Six Kinds of Roughly Convex Functions

This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.     

Semicontinuity and Quasiconvex Functions

Criteria are derived for quasiconvex functions under lower semicontinuity and upper semicontinuity conditions. The results thus obtained generalize earlier results for convex functions. We also give new conditions under which a given function is r-convex in the sense given by Avriel.     

An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator

A good robust functional should, if possible, be efficient at the model, smooth, and have a high breakdown point. M-estimators can be made efficient and Fréchet differentiable by choosing appropriate ψ-functions but they have a breakdown point of at most 1/(p + 1) in p dimensions. On the other hand, the local smoothness of known high breakdown functionals has not been investigated. It is known that Rousseeuw's minimum volume ellipsoid estimator is not differentiable and that S-estimators based on smooth functions force a trade-off between efficiency and breakdown point. However, by using a two-step M-estimator based on the minimum volume ellipsoid we show that it is possible to obtain a highly efficient, Fréchet differentiable estimator whilst still retaining the breakdown point. This result is extended to smooth S-estimators.     

Sur la Topologie m -Convexe d’une algèbre localement A -convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.

     

Each Invertible Sharply d-Transitive Finite Permutation Set with d ≥ 4 is a Group

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S ₄, S ₅, A ₆ and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.     

General decomposition theorems form-convex sets in the plane

A setS inR ^dis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm?1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom?1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane. IfS is a closedm-convex set in the plane,m ≧ 2, the first question may be answered in one way by the following result: If there is some lineH supportingS at a pointp in the kernel ofS, thenS is a union ofm ? 1 convex sets. Using this result, it is possible to prove several decomposition theorems forS under varying conditions. Finally, an answer to the second question is given: Ifm≧3, thenS is a union of (m?1)³2 ^m?3 or fewer convex sets.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Monitoring process variability using auxiliary information

In this study a Shewhart type control chart namely V _r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V _r chart is developed and its comparison is made with the well-known Shewhart control chart namely S ² chart used for the same purpose. Using power curves as a performance measure it is observed that V _r chart outperforms the S ² chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ _yx . These efficiency conditions on ρ _yx are also obtained for V _r chart in this study.     

关于图值随机元序列的强极限定理

本文研究在局部连通图中取值的随机元的r-阶均值集、r-阶广义样本均值集的基本性质及其极限性质.利用关于随机元的分离引理以及随机游动的常返性,得到了关于图值随机元序列广义强大数定理.推广了已有的结果.     

Integral representation for powers of elements in locally <Emphasis Type="Italic">m</Emphasis>-convex algebras and applications

This paper proves a new integral representation theorem for powers of elements in locally m-convex algebras. It is shown that the conditions handled in this theorem are satisfied for any sequentially complete locally m-convex Q-algebra. Various applications of this result concerning the spectrum, the spectral radius or the numerical radius are given.      

Variance estimation for sample quantiles using them out ofn bootstrap

We consider the problem of estimating the variance of a sample quantile calculated from a random sample of sizen. Ther-th-order kernel-smoothed bootstrap estimator is known to yield an impressively small relative error of orderO(n ^−r/(2r+1) ). It nevertheless requires strong smoothness conditions on the underlying density function, and has a performance very sensitive to the precise choice of the bandwidth. The unsmoothed bootstrap has a poorer relative error of orderO(n ^−1/4), but works for less smooth density functions. We investigate a modified form of the bootstrap, known as them out ofn bootstrap, and show that it yields a relative error of order smaller thanO(n ^−1/4) under the same smoothness conditions required by the conventional unsmoothed bootstrap on the density function, provided that the bootstrap sample sizem is of an appropriate order. The estimator permits exact, simulation-free, computation and has accuracy fairly insensitive to the precise choice ofm. A simulation study is reported to provide empirical comparison of the various methods. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7131/00P).     

Rotation set for maps of degree 1 on the graph sigma

For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e., the set of all rotation numbers) may not be connected and it is not known in general whether it is closed.     

Hölder-continuity of certain generalized convex functions

It is proved that for a rationally s-convex function continuity and local s-Hölder - Continuity are equivalent at each interior point of the domain of definition of the function. Furthermore, it is shown that a rationally s-convex function which is bounded on a nonempty open convex set is s-Hölder-continuous on every compact subset of this set.     

Alliance free and alliance cover sets

A defensive (offensive) k-alliance in Γ = (V,E) is a set S ⊆ V such that every υ in S (in the boundary of S) has at least k more neighbors in S than it has in V / S. A set X ⊆ V is defensive (offensive) k-alliance free, if for all defensive (offensive) k-alliance S, S/X ≠ ∅, i.e., X does not contain any defensive (offensive) k-alliance as a subset. A set Y ⊆ V is a defensive (offensive) k-alliance cover, if for all defensive (offensive) k-alliance S, S ∩ Y ≠ ∅, i.e., Y contains at least one vertex from each defensive (offensive) k-alliance of Γ. In this paper we show several mathematical properties of defensive (offensive) k-alliance free sets and defensive (offensive) k-alliance cover sets, including tight bounds on their cardinality.     

A process variability control chart

In this study a Shewhart type control chart namely the V _t chart, is proposed for improved monitoring of the process variability of a quality characteristic of interest Y. The proposed control chart is based on the ratio type estimator of the variance using a single auxiliary variable X. It is assumed that (Y, X) follows a bivariate normal distribution. The design structure of the V _t chart is developed for Phase-I quality control and its comparison is made with those of the S ² chart (a well-known Shewhart control chart) and the V _r chart (a Shewhart type control chart proposed by Riaz (Comput Stat, 2008a) used for the same purpose. It is observed that the proposed V _t chart outperforms the S ² and V _r charts, in terms of discriminatory power, for detecting moderate to large shifts in the process variability. It is observed that the performance of the V _t chart keeps improving with an increase in |ρ _yx | , where ρ _yx is the correlation between Y and X.     

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.     

Generalized ridge regression,least squares with stochastic prior information,and Bayesian estimators

The ridge estimator of the usual linear model is generalized by the introduction of an a priori vector r and an associated positive semidefinite matrix S. It is then shown that the generalized ridge estimator can be justified in two ways: (a) by the minimization of the residual sum of squares subject to a constraint on the length, in the metric S, of the vector of differences between r and the estimated linear model coefficients, (b) by incorporating prior knowledge, r playing the role of the vector of means and S proportional to the precision matrix. Both a Bayesian and an Aitken generalized least squares frameworks are used for the latter. The properties of the new estimator are derived and compared to the ordinary least squares estimator. The new method is illustrated with different assumptions on the form of the S matrix.     

On the E-Epigraph of an E-Convex Function

In Ref 1, Yang shows that some of the results obtained in Ref. 2 on E-convex programming are incorrect, but does not prove that the results which make the connection between an E-convex function and its E-epigraph are incorrect. In this note, we show that the results obtained in Ref. 2 concerning the characterization of an E-convex function f in terms of its E-epigraph are incorrect. Afterward, some characterizations of E-convex functions using a different notion of epigraph are given.     

Limiting -subgradient characterizations of constrained best approximation

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the set

K:=C∩{xX:-g(x)S},