Abstract

We propose a family of robust nonparametric estimators for robust regression function based on k-nearest neighbour (k-NN) method. We establish the asymptotic normality of the estimator under the concentration properties on small balls of the probability measure of the functional explanatory variables.

Keywords: Asymptotic distribution; functional data; k-nearest neighbour; robust estimation; small balls probability.

MSC: 62G05, 62G08, 62G20, 62G35

Abstract

Given non-negative integers $m$, $n$, $h$ and $k$ with $m\ge h\ge 1$ and $n\ge k\ge 1$, an $[h,k]$-bipartite multi hypertournament (or briefly $[h,k]$-BMHT) on $m+n$ vertices is a triple $(U,V,\bold A)$, where $U$ and $V$ are two sets of vertices with $|U|=m$ and $|V|=n$ and $\bold A$ is a set of $(h+k$)-tuples of vertices, called arcs with exactly $h$ vertices from $U$ and exactly $k$ vertices from $V$, such that for any $h+k$ subset $U_{1}\cup V_{1}$ of $U\cup V$, $\bold A$ contains at least one and at most $(h+k)!$ $(h+k)$-tuples whose entries belong to $U_{1}\cup V_{1}$. If $\bold A$ is a set of $(r+s)$-tuples of vertices, called arcs for $r$ ($1\leq r\leq h$) vertices from $U$ and $s$ ($1\leq s\leq k$) vertices from $V$ such that $\bold A$ contains at least one and at most $(r+s)!$ $(r+s)$-tuples, then the bipartite multi hypertournament is called an $(h,k)$-bipartite multi hypertournament (or briefly $(h,k)$-BMHT). We obtain necessary and sufficient conditions for a pair of sequences of non-negative integers in non-decreasing order to be losing score lists and score lists of $[h,k]$-BMHT and $(h,k)$-BMHT.

Keywords: Hypertournaments; bipartite hypertournaments; score; losing score.

MSC: 05C65

Abstract

In this paper, we give some characterizations of images of separable metric spaces under certain $\pi$-maps, and give an affirmative answer to the problem posed by Shou Lin in [Point-Countable Covers and Sequence-Covering Mappings, Chinese Science Press, Beijing, 2002].

Keywords: $cs^*$-network; $cs^*$-cover; $cs$-cover; $sn$-cover; separable metric space; Cauchy $sn$-symmetric; $\sigma$-strong network; compact map; $\pi$-map.

MSC: 54C10, 54D55, 54E40, 54E99

Abstract

In this paper we study the problem of partial global smoothness preservation in the cases of max-product Bernstein approximation operators, max-product Hermite-Féjer interpolation operators based on the Chebyshev nodes of first kind and max-product Lagrange interpolation operators based on the Chebyshev nodes of second kind.

Keywords: Max-product Bernstein approximation operator; max-product Hermite-Féjer interpolation operator; max-product Lagrange interpolation operator; Chebyshev nodes of the first and second kind; global smoothness preservation.

MSC: 41A05, 41A20, 41A17

Abstract

A family of harmonic starlike functions of complex order in the unit disc has been introduced and investigated by S.A. Halim and A. Janteng [Harmonic functions starlike of complex order, Proc. Int. Symp. on New Development of Geometric function Theory and its Applications, (2008), 132--140]. In this paper we consider a subclass consisting of harmonic parabolic starlike functions of complex order involving special functions and obtain coefficient conditions, extreme points and a growth result.

Keywords: Harmonic functions; harmonic starlike functions; hypergeometric functions; Dziok-Srivastava operator.

MSC: 30C45, 30C50

Abstract

In this paper, we prove that a space is a sequence-covering $\pi$-image of a metric space if and only if it has a $\sigma$-strong network consisting of $cs$-covers (or $sn$-covers) if and only if it is a Cauchy $sn$-symmetric space.

Keywords: Sequence-covering mappings; $\pi$-mappings; $cs$-covers; $sn$-covers; $\sigma$-strong networks; Cauchy $sn$-symmetric spaces.

MSC: 54D55, 54E40, 54E99

Abstract

In this note we show that weakly compact operators from a Banach space $X$ into a complete (LB)-space $E$ need not factorize through a reflexive Banach space. If $E$ is a Fréchet space, then weakly compact operators from a Banach space $X$ into $E$ factorize through a reflexive Banach space. The factorization of operators from a Fréchet or a complete (LB)-space into a Banach space mapping bounded sets into relatively weakly compact sets is also investigated.

Keywords: Weakly compact operators; reflexive operators; factorization; reflexive Banach spaces; (LB)-spaces; Fréchet spaces.

MSC: 46A04, 46A03, 46A25, 46B10, 47B07

Abstract

Here we present Ostrowski type inequalities on Cosine and Sine Operator Functions for various norms. At the end we give some applications.

Keywords: Ostrowski inequality; Cosine operator function; Sine operator function.

MSC: 26D99, 47D09, 47D99

Abstract

We introduce and investigate finite dimensions $(m,n)\text{-}\tx{\rm dim}$ defined by means of $m$-coverings. These dimensions generalize the Lebesgue dimension: $\tx{\rm dim}=(2,1)\text{-}\tx{\rm dim}$. If $n

Keywords: Dimension; dimension $(m,n)\text{-}\tx{\rm dim}$; metrizable space; hereditarily normal space.

MSC: 54F45