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167
The stationary behavior of ideal TCP congestion avoidance
, 1996
"... This note derives the stationary behavior of idealized TCP congestion avoidance. More specifically, it derives the stationary distribution of the congestion window size if loss of packets are independentevents with equal probability. The mathematical derivation uses a fluid flow, continuous time, ap ..."
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Cited by 129 (2 self)
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This note derives the stationary behavior of idealized TCP congestion avoidance. More specifically, it derives the stationary distribution of the congestion window size if loss of packets are independentevents with equal probability. The mathematical derivation uses a fluid flow, continuous time, approximation to the discrete time process #W n #, where W n is the congestion window after the nth packet. We derive explicit results for the stationary distribution and all its moments. Congestion avoidance is the algorithm used by TCP to set its window size (and indirectly its data rate) under moderate to light segment (packet) losses. The congestion avoidance mechanism we model is idealized in the sense that loss of multiple packets does not lead to timeout phenomena. Such idealized behavior can be implemented using Selective Acknowledgements (SACKs). As such, our model predicts behavior of TCP with SACKs. It also is an approximate model in other situations. Among the results are that if eve...
OneDimensional Quantum Walks
 STOC'01
, 2001
"... We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, ..."
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Cited by 90 (11 self)
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We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [\Gamma t= p
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 78 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 55 (7 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
Analytic Combinatorics of Noncrossing Configurations
, 1997
"... This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic c ..."
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Cited by 55 (8 self)
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This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.
Analytica  A Theorem Prover for Mathematica
 The Mathematica Journal
, 1993
"... Analytica is an automatic theorem prover for theorems in elementary analysis. The prover is written in Mathematica language and runs in the Mathematica environment. The goal of the project is to use a powerful symbolic computation system to prove theorems that are beyond the scope of previous automa ..."
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Cited by 36 (1 self)
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Analytica is an automatic theorem prover for theorems in elementary analysis. The prover is written in Mathematica language and runs in the Mathematica environment. The goal of the project is to use a powerful symbolic computation system to prove theorems that are beyond the scope of previous automatic theorem provers. The theorem prover is also able to guarantee the correctness of certain steps that are made by the symbolic computation system and therefore prevent common errors like division by a symbolic expression that could be zero. In this paper we describe the structure of Analytica and explain the main techniques that it uses to construct proofs. We have tried to make the paper as selfcontained as possible so that it will be accessible to a wide audience of potential users. We illustrate the power of our theorem prover by several nontrivial examples including the basic properties of the stereographic projection and a series of three lemmas that lead to a proof of Weierstrass's...
On the CharneyDavis and NeggersStanley Conjectures
"... For a graded naturally labelled poset P, it is shown that the PEulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the NeggersStanley conjecture on real zeroes for W (P, t) in t ..."
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Cited by 29 (3 self)
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For a graded naturally labelled poset P, it is shown that the PEulerian polynomial ... counting linear extensions of P by their number of descents has symmetric and unimodal coefficient sequence, verifying the motivating consequence of the NeggersStanley conjecture on real zeroes for W (P, t) in these cases. The result is deduced from McMullen's gTheorem, by exhibiting a simplicial polytopal sphere whose hpolynomial is W (P, t). Whenever this...
Onedimensional quantum walks with absorbing boundaries. arXiv.org ePrint quantph/0207008
, 2002
"... In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use ..."
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Cited by 22 (2 self)
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In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D − 1)dimensional wall in a Ddimensional space. 1
Exponential Error Terms For Growth Functions On Negatively Curved Surfaces
 Amer. J. Math
, 1998
"... In this paper we consider two counting problems associated with compact negatively curved surfaces and improve classical asymptotic estimates due to Margulis. In the first, we show that the number of closed geodesics of length at most T has an exponential error term. In the second we show that the ..."
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Cited by 22 (12 self)
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In this paper we consider two counting problems associated with compact negatively curved surfaces and improve classical asymptotic estimates due to Margulis. In the first, we show that the number of closed geodesics of length at most T has an exponential error term. In the second we show that the number of geodesic arcs (between two fixed points x and y) of length at most T has an exponential error term. The proof is based on a detailed study of the zeta function and Poincare series and benefits from recent work of Dolgopiat. 0.