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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.
Joint Iterative Decoding of LDPC Codes and Channels with Memory
, 2003
"... This paper considers the joint iterative decoding of irregular lowdensity paritycheck (LDPC) codes and channels with memory. It begins by introducing a new class of erasure channels with memory, known as generalizederasure channels. For these channels, a single parameter recursion for the density ..."
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Cited by 5 (2 self)
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This paper considers the joint iterative decoding of irregular lowdensity paritycheck (LDPC) codes and channels with memory. It begins by introducing a new class of erasure channels with memory, known as generalizederasure channels. For these channels, a single parameter recursion for the density evolution of the joint iterative decoder is derived. This provides a necessary and sucient condition for decoder convergence, and allows the algebraic construction of sequences of LDPC degree distributions. Under certain conditions, these sequences can achieve the symmetric information rate (SIR) of the channel using only iterative decoding. Example code sequences are given for two channels, and it is conjectured that they each achieve the respective SIR. Keywords: joint iterative decoding, erasure channel, capacityachieving, LDPC codes 1.
Complete monotonicity of some functions involving polygamma functions, submitted
"... Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) ..."
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Cited by 3 (3 self)
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Abstract. In the present paper, we establish necessary and sufficient conditions for the functions x α ˛ ˛ψ (i) (x + β) ˛ ˛ and α ˛ ˛ψ (i) (x + β) ˛ ˛ − x ˛ ˛ψ (i+1) (x + β) ˛ ˛ respectively to be monotonic and completely monotonic on (0, ∞), where i ∈ N, α> 0 and β ≥ 0 are scalars, and ψ (i) (x) are polygamma functions. 1.
MSC: Primary 33B15; Secondary 41A60
, 2009
"... Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function ..."
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Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function
DERIVATION OF INDEX THEOREM BY SUPERSYMMETRY
, 804
"... Abstract. In order to evaluate some quantities in Gaussian ensembles, we investigate ”vandermonde ensemble ” which represents the distribution of eigenvalues of GUE in the phase space. We derive the integral equation of the level density of vandermonde ensemble and we compute the level density numer ..."
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Abstract. In order to evaluate some quantities in Gaussian ensembles, we investigate ”vandermonde ensemble ” which represents the distribution of eigenvalues of GUE in the phase space. We derive the integral equation of the level density of vandermonde ensemble and we compute the level density numerically. 1.
A NEW PROOF OF SEMICIRCLE LAW OF FIXED TRACE SQUARE ENSEMBLE
, 804
"... Abstract. In the present paper, we give a simple proof of the level density of fixed trace square ensemble. We derive the integral equation of the level density of fixed trace square ensemble.Then we analyze the asymptotic behavior of the level density. 1. ..."
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Abstract. In the present paper, we give a simple proof of the level density of fixed trace square ensemble. We derive the integral equation of the level density of fixed trace square ensemble.Then we analyze the asymptotic behavior of the level density. 1.