On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2

For an integer $k\geq 2$, let $\{F^{(k)}_{n} \}_{n\geq 0}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ having at least two presentations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$

Precision measurement of the rotational energy-level structure of the three-electron molecule He2+_2^+

The term values of all rotational levels of the $^4$He{_2}^+\,X^+\,^2\Sigma_u^+\,(\nu^+=0) ground vibronic state with rotational quantum number $N^+\le 19$ have been determined with an accuracy of 8 x 10$^{-4}$ cm$^{-1}$ ($\sim{25}$ MHz) by MQDT-assisted Rydberg spectroscopy of metastable He$_2^*$. Comparison of these term values with term values recently calculated ab initio by Tung et al. [J. Chem. Phys. 136, 104309 (2012)] reveal discrepancies that rapidly increase with increasing rotational quantum number and reach values of 0.07 cm$^{-1}$ ($\sim{2.1}$ GHz) at $N^+=19$.Comment: 11 pages, 6 figure

The Role of Social Security in Household Decisions: Var Estimates of Saving and Fertility Behaviour in Germany

Estimating saving and fertility simultaneously by the VAR method, we find that social security cover has a positive effect on household saving, and a negative effect on fertility. In Germany, as in other countries where the hypothesis was tested, social security is thus good for growth. A possible explanation for this unconventional finding is that compulsory saving in the form of pension contributions tends to displace intra-family transfers, rather than asset formation. However, the negative effect of social security on fertility tends to erode the system’s own contributory base, because it reduces the number of future contributors. That is one of the reasons why, in Germany as elsewhere, pay-as-you-go pension systems tend to be financially unstable. To some extent, this is counteracted by child-related benefits, which tend to encourage fertility, but the effect appears to be weak.

Entanglement, Non-linear Dynamics, and the Heisenberg Limit

We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a $N$ qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle $\theta$. The analysis therefore singles out the class of entangled states which are {\it useful} to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.Comment: Phys. Rev. Lett. 102, 100401 (2009