Related papers: The 3-way intersection problem for S(2, 4, v) desi…

We solve the long-standing open problem of classifying all 3-(v,k,1) designs with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom. Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F.…

It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order…

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…

A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of…

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex…

Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for…

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…

In this paper we show that a nodal complete intersection threefold $X$ in $\mathbb{P}^{3+c}$ with defect, but without induced defect, has at least $\sum_{i\leq j} (d_i-1)(d_j-1)$ nodes, provided either $c=2$ or $d_c>\sum_{i=1}^{c-1} d_i$…

This article presents a coupling approach for the approximation of iterated stochastic integrals of length three. The generation of such integrals is the central problem of higher-order pathwise approximations for SDEs, which still lacks a…

A ${\rm{TS}}(v,\lambda)$ is a pair $(V,\mathcal{B})$ where $V$ contains $v$ points and $\mathcal{B}$ contains $3$-element subsets of $V$ so that each pair in $V$ appears in exactly $\lambda$ blocks. A $2$-block intersection graph ($2$-BIG)…

The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a…

Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(\Sigma;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing…

At $c=3$, two of the three integrable quantum $N=2$ supersymmetric Korteweg-de Vries equations become identical (SKdV$_1$ and SKdV$_4$). Quite remarkably, all their conservation laws can be written in closed form, which provides thus a…

Let $K_v$ denote the complete graph of order $v$ and $K_v - I$ denote $K_v$ minus a 1-factor. In this article we investigate uniformly resolvable decompositions of $K_v$ and $K_v-I$ into $r$ classes containing only copies of $3$-stars and…

A \textbf{single change covering design} is a $v$-set $X$ and an ordered list $\cL$ of $b$ blocks of size $k$ where every $t$-set must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. A single…

This paper considers the problem of many-to-many disjoint paths in the hypercube $Q_n$ with $f$ faulty vertices and obtains the following result. For any integer $k$ with $1\leq k\leq n-2$, any two sets $S$ and $T$ of $k$ fault-free…

We prove that every internally 4-connected non-planar bipartite graph has an odd K_3,3 subdivision; that is, a subgraph obtained from K_3,3 by replacing its edges by internally disjoint odd paths with the same ends. The proof gives rise to…

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a…

We discuss group divisible designs with block size four and type $g^u b^1 (gu/2)^1$, where $u = 5$, 6 and 7. For integers $a$ and $b$, we prove the following. (i) A 4-GDD of type $(4a)^5 b^1 (10a)^1$ exists if and only if $a \ge 1$, $b…

We give diagrammatic algorithms for computing the group trisection, homology groups, and intersection form of a closed, orientable, smooth 4-manifold, presented as a branched cover of a bridge-trisected surface in $\mathbb{S}^{4}$. The…