On separating systems whose elements are sets of at most k elements

A system A₁,…,A_m of subsets of X?{1,…,n} is called a separating system if for any two distinct elements of X there is a set A_i (1?i?m) that contains exactly one of the two elements. We investigate separating systems where each set A_i has at most k elements and we are looking for minimal separating systems, that means separating systems with the least number of subsets. We call this least number m(n,k). Katona has proved good bounds on m(n,k) but his proof is very complicated. We give a shorter and easier proof. In doing so we slightly improve the upper bound of Katona.     

Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions

A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed components. Some examples related to some lifetime distribution functions are given. We present a numerical example for evaluating the relationship between the mean residual life of the k-out-of-n: G system and that of its components.     

Reliability of a k out of n system of components sharing a common environment

This paper deals with the reliability function and the failure rate of the k out of n system of components, with and without incorporating the environmental effect. Special cases of the series system and the parallel system provide more compact expressions and interpretations. Some of the results in the literature follow as our special cases. Examples are given to illustrate the results.     

On the central limit theorem for f (n k x)

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) _n≥1 with 2π-periodic ${f\in L^2}$ this phenomenon is generally not valid and the asymptotic behavior of ( f (n _k x)) _k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n _k . By the classical theory, the central limit theorem holds for f (n _k x) if n _k  = 2 ^k , or if n _k+1/n _k → α with a transcendental α, but it fails e.g., for n _k  = 2 ^k  ? 1. The purpose of our paper is to give a necessary and sufficient condition for f (n _k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n _k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.     

Über (k,r)-Zahlen

For given integersr andk, 0<r<k, any integern>1 is uniquely representable in the formn=d ^k ·m withm k-free (that means there is nok-th prime powerp ^k dividingm);n is called a (k, r)-integer, ifm isr-free. In the present paper asymptotic formulae are derived for the number of (k, r)-integersn≤x contained in a given arithmetic progression and for the number of representations of a positive integer, as the sum of a (k ₁,r ₁)-integer and a (k ₂,r ₂)-integer.     

Decomposition of an M/D/r·k queue with FIFO into k Ek/D/r queues with FIFO

It is observed that the queuing system M/D/r·k with FIFO has the same waiting time distribution as the queuing system E_k/D/r with FIFO. Using this simple equivalence we can apply numerical methods and tables for M/Dn to solve E_k/D/r.     

P 4k−1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k ? 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k?1-factorization of λK _m,n is (1) (2k ? 1)m ? 2kn, (2) (2k ? 1)n ? 2km, (3) m + n ≡ 0 (mod 4k ? 1), (4) λ(4k ? 1)mn/[2(2k ? 1)(m + n)] is an integer.     

Approximation properties of sums of the form Σ k λ k h(λ k z)

A method for approximating functions f analytic in a neighborhood of the point z = 0 by finite sums of the form Σ _k λ _k h(λ _k z) is proposed, where h is a chosen function analytic on the unit disk and the approximation is carried out by choosing the complex numbers λ _k = λ _k (f). Some applications to numerical analysis are given.     

Necklaces of beads in k colors and k-ary de Bruijn sequences

This paper constructs a decreasing lexicographic list of necklaces of length n in beads of k colors. This list, somewhat modified, produces a k-ary de Bruijn sequence of length kⁿ.     

A note on marking lines in [k] n

In ${[k]^n=[k]{\times}[k]{\times}\cdots{\times}[k]}$ , a coordinate line consists of the collection of points where all but one coordinate is fixed and the unfixed coordinate varies over all possibilities. We consider the problem of marking (or designating) one point on each line in [k] ⁿ so that each point in [k] ⁿ is marked either a or b times, for some fixed a or b. This is equivalent to forming a strategy for a hat guessing game for n players with k different colors of hats where the number of correct guesses, regardless of hats placed, is either a or b. If we let s?≥?0 and t?≥?0 denote the number of vertices marked a and b times respectively, then we have the following obvious necessary conditions: s?+?t?=?k ⁿ (the number of points) and as?+?bt?=?nk ^n–1 (the number of lines). Our main result is to show for n?≤ 5, and k arbitrary, that these necessary conditions are also sufficient.     

Reliability and covariance estimation of weighted k-out-of-n multi-state systems

In the literature of reliability engineering, reliability of the weighted k-out-of-n system can be calculated using component reliability based on the structure function. The calculation usually assumes that the true component reliability is completely known. However, this is not the case in practical applications. Instead, component reliability has to be estimated using empirical sample data. Uncertainty arises during this estimation process and propagates to the system level. This paper studies the propagation mechanism of estimation uncertainty through the universal generating function method. Equations of the complete solution including the unbiased system reliability estimator and the corresponding unbiased covariance estimator are derived. This is a unified approach. It can be applied to weighted k-out-of-n systems with multi-state components, to weighted k-out-of-n systems with binary components, and to simple series and parallel systems. It may also serve as building blocks to derive estimators of system reliability and uncertainty measures for more complicated systems.     

On (k,n)*-visual cryptography scheme

Let P = {1, 2, . . . , n} be a set of elements called participants. In this paper we construct a visual cryptography scheme (VCS) for the strong access structure specified by the set Γ₀ of all minimal qualified sets, where ${\Gamma_0=\{S: S\subseteq P, 1\in S}$ and |S| = k}. Any VCS for this strong access structure is called a (k, n)*-VCS. We also obtain bounds for the optimal pixel expansion and optimal relative contrast for a (k, n)*-VCS.     

A Proof of KÜhnel’s Conjecture for n ⩾ k2 + 3k Eric Sparla

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ? k² + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.     

Selecting k objects from a cycle with p pairs of separation s

Recently Konvalina determined the number of ways of selecting k objects from a cycle of n objects with no two selected objects separated by exactly one object. For arbitrary integers p, k, and s, we show that a recent result of the author can be applied to determine the number of ways of selecting k objects (from a cycle of n objects) which contain exactly p pairs of selected objects separated by exactly s objects. Konvalina's result follows as the special case p = 0 and s = 1.     

The size of 3-compatible, weakly compatible split systems

A split system on a finite set X is a set of bipartitions of X. Weakly compatible and k-compatible (k??1) split systems are split systems which satisfy special restrictions on all subsets of a certain fixed size. They arise in various areas of applied mathematics such as phylogenetics and multi-commodity flow theory. In this note, we show that the number of splits in a 3-compatible, weakly compatible split system on a set X of size n is linear in?n.     

Cycle structures of automorphisms of 2-(v,k,1) designs

An automorphism of a 2?(v,k, 1) design acts as a permutation of the points and as another of the blocks. We show that the permutation of the blocks has at least as many cycles, of lengths n > k, as the permutation of the points. Looking at Steiner triple systems we show that this holds for all n unless n|Cp(n)| ? 3, where Cp(n) is the set of cycles of length n of the automorphism in its action on the points. Examples of Steiner triple systems for each of these exceptions are given. Considering designs with infinitely many points, but with k finite, we show that these results generalize. © 1995 John Wiley & Sons, Inc.     

On reliability analysis of consecutive k-out-of-n systems with arbitrarily dependent components

In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n?r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T_r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable.     

Symmetric solutions for (n,k) games

Symmetric solutions (or symmetric stable sets) and their uniqueness are investigated for some classes of symmetric,n-person, cooperative games in characteristic function form known as (n, k) games.     

Multipliers for ¦N,p n;δ¦ k summability of infinite series

In this paper a theorem on ¦N,p _n;δ¦_ksummability factors, which generalizes a theorem of Bor [3] on ¦N,p _n¦_ksummability factors, has been proved.     

Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities

The consecutive k-out-of-r-from-n: F system was generalized to multi-state case. This system consists of n linearly ordered components which are at state below j if and only if at least k_j components out of any r consecutive are in state below j. In this paper we suggest bounds of increasing multi-state consecutive-k-out-of-r-from-n: F system (k₁ ? k₂ ? ? ? k_M) by applying second order Boole–Bonferroni bounds and applying Hunter–Worsley upper bound. Also numerical results are given. The programs in V.B.6 of the algorithms are available upon request from the authors.