Two notes on notation

By Donald Ervin Knuth. 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 theoretical emphases. Our mathematical language continues to improve, just as "the d-ism 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 notations I'm talking about are (1) Iverson's convention for characteristic functions; and (2) the "right" notation for Stirling numbers, at last.

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Citations in the corpus

A practical tutorial on autoencoders for nonlinear feature fusion: Taxonomy, models, software and guidelines (2018, background): “Note that square brackets denote Iverson’s convention [46] and evaluate to 0 or 1 according to the truthiness of the proposition.”
Attribute-Based Classification for Zero-Shot Visual Object Categorization (methodology): “At test time, we assume that every class z induces its attribute vector a z in a deterministic way, i.e. p(a|z) = a = a z , where we have made use of Iverson's bracket notation [11]: P = 1 if the condition P is true and it is 0 otherwise. as the representation of the attribute–class layer.”
Information, Divergence and Risk for Binary Experiments (2009, background): “The relevant definitions and theory of convex functions will be introduced as required.”
Convolution identities and lacunary recurrences for Bernoulli numbers (2007, methodology): “In spite of some advantages to the bracket notation used in the book [21] (see also [29]), we chose the competing notation S(n, k) as used, for instance, in the book [10] or in recent papers such as [16] or [17].”
Learning to detect unseen object classes by between-class attribute transfer (2009, methodology): “p(a|z) = Ja = azK, making use of Iverson’s bracket notation: JP K = 1 if the condition P is true and it is 0 otherwise [19].”
Composite Binary Losses (2009, methodology): “When η : X → [0, 1] is an observation-conditional density, taking the M average of the point-wise risk gives the (full) risk of the estimator v, now interpreted as (1)This is the Iverson bracket notation as recommended by Knuth [27].”
Understanding Random Forests: From Theory to Practice (2014, methodology): “…the node samples Lt. Assuming distinct input values for all Nt node samples, the number of partitions of Lt into k non-empty subsets is given by the Stirling number of the second kind [Knuth, 1992] S(Nt,k) = 1 k! k∑ j=0 (−1)k−j ( k j ) jNt , (3.12) which reduces to 2Nt−1 − 1 for binary partitions.”
Fitting combinations of exponentials to probability distributions (2007, background): “(N.B. Knuth (1992) claims that Pochhammer did not use this symbol to mean the above ascending factorial, but rather the binomial coefficient ( z n ) .)”
Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals (2009, methodology): “rivatives The iteration formula (2.3) shows that all polynomials have positive integer coefficients. These coefficients are the Stirling numbers of the second kind (or ) - see [12], [14], [17], [22], [26], [35]. Given a set of elements, represents the number of ways this set can be partitioned into ”
On the Consistency of Ordinal Regression Methods (2014, methodology): “the components are increasing, that is, S:= n : 2Rk 1 and i i+1 for 1 ik 2 o : 2 pdenotes the p-dimensional simplex, which is dened as p:= 8 < : x2Rp: x i0 and Xp i=1 x i= 1 9 = ; : Following Knuth (1992) we use the Iverson bracket JK as JqK := ( 1 if q is true 0 otherwise : We will also make ref”