The spectrum for large sets of pure directed triple systems

An LPDTS(ν) is a collection of 3(ν-2) disjoint pure directed triple systems on the same set ofνelements. It is showed in Tian's doctoral thesis that there exists an LPDTS(ν) forν=0,4 (mod 6),ν≥4. In this paper, we establish the existence of an LPDTS(ν) forν= 1,3 (mod 6),ν> 3. Thus the spectrum for LPDTS(ν) is completely determined to be the set {ν:ν= 0, 1 (mod 3),ν≥4}.     

Large sets of pure directed triple systems with index λ

A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.     

The configuration polytope of ℓ-line configurations in Steiner triple systems

It has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≤ m ≤ ℓ; full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in v, the order of the Steiner triple system. Moreover, the counts of full configurations, together with v, form a linear basis for all of the configuration counts when ℓ ≤ 7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored. To Alex Rosa on the Occasion of his Seventieth Birthday     

Enclosings of λ-fold 4-cycle systems

In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K _v . This supplementary solution builds on a result of Raines and Staniszlo.     

Small embeddings of partial directed triple systems and partial triple systems with even λ

In 1976, Lindner and Rosa (Ars Combin. 1 (1976), 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3(λυ² + υ(2 ? λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ? 6λv² + 6v(1 ? λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).     

Steiner systems andn −1-isomorphisms

In this work we introduce the concept of n ^–1-isomorphism between Steiner systems (this coincides with the concept of isomorphism whenever n=1).Precisely two Steiner systems S₁ and S₂ are said to be n^–1-isomorphic if there exist n partial systems S _i ⁽¹⁾ ,...,S _i ⁽ⁿ⁾ contained in Si, i.{1,2},such that S ₁ ^(k) and S ₂ ^(k) are isomorphic for each k{1,..., n}.The n^–1-isomorphisms are also used to study nets replacements, see Ostrom [8], and to study the transformation methods of designs and other incidence structures introduced in [9] and generalized in [1] and [10].Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.     

λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Mathematical Notes - λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ > 1, any arbitrary finite union of...     

Painlevé-Kuratowski Convergences of the solution sets to perturbed generalized systems

In this paper,we obtain the Painlev’e-Kuratowski Convergence of the efficient solution sets,the weak efficient solution sets and various proper efficient solution sets for the perturbed generalized system with a sequence of mappings converging in a real locally convex Hausdorff topological vector spaces.     

λ5-geometry中的Steiner树问题( )

首先研究了λ5-geometry中4个点的Steiner最小树的某些特性,然后证明了对于λ5-geometry中的给定点集P,必有P的一个Steiner最小树,其Stein-er点在P的前2n/3代格点中.     

Complete orthonormal sets of polynomial solutions of the Riesz and Moisil-Teodorescu systems in ℝ3

As it is well-known, the generalization of the classical Cauchy-Riemann system to higher dimensions leads to the so-called Riesz and Moisil-Teodorescu systems. Rewriting these systems in quaternionic language and taking advantage of the underlying algebra, we construct complete sets of polynomials solutions of both systems that are orthonormal with respect to a certain inner product. The restrictions of those polynomials to the unit sphere can be viewed as analogues to the complex case of the Fourier exponential functions $\{e^{i n \theta}\}_{n\geq 0}$ on the unit circle and constitute a refinement of the well-known spherical harmonics.