Running Newton Constant, Improved Gravitational Actions, and Galaxy Rotation Curves

A renormalization group (RG) improvement of the Einstein-Hilbert action is performed which promotes Newton's constant and the cosmological constant to scalar functions on spacetime. They arise from solutions of an exact RG equation by means of a ``cutoff identification'' which associates RG scales to the points of spacetime. The resulting modified Einstein equations for spherically symmetric, static spacetimes are derived and analyzed in detail. The modifications of the Newtonian limit due to the RG evolution are obtained for the general case. As an application, the viability of a scenario is investigated where strong quantum effects in the infrared cause Newton's constant to grow at large (astrophysical) distances. For two specific RG trajectories exact vacuum spacetimes modifying the Schwarzschild metric are obtained by means of a solution-generating Weyl transformation. Their possible relevance to the problem of the observed approximately flat galaxy rotation curves is discussed. It is found that a power law running of Newton's constant with a small exponent of the order $10^{-6}$ would account for their non-Keplerian behavior without having to postulate the presence of any dark matter in the galactic halo.Comment: 72pp, to appear in Phys. Rev.

Overseas Property: An Answer to the Pensions Crisis

Two centuries ago the founding father of rigorous economic analysis, David Ricardo, argued that land is unique type of asset, subject to forces quite different from those determining other prices; and he sought to show that these forces make land a uniquely good long-term investment. In life, he even practised what he preached, using the profits that he had accumulated as a London banker to buy a large estate in Norfolk. In modern economic, Ricardo has often been deride for his theory. The theory of ‘efficient markets’ tells them that the underlying returns on all kinds of assets, adjusted for risk, will tend to equate; so land is just one asset amongst many. But most modern English men and women, inexpert in economics, seem to be closet Ricardians in their behaviour, believing that money tied up in ‘bricks and mortar’ is especially secure and profitable. And they too practise what they preach, regarding property investment as the best way of providing for their old age. Thus in the past decade growing numbers of people on quite modest incomes have purchased second properties within Britain , using the rental income to cover the mortgage payments; and even greater numbers have purchased houses for their own use which are far larger than they would otherwise want or require, believing that they can provide for their retirement by ‘trading down’. In addition, perhaps two or three millions – the exact number is unknown – have bought properties abroad, not only as holiday homes, but as long-term investments. This paper will argue that Ricardo and his present-day disciples are right. More particularly, it will propose that overseas property offers one answer – and perhaps the only answer – to the crisis that now confronts UK pensions. The first part will analyze the underlying causes of the pensions crisis, pointing to five paradoxes in which retirement saving is trapped. The second part will show how overseas property offers an escape from these paradoxes. The final part will describe a particular way in which the current law relating to pensions can be used to facilitate such an escape

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

Double Poisson Cohomology of Path Algebras of Quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson-Lichnerowicz cohomology groups for three types of (linear and non-linear) double Poisson brackets on the free algebra in two variables. This allows us to develop some useful techniques for the computation of the double Poisson-Lichnerowicz cohomology.Comment: 42 pages. Final version, to appear in Journal of Algebr

Formal structures and representation spaces

M. Kapranov introduced and studied in math.AG/9802041 the noncommutative formal structure of a smooth affine variety. In this note we show that his construction is a special case of microlocalization and extend it in a functorial way to representation schemes of affine algebras. We describe the formal completion in the case of path algebras of quivers and initiate the study of their finite dimensional representations

Compound droplet manipulations on fiber arrays

Recent works demonstrated that fiber arrays may constitue the basis of an open digital microfluidics. Various processes, such as droplet motion, fragmentation, trapping, release, mixing and encapsulation, may be achieved on fiber arrays. However, handling a large number of tiny droplets resulting from the mixing of several liquid components is still a challenge for developing microreactors, smart sensors or microemulsifying drugs. Here, we show that the manipulation of tiny droplets onto fiber networks allows for creating compound droplets with a high complexity level. Moreover, this cost-effective and flexible method may also be implemented with optical fibers in order to develop fluorescence-based biosensor