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74
On the Evolution of Random Graphs
 PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES
, 1960
"... his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_ ..."
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Cited by 1849 (7 self)
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his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_
Computing iceberg queries efficiently
 In Proc. of the 24th VLDB Conf
, 1998
"... Many applications compute aggregate functions... ..."
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 82 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Two notes on notation
 American Mathematical Monthly
, 1992
"... Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretic ..."
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Cited by 80 (2 self)
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Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretical emphases. Our mathematical language continues to improve, just as “the dism of Leibniz overtook the dotage of Newton ” in past centuries [4, Chapter 4]. In 1970 I began teaching a class at Stanford University entitled Concrete Mathematics. The students and I studied how to manipulate formulas in continuous and discrete mathematics, and the problems we investigated were often inspired by new developments in computer science. As the years went by we began to see that a few changes in notational traditions would greatly facilitate our work. The notes from that class have recently been published in a book [15], and as I wrote the final drafts of that book I learned to my surprise that two of the notations we had been using were considerably more useful than I had previously realized. The ideas “clicked ” so well, in fact, that I’ve decided to write this article, blatantly attempting to promote these notations among the mathematicians who have no use for [15]. I hope that within five years everybody will be able to use these notations in published papers without needing to explain what they mean.
The curious history of Faà di Bruno’s formula
 Amer. Math. Monthly
, 2002
"... the best known answer is Faà di Bruno’s Formula. If g and f are functions with a sufficient number of derivatives, then dm g ( f (t)) = dtm � m! b1! b2!···bm! g(k) � � ′ b1 � ′ ′ f (t) f (t) ( f (t)) ..."
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Cited by 46 (0 self)
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the best known answer is Faà di Bruno’s Formula. If g and f are functions with a sufficient number of derivatives, then dm g ( f (t)) = dtm � m! b1! b2!···bm! g(k) � � ′ b1 � ′ ′ f (t) f (t) ( f (t))
Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions
 Rev. Math Phys., Vol
, 1996
"... We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Sc ..."
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Cited by 17 (12 self)
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We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo–differential operators (− ∆ + m 2 0) −α for α ∈ (0,1) and m0> 0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X = Gα ∗ F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property.
On the distribution of the sum of n nonidentically distributed uniform random variables
 Annals of the Institute of Statistical Mathematics
"... integrals. Abstract. The distribution of the sum of independent identically distributed uniform random variables is wellknown. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit ..."
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Cited by 11 (1 self)
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integrals. Abstract. The distribution of the sum of independent identically distributed uniform random variables is wellknown. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulæ for the distribution of the sum of n nonidentically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulæ some special cases which have appeared in the literature.
Closed form summation of Cfinite sequences
 Transactions of the American Mathematical Society
, 2006
"... Abstract. We consider sums of the form n−1 ∑ j=0 ..."
Lattice Paths between Diagonal Boundaries
 Electronic J. Combinatorics
, 1998
"... A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n1])+# 1 (d[mr 1 ,ns 1 ]+d[ms 1 ,nr 1 ])++# k (d[mr k ,ns k ] +d[ms k ,nr k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetri ..."
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Cited by 7 (0 self)
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A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n1])+# 1 (d[mr 1 ,ns 1 ]+d[ms 1 ,nr 1 ])++# k (d[mr k ,ns k ] +d[ms k ,nr k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x + u<y<xl. With a solution we mean a formula that expresses d[m, n] as a sum of di#erences of recursions without the band restriction. Depending on the application, the boundary conditions can take di#erent forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x # 0, and d[x l, x] = 0 for all x # l (applies to the exact distribution of the KolmogorovSmirnov twosample statistic), d[x + u, x]=0 for all x # 0, and d[x  l +1,x]=d[xl+1,x1] for x # l (ordinary lattice paths with weighted left turns), and d[y, y  u +1]=d[y1,yu+1]for all y # u and d[x  l +1,x]=d[xl+1,x1] for x # l. The first theorem is a gene...