Riemann zeros, prime numbers and fractal potentials

Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one-dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be $D=1.5$ for the Riemann zeros and $D = 1.8$ for the prime numbers. This result is somewhat surprising since the nearest-neighbour spacings of the Riemann zeros are known to be chaotically distributed whereas the primes obey almost poisson-like statistics. Our findings show that the fractal dimension is dependent on both the level-statistics and spectral rigidity, $\Delta_3$, of the energy levels.Comment: Five postscript figures included in the text. To appear in Phys. Rev.

The Effect of Various Supplementary Irrigation Treatments on Plant and Soil Moisture Relationships in a Vineyard (Vitis Vinif era var. Chenin Blanc)

In a field plot trial with grape vines planted on a high potential soil, the soil moisture status on different irrigation treatment plots was compared with that of a dryland control. Soil moisture budgeting was based on physical soil properties and root distribution patterns before treatments commenced. A maximum concentration of roots occurred at the 300-450 mm depth zone, while approximately 90 per cent of the total number of roots was found above 900 mm. Tensiometric readings of soil moisture potential, supplemented by electrometric resistance readings from gypsum blocks, indicated complete depletion between veraison and harvesting of the total available moisture on dryland plots. Despite this severe moisture stress as illustrated by chrono-isopletes, a reasonably good grape yield was still produced by the dryland vines, indicating either exceptional drought resistance or the uptake of water from extraneous sources. Supplementary irrigations greatly improved soil moisture conditions during the important growth stages of (a) flowering and fruit set, (b) green berry stage and (c) veraison. Moisture depletion patterns changed markedly as the season progressed from November to January, and as naturally stored rain water became depleted. In contrast to the absorption pattern in November, the fastest rate of water loss occurred in the deepest root zones during January, because of unsaturated moisture flow into the dry partly decomposed parent material. The apparently unproductive subsoil might act as a large natural reservoir for superfluous winter rain, which in summer becomes available to the vines through unsaturated upward flow into the root zone. Determination of leaf water potential showed unexpectedly high values at night (minima of -283kPa) in plants growing in soil of which the major part was dried below wilting point. Vines were thus able to regain turgidity at night. Crop factors for use with the American Class A-pan for scheduling irrigations were calculated. Crop factors were low (November to February = 0,20-0,30), and again emphasised the ability of vines to use water frugally. Crop factors were strongly dependent upon soil moisture conditions. Empirical coefficients for use in the Blaney-Criddle formula are also presented

Molecular analysis of a Saccharomyces cerevisiae mutant with improved ability to utilize xylose shows enhanced expression of proteins involved in transport, initial xylose metabolism, and the pentose phosphate pathway.

Differences between the recombinant xylose-utilizing Saccharomyces cerevisiae strain TMB 3399 and the mutant strain TMB 3400, derived from TMB 3399 and displaying improved ability to utilize xylose, were investigated by using genome-wide expression analysis, physiological characterization, and biochemical assays. Samples for analysis were withdrawn from chemostat cultures. The characteristics of S. cerevisiae TMB 3399 and TMB 3400 grown on glucose and on a mixture of glucose and xylose, as well as of S. cerevisiae TMB 3400 grown on only xylose, were investigated. The strains were cultivated under chemostat conditions at a dilution rate of 0.1 h-1, with feeds consisting of a defined mineral medium supplemented with 10 g of glucose liter-1, 10 g of glucose plus 10 g of xylose liter-1 or, for S. cerevisiae TMB 3400, 20 g of xylose liter-1. S. cerevisiae TMB 3400 consumed 31% more xylose of a feed containing both glucose and xylose than S. cerevisiae TMB 3399. The biomass yields for S. cerevisiae TMB 3400 were 0.46 g of biomass g of consumed carbohydrate-1 on glucose and 0.43 g of biomass g of consumed carbohydrate-1 on xylose. A Ks value of 33 mM for xylose was obtained for S. cerevisiae TMB 3400. In general, the percentage error was <20% between duplicate microarray experiments originating from independent fermentation experiments. Microarray analysis showed higher expression in S. cerevisiae TMB 3400 than in S. cerevisiae TMB 3399 for (i) HXT5, encoding a hexose transporter; (ii) XKS1, encoding xylulokinase, an enzyme involved in one of the initial steps of xylose utilization; and (iii) SOL3, GND1, TAL1, and TKL1, encoding enzymes in the pentose phosphate pathway. In addition, the transcriptional regulators encoded by YCR020C, YBR083W, and YPR199C were expressed differently in the two strains. Xylose utilization was, however, not affected in strains in which YCR020C was overexpressed or deleted. The higher expression of XKS1 in S. cerevisiae TMB 3400 than in TMB 3399 correlated with higher specific xylulokinase activity in the cell extracts. The specific activity of xylose reductase and xylitol dehydrogenase was also higher for S. cerevisiae TMB 3400 than for TMB 3399, both on glucose and on the mixture of glucose and xylose

Geometric scaling in the spectrum of an electron captured by a stationary finite dipole

We examine the energy spectrum of a charged particle in the presence of a {\it non-rotating} finite electric dipole. For {\emph{any}} value of the dipole moment $p$ above a certain critical value p_{\mathrm{c}}$ an infinite series of bound states arises of which the energy eigenvalues obey an Efimov-like geometric scaling law with an accumulation point at zero energy. These properties are largely destroyed in a realistic situation when rotations are included. Nevertheless, our analysis of the idealised case is of interest because it may possibly be realised using quantum dots as artificial atoms.Comment: 5 figures; references added, outlook section reduce

Simulations of Spinodal Nucleation in Systems with Elastic Interactions

Systems with long-range interactions quenched into a metastable state near the pseudospinodal exhibit nucleation that is qualitatively different than the classical nucleation observed near the coexistence curve. We have observed nucleation droplets in our Langevin simulations of a two-dimensional model of martensitic transformations and have determined that the structure of the nucleating droplet differs from the stable martensite structure. Our results, together with experimental measurements of the phonon dispersion curve, allow us to predict the nature of the droplet. These results have implications for nucleation in many solid-solid transitions and the structure of the final state

Nucleation in Systems with Elastic Forces

Systems with long-range interactions when quenced into a metastable state near the pseudo-spinodal exhibit nucleation processes that are quite different from the classical nucleation seen near the coexistence curve. In systems with long-range elastic forces the description of the nucleation process can be quite subtle due to the presence of bulk/interface elastic compatibility constraints. We analyze the nucleation process in a simple 2d model with elastic forces and show that the nucleation process generates critical droplets with a different structure than the stable phase. This has implications for nucleation in many crystal-crystal transitions and the structure of the final state

Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed

Magnetoplasmon excitations in arrays of circular and noncircular quantum dots

We have investigated the magnetoplasmon excitations in arrays of circular and noncircular quantum dots within the Thomas-Fermi-Dirac-von Weizs\"acker approximation. Deviations from the ideal collective excitations of isolated parabolically confined electrons arise from local perturbations of the confining potential as well as interdot Coulomb interactions. The latter are unimportant unless the interdot separations are of the order of the size of the dots. Local perturbations such as radial anharmonicity and noncircular symmetry lead to clear signatures of the violation of the generalized Kohn theorem. In particular, the reduction of the local symmetry from SO(2) to $C_4$ results in a resonant coupling of different modes and an observable anticrossing behaviour in the power absorption spectrum. Our results are in good agreement with recent far-infrared (FIR) transmission experiments.Comment: 25 pages, 6 figures, typeset in RevTe

Plume Characterization of a Typical South African Braai

To braai is part of the South African heritage that transcends ethnic barriers and socio-economic groups. In this paper, a comprehensive analysis of atmospheric gaseous and aerosol species within a plume originating from a typical South African braai is presented. Braai experiments were conducted at Welgegund – a comprehensively equipped regional background atmospheric air quality and climate change monitoring station. Five distinct phases were identified during the braai. Sulphur dioxide (SO2), nitrogen oxides(NOx) and carbonmonoxide (CO) increased significantly, while ozone (O3) did not increase notably. Aromatic and alkane volatile organic compounds were determined, with benzene exceeding the 2015 South African one-year ambient air quality limit. A comparison of atmospheric PM10 (particulate matter of an aerodynamic diameter ≤10 μm) concentrations with the 24-hour ambient limit indicated that PM10 is problematic during the meat grilling phase. From a climatic point of view, relatively high single scattering albedo (ωo) indicated a cooling aerosol direct effect, while periods with lowerωo coincided with peak black carbon (BC) emissions. The highest trace metal concentrations were associated with species typically present in ash. The lead (Pb) concentration was higher than the annual ambient air quality limit. Sulphate (SO4 2–), calcium (Ca2+) and magnesium (Mg2+) were the dominant water-soluble species present in the aerosols. The largest number of organic aerosol compounds was in the PM 2.5–1 fraction, which also had the highest semi-quantified concentration. The results indicated that a recreational braai does not pose significant health risks. However, the longer exposure periods that are experienced by occupational vendors, will significantly increase health risks.KEYWORDS Braai (barbeque), atmospheric gaseous species, aerosols, atmospheric organic compounds, optical properties, chemical properties