K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Polybius’ ‘previous landslide’: Proof that Hannibal’s invasion route crossed the col de la traversette

Both Polybius and Livy described a landslide/landslip that blocked the Punic Army’s exfiltration from a high col on the water divide in the Western Alps. The landslide, more aptly termed rockfall, has been a source of contention amongst classicists for centuries despite the fact that only two cols—Clapier and Traversette—exhibit rockfall debris on the lee side of the Alps. While the Clapier rockfall is too small and too young to have provided blockage, the Traversette debris is nearly as Polybius described it when he retraced the invasion route some 60 years after the event. His ‘two-tier’ description of the deposit, a doublet of younger and older rock rubble, including measurements of width and volume are close to modern measurements and prove that he knew, in advance, the route Hannibal had followed. It would take a practiced eye to correctly identify the stratigraphic complexity inherent in the Traversette Rockfall. Here we present weathering ratios, soil stratigraphic, mineral, chemical and microbiological evidence in support of Polybius’ observations as a considerable background database for future geoarchaeological exploration.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Optimal 5-step nilpotent quadratic algebras

By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.

Scaling of domain periodicity with thickness measured in BaTiO3 single crystal lamellae and comparison with other ferroics

The focused ion beam microscope has been used to cut parallel-sided {100}-oriented thin lamellae of single crystal barium titanate with controlled thicknesses, ranging from 530 nm to 70 nm. Scanning transmission electron microscopy has been used to examine domain configurations. In all cases, stripe domains were observed with {011}-type domain walls in perovskite unit-cell axes, suggesting 90 degrees domains with polarization in the plane of the lamellae. The domain widths were found to vary as the square root of the lamellar thickness, consistent with Kittel's law, and its later development by Mitsui and Furuichi and by Roytburd. An investigation into the manner in which domain period adapts to thickness gradient was undertaken on both wedge-shaped lamellae and lamellae with discrete terraces. It was found that when the thickness gradient was perpendicular to the domain walls, a continuous change in domain periodicity occurred, but if the thickness gradient was parallel to the domain walls, periodicity changes were accommodated through discrete domain bifurcation. Data were then compared with other work in literature, on both ferroelectric and ferromagnetic systems, from which conclusions on the widespread applicability of Kittel's law in ferroics were made.

The effects of the specific adsorption of anion on the reactivity of the Ru(0001) surface towards CO adsorption and oxidation: in situ FTIRS studies

The dynamics of adsorption and oxidation of CO on Ru(0001) electrode in sulfuric acid solution have been studied using in situ FTIR spectroscopy under potential control and at open circuit, the latter at 20 and 55 degrees C. The in situ IR data show clearly that the bisulfate anion adsorbs on the Ru(0001) surface over the potential range from -200 mV to 350 mV (vs. Ag/AgCl) at 20 degrees C in the absence and presence of adsorbed CO; however, increasing the temperature to 55 degrees C and/ or increasing the concentration of dissolved O-2 reduces the bisulfate adsorption. The formation of surface (hydro-) oxide at higher potentials replaces the bisulfate adsorbates. Both linear (COL) and three-fold hollow bonded CO (COH) adsorbates were produced following CO adsorption at Ru(0001) in H2SO4, as was observed in our previous studies in HClO4. However, the amount of adsorbed CO observed in H2SO4 was ca. 10% less than that in HClO4; in addition, the COL and COH frequencies were higher in H2SO4, and the onset potential for COads oxidation 25 mV lower. These new results are interpreted in terms of a model in which the adsorbed bisulfate weakens the CO adlayer, allowing the active Ru oxide layer to form at lower potentials. Significantly different results were observed at open circuit in H2SO4 compared both to the data under potential control and to our earlier data in HClO4, and these observations were rationalized in terms of the adsorbed HSO4- anions (pre-adsorbed at -200 mV) inhibiting the oxidation of the surface at open circuit (after stepping from the initial potential of -200 mV), as the latter was no longer driven by the imposed electrochemical potential but via chemical oxidation by trace dissolved O-2. Results from experiments at open circuit at 55 degrees C and using oxygen-saturated H2SO4 supported this model. The difference in Ru surface chemistry between imposed electrochemical control and chemical control has potential implications with respect to fuel cell electrocatalysis.

Drifting periodic structures in a degenerate-planar bent-rod nematic liquid crystal beyond the dielectric inversion frequency

We report on the electric-field-generated effects in the nematic phase of a twin mesogen formed of bent-core and calamitic units, aligned homeotropically in the initial ground state and examined beyond the dielectric inversion point. The bend-Freedericksz (BF) state occurring at the primary bifurcation and containing a network of umbilics is metastable; we focus here on the degenerate planar (DP) configuration that establishes itself at the expense of the BF state in the course of an anchoring transition. In the DP regime, normal rolls, broad domains, and chevrons (both defect-mediated and defect-free types) form at various linear defect-sites, in different regions of the frequency-voltage plane. A significant novel aspect common to all these patterned states is the sustained propagative instability, which does not seem explicable on the basis of known driving mechanisms.

Stability and dynamical behavior in a lake-model and implications for regime shifts in real lakes

A modified abstract version of the Comprehensive Aquatic Simulation Model (CASM) is found to exhibit three types of folded bifurcations due to nutrient loading. The resulting bifurcation diagrams account for nonlinear dynamics such as regime shifts and cyclic changes between clear-water state and turbid state that have actually been observed in real lakes. In particular, pulse-perturbation simulations based on the model presented suggest that temporal behaviors of real lakes after biomanipulations can be explained by pulse-dynamics in complex ecosystems, and that not only the amplitude (manipulated abundance of organisms) but also the phase (timing) is important for restoring lakes by biomanipulation. Ecosystem management in terms of possible irreversible changes in ecosystems induced by regime shifts is also discussed. (c) 2007 Elsevier B.V All rights reserved.

On the limits of spectral methods for frequency estimation

An algorithm is presented which generates pairs of oscillatory random time series which have identical periodograms but differ in the number of oscillations. This result indicates the intrinsic limitations of spectral methods when it comes to the task of measuring frequencies. Other examples, one from medicine and one from bifurcation theory, are given, which also exhibit these limitations of spectral methods. For two methods of spectral estimation it is verified that the particular way end points are treated, which is specific to each method, is, for long enough time series, not relevant for the main result.

Pattern formation of chevrons in the conduction regime in homeotropically aligned liquid crystals

We report on chevrons (herringbonelike patterns) observed in homeotropically aligned liquid crystals with high electric conductivity. We focus our attention on two types of chevrons observed in the conduction regime. The threshold voltage and the characteristic double periodicity of chevrons (i.e., the short wavelength lambda(1) of the striated rolls and the long wavelength lambda(2) Of the chevron bands) have been measured as functions of the applied electric frequency f. With the aid of a crossed polarizer set, we have, in addition, determined the director field which shows a periodic in-plane rotation for our chevrons (with a wavelength lambda(2)) We arrived at the types of chevrons after qualitatively different bifurcation sequences with increasing voltage. The frequency dependence of lambda(2) also shows a qualitatively different behavior with respect to the two types of chevrons. The experimental results are discussed in terms of recent theoretical investigations.

Nonlinear excitations in strongly-coupled plasma lattices: envelope solitons, kinks and intrinsic localized modes

Ensembles of charged particles (plasmas) are a highly complex form of matter, most often modeled as a many-body system characterized by weak inter-particle interactions (electrostatic coupling). However, strongly-coupled plasma configurations have recently been produced in laboratory, either by creating ultra-cold plasmas confined in a trap or by manipulating dusty plasmas in discharge experiments. In this paper, the nonlinear aspects involved in the motion of charged dust grains in a one-dimensional plasma monolayer (crystal) are discussed. Different types of collective excitations are reviewed, and characteristics and conditions for their occurrence in dusty plasma crystals are discussed, in a quasi-continuum approximation. Dust crystals are shown to support nonlinear kink-shaped supersonic solitary longitudinal excitations, as well as modulated envelope localized modes associated with longitudinal and transverse vibrations. Furthermore, the possibility for intrinsic localized modes (ILMs) — Discrete Breathers (DBs) — to occur is investigated, from first principles. The effect of mode-coupling is also briefly considered. The relation to previous results on atomic chains, and also to experimental results on strongly-coupled dust layers in gas discharge plasmas, is briefly discussed.