In an earlier blog post, I showed the solution to a first-order linear system of two (ordinary) differential equations with constant coefficients for the case where the characteristic equation of the (constant coefficient) square matrix has repeated roots. In particular, I considered the system , where are real constants and are real variables, for the case where and . Now I look at the case where . Read the rest of this entry »
Here’s a funny comment on this answer at Mathematics Stack Exchange.
On being presented an equation where the left-hand side’s terms have a factor of 1/2 and the right-hand side is constant, a user asks “why not multiply through by 2?” Another user replies “Because this is physics.”
I was looking through the Rare Books section of the AbeBooks website a few days ago and searched for some books with the keyword math. The first entry is shown above, a book by Copernicus published in 1542. (It seems that the books with the highest prices are listed first.) A quick internet search reveals that the $350,000 price for this book is reasonable. (But I doubt that anyone interested in buying the book would have it shipped for $6.50.)
The second book on the list is shown above, a book published in 1990. There are three copies of this book for sale. Which one would you choose, the one priced at $149.90 or the one priced at $118,935.97?
From Sir Arthur Conan Doyle’s “Silver Blaze”:
Gregory (Scotland Yard detective): “Is there any other point to which you would wish to draw my attention?”
Holmes: “To the curious incident of the dog in the night-time.”
Gregory: “The dog did nothing in the night-time.”
Holmes: “That was the curious incident.”
My daughter found a copy of Mark Haddon’s The Curious Incident of the Dog in the Night-Time (Vintage, 2004) at a local BOOKSALE outlet and I bought it because she liked it. I very seldom look at novels, but I’m glad that I read this book.
This “murder mystery novel” is quite unusual in that (a) it starts at chapter 2, (b) it has footnotes, drawings, and an appendix consisting of a proof of a mathematical theorem, and (c) the murderer is revealed halfway through the novel.
So I think it is not a “real” murder mystery novel (hence the scare quotes in the previous paragraph), and, to paraphrase Sherlock Holmes, that is the curious incident.
Although the story is told from the viewpoint of a 15-year-old boy, I found the observations on language (literal and figurative), writing (how to write detective fiction), the nature of the mind (how a normal person’s way of thinking differs from that of an autistic person, or of an animal, or of a computer), and mathematics very deep. I particularly like how a wide variety of mathematics is presented (probability, chaos, games, tessellations).
A quote from Wikipedia mentions that the book “was published simultaneously in separate editions for adults and children.” It seems that my copy has a few differences from the version that Wikipedia refers to. For example, Wikipedia mentions that the lead character is given a Cocker Spaniel puppy at the end, but in my copy of the book, the puppy is a Golden Retriever.
I found the ending quite sad, although most people would probably consider it a happy ending.
I’m currently teaching an undergraduate course on ordinary differential equations using the 8th edition of Elementary Differential Equations by Rainville, Bedient, and Bedient (published in 1996 by Prentice Hall). I’ve always wanted to have a blog post containing some LaTeX (using the plug-in WP LaTeX), so in this blog post I’ll be showing my solution to one of the exercises in the book.
I will derive the solution to a first-order linear system of two (ordinary) differential equations with constant coefficients for the case where the characteristic equation of the (constant coefficient) square matrix has repeated roots.
Read the rest of this entry »
In Edward Nelson’s (2007) review of the book 18 Unconventional Essays on the Nature of Mathematics (American Mathematical Monthly, vol. 114, pp. 843–848), he tries to answer the question in the title.
We [mathematicians] are no respecters of persons (in that curious phrase that means we do respect persons but pay little attention to the trappings of age, position, or prestige), we take equal delight in fierce competition and collaborative effort, and we are quick to say “I was wrong.” Perhaps some of us know an exception that proves the rule, but by and large I speak sooth, especially when one compares mathematicians to our colleagues in the humanities.
How does one explain that we are so lovable? Is there something in the nature of mathematics that attracts gentle souls? Possibly, but another explanation is more convincing. We are singularly blessed in that the worth of a mathematical work is judged largely by whether the proof is correct, and this is something on which we all agree (eventually), despite the fact that we may have divergent views on the nature of mathematics [...]. This is a singular fact. In art, projection of personality may prevail; in the humanities, the power of position may prevail; in science, the prevailing fad may prevent the publication even of excellent work—but we are extraordinarily fortunate that in our field none of this matters.
One very interesting cube-free infinite binary word is the Kolakoski word (also known as the Kolakoski sequence). The sequence is named after William Kolakoski, who introduced it in a problem published in the American Mathematical Monthly in 1965.
It seems that it was recently (last year?) discovered that the sequence was published earlier, in 1939, by Rufus Oldenburger in his paper Exponent trajectories in symbolic dynamics (Transactions of the American Mathematical Society, vol. 46, pp. 453-466).
One currently open problem (see, for example, problem 10 here) is to prove (or disprove) that the limiting frequency of each of the two characters exists and is equal to 1/2. Perhaps some insight can be found in Oldenburger’s paper?
Any competent mathematician nowadays will agree that two is the smallest prime number, but this was not always the case. There were instances when one (or even three) was considered the smallest prime number. Two recently published papers survey the history of how prime numbers are defined: “What is the Smallest Prime?” by Chris K. Caldwell and Yeng Xiong (Journal of Integer Sequences, vol. 15 (2012), Article 12.9.7) and “The History of the Primality of One: A Selection of Sources” by Chris K. Caldwell, Angela Reddick, Yeng Xiong, and Wilfrid Keller (Journal of Integer Sequences, vol. 15 (2012), Article 12.9.8). I think these two papers will be of great use to mathematics educators.
(Originally posted at http://joelnoche.multiply.com/journal/item/113/Thomas-Harriots-Artis-Analyticae-Praxis-A-first-look on
September 14, 2012 9:56 AM)
A few months ago, I bought a copy of Seltman and Goulding’s Thomas Harriot’s Artis Analyticae Praxis: An English Translation with Commentary (Springer, 2007) at a BOOKSALE outlet.
From Seltman and Goulding’s Introduction (p. 1):
In the algebraic work of Thomas Harriot, it was above all his notation that was revolutionary. His algebra was the first to be totally expressed in a purely symbolic notation [...].
Yet, Harriot is known in general histories of mathematics principally for certain technical innovations in algebra—for the invention of the inequality signs, for equating the terms of a polynomial equation to zero, and for generating such equations from the product of binomial factors, thereby displaying their structure.
I have not read much of Seltman and Goulding’s book, but as a mathematics educator I find Harriot’s work very interesting. It seems that he was the originator of the decimal digit-by-digit method of obtaining square roots that I was taught in elementary school.
The Artis Analyticae Praxis has two parts; the first has six sections serving as preparatory material for the second part, which is about “the numerical solution of [polynomial] equations by the method of successive approximation” (p. 13). Harriot builds upon previous work by François Viète, making it “more convenient and practical” (p. 4).
It is quite unfortunate that Seltman and Goulding’s book has numerous typographical errors, many to do with incorrect hyphenation (for example, “The-re” on p. 2, “ha-ve” on p. 12, “tho-se” on p. 20, and so on) and alignment of columns in tables (such as on p. 132).
When I have the time I would like to study this book in more detail. I feel that it has some exciting insights that can easily be explained by college teachers to their undergraduate students.