On symmetric configurations in some problems on extremal decomposition

Some problems on extremal decomposition in families of systems of nonoverlapping domains of various types with free parameters are studied. The functionals considered contain a sum of the reduced modules of biangles. For the problems under consideration, the existence of extremal configurations with appropriate symmetry is investigated. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 160–172.     

On large sets of resolvable and almost resolvable oriented triple systems

An MTS(v) [or DTS(v)] is said to be resolvable, denoted by RMTS(v) [or RDTS(v)], if its block set can be partitioned into parallel classes. An MTS(v) [or DTS(v)] is said to be almost resolvable, denoted by ARMTS(v) [or ARDTS(v)], if its block set can be partitioned into almost parallel classes. The large set of RMTS(v) [or RDTS(v) or ARMTS(v) or ARDTS(v)] is denoted by LRMTS(v) [or LRDTS(v) or LARMTS(v) or LARDTS(v)]. In this article we do some preliminary study for their existence, and give several recursive theorems using other combinatorial structures. © 1996 John Wiley & Sons, Inc.     

On symmetric configurations in some problems on extremal decomposition. II

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 158–179.     

On symmetric configurations in some problems on extremal decomposition. III

Some problems on extremal decomposition in families of non overlapping domains containing systems of biangles with distinguished vertices are considered. The present work continues earlier author’s investigations. Bibliography: 8 titles.     

On the upper embedding of symmetric configurations with block size 3

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

On resolvable designs S3(3; 4, v)

We study a method of Lonz and Vanstone which constructs an S₃(3, 4, 2n) from any given 1-factorization of K_2n. We show that the resulting designs admit at least 3 mutually orthogonal resolutions whenever n ⩾ 4 is even. In particular, the necessary conditions for the existence of a resolvable S₃(3, 4, ν) are also sufficient. Examples without repeated blocks are shown to exist provided that n ≢ 2 mod 3.     

On the dimension of affine resolvable designs and hypercubes

The equivalence between complete sets of mutually orthogonal hypercubes and affine resolvable designs, which generalizes the well-known equivalence between complete sets of mutually orthogonal latin squares and affine planes, is used to examine the dimension of designs by studying the prime classes in the associated hypercubes. Particular attention is given to designs of order n=9 including a design which is nonisomorphic to AG(3, 9) even though it possesses the same parameters and three prime classes. © 1996 John Wiley & Sons, Inc.     

Linear,non-homogeneous,symmetric patterns and prime power generators in numerical semigroups associated to combinatorial configurations

We prove that the numerical semigroups associated to the combinatorial configurations satisfy a family of linear, non-homogeneous, symmetric patterns. We use these patterns to prove an upper bound of the conductor and we also give an upper bound of the multiplicity. Also, we compare bounds of the conductor of numerical semigroups associated to balanced configurations, and to configurations with coprime parameters. The proof of the latter involves a bound of the conductor of prime power generated numerical semigroups.     

On uniformly resolvable designs with block sizes 3 and 4

A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r _k . In this paper we consider the case of block sizes 3 and 4. The cases r ₃ = 1 and r ₄ = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h ^u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ≡ 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r ₃ = 4 exist, and that those with r ₃ = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r ₃ ≤ 10. Furthermore, we prove that URDs with r ₄ = 7 exist and that those with r ₄ = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 ¹⁴², 2 ³⁴⁶ and 6 ⁵⁴. Finally, we provide four {3,5}-URDs with 105 points.     

Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

We consider the Schrödinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran (1995) [17] to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega (2003) [16] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N   nearly parallel filaments for any N?2$N?2$. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.     

On resolvable Steiner 2-designs and maximal arcs in projective planes

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order \((v-k)/(k-1)\) is proved, and a generalization of a conjecture by Andries Brouwer (Geometries and groups, Springer, Heidelberg, 1981) is formulated.     

On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays

The purpose of this paper is twofold. A doubly resolvable Kirkman system is a (v, k, 1)-BIBD whose blocks can be resolved into two resolutions R and R' such that any resolution class from R has at most one block in common with any resolution class from R'. Room squares are examples of such systems in the case k = 2. It was not known whether such systems existed for k ? 3. In this paper, we construct infinite classes of such designs. In particular, we display several doubly resolvable Kirkman systems with v = 27 and k = 3.An equidistant permutation array (EPA) is a v × r array defined on an r-set V of symbols such that each row is a permutation of V and any two distinct rows have Hamming distance r ? 1. Such arrays are closely related to the problem described above and have attracted much interest recently. A central problem of EPA's is to determine the maximum value of v for a fixed value of r. Until now, there has been only one value of r for which a direct construction existed to produce an array having v=2r+1. The present paper provides a direct construction for these arrays which have v the order of $\text{r}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$. Hence, this establishes that the maximum value of v grows non-linearly with r. In the case of r = 13, the largest value of v previously known was 27. We display such an array with v = 36.     

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.     

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

On some points-and-lines problems and configurations

Summary We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.     

On quasiaffine spaces and coherent configurations

In [2] André deduced a (1?1) correspondence between the class of homogeneous coherent configurations and the class of certain noncommutative spaces which he called quasiaffine. In this note we establish a (1?1) correspondence between (not necessarily homogeneous) coherent configurations and weakly quasiaffine spaces which generalizes André's. Furthermore we consider some applications of this correspondence to quasiaffine spaces; especially we characterize such spaces with maximal diameter with respect to one direction (compare with [5]).