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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 Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 15 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
A symplectic test of the Lfunctions ratios conjecture
 Int. Math. Res. Notices, 2008, article ID rnm
"... ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and den ..."
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Cited by 9 (4 self)
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ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The Lfunctions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d ≤ X. For test functions supported in (−1/3, 1/3) we calculate all the lower order terms up to size O(X −1/2+ǫ) and observe perfect agreement with the conjecture (for test functions supported in (−1, 1) we show agreement up to errors of size O(X −ǫ) for any ǫ). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture’s prediction for the 1level density. 1.
Theory of Bergman spaces in the unit ball
 Memoires de la SMF 115, 2008. MR2537698 (2010g:32010), Zbl 1176.32001
"... ABSTRACT. There has been a great deal of work done in recent years on weighted Bergman spaces A p α on the unit ball Bn of C n, where 0 < p < ∞ and α> −1. We extend this study in a very natural way to the case where α is any real number and 0 < p ≤ ∞. This unified treatment covers all classical Berg ..."
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ABSTRACT. There has been a great deal of work done in recent years on weighted Bergman spaces A p α on the unit ball Bn of C n, where 0 < p < ∞ and α> −1. We extend this study in a very natural way to the case where α is any real number and 0 < p ≤ ∞. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space H 2, and the socalled Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk. 1.
On the analyticity of laguerre series
 Journal of Physics A: Mathematical and Theoretical
, 2008
"... The transformation of a Laguerre series f(z) = ∑ ∞ n=0 λ(α) n L (α) n (z) to a power series f(z) = ∑ ∞ n=0 γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a ..."
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Cited by 3 (1 self)
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The transformation of a Laguerre series f(z) = ∑ ∞ n=0 λ(α) n L (α) n (z) to a power series f(z) = ∑ ∞ n=0 γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patters of the Laguerre series coefficients λ (α) n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ (α) n either decay exponentially or factorially as n → ∞. The situation is much more complicated – but also much more interesting – if the λ (α) n decay only algebraically as n → ∞. If the
Averagecase analysis of approximate trie search
 IN PROC. 15TH SYMP. ON COMBINATORIAL PATTERN MATCHING (CPM), VOLUME 3109 OF LNCS
, 2004
"... ..."
Generalized jFactorial Functions, Polynomials, and Applications
"... The paper generalizes the traditional single factorial function to integervalued multiple factorial (jfactorial) forms. The generalized factorial functions are defined recursively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial fu ..."
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The paper generalizes the traditional single factorial function to integervalued multiple factorial (jfactorial) forms. The generalized factorial functions are defined recursively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial functions. The resulting coefficient triangles are similar to the classical sets of Stirling numbers and satisfy many analogous finitedifference and enumerative properties as the wellknown combinatorial triangles. The generalized triangles are also considered in terms of their relation to elementary symmetric polynomials and the resulting symmetric polynomial index transformations. The definition of the Stirling convolution polynomial sequence is generalized in order to enumerate the parametrized sets of jfactorial polynomials and to derive extended properties of the jfactorial function expansions. The generalized jfactorial polynomial sequences considered lead to applications expressing key forms of the jfactorial functions in terms of arbitrary partitions of the jfactorial function expansion triangle indices, including several identities related to
1/2BPS Correlators as c = 1 Smatrix
, 2006
"... We argue from two complementary viewpoints of Holography that the 2point correlation functions of 1/2BPS multitrace operators in the largeN (planar) limit are nothing but the (Wickrotated) Smatrix elements of c = 1 matrix model. On the bulk side, we consider an Euclideanized version of the so ..."
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We argue from two complementary viewpoints of Holography that the 2point correlation functions of 1/2BPS multitrace operators in the largeN (planar) limit are nothing but the (Wickrotated) Smatrix elements of c = 1 matrix model. On the bulk side, we consider an Euclideanized version of the socalled bubbling geometries and show that the corresponding droplets reach the conformal boundary. Then the scattering matrix of fluctuations of the droplets gives directly the twopoint correlators through the GKPW prescription. On the YangMills side, we show that the twopoint correlators of holomorphic and antiholomorphic operators are essentially equivalent with the transformation functions between asymptotic in and outstates of c = 1 matrix model. Extension to nonplanar case is also discussed.