Thursday, March 31, 2011

Joey's Kids - Fourier Series - From Sines to Sawteeth

Joe Fourier gave us Fourier Analysis, and I can prove it because it's named after him not after Euler or Gauss or anyone else. You see? Not every proof is hard. The following is still my favorite though:

Did you know that all numbers are interesting? What’s that? You don’t believe me? Well I have a proof. Suppose not every number is interesting. Then let n be the smallest uninteresting number. That’s a rather interesting property isn’t it?
... Ron Graham

OK, that's enough silliness for one day, let's take out our notebooks, computer or "old school" spiral (for the young - that means a spiral notebook made with paper and cardboard. Back in my day we used a writing instrument known as a pencil or pen to make marks on the pages. You've probably seen them in a museum or your grandparent's attic), and get down to brass tacks.

Fourier Analysis. What is it? Fortunately, Mathematicians write things down, and from the book Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations we see that Fourier Analysis is part of the global Mathematical field known as "Analysis" (which includes Calculus), and in that book anyway is the most advanced topic, at least for Introductory purposes. Fourier Analysis can be broken down as follows:

1) Sine waves
2) Building waveforms
3) Fourier series
4) Fourier's theorem
5) Fourier's formulas
6) Complex Fourier series
7) The Fourier transform

Eventually we will get to #7, but not today. Today we will give a very short example of #3 above. Everything in stages. But you should know why we study this stuff, so we will get to #7 and here's why from the book:

The Fourier transform is a powerful weapon in the mathematician's arsenal, and has wide applications, from representation theory to quantum mechanics.

So yesterday we talked about "The Math Grenade" and today we talk about a powerful weapon. Why mathematicians don't organize like a modern military is beyond me. But whatever, here goes (and from Wiki) ...

HOW CAN WE TURN A SINE WAVE INTO A SAWTOOTH WAVE ?
Animated plot of the first five successive partial Fourier series.

Plot of a periodic identity function—a sawtooth wave.


f(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, π], as Fourier did, and we will then discuss different formulations and generalizations.

Fourier's formula for 2π-periodic functions using sines and cosines

For a periodic function ƒ(x) that is integrable on [−π, π], the numbers
a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0
and
b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1
are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by
(S_N f)(x) = \frac{a_0}{2} + \sum_{n=1}^N \, [a_n \cos(nx) + b_n \sin(nx)], \quad N \ge 0.
The partial sums for ƒ are trigonometric polynomials. One expects that the functions SN ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum
\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]
is called the Fourier series of ƒ.

We can now use the formula  above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave
f(x) = x, \quad \mathrm{for } -\pi < x < \pi,
f(x + 2\pi) = f(x), \quad \mathrm{for } -\infty < x < \infty.
In this case, the Fourier coefficients are given by
\begin{align} a_0 &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x\,dx = 0. \\ a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0, \quad n \ge 0. \\ b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) + \frac{2}{\pi n^2}\sin(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}
It can be proved that the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, and therefore:
\begin{align} f(x) &= \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\ &=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad x - \pi \notin 2 \pi \mathbf{Z}. \end{align}




When x = π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ƒ at x = π. This is a particular instance of the Dirichlet theorem for Fourier series.

And there you have it folks, a real honest-to-God application of Calculus. That's right, there's more than one reason to learn Math other than as one sage put it: "So you can tell when your parents are lying." 
:-)
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Wednesday, March 30, 2011

Do NOT Take Potassium Iodide ! Unless You're Stupid.

Fukushima, 3 minutes after the Disaster

Folks, I was going to let this roll around until my "Engineering" cycle, but I feel it's too important to NOT get the word out immediately.

Do NOT take Potassium Iodide pills. They're poison. OK, if you live within 10 miles of the Japanese Fukushima  plant and the skin on your bones and muscles is warping, DO take them. It's a pill of last resort, and in that sense, sure, take them if you must.

But for EVERYONE ELSE? Do NOT take these pills!

Wanna have em on hand, if say you live 10 miles within say one of Salem Co., NJ's 3 Nuclear Power reactors? Fine. Have em and hold on to em. But really, can we get any MORE paranoid? Because I honestly don't think we can.

GW Johnson, Professional Consulting Engineer, goes into the explicit details: here.

A  two-headed calf. Just one of the effects that may occur by ingesting the "emergency ONLY"  teratogen that is Potassium Iodide. Although I really don't think humans should worry about giving birth to calves, one-headed or otherwise.

"Potassium iodide is a mild irritant and should be handled with gloves. Chronic overexposure can have adverse effects on the thyroid. Potassium iodide is a possible teratogen."


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From Buchenwald to e-Bay - The Curta Calculator

Invented by a Jewish chap in the WWII Buchenwald Death Camp, Curtas were widely considered the best portable calculators available until they were displaced by electronic calculators in the 1970s. I just went to e-Bay and they're being offered for sale in a price range from $348 to $1500.


The Curta is a small, hand-cranked mechanical calculator introduced in 1948. It has an extremely compact design: a small cylinder that fits in the palm of the hand. It can be used to perform addition, subtraction, multiplication, division, and —with more difficulty— square roots and other operations.

The Curta's design is a descendant of Gottfried Leibniz's Stepped Reckoner and Thomas's Arithmometer, accumulating values on cogs, which are added or complemented by a stepped drum mechanism.


History of the Invention

The Curta was conceived by Curt Herzstark (1902–1988) in the 1930s in Vienna. By 1938, he had filed a key patent, covering his complemented stepped drum, Deutsches Reichspatent (German Empire Patent) No. 747073. This single drum replaced the multiple drums, typically around 10 or so, of contemporary calculators, and it enabled not only addition, but subtraction through nines complement math, essentially subtracting by adding. The nines' complement math breakthrough eliminated the significant mechanical complexity created when "borrowing" during subtraction. This drum would prove to be the key to the small, hand-held mechanical calculator the Curta would become.
His work on the pocket calculator stopped in 1938 when the Nazis forced him and his company to concentrate on manufacturing measuring instruments and distance gauges for the German army.
Herzstark, the son of a Catholic mother but Jewish father, was taken into custody in 1943, eventually finding himself at the Buchenwald concentration camp. Ironically, it was in the concentration camp that he was encouraged to continue his earlier research: "While I was imprisoned inside [Buchenwald] I had, after a few days, told the [people] in the work production scheduling department of my ideas. The head of the department, Mr. Munich said, 'See, Herzstark, I understand you've been working on a new thing, a small calculating machine. Do you know, I can give you a tip. We will allow you to make and draw everything. If it is really worth something, then we will give it to the Führer as a present after we win the war. Then, surely, you will be made an Aryan.' For me, that was the first time I thought to myself, my God, if you do this, you can extend your life. And then and there I started to draw the CURTA, the way I had imagined it."[1]
Herzstark worked furiously to move his invention from his knowing how to build the device "in principle"[1] to concise working drawings for a manufacturable device.
The department head's celebration plan didn't work out, but Herzstark's construction plans did: Between April 11, 1945, when Buchenwald was liberated by the Americans, and the following November, Herzstark was, after making only a few "detailed improvements"[1] to the design, able to locate a factory in Sommertal, near Weimar, where machinists were skilled enough to work at the necessary level of precision, and walk away with three working models of the calculator.
The Russians had arrived in July, and Hertstark feared being sent to Russia, so, later that same month, he fled to Austria. He began to look for financial backers, at the same time filing continuing patents as well as several additional patents to protect his work. The Prince of Liechtenstein eventually showed interest in the manufacture of the device, and soon a newly-formed company, Contina AG Mauren, (aka, Contina Ltd Mauren) began production in Liechtenstein.
It wasn't long before the money men, apparently having gotten from him all they thought they needed, contrived to force him out by reducing the value of all existing stock to zero, including his one-third interest in the company. These were the same money men who, earlier, had elected not to transfer ownership of the Herzstark's patents to the company, so that, should anyone sue, they'd be suing Herzstark, not the company, thereby protecting themselves at Herzstark's expense. This ploy now backfired: Without the patent rights, they could manufacture nothing. Herzstark was able to negotiate a new agreement, and money continued to flow to him.
Curtas were widely considered the best portable calculators available until they were displaced by electronic calculators in the 1970s. Herzstark continued to make money from his invention until that time, although, like many inventors before him, he was not among those who profited the most from his invention. The Curta, however, lives on, being a highly-popular collectible, with thousands of machines working just as smoothly as they did 40, 50, and 60 years ago when they were manufactured.[1][2][3]

Description and use

Numbers are entered using slides (one slide per digit) on the side of the device. The revolution counter and result counter appear on the top. A single turn of the crank adds the input number to the result counter, at any position, and increments the revolution counter accordingly. Pulling the crank upwards slightly before turning it performs a subtraction instead of an addition. Multiplication, division, and other functions require a series of crank and carriage-shifting operations.
The Curta was affectionately known as the "pepper grinder" or "peppermill" due to its shape and means of operation. It was also termed the "math grenade", due to its superficial resemblance to a certain type of hand grenade.

Curta Type I and Type II

The Type I Curta has 8 digits for data entry (known as "setting sliders"), a 6-digit revolution counter, and an 11-digit result counter. According to the advertising literature, it weighs only 8 ounces (about 230 grams). Serial number 70154, produced in 1969, weighs 245 grams.
The larger Type II Curta, introduced in 1954, has 11 digits for data entry (known as "setting sliders"), an 8-digit revolution counter, and a 15-digit result counter. It weighs 13.15 ounces or 373 grams, based on weighing serial number 550973, produced in early 1966.
An estimated 140,000 Curta calculators were made (80,000 Type I and 60,000 Type II). According to Curt Herzstark, the last Curta was produced in 1972.[1]

Gallery

  • Curta Mechanical computer img 1651.jpg
    •  
  • A partially disassembled Curta calculator, showing the digit slides and the stepped drum behind them.
    •  
  • Curta Type I calculator showing view from top.
    •  
  • Curta Type I calculator showing view from bottom.

Use in car rallies

The Curta was popular among contestants in sports car rallies during the 1960s, 1970s and into the 1980s. Even after the introduction of the electronic calculator for other purposes, they were used in time-speed-distance (TSD) rallies to aid in computation of times to checkpoints, distances off-course, etc., since the early electronic calculators did not fare well with the bounces and jolts of rally racing.[citation needed]
Contestants who used such calculators were often called "Curta-crankers" by those who were limited to paper and pencil, or who used computers linked to the car's wheels.[citation needed]
Curta calculators contributed to the saying when describing the process of calculating, "Cranking out the answer."[citation needed]

Use by pilots

The Curta was also favored by both commercial and general-aviation pilots, before the advent of electronic calculators, because of both its precision and the user's ability to confirm the accuracy of his or her manipulations via the revolution counter. Because calculations such as weight and balance are a matter of life and death, precise results free of pilot error are essential.

The real cost of a Curta

While only 3% of Curtas were returned to the factory because of repairs under warranty[1], a small, but significant stream returned the Curta in pieces. Many purchasers attempted to disassemble the Curta. Reassembling the machine was more difficult, as assembly required intimate knowledge of the orientation and installation order for each part and sub-assembly. Many identical looking parts, each with slightly different dimensions, required test fitting and selection as well as special tools to adjust tolerances. One chagrined owner, on showing up to the dealer to retrieve his $600 Curta, now reassembled for an additional $300, was told, "don't feel bad. Curtas really cost $900. Everyone takes them apart."[4]

The Curta in print

The Curta is featured in the novel Pattern Recognition by William Gibson, where one of the minor characters has an interest in them.
The Curta is also the subject of an article by Cliff Stoll in the January 2004 edition of Scientific American.
Lastly, the Curta is on the Popular Science magazine list of top 100 pieces of retro technology.

References

  1. ^ a b c d e f "An Interview with Curt Hertstark" conducted by Erwin Tomash, Charles Babbage Institute, University of Minnesota, Minneapolis. Hertstark interview in German
  2. ^ Curt Herzstark and his Pocket Calculator, Peter Kradolfer, backup 6/88 pp. 5–9
  3. ^ Stoll, Cliff (January 2004). "The Curious History of the First Pocket Calculator". Scientific American 290 (1): 92–9.
  4. ^ As related to poster, Bruce Tognazzini, at the time by the chagrined owner

External links

Wikimedia Commons has media related to: Curta calculator
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Tuesday, March 29, 2011

One-Tenth Light Speed and Jordin Kare

In what is the Lunar Colonization/Space Exploration part of my cycle I was going to discuss Bigelow Aerospace but an article at Centauri Dreams (currently hosting The Carnival of Space #190) by Paul Gilster caught my eye, regarding another dream of many of us: Interstellar Exploration.

(Don't worry Pat Ballew, that means we cycle to Math tomorrow :-)  )

As Paul writes in the article "Pedal to the Metal", here :

It’s fun to juggle these numbers even as we think about how far we have to go before an interstellar probe becomes a possibility. If the goal is to reach Alpha Centauri with a mission lasting, say, forty years, then we need a tenth of lightspeed, or roughly 30,000 kilometers per second.

[ .... 17.05 kilometers per second, which is faster than any of our outward bound spacecraft but would take well over 70,000 years to reach Alpha Centauri, assuming Voyager 1 were pointed in that direction. New Horizons is currently making 15.73 kilometers per second on its way to a Pluto/Charon flyby in July of 2015, impressive but not the kind of speed that would get us to interstellar probe territory.

Interestingly, the fastest spacecraft ever built weren’t headed out of the Solar System at all, but in toward the Sun. The Helios probes were West German vehicles launched by NASA, one in 1974, the other in 1976, producing successful missions to study conditions close to the Sun for a period of over ten years. The orbits of these two craft were highly elliptical, and at closest approach to the Sun, each reached speeds in the range of 70 kilometers per second. Helios II, marginally faster, lays claim to being the fastest man-made object in history.]

That makes .10c a figure of distinction, because it creates a mission that can be built, flown and studied to completion by the same team.

In the replies section, reader Adam says :

For just 30 GW of in-space laser power, Jordin Kare’s Sail-Beam can push a 1 ton probe to 0.1c. Whether we can give a 1 ton probe enough power to do anything useful at Alpha Centauri when it gets there is a whole other question.

Who the heck is Jordin Kare? I must confess my membership papers in the Interstellar community are dusty and unused of late, as I feel I have bigger fish to fry at the moment, but still, it's intriguing.

I do recall having similar yet different thoughts in my Railroad to the Stars article of this past December, but Jordin is a Professional in the field and I think we should listen to him first:

Jordin Kare

Jordin Kare (born 1956) is a physicist and aerospace engineer known for his research on laser propulsion. In particular, he was responsible for Mockingbird, a conceptual design for an extremely small (75 kg dry mass) reusable launch vehicle, and was involved in the Clementine lunar mapping mission.[2][3] Kare is also known as developer of the Sailbeam interstellar propulsion concept and, in the science fiction fan community, as a composer, performer and recording artist of filk music parodies.

Background

He received his B.S. in Electrical Engineering and Physics from the Massachusetts Institute of Technology in 1978 and Ph.D. in Astrophysics from University of California, Berkeley in 1984.[2][4] Kare is the brother of Susan Kare, designer of the fonts and icons of the original Apple Macintosh user interface.[5][6]

Career

Kare worked for many years at Lawrence Livermore National Laboratory. In 1996, he left LLNL and, after working briefly for a small space-related startup company, became an independent consultant specializing in advanced space system design in 1997,[4] and started his own company.[5] He is a leading advocate of laser propulsion for space launch and in-space propulsion. He organized a 1986 Workshop on Laser Propulsion at LLNL and later led a development program for ground to orbit laser launch supported by SDIO. He has received a NASA Institute for Advanced Concepts grant to study a near-term form of laser launch using arrays of relatively low powered lasers.[4][7][8][9] He is a team member of LaserMotive,[10] a laser power beaming entrant in the Elevator:2010 Beam Power Challenge.

Sailbeam

Kare initially presented the concept of a SailBeam Boosted Magsail in a report prepared for NASA’s Institute for Advanced Concepts called “High-Acceleration Micro-Scale Laser Sails for Interstellar Propulsion”. A key idea is that if you accelerate vast numbers of tiny sails rather than one enormous one, you can bring the same amount of mass to high speeds with a less complex optical system. Unlike particle beam propulsion where the beam disperses as it travels, a stream of low-mass microsails is not limited by such diffraction. Using dielectric rather than metal sails, you can also accelerate the sails much closer to their power source. The stream of microsails then becomes a source of propulsion to a starship as particle beams mounted on the starship vaporize the incoming sails into plasma.

Filk music and science fiction

Kare is also known as a science fiction fan and filksinger. He has been a regular attendee and program participant at science fiction conventions since 1975.[1][3] He was an editor of The Westerfilk Collection: Songs of Fantasy and Science Fiction, an important filksong collection, and later a partner in Off Centaur Publications, the first commercial publisher specializing in filk songbooks and recordings.[1][11][12] An astrophysicist character with his name appears in War of Honor, a military science fiction novel in the Honor Harrington series by David Weber.

Publications

Musical

  • Self-published two albums of his songs, Fire in the Sky (1991; distributed by Wail Songs) and Parody Violation: Jordin Kare Straight and Twisted (2000)[1]

Honors

References

  1. ^ a b c d Filk biography in CopperCon 22 Filking News
  2. ^ a b Jordin Kare. "Intersection Science Programme Participants: Jordin Kare". Intersection Science Programme Participants. John Bray. http://www.johnbray.org.uk/intersection/people.html#Jordin%20Kare. Retrieved 2007-08-13. 
  3. ^ a b Capclave 2005: Confirmed Program Participants: Jordin Kare
  4. ^ a b c "Space Access Update #93". Space Access Society. 2000-04-13. http://www.space-access.org/updates/sau93.html. Retrieved 2007-08-13. 
  5. ^ a b "The Monell Connection, Winter 2003" (PDF). Monell Chemical Senses Center. 2003. pp. 9. http://www.monell.org/Newsletters/Monell_Fall03.pdf. Retrieved 2007-08-13. 
  6. ^ Alex Soojung-Kim Pang (2001-02-19). "Interview with Susan Kare". Making the Macintosh. Stanford University. http://library.stanford.edu/mac/primary/interviews/kare/index.html. Retrieved 2007-08-13. 
  7. ^ Jordin Kare Laser Launch Bibliography (1986-1992)
  8. ^ Dr. Jordin T. Kare (2004-05-18). "Modular Laser Launch Architecture: Analysis and Beam Module Design. Final Report." (PDF). Kare Technical Consulting. http://www.niac.usra.edu/files/studies/final_report/897Kare.pdf. Retrieved 2007-08-12. 
  9. ^ http://www.foxnews.com/scitech/2011/01/25/nasa-exploring-lasers-beams-zap-ro ckets-outer-space/
  10. ^ "LaserMotive Team Bios". http://www.lasermotive.com/blog/?page_id=6. Retrieved 2007-09-06. 
  11. ^ Jordin Kare. "Filk music?". Sing Out!. http://www.interfilk.org/interfilk/singout.htm. Retrieved 2007-08-13. 
  12. ^ "Jordin Kare". Fan Gallery. SCIFI Inc.. http://www.scifiinc.net/scifiinc/gallery/bio/Kare,_Jordin.htm. Retrieved 2007-08-16. 
  13. ^ Hertz Foundation. "Hertz Foundation Fellows". http://www.hertzfoundation.org/dx/fellowships/fellows.aspx. Retrieved 23 January 2011. 

External links

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