English funding of the Scottish armies in England and Ireland 1640-1648

The rebellion against Charles I's authority that began in Edinburgh in 1637 involved the Scots in successive invasions of England and armed intervention in Ireland. Historians have almost universally taken a negative view of Scottish involvement in these wars, because it has been assumed that the Scottish political leadership sacrificed all other considerations in order to pursue an unrealistic religious crusade. This article suggests that aspects of the Anglo-Scottish relationship need to be reappraised. Using estimates of English payments to the Scots during the 1640s, it will be argued that the Scottish leadership made pragmatic political decisions based on a practical appreciation of the country's military and fiscal capacity. Substantial payouts from the English parliament enabled the Scottish parliamentary regime to engage in military and diplomatic activities that the country could not otherwise have afforded. The 1643 treaty that brought the Scots into the English Civil War on the side of parliament contrasts favourably with the 1647 Engagement in support of the king. It will be shown that, although the English parliament did not honour all of its obligations to the Scots, it does not automatically follow that the alliance was a failure in financial terms

Exploring arbitrarily high orders of optimized perturbation theory in QCD with nf -> 16.5

Perturbative QCD with nf flavours of massless quarks becomes simple in the hypothetical limit nf -> 16.5, where the leading beta-function coefficient vanishes. The Banks-Zaks (BZ) expansion in a0=(8/321)(16.5-nf) is straightforward to obtain from perturbative results in MSbar or any renormalization scheme (RS) whose nf dependence is `regular.' However, `irregular' RS's are perfectly permissible and should ultimately lead to the same BZ results. We show here that the `optimal' RS determined by the Principle of Minimal Sensitivity does yield the same BZ-expansion results when all orders of perturbation theory are taken into account. The BZ limit provides an arena for exploring optimized perturbation theory at arbitrarily high orders. These explorations are facilitated by a `master equation' expressing the optimization conditions in the fixed-point limit. We find an intriguing strong/weak coupling duality a -> a*^2/a about the fixed point a*.Comment: 32 pages, 4 figure

The lambda Phi^4 Miracle: lattice data and the zero-point potential

Recent lattice data for the effective potential of lambda Phi^4 theory fits the massless one-loop formula with amazing precision. Any corrections are at least 100 times smaller than is reasonable, perturbatively. This is strong evidence for the "exactness conjecture" of Consoli and Stevenson.Comment: 6 pages, LaTe