Dr. Dobb's Math Power Newsletter - December 2005

Official Publication of Math Power Club
Incorporating LIGHT WORK and COMPUTER WRANGLER
Math Power Club is an Unofficial Club of Pima CC East
Editor and Publisher Emeritus: Homer B. Tilton

Mathematics for Engineers and Scientists, sixth edition
Alan Jeffrey, Chapman & Hall / CRC, 2005, ISBN 1-58488-488-6 994 plus xvi pages

Organization/Contents (as if split into three parts):

Part 1. Numbers, Functions & Differentiation

1. Numbers,trigonometric functions & coordinate geometry
2. Variables, Functions and mappings
3. Sequences, limits and continuity
4. Complex numbers and vectors
5. Differentiation of functions of one or more real variables
6. Exponential, logarithmic and hyperbolic functions and an introduction to complex functions

Part 2. Integration, Fields & Series

7. Fundamentals of integration
8. Systematic integration
9. Double integrals in Cartesian and plane polar coordinates
10. Matrices and linear transformations
11. Scalars, vectors and fields
12. Series, Taylor's theorem and its uses

Part 3. Differential Equations, Statistics & Computer Algebra / Index

13. Differential equations and geometry
14. First-order differential equations
15. Higher-order linear differential equations
16. Fourier series
17. Numerical analysis
18. Probability and statistics
19. Symbolic algebraic manipulation by computer software
20. Answers, Index

A few observations now.

• Page 75, even and odd functions: "Most functions are neither even nor odd." Yes, but I didn't find it mentioned that any function can be expressed as the sum of an even and an odd function. (See p. 2 of Special Calculus.)
• Page 97, signum function: Jeffrey gives the classical sgn function which has only limited usefulness. If we introduce the concept of infinitesimal hysteresis we have a new signum function sgn which is either +1 or -1 everywhere, is never zero, and is still an odd function. It has the properties that 1/sgn(x) = sgn(x) and |sgn(x)| = 1. (See p. 3 of Special Calculus.)
• Sign of zero: (The complex zero is referred to on page 146.) From that definition it is seen that under infinitesimal hysteresis, the sign of zero can be +, i, -, -i, or anything in between depending on what direction is it approached from. (Yes, "i" as a sign makes very good sense.)

The real number zero is commonly said to be neither + nor -, or to have no sign. I suggest it is both + and - as a neutral atom is, and that |0| is positive while still being zero. This conforms to the definition of sgn using infinitesimal hysteresis. (See p. 4 of Special Calculus.)

• Definition of absolute value, not found in Jeffreys' book, but found referenced on p. 75: "From the definition of the absolute value of x, it is seen graph of |x| has the form shown..." I suggest that the classical definition as the distance from zero on the number line be abandoned, and we go exclusively with |x| = (x²). (See p. 1 of Special Calculus.)
• Complex numbers: The complex number plane is not introduced until late (p. 153) in the discussion in Chapter 4 (pages 144-161). But trying to explain and discuss complex numbers without using the complex number plane is like trying to explain real numbers without using the real number line. Needlessly abstruse. Should introduce the complex number plane right up front.
• Page 77, [x], bracketed x: Compare x-Arctan(tan x) where Arctan(tan x) is a linear sawtooth wave; it is not x, because Arctan is the principle value of arctan.
• It is high time to include the analytical expressions for the "generalised functions" of Sir James Lighthill. One example is given above, x-Arctan(tan x).

A classical treatment, but comprehensive and highly useful as are all of Jeffrey's works that I've seen. Highly recommended.

For your own copy of the book, Special Calculus, referred to in the above book review, write by US mail to me, Homer Tilton/ 8401 E. Desert Steppes Dr./ Tucson, AZ 85710-4207.

What's new -

We wrote Dr. Alan Jeffrey concerning the new, sixth edition of his book, Mathematics for Engineers and Scientists. See review which starts on page 1. A short time later we received this reply:

Dear Homer,

Many thanks for you useful comments about my book that I will certainly act on when a new edition is due - in fact the publisher is looking into the timing of this right now. I agree that in many ways it could be a good thing to publish a future edition in two parts, but the decision to publish as a single volume is that of the publisher.

This decision is motivated partly by the fact that competing books are all equally large and cover much the same range of topics, but also because students seem to prefer to buy one large book that will serve them during all of their studies, rather than two smaller ones. Another reason is the "folk law" in publishing that predicts when a book is published as a two volume set, sales of the second volume will never be anything like those of the first volume!

It was thoughtful of you to send me a copy of your Special Calculus. It arrived today so I have only just had time to glance through it, but I can see it will repay careful study.

Your relativity convention sounded interesting, and I am glad it proved to be such a success. Having myself been involved with the organization of a few conferences I know all too well how much work is involved, but also that when things go well the effort all seems worth while.

It is good to know your open heart surgery went well - I had no idea you had trouble with your mitral valve, or that you are now so senior! From your letter it would seem you are functioning normally again, and that is excellent news. I have two friends who have undergone similar surgery with great success, and in both cases the result has been that their life is back to normal again - so I hope and expect the same will be true for you.

Sincerely, Alan Jeffrey/ University of Newcastle/ England

Dear Alan,

I was thinking more along the lines of a three volume boxed set for your book, much like an encyclopedia where the purchaser buys the whole thing, not just one or two volumes. Of course with an encyclopedia you have an alphabeti- cal organization. I'm not sure how one would get that kind of connection and interdependence with a math reference. One suggestion may be apparent from the "Book review" on page 1.

Thousand-page books can be a bit unweildy; and may have a tendency, after some use, to end up with a broken spine somewhere in the middle. My copy of the MS-DOS Encyclopedia (over 1500 pages) did that; and now it is in two volumes, like it or not, simply because it came apart after much use!

The story of my planned heart surgery is told elsewhere in this issue. Thank you for your concern. Yes, you could say I am senior. That that sur- prises you is most flattering. I was born just two years after our recently- deceased colleague Sir James Lighthill of whom you have written so highly, and whose pioneering work on discontinuous generalised functions so impressed me. He was an inspiration and role model and I treasure his letters to me.

Thank your for your kind words about my book, Special Calculus. It is an outgrowth of my 1986 book from Prentice-Hall, Waveforms: A modern guide to nonsinusoidal functions and nonlinear processes. There has been some interest in Special Calculus here at Pima Community College. The problem is, it is of interest mainly to engineers - the academic mathematician not yet picking up on the need for it.

For your further indulgence, I've sent you the October issue of MATH POWER journal. It outlines a cutting-edge space proposal, The John McCain Southern Arizona Starport Corridor. While some may think such a proposal premature, I feel a certain urgency in getting these thoughts out there, in the public domain. (I wonder why that is?) ...HBT

3-D again -

Dear Homer Tilton, (e-mail dated 6 Jul '05, reviewed 7 Sep '05)

I am a contributing editor for Laser Focus World magazine. (www.laserfocusworld.com) and am working on an article about 3-D displays to run in the September issue.

I know that you've been involved in 3-D displays for decades now. Is Visonics Labs still making stereo displays? If so, could you provide a photo of your system in operation and a description of how it works?

Even if you are not still selling displays, I'd love to talk to you for some perspective about the 3-D display industry. Is there a good time for me to call you?

My deadline is next Monday, July 11. Thank you for any help you can give me./ Sincerely, Yvonne Carts-Powell/ Science Writer, specializing in photonics, imaging, micro- and nano-technology, robotics/ Belmont, MA 02478/ www.nasw.org

Dear Yvonne,

As you can see from the article "Under the Knife" referenced elsewhere in this issue of MATH POWER, I've been in the hospital, and have only now been able to respond to your most gracious e-mail. I am sorry to be so late.

Two of the last 3-D units we manufactured were built for British Telecom as ordered by Ian Sexton. A large-screen unit was also built for Walter Funk, an artist/engineer in Silicon Valley. I call those devices, my most recent, the parallactiscope.

Those units present parallactic displays which show much more than simple stereo images; they produce movement parallax as well. Multiple observers can view the displayed images, each being able to "look around" displays according to their position. Each viewer receives images appropriate for their position; there is no "observer tracker." I believe that these devices, and the technology they implement to still be unique.

No viewing aid is required. Use your naked eyeballs, and move side-to-side as you wish to gain an appreciation for the form of the any of the virtually infinite number of virtual 3-D "objects." Ask your buddy to view along with you; he will receive his own, appropriate and distinctive images. [As you may have by now guessed, almost no one who had not seen the displays believed any of this, perhaps explaining why we went out of business.]

As for your request for a photo, I've sent you copies of pages from my book,[1] of a large-screen unit. Photos cannot be taken of the displays that do them justice, as these are live, dynamic displays. Instead, multiscopic, monocular photos appear in sequences of four each in the book. View them stereoscopically in pairs; move to the next pair to get a sense of the "look- around" property of the live displays. Since then I have constructed a still larger-screen (17-inch) model and can arrange a demonstration of that for you at my lab, given adequate notice.

Another book contains chapters of various recent 3-D technologies, including mine.[2]

If you still wish to call me at home, any Friday, between 10-5 Arizona time would probably yield results. Thank you very much for your interest. ...HBT

References -

[1] Homer B. Tilton The 3-D oscilloscope/ A Practical Manual and Guide, Prentice-hall, 1986, ISBN 0-13-920240-4; The book gives complete construction plans for building your own parallactiscope.

[2] Stereo Computer Graphics and Other True 3D Technologies, Edited by David F. McAllister, Princeton U.P., 1993, ISBN 0-691-08741-5

Definition of the month - SI units - System of coherent metric units (Système International d'Unités) ... Chambers Dictionary of Science and Technology

Reference of the month - Wildi, Theordore, Metric Units and Conversion Charts, 2nd edition, IEEE Press, 1995, ISBN 0-7803-1050-0

An authoritative reference (Wildi, page 25) defines the "light year" as a non-SI unit equal to 9.460 528 petameters. (A petameter is 10¹⁵ meters.) It also gives (page 42) the number of seconds in the year as (365)X(86 400) (non- SI) and the speed of light, c, as exactly (page 48) 299 792 458 m/s. That reference gives no abbreviation for light year. Since the second is now (since 1 Jan 1990) an SI unit, that would tie the light year into Earth years, putting the nearest star, Alpha Centauri, at 4.3 light years distant; but in terms of Mars years, it would be only only a little more than half that. So we now see the need to define an SI year and an SI light year.

The plan: That number given by Wildi, (365)X(86 400), the number of seconds in an Earth year, is 3.1536X10⁷ which is only a little larger than X10⁷. Therefore it is proposed that t, the SI year, be defined as X10⁷ SI seconds. It is further proposed that the SI light year, a measure of distance or length, s, be abbreviated lt^.yr with the raised dot (^.) being a multiplication sign. Since the speed of light is "one lt^.yr/yr" = one lt, we have that c=s/t or s=ct. It follows that the SI lt^.yr would be s=ct, and one light year would then be 299 792 458 m/s times X10⁷ s, or (2.997 924 58X10⁸)(X10⁷) meters. Note that "lt" is a measure of speed, , with the speed of light being specified as = 1 lt.

Bottom line: The proposed SI light year is exactly (2.997 924 58) peta- meters (Pm). Rounding to 9.418 259 Pm, we can compare that to Wildi's "civil" light year given at the top of this piece, 9.460 528 Pm.

The postal bar code for a nine-digit ZIP consists of 5(9+1)+2 = 52 bars, some short some long. The first and last bars (always long bars) mark the beginning and end of the code. The remaining 50 bars are read in groups of five, with each dit (decimal digit) being coded as shown here in the chart:

In the chart, "1" represents a long bar; "0" represents a short bar. Note that every dit has two long and three short bars. The nine dits of the ZIP then occupy 45 bars; the tenth dit is a parity or error-checking dit, P.

Thus, the ZIP 85709-4000 is coded like this:

             ||  |  | | |   |||   | |   |  |||   ||   ||   |   ||
             ||||||||||||||||||||||||||||||||||||||||||||||||||||
              -----     -----     -----     -----     -----      
                   -----     -----     -----     -----     -----      

                8    5    7    0    9    4    0    0    0   P=7

P is found as the tens complement of n modulus 10, or in BASIC notation,

        P = 10 - (n MOD 10)                                     (1)

where n is the sum of all digits. Here n MOD 10 is the remainder after dividing n by 10. In this instance

        n = 8+5+7+9+4 = 33,                                     (2)

so that

        P = 10 - (33 MOD 10) = 10 - 3 = 7.                      (3)

Thanks to Dr. Theresa Riel for sharing this secret with us.

Coming soon: Other bar codes examined.

Copy of the Proceedings of the first conference on relativity are available from me (Homer) at $29.95 ea payable to the Art Alberding Mathematics Scholarship Fund at PCC-EC. Or you can download a free copy from:

www.gallup.unm.edu/~smarandache/ProceedingsTucson.pdf

The file length is 1.9 MB and it is in pdf format.

In June I went into the hospital for open-heart surgery to replace a badly leaking mitral valve. Version 1 of the December MATH POWER, available only directly from me in hard copy, describes my three hospital stays for a total of 13 days spread out over the period from June 29th to October 4th. I remain on heavy medication as of this writing, November 19th. It describes the first two of those visits in some detail. The writeup tells of my experiences there and is meant to be humorous; but since it is a bit off the beaten path for a math journal, my DDJ editor and I have agreed that it should not appear in this on- line version, Version 2.

Write me via US mail, for your printed copy of version 1. It contains the aforesaid seven-page article, "Under the Knife." Address your request to Homer Tilton, at 8401 E. Desert Steppes Dr./Tucson, AZ 85710-4207.

Look for "Under the Knife" in three parts corresponding to my three hospital visits to possibly appear, illustrated, in the weekly journal NurseWeek. See above. Ask your friendly neighborhood RN to see her (or his) copy of NurseWeek.

Written reader comments are invited on all material. Those intended for my attention must be submitted by US Mail to my Tucson address. All such comments are subject to being published unless requested otherwise. They may also be subject to editing. -HBT

The hard copy version of MATH POWER is published as a shareletter; that means you are permitted to make not-for-profit copies of it for distribution to your colleagues and students.

MATH POWER is published monthly. It is published and edited by Homer B. Tilton under the auspices of Pima Community College, East Campus, 8181 E. Irvington Rd., Tucson AZ 85709-4000. Editorial Assistant, Jo Taylor. All material is copyright Homer B. Tilton unless otherwise noted. A limited number of copies may be made at educational institutions for internal use of faculty and students. For more extended copying or to request additional copies contact the Editor at the above address. Letters and editorial material are welcome. All submitted material may be published in MATH POWER, and edited, unless specifically requested otherwise.