Sharkovskiĭ's theorem holds for some discontinuous functions

Piotr Szuca

Displaying similar documents to “Sharkovskiĭ's theorem holds for some discontinuous functions”

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

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A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f : ℝ \to ℝ$ is of the first Baire class if and only if for each $ϵ > 0$ there is a sequence of closed sets ${C_{n}}_{n = 1}^{\infty}$ such that $D_{f} = ⋃_{n = 1}^{\infty} C_{n}$ and $ω_{f} (C_{n}) < ϵ$ for each $n$ where $ω_{f} (C_{n}) = sup {| f (x) - f (y) | : x, y \in C_{n}}$ and $D_{f}$ denotes the set of points of discontinuity of $f$ . The proof of the main theorem is based on a recent $ϵ$ - $δ$ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

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A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$ , denoted by $α^{'} (G)$ , is the number of edges in a maximum matching of $G$ . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...

On the bounds of Laplacian eigenvalues of $k$ -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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Let $μ_{n - 1} (G)$ be the algebraic connectivity, and let $μ_{1} (G)$ be the Laplacian spectral radius of a $k$ -connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $μ_{n - 1} (G) \geq \frac{2 n k^{2}}{(n (n - 1) - 2 m) (n + k - 2) + 2 k^{2}},$ with equality if and only if $G$ is the complete graph $K_{n}$ or $K_{n} - e$ . Moreover, if $G$ is non-regular, then $μ_{1} (G) < 2 Δ - \frac{2 (n Δ - 2 m) k^{2}}{2 (n Δ - 2 m) (n^{2} - 2 n + 2 k) + n k^{2}},$ where $Δ$ stands for the maximum degree of $G$ . Remark that in some cases, these two inequalities improve some previously known results.

Baire classes of complex $L_{1}$ -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

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Let $X$ be a complex $L_{1}$ -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ of the extreme points of the dual unit ball $B_{X^{*}}$ to the whole unit ball $B_{X^{*}}$ . As a corollary we show that, given $α \in [1, ω_{1})$ , the intrinsic $α$ -th Baire class of $X$ can be identified with the space of bounded homogeneous Baire- $α$ functions on the set $ext B_{X^{*}}$ when $ext B_{X^{*}}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...

On D-connected sets in the space $V^{q}$

S. Topa (1976)

Annales Polonici Mathematici

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Domination numbers in graphs with removed edge or set of edges

Magdalena Lemańska (2005)

Discussiones Mathematicae Graph Theory

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It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_{w}$ and the connected domination number $γ_{c}$ , i.e., we show that $γ_{w} (G) \leq γ_{w} (G - e) \leq γ_{w} (G) + 1$ and $γ_{c} (G) \leq γ_{c} (G - e) \leq γ_{c} (G) + 2$ if G and G-e are connected. Additionally we show that $γ_{w} (G) \leq γ_{w} (G - E ₚ) \leq γ_{w} (G) + p - 1$ and $γ_{c} (G) \leq γ_{c} (G - E ₚ) \leq γ_{c} (G) + 2 p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

Some remarks on α-domination

Franz Dahme, Dieter Rautenbach, Lutz Volkmann (2004)

Discussiones Mathematicae Graph Theory

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Let α ∈ (0,1) and let $G = (V_{G}, E_{G}$ ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D \subseteq V_{G}$ is called an α-dominating set of G, if $| N_{G} (u) \cap D | \geq α d_{G} (u)$ for all $u \in V_{G} ∖ D$ . We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar, Manoharan Subajini (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity and $J (R)$ the Jacobson radical of $R$ . The Jacobson graph of $R$ , denoted by $𝔍_{R}$ , is defined as the graph with vertex set $R ∖ J (R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1 - x y$ is not a unit of $R$ . The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_{n}$ . In this paper, we investigate the genus number of the compact Riemann surface in which $𝔍_{R}$ can be embedded and...

Diagonals of separately continuous functions of $n$ variables with values in strongly $σ$ -metrizable spaces

Olena Karlova, Volodymyr Mykhaylyuk, Oleksandr Sobchuk (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove the result on Baire classification of mappings $f : X \times Y \to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $P P$ -space, $Y$ is a topological space and $Z$ is a strongly $σ$ -metrizable space with additional properties. We show that for any topological space $X$ , special equiconnected space $Z$ and a mapping $g : X \to Z$ of the $(n - 1)$ -th Baire class there exists a strongly separately continuous mapping $f : X^{n} \to Z$ with the diagonal $g$ . For wide classes...

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.