Some new BIBDS with λ = 1 and 6 ≤ k ≤ 10

This article is in two main parts. The first gives some (q,k, 1) difference families with q a prime power and 7 ≤ k ≤ 9; it also gives some GD(k, 1, k,kq)s which are extendable to resolvable (kq,k, 1) BIBDs for k E {6,8,10} and q a prime power equal to 1 mod 2(k − 1). The second uses some of these plus several recursive constructions to obtain some new (v,k,, 1) BIBDs with 7 ≤ k ≤ 9 and some new (v,8,1) resolvable BIBDs. © 1996 John Wiley & Sons, Inc.     

Virtually balanced incomplete block designs for v = 22, k = 8, and λ = 4

When basic necessary conditions for the existence of a balanced incomplete block design are satisfied, the design may still not exist or it may not be known whether it exists. In either case, other designs may be considered for the same parameters. In this article we introduce a class of alternative designs, which we will call virtually balanced incomplete block designs. From a statistical point of view these designs provide efficient alternatives to balanced incomplete block designs, and from a combinatorial point of view they offer challenging new questions. © 1995 John Wiley & Sons, Inc.     

On the existence of pure (v, 4, λ)-PMD with λ = 1 and 2

In this article, we investigate the existence of pure (v, 4, λ)-PMD with λ = 1 and 2, and obtain the following results: (1) a pure (v, 4, 1)-PMD exists for every positive integer v = 0 or 1 (mod 4) with the exception of v = 4 and 8 and the possible exception of v = 12; (2) a pure (v, 4, 2)-PMD exists for every integer v ≥ 6. © 1994 John Wiley & Sons, Inc.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

The λ-complete multigraphs

Let E₁……E_m be the edges of a hypergraph. Assume each element in the union of the edges occurs in at least two of E₁……E_m. Assume further that every pair of distinct edges, E₁, and E₁, intersect in at most one element and that for each such pair there are exactly λ other edges E_k such that E₁ and E₁ both intersect E_k. We characterize the hypergraphs for which each |E₁| ≤ 2. These are the λ-complete multigraphs.     

On Symmetric Association Schemes with k 1=3

In this paper, we have a classification of primitive symmetric association schemes with k ₁ = 3.     

On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ

The inverse scattering problem for cubic eigenvalue equations of the form ψ_xxx + 6Qψ_x + 6R_ψ = λψ is outlined and formally solved. Many properties of the scattering data are obtained, the continuous spectrum is briefly discussed, special one soliton solutions are obtained, and the infinity of conserved quantities are determined in terms of the scattering data.     

On the series ∑ k=1 ∞ ( k 3k )−1 k −n x k

In this paper we investigate the series ∑ _k=1 ^∞ ( _k ^3k )⁻¹ k ⁻ⁿ x ^k . Obtaining some integral representations of them, we evaluated the sum of them explicitly forn = 0, 1, 2.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

On the Distributions δk and (δ')k

Powers and products of distributions have not as yet been defined to hold true in general. In this paper, we choose a fixed δ-sequence and use the concept of neutrix limit to give meaning to the distributions δ^k and (δ')^k for some k. These may be regarded as powers of Dirac delta functions.     

Embedding directed and undirected partial cycle systems of index λ > 1

Recent results have found small embeddings for partial m-cycle systems of order n with λ= 1. However, if λ> 1 then the best known techniques produce embeddings that are often quadratic functions of both m and n and linear functions of λ. In this article we obtain embeddings for partial m-cycle systems of order n, and of partial directed m-cycle systems, for all values of m. These embeddings are independent of λ and linear in both n and m. © 1993 John Wiley & Sons, Inc.     

Non Deterministic Classical Logic: The λμ++ ‐calculus

In this paper, we present an extension of λμ‐calculus called λμ⁺⁺‐calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel‐or.     

The basis decision problem in λ-calculus

We show that the problem of deciding if a finite set of closed terms in normal form is a basis is recursively unsolvable. The restricted problem concerning one element sets is still recursively unsolvable. MSC: 03B40, 03D35.     

On the λY calculus

The λY calculus is the simply typed λ calculus augmented with the fixed point operators. We show three results about λY: (a) the word problem is undecidable, (b) weak normalisability is decidable, and (c) higher type fixed point operators are not definable from fixed point operators at smaller types.