Simple constructions for balanced incomplete block designs with block size three

AbstractLet S be a finite set with v elements. It is known that there exists a sequence of three-element subsets of S such that each two-element subset of S is contained in exactly λ terms of the sequence if and only if λ(v − 1)2 and λv(v − 1)6 are integers. The known proof is somewhat complicated when v ≡ 2 (mod 6), and this paper provides a simpler proof for this case. Proofs are also given for all other values of v by reviewing known constructions or providing new ones

Experimental study of the effects of flameholder geometry on emissions and performance of lean premixed combustors

Emissions of NOx, CO, and unburned hydrocarbons (UHC) are reported for a lean premixed propane-air system at inlet conditions of 800K and 1MPa using twelve flameholder designs. The flameholders tested represent six design concepts with two values of blockage for each concept. Data were obtained at reference velocities of 35 m/s, 25 m/s and 20 m/s at combustor stations 10 cm and 30 cm downstream of the flameholders. Flameholder pressure drop was found to be a principal determinant of emissions performance. Designs producing larger pressure drops also produced less NOx, CO, and UHC emissions. The lean stability limit equivalence ratio was found to be approximately 0.35 for all designs. Flashback velocities (axial components in the flameholder passages) varied between 30 m/s and 40 m/s. A perforated plate flameholder was operated with a velocity as low as 23 m/s through the perforations at equivalence ratio 0.7 without producing flashback

Mixed group divisible designs with three groups and block size 4

AbstractA group divisible design (GDD) with three groups and block size 4 is called even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups are always even, always odd, or always mixed. It has been shown that the necessary conditions for the existence of GDDs of these three types are also sufficient except possibly for the minimal case of mixed designs for group size 5t (t>1). In this paper, we complete the undetermined families of mixed GDDs using two constructions based on idempotent self-orthogonal Latin squares and skew Room squares

Large Cross-free sets in Steiner triple systems

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\lfloor{n-3\over 3}\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets of size close to $6k$ (but cannot have size exactly $6k$). The constructed STS$(18k+3)$ shows that equality is possible for $n=18k+3$ in the following result: in every $3$-coloring of the blocks of any Steiner triple system STS$(n)$ there is a monochromatic connected component of size at least $\lceil{2n\over 3}\rceil+1$ (we conjecture that equality holds for every admissible $n$). The analogue problem can be asked for $r$-colorings as well, if r-1 \equiv 1,3 \mbox{ (mod 6)} and $r-1$ is a prime power, we show that the answer is the same as in case of complete graphs: in every $r$-coloring of the blocks of any STS$(n)$, there is a monochromatic connected component with at least ${n\over r-1}$ points, and this is sharp for infinitely many $n$.Comment: Journal of Combinatorial Designs, 201

Modified group divisible designs with block size four

AbstractThe existence of modified group divisible designs with block size four is settled with a handful of possible exceptions