The Split-Algebras and Non-compact Hopf Maps

We develop a noncompact version of the Hopf maps based on the split algebras. The split algebras consist of three species: split-complex numbers, split quaternions, and split octonions. They correspond to three noncompact Hopf maps that represent topological maps between hyperboloids in different dimensions with hyperboloid bundle. We realize such noncompact Hopf maps in two ways: one is to utilize the split-imaginary unit, and the other is to utilize the ordinary imaginary unit. Topological structures of the hyperboloid bundles are explored, and the canonical connections are naturally regarded as noncompact gauge field of monopoles.Comment: 40 pages, no figures, + 8 pages, discussions and references added, published versio

Effect of multivitamin and multimineral supplementation on cognitive function in men and women aged 65 years and over : a randomised controlled trial

Background: Observational studies have frequently reported an association between cognitive function and nutrition in later life but randomised trials of B vitamins and antioxidant supplements have mostly found no beneficial effect. We examined the effect of daily supplementation with 11 vitamins and 5 minerals on cognitive function in older adults to assess the possibility that this could help to prevent cognitive decline. Methods: The study was carried out as part of a randomised double blind placebo controlled trial of micronutrient supplementation based in six primary care health centres in North East Scotland. 910 men and women aged 65 years and over living in the community were recruited and randomised: 456 to active treatment and 454 to placebo. The active treatment consisted of a single tablet containing eleven vitamins and five minerals in amounts ranging from 50–210 % of the UK Reference Nutrient Intake or matching placebo tablet taken daily for 12 months. Digit span forward and verbal fluency tests, which assess immediate memory and executive functioning respectively, were conducted at the start and end of the intervention period. Risk of micronutrient deficiency at baseline was assessed by a simple risk questionnaire. Results: For digit span forward there was no evidence of an effect of supplements in all participants or in sub-groups defined by age or risk of deficiency. For verbal fluency there was no evidence of a beneficial effect in the whole study population but there was weak evidence for a beneficial effect of supplementation in the two pre-specified subgroups: in those aged 75 years and over (n 290; mean difference between supplemented and placebo groups 2.8 (95% CI -0.6, 6.2) units) and in those at increased risk of micronutrient deficiency assessed by the risk questionnaire (n 260; mean difference between supplemented and placebo groups 2.5 (95% CI -1.0, 6.1) units). Conclusion: The results provide no evidence for a beneficial effect of daily multivitamin and multimineral supplements on these domains of cognitive function in community-living people over 65 years. However, the possibility of beneficial effects in older people and those at greater risk of nutritional deficiency deserves further attention.Peer reviewedPublisher PD

On complexified mechanics and coquaternions

While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced, and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterised by complex Hamiltonians invariant under space-time reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the space-time dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.Comment: 11 pages, version to appear in Journal of Physics

Human and mouse essentiality screens as a resource for disease gene discovery.

The identification of causal variants in sequencing studies remains a considerable challenge that can be partially addressed by new gene-specific knowledge. Here, we integrate measures of how essential a gene is to supporting life, as inferred from viability and phenotyping screens performed on knockout mice by the International Mouse Phenotyping Consortium and essentiality screens carried out on human cell lines. We propose a cross-species gene classification across the Full Spectrum of Intolerance to Loss-of-function (FUSIL) and demonstrate that genes in five mutually exclusive FUSIL categories have differing biological properties. Most notably, Mendelian disease genes, particularly those associated with developmental disorders, are highly overrepresented among genes non-essential for cell survival but required for organism development. After screening developmental disorder cases from three independent disease sequencing consortia, we identify potentially pathogenic variants in genes not previously associated with rare diseases. We therefore propose FUSIL as an efficient approach for disease gene discovery