A zero-sqrt(5)/ 2 law for cosine families

Let $a \in \R,$ and let $k(a)$ be the largest constant such that sup\vert cos(na)-cos(nb)\vert \textless{} k(a) for $b\in \R$ implies that $b \in \pm a+2\pi\Z.$ We show that if a cosine sequence $(C(n))\_{n\in \Z}$ with values in a Banach algebra $A$ satisfies sup\_{n\ge 1}\Vert C(n) -cos(na).1\_A\Vert \textless{} k(a), then $C(n)=cos(na)$ for $n\in \Z.$ Since ${\sqrt 5\over 2} \le k(a) \le {8\over 3\sqrt 3}$ for every $a \in \R,$ this shows that if some cosine family $(C(g))\_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies sup\_{g\in G}\Vert C(g)-c(g)\Vert \textless{} {\sqrt 5\over 2} for some scalar cosine family $(c(g))\_{g\in G},$ then $C(g)=c(g)$ for $g\in G,$ and the constant ${\sqrt 5\over 2}$ is optimal. We also describe the set of all real numbers $a \in [0,\pi]$ satisfying $k(a)\le {3\over 2}.

From Specifications to Behavior: Maneuver Verification in a Semantic State Space

To realize a market entry of autonomous vehicles in the foreseeable future, the behavior planning system will need to abide by the same rules that humans follow. Product liability cannot be enforced without a proper solution to the approval trap. In this paper, we define a semantic abstraction of the continuous space and formalize traffic rules in linear temporal logic (LTL). Sequences in the semantic state space represent maneuvers a high-level planner could choose to execute. We check these maneuvers against the formalized traffic rules using runtime verification. By using the standard model checker NuSMV, we demonstrate the effectiveness of our approach and provide runtime properties for the maneuver verification. We show that high-level behavior can be verified in a semantic state space to fulfill a set of formalized rules, which could serve as a step towards safety of the intended functionality.Comment: Published at IEEE Intelligent Vehicles Symposium (IV), 201

Lower estimates near the origin for functional calculus on operator semigroups

This paper provides sharp lower estimates near the origin for the functional calculus F(-uA) of a generator A of an operator semi- group defined on the (strictly) positive real line; here F is given as the Laplace transform of a measure or distribution. The results are linked to the existence of an identity element or an exhaustive sequence of idempotents in the Banach algebra generated by the semigroup. Both the quasinilpotent and non-quasinilpotent cases are considered, and sharp results are proved extending many in the literature

Centralised, Decentralised, and Self-Organised Coverage Maximisation in Smart Camera Networks

When maximising the coverage of a camera network, current approaches rely on a central approach and rarely consider the decentralised or even self-organised potential. In this paper, we study the performance of decentralised and self-organised approaches in comparison to centralised ones in terms of geometric coverage maximisation. We present a decentralised and self-organised algorithm to maximise coverage in a camera network using a Particle Swarm Optimiser (PSO) and compare them to a centralised version of PSO. Additionally, we present a decentralised and self-organised version of ARES, a centralised approximation algorithm for optimal plans combining PSO, Importance Splitting, and an adaptive receding horizons at its core. We first show the benefits of ARES over using PSO as a single, centralised optimisation algorithm when used before deployment time. Second, since cameras are not able to change instantaneously, we investigate gradual adaptation of individual cameras during runtime. Third, we compare achieved geometrical coverage of our decentralised approximation algorithm against the centralised version of ARES. Finally, we study the benefits of a self-organised version of PSO and ARES, allowing the system to improve its coverage over time. This allows the system to deal with quickly unfolding situations

CHAINMAIL:Distributed coordination for multi-task κ-Assignment using autonomous mobile IoT devices

The Internet-of-Things (IoT) becomes more and more pervasive and supports us in our daily activities. However, when individual devices struggle in accomplishing certain tasks, they have to cooperate in order to achieve desired outcomes. In the absence of a central controller, devices have to coordinate autonomously within the network in order to attain and complete as many tasks as possible. We propose CHAINMAIL, a novel, distributed approach to coordinate sensors to attain tasks that cannot be accomplished by single, but only by cooperation of multiple devices. We demonstrate our approach with an IoT case study on multi-object k-coverage with autonomously operating mobile cameras and show that our approach does not over-provision tasks, allowing the remaining devices to attain other duties. This enables our network to provision more tasks in the same time as other comparable solutions