On the existence of partitionable skew Room frames

Partitionable skew Room frames of type hⁿ have played an important role in the constructions of 4-frames, (K₄-e)-frames and super-simple (4,2)-frames. In this paper, we investigate the existence of partitionable skew Room frames of type hⁿ. The necessary conditions for the existence of such a design are that and h?5. It is proved that these necessary conditions are also sufficient with a few possible exceptions. As a byproduct, the known results on the existence of skew Room frames and uniform 4-frames are both improved.     

On the existence of super-simple designs with block size 4

Summary We prove that forv = 1 and for allv 1 (mod 3),v 10, there is a (v, 4, 4) design with the property that no triple appears in more than one block. The proof of this result is made more difficult by the non-existence of a GDD (4, 4, 3; 15) with no triple appearing in more than one block. We also show that forv = 1 and for allv 1, 4 (mod 12),v 13, there is a (v, 4, 2) design with this property, and with the additional property that the design is the union of two (v, 4, 1) designs.     

On the existence of nested orthogonal arrays

A nested orthogonal array is an OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray. Here r<s and M<N. Necessary conditions for the existence of such arrays are obtained in the form of upper bounds on k, given N, M, s, r and g. Examples are given to show that these bounds are quite powerful in proving nonexistence. The link with incomplete orthogonal arrays is also indicated.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

An even side analogue of Room squares

A Room square of even side cannot exist. We study arrays of even side which very closely resemble Room squares. The spectrum is determined: the desired arrays exist for all even sides exceeding 4. These arrays are useful in a construction for Room squares containing subsquares. It is shown that, for alln4, and all oddr7 (exceptr=11), there is a Room square of sidenr +n – 1 which contains subsquares of sidesr and 2n – 1.     

On the existence of two non-neighboring subgraphs in a graph

Does there exist a functionf(r, n) such that each graphG with _Z (G)≧f(r, n) contains either a complete subgraph of orderr or else two non-neighboringn-chromatic subgraphs? It is known thatf(r, 2) exists and we establish the existence off(r, 3). We also give some interesting results about graphs which do not contain two independent edges.     

On the chromatic number of rational five-space

Summary The chromatic number of rational five-space is the chromatic number of the infinite graph whose vertex set is the set of all those five-dimensional vectors with all the coordinates being rational numbers and with two vertices forming an edge iff the Euclidean distance is exactly one. In this paper it is shown that the chromatic number of rational five-space is at least six.     

On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

On the existence of difference sets in groups of order 96

The correspondence between a (96,20,4) symmetric design having regular automorphism group and a difference set with the same parameters has been used to obtain difference sets in groups of order 96. Starting from eight such symmetric designs constructed by the tactical decomposition method, 55 inequivalent (96,20,4) difference sets are distinguished. Thereby the existence of difference sets in 22 nonabelian groups of order 96 is proved.     

The existence of Room squares

The authors give a condensed proof of the existence of Room squares for positive odd sides except 3 and 5. Some areas of current research on Room squares are also discussed.Aequationes Mathematicae launches a systematic program of expository papers. We will endeavour to publish at least one in every volume.     

On the degrees of vertices in A bichromatic random graph

The asymptotic probability distribution of the number of vertices of a given degree in a random bichromatic graph is presented.     

Some existence results for a periodic problem with non-smooth potential

In this paper, two existence results for a class of second order periodic boundary value problems with non-smooth potential are obtained. We extend the Castro-Lazer-Thews reduction method to non-smooth functionals, the obtained result is then exploited to prove the existence of a nontrivial solution. Furthermore, we prove the existence of multiple solutions by using a multiplicity result based on local linking.     

On existence of sets of distinct representatives for families of subsets of a multiset

IfX is a family ofl-subsets of a multiset, then there is a set of distinct representatives of the members ofX among the (l-k)-subsets of the members ofX provided the cardinality ofX does not exceed a certain bound. An algorithm for calculating this bound is given.     

Character tables and the problem of existence of finite projective planes

We use the connection between positive definite functions and the character table of the symmetric group S₆ to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research.     

Existence of Five MOLS of Orders 18 and 60

In this article, we provide direct constructions for five mutually orthogonal Latin squares (MOLS) of orders and 60. For , these come from a new (60, 6, 1) difference matrix. For , the required construction is obtained by combining two different methods that were used in the constructions of four MOLS(14) and eight MOLS(36).     

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).     

On associated quasi connections

The purpose of this paper is to give a necessary and sufficient condition on the existence of associated splittings (defined in this paper) and to consider some applications to associated quasi-connections on fibred manifolds and vector bundles, using the idea and extending Theorem 1 from [2]. In Section 1, a general condition on the existence of associated splittings is given. In Section 2, the basic constructions concerning q.c.s. used in the next Section are briefly described following [7]; they extend the q.c.s. of Wang [8, 1, 2]. In Section 3 there are proved two theorems on associated q.c.s. using essentially the main theorem from Section 1.     

On the Number of Latin Squares

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f! where f is a particular integer close to (3) provide a formula for the number of Latin squares in terms of permanents of (+1, −1)-matrices, (4) find the extremal values for the number of 1-factorisations of k-regular bipartite graphs on 2n vertices whenever 1 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases. Received September 3, 2004     

On the enumeration of a class of plane multigraphs

A class of plane multigraphs having a series-parallel structure is defined and enumerated. The enumeration is carried out both for the rooted graphs as originally defined, and also for those where no distinctions are made between the vertices. Some other related problems are also discussed.