On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields

P. E. Conner; J. Hurrelbrink

Displaying similar documents to “On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields”

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

Hisao Kato (1994)

Fundamenta Mathematicae

Similarity:

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...

Convexity ranks in higher dimensions

Menachem Kojman (2000)

Fundamenta Mathematicae

Similarity:

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a...

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

Similarity:

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...

Entropy and growth of expanding periodic orbits for one-dimensional maps

A. Katok, A. Mezhirov (1998)

Fundamenta Mathematicae

Similarity:

Let f be a continuous map of the circle $S^{1}$ or the interval I into itself, piecewise $C^{1}$ , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h - ε) n_{k}}$ periodic points of period $n_{k}$ with large derivative along the period, $| {(f^{n_{k}})}^{'} | > e^{(h - ε) n_{k}}$ for some subsequence $n_{k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1 - ε) e^{h n}$ such points.

Gδ -sets in topological spaces and games

Winfried Just, Marion Scheepers, Juris Steprans, Paul Szeptycki (1997)

Fundamenta Mathematicae

Similarity:

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_{n}$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_{n}$ of X. The players must obey the rule that $O_{n} \subseteq O_{n + 1} \subseteq T_{n + 1} \subseteq T_{n}$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a $G_{δ}$ -set. To what extent is the converse true? We show that: (A) For ℱ the collection of countable subsets of X: 1....

Strong Fubini properties of ideals

Ireneusz Recław, Piotr Zakrzewski (1999)

Fundamenta Mathematicae

Similarity:

Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_{x} = y : ⟨ x, y ⟩ \in D$ are in J, then the sections $D^{y} = x : ⟨ x, y ⟩ \in D$ are in I for every y outside a set from J (“measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this...

Rigid $ℵ_{ε}$ -saturated models of superstable theories

Ziv Shami, Saharon Shelah (1999)

Fundamenta Mathematicae

Similarity:

In a countable superstable NDOP theory, the existence of a rigid $ﬡ_{ε}$ -saturated model implies the existence of $2^{λ}$ rigid $ﬡ_{ε}$ -saturated models of power λ for every $λ > 2^{ﬡ_{0}}$ .

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

Ramez Sami (1999)

Fundamenta Mathematicae

Similarity:

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

Similarity:

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.

If it looks and smells like the reals...

Franklin Tall (2000)

Fundamenta Mathematicae

Similarity:

Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_{M}$ is homeomorphic to ℝ, then $X = X_{M}$ . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

Similarity:

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,...

Standardness of sequences of σ-fields given by certain endomorphisms

Jacob Feldman, Daniel Rudolph (1998)

Fundamenta Mathematicae

Similarity:

Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields $ℱ, E^{- 1} ℱ, E^{- 2} ℱ, . . .$ A central example is the one-sided shift σ on $X = {0, 1}^{ℕ}$ with $\frac{1}{2}, \frac{1}{2}$ product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by $(x, y) \to (σ (x), T^{x (1)} (y))$ . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of...

Borel partitions of unity and lower Carathéodory multifunctions

S. Srivastava (1995)

Fundamenta Mathematicae

Similarity:

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...

Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik (1998)

Fundamenta Mathematicae

Similarity:

Let α be an ergodic rotation of the d-torus $𝕋^{d} = ℝ^{d} / ℤ^{d}$ . For any piecewise smooth function $f : 𝕋^{d} \to ℝ$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on $L^{2} (𝕋^{d})$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product $S_{f} : 𝕋^{d + 1} \to 𝕋^{d + 1}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum...

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

Similarity:

Let $X \subseteq 2^{w}$ . Consider the class of all Borel $F \subseteq X \times 2^{w}$ with null vertical sections $F_{x}$ , x ∈ X. We show that if for all such F and all null Z ⊆ X, $\cup_{x \in Z} F_{x}$ is null, then for all such F, $\cup_{x \in X} F_{x} \neq 2^{w}$ . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Misiurewicz maps unfold generically (even if they are critically non-finite)

Sebastian van Strien (2000)

Fundamenta Mathematicae

Similarity:

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if $f_{λ_{0}}$ is critically finite with non-degenerate critical point $c_{1} (λ_{0}), . . ., c_{n} (λ_{0})$ such that $f_{λ_{0}}^{k_{i}} (c_{i} (λ_{0})) = p_{i} (λ_{0})$ are hyperbolic periodic points for i = 1,...,n, then IV-1. Age impartible......................................................................................................................................................................... 31 $λ \mapsto (f_{λ}^{k_{1}} (c_{1} (λ)) - p_{1} (λ), . . ., f_{λ}^{k_{d - 2}} (c_{d - 2} (λ)) - p_{d - 2} (λ))$ is a local diffeomorphism...

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

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

Acta Arithmetica

Similarity:

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...

The dimension of X^n where X is a separable metric space

John Kulesza (1996)

Fundamenta Mathematicae

Similarity:

For a separable metric space X, we consider possibilities for the sequence $S (X) = d_{n} : n \in ℕ$ where $d_{n} = d i m X^{n}$ . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is $X_{n}$ such that $S (X_{n}) = n, n + 1, n + 2, . . .$ , $Y_{n}$ , for n >1, such that $S (Y_{n}) = n, n + 1, n + 2, n + 2, n + 2, . . .$ , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of $ℝ^{2}$ is shown to exist which satisfies $1 = d i m X = d i m X^{2}$ and $d i m X^{3} = 2$ .

A complement to the theory of equivariant finiteness obstructions

Paweł Andrzejewski (1996)

Fundamenta Mathematicae

Similarity:

It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family $w_{α}^{H} (X)$ of elements of the groups $K_{0} (ℤ [π_{0} {(W H (X))}_{α}^{*}])$ . We prove that every family $w_{α}^{H}$ of elements of the groups $K_{0} (ℤ [π_{0} {(W H (X))}_{α}^{*}])$ can be realized as the family of equivariant finiteness obstructions $w_{α}^{H} (X)$ of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s...