Higher order monotonic functions of several variables

Multivariate functions with a specific degree of higher order monotonicity in each variable are introduced. When normalized, they turn out to form a simplex whose extreme points are precisely the tensor products of their univariate counterparts. Under natural conditions on the degrees these functions will be shown to operate on each other.     

Higher order alternating sequences

Sequences which are alternating of a (finite) higher order n, appropriately normalized, are shown to form a Bauer simplex, and its countably many extreme points are identified. For \(\, n = 2 \,\) we are dealing with increasing concave sequences. The proof makes use of multivariate co-survival functions of (not necessarily finite) Radon measures.     

Higher order estimation at Lebesgue points

The symmetric derivative of a probability measure at a Lebesgue point can often be specified by an exact relation involving a regularity index. Knowledge of this index is of practical interest, for example to specify the local behavior of the measure under study and to evaluate bandwidths or number of neighbors to take into account in smoothing techniques. This index also determines local rates of convergence of estimators of particular points of curves and surfaces, like minima and maxima. In this paper, we consider the estimation of the d-dimensional regularity index. We introduce an estimator and derive the basic asymptotic results. Our estimator is inspired by an estimator proposed in Drees and Kaufmann (1998, Stochastic Processes and their Applications, 75, 149–172) in the context of extreme value statistics. Then, we show how (estimates of) the regularity index can be used to solve practical problems in nearest neighbor density estimation, such as removing bias or selecting the number of neighbors. Results of simulations are presented.     

多项式函数的极值点与拐点判别及个数公式

本文讨论了多项式函数(x-a)~n,(x-a)~ng(x)(n∈N,n1,a∈R)和∏ki=1(x-a_i)~(n_i)(n_i∈N,n_i0,a_i∈R)的极值点和拐点,并给出了函数∏ki=1(x-a~i)~(n_i)(n_i∈N,n_i0,a_i∈R)所有极值点和拐点的个数公式.     

Estimation of the extreme value and the extreme points

Summary Letf be a continuous function defined on some domainA andX ₁,X ₂, ... be iid random variables. We estimate the extreme value off onA by studying the limiting distribution of min {f(X ₁), ...,f(X _n )} or max {f(X ₁), ...,f(X _n )} properly normalized. Sufficient conditions for the existence of the limiting distribution as well as a characterization of the limiting distribution relative to the extreme points off will be provided. A discussion of the multidimensional case is also carried out. Partially supported by CNPq-No. 301508/84.     

On perfect graphs and polyhedra with (0, 1)-valued extreme points

LetC andA be (0, 1)-valued matrices with no zero columns. Fulkerson has shown that the extreme points of {x: Cx 1,x 0} are given by the rows ofA and their projections and the extreme points of {x: Ax 1,x 0} are given by the rows ofC and their projections if and only if the maximal rows ofC andA are the incidence vectors of maximal cliques and anticliques, respectively, of a perfect graph. This theorem is discussed and a new proof is given for the only if implication.Research partially supported by grant ENG 76-09936 from the National Science Foundation to Cornell University and by Sonderforschungsbereich 21 (DFG), Institut für Operations Research, Universität Bonn.     

Koenigs problem and extreme fixed points

This note continues some previous studies by the authors. We consider a linear-fractional mapping $ F_A :K \to K $ F_A :K \to K generated by a triangular operator, where $ K $ K is the unit operator ball and the fixed point C of the extension of $ F_A $ F_A to $ \overline K $ \overline K is either an isometry or a coisometry. Under some natural restrictions on one of the diagonal entries of the operator matrix A, the structure of the other diagonal entry is investigated completely. It is shown that generally C cannot be replaced in all these considerations by an arbitrary point of the unit sphere. Some special cases are studied in which this is nevertheless possible.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function f admits the double Jordan-type inequality if and only if f is analytic and even. Associated with f is the function g with f(x)=g(x²). In this short note, based on this association, and using properties of absolutely and/or (completely) monotonic functions, we propose a concise method to derive the inequality from the coefficients in the Taylor’s series of f. The results include some existing ones as special cases.     

Means of extreme points and F-spaces

Given a topological space T and a strictly convex real normed space X, let be the space of continuous and bounded functions from T into X, with its uniform norm. This paper is devoted to the study of the relation between the fact of T being an F-space and the property that every element in the unit ball of has a representation as a mean of two extreme points.