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

Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments TWh(v)

AbstractA necessary condition for the existence of a triplewhist tournament TWh(v) is v≡0 or 1(mod4); this condition is known to be sufficient except for v=5,9,12,13 and possibly v=17,57,65,69,77,85,93,117,129,153. In this paper, we remove all the possible exceptions except v=17. This provides an almost complete solution for the more than 100 year old problem on the existence of triplewhist tournaments TWh(v). By applying frame constructions and product constructions, several new infinite classes of Z-cyclic triplewhist tournaments are also obtained. A couple of new cyclic difference matrices are also obtained

A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

AbstractA Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v;μ,λ), is a t×t array (t=λ(v-1)/μ(k-1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v,k,λ)-BIBD. In a series of papers, Lamken established the existence of the following designs: KS3(v;1,2) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable (v,3,2)-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50–76], KS3(v;2,4) with two possible exceptions [E.R. Lamken, The existence of KS3(v;2,4)s, Discrete Math. 186 (1998) 195–216], and doubly near resolvable (v,3,2)-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable (v,3,2)-BIBDs, J. Combin. Designs 2 (1994) 427–440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS3(v;1,1) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases

New Z-cyclic triplewhist frames and triplewhist tournament designs

AbstractTriplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on v players is nearly complete. It is now known that triplewhist designs do not exist for v=5,9,12,13 and do exist for all other v≡0,1(mod4) except, possibly, v=17. Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z-cyclic if the players are elements in Zm∪A where m=v, A=∅ when v≡1(mod4) and m=v-1, A={∞} when v≡0(mod4) and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R1,R2,… in such a way that Rj+1 is obtained by adding +1(modm) to every element in Rj. The production of Z-cyclic triplewhist designs is particularly challenging when m is divisible by any of 5,9,11,13,17. Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m being divisible by at least one of 5,9,11,13,17

Group divisible designs with block size 4 and group sizes 4 and 7

In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t, s)$.Comment: arXiv admin note: substantial text overlap with arXiv:2109.1122

GBRDs over groups of orders ≤100 or of order pq with p, q primes

AbstractThere are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G, with block size k=3. It has been conjectured that these necessary conditions are indeed sufficient. We prove that they are sufficient for groups G of order pq where p,q are primes and for groups of all orders ≤100 except possibly 32, 36, 48, 54, 60, 64, 72, 96

Recognising the importance of 'family time-out' in consultations: An exploratory qualitative study

Objectives: Patients are often accompanied by family or companions during consultations, but little is known about how this might influence the process. We explored how the presence of a companion in a consultation contributes to communication and the decision-making process. Design: Observational study. Setting: A teaching hospital and a district general hospital in south-west England. Participants: 31 patients and their physicians were observed during consultations in which decisions to undergo palliative chemotherapy were made. Each patient was accompanied by at least one companion. Outcome measures: Communication patterns between physicians, patients and companions. Results: In addition to standard patient/physician interactions, patients and companions were often found to discuss medical information and exchange opinions between themselves without the physician actively participating. We called these instances 'family timeout'. On the occasion of disagreement between patients and companions about preferred treatment options, physicians and patients were able to agree the decision while acknowledging the differences in opinion. Conclusions: Instances of 'family time-out' may contribute to better consultation outcomes because they are understood and supported by the patient's social system. This study highlights the potentially important role of exchanges between patients and companions during consultations and how physicians may benefit from observation of such exchanges. We recommend testing the value of making space for family time-out during consultations. Also, we recommend further study into the medical ethics of family time-out. While the focus here is on palliative chemotherapy, this finding has implications for other consultations, particularly those involving difficult treatment decisions

Nox4 reprograms cardiac substrate metabolism via protein O-GlcNAcylation to enhance stress adaptation.

Cardiac hypertrophic remodeling during chronic hemodynamic stress is associated with a switch in preferred energy substrate from fatty acids to glucose, usually considered to be energetically favorable. The mechanistic interrelationship between altered energy metabolism, remodeling, and function remains unclear. The ROS-generating NADPH oxidase-4 (Nox4) is upregulated in the overloaded heart, where it ameliorates adverse remodeling. Here, we show that Nox4 redirects glucose metabolism away from oxidation but increases fatty acid oxidation, thereby maintaining cardiac energetics during acute or chronic stresses. The changes in glucose and fatty acid metabolism are interlinked via a Nox4-ATF4-dependent increase in the hexosamine biosynthetic pathway, which mediates the attachment of O-linked N-acetylglucosamine (O-GlcNAcylation) to the fatty acid transporter CD36 and enhances fatty acid utilization. These data uncover a potentially novel redox pathway that regulates protein O-GlcNAcylation and reprograms cardiac substrate metabolism to favorably modify adaptation to chronic stress. Our results also suggest that increased fatty acid oxidation in the chronically stressed heart may be beneficial

Exploring miniature insect brains using micro-CT scanning techniques

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