Related papers: Group divisible designs with block size 4 and type…

In this note, we give direct constructions of some group divisible designs (GDDs) with block size $4$ that have up to $50$ points.

Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular…

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…

Composite Higgs models can trivially satisfy precision-electroweak and flavour constraints by simply having a large spontaneous symmetry breaking scale, f > 10 TeV. This produces a 'split' spectrum, where the strong sector resonances have…

A regular-graph design is a block design for which a pair $\{a,b\}$ of distinct points occurs in $\lambda+1$ or $\lambda$ blocks depending on whether $\{a,b\}$ is or is not an edge of a given $\delta$-regular graph. Our paper describes a…

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i, j, k, l\} = \{3,4,5,6\}.$ For…

A skew Hadamard difference set (SHDS) is a difference set that satisfies the skew condition. It is known that if a group G admits a skew hadamard difference set, then G is a p-group with order congruent to 3 modulo 4. We will generalize…

Recently, a construction of group divisible designs (GDDs) derived from the decoding of quadratic residue (QR) codes was given. In this paper, we extend the idea to obtain a new family of GDDs, which is also involved with a well-known…

Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \mathcal F\}=G\setminus H$. Let $g\mathbb Z_{gh}$ is the subgroup of order…

There is a one-to-one correspondence between the point set of a group divisible design (GDD) with $v_1$ groups of $v_2$ points and the edge set of a complete bipartite graph $K_{v_1,v_2}$. A block of GDD corresponds to a subgraph of…

Let $G$ be a noncompact connected simple Lie group, and $(G,G^\Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for…

A weak $c$-colouring of a design is an assignment of colours to its points from a set of $c$ available colours, such that there are no monochromatic blocks. A colouring of a design is block-equitable, if for each block, the number of points…

The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $\tau$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring,…

Let $K$ be a set of positive integers and let $G$ be an additive group. A $(G, K, 1)$ difference packing is a set of subsets of $G$ with sizes from $K$ whose list of differences covers every element of $G$ at most once. It is balanced if…

Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers…

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes…

For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$,…

Let $p$ be a prime such that $p \geq 5$. Let $G$ be a finite $p$-solvable group and let $p^a$ be the largest power of $p$ dividing $\chi(1)$ for an irreducible character $\chi$ of $G$, we show that $|G:F(G)|_p \leq p^{5.5a}$. Let $G$ be a…

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$…

Let $(G,G^\Gamma)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^\Gamma)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $\pi_K$ that is…