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49
HopfAlgebraic Structures of Families of Trees
 J. Algebra
, 1987
"... this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to e#ciently compute certain di#erential operators. Given ..."
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Cited by 62 (17 self)
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this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to e#ciently compute certain di#erential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivations on an algebra, one expects to find a Hopf algebra of some sort. We were pleased to find that not only was there a Hopf algebra associated with these data structures, but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees. Previous work applying Hopf algebras to combinatorial objects (such as [10], [13] or [14]) has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets, rather than on trees themselves. # The first author is a National Science Foundation Postdoctoral Research Fellow
RungeKutta methods and renormalization
, 1999
"... A connection between the algebra of rooted trees used in renormalization theory and RungeKutta methods is pointed out. Butcher’s group and Bseries are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally Bseries are used to sol ..."
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Cited by 33 (0 self)
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A connection between the algebra of rooted trees used in renormalization theory and RungeKutta methods is pointed out. Butcher’s group and Bseries are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally Bseries are used to solve a class of nonlinear partial differential equations. 1
Conjunctive selection conditions in main memory
 In Proc. PODS
, 2002
"... We consider the fundamental operation of applying a compound filtering condition to a set of records. With large main memories available cheaply, systems may choose to keep the data entirely in main memory, in order to improve query and/or update performance. The design of a dataintensive algorithm ..."
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Cited by 26 (2 self)
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We consider the fundamental operation of applying a compound filtering condition to a set of records. With large main memories available cheaply, systems may choose to keep the data entirely in main memory, in order to improve query and/or update performance. The design of a dataintensive algorithm in main memory needs to take into account the architectural characteristics of modern processors, just as a diskbased method needs to consider the physical characteristics of disk devices. An important architectural feature that influences the performance of main memory algorithms is the branch misprediction penalty. We demonstrate that branch misprediction has a substantial impact on the performance of an algorithm for applying selection conditions. We describe a space of “query plans ” that are logically equivalent, but differ in terms of performance due to variations in their branch prediction behavior. We propose a cost model that takes branch prediction into account, and develop a query optimization algorithm that chooses a plan with optimal estimated cost for conjunctive conditions. We also develop an efficient heuristic optimization algorithm. We also show how records can be ordered to further reduce branch misprediction effects.
Convolution polynomials
 The Mathematica Journal 2,4 (Fall
, 1992
"... Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about ..."
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Cited by 21 (1 self)
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Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F0(x),F1(x),F2(x),... forms a convolution family if Fn(x) has degree ≤ n and if the convolution condition Fn(x + y) = Fn(x)F0(y) + Fn−1(x)F1(y) + · · · + F1(x)Fn−1(y) + F0(x)Fn(y) holds for all x and y and for all n ≥ 0. Many such families are known, and they appear frequently in applications. For example, we can let Fn(x) = x n /n!; the condition (x + y) n n! n∑ k=0 x k k! y n−k (n − k)! is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial coefficient () x n; the corresponding identity ( ) n∑ x + y x y n k n − k is commonly called Vandermonde’s convolution. k=0 How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 ≤ n ≤ 4: F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}]
Some Canonical Sequences of Integers
, 1995
"... Introduction to Combinatorial Analysis, Wiley, NY, 1958. [46] J. Riordan, Combinatorial Identities, Wiley, NY, 1968. [47] J. Riordan and C. E. Shannon, The number of twoterminal seriesparallel networks, J. Math. Phys., 21 (1942), 8393. Reprinted in C. E. Shannon, Collected Ppaers, edited N. J. A ..."
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Cited by 20 (3 self)
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Introduction to Combinatorial Analysis, Wiley, NY, 1958. [46] J. Riordan, Combinatorial Identities, Wiley, NY, 1968. [47] J. Riordan and C. E. Shannon, The number of twoterminal seriesparallel networks, J. Math. Phys., 21 (1942), 8393. Reprinted in C. E. Shannon, Collected Ppaers, edited N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993. [48] F. S. Roberts, Applied. Combinatorics, PrenticeHall, Englewood Cliffs, NJ, 1984. [49] D. G. Rogers, A Schroder triangle: three combinatorial problems, in Combinatorial Mathematics V (Melbourne 1976), ed. C. H. C. Little, Lecture Notes in math., Vol. 622 (1977), pp. 175196. [50] G.C. Rota, The number of partitions of a set, Amer. Math. Monthly, 71 (1964), 498504. [51] G.C. Rota et al., Finite Operator Calculus, Academic Press, NY, 1975. [52] M. Schader,
The Narayana Distribution
"... : The Narayana distribution is N n;k = 1 n i n k\Gamma1 ji n k j for 1 k n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining t ..."
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Cited by 19 (6 self)
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: The Narayana distribution is N n;k = 1 n i n k\Gamma1 ji n k j for 1 k n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the n th Narayana polynomial as N n (w) = P 1kn N n;k w k , for n 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schroder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schroder paths. keywords: Catalan numbers, lattice paths, Schroder numbers AMS Subject Classification: 05A15 email: sulanke@math.idbsu.edu Note to typesetter: the symbol L is a calligraphic L. It should be set as either a calligraphic L or a script L. Thanks. 1 2 1. introduction Using the steps V = (0; 1) and H = (1; 0), the set of Catalan paths, Cat n , is the set of lattice paths from (0; ...
Multicast Routing and Bandwidth Dimensioning in Overlay Networks
 IEEE Journal on Selected Areas in Communications
, 2002
"... Multicast services can be provided either as a basic network service or as an applicationlayer service. Higher level multicast implementations often provide more sophisticated features, and can provide multicast services at places where no network layer support is available. Overlay multicast netwo ..."
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Cited by 19 (2 self)
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Multicast services can be provided either as a basic network service or as an applicationlayer service. Higher level multicast implementations often provide more sophisticated features, and can provide multicast services at places where no network layer support is available. Overlay multicast networks offer an intermediate option, potentially combining the flexibility and advanced features of application layer multicast with the greater efficiency of network layer multicast. In this paper, we introduce the multicast routing problem specific to the overlay network environment and the related capacity assignment problem for overlay network planning. Our main contributions are the design of several routing algorithms that optimize the endtoend delay and the interface bandwidth usage at the multicast service nodes within the overlay network. The interface bandwidth is typically a key resource for an overlay network provider, and needs to be carefully managed in order to maximize the number of users that can be served. Through simulations, we evaluate the performance of these algorithms under various traffic conditions and on various network topologies. The results show that our approach is costeffective and robust under traffic variations.
The Cycle Lemma and Some Applications
, 1990
"... Two proofs of a frequently rediscovered combinatorial lemma are presented. Using the lemma, a combinatorial proof is given that the average height of an ordered (planeplanted) tree is approximately twice the average node (vertex) level. ..."
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Cited by 17 (2 self)
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Two proofs of a frequently rediscovered combinatorial lemma are presented. Using the lemma, a combinatorial proof is given that the average height of an ordered (planeplanted) tree is approximately twice the average node (vertex) level.
Counting rooted trees: The universal law t(n) cρ −n n −3/2
, 2005
"... Abstract. Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quic ..."
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Cited by 13 (1 self)
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Abstract. Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: Cρ −n n −3/2, where ρ is the radius of convergence of T. 1.
Popa M.: Feynman Diagrams and Wick products associated with qFock space
 Proc. Natl. Acad. Sci. USA 100
, 2003
"... Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1. ..."
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Cited by 11 (1 self)
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Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute qWick products and normal products in terms of each other. 1.