On the Lambert W function
01 Dec 1996-Advances in Computational Mathematics (Baltzer Science Publishers, Baarn/Kluwer Academic Publishers)-Vol. 5, Iss: 1, pp 329-359
TL;DR: A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented.
Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we
w
. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.
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TL;DR: In this paper, an extension of sinc interpolation to algebraically decaying functions is presented, where the algebraic order of decay of a function's decay can be estimated everywhere in the horizontal strip of complex plane around the complex plane.
Abstract: An extension of sinc interpolation on $\mathbb{R}$ to the class of
algebraically decaying functions is developed in the paper. Similarly to the
classical sinc interpolation we establish two types of error estimates. First
covers a wider class of functions with the algebraic order of decay on
$\mathbb{R}$. The second type of error estimates governs the case when the
order of function's decay can be estimated everywhere in the horizontal strip
of complex plane around $\mathbb{R}$. The numerical examples are provided.
1,000 citations
07 Mar 2004
TL;DR: This work presents a new congestion control scheme that alleviates RTT unfairness while supporting TCP friendliness and bandwidth scalability, and uses two window size control policies called additive increase and binary search increase.
Abstract: High-speed networks with large delays present a unique environment where TCP may have a problem utilizing the full bandwidth. Several congestion control proposals have been suggested to remedy this problem. The existing protocols consider mainly two properties: TCP friendliness and bandwidth scalability. That is, a protocol should not take away too much bandwidth from standard TCP flows while utilizing the full bandwidth of high-speed networks. This work presents another important constraint, namely, RTT (round trip time) unfairness where competing flows with different RTTs may consume vastly unfair bandwidth shares. Existing schemes have a severe RTT unfairness problem because the congestion window increase rate gets larger as the window grows ironically the very reason that makes them more scalable. RTT unfairness for high-speed networks occurs distinctly with drop tail routers for flows with large congestion windows where packet loss can be highly synchronized. After identifying the RTT unfairness problem of existing protocols, This work presents a new congestion control scheme that alleviates RTT unfairness while supporting TCP friendliness and bandwidth scalability. The proposed congestion control algorithm uses two window size control policies called additive increase and binary search increase. When the congestion window is large, additive increase with a large increment ensures square RTT unfairness as well as good scalability. Under small congestion windows, binary search increase supports TCP friendliness. The simulation results confirm these properties of the protocol.
984 citations
01 Jan 2017
TL;DR: This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.
Abstract: This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.
934 citations
Cites background from "On the Lambert W function"
...The branches of W are described in [87], where also the fact that...
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19 Apr 2012TL;DR: In this paper, the authors introduce the study of differential difference equations and discuss some of the main features of the theory, and discuss the asymptotic behavior of solutions and the problem of stability.
Abstract: Publisher Summary A systematic development of the theory of differential–difference equations was not begun until E. Schimdt published an important paper about fifty years ago. The subsequent gradual growth of the field has been replaced, in the last decade or so, by a rapid expansion due to the stimulus of various applications. This chapter introduces the study of differential–difference equations and discusses some of the main features of the theory. The role of differential–difference equations is vital in some areas, such as engineering problem and fluid mechanics. In engineering problem, the problem of controlling the temperature in a reaction tank is addressed using differential difference equations. The temperature variation is reported because of random disturbances, inherent effects due to u being non-zero, and the operation of the control device. The chapter discusses the asymptotic behavior of solutions and the problem of stability.
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"On the Lambert W function" refers background in this paper
...essentially in terms of the W function (see equations (11) and (12) on page 104 of [22])....
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