Showing papers in "American Mathematical Monthly in 1973"
444 citations
TL;DR: In this article, Hilbert's Tenth Problem is shown to be unsolvable, and the paper concludes that Hilbert's problem cannot be solved by any number of solutions, e.g.,
Abstract: (1973). Hilbert's Tenth Problem is Unsolvable. The American Mathematical Monthly: Vol. 80, No. 3, pp. 233-269.
402 citations
TL;DR: Differentiation under the Integral Sign (DOS) as mentioned in this paper is a special case of the DOS problem, and it can be seen as an instance of DOS in the classical setting.
Abstract: (1973). Differentiation Under the Integral Sign. The American Mathematical Monthly: Vol. 80, No. 6, pp. 615-627.
344 citations
336 citations
TL;DR: In this paper, the probability that two group elements commute is investigated in terms of the probability of two groups of elements commuting, and the results show that the probability is high.
Abstract: (1973). What is the Probability that Two Group Elements Commute? The American Mathematical Monthly: Vol. 80, No. 9, pp. 1031-1034.
247 citations
TL;DR: In this article, the crossing number problem is formulated as a set of crossing number problems, and solved in a linear fashion, with a fixed number of crossings per number of columns.
Abstract: (1973). Crossing Number Problems. The American Mathematical Monthly: Vol. 80, No. 1, pp. 52-58.
177 citations
TL;DR: The Quaternion Calculus has been studied extensively in the literature, see as mentioned in this paper for an overview of some of the major works. The American Mathematical Monthly: Vol. 80, No. 9, pp. 995-1008.
Abstract: (1973). The Quaternion Calculus. The American Mathematical Monthly: Vol. 80, No. 9, pp. 995-1008.
146 citations
104 citations
TL;DR: Nonstandard analysis (NSA) as mentioned in this paper is a framework for systematically applying some of the basic ideas of model theory to all areas of mathematics, and it is especially effective in analysis, geometry, topology, and related areas of Mathematics where the concept of limit is central.
Abstract: Nonstandard Analysis (NSA) is a framework for systematically applying some of the basic ideas of model theory to all areas of mathematics. It is especially effective in analysis, geometry, topology, and related areas of mathematics where the concept of limit is central. Forty years ago, the logician Abraham Robinson observed that the construction of nonstan-dard extensions could provide a rigorous foundation for the use of infinitesimals in basic analysis. 1 Since then, applications of this set of ideas have spread through all of mathematics , greatly extending Robinson's original use of infinitely small and infinitely large numbers, and NSA has become an active branch of research in its own right. In order to reach advanced applications of NSA in this course, we will assume a knowledge of first-order logic extending at least through the compactness theorem. Students should be able to formulate mathematical statements within first-order logic and should have some experience with nonstandard models. We will also use some tools (such as the construction of saturated models) from the beginning parts of model theory. After developing the basic framework of NSA we will give a substantial indication of how NSA is developed within two areas of advanced mathematics: • probability and stochastic analysis (based on the Loeb measure construction); • geometry and functional analysis (based on the nonstandard hull construction). Prerequisites: A knowledge of first-order logic through the compactness theorem; what is covered in the first half of Math 570 at UIUC or in a good undergraduate course in logic will be sufficient. References: There will be no text for the course, and a set of class notes will be distributed during the semester. The following books give an introduction to NSA as well as an indication of the range of its applications: Grading: This is primarily a lecture course. There will be regular assignments of homework problems, on the understanding that such work is essential to learning any area of mathematics. Students will be encouraged to give a lecture on a topic or project of their choice.
62 citations
TL;DR: Another elementary proof of Euler's formula for ε(2n) was given in this article. But the proof was based on the Euler formula for the ε-approximation.
Abstract: (1973). Another Elementary Proof of Euler's Formula for ζ(2n) The American Mathematical Monthly: Vol. 80, No. 4, pp. 425-431.
62 citations
TL;DR: In this article, the Equation x′(t) = ax(t + bx(t − τ) with small delay with "small" delay was introduced, where τ is the length of the smallest delay.
Abstract: (1973). The Equation x′(t) = ax(t) + bx(t − τ) With “Small” Delay. The American Mathematical Monthly: Vol. 80, No. 9, pp. 990-995.
TL;DR: The Lagrange Multiplier Rule as discussed by the authors is an extension of the Lagrange multiplier rule, and it has been used in many applications in the mathematical community, e.g.
Abstract: (1973). The Lagrange Multiplier Rule. The American Mathematical Monthly: Vol. 80, No. 8, pp. 922-925.
TL;DR: In this article, a unified theory of integration is presented, which is based on the notion of integration of a set of points of a hierarchy, and a unified hierarchy of points in a hierarchy.
Abstract: (1973). A Unified Theory of Integration. The American Mathematical Monthly: Vol. 80, No. 4, pp. 349-359.
TL;DR: A history of the prime number theorem can be found in this paper, where the author presents a history of prime number theory. But this history is limited to the first order prime numbers.
Abstract: (1973). A History of the Prime Number Theorem. The American Mathematical Monthly: Vol. 80, No. 6, pp. 599-615.
TL;DR: The Legend of John Von Neumann as discussed by the authors is a popular work in the literature about the VonNeumann family. The American Mathematical Monthly: Vol. 80, No. 4, pp. 382-394.
Abstract: (1973). The Legend of John Von Neumann. The American Mathematical Monthly: Vol. 80, No. 4, pp. 382-394.
TL;DR: In this paper, the prime numbers and Brownian motion were studied in the context of Brownian Motion, and they were shown to be a special case of the Brownian Process.
Abstract: (1973) Prime Numbers and Brownian Motion The American Mathematical Monthly: Vol 80, No 10, pp 1099-1115
TL;DR: In this paper, the history of Spectral Theory is discussed and highlights in the history are discussed, with a focus on Spectral Spectral Models and their applications in the field of Artificial Intelligence.
Abstract: (1973). Highlights in the History of Spectral Theory. The American Mathematical Monthly: Vol. 80, No. 4, pp. 359-381.
TL;DR: The first rigorous proof of the Riemann mapping theorem was given in 1900 by W. Fogg Osgood as discussed by the authors, which represented the "coming of age" of mathematics in America.
Abstract: The Riemann mapping theorem, that an arbitrary simply connected region of the plane can be mapped one-to-one and conformally onto a circle, first appeared in the Inaugural dissertation of Riemann (1826-1866) in 1851. The theorem is important, for by it a result proved for the circle can often be transformed from the circle to a more general region. The proof is difficult, as involving both behavior of a function in the small (conformal mapping) and behavior in the large (one-toone mapping). Riemann's proof was open to criticism and in the following decades numerous mathematicians sought for a proof, e.g., H. A. Schwarz (1843-1921), A. Harnack (1851-1888), H. Poincare (1854-1912), etc., until the first rigorous proof was given in 1900 by W. F. Osgood. The proof of Osgood represented, in my opinion, the "coming of age" of mathematics in America. Until then, numerous American mathematicians had gone to Europe for their doctorates, or for other advanced study, as indeed did Osgood. But the mathematical productivity in this country in quality lagged behind that of Europe, and no American before 1900 had reached the heights that Osgood then reached. William Fogg Osgood (1864-1943) was born in Boston in 1864, graduated from Harvard College in 1886, stayed in Cambridge for a year of graduate work, and then went to Gottingen with a Harvard fellowship for further study, especially with Felix Klein (1849-1925). According to gossip, Osgood became so enamored of a Gottingen lady that his work suffered and Klein sent him to Erlangen for his doctorate. In any case, he was accorded the degree from Erlangen in 1890 for a thesis on Abelian integrals, and one or two days later he married the girl in G6ttingen, and one or two days still later they sailed for the United States of America. His
TL;DR: The circle groups of nilpotent rings as discussed by the authors have been studied extensively in the literature, see, e.g., the American Mathematical Monthly: Vol. 80, No. 1, pp. 48-52.
Abstract: (1973). Circle Groups of Nilpotent Rings. The American Mathematical Monthly: Vol. 80, No. 1, pp. 48-52.
TL;DR: The Hamel Dimension of Any Infinite Dimensional Separable Banach Space is c. c. as discussed by the authors, and the Hamel dimension of any infinite dimension separable space can be found in this paper.
Abstract: (1973). The Hamel Dimension of Any Infinite Dimensional Separable Banach Space is c. The American Mathematical Monthly: Vol. 80, No. 3, pp. 298-298.
TL;DR: In this article, a simple proof of the formula is given. But the proof is based on a simple formula and it is not provable that the formula can be proven. The American Mathematical Monthly: Vol 80, No. 4, pp. 424-425.
Abstract: (1973). A Simple Proof of the Formula . The American Mathematical Monthly: Vol. 80, No. 4, pp. 424-425.
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