Displaying similar documents to “Distribution of lattice points on hyperbolic surfaces”

Shift spaces and attractors in noninvertible horseshoes

H. Bothe (1997)

Fundamenta Mathematicae

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As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square I 2 in 2 (or more generally, of the cube I m in m ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of I 2 (or I m ). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

The space of ANR’s in n

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

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The hyperspaces A N R ( n ) and A R ( n ) in 2 n ( n 3 ) consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute G δ σ δ -spaces and that, indeed, they are not F σ δ σ -spaces. The main result is that A N R ( n ) is an absorber for the class of all absolute G δ σ δ -spaces and is therefore homeomorphic to the standard model space Ω 3 of this class.

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension ω 0 , or, equivalently, of small transfinite dimension ω 0 ; that is, the family consists of compact metrizable spaces whose transfinite dimension is ω 0 , and every compact metrizable space with transfinite dimension ω 0 is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the...

Length of continued fractions in principal quadratic fields

Guillaume Grisel (1998)

Acta Arithmetica

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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l ( ( d ) 2 n + 1 ) be the length of the continued fraction expansion of ( d ) 2 n + 1 . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that C ( d ) 2 n + 1 l ( ( d ) 2 n + 1 ) C ( d ) 2 n + 1 for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

Countable partitions of the sets of points and lines

James Schmerl (1999)

Fundamenta Mathematicae

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The following theorem is proved, answering a question raised by Davies in 1963. If L 0 L 1 L 2 . . . is a partition of the set of lines of n , then there is a partition n = S 0 S 1 S 2 . . . such that | S i | 2 whenever L i . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

Operators on C(ω^α) which do not preserve C(ω^α)

Dale Alspach (1997)

Fundamenta Mathematicae

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It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from C ( ω ω α ) onto itself such that if Y is a subspace of C ( ω ω α ) which is isomorphic to C ( ω ω α ) , then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from C ( ω ω α ) onto itself there is a subspace of C ( ω ω α ) which is isomorphic to C ( ω ω α ) on which the operator is an isomorphism.

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

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Let X be a Polish space, and let ( A p ) p ω be a sequence of G δ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether p ω A p is a true 3 0 subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true 3 0 .

A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

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Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let p k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then p * k , m ( j , n ) = p k , m ( j , n ) .

Decomposing Baire class 1 functions into continuous functions

Saharon Shelah, Juris Steprans (1994)

Fundamenta Mathematicae

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It is shown to be consistent that every function of first Baire class can be decomposed into 1 continuous functions yet the least cardinal of a dominating family in ω ω is 2 . The model used in the one obtained by adding ω 2 Miller reals to a model of the Continuum Hypothesis.

Dugundji extenders and retracts on generalized ordered spaces

Gary Gruenhage, Yasunao Hattori, Haruto Ohta (1998)

Fundamenta Mathematicae

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an L c h -extender (resp. L c c h -extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous L c h -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology...

The Zahorski theorem is valid in Gevrey classes

Jean Schmets, Manuel Valdivia (1996)

Fundamenta Mathematicae

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Let Ω,F,G be a partition of n such that Ω is open, F is F σ and of the first category, and G is G δ . We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.

An ordinal version of some applications of the classical interpolation theorem

Benoît Bossard (1997)

Fundamenta Mathematicae

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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space E 1 with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space E 2 with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of E 1 and E 2 can be controlled by the Szlenk index of E, where the...

The homotopy groups of the L2 -localization of a certain type one finite complex at the prime 3

Yoshitaka Nakazawa, Katsumi Shimomura (1997)

Fundamenta Mathematicae

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For the Brown-Peterson spectrum BP at the prime 3, v 2 denotes Hazewinkel’s second polynomial generator of B P * . Let L 2 denote the Bousfield localization functor with respect to v 2 - 1 B P . A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups π * ( L 2 M X ) for the 8 skeleton X of BP.

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

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Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by U S C C B ( X ) the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify φ U S C C B ( X ) with its graph which is a closed subset of X × ℝ. The space U S C C B ( X ) admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then U S C C B ( X ) is homeomorphic...

Property C'', strong measure zero sets and subsets of the plane

Janusz Pawlikowski (1997)

Fundamenta Mathematicae

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Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections F x (x ∈ X) null, x X F x is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections F x (x ∈ ℝ) null, x X F x is null.

Homotopy orbits of free loop spaces

Marcel Bökstedt, Iver Ottosen (1999)

Fundamenta Mathematicae

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Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors Ω λ ( R ) and ℓ(R) together with algebra homomorphisms e : Ω λ ( R ) H * ( Λ X ) and ψ : ( R ) H * ( E S 1 × S 1 Λ X ) . When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

Cantor manifolds in the theory of transfinite dimension

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Z α such that i n d Z α = α , and no closed subset L of Z α with ind L less than the predecessor of α is a partition in Z α . An α-dimensional Cantor Ind-manifold can be constructed similarly.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

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Let f be a measurable function such that Δ k ( x , h ; f ) = O ( | h | λ ) at each point x of a set E, where k is a positive integer, λ > 0 and Δ k ( x , h ; f ) is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative f ( k ) exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative f ( [ λ ] ) exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano...