Proper translation

Heike Mildenberger; Saharon Shelah

Displaying similar documents to “Proper translation”

Weak square sequences and special Aronszajn trees

John Krueger (2013)

Fundamenta Mathematicae

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A classical theorem of set theory is the equivalence of the weak square principle $□_{μ} *$ with the existence of a special Aronszajn tree on μ⁺. We introduce the notion of a weak square sequence on any regular uncountable cardinal, and prove that the equivalence between weak square sequences and special Aronszajn trees holds in general.

Closure for spanning trees and distant area

Jun Fujisawa, Akira Saito, Ingo Schiermeyer (2011)

Discussiones Mathematicae Graph Theory

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A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $d e g_{G} u + d e g_{G} v \geq n - 1$ , G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $d e g_{G} u + d e g_{G} v$ and the structure of the distant area for u and v. We prove...

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)

Czechoslovak Mathematical Journal

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The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^{n}$ vertices, $n \geq 1$ , is $n$ . The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...

A partition of the Catalan numbers and enumeration of genealogical trees

Rainer Schimming (1996)

Discussiones Mathematicae Graph Theory

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A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers $C_{n, k}$ of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the $C_{n, k}$ form a partition of the n-th Catalan numer Cₙ, that means $C_{n, 1} + C_{n, 2} + . . . + C_{n, n} = C ₙ$ .

Iterating along a Prikry sequence

Spencer Unger (2016)

Fundamenta Mathematicae

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We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, $ℵ_{ω}$ is strong limit and $2^{ℵ_{ω}} = ℵ_{ω + 2}$ .

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

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We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

Compactness properties of weighted summation operators on trees

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

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We investigate compactness properties of weighted summation operators $V_{α, σ}$ as mappings from ℓ₁(T) into $ℓ_{q} (T)$ for some q ∈ (1,∞). Those operators are defined by $(V_{α, σ} x) (t) : = α (t) \sum_{s ⪰ t} σ (s) x (s)$ , t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $e ₙ (V_{α, σ})$ , the (dyadic) entropy numbers of $V_{α, σ}$ . The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights...

On the (2,2)-domination number of trees

You Lu, Xinmin Hou, Jun-Ming Xu (2010)

Discussiones Mathematicae Graph Theory

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Let γ(G) and $γ_{2, 2} (G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2 (γ (T) + 1)) / 3 \leq γ_{2, 2} (T) \leq 2 γ (T)$ . Moreover, we characterize all the trees achieving the equalities.

Note about sequences of extrema $(A, 2 B)$ -edge coloured trees

Urszula Bednarz, Iwona Włoch (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we determine successive extremal trees with respect to the number of all $(A, 2 B)$ -edge colourings.

Trees and the dynamics of polynomials

Laura G. DeMarco, Curtis T. McMullen (2008)

Annales scientifiques de l'École Normale Supérieure

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In this paper we study branched coverings of metrized, simplicial trees $F : T \to T$ which arise from polynomial maps $f : ℂ \to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $ℙ T_{D}$ compactifying the moduli space of polynomials of degree $D$ ; that $F$ records the asymptotic behavior of the multipliers of $f$ ; and that any meromorphic family of polynomials over $Δ^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a...

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. Dijkstra, Kirsten I. S. Valkenburg (2010)

Fundamenta Mathematicae

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One way to generalize complete Erdős space $_{c}$ is to consider uncountable products of zero-dimensional $G_{δ}$ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng (2008)

Studia Mathematica

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Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower- $ℓ_{q}$ -tree estimate and let T be a bounded linear operator from X which satisfies an upper- $ℓ_{p}$ -tree estimate. Then T factors through a subspace of ${(\sum F ₙ)}_{ℓ_{r}}$ , where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $(ℓ_{p}, ℓ_{q})$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic...

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

On operators which factor through $l_{p}$ or c₀

Bentuo Zheng (2006)

Studia Mathematica

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Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${(\sum F ₙ)}_{l_{p}}$ , where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_{p}$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_{p}$ . This gives an answer to a question of W. B. Johnson. We also prove...

Turán's problem and Ramsey numbers for trees

Zhi-Hong Sun, Lin-Lin Wang, Yi-Li Wu (2015)

Colloquium Mathematicae

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Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V = v ₀, v ₁, . . ., v_{n - 1}$ , $E ₁ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 4} v_{n - 2}, v_{n - 3} v_{n - 1}$ and $E ₂ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 3} v_{n - 2}, v_{n - 3} v_{n - 1}$ . For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r (T ₘ, T ₙ^{i})$ , where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

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The topology and combinatorial structure of the Mandelbrot set $ℳ^{d}$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in $ℳ^{d}$ . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$ . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

Weak roman domination in graphs

P. Roushini Leely Pushpam, T.N.M. Malini Mai (2011)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph and f be a function f:V → 0,1,2. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f’: V → 0,1,2 defined by f’(u) = 1, f’(v) = f(v)-1 and f’(w) = f(w) if w ∈ V-u,v, has no undefended vertex. The weight of f is $w (f) = \sum_{v \in V} f (v)$ . The weak Roman domination...

Pruning Galton–Watson trees and tree-valued Markov processes

Romain Abraham, Jean-François Delmas, Hui He (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process ${𝒢 (u)}$ by pruning Galton–Watson trees and an analogous process ${𝒢^{*} (u)}$ by pruning a critical or subcritical Galton–Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process ${𝒢 (u)}$ run until its ascension time has a representation in terms of ${𝒢^{*} (u)}$ . A similar result was obtained by...

The tree property at the double successor of a measurable cardinal κ with $2^{κ}$ large

Sy-David Friedman, Ajdin Halilović (2013)

Fundamenta Mathematicae

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Assuming the existence of a λ⁺-hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ⁺⁺ has the tree property and $2^{κ} = κ ⁺ ⁺ ⁺$ . If the assumption is strengthened to the existence of a θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get $2^{κ} = θ$ .

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${(A ₙ)}_{n \in ω}$ of X such that $f_{| A ₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Combinatorics of open covers (VII): Groupability

Ljubiša D. R. Kočinac, Marion Scheepers (2003)

Fundamenta Mathematicae

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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31 / 2}$ -space. In [9] we showed that $C_{p} (X)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_{p} (X)$ has countable fan tightness and the Reznichenko...

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

On locating and differentiating-total domination in trees

Mustapha Chellali (2008)

Discussiones Mathematicae Graph Theory

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A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γ ₜ^{L} (G)$ and $γ ₜ^{D} (G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total...