Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

LAN of extreme order statistics

Consider an iid sampleZ ₁,...,Z _n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F : }. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z _n–i+1n ) _i=1 ^k of the upperk=k(n) _n order statistics in the sample, if the family {F :} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZ _n–k+1n is the central sequence generating LAN. This implies thatZ _n–k+1n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter can be based on the single order statisticZ _n–k+1n . The rate at whichZ _n–k+1n becomes asymptotically sufficient is however quite poor.     

A comparison theorem for the mean exit time from a domain in a Kähler manifold

Let M be a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary . The main result of this paper is a comparison theorem between the Mean Exit Time function defined on and the Mean Exit Time from a geodesic ball of the complex projective space ⁿ () which involves a characterization of the geodesic balls among the domain . In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of , using the Index Lemma for Submanifolds.Work partially supported by a DGICYT Grant No. PS87-0115-C03-01.     

On the Influence of Weights on Extremes

Let X ₁, ..., X_n be an i.i.d. sequence of random variables, from an unknown distribution F, and X ₁ ^W , ... X _n ^W be a sample from , the weighted empirical distribution function of X ₁, ..., X_n. We define the order statistics X _1,n ^W ... X _n,n ^W of X ₁ ^W , ..., X _n ^W . Under suitable assumptions on weights, we study the influence of the maxima in the construction of limit theorems. We choose a resample size m(n) and we derive conditions on m(n) for the in probability and with probability 1 consistency of X _m(n),m(n) ^W . The presence of weights has an influence on the resample size and requires the use of new tools. When X _n,n is in the domain of attraction of an extreme value distribution, m(n) , and , as n , all our results hold.     

Un théorème sur les séries de Dirichlet

We considerk Dirichlet series a _j (n)n ^–s (1jk),k2. We suppose that for eachj the series a _j (n)n ^–s converges fors=s _j =j+it _j , and that Max j<1/(k–1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line es = (₁+...+ _k )/k+1–1/k and we estimate the rate of convergence. The number 1–1/k cannot be replaced by a smaller one.     

Cross-monotone subsequences

Two finite real sequences (a ₁,...,a _k ) and (b ₁,...,b _k ) are cross-monotone if each is nondecreasing anda _i+1–a _i b _i+1–b _i for alli. A sequence (₁,..., _n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (₁,..., _n ) is in CM(k), is bounded between aboutk ²/4 andk ₂/2. It also shows thatg(k), the smallestn for which all (₁,..., _n ) are in CM(k)and eithera _k b ₁ orb _k a ₁, equalsk(k–1)+2, and thath(k), the smallestn for which all (₁,..., _n ) are in CM(_k)and eithera ₁b ₁...a _k b _k orb ₁a ₁...b _k a _k , equals 2(k–1)²+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark ²×k ² (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk ² is replaced byk ²–1.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Characterizations of absolute F σδ-sets

We give several internal characterizations for the metrizable absolute F -spaces. The characterizing conditions involve the existence of compatible bicomplete quasi-metrics, of complete sequences of -discrete closed covers and of compact -discrete closed networks.     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

The embedding of linearly ordered sets

It is shown that if a linearly ordered set B does not contain as subsets sets of order type and ^* then B can be embedded in 2 . We construct an example of a set satisfying the above conditions which cannot be embedded in any 2 if < . Simultaneously we show that for any ordinal, 2 ⁺¹ cannot be embedded in 2 and that there exists at least ₊₁ distinct dense order types of cardinality 2 .Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 83–88, January, 1972.In conclusion, I wish to take the opportunity to thank Yu. L. Ershov for kindness and assistance in this work.     

Improvement of Bruck's completion theorem

Improving Bruck's Completion-Theorem for nets, we show that a net of order k and degree k + 1 – can be extended to an affine plane, if 3k > 8³ – 18² + 8 + 4. As applications we obtain the following two theorems: A maximal partial t-spread in PG(2t + 1, q), q not a square, with deficiency > 0 satisfies 8³ – 18² + 8 + 4 3q ². There exists an absolute constant c such that every linear space with constant point degree n + 1 and minimum line degree n + 1 – a can be embedded in a protective plane of order n provided that n > ca ³.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

On Resolvable Multipartite <Emphasis Type="Italic">G</Emphasis>-Designs

A necessary and sufficient condition for the existence of a _km–factorization of the complete symmetric k–partite multi-digraph K*(n₁,n₂,...,n_k) is obtained for odd k. As a consequence, a resolvable (k,n,km,) multipartite _km–design exists for odd k if and only if m|n. This deduces a result of Ushio when m=1 and k=3. Further, a necessary and sufficient condition for the existence of a _km–factorization of is established for even k, where denotes the wreath product of graphs. Finally, a simple and short proof for the non-existence of a _k–factorization of is obtained for odd k.Acknowledgments.The author thanks Dr. P. Paulraja for his useful ideas in writing this paper and the Department of Science and Technology, New Delhi, for its support (Project Grant No. DST/MS/103/99).Final version received: November 17, 2003     

Maximal order for quadratures usingn evaluations

The problem is to find approximationsI (f; h) to the integralI(f; h)= ₀ ^h f. Such an approximation has local orderp ifI(f; h)–I (f; h)=O(h ^p ) ash0. Let(n) denote the maximal local order possible for a method usingn evaluations of a function or its derivatives. We show that (n)=2n+1 if the information used is Hermitian. This is conjectured to be true in general. The conjecture is established for all methods using three or fewer evaluations.This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. Numerical results reported in this paper were obtained through the computing facilities of the University of Maryland.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Defining relations of semigroups of all directed transformations of an ordered finite set

The semigroup of all transformations X of a finite (partially) ordered set , such that X for all , is considered. All possible generating sets of a are elucidated. Only one of those sets is irreducible. A system of defining relations is found for that generating set.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 657–662, June, 1968.