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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, Γ ..."
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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.
Uniform Asymptotic Expansions of Integrals: A Selection of Problems
, 1995
"... On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which ..."
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On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.
Uniform asymptotic analysis for waves in an incompressible elastic rod I. Disturbances superimposed on an initially stressfree state
, 1996
"... This paper studies the propagation of disturbances in an initially stressfree elastic circular rod. Starting from the MindlinHermann equations, we derive a fourthorder partial differential equation as the governing equation for small axialradial deformations in a rod composed of an incompressibl ..."
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This paper studies the propagation of disturbances in an initially stressfree elastic circular rod. Starting from the MindlinHermann equations, we derive a fourthorder partial differential equation as the governing equation for small axialradial deformations in a rod composed of an incompressible material. Then, we consider an initialvalue problem with an initial singularity in the shear strain. Using the technique of Fourier transforms, we manage to express the physical quantities in terms of integrals. The classical method of stationary phase is first used to obtain some important information. However, for material points in a neighbourhood behind the shearwave front, the phase function of the integrals has a stationary point which approaches positive infinity. Consequently, the classical method of stationary phase fails completely. Here, instead, we use a new method developed by us (Dai & Wong 1994 Wave Motion 19, 293308) to handle this case. An asymptotic expansion for the shear strain, which is uniformly valid in a neighbourhood behind the shearwave front, is derived. This uniform asymptotic expansion reveals that for the shear strain there is a transition from 0(1) disturbance to O(t~*) disturbance as the distance to the shearwave front increases. We also find that the shear strain has three other jumps in terms of asymptotic orders. The first jump is from a larger O(t~$) disturbance to a smaller O(t ~ ) disturbance behind and ahead of the barwave front. The second jump is from a larger 0(1) disturbance to a smaller O(t~l) disturbance behind and ahead of the barwave front. This also implies that in an asymptotic sense the initial singularity in the shear strain will be preserved and propagates with the shearwave speed as time progresses. The third jump is from a smaller O(f~') disturbance to a larger O(t~i) disturbance behind and ahead of a third wave front. 1.
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"... We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence int ..."
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We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interval of T symbol periods, after which they change to new independent values which they maintain for another T symbol periods, and so on. Computing the link capacity, associated with channel coding over multiple fading intervals, requires an optimization over the joint density of T M complex transmitted signals. We prove that there is no point in making the number of transmitter antennas greater than the length of the coherence interval: the capacity for M> Tis equal to the capacity for M = T. Capacity is achieved when the T M transmitted signal matrix is equal to the product of two statistically independent matrices: a T T isotropically distributed unitary matrix times a certain T M random matrix that is diagonal, real, and nonnegative. This result enables us to determine capacity for many interesting cases. We conclude that, for a fixed number of antennas, as the length of the coherence interval increases, the capacity approaches the capacity obtained as if the receiver knew the propagation coefficients. Index Terms—Multielement antenna arrays, wireless communications, spacetime modulation 1
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"... 1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathemati ..."
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1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the
4 Examples of Point Processes Associated to Weights
, 2008
"... We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2×2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scal ..."
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We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2×2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.
Cooperative Training in Wireless Sensor and Actor Networks
"... Abstract. Exploiting features of high density wireless sensor networks represents a challenging issue. In this work, the training of a sensor network which consists of anonymous and asynchronous sensors, randomly and massively distributed in a circular area around a more powerful device, called acto ..."
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Abstract. Exploiting features of high density wireless sensor networks represents a challenging issue. In this work, the training of a sensor network which consists of anonymous and asynchronous sensors, randomly and massively distributed in a circular area around a more powerful device, called actor, is considered. The aim is to partition the network area in concentric coronas and sectors, centered at the actor, and to bring each sensor autonomously to learn to which corona and sector belongs. The new protocol, called Cooperative, is the fastest training algorithm for asynchronous sensors, and it matches the running time of the fastest known training algorithm for synchronous sensors. Moreover, to be trained, each sensor stays awake only a constant number of time slots, independent of the network size, consuming very limited energy. The performances of the new protocol, measured as the number of trained sensors, the accuracy of the achieved localization, and the consumed energy, are also experimentally tested under different network density scenarios. Key words: : wireless sensor network, training, localization, distributed algorithms. 1.1