Counting Rational Points on K3 Surfaces

For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function N_V(B) = #{P\in V(k) \mid H(P)\leq B}. In this paper, we calculate the counting function for Kummer surfaces $V$ whose associated abelian surface is the product of elliptic curves. In particular, we effectively construct a finite union $C = \cup C_i$ of curves $C_i$ on $V$ such that $N_{V-C}(B)\ll N_C(B)$; that is, $C$ is an accumulating subset of $V$. In the terminology of Batyrev and Manin, this amounts to proving that $C$ is the first layer of the arithmetic stratification of $V$.Comment: LaTeX, 9 pages, no figures. Typo corrected, acknowledgements added, a few minor clarification

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $\alpha_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $\alpha$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $\epsilon_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.Comment: 55 pages, published versio

Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by $\frac{\pi}{n}$. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of $\frac{\pi}{n}$ z-rotations. For the Clifford+T case $n=4$ the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring $\mathbb{Z}[e^{i\frac{\pi}{n}},1/2]$. We prove that this characterization holds for a handful of other small values of $n$ but the fraction of positive even integers for which it fails to hold is 100%.Comment: v2: published versio

Introduction: Social security and the challenge of demographic change

From 29 November to 4 December 2010, the International Social Security Association (ISSA) will meet in Cape Town, Republic of South Africa, to mark the event of the ISSA World Social Security Forum. The Forum provides a unique opportunity for decision-makers from all regions to share knowledge, recognize good practices and discuss key policy challenges as these relate to the design and delivery of national social security programmes. One key policy challenge identified by the ISSA's worldwide membership is demographic change. For this important reason, among the events planned for the Cape Town Forum, a plenary will focus specifically on demography. To coincide with the preparations for the World Forum, and to complement the wider and longer-term endeavours of the ISSA to promote knowledge sharing, the International Social Security Review has chosen to produce this double special issue on "Social security and the challenge of demographic change". The expectation is that this set of papers will make a contribution to supporting social security policy-makers, practitioners, analysts and researchers in all countries as they work towards developing and implementing tailored policy responses to the multifaceted challenge of demographic change.social security, demographic change, demography, policy