Latin bitrades derived from groups

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is equivalent to three derangements whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some latin bitrades may be derived from groups without specifying an independent group action. Properties of latin trades such as homogeneousness, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown latin bitrades. In particular, we show the existence of minimal, $k$-homogeneous latin trades for each odd $k\geq 3$. In some cases these are the smallest known such examples.Comment: 23 page

Decomposing dense bipartite graphs into 4-cycles

Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles

Maximal partial Latin cubes

We prove that each maximal partial Latin cube must have more than 29.289% of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders n the numbers k for which there exists a maximal partial Latin cube of order n with exactly k entries. Finally, we prove that maximal partial Latin cubes of order n exist of each size from approximately half-full (n3/2 for even n ≥ 10 and (n3 + n)/2 for odd n ≥21) to completely full, except for when either precisely 1 or 2 cells are empty

Induced subarrays of Latin squares without repeated symbols

We show that for any Latin square L of order 2m, we can partition the rows and columns of L into pairs so that at most (m+3)/2 of the 2x2 subarrays induced contain a repeated symbol. We conjecture that any Latin square of order 2m (where m ≥ 2, with exactly five transposition class exceptions of order 6) has such a partition so that every 2x2 subarray induced contains no repeated symbol. We verify this conjecture by computer when m ≤ 4

A lower bound for the size of the smallest critical set in the back circulant latin square

The back circulant latin square of order n is the latin square based on the addition table for the integers modulo n. A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. In this note we show that the size of a critical set in the back circulant latin square of order n is at least n ⁴/³/2 - n - n²/³/2 + 2n¹/³ - 1

An exploration of attributional style using the Descriptive Experience Sampling method

This study explored the association between cognitive style and depression. Phase I: 195 participants completed questionnaires that assessed cognitive style and psychological distress. Among these participants the expected relationships were found between depression, anxiety, interpersonal sensitivity, attributional style, and rumination; Phase II: Six participants selected based on their attributional style engaged in Descriptive Experience Sampling, which entails wearing a small device that will beep at random intervals, recording their awareness at the sound of the beep, and being interviewed within twenty-four hours of doing so. The inner experiences revealed by DES did not generally show the patterns suggested by current theories of depression. Two individuals who appeared to be at greatest risk for depression, one of whom was in the low risk group based on the Phase I screening data, did show signs of more negatively valenced inner experience and more thoughts consistent with Beck\u27s cognitive theory of depression