Tight t-Designs and Squarefree Integers

The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5.Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F=ℝ,O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the author's result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1

Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k. We show how to construct these difference sets from commutative semifields and that several known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in arxiv.org:1104.337

Climate change and agriculture: modelling the impact of carbon dioxide emission on cereal yield in Ghana

The objective of the paper is to contribute to the body of knowledge in the area of climate change and agriculture by examining the effect of carbon dioxide concentration (CO2) on cereal yield using autoregressive distributed lag models (ARDL). The research is based on quantitative, descriptive and cross-sectional research using secondary data obtained from World Bank data base for the period of 1961-2010. The co-integration test indicates the series are co-integrated. The results on the long run and shorts run elastically co-efficient indicate that there is significant negative link between CO2 and cereal yield. There significant positive long run and short run link between cereal yield and income (proxied by real gross domestic product). Policy makers and agriculture scientists and environmental scientists should put in place policies to reduce atmospheric temperature increase and pollution to benefit from CO2 fertilization in order to ensure food security. The findings indicate that income (proxied by real gross domestic product) positively affect cereal yield. The link between CO2 and cereal production should be examine in future studies current study considered cereal yield

A special irreducible matrix representation of the real Clifford algebra C(3,1)

4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g. due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4x4 Dirac matrices (Majorana representation) which may prove useful in the future. As a byproduct, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras.Comment: 23 pages LaTeX, to appear in the J. Math. Phys. (v2: appendix B on Pauli matrices and references are added, minor other changes

Climate change and agriculture: modelling the impact of carbon dioxide emission on cereal yield in Ghana

The objective of the paper is to contribute to the body of knowledge in the area of climate change and agriculture by examining the effect of carbon dioxide concentration (CO2) on cereal yield using autoregressive distributed lag models (ARDL). The research is based on quantitative, descriptive and cross-sectional research using secondary data obtained from World Bank data base for the period of 1961-2010. The co-integration test indicates the series are co-integrated. The results on the long run and shorts run elastically co-efficient indicate that there is significant negative link between CO2 and cereal yield. There significant positive long run and short run link between cereal yield and income (proxied by real gross domestic product). Policy makers and agriculture scientists and environmental scientists should put in place policies to reduce atmospheric temperature increase and pollution to benefit from CO2 fertilization in order to ensure food security. The findings indicate that income (proxied by real gross domestic product) positively affect cereal yield. The link between CO2 and cereal production should be examine in future studies current study considered cereal yield

Symmetric Informationally Complete Quantum Measurements

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.Comment: 8 page

Symmetric Informationally Complete Measurements of Arbitrary Rank

There has been much interest in so-called SIC-POVMs: rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs which are symmetric and informationally complete but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM which is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1 symmetric informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.Comment: Contribution to proceedings of International Conference on Quantum Optics, Minsk, 200

Tight informationally complete quantum measurements

We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, "as close as possible" to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows "painless" quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.Comment: 20 pages. Final versio