On the Displacement for Covering a d−d-dimensional Cube with Randomly Placed Sensors

Consider $n$ sensors placed randomly and independently with the uniform distribution in a $d-$dimensional unit cube ($d\ge 2$). The sensors have identical sensing range equal to $r$, for some $r >0$. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the $d-$dimensional unit cube is completely covered, i.e., every point in the $d-$dimensional cube is within the range of a sensor. If the $i$-th sensor is displaced a distance $d_i$, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the $a$-total movement defined as the sum $M_a:= \sum_{i=1}^n d_i^a$, for some constant $a>0$. We assume that $r$ and $n$ are chosen so as to allow full coverage of the $d-$dimensional unit cube and $a > 0$. The main contribution of the paper is to show the existence of a tradeoff between the $d-$dimensional cube, sensing radius and $a$-total movement. The main results can be summarized as follows for the case of the $d-$dimensional cube. If the $d-$dimensional cube sensing radius is $\frac{1}{2n^{1/d}}$ and $n=m^d$, for some $m\in N$, then we present an algorithm that uses $O\left(n^{1-\frac{a}{2d}}\right)$ total expected movement (see Algorithm 2 and Theorem 5). If the $d-$dimensional cube sensing radius is greater than $\frac{3^{3/d}}{(3^{1/d}-1)(3^{1/d}-1)}\frac{1}{2n^{1/d}}$ and $n$ is a natural number then the total expected movement is $O\left(n^{1-\frac{a}{2d}}\left(\frac{\ln n}{n}\right)^{\frac{a}{2d}}\right)$ (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations

A multiplier theorem for the Hankel transform

Sin resume

Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors

Consider $n$ mobile sensors placed randomly in $m-$dimensional unit cube for fixed $m\in\{1,2\}.$ The sensors have identical sensing range, say $r.$ We are interested in moving the sensors from their initial random positions to new locations so that every point in the unit cube is within the range of at least one sensor, while at the same time each pair of sensors is placed at interference distance greater or equal to $s.$ Suppose the displacement of the $i-$th sensor is a distance $d_i$. As a \textit{energy consumption} for the displacement of a set of $n$ sensors we consider the $a-$total displacement defined as the sum $\sum_{i=1}^n d_i^a,$ for some constant $a> 0.$ The main contribution of this paper can be summarized as follows. For the case of unit interval we \textit{explain a threshold} around the sensing radius equal to $\frac{1}{2n}$ and the interference distance equal to $\frac{1}{n}$ for the expected minimum $a-$total displacement. For the sensors placed in the unit square we \textit{explain a threshold} around the square sensing radius equal to $\frac{1}{2 \sqrt{n}}$ and the interference distance equal to $\frac{1}{\sqrt{n}}$ for the expected minimum $a-$total displacement

A multiplier theorem for the Hankel transform

Riesz function technique is used to prove a multiplier theorem for the Hankel transform, analogous to the classical Hörmander-Mihlin multiplier theorem [6]

On the Range Assignment in Wireless Sensor Networks for Minimizing the Coverage-Connectivity Cost

This article deals with reliable and unreliable mobile sensors having identical sensing radius r, communication radius R, provided that r ≤ R and initially randomly deployed on the plane by dropping them from an aircraft according to general random process. The sensors have to move from their initial random positions to the final destinations to provide greedy path k1-coverage simultaneously with k2-connectivity. In particular, we are interested in assigning the sensing radius r and communication radius R to minimize the time required and the energy consumption of transportation cost for sensors to provide the desired k1-coverage with k2-connectivity. We prove that for both of these optimization problems, the optimal solution is to assign the sensing radius equal to r = k1||E[S]||/2 and the communication radius R = k2||E[S]||/2, where ||E[S]|| is the characteristic of general random process according to which the sensors are deployed. When r\u3c k1||E[S]||/2 or R\u3c k2||E[S]||/ 2, and sensors are reliable, we discover and explain the sharp increase in the time required and the energy consumption in transportation cost to ensure the desired k1-coverage with k2-connectivity