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Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v wherewith p1 p2, … , Pr being distinct primes and

Short Communications COROLLARY 1. A near-domain whose center is of finite index is a near-field.COROLLARY 2. A near-domain with a finite commutator group is a near-field. The latter two results are then interpreted in terms of the associated sharply 2-transitive groups. Extended Triple Systems*)D. M. JOHNSON and N. S. MENDELSOHN An extended triple system on n elements is a collection of unordered triples (whose elements are not necessarily distinct) such that every pair of elements (not necessarily distinct) lies in exactly one triple. For example, if n = 3 there are two non-isomorphic extended triple systems.I: {1, 2, 3}, {1, 1, 1), {2, 2, 2}, {3, 3, 3} II: {1, 1,2}, {2,2,3}, {3,3,1}.Algebraically, an extended triple system on n elements is equivalent to a groupoid on the n elements satisfying the laws y(yx) = x and (xy) y = x.In connection with extended triple systems it may be noted that if all elements of the groupoid are idempotent the system is equivalent to a Steiner Triple System. Let {n; b} represent the class of all extended triple systems on n elements with b idempotent elements. If {n, b} is non-empty we say {n; b} exists. The main theorem is the following. The authors conjecture that the above necessary conditions for the existence of {n; a} are also sufficient. In fact, sufficiency has been proved in most cases and will appear in a subsequent paper. Amongst cases of sufficiency proved by the authors but not appearing in this paper are: (1) all appropriate {n; a} with n=6 mod 12; (3) all appropriate {n; a} with n = 7 mod 12 and (4) all appropriate {n; a} with n = 14 mod 24.The paper also shows how extended triple systems can be used for the construction of other extended triple systems including Steiner triple systems, by means of a singular direct product. *) Received June 25, 1971.

Despite multiple biases that favored effectiveness, the maximum potential benefit of a decision support system to mitigate PACU delays on the day on the surgery was below the 70% minimum threshold for utility of automated decision support messages, previously established via meta-analysis. Neither reduction in PACU delays nor reassigning promised PACU slots based on reducing over-utilized OR time were realized sufficiently to warrant further development of the system. Based on these results, the only evidence-based method of reducing PACU delays is to adjust PACU staffing and staff scheduling using computational algorithms to match the historical workload (e.g., as developed in 2001).

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