Trades in complex Hadamard matrices

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order $n$ all trades contain at least $n$ entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar $c \neq 1$ to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order $n$ and show that they all contain at least $n$ entries. We conjecture that all trades in complex Hadamard matrices contain at least $n$ entries.Comment: 9 pages, no figure

The Physics of Star Cluster Formation and Evolution

© 2020 Springer-Verlag. The final publication is available at Springer via https://doi.org/10.1007/s11214-020-00689-4.Star clusters form in dense, hierarchically collapsing gas clouds. Bulk kinetic energy is transformed to turbulence with stars forming from cores fed by filaments. In the most compact regions, stellar feedback is least effective in removing the gas and stars may form very efficiently. These are also the regions where, in high-mass clusters, ejecta from some kind of high-mass stars are effectively captured during the formation phase of some of the low mass stars and effectively channeled into the latter to form multiple populations. Star formation epochs in star clusters are generally set by gas flows that determine the abundance of gas in the cluster. We argue that there is likely only one star formation epoch after which clusters remain essentially clear of gas by cluster winds. Collisional dynamics is important in this phase leading to core collapse, expansion and eventual dispersion of every cluster. We review recent developments in the field with a focus on theoretical work.Peer reviewe

Balanced ternary designs with block size three, any Λ and any R

A balanced ternary design on V elements is a collection of B blocks (which are multisets) of size K, such that each element occurs 0, 1 or 2 times per block and R times altogether, and such that each unordered pair of distinct elements occurs Λ times. (For example, in the block xxyyz, the pair xy is said to occur four times and the pairs xz, yz twice each.) It is straightforward to show that each element has to occur singly in a constant number of blocks, say ρ1, and so each element also occurs twice in a constant number of blocks, say ρ2, where R=ρ1+2 ρ2. If ρ2=0 the design is a balanced incomplete block design (binary design), so we assume ρ2>0, and K1 if ρ2>0 (and K>2). In 1980 and 1982 the author gave necessary and sufficient conditions for the existence of balanced ternary designs with K=3, Λ=2 and ρ2=1, 2 or 3. In this paper work on the existence of balanced ternary designs with block size three is concluded, in that necessary and sufficient conditions for the existence of a balanced ternary design with K=3, any Λ>1 and any ρ2 are given

Manufacturing sequences for the Economic Lot Scheduling problem

In the basic Economic Lot Scheduling problem, a production schedule is required to manufacture sequentially a number of products on a single machine, with the schedule chosen to minimize set-up and inventory costs. The products suffer continuous demand, and no shortfall is allowed. A recent approach involves repetitions of a production cycle (such as ABCBC for three products A, B and C, with manufacturing times chosen to prevent shortage occurring); an exhaustive search is performed over a large set of possible cycles to discover the optimal schedule. This paper discusses the question “How many such sycles need to be examined?”, Since the answer is very relevant to practical application of the method. The case of three products is considered. Complete information is obtained for cycles up to length 12 (that is, 12 production switch overs), and partial results for longer ones. An estimate, apparently reasonable, is obtained for cycles of any length. The major trend to emerge is that surprisingly few cycles are involved

On the spectrum for K_{m+2}/K_m designs

The spectrum problem for the decomposition of K-n into copies of the graph K_{m+2}\K_m is solved for n = 0 or 1 (mod 2m + 1). (C) 1997 John Wiley & Sons, Inc

Decomposing complete equipartite graphs into closed trails of length k

Necessary conditions for a simple connected graph G to admit a decomposition into closed trails of length k a parts per thousand yen 3 are that G is even and its total number of edges is a multiple of k. In this paper we show that these conditions are sufficient in the case when G is the complete equipartite graph having at least three parts, each of the same size