Effect of the APD Receiver’s Tilted Angle on Channel Capacity for Underwater Wireless Optical Communications

This paper focuses on investigating the effect of the receiver’s tilted angle on the channel capacity of an underwater wireless optical communication (UWOC) system, in which an avalanche photodiode (APD) detector is adopted as the receiver. Under the non-negativity, peak power, and average power constraints, the lower bounds on the capacity of UWOC are derived in detail according to different average-to-peak power ratios. With modeling achieving the maximum of the lower bounds of the capacity as an optimization object, we prove that the proposed optimization issue is in fact a simple convex optimization about the tilted angle of the APD receiver, and then present related theoretical solution for it. Both theoretical analysis and simulation results show that by appropriately tilting the receiver, we can significantly enhance the final capacity performance of the UWOC with APD receiver.

The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and β--athermality.

Continuous-domain assignment flows

Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

Identifying Hazardous Shapes in the Plane

This paper explores the problem of identifying the shapes of invisible hazardous entities in R2 by a set S = {s1, s2, . . . , sk} of mobile sensors (autonomous robots). A hazardous entity, H, is a region that affects the operation of robots that either penetrate the area or come in contact with it. In this paper, we propose algorithms for searching a rectangular region for a stationary hazardous entity, where some a priori geometrical knowledge is given (e.g., edge size range), and if such an entity exists, then determine the area that it occupies. We explore entities that are convex in nature such as line segment, circles (discs), and simple convex shapes. The objectives are to minimize the distance travelled by the robots during the search phase, and to minimize the number of robots that are required to identify the region covered by the hazardous entity. The number of robots required to locate H is three or four robots when H is a line segment, two or three robots when H is a circle, and seven robots are sufficient when H is a triangle. Our results extend to n-vertex convex shapes and we show that 2n + 1 robots are sufficient to determine the coverage of H.

The Kinematics of Containment for N-Dimensional Ellipsoids

Knowing the set of allowable motions of a convex body moving inside a slightly larger one is useful in applications such as automated assembly mechanisms, robot motion planning, etc. The theory behind this is called the “kinematics of containment (KC).” In this article, we show that when the convex bodies are ellipsoids, lower bounds of the KC volume can be constructed using simple convex constraint equations. In particular, we study a subset of the allowable motions for an n-dimensional ellipsoid being fully contained in another. The problem is addressed in both algebraic and geometric ways, and two lower bounds of the allowable motions are proposed. Containment checking processes for a specific configuration of the moving ellipsoid and the calculations of the volume of the proposed lower bounds in the configuration space (C-space) are introduced. Examples for the proposed lower bounds in the 2D and 3D Euclidean space are implemented, and the corresponding volumes in C-space are compared with different shapes of the ellipsoids. Practical applications using the proposed theories in motion planning problems and parts-handling mechanisms are then discussed.

Gradient Descent is a Technique for Learning to Learn

In machine learning, the transition from hand-designed features to learned features has been a huge success. Regardless, optimization methods are still created by hand. In this study, we illustrate how an optimization method's design can be recast as a learning problem, allowing the algorithm to automatically learn to exploit structure in the problems of interest. On the tasks for which they are taught, our learning algorithms, implemented by LSTMs, beat generic, hand-designed competitors, and they also adapt well to other challenges with comparable structure. We show this on a variety of tasks, including simple convex problems, neural network training, and visual styling with neural art.

Projective bundles over small covers and the bundle triviality problem

The present paper investigates the projective bundles over small covers. We first give a necessary and sufficient condition for the projectivization of a real vector bundle over a small cover to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover by introducing a new characteristic function on simple convex polytopes. As an application, we characterize the real projective bundles over 2-dimensional small covers by interpreting the fiber sum operation to some combinatorial operation. We next determine when the projectivization of Whitney sum of the tautological line bundle and the tangent bundle over real projective space is diffeomorphic to the product of two real projective spaces. This answers an open question regarding the topology of the fiber of the Monster-Semple tower.