Self‐converse Mendelsohn designs with block size 4 and 5

A (v, k, λ)‐Mendelsohn design(X, ℬ︁) is called self‐converse if there is an isomorphic mapping ƒ from (X, ℬ︁) to (X, ℬ︁⁻¹), where ℬ︁⁻¹ = {B⁻¹ = 〈x_k, x_k−1,…,x₂, x₁〉: B = 〈x₁, x₂,…,x_k−1, x_k〉 ϵ ℬ︁}. In this paper, we give the existence spectrum for self‐converse (v, 4, 1)– and (v, 5, 1)– Mendelsohn designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 411–418, 2000     

Perfect Mendelsohn designs with equal-sized holes and block size four

Let M = {m₁, m₂, …, m_h} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m₁, m₂, …, m_h) is called the type of the HPMD. If m₁ = m₂ = … = m_h = m, we write briefly m^h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type m^h, namely, is also sufficient with the exception of types 2⁴ and 1⁸ with λ = 1, and type m⁴ for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Direct construction methods for incomplete perfect Mendelsohn designs with block size four

Let v, k, λ, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD, is a triple (X,Y,B) where X is a v-set (of points), Y is an n-subset of X, and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a,b) E (X × X)\(Y × Y) appears t-apart in exactly λ blocks of B and no ordered pair (a,b) E Y × Y appears in any block of B for any t, where 1 ≤ t ≤ k − 1. In this article, we introduce an effective and easy way to construct IPMDs for k = 4 and even v − n, and use it to construct some small examples for λ = 1 and 2. Obviously, these results will play an important role to completely solve the existence of (v,n,4,λ)-IPMDs. Furthermore, we also use this method to construct some small examples for HPMDs. © 1996 John Wiley & Sons, Inc.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Old and new designs via difference multisets and strong difference families

Let X =((x_1,1,x_1,2,…,x_1,k),(x_2,1,x_2,2,…,x_2,k),…,(x_t,1,x_t,2,…,x_t,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (x_h,i-x_h,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Hamiltonian paths and cycles in hypertournaments

Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, |V| = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V, A$ contains exactly one of the k! k-tuples whose entries belong to S. A 2-hypertournament is merely an (ordinary) tournament. A path is a sequence v₁a₁v₂v₃···v_t−1v_t of distinct vertices v₁, v₂,⋖, v_t and distinct arcs a₁, ⋖, a_t−1 such that v_i precedes v_t−1 in a, 1 ≤ i ≤ t − 1. A cycle can be defined analogously. A path or cycle containing all vertices of T (as v_i's) is Hamiltonian. T is strong if T has a path from x to y for every choice of distinct x, y ≡ V. We prove that every k-hypertournament on n (k) vertices has a Hamiltonian path (an extension of Redeis theorem on tournaments) and every strong k-hypertournament with n (k + 1) vertices has a Hamiltonian cycle (an extension of Camions theorem on tournaments). Despite the last result, it is shown that the Hamiltonian cycle problem remains polynomial time solvable only for k ≤ 3 and becomes NP-complete for every fixed integer k ≥ 4. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 277–286, 1997     

Upper bounds on the general covering number Cλ(v,k, t,m)

A collection of k‐subsets (called blocks) of a v‐set X (v) = {1, 2,…, v} (with elements called points) is called a t‐(v, k, m, λ) covering if for every m‐subset M of X (v) there is a subcollection of with such that every block K ∈ has at least t points in common with M. It is required that v ≥ k ≥ t and v ≥ m ≥ t. The minimum number of blocks in a t‐(v, k, m, λ) covering is denoted by C_λ(v, k, t, m). We present some constructions producing the best known upper bounds on C_λ(v, k, t, m) for k = 6, a parameter of interest to lottery players. © 2004 Wiley Periodicals, Inc.     

Self-Converse Directed BIBDs with Block Size Four

A directed balanced incomplete block design (or D B(k,;v)) (X,) is called self-converse if there is an isomorphic mapping f from (X,) to (X,^–1), where ^–1={B ^–1:B} and B ^–1=(x _k ,x _{k –1},,x ₂,x ₁) for B=(x ₁,x ₂,,x _{k –1},x _k ). In this paper, we give the existence spectrum for self-converse D B(4,1;v). AMS Classification:05BResearch supported in part by NSFC Grant 10071002 and SRFDP under No. 20010004001     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

Disproof of a conjecture about independent branchings in k-connected directed graphs

For each k ≥ 3, we construct a finite directed strongly k-connected graph D containing a vertex t with the following property: For any k spanning t-branchings, B₁, …, B_k in D (i. e., each B_i is a spanning tree in D directed toward t), there exists a vertex x ≠ t of D such that the k, x, t-paths in B₁, …, B_k are not pairwise openly disjoint. This disproves a well-known conjecture of Frank. © 1995, John Wiley & Sons, Inc.     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

Covering arrays of strength 3 and 4 from holey difference matrices

A covering array CA(N; t, k, v) is an N × k array with entries from a set X of v symbols such that every N × t sub-array contains all t-tuples over X at least once, where t is the strength of the array. The minimum size N for which a CA(N; t, k, v) exists is called the covering array number and denoted by CAN(t, k, v). Covering arrays are used in experiments to screen for interactions among t-subsets of k components. One of the main problems on covering arrays is to construct a CA(N; t, k, v) for given parameters (t, k, v) so that N is as small as possible. In this paper, we present some constructions of covering arrays of strengths 3 and 4 via holey difference matrices with prescribed properties. As a consequence, some of known bounds on covering array number are improved. In particular, it is proved that (1) CAN(3, 5, 2v) ≤ 2v ²(4v + 1) for any odd positive integer v with gcd(v, 9) ≠ 3; (2) CAN(3, 6, 6p) ≤ 216p ³ + 42p ² for any prime p > 5; and (3) CAN(4, 6, 2p) ≤ 16p ⁴ + 5p ³ for any prime p ≡ 1 (mod 4) greater than 5.     

Hypersurfaces containing flat subspaces

Lek k be an infinite field and suppose m.i. and n are positive integers such that t m We study the subset of k[x ₁,x ₂, … x_m ] which consists of 0 and the homogeneous members t of f of k[x ₁,x ₂, … x_m ] of fixed degree n such that there exists homogeneous F ₁, F ₂, … F_t in k[x ₁,x ₂, … x_m ] of degree one and homogenous g ₁ g ₂, …g_t , in k[x ₁,x ₂, … x_m ] such that f(x) = F ₁(x)g ₁(x) + F ₂(x)g ₂(x) + … + F _t (x)g _t (x) for each x in k ^m. In case k is algebrarcally closed we are able to prove that this set is an algebraic variety. Consequently. if k is also of characteristic 0 then we are able to prove that certain collections of symmetric k-valued multilinear functions are algebraic varieties.     

The kernel,the bargaining set and the reduced game

We present an example showing that forxK(N, v, B) the section ofK(N, v, B) atx| _{N-B k} may be a proper subset ofK(B _k, v_x, X_k). Further we prove that under appropriate conditions these two sets coincide. For the bargaining set we prove a similar result.We are grateful to an anonymous referee for valuable comments.     

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.     

Some Indecomposable t-Designs

The existence of large sets of 5-(14,6,3) designs is in doubt. There are five simple 5-(14,6,6) designs known in the literature. In this note, by the use of a computer program, we show that all of these designs are indecomposable and therefore they do not lead to large sets of 5-(14,6,3) designs. Moreover, they provide the first counterexamples for a conjecture on disjoint t-designs which states that if there exists a t-(v, k, λ) design (X, D) with minimum possible value of λ, then there must be a t-(v, k, λ) design (X, D′) such that D∩ D′ = Ø.     

Legendre polynomial kernel estimation of a density function with censored observations and an application to clinical trials

Let f(x), x ∈ ?^M, M ≥ 1, be a density function on ?^M, and X₁, …., X_n a sample of independent random vectors with this common density. For a rectangle B in ?^M, suppose that the X's are censored outside B, that is, the value X_k is observed only if X_k ∈ B. The restriction of f(x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {(u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let K_{m, n}(x) be the classical kernel estimator of f(x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|K_{m, n}(x) ? f(x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc.