Related papers: Boolean 2-designs and the embedding of a 2-design …
For a countable group G = <A | R> presented by its generators A and defining relations R we discuss a simple method to embed G into such a 2-generator group T that the images of generators from A are explicitly given in T, and the defining…
The paper is an investigation of the structure of block-transitive automorphism groups of a 3-design with small block size. Let $G$ be a block-transitive automorphism group of a nontrivial $3$-$(v,k,\lambda)$ design $\mathcal{D}$ with $k\le…
We compute the invariants so of Weyl groups in mod 2 Milnor K-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the…
An incidence structure consists simply of a set P of points and a set B of blocks, with a relation of incidence between points and blocks.A symmetric (v,k,\lambda) block design is the subject of this paper. The symmetric (n^2+n+1, n+1,1)…
The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a…
Optimal block designs in small blocks are explored when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first develop an approximate theory which leads to a convenient multiplicative…
The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in…
We classify the Boolean degree $1$ functions of $k$-spaces in a vector space of dimension $n$ (also known as Cameron-Liebler classes) over the field with $q$ elements for $n \geq n_0(k, q)$. This also implies that two-intersecting sets with…
We devise a method for designing materials that will have some desired structural characteristics. We apply it to multiblock copolymers that have two different types of monomers, A and B. We show how to determine what sequence of A's and…
We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and…
Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and…
AG codes correspond geometrically to points in the Grassmannian of k-planes in an n-dimensional projective space PG(n, F_q) defined over a finite field F_q of q elements. We prove that invariant subgrassmannians by the action of a triangle…
Suppose we take an abelian group G and quotient it by the action of negation. What structure does the quotient K inherit from the group structure of G? We describe this structure (which we call the Kummer of G) in terms of a map from the…
We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this…
Superembeddings which have bosonic codimension zero are studied in 3,4 and 6 dimensions. The worldvolume multiplets of these branes are off-shell vector multiplets in these dimensions, and their self-interactions include a Born-Infeld term.…
A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points satisfying (i) every two blocks intersect each other in $\lambda$ points for a fixed $\lambda < v$ (ii) there…
We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a…
The feedback class of a locally Brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants [4]. In this paper we attack the problem of enumerating all feedback classes of locally…