In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using time-sharing to the strongest user, dirty paper coding (DPC), and beamforming when the number of users (receivers) n is large. Throughout the paper, we assume a fix av-erage transmit power and consider a block fading Rayleigh channel. First, we show that for a system with M transmit antennas and users equipped with N antennas, the sum rate scales like M log log nN for DPC and beamforming when M is fixed and for any N (either growing to in-finity or not). On the other hand, when both M and N are fixed, the sum rate of time-sharing to the strongest user scales like min(M, N) log log n. Therefore, the asymptotic gain of DPC over time-sharing for the sum rate is M min(M,N) when M and N are fixed. It is also shown that if M grows as log n, the sum rate of DPC and beamforming will grow linearly in M, but with different constant multiplicative factors. In this region, the sum rate capacity of time-sharing scales like N log log n.

In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems: Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3-dimensional region R of volume V, how large must the sensing range rSense be to ensure a given degree of coverage of the region to monitor? For a given transmission range rTrans, what is the minimum (resp. maximum) degree of the network? What is then the typical hop-diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks. Keywords Sensor networks, ad hoc networks; coverage, connectivity; hop-diameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.

Explicit formulae are given for the Hilbert transform Z R \Gamma w(t)dt=(t \Gamma x), where w is either the generalized Laguerre weight function w(t) = 0 if t 0, w(t) = t ff e \Gammat if 0 ! t ! 1, and ff ? \Gamma1, x ? 0, or the Hermite weight function w(t) = e \Gammat 2 , \Gamma1 ! t ! 1, and \Gamma1 ! x ! 1. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddle-point asymptotics, and series expansions. We also study the numerical stability of the three-term recurrence relation satisfied by the integrals Z R \Gamma n (t; w)w(t)dt=(t \Gamma x), n = 0; 1; 2; : : : , where n ( \Delta ; w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n. AMS subject classification: 65D30, 65D32, 65R10. Key words: Hilbert transform, classical weight functions, computational methods. 1

Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z |> c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z) | is of order |z | a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.

Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.

Some monotonicity and limit results for the regularised incomplete gamma function by Wojciech Chojnacki (Adelaide and Warszawa) Abstract. Letting P (u, x) denote the regularised incomplete gamma function, it is shown that for each α ≥ 0, P (x, x + α) decreases as x increases on the positive real semiaxis, and P (x, x + α) converges to 1/2 as x tends to infinity. The statistical significance