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The Institute of Fundamental Sciences (IFS) is an academic unit embracing the four disciplines of chemistry, mathematics, physics and statistics. Each has its own distinctive identity within the Institute. Interdisciplinary teaching is encouraged and promoted through programmes drawing upon expertise as appropriate and required from the individual disciplines. The nanoscience major for example, draws upon chemists and physicists, as well as colleagues from other Institutes.

The fundamental sciences of necessity underpin all scientific endeavour, but more significant is the increasingly important role of these sciences as cutting edges at the forefront of science. Research within the Institute is exemplified by the latest publications.
(See Latest 5 list in right column and below.)

Generalized knot groups distinguish the square and granny knots (with an appendix by David Savitt)   One of the goals of knot theory is to find mathematical methods of distinguishing knots from each other. Surprisingly, one of the most important tools for understanding knots, the fundamental group , cannot tell the difference between a reef knot and a granny knot! This is because it has trouble with mirror images, and these two knots differ only by a reflection of half of the knot (some other knot invariants, such as the famous Jones polynomial, don't suffer from this problem). This paper shows that strengthened versions of the fundamental group, so called generalised knot groups, can in fact detect the difference between a reef knot and a granny knot.

Nelson and Neumann have since shown the stronger result that the generalised knot groups can detect the difference between any two knots that are not simply reflections of each other.

Identifying Health Inequalities between Mâori and Non-Mâori using Mortality Tables   While there is a need for more detailed information on health inequality to guide public health policy, the most complete and easily available data remains that in mortality tables. We investigate, via a comparative analysis of data from New Zealand on M ori and non-M ori mortality, whether more detailed information than raw life expectancy may be extracted from the mortality tables. Given a parametric distribution for the mortality capable of fitting irregularities in mortality table data, the curvature of the survival and hazard rates can identify changes in mortality rates, such as infant and late-life adult mortality, which allows for straightforward comparisons between the two sub-populations. Our results identify an exogenous effect in earlier mortality among Māori, which correlates well with many published observations of health and health-care inequalities between Māori and non-Māori. This “proof of concept” for our method of analysis indicates that examination of bulk data such as those in mortality tables has a potential role in the design of more detailed studies involving causes of mortality.

Modelling N and W shaped hazard rate functions without mixing distributions   The presence of non-conforming components instead of, or in addition to, the usual assembly errors, results in N or W shaped hazard rate (HR) functions rather than the usual bathtub (i.e., U shaped) ones. Although there have been numerous models for bathtub shaped HR functions, N and W shaped HR functions are usually modelled using mixtures of two or more distributions. While this approach does sometimes lead to tidy interpretation, there can be a degree of over-parameterization, with consequent problems in stability and fitting. For this reason, the present paper revisits the natural approach of modelling N and W shaped HR functions using polynomial functions of degree three or four. Although the non-negativity of the hazard rate function becomes non-trivial, this ensures a minimal number of parameters. The polynomial approach also allows the use of a parametric model without imposing a particular shape of hazard rate function on the data, which usually requires a non-parametric approach. The possible hazard rate shapes obtainable are characterized, and detailed formulae for local minima and maxima of the functions provided. The performance of the models is compared to that of several generalizations of the Weibull distribution, with promising results.

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