How many clouds cover the plane?
James H. Schmerl (2003)
Fundamenta Mathematicae
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The plane can be covered by n + 2 clouds iff .
James H. Schmerl (2003)
Fundamenta Mathematicae
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The plane can be covered by n + 2 clouds iff .
M. Jelić (1987)
Matematički Vesnik
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Yasunao Hattori, Jan van Mill (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove that , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.
Ludwik Jaksztas (2007)
Bulletin of the Polish Academy of Sciences. Mathematics
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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of , given by Urbański and Zinsmeister. The closure of the limit set of our IFS is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...
Michael Levin (2003)
Fundamenta Mathematicae
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We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that we have and r is G-acyclic.
Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)
Mathematica Bohemica
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We prove that the Niemytzki plane is -metrizable and we try to explain the differences between the concepts of a stratifiable space and a -metrizable space. Also, we give a characterisation of -metrizable spaces which is modelled on the version described by Chigogidze.
Ludwik Jaksztas (2011)
Fundamenta Mathematicae
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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set is continuous at σ₀ as the function of the parameter if and only if . Since on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of on an open and dense subset of...
H. Murat Tuncali, Vesko Valov (2002)
Fundamenta Mathematicae
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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an -subset of X such that and the restriction is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...
Nicolas Monod, Henrik Densing Petersen (2014)
Annales de l’institut Fourier
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Let be any group containing an infinite elementary amenable subgroup and let . We construct an exhaustion of by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to -dimension and gives an answer to a question of Gaboriau.
Chun-Yun Cao, Bao-Wei Wang, Jun Wu (2013)
Studia Mathematica
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Let be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence of integers, called the digit sequence of x, such that . We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set for any infinite subset B ⊂ ℕ, a question posed by...
Fan Lü, Bo Tan, Jun Wu (2014)
Fundamenta Mathematicae
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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with . We prove that for any x ∈ (0,1), (x) contains a sequence increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure are nowhere dense.
Behrouz Taji (2014)
Annales de l’institut Fourier
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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an provided that has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.
Tomasz Szarek, Maciej Ślęczka (2006)
Bulletin of the Polish Academy of Sciences. Mathematics
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It is shown that every Polish space X with admits a compact subspace Y such that where and denote the topological and Hausdorff dimensions, respectively.
Mikhail Skopenkov (2003)
Fundamenta Mathematicae
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For any collection of graphs we find the minimal dimension d such that the product is embeddable into (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and are not embeddable into , where K₅ and are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.
L. Olsen (2005)
Colloquium Mathematicae
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For a subset and , the local Hausdorff dimension function of E at x is defined by where denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.
Min-wei Tang, Zhi-Yi Wu (2020)
Czechoslovak Mathematical Journal
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It is known that a set of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if , where and denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set by replacing with . It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.
G. A. Karagulyan (2007)
Studia Mathematica
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Let be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis differentiates the integral of f if s ∉ S, and almost everywhere if s ∈ S. If the condition holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a (resp. a ).
Juan Rivera-Letelier (2001)
Fundamenta Mathematicae
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Given d ≥ 2 consider the family of polynomials for c ∈ ℂ. Denote by the Julia set of and let be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters : those for which the critical point 0 is not recurrent by and without parabolic cycles. The Hausdorff dimension of , denoted by , does not depend continuously on c at such ; on the other hand the function is analytic in . Our first result asserts that there is still some...
Werner Klöckl (2008)
Discussiones Mathematicae Graph Theory
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The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number of G. Extending these concepts to infinite graphs we prove that and , where denotes the hypercube of countable dimension. We also show that , thereby completing the investigation of finite hypercubes with respect to . Our...
Erica Flapan, Blake Mellor, Ramin Naimi (2008)
Fundamenta Mathematicae
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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, and , where denotes the second coefficient of the Conway polynomial of .
Dejun Wu (2015)
Czechoslovak Mathematical Journal
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We study the relations between finitistic dimensions and restricted injective dimensions. Let be a ring and a left -module with . If is selforthogonal, then we show that . Moreover, if is a left noetherian ring and is a finitely generated left -module with finite injective dimension, then . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.
Włodzimierz M. Mikulski (2024)
Archivum Mathematicum
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We classify classical linear connections on the total space of a fibred manifold induced in a natural way by the following three objects: a general connection in , a classical linear connection on and a linear connection in the vertical bundle . The main result says that if and then the natural operators under consideration form the dimensional affine space.
Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)
Czechoslovak Mathematical Journal
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We study the generalized -connectivity as introduced by Hager in 1985, as well as the more recently introduced generalized -edge-connectivity . We determine the exact value of and for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case .
Ilija S. Vrećica (2020)
Czechoslovak Mathematical Journal
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We determine the distribution over square-free integers of the pair , where is a curve in the congruent number curve family, is the image of isogeny , , and is the isogeny dual to .