Transformations on [0, 1] with infinite invariant measures

Under certain regularity conditions a real transformation with indifferent fixed points has an infinite invariant measure equivalent to Lebesgue measure. In this paper several ergodic properties of such transformations are established.     

A matroid generalization of a result of Dirac

This paper generalizes a theorem of Dirac for graphs by proving that ifM is a 3-connected matroid, then, for all pairs {a,b} of distinct elements ofM and all cocircuitsC ^* ofM, there is a circuit that contains {a,b} and meetsC ^*. It is also shown that, although the converse of this result fails, the specified condition can be used to characterize 3-connected matroids.The author's research was partially supported by a grant from the National Security Agency.     

A generalization of a result of Dembowski and Wagner

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG _n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG _d (n, q) with 2 ≤ d ≤  n − 1. We establish this conjecture, hereby improving various previous results.     

A generalization of a result of Kazhdan and Lusztig

Kazhdan and Lusztig showed that every topologically nilpotent, regular semisimple orbit in the Lie algebra of a simple, split group over the field is, in some sense, close to a regular nilpotent orbit. We generalize this result to a setting that includes most quasisplit -adic groups.

     

A generalization of a result of Bauer and Schmeichel

We prove that if G is a 1-tough graph withn ≥ 3 vertices such thatd(u) + d(v) +d(w) ≥n+ κ —2 holds for any triple of independent verticesu, v andw ofG, thenG is hamiltonian, wherek is the vertex connectivity ofG. This generalizes a recent result of Baur and Schmeichel.     

A generalization of a result of Tran Van Trung on t-designs

We prove that if there exists a t − (v, k, λ) design satisfying the inequality for some positive integer j (where m = min{j, v − k} and n = min {i, t}), then there exists a t − (v + j, k, λ ()) design. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 107–112, 1999     

On a result of R. R. Hall

An upper bound for the mean value of a non-negative submultiplicative function by R. R. Hall [3] is sharpened and generalised. Hall's inequality implies a certain rather accurate upper sieve estimate, and this aspect of Hall's result is exploited in deriving good lower bounds for π(x) via the sieve.     

A generalization of a result of Sylvester's

Let v₁,…,v_n be vectors in Zⁿ with D = det(v₁,…,v_n) > 0. Let v_{n + 1} be in the cone generated by v₁,…,v_n and such that v₁,…,v_v, v_{n + 1} generate Zⁿ as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v₁,…,v_n, v_{n + 1} and χ = Dv_{n + 1} ? (v₁ + … v_n + v_{n + 1}).     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

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has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

On a Property of the Franklin System in C[0, 1] and L1[0, 1]

Mathematical Notes - A problem posed by J. R. Holub is solved. In particular, it is proved that if $$\left\{ {{{\tilde f}_n}} \right\}$$ is the normalized Franklin system in L1[0, 1], {an} is a...     

A note on a result of Kong about p-nilpotency

Monatshefte für Mathematik - We point out an error in the paper ‘On an extension of the Frobenius’ theorem about p-nilpotency of a finite group’ (Monatsh Math...     

R（0，θ）分布族参数的矩估计与最大似然估计

本文讨论了均匀分布族｛Ｒ（０，θ），θ＞０｝中参数θ的矩估计及最大似然估计的效率及相合性，结果表明θ修正后的最大似然估计要优于θ的矩估计。     

A generalization of Polyak's convergence result for subgradient optimization

This paper generalizes a practical convergence result first presented by Polyak. This new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach.     

A generalization of von Neumann's assignment problem and K. Fan optimization result

Summary In this paper we generalize the assignment problem in higher dimensions, referring at to another study by the authors. The hide-and-seek game, which is intimately related to the assignment problem, is extended, and an elegant result due to K. Fan about extrema is generalized.