The integral logarithm in Iwasawa theory : an exercise

Jürgen Ritter; Alfred Weiss

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Displaying similar documents to “The integral logarithm in Iwasawa theory : an exercise”

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...

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...

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.

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...

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...

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

Chunmei Wang (2014)

Applications of Mathematics

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In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by ${(1 + log (H / h))}^{2}$ , where $H$ and $h$ are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.

On the restricted Waring problem over $_{2^{n}} [t]$

Luis Gallardo (2000)

Acta Arithmetica

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1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, $_{16}$ , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is...

Theta functions of quadratic forms over imaginary quadratic fields

Olav K. Richter (2000)

Acta Arithmetica

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1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function $ϑ_{Q} (z) : = \sum_{g \in ℤ^{n}} e x p π i^{t} g Q g z$ , Im z > 0, is a modular form of weight n/2 on $Γ_{0} (N)$ , where N is the level of Q, i.e. $N Q^{- 1}$ is integral and $N Q^{- 1}$ has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier...

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...

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

P. E. Conner, J. Hurrelbrink (1995)

Acta Arithmetica

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A large number of papers have contributed to determining the structure of the tame kernel $K ₂_{F}$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K ₂_{F}$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]). We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic...

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...