Related papers: Design Lines
A generalization of modularity, called block modularity, is defined. This is a quality function which evaluates a label assignment against an arbitrary block pattern. Therefore, unlike standard modularity or its variants, arbitrary network…
The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs…
Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets:…
We present a technique to produce arrangements of lines with nice properties. As an application, we construct $(22_4)$ and $(26_4)$ configurations of lines. Thus concerning the existence of geometric $(n_4)$ configurations, only the case…
In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t,…
A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of…
Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to q-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. We also define the corresponding concept of $n$-dimensional difference…
Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in…
We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…
We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of cardinalities of intersections of four blocks.
In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been…
Collective classification models attempt to improve classification performance by taking into account the class labels of related instances. However, they tend not to learn patterns of interactions between classes and/or make the assumption…
We take points and planes as fundamental, lines as derived, in an axiomatic formulation of three-dimensional projective space, the self-dual nature of which formulation renders automatic the principle of duality.
A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which…
A statistician designing an experiment wants to get as much information as possible from the data gathered. Often this means the most precise estimate possible (that is, an estimate with minimum possible variance) of the unknown parameters.…