On Zudilin's q-question about Schmidt's problem

We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.Comment: 5 page

A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical change

Some q-analogues of supercongruences of Rodriguez-Villegas

We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv (-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \varepsilon=\pm1. Several related conjectures are also proposed.Comment: 14 pages, to appear in J. Number Theor

Association of late childbearing with healthy longevity among the oldest-old in China

Statistical analysis of a large and unique longitudinal data set demonstrates that late childbearing after age 35 or 40 is significantly associated with survival and healthy survival among very old Chinese women and men. The association is stronger in oldest-old women than men. The estimates are adjusted for a variety of confounding factors of demographic characteristics, family support, social connections, health practices, and health conditions. Further analysis based on an extension of the Fixed Attribute Dynamics method shows that late childbearing is positively associated with long-term survival and healthy survival from ages 80-85 to 90-95 and 100-105. This association exists among oldest-old women and men, but, again, the effects are substantially stronger in women than men. We discuss four possible factors which may explain why late childbearing affects healthy longevity at advanced ages: (1) social factors; (2) biological changes caused by late pregnancy and delivery; (3) genetic and other biological characteristics; and (4) selection.

Oldest Old Mortality in China

We find that the Kannisto model, a two-parameter logistic formula, fits Han Chinese death rates at oldest-old ages better than the Gompertz and four other models. Chinese death rates appear to be roughly similar to Swedish and Japanese rates after age 97 for both males and females. Because reports of age seem to be serviceably reliable up to age 100 and perhaps age 105 in China, we think that this convergence may be mainly due to mortality selection in the heterogeneous Chinese population. We show that in China, as in developed countries, the rate of increase in mortality with age decelerates at very old ages.age, China, models, mortality

Collinear antiferromagnetic state in a two-dimensional Hubbard model at half filling

In a half-filled Hubbard model on a square lattice, the next-nearest-neighbor hopping causes spin frustration, and the collinear antiferromagnetic (CAF) state appears as the ground state with suitable parameters. We find that there is a metal-insulator transition in the CAF state at a critical on-site repulsion. When the repulsion is small, the CAF state is metallic, and a van Hove singularity can be close to the Fermi surface, resulting in either a kink or a discontinuity in the magnetic moment. When the on-site repulsion is large, the CAF state is a Mott insulator. A first-order transition from the CAF phase to the antiferromagnetic phase and a second-order phase transition from the CAF phase to the paramagnetic phase are obtained in the phase diagram at zero temperature.Comment: 5 pages, 5 figures, two column