Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation

Holonomy groups and W-symmetries

Irreducible sigma models, i.e. those for which the partition function does not factorise, are defined on Riemannian spaces with irreducible holonomy groups. These special geometries are characterised by the existence of covariantly constant forms which in turn give rise to symmetries of the supersymmetric sigma model actions. The Poisson bracket algebra of the corresponding currents is a W-algebra. Extended supersymmetries arise as special cases.Comment: pages 2

Cystic ovarian follicles and thyroid activity in the dairy cow

Thyroid activity affects the functionality of the reproductive axis and thyroid dysfunction has been associated with ovarian hyperstimulation syndrome and polycystic ovarian syndrome, in human medicine. This study investigates serum17- < beta> estradiol, progesterone, thyrotropic and thyroid hormone levels, in cyclic dairy cows on heat (Group H) and in dairy cows with ovarian follicular cysts (Group FC). Both 17- < beta> estradiol and progesterone serum concentrations were statistically higher in cystic than in cyclic cows (estradiol: 8.51 ± 1.91 vs 6.32 ± 1 pg/mL) (progesterone: 0.49 ± 0.17 vs 0.13 ± 0.03 ng/mL), whereas TSH and fT4 serum concentrations were statistically lower in cows with cystic ovarian follicles (COF), compared to cyclic ones (TSH: 2.48 ± 1.31 vs 3.56 ± 1.03 ng/mL) (fT4: 5,86 ± 1.69 vs 8.63 ± 1.08). fT3 serum levels were similar, in both cystic and cyclic subjects (2.94 ± 0.65 vs 3.02 ± 0.9, respectively). Standing on these results, a look to the functioning of thyrotropic axis should be given in dairy cows with COF, just like it is routinely performed in human gynaecology, in case of ovarian diseases. If severe hypothyroidism should be found, a hormone replacement therapy could be attempted in cystic cows refractory to “ordinary” therapies