On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng

Displaying similar documents to “On operators from separable reflexive spaces with asymptotic structure”

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...

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...

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

Yaping Mao, Christopher Melekian, Eddie Cheng (2023)

Czechoslovak Mathematical Journal

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For a connected graph $G = (V, E)$ and a set $S \subseteq V (G)$ with at least two vertices, an $S$ -Steiner tree is a subgraph $T = (V^{'}, E^{'})$ of $G$ that is a tree with $S \subseteq V^{'}$ . If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$ -Steiner tree. Two $S$ -Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S \subseteq V (G)$ and $| S | \geq 2$ , the pendant tree-connectivity $τ_{G} (S)$ is the maximum number of internally disjoint pendant $S$ -Steiner trees in $G$ , and for $k \geq 2$ , the $k$ -pendant tree-connectivity $τ_{k} (G)$ is the...

Quasi-tree graphs with the minimal Sombor indices

Yibo Li, Huiqing Liu, Ruiting Zhang (2022)

Czechoslovak Mathematical Journal

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The Sombor index $S O (G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)}$ of all edges $u v$ of $G$ , where $d_{G} (u)$ denotes the degree of the vertex $u$ in $G$ . A connected graph $G = (V, E)$ is called a quasi-tree if there exists $u \in V (G)$ such that $G - u$ is a tree. Denote $𝒬 (n, k) = {G : G$ is a quasi-tree graph of order $n$ with $G - u$ being a tree and $d_{G} (u) = k}$ . We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬 (n, k)$ . Furthermore, we characterized the corresponding extremal graphs, respectively.

The relation between the number of leaves of a tree and its diameter

Pu Qiao, Xingzhi Zhan (2022)

Czechoslovak Mathematical Journal

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Let $L (n, d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d .$ Lesniak (1975) gave the lower bound $B (n, d) = ⌈ 2 (n - 1) / d ⌉$ for $L (n, d) .$ When $d$ is even, $B (n, d) = L (n, d) .$ But when $d$ is odd, $B (n, d)$ is smaller than $L (n, d)$ in general. For example, $B (21, 3) = 14$ while $L (21, 3) = 19 .$ In this note, we determine $L (n, d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

On a characterization of $k$ -trees

De-Yan Zeng, Jian Hua Yin (2015)

Czechoslovak Mathematical Journal

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A graph $G$ is a $k$ -tree if either $G$ is the complete graph on $k + 1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$ -tree. Clearly, a $k$ -tree has at least $k + 1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$ -connected graph and has no $K_{3}$ -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$ -trees as follows: if $G$ is a graph with at least $k + 1$ vertices, then $G$ is...

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.

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...

Spanning trees whose reducible stems have a few branch vertices

Pham Hoang Ha, Dang Dinh Hanh, Nguyen Thanh Loan, Ngoc Diep Pham (2021)

Czechoslovak Mathematical Journal

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Let $T$ be a tree. Then a vertex of $T$ with degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$ . The set of leaves of $T$ is denoted by $L (T)$ and the set of branch vertices of $T$ is denoted by $B (T)$ . For two distinct vertices $u$ , $v$ of $T$ , let $P_{T} [u, v]$ denote the unique path in $T$ connecting $u$ and $v .$ Let $T$ be a tree with $B (T) \neq \emptyset$ . For each leaf $x$ of $T$ , let $y_{x}$ denote the nearest branch vertex to $x$ . We delete $V (P_{T} [x, y_{x}]) ∖ {y_{x}}$ from $T$ for all $x \in L (T)$ . The resulting subtree of $T$ is called the reducible stem...

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...

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...

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.

On graceful colorings of trees

Sean English, Ping Zhang (2017)

Mathematica Bohemica

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A proper coloring $c : V (G) \to {1, 2, ..., k}$ , $k \geq 2$ of a graph $G$ is called a graceful $k$ -coloring if the induced edge coloring $c^{'} : E (G) \to {1, 2, ..., k - 1}$ defined by $c^{'} (u v) = | c (u) - c (v) |$ for each edge $u v$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$ -coloring is the graceful chromatic number $χ_{g} (G)$ . It is known that if $T$ is a tree with maximum degree $Δ$ , then $χ_{g} (T) \leq ⌈ \frac{5}{3} Δ ⌉$ and this bound is best possible. It is shown for each integer $Δ \geq 2$ that there is an infinite class of trees $T$ with maximum degree $Δ$ such that $χ_{g} (T) = ⌈ \frac{5}{3} Δ ⌉$ . In particular, we investigate for each...

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of a positive solution of the system ⎧ $- ∆_{p (x)} u = F (x, u, v)$ , x ∈ Ω, ⎨ $- ∆_{q (x)} v = H (x, u, v)$ , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded domain with C² boundary, $F (x, u, v) = λ^{p (x)} [g (x) a (u) + f (v)]$ , $H (x, u, v) = λ^{q (x}) [g (x) b (v) + h (u)]$ , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $- ∆_{p (x)} u = - {d i v (| \nabla u |}^{p (x) - 2} \nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.