Orthogonal trades in complete sets of MOLS

Let Bₚ be the Latin square given by the addition table for the integers modulo an odd prime p (i.e. the Cayley table for (Zₚ, +)). Here we consider the properties of Latin trades in Bₚ which preserve orthogonality with one of the p−1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (log p)² / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in Bₚ hits the main diagonal either p or at most p − log₂ p – 1 times. Finally, if p ≡ 1 (mod 6) we show the existence of a Latin square which is orthogonal to Bₚ and which contains a 2 × 2 subsquare

Cyclic cycle systems of the complete multipartite graph

In this paper, we study the existence problem for cyclic $\ell$-cycle decompositions of the graph $K_m[n]$, the complete multipartite graph with $m$ parts of size $n$, and give necessary and sufficient conditions for their existence in the case that $2\ell \mid (m-1)n$

Nonextendible Latin Cuboids

We show that for all integers m >= 4 there exists a 2m x 2m x m latin cuboid that cannot be completed to a 2mx2mx2m latin cube. We also show that for all even m > 2 there exists a (2m-1) x (2m-1) x (m-1) latin cuboid that cannot be extended to any (2m-1) x (2m-1) x m latin cuboid

There are asymptotically the same number of Latin squares of each parity

A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order n there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order n → ∞

Identifying flaws in the security of critical sets in latin squares via triangulations

In this paper we answer a question in theoretical cryptography by reducing it to a seemingly unrelated geometrical problem. Drápal (1991) showed that a given partition of an equilateral triangle of side n into smaller, integer-sided equilateral triangles gives rise to, under certain conditions, a latin trade within the latin square based on the addition table for the integers (mod n). We apply this result in the study of flaws within certain theoretical cryptographic schemes based on critical sets in latin squares. We classify exactly where the flaws occur for an infinite family of critical sets. Using Drápal's result, this classification is achieved via a study of the existence of triangulations of convex regions that contain prescribed triangles

Biembeddings of cycle systems using integer Heffter arrays

In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in such a way that the group, (Z₂ₙₖ₊₁, +) acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays, H (n ; k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk + 1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2)

Retinoic acid-responsive CD8 effector T-cells are selectively increased in IL-23-rich tissue in gastrointestinal GvHD.

Gastrointestinal (GI) graft-versus-host disease (GvHD) is a major barrier in allogeneic hematopoietic stem-cell transplantation (AHST). The metabolite retinoic acid (RA) potentiates GI-GvHD in mice via alloreactive T-cells expressing the RA-receptor-alpha (RARα), but the role of RA-responsive cells in human GI-GvHD remains undefined. We therefore used conventional and novel sequential immunostaining and flow cytometry to scrutinize RA-responsive T-cells in tissues and blood of AHST patients and characterize the impact of RA on human T-cell alloresponses. Expression of RARα by human mononuclear cells was increased after RA exposure. RARαhi mononuclear cells were increased in GI-GvHD tissue, contained more cellular RA-binding proteins, localized with tissue damage and correlated with GvHD severity and mortality. Using a targeted candidate protein approach we predicted the phenotype of RA-responsive T-cells in the context of increased microenvironmental IL-23. Sequential immunostaining confirmed the presence of a population of RARahi CD8 T-cells with the predicted phenotype, co-expressing the effector T-cell transcription factor T-bet and the IL-23-specific receptor. These cells were increased in GI- but not skin-GvHD tissues and were also selectively expanded in GI-GvHD patient blood. Finally, functional approaches demonstrated RA predominantly increased alloreactive GI-tropic RARahi CD8 effector T-cells, including cells with the phenotype identified in vivo. IL-23-rich conditions potentiated this effect by selectively increasing b7 integrin expression on CD8 effector T-cells and reducing CD4 T-cells with a regulatory cell phenotype. In conclusion we have identified a population of RA-responsive effector T-cells with a distinctive phenotype which are selectively expanded in human GI-GvHD and represent a potential new therapeutic target

Criteria for the use of omics-based predictors in clinical trials.

The US National Cancer Institute (NCI), in collaboration with scientists representing multiple areas of expertise relevant to 'omics'-based test development, has developed a checklist of criteria that can be used to determine the readiness of omics-based tests for guiding patient care in clinical trials. The checklist criteria cover issues relating to specimens, assays, mathematical modelling, clinical trial design, and ethical, legal and regulatory aspects. Funding bodies and journals are encouraged to consider the checklist, which they may find useful for assessing study quality and evidence strength. The checklist will be used to evaluate proposals for NCI-sponsored clinical trials in which omics tests will be used to guide therapy