Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

Can a Lattice String Have a Vanishing Cosmological Constant?

We prove that a class of one-loop partition functions found by Dienes, giving rise to a vanishing cosmological constant to one-loop, cannot be realized by a consistent lattice string. The construction of non-supersymmetric string with a vanishing cosmological constant therefore remains as elusive as ever. We also discuss a new test that any one-loop partition function for a lattice string must satisfy.Comment: 14 page

Maslov Indices and Monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived.Comment: 6 page

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page

Complete intersections: Moduli, Torelli, and good reduction

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.Comment: 37 pages. Typo's fixed. Expanded Section 2.

Pion-Muon Asymmetry Revisited

Long ago an unexpected and unexplainable phenomena was observed. The distribution of muons from positive pion decay at rest was anisotropic with an excess in the backward direction relative to the direction of the proton beam from which the pions were created. Although this effect was observed by several different groups with pions produced by different means, the result was not accepted by the physics community, because it is in direct conflict with a large set of other experiments indicating that the pion is a pseudoscalar particle. It is possible to satisfy both sets of experiments if helicity-zero vector particles exist and the pion is such a particle. Helicity-zero vector particles have direction but no net spin. For the neutral pion to be a vector particle requires an additional modification to conventional theory as discussed herein. An experiment is proposed which can prove that the asymmetry in the distribution of muons from pion decay is a genuine physical effect because the asymmetry can be modified in a controllable manner. A positive result will also prove that the pion is NOT a pseudoscalar particle.Comment: 9 pages, 3 figure

Asymptotics of orthogonal polynomials for a weight with a jump on [−1,1]

We consider the orthogonal polynomials on [-1, 1] with respect to the weight w(c)(x) = h(x)(1 - x)(alpha) (1+ x)beta Xi(c)(x), alpha, beta > -1, where h is real analytic and strictly positive on [-1, 1] and Xi(c) is a step-like function: Xi(c)(x) = 1 for x is an element of [-1, 0) and Xi(c) (x) = c(2), c > 0, for x is an element of [0, 1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in C, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n -> infinity. In particular, we prove for w(c) a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at x = 0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.Junta de Andalucía-Spain- FQM-229 and P06- FQM-01735.Ministry of Science and Innovation of Spain - MTM2008-06689-C02-01FCT -SFRH/BD/29731/200

Theory and practice – a case study of coordination and ownership in the Bangladesh health SWAp

BACKGROUND: In the past decade the sector-wide approach (SWAp) model has been promoted by donors and adopted by governments in several countries. The purpose of this study is to look at how partners involved in the health SWAp in Bangladesh define ownership and coordination, in their daily work and to analyse the possible implications of these definitions. METHODOLOGY: The study object was a process of decision-making in the Government of Bangladesh in 2003. Information was collected through participant observations, interviews and document review. RESULTS: During the study period the Government of Bangladesh decided to reverse a decision to unify the two wings of the Ministry of Health and Family Welfare. The decision led to disagreements with development partners, which had serious implications for cooperation between key actors in the Bangladesh health sector leading to deteriorated relationships and suspension of donor funds. The donor community in itself was also in disagreement which led to inconsistencies in the dialogue between the development partners and the Government of Bangladesh. CONCLUSION: The case shows that main actors in the Bangladesh health SWAp interpret ownership and coordination, fundamental aspects of SWAp, differently. As long as work ran smoothly, the different definitions did not create any problems, but when disagreements arose they became an obstacle. It is concluded that partners in development should devote more effort to their working relationships and that responsibilities within a SWAp need to be more clearly delineated

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat