Points at rational distances from the vertices of certain geometric objects

We consider various problems related to finding points in \Q^{2} and in \Q^{3} which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in \Q^{2}, and a cube or tetrahedron in \Q^{3}.Comment: 23 pages, submitte

On the reducibility type of trinomials

Say a trinomial x^n+A x^m+B \in \Q[x] has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in \Q[x] of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq ... \leq n_k$. Specifying the reducibility type of a monic polynomial of fixed degree is equivalent to specifying rational points on an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable hope that all its rational points may be described; and techniques are available that may also find all points when the genus is 2. Thus all corresponding reducibility types may be described. These low genus instances are the ones studied in this paper.Comment: to appear in Acta Arithmetic

On certain diophantine equations of diagonal type

In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\in\{(2,4,8,8), (2,8,4,8)\}$. In the case $(p,q,r,s)=(4,4,4,4)$, we present some new parametric solutions of the equation $x^4-y^4=4(z^4-w^4)$.Comment: 16 pages, revised version will appear in the Journal of Number Theor

The World Bank's health projects in Timor-Leste: The political economy of effective aid

The World Bank's health sector projects in Timor-Leste - the Health Sector Rehabilitation and Development Project and the Second Health Sector Rehabilitation and Development Project - have been among the few successful operations it has funded in that country. This paper examines the factors underpinning their relative success and considers the wider lessons of the Bank's experience for our understanding of the conditions that lead to effective aid in fragile contexts. Much commentary on these projects has suggested, either implicitly or explicitly, that good design and management were key factors in their success. We argue that political economy factors also played an important role, extending and revising an earlier analysis. In particular, we suggest that these rehabilitation and development projects benefitted from (i) a political economy context that was relatively conducive to aid effectiveness in general and (ii) the fact that there was relatively little elite resistance to the World Bank's health policy agenda compared to its policy agenda in other sectors. In terms of wider lessons, we argue for a more political understanding of the determinants of aid effectiveness. Specifically we suggest that aid effectiveness needs to be seen as a function not just of the technical quality of project design and the administrative competence of project managers but also the extent to which there is congruence between donor and local elites' agendas

Multisensory perception of looming and receding objects in human newborns

When newborns leave the enclosed spatial environment of the uterus and arrive in the outside world, they are faced with a new audiovisual environment of dynamic objects, actions and events both close to themselves and further away. One particular challenge concerns matching and making sense of the visual and auditory cues specifying object motion [1-5]. Previous research shows that adults prioritise the integration of auditory and visual information indicating looming (for example [2]) and that rhesus monkeys can integrate multisensory looming, but not receding, audiovisual stimuli [4]. Despite the clear adaptive value of correctly perceiving motion towards or away from the self - for defence against and physical interaction with moving objects - such a perceptual ability would clearly be undermined if newborns were unable to correctly match the auditory and visual cues to such motion. This multisensory perceptual skill has scarcely been studied in human ontogeny. Here we report that newborns only a few hours old are sensitive to matches between changes in visual size and in auditory intensity. This early multisensory competence demonstrates that, rather than being entirely na\uefve to their new audiovisual environment, newborns can make sense of the multisensory cue combinations specifying motion with respect to themselves

Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over \Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor

Measuring Controlled-NOT and two-qubit gate operation

Accurate characterisation of two-qubit gates will be critical for any realisation of quantum computation. We discuss a range of measurements aimed at characterising a two-qubit gate, specifically the CNOT gate. These measurements are architecture-independent, and range from simple truth table measurements, to single figure measures such as the fringe visibility, parity, fidelity, and entanglement witnesses, through to whole-state and whole-gate measures achieved respectively via quantum state and process tomography. In doing so, we examine critical differences between classical and quantum gate operation.Comment: 10 pages (two-column). 1 figur