Finite-state Markov Chains obey Benford's Law

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.Comment: 31 pages, no figure

Influence of mutations of Val226 on the catalytic rate of haloalkane dehalogenase

Haloalkane dehalogenase converts haloalkanes to their corresponding alcohols. The 3D structure, reaction mechanism and kinetic mechanism have been studied. The steady state kcat with 1,2-dichloroethane and 1,2-dibromoethane is limited mainly by the rate of release of the halide ion from the buried active-site cavity. During catalysis, the halogen that is cleaved off (Clα) from 1,2-dichloroethane interacts with Trp125 and the Clβ interacts with Phe172. Both these residues have van der Waals contacts with Val226. To establish the effect of these interactions on catalysis, and in an attempt to change enzyme activity without directly mutating residues involved in catalysis, we mutated Val226 to Gly, Ala and Leu. The Val226Ala and Val226Leu mutants had a 2.5-fold higher catalytic rate for 1,2-dibromoethane than the wild-type enzyme. A pre-steady state kinetic analysis of the Val226Ala mutant enzyme showed that the increase in kcat could be attributed to an increase in the rate of a conformational change that precedes halide release, causing a faster overall rate of halide dissociation. The kcat for 1,2-dichloroethane conversion was not elevated, although the rate of chloride release was also faster than in the wild-type enzyme. This was caused by a 3-fold decrease in the rate of formation of the alkyl-enzyme intermediate for 1,2-dichloroethane. Val226 seems to contribute to leaving group (Clα or Brα) stabilization via Trp125, and can influence halide release and substrate binding via an interaction with Phe172. These studies indicate that wild-type haloalkane dehalogenase is optimized for 1,2-dichloroethane, although 1,2-dibromoethane is a better substrate.

Towards the re-verification of process tank calibrations

Re-verification is needed to ensure that the calibration (the relationship between measured level and measured volume) that is obtained during commissioning hasn’t changed over time. This can be achieved, in part, by metering in solution and correlating with marks identified a priori. Mark identification and correlation are discussed and possible error sources are outlined

The diurnal evolution of the urban heat island of Paris: a model-based case study during Summer 2006

The urban heat island (UHI) over Paris during summer 2006 was simulated using the Advanced Regional Prediction System (ARPS) updated with a simple urban parametrization at a horizontal resolution of 1 km. Two integrations were performed, one with the urban land cover of Paris and another in which Paris was replaced by cropland. The focus is on a five-day clear-sky period, for which the UHI intensity reaches its maximum. The diurnal evolution of the UHI intensity was found to be adequately simulated for this five day period. The maximum difference at night in 2 m temperature between urban and rural areas stemming from the urban heating is reproduced with a relative error of less than 10%. The UHI has an ellipsoidal shape and stretches along the prevailing wind direction. The maximum UHI intensity of 6.1 K occurs at 23:00 UTC located 6 km downstream of the city centre and this largely remains during the whole night. An idealized one-column model study demonstrates that the nocturnal differential sensible heat flux, even though much smaller than its daytime value, is mainly responsible for the maximum UHI intensity. The reason for this nighttime maximum is that additional heat is only affecting a shallow layer of 150 m. An air uplift is explained by the synoptic east wind and a ramp upwind of the city centre, which leads to a considerable nocturnal adiabatic cooling over cropland. The idealized study demonstrates that the reduced vertical adiabatic cooling over the city compared to cropland induces an additional UHI build-up of 25%. The UHI and its vertical extent is affected by the boundary-layer stability, nocturnal low-level jet as well as radiative cooling. Therefore, improvements of representing these boundary-layer features in atmospheric models are important for UHI studies

Drivers of land use change and household determinants of sustainability in smallholder farming systems of Eastern Uganda

Smallholder farming systems in sub-Saharan Africa have undergone changes in land use, productivity and sustainability. Understanding of the drivers that have led to changes in land use in these systems and factors that influence the systems’ sustainability is useful to guide appropriate targeting of intervention strategies for improvement. We studied low input Teso farming systems in eastern Uganda from 1960 to 2001 in a place-based analysis combined with a comparative analysis of similar low input systems in southern Mali. This study showed that policy-institutional factors next to population growth have driven land use changes in the Teso systems, and that nutrient balances of farm households are useful indicators to identify their sustainability. During the period of analysis, the fraction of land under cultivation increased from 46 to 78%, and communal grazing lands nearly completely disappeared. Cropping diversified over time; cassava overtook cotton and millet in importance, and rice emerged as an alternative cash crop. Impacts of political instability, such as the collapse of cotton marketing and land management institutions, of communal labour arrangements and aggravation of cattle rustling were linked to the changes. Crop productivity in the farming systems is poor and nutrient balances differed between farm types. Balances of N, P and K were all positive for larger farms (LF) that had more cattle and derived a larger proportion of their income from off-farm activities, whereas on the medium farms (MF), small farms with cattle (SF1) and without cattle (SF2) balances were mostly negative. Sustainability of the farming system is driven by livestock, crop production, labour and access to off-farm income. Building private public partnerships around market-oriented crops can be an entry point for encouraging investment in use of external nutrient inputs to boost productivity in such African farming systems. However, intervention strategies should recognise the diversity and heterogeneity between farms to ensure efficient use of these external inputs

Impacts of heterogeneity in soil fertility on legume-finger millet productivity, farmers ' targeting and economic benefits

Targeting of integrated management practices for smallholder agriculture in sub-Saharan Africa is necessary due to the great heterogeneity in soil fertility. Experiments were conducted to evaluate the impacts of landscape position and field type on the biomass yield, N accumulation and N2-fixation by six legumes (cowpea, green gram, groundnut, mucuna, pigeonpea and soyabean) established with and without P during the short rain season of 2005. Residual effects of the legumes on the productivity of finger millet were assessed for two subsequent seasons in 2006 in two villages in Pallisa district, eastern Uganda. Legume biomass and N accumulation differed significantly (P <0.001) between villages, landscape position, field type and P application rate. Mucuna accumulated the most biomass (4.8–10.9 Mg ha-1) and groundnut the least (1.0–3.4 Mg ha-1) on both good and poor fields in the upper and middle landscape positions. N accumulation and amounts of N2-fixed by the legumes followed a similar trend as biomass, and was increased significantly by application of P. Grain yields of finger millet were significantly (P <0.001) higher in the first season after incorporation of legume biomass than in the second season after incorporation. Finger millet also produced significantly more grain in good fields (0.62–2.15 Mg ha-1) compared with poor fields (0.29–1.49 Mg ha-1) across the two villages. Participatory evaluation of options showed that farmers preferred growing groundnut and were not interested in growing pigeonpea and mucuna. They preferentially targeted grain legumes to good fields except for mucuna and pigeonpea which they said they would grow only in poor fields. Benefit-cost ratios indicated that legume-millet rotations without P application were only profitable on good fields in both villages. We suggest that green gram, cowpea and soyabean without P can be targeted to good fields on both upper and middle landscape positions in both villages. All legumes grown with P fertiliser on poor fields provided larger benefits than continuous cropping of millet