Balanced Incomplete Block Designs with Block Size 7

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with =6, 7, 21 and 42. The construction problem is solved here for designs with > 2 except for v=253, = 4,5 ; also for = 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown).     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday     

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ₁, ₂ satisfyingrk – ₂ v + 1 and ₂ = ₁ + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.Supported in part by Grant 59540043 (C), Japan.     

Resolvable Balanced Incomplete Block Designs with Block Size 8

The necessary condition for the existence of a resolvable balanced incomplete block design on v points, with = 1 and k = 8, is that v 8 mod 56. With the exception of 66 values of v, this condition is shown to be sufficient. The largest exceptional value of v is 24480.     

Constructions for resolvable and related designs

Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with ₁=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

A note on square divisible designs

We consider square divisible designs with parameters n, m, k=r, 0 and . We show that being disjoint induces an equivalence relation on the block set of such a design and that any two disjoint blocks meet precisely the same point classes. Also, the intersection number of two blocks depends only on their equivalence classes. The number of blocks disjoint with a given block is at most n–1; equality holds for all blocks iff the dual of the given design is also divisible with the same parameters. We then give a few applications.The author gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft via a Heisenberg grant during the time of this research.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

Balanced Incomplete Block Designs with Block Size 9 and λ=2,4,8

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: $$\begin{gathered} {\text{ }}\lambda (v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \lambda v(v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \end{gathered} $$ In this paper we study k=9 with λ=2,4 or 8. For λ=8, we show these conditions on v are sufficient, and for λ=2, 4 respectively there are 8 and 3 possible exceptions the largest of which are v=1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.     

A family of relative difference sets associated with Hjelmslev structures over finite projective spaces

In this note, we construct a new family of relative difference sets, with parameters n=q^d, m=q^d+...+q+1, k=q^d-1(q^d-1), ₁ =q^d-1(q^d-q^d-1-1) and ₂ =q^d-2(q-1)(q^d-1-1) where q is a prime power and d 2 an integer. The associated symmetric divisible designs admit natural epimorphisms onto the symmetric designs formed by points and hyperplanes in the corresponding projective spaces PG(d,q). As in the theory of Hjelmslev planes, points with the same image can be recognized from having the larger of the two possible joining numbers, and dually. More formally, these symmetric divisible designs are balanced c-H-structures (in the sense of Drake and Jungnickel [2]) with parameters c=q^d-2 (q-1)² and t=q^d-1 (q-1) over PG_d-1(d,q). These are the first examples of balanced non-uniform c-H-structures of type 2; they can be used in known constructions to obtain new balanced c-H-structures (for suitable c) of arbitrary type. In fact, all these results are special cases of a more general construction involving arbitrary difference sets.The author gratefully acknowledges the hospitality of the University of Waterloo and the financial support of NSERC under grant IS-0367.     

Some imbedding theorems for block designs balanced with respect to pairs

We prove imbedding theorems for block designs balanced with respect to pairs, and with the aid of these theorems we establish the existence of (v, k, )-resolvable BIB block designs with parameters v, k, such that =k–1 [and also such that =(k–l)/2 if k is odd], k ¦(p–1) for each prime divisor p of the number v/k; we also establish an imbedding theorem for Kirkman triple systems.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 173–184, July, 1974.     

On the non-existence of a class of symmetric group divisible partial designs

Many non-existence theorems are known for symmetric group divisible partial designs. In the case that these partial designs are auto-dual with ₁=0, an ideal incidence structure can be defined whose elements are the equivalence-classes of non-collinear points and parallel blocks. Except for some trivial cases this incidence structure turns out to be a symmetric design, and by studying its existence we can prove much more powerful non-existence theorems.     

(C_k \oplus G,k,\lambda ) Difference Families

Let G be an additive group and C _k be the additive group of the ring Z _k of residues modulo k. If there exist a (G, k, ) difference family and a (G, k, ) perfect Mendelsohn difference family, then there also exists a difference family. If the (G, k, ) difference family and the (G, k, ) perfect Mendelsohn difference family are further compatible, then the resultant difference family is elementary resolvable. By first constructing several series of perfect Mendelsohn difference families, many difference families and elementary resolvable difference families are thus obtained.     

On the estimation of ordered means of two exponential populations

Let random samples of equal sizes be drawn from two exponential distributions with ordered means _i . The maximum likelihood estimator _i ^* of _i is shown to have a smaller mean square error than that of the usual estimator X_i, for each i=1,2. The asymptotic efficiency of _i ^* relative to X_i has also been found.     

Efficiency in integral facility design problems

An example of design might be a warehouse floor (represented by a setS) of areaA, with unspecified shape. Givenm warehouse users, we suppose that useri has a known disutility functionf _isuch thatH _i(S), the integral off _iover the setS (for example, total travel distance), defines the disutility of the designS to useri. For the vectorH(S) with entriesH _i(S), we study the vector minimization problem over the set {H(S) :S a design} and call a design efficient if and only if it solves this problem. Assuming a mild regularity condition, we give necessary and sufficient conditions for a design to be efficient, as well as verifiable conditions for the regularity condition to hold. For the case wheref _iis thel _p-distance from warehouse docki, with 1<p<, a design is efficient if and only if it is essentially the same as a contour set of some Steiner-Weber functionf =₁ f ₁++ _m f _m ,when the _i are nonnegative constants, not all zero.This research was supported in part by the Interuniversity College for PhD Studies in Management Sciences (CIM), Brussels, Belgium; by the Army Research Office, Triangle Park, North Carolina; by a National Academy of Sciences-National Research Council Postdoctorate Associateship; and by the Operations Research Division, National Bureau of Standards, Washington, D.C. The authors would like to thank R. E. Wendell for calling Ref. 16 to their attention.     

The dynamics of perturbations of the contracting Lorenz attractor

We show here that by modifying the eigenvalues ₂ < ₃ < 0 < ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition ₃+₁ > 0 by acontracting condition ₃+₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.     

Equivalent conditions for representing analytic functions by exponential series

Let L() be an entire function of exponential type with simple zeros ₁, ₂, ...; let ¯D be the smallest closed convex set which contains all of the singularities of the function which is associated with L() in the sense of Borel. In [1] there are necessary and sufficient conditions on L() under which a function f(z) which is analytic in ¯D can be represented in D by a Dirichlet series with exponents ₁, ₂, ... We obtain new equivalent conditions on L().Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 91–104, July, 1976.     

Infinite families of non-embeddable quasi-residual hadamard designs

LetD be a quasi-residual Hadamard design with =(2m + 1)2ⁿ–1, wherem andn are positive integers. IfD contains a pair of blocks intersecting in m2ⁿ⁺¹ points together with a third block intersecting each of the first two blocks in (m + 1)2ⁿ points thenD is non-embeddable. Using this result together with a recursive construction for quasi-residual Hadamard designs the existence of a previously unknown infinite family of non-embeddable quasi-residual Hadamard designs with =5(2ⁿ)–1 is established. An additional infinite family of non-embeddable quasi-residual Hadamard designs is given. This family has = 2ⁿ–1 with each design in the family having a pair of blocks meeting in (3 + 3)/4 points and a third block meeting each of the first two blocks in (5 + 5)/8 points.