New 2--critical sets in the abelian 2--group

In this paper we determine a class of critical sets in the abelian {2--group} that may be obtained from a greedy algorithm. These new critical sets are all 2--critical (each entry intersects an intercalate, a trade of size 4) and completes in a top down manner.Comment: 25 page

An enumeration of equilateral triangle dissections

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove W. T. Tutte's conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.Comment: Final version sent to journal

Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, every $k$-latin square of order $n$ is separable. We discuss the connection between $k$-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and $k$-latin trades. We also enumerate and classify $k$-latin squares of small orders.Comment: Final version as sent to journa

An ontologically consistent MRI-based atlas of the mouse diencephalon

In topological terms, the diencephalon lies between the hypothalamus and the midbrain. It is made up of three segments, prosomere 1 (pretectum), prosomere 2 (thalamus), and prosomere 3 (the prethalamus). A number of MRI-based atlases of different parts of the mouse brain have already been published, but none of them displays the segments the diencephalon and their component nuclei. In this study we present a new volumetric atlas identifying 89 structures in the diencephalon of the male C57BL/6J 12 week mouse. This atlas is based on an average of MR scans of 18 mouse brains imaged with a 16.4T scanner. This atlas is available for download at www.imaging.org.au/AMBMC. Additionally, we have created an FSL package to enable nonlinear registration of novel data sets to the AMBMC model and subsequent automatic segmentation

Latin Bitrades And Related Structures

Let Ll and La be two latin squares. If we remove from Ll and La the cells ˆi, j[?] where i l j 􀀀 i a j then the resulting partial latin squares Tl, Ta form a bitrade. Chapter ò introduces the theory of bitrades and presents the hypermap representation. [?]is puts the study of bitrades in a more algebraic setting. For example, meaning can be given to the notion of a universal covering of a bitrade. Chapter çmakes use of the universal covering of a bitrade to give an alternative (and geometric) proof of a result of Cavenagh that any ç-homogeneous bitrade can be partitioned into three transversals. Chapter ¥ looks at the case where the permutation group G 􀀀 `0Ô, 0ò, 0çeassociated with a bitrade acts on a set of size STlS. A number of constructions are given using properties of groups of order pç and pq for primes p, q and also using the alternating group Açm[?]Ô. Finally, Chapter gives a new class of critical sets in the abelian ò-group Ls of order òs and in isotopisms of Ls. [?]e symmetry of Ls is key to identifying bitrades to ensure that the constructed partial latin squares are critical sets