Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. Let $q_{1} = 2, q_{2} = 3, \ldots, q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann Hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n > 5040$ such that Robin inequality does not hold and $n < (4.48311)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$.

The use of Smart Antenna Systems (SAS) in pervasive environments such the Mobile Ad hoc Networks (MANET) has been promoted as the best choice to improve Spatial Division Multiple Access (SDMA) and throughput. Although directional communications are expected to provide great advantages in terms of network performance, directional MAC (Medium Access Control) protocols introduce several issues. One of the most known problems in this context is represented by the fact that, attempting to solve or at least mitigate the problems introduced by these kinds of antennas especially at MAC layer, a large amount of energy consumption is achieved ; for example, due to excessive retransmissions introduced by very frequently issue such as deafness and handoff. The expedients proposed in order to reduce these drawbacks attempting to limit beamforming time of nodes in cooperation with a Round-Robin scheduling, can grant high performance in terms of fairness and throughput. However the overall energy consumption in the network is not efficent due to the static approach. In view of this, we propose an Adaptive Beamforming Time with Round-Robin MAC providing for a dynamic assignment of the beamforming time with the purpose to limit the waste of energy of nodes.

Robin sequence is a congenital anomaly characterized by a triad of features that include micrognathia, glossoptosis, and airway obstruction. This comprehensive historical review maps the evolution of approaches and appliances for its treatment from the past until current modern possibilities of interdisciplinary combination of modern engineering, medicine, materials, and computer science combined approach with emphasis to design appliances inspired by nature - individual human anatomy. Current biomimetic designs are clinically applied resulting in appliances that are more efficient, comfortable, sustainable, and safer than legacy traditional designs. Review maps the treatment modalities that have been used for patients with Robin sequence over the years. Early management of Robin sequence focused primarily on airway maintenance and feeding support, while current management strategies involve both nonsurgical and surgical interventions and biomimetic biocompatible personalized appliances. The goal of this paper was to provide a review of the evolution of management strategies for patients with Robin sequence that led to the current interdisciplinary biomimetic approaches impacting the future of Robin Sequence treatment with biomimetics at the forefront.

In this study, we consider a Lambert series whose coefficients are the sum of divisors function. Utilizing the Lambert series in the sequel we introduce a normalized linear operator JR_(α,β) (z) by applying the convolution with Rabotnov function. We then, acquire sufficient conditions for JR_(α,β) (z) to be Univalent, Starlike and Convex respectively. In each component of this study, we expand the derived results by applying two Robin's inequalities, one of which is equivalent to the Riemann hypothesis.