Some comments on a paper onk-HNBUE life distributions

Summary In a recent paper by A. P. Basu and N. Ebrahimi (1984, Ann. Inst. Statist. Math., A, 36, 87–100) a new class of life distributions calledk-HNBUE (with dualk-HNWUE) is introduced. Closure properties and bounds on the moments and on the survival function to ak-HNBUE (k-HNWUE) life distribution are presented. However, some of the results presented are incorrect. This research was supported by Swedish Natural Science Research Council Post Doctoral Fellowship F-PD 1564-100.     

Comportement moyen du cardinal de certains ensembles de facteurs premiers

Letn be a positive integer andS _n a particular set of prime divisors ofn. We establish the average order off(n) wheref(n) stands for the cardinality ofS _n . Thek-ary,k-free, semi-k-ary prime factors ofn are some of the classes of prime divisors studied in this paper.

     

Dualisierung der Hauptkurven in allgemeinen Cayley/Klein-Räumen

Dualizing the osculating hyperplanes of a curvek of an n-dimensional Cayley/Klein space (CK space) according to the duality principle of projective spaces yields a curve of the dual CK space, the dual curvek _d ofk. A point ofk with torsion 0 corresponds to a stationary point ofk _d . The dualization of the osculating hyperplanes ofk _d yieldsk. There are simple relations between the moving simplexes, the CK arc lengths and the CK curvatures of the two curves which are valid for all CK spaces, even for those which are not self-dual.

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Herrn Prof. Dr.Johannes Böhm zum 70. Geburtstag gewidmet     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.     

An algorithm recursively computing the exact fourier coefficients ofB-splines with nonequidistant knots

An algorithm has been developed for the numerical computation of the Fourier coefficients ofB-splinesM of arbitrary order and possibly nonequidistant and coincident knots. Ifk is the order ofM, then there is a matrixA, depending only onM such that each Fourier coefficient ofM can be obtained fromA by 0(k ²) operations. The computation ofA itself needs a number of operations of the order ofk ⁴. A program of the algorithm is given in an informal language and some numerical results are presented.

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Zusammenfassung Es wird ein Algorithmus beschrieben, der es gestattet, die Fourierkoeffizienten vonB-SplinefunktionenM von beliebiger Ordnung und mit möglicherweise nichtäquidistanten und zusammenfallenden Knoten numerisch exakt zu berechnen. Istk die Ordnung vonM, so gibt es eine nur vonM abhängige Matrix derart, dass jeder Fourierkoeffizient vonM ausA mit 0(k ²) Operationen erhalten werden kann. Die Berechnung vonA selbst benötigt eine Anzahl Operationen von der Ordnungk ⁴. Ein Rechenprogramm für den Algorithmus wird mit Hilfe einer informellen Programmiersprache (INFL) gegeben. Schliesslich werden die mit dem Programm berechneten Resultate von sechs verschiedenen Beispielen dargestellt.

     

Die Verteilung der k-vollen und l-freien Zahlen

Zusammenfassung  We consider the distribution ofk-full numbers. But we go into more detail and investigate suchk-full integers which are at the same timel-free. We give asymptotic results for the numberN _k,l (x) ofk- full andl-free integers not exceedingx in cases ofl =k + r with 2 ≤r ≤ 5. Moreover, we consider these cases and the casesk = 2, 3,l ≥k + 2 also under the assumption of Riemann’s Hypothesis.      

Ramsey numbers for local colorings

The concept of a localk-coloring of a graphG is introduced and the corresponding localk-Ramsey numberr _loc ^k (G) is considered. A localk-coloring ofG is a coloring of its edges in such a way that the edges incident to any vertex ofG are colored with at mostk colors. The numberr _loc ^k (G) is the minimumm for whichK _m contains a monochromatic copy ofG for every localk-coloring ofK _m . The numberr _loc ^k (G) is a natural generalization of the usual Ramsey numberr ^k (G) defined for usualk-colorings. The results reflect the relationship betweenr ^k (G) andr _loc ^k (G) for certain classes of graphs.This research was done while under an IREX grant.     

The normalized matching property from the generalized Macaulay theorem

IfA is a family ofk-element subsets of a finite setM having elements of several different types (i.e., amultiset) and A is the set of all (k–1)-element subsets ofM obtainable by removing a single element ofM from a single member ofA, then, according to the well known normalized matching condition, the density ofA among thek-element subsets ofM never exceeds the density of A among the (k–1)-element subsets ofM. We show that this useful fact can be regarded as yet another corollary of the generalized Macaulay theorem.     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

On the graph of large distances

For a setS of points in the plane, letd ₁>d ₂>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd _k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|constk ². IfS consists of the vertices of a convex polygon and |S|constk ², then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k – 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.     

Varieties with many lines

This paper contains a classification ofk-dimensional algebraic varieties,even non-smooth, containing a family of lines of dimension (2k−4). If the codimension of the varieties is greater than two, these are: bidimensional families ofP ^k−2 ; one dimensional families of irreducible quadrics; linear sections of the grassmannian of lines ofP ⁴.     

A note onC o Galerkin methods for two-point boundary problems

Summary As is known [4]. theC ^o Galerkin solution of a two-point boundary problem using piecewise polynomial functions, hasO(h ^2k ) convergence at the knots, wherek is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient hasO(h ^k+1) convergence, instead ofO(h ^k ). In this note, it is proved that on any segment there arek–1 interior points where the Galerkin solution is ofO(h ^k+2), one order better than the global order of convergence. These points are the Lobatto points.     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

On maximal (k,t)-sets

Summary This investigation was originally motivated by the problem of determining the maximum number of points in finiten-dimensional projective spacePG(n, s) based on the Galois fieldGF(s) of orders=p ^h (wherep andh are positive integers andp is the prime characteristic of the field), such that not of these chosen points are linearly dependent. A set ofk distinct points inPG(n, s), not linearly dependent, is called a (k, t)-set fork ₁ >k. The maximum value ofk is denoted bym _t (n+1, s). The purpose of this paper is to find new upper bounds for some values ofn, s andt. These bounds are of importance in the experimental design and information theory problems.     

Onk-step methods with almost constant coefficients

A class ofk-step methods where the coefficients are functions ofh andf/y is studied. It is shown that stability and consistency are necessary and sufficient for convergence, and that we can find stable methods of any order k+1 (ork+2 ifk is even) that are not weakly stable.     

On skew field coproducts with a finite extension

It is shown that the field coproduct of any skew fieldE with a binomial (commutative) field extensionF/k overk can be expressed as a cyclic extension of a skew fieldK (theE-socle), itself the field coproduct of [F:k] copies ofE overk. Qua vector space the coproduct may also be expressed as a tensor product ofE andK overk. To the memory of Shimshon Amitsur     

Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one

Summary Letk be a perfect field with cdk(i)≤1. It has recently been proved by the author that homogeneous spaces under connected linear groups overk satisfy a Hasse principle with respect to the real closures ofk. Using this result we classify the semisimple algebraic groups overk and, in particular, characterize the anisotropic ones. Similarly we classify the various types of hermitian forms over skew fields overk and exhibit to what extent weak or strong local-global principles hold. In the case wherek is the function field of a smooth projective curveX over ℝ, we also cover the local-global questions vis-à-vis the completions ofk at the points ofX.     

Fast formula of a reliability ofm-consecutive-k-out-of-n:F system with cyclek

Anm-consecutive-k-out-of-n:F system is a system ofn linearly arranged components which fails if and only if at leastm non-overlapping sequences ofk components fail, when there arek distinct components with failure probabilitiesq _i fori=1,...,k and where the failure probability of thej-th component (j=rk+i (1 ≤i ≤k) isq _j =q _i , we call this system by anm-consecutive-k-out-of-n:F system with cycle (or period)k. In this paper we give a formula of the failure probability ofm-consecutive-k-out-of-n:F system with cyclek via the failure probability of consecutive-k-out-of-n:F system.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC