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17
A comparison of timesharing, DPC, and beamforming for MIMO broadcast channels with many users
 IEEE Trans. Commun
, 2007
"... In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using timesharing 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 average transmit power and ..."
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In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using timesharing 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 average 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 infinity or not). On the other hand, when both M and N are fixed, the sum rate of timesharing to the strongest user scales like min(M, N) log log n. Therefore, the asymptotic gain of DPC over timesharing 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 timesharing scales like N log log n.
Extremal properties of threedimensional sensor networks with applications
 IEEE Transactions on Mobile Computing
"... In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a cri ..."
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Cited by 20 (1 self)
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In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale 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 3dimensional 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 hopdiameter 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; hopdiameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.
On the uniform quantization of a class of sparse sources
 IEEE TRANS. ON INFORMATION THEORY
, 2009
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SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS
"... 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 wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesu ..."
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Cited by 5 (0 self)
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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 wellknown 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.
Computing The Hilbert Transform Of The Generalized Laguerre And Hermite Weight Functions
, 2000
"... 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 ! ..."
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Cited by 2 (1 self)
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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 saddlepoint asymptotics, and series expansions. We also study the numerical stability of the threeterm 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
Crandall’s computation of the incomplete Gamma Function and the Hurwitz Zeta Function with applications to Dirichlet Lseries
, 2014
"... This paper extends tools developed by Richard Crandall in [16] to provide robust, highprecision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet Lseries and cha ..."
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This paper extends tools developed by Richard Crandall in [16] to provide robust, highprecision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet Lseries and character polylogarithms.
Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335
"... 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 > ca − 1} with a real and z complex. Our results show that within the hyperboloid, ..."
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Cited by 1 (1 self)
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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 > ca − 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.
INEQUALITIES FOR 3LOGCONVEX FUNCTIONS
, 2008
"... ABSTRACT. This note gives a simple method for obtaining inequalities for ratios involving 3logconvex functions. As an example, an inequality for Wallis’s ratio of GautchiKershaw type is obtained. Inequalities for generalized means are also considered. ..."
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ABSTRACT. This note gives a simple method for obtaining inequalities for ratios involving 3logconvex functions. As an example, an inequality for Wallis’s ratio of GautchiKershaw type is obtained. Inequalities for generalized means are also considered.
ANNALES POLONICI MATHEMATICI
"... 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 sem ..."
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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
Asymptotic and exact series representations for the incomplete Gamma function
, 2005
"... Abstract. Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can b ..."
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Abstract. Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a, x) for any value of a or x. Applications of these formulas are discussed.