Integral points on Markoff type cubic surfaces

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if $V_{k}(\mathbb{Z}_p)$ is non-empty for all primes $p$, and that there are infinitely many $k$'s for which it fails. The Markoff morphisms act on $V_{k}(\mathbb{Z})$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.Comment: 57 pages, many figures, revised Introduction, Sec. 5 and Appendi

Real zeros of holomorphic Hecke cusp forms

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity

Trade and Expropriation: A Factor Proportions Approach

An extended small open economy model is developed and used to examine the effect of trade on the illicit expropriation of incomes and the provision of legal services. We derive conditions under which trade liberalization will reduce expropriation activities. We also derive sufficient conditions for the gains from trade to be amplified or muted relative to the standard model. The signs of these effects depend on factor intensity rankings and factor abundance ratios. Thus the results show that trade liberalization will be beneficial to countries that export labor intensive goods by reducing the incentives for illicit expropriation and reducing the costs of providing legal services. The model also shows that trade liberalization can increase expropriation, particularly for countries that import labor intensive goods and have labor intensive crime problems.Expropriation; Factor Proportions; Gains from Trade; Legal Services

Towards human technology symbiosis in the haptic mode

Search and rescue operations are often undertaken in dark and noisy environments in which rescue teams must rely on haptic feedback for exploration and safe exit. However, little attention has been paid specifically to haptic sensitivity in such contexts or to the possibility of enhancing communicational proficiency in the haptic mode as a life-preserving measure. Here we discuss the design of a haptic guide robot, inspired by careful study of the communication between blind person and guide dog. In the case of this partnership, the development of a symbiotic relationship between person and dog, based on mutual trust and confidence, is a prerequisite for successful task performance. We argue that a human-technology symbiosis is equally necessary and possible in the case of the robot guide. But this is dependent on the robot becoming 'transparent technology' in Andy Clark's sense. We report on initial haptic mode experiments in which a person uses a simple mobile mechanical device (a metal disk fixed with a rigid handle) to explore the immediate environment. These experiments demonstrate the extreme sensitivity and trainability of haptic communication and the speed with which users develop and refine their haptic proficiencies in using the device, permitting reliable and accurate discrimination between objects of different weights. We argue that such trials show the transformation of the mobile device into a transparent information appliance and the beginnings of the development of a symbiotic relationship between device and human user. We discuss how these initial explorations may shed light on the more general question of how a human mind, on being exposed to an unknown environment, may enter into collaboration with an external information source in order to learn about, and navigate, that environment

Density of states of colloidal glasses

Glasses are structurally liquid-like, but mechanically solid-like. Most attempts to understand glasses start from liquid state theory. Here we take the opposite point of view, and use concepts from solid state physics. We determine the vibrational modes of a colloidal glass experimentally, and find soft low-frequency modes that are very different in nature from the usual acoustic vibrations of ordinary solids. These modes extend over surprisingly large length scales

Active Brownian motion in a narrow channel

We review recent advances in rectification control of artificial microswimmers, also known as Janus particles, diffusing along narrow, periodically corrugated channels. The swimmer self-propulsion mechanism is modeled so as to incorporate a nonzero torque (propulsion chirality). We first summarize the effects of chirality on the autonomous current of microswimmers freely diffusing in channels of different geometries. In particular, left-right and upside-down asymmetric channels are shown to exhibit different transport properties. We then report new results on the dependence of the diffusivity of chiral microswimmers on the channel geometry and their own self-propulsion mechanism. The self-propulsion torque turns out to play a key role as a transport control parameter.Comment: to be published in Eur. Phys. J Special Topic

Variation in proteoglycan metabolism by articular chondrocytes in different joint regions is determined by post-natal mechanical loading

SummaryIn this study we investigated the hypothesis that cartilage from defined regions of ovine stifle joints, which were subjected to differing mechanical stresses, contained phenotypically distinct chondrocyte populations. Chondrocyte phenotypes were identified by the relative biosynthesis of the proteoglycans (PGs) aggrecan, biglycan and decorin. Articular cartilage (AC) from adult and neonatal ovine stifle joints were examined. Cells were cultured as both full-depth AC explants and in alginate beads after their isolation from the AC matrix. When chondrocytes from the various topographical regions of adult ovine knee joints were cultured as explants they demonstrated a consistent difference with regard to the metabolism of aggrecan and decorin. Significantly, this topographically-dependent phenotypic expression of PGs was preserved when the chondrocytes were cultured in alginate beads. In adult joints, chondrocytes from the central region of the tibial plateau not covered by the meniscus, which is subjected to high mechanical loads in-vivo, synthesized less aggrecan but more decorin than cells from regions covered by the meniscus. When chondrocytes from identical AC regions of neonatal ovine joints were cultured as explants, no topographical difference in aggrecan nor decorin metabolism could be detected. The results of this study, in association with the existing literature, lead us to propose that post-natal mechanical loading of AC could select for chondrocyte clones or induce a lasting modulation of chondrocyte phenotypic expression in different joint regions. Such cellular changes could result in the synthesis of PG populations that confer properties to AC most suited to resist the variable mechanical stresses in the different joint regions. This study serves to emphasize the importance of using cartilage from identical joint areas when examining PG metabolism by chondrocytes. Further investigation into the relationship between mechanical loading, regional chondrocyte phenotype selection and the response of these cells to anabolic and catabolic factors may provide important insights into the focal nature of AC degeneration in osteoarthritis

Geometric quantum gate for trapped ions based on optical dipole forces induced by Gaussian laser beams

We present an implementation of quantum logic gates via internal state dependent displacements of ions in a linear Paul trap caused by optical dipole forces. Based on a general quantum analysis of the system dynamics we consider specific implementations with alkaline earth ions. For experimentally realistic parameters gate infidelities as low as $10^{-4}$ can be obtained.Comment: 10 pages, 4 figure