An effective Ratner equidistribution theorem for multiplicative Diophantine approximation on planar lines

In this paper, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$. This enables us to establish a strengthening of the Littlewood conjecture, valid for almost every point on a planar line, subject to a Diophantine condition. We also provide a complementary convergence theory, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock's from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.Comment: 32 pages. Comments are welcom

Making the decision to purchase health supplement products: a study of consumer behaviour

Health Supplement Products (HSPs), a form of Complementary and Alternative Medicine (CAM), have become increasingly popular in Western societies. Despite their widespread usage, the issue of why some people purchase HSPs and others do not is poorly understood. In part, this is due to limited research: few CAM studies have utilised consumer behaviour perspectives or have applied the Health Belief Model (HBM) in explaining and predicting the purchase of HSPs. A conceptual model and twelve research hypotheses were developed from the findings of preliminary qualitative research (in-depth interviews with 65 adults) and a review of the literature regarding CAM and the HBM. The criterion variable used in the conceptual model was the \u27likelihood of a consumer purchasing an HSP\u27. The predictor variables comprise \u27perceived value of HSPs\u27, \u27perceived general health threats\u27, \u27external cues to action\u27, five personality traits, and three socio-demographic variables

Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach

10.1109/TNN.2008.2003290IEEE Transactions on Neural Networks19111873-1886ITNN

High-throughput screening of encapsulated islets using wide-field lens-free on-chip imaging

Islet microencapsulation is a promising solution to diabetes treatment, but its quality control based on manual microscopic inspection is extremely low-throughput, highly variable and laborious. This study presents a high-throughput islet-encapsulation quality screening system based on lens-free on-chip imaging with a wide field-of-view of 18.15 cm^2, which is more than 100 times larger than that of a lens-based optical microscope, enabling it to image and analyze ~8,000 microcapsules in a single frame. Custom-written image reconstruction and processing software provides the user with clinically important information, such as microcapsule count, size, intactness, and information on whether each capsule contains an islet. This high-throughput and cost-effective platform can be useful for researchers to develop better encapsulation protocols as well as perform quality control prior to transplantation