An Efficient Geometric Search Algorithm of Pandemic Boundary Detection

We consider a scenario where the pandemic infection rate is inversely proportional to the power of the distance between the infected region and the non-infected region. In our study, we analyze the case where the exponent of the distance is 2, which is in accordance with Reilly’s law of retail gravitation. One can test for infection but such tests are costly so one seeks to determine the region of infection while performing few tests. Our goal is to find a boundary region of minimal size that contains all infected areas. We discuss efficient algorithms and provide the asymptotic bound of the testing cost and simulation results for this problem.

An asymptotic bound for the strong chromatic number

The strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

Asymptotic Bound of Secrecy Capacity for MIMOME-Based Transceivers: A Suboptimally Tractable Solution for Imperfect CSI

The paper principally proposes a suboptimally closed-form solution in terms of a general asymptotic bound of the secrecy capacity in relation to MIMOME-based transceivers. Such pivotal solution is essentially tight as well, fundamentally originating from the principle convexity. The resultant novelty, per se, is strictly necessary since the absolutely central criterion imperfect knowledge of the wiretap channel at the transmitter should also be highly regarded. Meanwhile, ellipsoidal channel uncertainty set-driven strategies are physically taken into consideration. Our proposed solution is capable of perfectly being applied for other general equilibria such as multiuser ones. In fact, this in principle addresses an entirely appropriate alternative for worst-case method-driven algorithms utilising some provable inequality-based mathematical expressions. Our framework is adequately guaranteed regarding a totally acceptable outage probability (as 1 − preciseness coefficient). The relative value is almost 10% for the estimation error values (EEVs) ⩽0.5 for 2×2-based transceivers, which is noticeably reinforced at nearly 5% for EEVs ⩽0.9 for the case 4×4. Furthermore, our proposed scheme basically guarantees the secrecy outage probability (SOP) less than 0.05% for the case of having EEVs ⩽0.3, for the higher power regime.

The critical dimension for -measures

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.