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27522. On sequential monitoring for change in scale
- Creator:
- Chochola, Ondřej
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- sequential test for change in scale
- Language:
- English
- Description:
- We propose a sequential monitoring scheme for detecting a change in scale. We consider a stable historical period of length m. The goal is to propose a test with asymptotically small probability of false alarm and power 1 as the length of the historical period tends to infinity. The asymptotic distribution under the null hypothesis and consistency under the alternative hypothesis is derived. A small simulation study illustrates the finite sample performance of the monitoring scheme.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27523. On sequential properties of Banach spaces, spaces of measures and densities
- Creator:
- Borodulin-Nadzieja , Piotr and Plebanek, Grzegorz
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Gelfand-Phillips property, Mazur property, and generalized density
- Language:
- English
- Description:
- We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space $E$ can be naturally expressed in terms of {\it weak}* continuity of seminorms on the unit ball of $E^*$. \endgraf We attempt to carry out a construction of a Banach space of the form $C(K)$ which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the {\it weak}* sequential closure of atomic measures, and the set-theoretic properties of generalized densities on the natural numbers.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27524. On set covariance and three-point test sets
- Creator:
- Rataj, Jan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- convex body, set with positive reach, normal measure, and set covariance
- Language:
- English
- Description:
- The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of $\mathbb{R}^d $ is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27525. On sets of non-differentiability of Lipschitz and convex functions
- Creator:
- Zajíček, Luděk
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Lipschitz function, convex function, Gˆateaux differentiability, Fréchet differentiability, Γ-null sets, ball small sets, δ-convex surfaces, and strong porosit
- Language:
- English
- Description:
- We observe that each set from the system A˜ (or even C˜) is Γ-null; consequently, the version of Rademacher’s theorem (on Gˆateaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on n is σ-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27526. On sets with Baire property in topological spaces
- Creator:
- Basu, S.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Baire property, first category, and second category
- Language:
- English
- Description:
- Steinhaus [9] prove that if a set $A$ has a positive Lebesgue measure in the line then its distance set contains an interval. He obtained even stronger forms of this result in [9], which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar [1]. In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus [9] and their corresponding form for ratios by Bose-Majumdar [1].
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27527. On signed distance-$k$-domination in graphs
- Creator:
- Xing, Huaming, Sun, Liang, and Chen, Xuegang
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- signed distance-$k$-domination number, signed distance-$k$-dominating function, and signed domination number
- Language:
- English
- Description:
- The signed distance-$k$-domination number of a graph is a certain variant of the signed domination number. If $v$ is a vertex of a graph $G$, the open $k$-neighborhood of $v$, denoted by $N_k(v)$, is the set $N_k(v)=\lbrace u\mid u\ne v$ and $d(u,v)\le k\rbrace $. $N_k[v]=N_k(v)\cup \lbrace v\rbrace $ is the closed $k$-neighborhood of $v$. A function $f\: V\rightarrow \lbrace -1,1\rbrace $ is a signed distance-$k$-dominating function of $G$, if for every vertex $v\in V$, $f(N_k[v])=\sum _{u\in N_k[v]}f(u)\ge 1$. The signed distance-$k$-domination number, denoted by $\gamma _{k,s}(G)$, is the minimum weight of a signed distance-$k$-dominating function on $G$. The values of $\gamma _{2,s}(G)$ are found for graphs with small diameter, paths, circuits. At the end it is proved that $\gamma _{2,s}(T)$ is not bounded from below in general for any tree $T$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27528. On signed edge domination numbers of trees
- Creator:
- Zelinka, Bohdan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- tree, signed edge domination number, and signed edge total domination number
- Language:
- English
- Description:
- The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a mapping of the edge set E(G) of G into the set {−1, 1}. If ∑ x∈N[e] f(x) ≥ 1 for each e ∈ E(G), then f is called a signed edge dominating function on G. The minimum of the values ∑ x∈E(G) f(x), taken over all signed edge dominating function f on G, is called the signed edge domination number of G and is denoted by γ s(G). If instead of the closed neighbourhood NG[e] we use the open neighbourhood NG(e) = NG[e] − {e}, we obtain the definition of the signed edge total domination number γ st(G) of G. In this paper these concepts are studied for trees. The number γ s(T) is determined for T being a star of a path or a caterpillar. Moreover, also γ s(Cn) for a circuit of length n is determined. For a tree satisfying a certain condition the inequality γ s(T) ≥ γ (T) is stated. An existence theorem for a tree T with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for γ st(T).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27529. On signed majority total domination in graphs
- Creator:
- Xing, Hua-Ming, Sun, Liang, and Chen, Xue-Gang
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- signed majority total dominating function and signed majority total domination number
- Language:
- English
- Description:
- We initiate the study of signed majority total domination in graphs. Let $G=(V,E)$ be a simple graph. For any real valued function $f\: V \rightarrow \mathbb{R}$ and ${S\subseteq V}$, let $f(S)=\sum _{v\in S}f(v)$. A signed majority total dominating function is a function $f\: V\rightarrow \lbrace -1,1\rbrace $ such that $f(N(v))\ge 1$ for at least a half of the vertices $v\in V$. The signed majority total domination number of a graph $G$ is $\gamma _{{\mathrm maj}}^{{\,\mathrm t}}(G)=\min \lbrace f(V)\mid f$ is a signed majority total dominating function on $G\rbrace $. We research some properties of the signed majority total domination number of a graph $G$ and obtain a few lower bounds of $\gamma _{{\mathrm maj}}^{{\,\mathrm t}}(G)$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
27530. On signpost systems and connected graphs
- Creator:
- Nebeský, Ladislav
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- connected graph and signpost system
- Language:
- English
- Description:
- By a signpost system we mean an ordered pair $(W, P)$, where $W$ is a finite nonempty set, $P \subseteq W \times W \times W$ and the following statements hold: \[ \text{if } (u, v, w) \in P, \text{ then } (v, u, u) \in P\text{ and } (v, u, w) \notin P,\text{ for all }u, v, w \in W; \text{ if } u \ne v,i \text{ then there exists } r \in W \text{ such that } (u, r, v) \in P,\text{ for all } u, v \in W. \] We say that a signpost system $(W, P)$ is smooth if the folowing statement holds for all $u, v, x, y, z \in W$: if $(u, v, x), (u, v, z), (x, y, z) \in P$, then $(u, v, y) \in P$. We say thay a signpost system $(W, P)$ is simple if the following statement holds for all $u, v, x, y \in W$: if $(u, v, x), (x, y, v) \in P$, then $(u, v, y), (x, y, u) \in P$. By the underlying graph of a signpost system $(W, P)$ we mean the graph $G$ with $V(G) = W$ and such that the following statement holds for all distinct $u, v \in W$: $u$ and $v$ are adjacent in $G$ if and only if $(u,v, v) \in P$. The main result of this paper is as follows: If $G$ is a graph, then the following three statements are equivalent: $G$ is connected; $G$ is the underlying graph of a simple smooth signpost system; $G$ is the underlying graph of a smooth signpost system.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public