Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the $j$-function (which defines the $\hat A$-class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain a deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper.Comment: 44 pages, 11 figures, some minor updates from the previous versio

Function-led design of multifunctional stimuli-responsive superhydrophobic surface based on hierarchical graphene-titania nanocoating

Multifunctional smart superhydrophobic surface with full-spectrum tunable wettability control is fabricated through the self-assembly of the graphene and titania nanofilm double-layer coating. Advanced microfluidic manipulative functions, including directional water transport, adhesion & spreading controls, droplet storage & transfer, and droplet sensing array, can be readily realized on this smart surface. An in-depth mechanism study regarding the underlying secrets of the tunable wettability and the UV-induced superhydrophilic conversion of anatase titania are also presented

Group Learning and Opinion Diffusion in a Broadcast Network

We analyze the following group learning problem in the context of opinion diffusion: Consider a network with $M$ users, each facing $N$ options. In a discrete time setting, at each time step, each user chooses $K$ out of the $N$ options, and receive randomly generated rewards, whose statistics depend on the options chosen as well as the user itself, and are unknown to the users. Each user aims to maximize their expected total rewards over a certain time horizon through an online learning process, i.e., a sequence of exploration (sampling the return of each option) and exploitation (selecting empirically good options) steps. Within this context we consider two group learning scenarios, (1) users with uniform preferences and (2) users with diverse preferences, and examine how a user should construct its learning process to best extract information from other's decisions and experiences so as to maximize its own reward. Performance is measured in {\em weak regret}, the difference between the user's total reward and the reward from a user-specific best single-action policy (i.e., always selecting the set of options generating the highest mean rewards for this user). Within each scenario we also consider two cases: (i) when users exchange full information, meaning they share the actual rewards they obtained from their choices, and (ii) when users exchange limited information, e.g., only their choices but not rewards obtained from these choices