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We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.
Euclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.
A group of lawyers, psychiatrists, best interest assessors and independent mental capacity advocates were asked to make binary judgements about whether real-life situations in 12 vignettes amounted to deprivation of liberty. Kappa coefficients were calculated to describe the level of agreement within each professional group and for the total group of professionals.
There was total agreement between all professionals about deprivation of liberty in only 1 of the 12 cases. The overall level of agreement for judgements made by all professionals was ‘slight’ (κ=0.16, P < 0.01).
There are practical difficulties involved in making reliable deprivation of liberty judgements within the Deprivation of Liberty Safeguards (DoLS) legislation. A clear interpretation of deprivation of liberty is necessary to facilitate professionals' decision-making in this area.
There is currently no generally accepted formula for the optimal timing of health technology assessments (HTAs). This paper presents some of the relevant issues and then reviews the existing literature on timing of HTAs. It finds that the literature that specifically addresses these issues is limited. There is a consensus that HTAs should be initiated at an early stage of the development of a new health technology, and repeated during the life cycle of the technology. However, the questions of reliably identifying new technologies at an early stage in their development and of deciding on a detectable critical point for starting evaluation are not resolved. It is proposed that a system of categorization and prioritization of health technologies should be developed to allow decisions to be made as to when a strongly precautionary approach is required and how the limited resources available for HTA could be optimally deployed.
Using Moore's ergodicity theorem, S.G. Dani and S. Raghavan proved that the linear action of SL(n, ℤ) on ℝn is topologically (n − l)-transitive; that is, topologically transitive on the Cartesian product of n − 1 copies of ℝn. In this paper, we give a more direct proof, using the prime number theorem. Further, using the congruence subgroup theorem, we generalise the result to arbitrary finite index subgroups of SL(n, ℤ).
Let T be a subgroup of PSL (2, ℚ) generated by a pair of rational parabolic matrices P1, P2, and let ℐ be the Jøgensen number. We prove that T has a non-trivial element of finite order if and only if ℐ = 4/n2 or ℐ = 9/n2 for some non-zero integer n.
For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.
D. Gromoll and K. Grove showed that metric flows on constant curvature spaces are either flat or locally spanned by Killing vector fields. We generalise this result to certain flows on manifolds of variable curvature.
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