Diophantine conditions and real or complex Brjuno functions

The continued fraction expansion of the real number x=a_0+x_0, a_0\in {\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN}, for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$ and the number $x$ satisfies the Brjuno diophantine condition whenever $B(x)$ is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value $+\infty$ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10 April 5-10, 1999 [SPhT-T99/116

Proof of the cases p≤7p \leq 7 of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture

It is shown that the polynomial $\lambda(t) = {\rm Tr}[(A + tB)^p]$ has nonnegative coefficients when $p \leq 7$ and A and B are any two complex positive semidefinite $n \times n$ matrices with arbitrary $n$. This proofs a general nontrivial case of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture which is a long standing problem in theoretical physics.Comment: 5 pages; typos corrected; accepted for publication in Journal of Statistical Physic

Policies to favour crop intensification and farm income under climatic risk in West Africa. [P-3330-32]

In West African countries, agricultural production per capita has decreased over the past half century. With continued population growth and the diminishing availability of marginal arable land, pressure on land is rapidly increasing and there is now a common view that crop yield must be increased in this region, especially as there is a wide gap between actual and potential yields. Although there are several factors which may explain this yield gap, the fact that agricultural production takes place in resource-constrained farm households exposed to risk is widely recognized as being important. Indeed, risk discourages the adoption of high-risk, high-return agricultural technologies, which in turn impedes the improvement of yields. In order to assess how climatic risk constrains intensification strategy in West Africa, we built and calibrated a bioeconomic farm simulation model predicting the choice to intensify crops or livestock as depending on the availability of key policies in the economic environment of farms, for typical cases in the groundnut basin of Senegal. These cases include two regions contrasted in terms of rainfall (Sine and Saloum) and in each region two typical farms, representing poor and less poor farmers. The model features uncertainty in weather (hence yields) and crop prices, farmer's risk aversion, nine cropping systems representing millet, maize and groundnut with various intensification levels, and the main interactions between crop and livestock: draught animal power, the feeding of animals with suitable crop products (groundnut haulms, cereal straw) and the production of farm manure. Farmers are constraint by land, labour, cash and credit availability. 180 households were surveyed to build the socio-demographic and economic dataset used by the model, and agronomic data were collected from 206 fields. These key policies analysed are (i) weather index insurances against drought impact on crop yields, either subsidised or not, (ii) subsidies to short term credit for purchasing farm inputs, (iii) subsidies to fertilizer, and (iv) direct payments to farmers. In our simulations, under the current climate and prices of agricultural products and inputs, all these policies appear favourable to the increase of farmers' expected utility for typical farms representing the vast majority of farms in the groundnut basin. Apart for insurance, all of them appear also favourable to intensification of coupled crop and livestock activities for those typical farms. Insurance appears favourable to this intensification strategy only for farms located in the northern part of the region studied, where climatic risk is higher. Among the scenario tested, for most typical farms, combining unsubsidized insurance with subsidized credit appeared as the best use of a given amount of public funds in support of crop intensification: subsidized credit allows the farmers to buy costly inputs while insurance reduces the risk that a drought prevents them from reimbursing the credit. Direct payments also rank high in this respect, because they efficiently mitigate the cash constraint. The amount of subsidies required to obtain a given increase of the value of farm production varies across farm types and subzones in the region, but is relatively reasonable. These results suggest that crop intensification is currently relatively close to becoming a relevant option for farmers and that public policies may favour it by improving the economic environment of farms. (Texte intégral

Confined spin waves reveal an assembly of nanosize domains in ferromagnetic La(1-x)CaxMnO3 (x=0.17,0.2)

We report a study of spin-waves in ferromagnetic La$_{1-x}$Ca$_{x}$MnO$_3$, at concentrations x=0.17 and x=0.2 very close to the metallic transition (x=0.225). Below T$_C$, in the quasi-metallic state (T=150K), nearly q-independent energy levels are observed. They are characteristic of standing spin waves confined into finite-size ferromagnetic domains, defined in {\bf a, b) plane for x=0.17 and in all q-directions for x=0.2. They allow an estimation of the domain size, a few lattice spacings, and of the magnetic coupling constants inside the domains. These constants, anisotropic, are typical of an orbital-ordered state, allowing to characterize the domains as "hole-poor". The precursor state of the CMR metallic phase appears, therefore, as an assembly of small orbital-ordered domains.Comment: 4 pages, 5 figure

Experimental approximation of the Jones polynomial with DQC1

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors corrected, slight changes to improve readabilit

Seasonal variation in species composition and frequency of insecticide resistance alleles (kdr and ace-1r) in the Anopheles gambiae complex from an irrigated rice fields area in Western Burkina Faso

Monitoring of the spread of insecticide resistance in field vector populations is a prerequisite for the implementation of efficient and sustainable vector control strategies based on the use of insecticides. Screening for resistance alleles in Anopheles gambiae populations is facilitated by the availability of molecular diagnostics to detect major target-site mutations, such as knock-down resistance (kdr) and insensitive acetylcholinesterase (ace-1R). Anopheles gambiae mosquitoes were collected resting indoors in two villages within a rice cultivation area in western Burkina Faso, from January to December 2007. Specimens were identified to species and molecular form and their genotype at the kdr and ace-1 locus was determined using PCR and RFLP protocols. The M form was largely predominant in our samples and was present all year round in both villages. S-form mosquitoes gradually appeared during the rainy season in the village at the margins of the rice fields (VK7) whereas it was very rare in the center of the rice cultivation area (VK5) throughout the survey. The frequency of both kdr and ace-1R mutations was higher in the S than in the M form at any time. In the M form, frequency of the kdr mutation was higher during the rainy season in both villages (P&lt;0.005). We report occurrence of the ace-1R mutation in the M form, albeit at a low frequency (&lt;1%). Our results highlight the preoccupying status of insecticide resistance in An. gambiae populations from Burkina Faso, and suggest that comprehensive monitoring strategies need to consider population dynamics. (Texte intégral

Financial Constraints and Girls' Secondary Education: Evidence from School Fee Elimination in The Gambia

We assess the impact of large-scale fee elimination for secondary school girls in The Gambia on the quantity, composition, and achievement of students. The gradual rollout of the program across geographic regions provides identifying variation in the policy. The program increased access to secondary education substantially without harming learning outcomes. We find an increase of around 50% in the number of girls and boys taking the high school exit exam from a low baseline, as well as a 0.1 standard deviations gain in test scores in response to the program. This result is notable in a setting where expanded access could put additional strains on limited resources and the quality of schools. These findings suggest that financial constraints remain serious barriers to post-primary education and that efforts to expand access to secondary education need not come at the expense of learning in low-income countries like The Gambia