k}$ or when $a_{j}$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_{j}$ is j. In contrast we show, under fairly general conditions, that the maximum is less than $2^{n} / n^{r}$ , where r is an arbitrary positive number. One consequence...

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

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Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $α < ω_{1}$ then f ○ g ∈ $B_{α} (Z)$ for every $g \in B_{α} (Z) \cap^{Z} I$ if and only if f is continuous on I, where $B_{α} (Z)$ stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes $S_{α} (Z) (α > 0)$ in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally...

A note on strange nonchaotic attractors

Gerhard Keller (1996)

Fundamenta Mathematicae

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For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ $T^{1} \times ℝ_{+}$ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties: 1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in $T^{1} \times ℝ_{+}$ . The set θ:ϕ(θ) ≠ 0 is meager but has full 1-dimensional Lebesgue measure. 2. The omega-limit of Lebesgue-a.e point in $T^{1} \times ℝ_{+}$ is $Γ ̅$ , but for a residual set of points in $T^{1} \times ℝ_{+}$ the omega...

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

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For X ⊆ [0,1], let $D_{X}$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of $D_{X}$ . For α ≥ 3, X is properly $D_{ξ} (Π_{α}^{0})$ iff $D_{X}$ is properly $D_{ξ} (Π_{1 + α}^{0})$ . We also show that for every nonempty set X ⊆[0,1], $D_{X}$ is $Π_{3}^{0}$ -hard. For each nonempty $Π_{2}^{0}$ set X ⊆ [0,1], in particular for X = x, $D_{X}$ is $Π_{3}^{0}$ -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is $Π_{3}^{0}$ -complete. Moreover,...

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Hisao Kato (1994)

Fundamenta Mathematicae

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A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d (f^{n} (x), f^{n} (y)) > c$ (resp. $d i a m f^{n} (A) > c$ ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate...