Showing papers in "American Mathematical Monthly in 1960"

Linear Operators. Part I: General Theory.

Abstract: This classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician—treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes.

Finite Markov Chains.

Abstract: Markov chains are one of the most useful classes of stochastic processes, being • simple, flexible and supported by many elegant theoretical results • valuable for building intuition about random dynamic models • central to quantitative modeling in their own right You will find them in many of the workhorse models of economics and finance. In this lecture we review some of the theory of Markov chains. We will also introduce some of the high quality routines for working with Markov chains available in QuantEcon.jl. Prerequisite knowledge is basic probability and linear algebra.

Ordinary Differential Equations.

Abstract: together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values for the dependent variable, y0, y1, . . . , so that each yn approximates the solution at tn yn ≈ y(tn), n = 0, 1, . . . Modern numerical methods automatically determine the step sizes hn = tn+1 − tn so that the estimated error in the numerical solution is controlled by a specified tolerance. The Fundamental Theorem of Calculus gives us an important connection between differential equations and integrals.

Topologische lineare Räume

Abstract: § 14 enthalt die elementare Theorie der normierten Raume und der Banachraume. Eine Anzahl klassischer Beispiele wird diskutiert, die in den spateren Teilen des Buches immer wieder herangezogen werden.

Fibonacci Series Modulo m

Abstract: (1960). Fibonacci Series Modulo m. The American Mathematical Monthly: Vol. 67, No. 6, pp. 525-532.

Generalized Fibonacci Numbers and Associated Matrices

Abstract: (1960). Generalized Fibonacci Numbers and Associated Matrices. The American Mathematical Monthly: Vol. 67, No. 8, pp. 745-752.

The Double Dixie Cup Problem

Abstract: The familiar childhood occupation of obtaining a complete set of pictures of baseball players, movie stars, etc., which appear on the covers of dixie cups raises some interesting questions. One, which has already been answered, is the "single dixie cup problem," that of determining the expected number, E(n), of dixie cups which must be purchased before a complete set of n pictures is obtained: E(n) = n(I + 1/2 + * * * + 1 /n) ( [1 ] p. 213). Some time ago W. Weissblum asked how long, on the average, it would take to obtain two complete sets of n pictures. This corresponds to the situation observed when two tots collect cooperatively, i.e., "trading" takes place. This "double dixie cup" problem cannot be handled by the same device used for the problem of the single set and in this paper we find a new method which allows us to write down the solution, Em(ln), (as an easily evaluated definite integral) for the problem of collecting m sets. For m fixed and n large the expected number of dixie cups turns out to be n(log n+(m1) loglog n+o(l)). Thus, although the first set "costs" n log n, all further sets cost n loglog n. Suppose m sets are desired. Let pi be the probability of failure of obtaining m sets up to and including the purchase of the ith dixie cup. Then the expected number of dixie cups Em(ln) = Z%=0 pi, by a well-known argument ([1] p. 211). Now pi= Ni/ni where Ni is the number of ways that the purchase of i dixie cups can fail to yield m copies of each of the n pictures in the set. If we represent the pictures by xi, * , xn, then Ni is simply (xi + * * * +xn)i expanded and evaluated at (1, . . ., 1) after all the terms have been removed which have each exponent for each variable larger than m -1. Now consider m fixed and introduce the following notation. If P(x1, . . . , x,,) is a polynomial or power series we define { P(xi, . . . , x,C) } to be the polynomial, or series, resulting when all terms having all exponents _ m have been removed. In terms of this notation pi is { (xi + * +x.) } /ni evaluated at x = *= -1. If we now make the definition

Embedding Incomplete Latin Squares

Abstract: (1960). Embedding Incomplete Latin Squares. The American Mathematical Monthly: Vol. 67, No. 10, pp. 958-961.

A Generalization of Hermite's Interpolation Formula

Abstract: (1960). A Generalization of Hermite's Interpolation Formula. The American Mathematical Monthly: Vol. 67, No. 1, pp. 42-46.