Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle

It is shown that a $N\times N$ real symmetric [complex hermitian] positive definite matrix $V$ is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in $SO(m,n)$ [ $SU(m,n)$], for any choice of partition $N=m+n$. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson's theorem which states that if $N$ is even then $V$ is congruent also to a diagonal matrix modulo a symplectic matrix in $Sp(N,{\cal R})$ [$Sp(N,{\cal C})$]. Applications of these results considered include a generalization of the Schweinler-Wigner method of `orthogonalization based on an extremum principle' to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors.Comment: 7 pages, latex, no figure

Wigner distributions for finite state systems without redundant phase point operators

We set up Wigner distributions for $N$ state quantum systems following a Dirac inspired approach. In contrast to much of the work on this case, requiring a $2N\times 2N$ phase space, particularly when $N$ is even, our approach is uniformly based on an $N\times N$ phase space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both $N$ odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the $N$ odd case permits full implementation of the marginals property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.Comment: Latex, 14 page

Classical Light Beams and Geometric Phases

We present a study of geometric phases in classical wave and polarisation optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarisation optics, and the behaviour of polarisation in the eikonal or ray limit of Maxwell's equations in a transparent medium are considered. The case of a beam of light whose propagation direction and polarisation state are both subject to change is dealt with, attention being paid to the validity of Maxwell's equations at all stages. Global topological aspects of the space of all propagation directions are discussed using elementary group theoretical ideas, and the effects on geometric phases are elucidated.Comment: 23 pages, 1 figur

The Sampling Theorem and Coherent State Systems in Quantum Mechanics

The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on properties of von Neumann and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late

Hamilton's theory of turns revisited

We present a new approach to Hamilton's theory of turns for the groups SO(3) and SU(2) which renders their properties, in particular their composition law, nearly trivial and immediately evident upon inspection. We show that the entire construction can be based on binary rotations rather than mirror reflections.Comment: 7 pages, 4 figure

Wigner-Weyl isomorphism for quantum mechanics on Lie groups

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space $T^*G$ and the Hilbert space of square integrable functions on $G$. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.Comment: 32 pages, Latex2

Assessing the climate impacts of Chinese dietary choices using a telecoupled global food trade and local land use framework

Global emissions trajectories developed to meet the 2⁰C temperature target are likely to rely on the widespread deployment of negative emissions technologies and/or the implementation of substantial terrestrial carbon sinks. Such technologies include afforestation, carbon capture and storage (CCS) and bioenergy with carbon capture and storage (BECCS), but mitigation options for agriculture appear limited. For example, using the Global Calculator tool (http://www.globalcalculator.org/), under a 2⁰C pathway, the ‘forests and other land use’ sector is projected to become a major carbon sink, reaching -15 GtCO2e yr-1 by 2050, compared to fossil emissions of 21 GtCO2e yr-1. At the same time, rates of agricultural emissions remain static at about 6 GtCO2e yr-1, despite increasing demands for crop and livestock production to meet the forecast dietary demands of the growing and increasingly wealthy global population. Emissions in the Global Calculator are sensitive to the assumed global diet, and particularly to the level and type of meat consumption, which in turn drive global land use patterns and agricultural emissions. Here we assess the potential to use a modified down-scaled Global Calculator methodology embedded within the telecoupled global food trade framework, to estimate the agricultural emissions and terrestrial carbon stock impacts in China and Brazil, arising from a plausible range of dietary choices in China. These dietary choices are linked via telecoupling mechanisms to Brazilian crop production (e.g. Brazilian soy for Chinese animal feed provision) and drive land and global market dynamics. ‘Spill-over’ impacts will also be assessed using the EU and Malawi as case studies

Prevalence of β thalassemia carrier state in Sindhi community of Wardha and evaluation of risk factors for β thalassemia trait

Objective: To determine the prevalence of β thalassemia in Sindhi community of Wardha and evaluation of risk factors. To give genetic counseling to those diagnosed as carriers.Study Design: Cross‑sectional (prevalence study).Setting: Sindhi community residing in and around Wardha.Study Period: 18 months.Materials and Methods: A total of 578 individuals belonging to Sindhi community residing in and around Wardha in India were selected by systemic randomization. Those who fulfilled the inclusion criteria and voluntarily gave consent were subject to Naked Eye Single Tube Rapid Osmotic Fragility Test (Nestroft). Those found positive by Nestroft were labeled carriers and advised to undergo Hemoglobin A2(HbA2) quantification for confirmation of carrier status. Carriers thus identified were given genetic counseling.Result: The mean age of study population was 15.06 years with a range of 3‑28 years. The largest group consisted of those between 12 and 18 years. The prevalence of β thalassemia carrier state as diagnosed by Nestroft is 36.36% (200/550) and incidence of carrier state by HbA2 quantification in the study population was 17.2% (95/550).Conclusion: Prevalence of β thalassemia in the Sindhi community of Wardha is higher than in other studies and equal to the highest prevalent communities in India. Thus, we recommend that urgent measures to increase awareness and control the disease be taken.Key words: Hemoglobin A2 quantification, naked eye single tube rapid osmotic fragility test, prevalence, Sindhi community, β thalassemi