olliemath

Math Matters, December 2006

2006-12-18T14:59:00.000-08:00

Unfortunately, that is the only kind of math that I have given any thought to lately.

But here are some things to think about over break, (while I am finishing a referee job)

Geometric Topology: there is a way to form a wild knot as an intersection of nested solid tori, and I think that I can prove that if two knots from this class are equivalent, then the defining sequences must also be equivalent up to some sort of cofinality condition. If I can do this, then perhaps I can come up with some invariants that distinguish very ill behaved will knots (that is, knots that are so wild that they fail to pierce a disk anywhere)

College level mathematics: suppose one has the differential equation y”+y = f(t) and f(t) is both bounded and periodic of period 2pi as well as piecewise smooth. Then any solution to this differential equation is bounded on [0, infinity) if and only if the k = 1 term of the Fourier series expansion vanishes. Sufficiency is easy to show; necessity is where the work lies.
Of course, this is just a mild extension of the concept of resonance; for example the solution to y” + y = sin(t), y(0)=0, y’(0)= -1/2 is y(t) = -(1/2)tcos(t), which has unbounded oscillations as t goes to infinity, even though the driving function is bounded. For those who like to see examples, I suggest surfing to the interactive differential equations website. Click enter, and then go to the menu; there is tons of stuff there. You can use sliders to vary parameters and you can watch the phase plane as the systems evolve with time.

Ok, now I’ve said it. Now what am I going to do about it?

Topic: math teaching

2006-10-02T10:28:00.000-07:00

Math and People Skills

It appears that the students consider me to be a bit more approachable this year. Hence, they ask more questions and therefore I can see what confuses them.

Of course, some of that startles me.

I am teaching differential equations and a current topic is linear second order differential equations.

For those who have had some calculus, a differential equation is an equation involving a function and its derivatives (possibly some of higher order).

Example: y' + y = 0 is an example of a differential equation; this reads: a function plus its derivative is always zero. A general solution is y = a*exp(-t) because
y' + y = -a*exp(-t) + a*exp(-t) = 0.

Now if we had instead: y' + y = cos(t) we would obtain:

y = a*exp(-t) + (1/2)(cos(t) + sin(t)) as a solution. The solution with the arbitrary constant is called the homogenous solution, and the one without is (the trig part, in this case) is called the particular solution.

There are many techniques for solving these; you know that you are done by the various existance and uniqueness theorems (e. g. differential equations meeting certain criteria have a unique solution, so if you get a solution by "any means necessary" (e. g. good guessing) then you have all of the solutions.)

So, we need to teach "good guessing" which is sometimes known as "the method of undetermined coefficients".

The basic principle is that

if yp(t) = exp(a*t) then all derivatives are of (something)*exp(a*t)
if yp(t) = (b*cos(r*t) + d* sin(r*t)) then all derivatives are sums of sin(r*t) and cos(r*t)
if yp(t) = polynomial, then all derivatives are polynomials.

That is, certain types of functions formed a closed class with respect to taking derivatives.

To see this, consider a function that does not fall into this category: y(t) = sec(t)
taking derivatives yields tan(t)sec(t), then sec^3(t) + tan^2(t)sec(t), etc., which are not of the form (some constant) * sec(t)

So the good guessing technique doesn't work with sec(t).

This is something I understood from day one as a student.

Most of my students (or at least many) don't have a natural understanding of this.

I never knew that.

Math Fun From An Old Friend

One of my old graduate student buddies is a professor at Pittsburg State University in Kansas.
He calls himself The Okie in Exile.

His reasearch area is geometric topology; he has published in the area of non-compact three manifolds. He is one of the world's experts in Whitehead manifolds.

He has put out a couple of Youtube intervals in which he teaches math topics via folksy, funny stories.

The first of these deals with the Chinese Remainder Theorem in a clever way; the second one talks about Egyptian fractions. Enjoy!

Moral: When you write an exam, check your problems.

2006-09-17T12:53:00.000-07:00

Mathematics: Differential Equations

I almost had some egg on my face. In differential equations class, we had just covered the basic existance and uniqueness theorems for linear first order differential equations. That is, if one is given y' = f(t, y) where f is continuous at (t0, y0) and continuous on some open rectangle containing (to, yo) then the differential equation y'=f(t,y), y(t0) = y0 has at least one solution. Furthermore, if the partial derivative of f(t,y) with respect to y is similarly continuous at (t0, y0), then the solution is unique.

Example: y' = t*exp(y), y(t0) = y0 has a unique solution for any initial condition,
y' = t*y^(2/3) has a soluton of all initial conditions, but fails to have a unique solution for any initial condition y(t0) = 0, as the partial derivative with respect to y is t*(2/3)*y^(-1/3) which fails to be continuous when y = 0.

So, on the test I gave y' = y^(2/3)*ln(t)*tan(y) and asked if there was a guarantee for a unique solution at y(2) = 0.

Answer: not that obvious! taking the partial derivative with respect to y we get:

ln(t) (y^(2/3) * sec^2(y) + (2/3)^y(-1/3)*tan(y)). Pluggin in t = 2 causes no difficulty, but what about y = 0? The first term gives 0 (no problems) but what about the second?

Actually, if you go through the proof (of the uniqueness theorem), it is the fact that we have some boundedness condition for the partial with respect to y (or a Lipshitz condition, at least).

Note what happens if we apply L'Hopital's rule to the second term:
lim as y goes to zero of (2/3)*(tan(y)/y^(1/3)) = (2/3)(sec^2(y)/ (1/3)y^(-2/3)) = (2/3)(3)sec^2(y)y^(2/3) = 2*1*0 = 0 !!!

In other words, technically speaking, the partical derivative may not have been continuous, but the singularity is removable and hence the uniqueness condition holds.

I didn't anticipate this at all and was lucky that I caught this prior to passing out the exams. Moral: test your problems ahead of time.

New Calculators: Brain still required

2006-09-01T13:09:00.000-07:00

I teach lots of calculus as I am a small-time college mathematics professor.

By "small time" I mean that my research never put me in contention for an endowed professorship anywhere!

One of the biggest changes we've had to calculus teaching over the past 15 years or so is the introduction of calculators that do symbolic calculus; for exampe, if you wanted to find:
integral(e^2*x dx) you could just enter it and obtain (1/2)e^(2x). You'd have to add the "c", of course.

So, after a recent class, one student brought me the following:

integral((e^x )*sin(2*x) dx) and obtained:
(1/pi^2 + 8100)*e^x*(-90*pi*cos(2*x) + 8100*sin(2*x)) + C

What did the student do wrong?

Daily Kos and Mathematics?

2006-08-19T11:30:00.000-07:00

What does the Daily Kos have to do with mathematics? Well, we have a "Science Friday" and this past Friday we had a gem of an article:

http://www.dailykos.com/story/2006/8/18/10571/4335

The Poincaré Conjecture
by angrytoyrobot [Subscribe]
Fri Aug 18, 2006 at 07:57:01 AM PDT
It's Science Friday!
We are here to blow up the brains of any Republicans who may be sneaking, yes SSSNEAKING Precious, around this board.
It was mentioned earlier, in another post (on cats) that there had been no mention of the Poincaré Conjecture Proved--This Time for Real
In mathematics, a 3-manifold is a 3-dimensional manifold. A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional.
angrytoyrobot's diary :: ::

In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and a pair of circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.
Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.
Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.
http://mathworld.wolfram.com/...
Russian mathematician Dr. Grigori (Grisha) Perelman of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg) gave a series of public lectures at the Massachusetts Institute of Technology last week. These lectures, entitled "Ricci Flow and Geometrization of Three-Manifolds," were presented as part of the Simons Lecture Series at the MIT Department of Mathematics on April 7, 9, and 11. The lectures constituted Perelman's first public discussion of the important mathematical results contained in two preprints, one published in November of last year and the other only last month.
Perelman, who is a well-respected differential geometer, is regarded in the mathematical community as an expert on Ricci flows, which are a technical mathematical construct related to the curvatures of smooth surfaces. Perelman's results are clothed in the parlance of a professional mathematician, in this case using the mathematical dialect of abstract differential geometry. In an unusally explicit statement, Perleman (2003) actually begins his second preprint with the note, "This is a technical paper, which is the continuation of [Perelman 2002]." As a consequence, Perelman's results are not easily accessible to laypeople. The fact that Perelman's preprints are intended only for professional mathematicians is also underscored by the complete absence of a single reference to Poincaré in either paper and by the presence of only a single reference to Thurston's conjecture.
Stripped of their technical detail, Perelman's results appear to prove a very deep theorem in mathematics known as Thurston's geometrization conjecture.
LINK
Thurston's conjecture has to do with geometric structures on mathematical objects known as manifolds, and is an extension of the famous Poincaré conjecture. Since Poincaré's conjecture is a special case of Thurston's conjecture, a proof of the latter immediately establishes the former.
Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.
Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.
The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."
Thurston's conjecture is that, after you split a three-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:
1. Euclidean geometry,
2. Hyperbolic geometry,
3. Spherical geometry,
4. The geometry of S^2xR,
5. The geometry of H^2xR,
6. The geometry of the universal cover SL_2R^~ of the Lie group SL_2R,
7. Nil geometry, or
8. Sol geometry.
Here, S^2 is the 2-sphere (in a topologist's sense) and H^2 is the hyperbolic plane. If Thurston's conjecture is true, the truth of the Poincaré conjecture immediately follows. Thurston shared the 1982 Fields Medal for work done in proving that the conjecture held in a subset of these cases.
Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry (the geometry of constant negative scalar curvature). However, the geometry of constant positive curvature is still poorly understood, and in this geometry, the Thurston elliptization conjecture extends the Poincaré conjecture (Milnor).
Results due to Perelman (2002, 2003) appear to establish the geometrization conjecture, and thus also the Poincaré conjecture. Unlike a number of previous manuscripts attempting to prove the Poincaré conjecture, mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any mistakes (Robinson 2003).
In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953, pp. 486 and 498), Poincaré's conjecture stated that every closed simply connected three-manifold is homeomorphic to the three-sphere. Here, the three-sphere (in a topologist's sense) is simply a generalization of the familiar two-dimensional sphere (i.e., the sphere embedded in usual three-dimensional space and having a two-dimensional surface) to one dimension higher. More colloquially, Poincaré conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes. This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3.
The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).
Renewed interest in the Poincaré conjecture was kindled among the general public when the Clay Mathematics Institute included the conjecture on its list of million-dollar-prize problems. According to the rules of the Clay Institute, any purported proof must survive two years of academic scrutiny before the prize can be collected. A recent example of a proof that did not survive even this long was a five-page paper presented by M. J. Dunwoody in April 2002 (MathWorld news story, April 18, 2002), which was quickly found to be fundamentally flawed.
Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes.
References
Clay Mathematics Institute. "The Poincaré Conjecture."
http://www.claymath.org/...
Johnson, G. "A Mathematician's World of Doughnuts and Spheres." The New York Times, April 20, 2003, p. 5.
Perelman, G. "Ricci Flow and Geometrization of Three-Manifolds." Massachusetts Institute of Technology Department of Mathematics Simons Lecture Series.
http://www-math.mit.edu/...
Perelman, G. "The Entropy Formula for the Ricci Flow and Its Geometric Application." November 11, 2002.
http://www.arxiv.org/...
Perelman, G. "Ricci Flow with Surgery on Three-Manifolds." March 10, 2003.
http://www.arxiv.org/...
Poincaré, H. Oeuvres de Henri Poincaré, tome VI. Paris: Gauthier-Villars, pp. 486 and 498, 1953.
Robinson, S. "Russian Reports He Has Solved a Celebrated Math Problem." The New York Times, April 15, 2003, p. D3.

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Go to the link to see the rest of the article, where the author applies this to politics and to see the comments.

Class Size: whose perspective?

2006-08-17T13:13:00.000-07:00

I was reading the most recent College Mathematics Journal and found an article written by a former professor of mine, Allen Schwenk.

Yes, I had this guy for third semester calculus, freshman year. And yeah, I made a "C"; "weak" academic performance he said.

Why? Well, at that time, I simply wasn't intellectually mature enough to learn from the way I was being taught; I'd have loved him for my upper division courses.

Lesson: don't judge a professor by the opinion of a freshman! Other students did quite well in his classes.

Eventually, he moved from Annapolis to Western Michigan University.

Anyway, back to his article (and I always enjoy his articles):

When one shops for universities, one often encounters the statistic "average class size"; here is an example.

So, what does this mean?

Let's look at a hypothetical example: we have a small school of 500 students. Each student takes 5 classes: math, language, history, political science, and science.

Math is taught in 25 sections of 20 students each (500)
Language is taught in 50 sections of 10 students each (500)
History is taught in 20 sections of 25 students each (500)
Political Science is taught in 5 sections of 100 students each (500)
Science is taught in 2 sections of 250 students each (500)

So the school would say: we have 25 + 50 + 20 + 5 + 2 = 102 sections
And the class enrollment is: 500 * 5 = 2500
So the "average class size" is 2500/102 = 24.5 students per class.

And, this is what the faculty (one per section) would experience.

But what is it like for the student? In this simple example, the student would see:
5 classes, of size 20, 10, 25, 100, and 250, or each student "sees" 405/5 = 81 students per class.
If this seems strange, think of it this way: those 2 sections of 250 students each are experienced by 500 students, but only by 2 faculty members. Hence this receives "heavier weight" when one does the calculation from the student's perspective.

In the article, Dr. Schwenk takes this beyond the hypothetical example, and shows that the student will always see a class size at least as large, or larger than the faculty does.

He gives a couple of proofs; if you wish to try this yourself (and this is an elementary, but tricky problem; it requires some cleverness to set up) and you want a hint, think: "Cauchy Schwartz" inequality, or think of the proof from statistics that says the sum of the squares is always at least as great, or greater than the square of the sums.

Two cool articles

2006-08-15T12:55:00.000-07:00

I commend Mark Johnson for alerting me to two excellent articles that contain stuff about mathematics.

The first one deals with my research area: topology.

The second one is an interview with Field's Medalist Edward Witten on String Theory.

Topology: the Poincare Conjecture and Pearlman's proof

http://www.nytimes.com/2006/08/15/science/15math.html?_r=1&pagewanted=print&oref=slogin

August 15, 2006

Elusive Proof, Elusive Prover: A New Mathematical Mystery

By DENNIS OVERBYE

Grisha Perelman, where are you?

Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.

After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.

Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.

As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.

“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”

In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.

Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”

But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.

Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.

“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.

Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.

Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”

William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.

“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”

Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.

The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.

In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.

In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.

Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.

“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.

In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.

Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.

“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1986.

Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.

In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.

Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.

Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.

“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.

It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.

Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.

Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.

In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.

“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.

Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.

Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.

Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.

But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.

“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”

Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.

In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures..

Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”

Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).

Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.

James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.

“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.

In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.

Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.

Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”

PBS inverview with Edward Witten on String Theory

http://www.pbs.org/wgbh/nova/elegant/view-witten.html

Viewpoints on String Theory
Edward Witten

The Elegant Universe homepage

Many physicists consider Ed Witten to be Einstein's true successor. A mathematical physicist at the Institute for Advanced Study in Princeton, New Jersey, Witten has been awarded everything from a MacArthur "genius grant" to the Fields Medal, the highest honor in the world of mathematics. His contributions to string theory have been myriad, including the time in 1995 when he gave the then somewhat moribund field a much-needed boost by showing how the five different variations of the theory then competing with one another actually all belonged under one umbrella. In this interview, Witten talks about how the big bang could have coughed up a string so large that it might still survive in the universe and be visible with telescopes—and other ideas to make your head spin.

Note: For a definition of unfamiliar terms, see our glossary.

Spreading out particles

NOVA: What is string theory?

Witten: String theory is an attempt at a deeper description of nature by thinking of an elementary particle not as a little point but as a little loop of vibrating string. One of the basic things about a string is that it can vibrate in many different shapes or forms, which gives music its beauty. If we listen to a tuning fork, it sounds harsh to the human ear. And that's because you hear a pure tone rather than the higher overtones that you get from a piano or violin that give music its richness and beauty.

So in the case of one of these strings it can oscillate in many different forms—analogously to the overtones of a piano string. And those different forms of vibration are interpreted as different elementary particles: quarks, electrons, photons. All are different forms of vibration of the same basic string. Unity of the different forces and particles is achieved because they all come from different kinds of vibrations of the same basic string. In the case of string theory, with our present understanding, there would be nothing more basic than the string.

NOVA: Why does something as simple as replacing points with strings make such a huge difference?

Witten: It's indeed surprising that replacing the elementary particle with a string leads to such a big change in things. I'm tempted to say that it has to do with the fuzziness it introduces. So the particle is spread out. But it turns out that everything about spacetime is a little bit spread out; it's blurred. You have to start doing some calculations to really see it. It's hard to explain it just in words or by drawing pictures.

“You enter a completely new world where things aren’t at all what you’re used to.”

Spreading out the particle into a string is a step in the direction of making everything we're familiar with fuzzy. You enter a completely new world where things aren't at all what you're used to. It's as surprising in its own way as the fuzziness that much of physics acquired in light of quantum mechanics and the Heisenberg uncertainty principle.

NOVA: In reading about string theory and in talking to people about physics in general, we hear a lot about string theory being beautiful, but what does that mean? What's beautiful about it?

Witten: Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle. It's the kind of beauty that might be hard to explain to a person from a different walk of life who doesn't deal with science or math professionally. But the beauty of Einstein's equations, for example, is just as real to anyone who's experienced it as the beauty of music. We've learned in the 20th century that the equations that work have inner harmony.

Now there must be skeptics out there who will tell you that these beautiful equations might have nothing to do with nature. That's possible, but it's uncanny that they are so graceful and that they capture so much of what we already know about physics while shedding so much light on theories that we already have.

Enter fuzziness

NOVA: Can you give us an example of something concrete that string theory does for physics that goes beyond previous theories?

Witten: In Einstein's general relativity the structure of space can change but not its topology. Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything. You can imagine a bowling ball and you can imagine a coffee cup that has a handle—the coffee cup is different topologically because there's a handle. Even if you could bend it or stretch it, as long as you don't break it, it's still got that handle, which makes it topologically different.

There was a long history of speculation that in quantum gravity, unlike Einstein's classical theory, it might be possible for the topology of spacetime to change. And it turned out in string theory in the late '80s and early '90s we actually were able to calculate examples where you could really see changes in the topology of spacetime. That was fun because it was very concrete—you could understand it pretty well. And it illustrated how the theory went beyond Einstein's general relativity as understood before in a very nice and down-to-earth way.

NOVA: How does string theory allow you to change the topology of spacetime?

Witten: Quantum mechanics brought an unexpected fuzziness into physics because of quantum uncertainty, the Heisenberg uncertainty principle. String theory does so again because a point particle is replaced by a string, which is more spread out. And even though it's a naïve statement, it leads in the right direction: when we study it more deeply, we find that in string theory, spacetime becomes fuzzy.

So imagine now we have this coffee cup. If the handle is big enough, you can see it's there. But if you had a very small handle, because of the fuzziness of spacetime, you couldn't tell if it was there or not. Then it could disappear. That fuzziness of spacetime leads to the possibility that the topology can change.

Extra dimensions required

NOVA: String theory requires 10 dimensions of space. Does that complicate the theory or does it solve problems?

Witten: Technically you need the extra dimensions. At first people didn't like them too much, but they've got a big benefit, which is that the ability of string theory to describe all the elementary particles and their forces along with gravity depends on using the extra dimensions. You have that one basic string, but it can vibrate in many ways. But we're trying to get a lot of particles because experimental physicists have discovered a lot of particles. The electron and its heavy cousins the neutrinos, the quarks, photons, gravitons, and so on. There is really a big zoo of elementary particles that you're trying to explain. Having those extra dimensions and therefore many ways the string can vibrate in many different directions turns out to be the key to being able to describe all the particles that we see.

NOVA: But why do we think that they might actually exist? We certainly can't see them.

Witten: We see light waves with our eyes, but most of the other particles take 20th-century equipment to discover them. As for the forces, electromagnetism and gravity we experience in everyday life. But the weak and strong forces are beyond our ordinary experience. So in physics, lots of the basic building blocks take 20th- or perhaps 21st-century equipment to explore.

“I would conclude that extra dimensions really exist. They’re part of nature.”

As far as extra dimensions are concerned, very tiny extra dimensions wouldn't be perceived in everyday life, just as atoms aren't: we see many atoms together but we don't see atoms individually. In a somewhat similar fashion, our experiences and our observations would average over the extra dimensions if those were small enough.

NOVA: So just because we don't see them doesn't mean the theory is wrong?

Witten: The theory has to be interpreted that extra dimensions beyond the ordinary four dimensions the three spatial dimensions plus time are sufficiently small that they haven't been observed yet. So we would hope to test the theory, conceivably directly at accelerators. I suspect that's a long shot. More likely we'll do it indirectly by making more precise calculations about elementary particles based on the existence of extra dimensions.

NOVA: Do you think extra dimensions actually exist, or are they a mathematical device?

Witten: If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways. They're beyond our ordinary experience just like atomic nuclei are. On the other hand, we don't understand the theory too completely, and because of this fuzziness of spacetime, the very concept of spacetime and spacetime dimensions isn't precisely defined. I suspect that the fuzziness of spacetime will play more of a role in the eventual answer than we understand now. [To try to picture a fourth spatial dimension, see Imagining Other Dimensions.]

NOVA: If these extra dimensions exist, does string theory offer any explanation of why there are apparently three space dimensions larger than the rest?

Witten: That's a big problem that has to be explained. As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map.

Verifying string theory

NOVA: It seems like the standard criticism of string theory is that it isn't testable. How do you respond to that criticism?

Witten: One very important aspect of string theory is definitely testable. That was the prediction of supersymmetry, which emerged from string theory in the early '70s. Experimentalists are still trying to test it. It hasn't been proved that supersymmetry is right. But there is a very precise relationship among the interaction rates of different kinds of particles which follows from supersymmetry and which has been tested successfully. Because of that and a variety of other clues, many physicists do suspect that our present decade is the decade when supersymmetry will be discovered. Supersymmetry is a very big prediction; it would be interesting to delve into history and try to see any theory that ever made as big a prediction as that.

NOVA: What are some of the other ways that string theory could be confirmed experimentally?

Witten: There are a lot of conceivable ways we could get experimental information that would help with string theory. Explorations of cosmology, studying the cosmic background microwave radiation and hopefully finding gravitational waves left over from the big bang and studying their properties are very plausible avenues for eventually testing string theory, although there isn't yet to my thinking a satisfactory theoretical understanding of what to expect.

But it's conceivable that the big bang could have produced a string so large that it would be present in today's universe and visible in telescopes, perhaps discoverable by the satellites that are now mapping out the microwave sky. If that were discovered, it would be a dramatic confirmation of the existence of strings. Still, that's a story that will develop over the next decade or two as the experiments progress and conceivably as the theory progresses.

NOVA: How likely do you think it is that string theory will be proven correct?

Witten: Well, I don't have a crystal ball. You know, the theory of neutron stars was tested and the same is true of the theory of black holes and the theory of gravitational waves. A lot of the theories that were there in the '20s and '30s that looked like they were way beyond reach were eventually tested. They were tested because there were new technologies, there were new instruments, there were newer things found in the sky. Things happened that you couldn't foresee. That's what happens in science.

“I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past.”

So when you ask me how string theory might be tested, I can tell you what's likely to happen at accelerators or some parts of the theory that are likely to be tested. But I also have to point out that part of the answer is the unknown. Just as the theory of neutron stars, black holes, gravity waves, and so many other things were tested because of things that nobody foresaw, there are just so many ways that nice surprises could happen that would lead to new advances in string theory. There are all kinds of possibilities, like literally seeing a string in a telescope if nature has chosen to be kind to us in that particular way. I think that nature will turn out to be kind to us and that there will be some nice surprises, as there have been so many times in the past in science. But if I could tell you what they were, they wouldn't be surprises.

NOVA: Do you think it's possible that string theory will turn out to be wrong, or at least some branch of knowledge that just isn't connected to nature?

Witten: I guess it's possible that string theory could be wrong. But if it is in fact wrong, it's amazing that it's been so rich and has survived so many brushes with catastrophe and has linked up with the established physical theories in so many ways, providing so many new insights about them. I wouldn't have thought that a wrong theory should lead us to understand better the ordinary quantum field theories or to have new insights about the quantum states of black holes.

The question reminds me a little bit of the question about interpreting fossils. When fossils were first explored 100 or 200 years ago, some people thought they were traces of past life that had survived in the rocks and others thought that they had been placed there at the creation of the universe by the creator in order to test our faith. So I guess string theory might be wrong, but it would seem like a kind of cosmic conspiracy.

NOVA: It's been said that string theory really belongs to the 21st century. Do you agree?

Witten: Back in the early '70s, the Italian physicist, Daniele Amati reportedly said that string theory was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark. How wise it was is so clear from the fact that 30 years later we're still trying to understand what string theory really is. What Amati meant was that usually the physical theory isn't developed until there are more or less the concepts and ideas in hand for making sense out of it. By the time Einstein developed general relativity, he actually knew what he was doing.

But string theory wasn't like that. The first traces appeared in 1968 with the Venetziano model. Nobody at the time had the conception that could have led to string theory in a clear way or understood what it was. It was something incredibly beautiful, a trail that people followed without understanding what it was. We've come through 30 years of remarkable discoveries, and we can see a lot of puzzles still ahead.

NOVA: Where does inspiration in this field come from for you?

Witten: You have to be open-minded because ideas come from different places. You can think about something in one way for a long time and it seems like the only way to think about it, but it really isn't. Somebody could make a suggestion that really sounds naïve. It might even be naïve, but it could have an important element of the truth in it. And it could be truth that one's overlooking. So it's really hard to state a general rule. If one could say the general rule about where to find inspiration, we would just teach it to our students and then science would be much more straightforward.

Math Fest, 2006

2006-08-15T12:46:00.000-07:00

This is a summary from my obeservations from the Mathematical Association of America's Math Fest 2006.

Everything here has been included in my personal blog, blueollie. Here I've attempted to strip off irrelevant materal. I am starting with a little bit about my adventure getting to the conference; those who want to read only about mathematics should scroll down to the next section.

Day One; getting there, getting situated.

Today, I work up a bit later than normal due to being very tired. Hence, I limited my workout to a 3 mile run on the treadmill followed by 15 minutes of yoga.

This hotel (Crown Plaza) has an excellent workout facility and a reasonably well stocked weight room; I'll use it a bit more extensively tomorrow morning, I hope.

The trip itself: interesting. First of all, I noticed that in Chicago (the day prior to this trip) and at times on this trip, more people than normal have given me nods of approval.

I wonder why; it isn't as if I've trimmed down that much; then I realized: I now have a very close cropped crew cut. My guess is that many think that I am military or perhaps police.

Go figure; if only they knew that I was a Kos reading lefty.

As far as the road trip, here is the good: my Prius got 45.8 miles to the gallon, and that included the stretch through big hills. The bad: I left my credit card in a Peoria drugstore (CVS at Campus town). Fortunately, they still had my card, but I didn't know that at the time. I had to call after hours to get this card cancelled and that was a trip. I highly suggest carrying the numbers of your credit card companies with you for after hours emergencies such as this one.

Numbers: VISA (800) 847-2911
Master Card: 1-800-MC-ASSIST (1-800-622-7747)
Discover: 1-800-DISCOVER (1-800-347-2683)
American Express: 1-888-412-6945

Anyway, having these numbers would have helped me some.

But, between playing with Vickie, hitting an early rain storm and discovering that my check card was lost, and having to stop almost every hour to relieve my bladder, I didn't get in until after 9 pm central time, or after 10 pm local time.

I was very tired.

Day One: The Mathematics

The good news is that the talks so far have been outstanding.

Dorothy Buck lead off with an excellent talk about knot theory and its role in understanding the chemical reactions in DNA. Basically, Dr. Buck and Erica Flapan have shown that, subject to three biological assumptions, that only 6 types of links need to be considered (when one wants to understand the cross changing operations that DNA undergoes) and, if one uses tangle theory, one can pass to the theory of knots and lens spaces by passing to the double branched cover of a tangle (which is a solid torus) and studying Dehn surgery.

Next, Fields Medalist W. T. Gowers gave a nice talk about some "easy to state, hard to solve" combinatorial problems. Exampe: let R be a commutative ring and A a subset. The sum set is the set of all elements of R of the form x + y where x and y are both in A (note: x + x is permitted). The multiplicative set is defined in a similar way.

So, suppose that you know that the additive and multiplicative sets are of a certain size: what can you say about the given set to begin with?

Next, Jesus De Loera gave a nice lecture on convex polytopes and gave some simple to state, yet still unsolved conjectures. Here is one:

in dimension three, it is known that if you are given a triple of integers x, y, z such that

x -y + z = 2
2y >= 3z
2y >= 3x

Then there is a unique convex polytope with x vertices, y edges and z faces. For example, for a tetrahedron, x = 4, y = 6, z = 4,for a cube: x = 8, y = 12, z = 6, for a pyramid: x = 5, y = 8, z = 5.

Such a formula, if one exists, is completely unknown for 4 dimensional polytopes.

Of course, these were some of the major invited addresses; I expected these to be good. But the contributed talks went well as well.

Among those: I head a nice talk about Hill's cipher algorithm (given by Bill Wardlaw, who was my first abstract mathematics professor at Annapolis), check digits of codes which detect errors in digits as well as order of digits (using the action of permutation groups on dihedral groups) , using magic squares (and their relatives) as examples of vector spaces and ideals in rings, enumeration of sudoku puzzles. I also heard a nice talk by Ed Burger on how to (not) teach a class that introduces the idea of proofs to undergraduates. Ed was a friend in graduate school; he now has a list of honors 10 miles long; included in these was the Chauvenet Prize for expository writing. Sigh...I knew him when...

This is Ed Burger, answering questions after his well received talk. If someone thinks that stellar teaching and good research can't go hand in hand, think again! Ed is proof that it can.

Though this shot leaves something to be desired, this is Bill Wardlaw of the U. S. Naval Academy, who taught me my first course on Modern Algebra back in the spring of 1979.

Day Two: I give my talk (ok, it is a 10 minute talk)

Personal Journal: I started the day with a short walk on the treadmill (3 miles, 37 minutes; 24 minutes for the last two miles) followed by weights and yoga.

Today I gave my talk; it seemed to go ok. Here is what I talked about: I presented a paper which said, in effect, that if one attempted to evaluate the limit of a two variable function by evaluating it over all curves that had continuous first derivative and ran through a given point and if one obtained the same limit over all such curves, then the limit of the function exists there.

Note that this is false if one replaces "all curves with continuous first derivatives" by "all lines" or even by "all real analytic functions" (functions which have a power series expansion at every point in their domain which is valid on some open set).

In addition to giving my talk, I also dropped some dollars on math books. This is to be expected.

Notable talks, in addition to the Gowers talk, were the series on Physical Knot Theory.

First, Ken Millett talked about using the computer to make random knots (physical) and to see what knots were produced. Note that he wasn't merely interested in knot type, but also the various physical qualities that a given knot representation had. He was especially interested in "equilateral polygonal knots" (knots formed by line segments, each of the same length).

Eric Rawdon informed me that, up to now, equilateral polygonal knots were realized at their (known) minimum stick number presentation, though the knot 8-19 might not have this happen.

Next Lou Kauffman talked about knots and rational tangles. He described work with Sofia Lambropoulou. In particular, he used the theory of rational tangles to find a way to recognize the unknot. Note that recognizing the unknot isn't always easy on computational grounds; a Theorem of Hass and Lagarias shows that one can always unknot a diagram by using 2^cn Reidermeister moves where c is some fixed constant and n is the number of crossings of the diagram. Unfortunately, the lowest known value of c is 10¹¹.

Tom Banchoff then talked about Piecewise Circular Space Curves; that is, curves that are formed out of pieces of honest to goodness circles. He talked about various properties (e. g., rigidity) and about various kinds of "dual polygons" associated with such curves.

Eric Rawdon gave an interesting talk about tight knots. Here, the definition of "tight knot" means the following: Suppose you have a regular neighborhood of a smooth knot, where one, say, ensures that the regular neighborhood (a knotted solid torus) has some fixed radius as a diameter. Call this diameter "1". Then, using this scale, what is the shortest "length" that one needs to obtain a given knot?

Eric's papers can be found here: http://www.mathcs.duq.edu/~rawdon/Preprints/
There is enough here to keep you busy for a very long time.

Day Three: Another good session.

This post will start with the personal and then get to the mathematics. Again, those who wish can scroll down a bit.

Athletically: 4 miles of "running" on the treadmill; the first mile was a 10:10 warm up (5:30 first half mile). I admit that I was a bit distracted when writing this post as I am in a hotel lobby and a tall lady in a pretty, loose but sort of clingy dress just got on the elevator.

Hey, if I didn't like women, I wouldn't like my wife.

Now back to the post: my run was both good (got some exercise) and bad (couldn't hold a 8:30 pace for 2.5 miles; that used to be my marathon pace). The road back is very long; the good news is that my hip/back/IT band is slowly getting better.

While in Knoxville, I found a nice place to eat called The Tomato Head. It is in the Market Square. They serve good pizza as well as other types of food (salads, sandwiches, etc.) The servers and workers are from the "tye dye" set and are downright pleasant. One even asked me about my ultramarathon t-shirt (no, she wasn't my server so she wasn't sucking up for a tip). This is the kind of place you go to after yoga class; it is the type of place you go to in order to forget that George W. Bush is our President.

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The Mathematics from Day Three

The morning invited talks were outstanding. In the second talk of the day, Gowers finished his three part lecture on analytic combinatorics by showing the utility of the Discrete Fourier Transform, among other things.

In the first talk of the day, Trachette Jackson, who is one of the hottest young researchers, gave an excellent talk on the mathematics of cancer cell modeling.

The following is a very incomplete sketch of what she talked about: first, she gave the basics of cancer cells by giving a basic definition: what makes a cell cancerous? In a nutshell, a cancer cell is one which gets around dying naturally (evading apoptosis), doesn't take growth/death cues from its environment, can initiate and sustain angiogenesis (grows blood vessels to take blood from the host body), is invasive and has nearly limitless reproductive potential.

She said that the cells first start out as a somewhat spherical cluster which gets nutrition via its boundary. Eventually, the inner core starts to die off (gets starved; this is stage I); at that point, the cell starts growing capillaries to get a blood supply (this is stage II). At stage three, it is growing and has established its own blood supply.

There are ways to attacking cancer cells; some deal with trying to shrink the cell itself, some deal with trying to kill off the blood vessels supplying the cell (this was Judah Folkmin's 1998 breakthrough in mice that failed to be replicated in humans).

Her research deals with using partial differential equations to model the situation at various stages, and to see what happens to growth when various parameters are changed (reducing the ability to obtain oxygen, shrinking the newly made blood vessels, limiting the production of certain secreted chemicals, for example).

The partial differential equations that are obtained are not for the faint of heart; one such is
dN/dtt = phiv(V)N(1-N/N0) where N is the number of cells in the larger cancer cell, V is a measure of the blood vessel density, and phi is a function:

(r1c^2)/(c1^2 + c^2) - r2(1 - sigmac^2/(c2^2 + C^2)) where C is the oxygen concentration which is itself a function of v:
c(V) = CmV/(k+V) and Cm, sigma, k, c1, r1 and r2 are parameters which must be determined from experiment.

Of course, these equations must be solved numerically.

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The next invited talk was one about math history and the current teaching of calculus by David Bressoud. His talk centered around some of the basic theorems of real analysis and calculus and their historical context. What made this talk a bit different is that he used the history to explain why some of these theorems are difficult for the average undergraduate to understand or even appreciate.

The talk he gave can be accessed here: http://www.macalester.edu/~bressoud/talks/Knoxville.ppt

Some examples: take how we teach the Fundamental Theorem of Calculus. It wasn't even published in its current form until 1907; prior to that, various forms of "integral as some sort of an anti-derivative" definition was used rather than the traditional Riemann sums. The power of the fundamental theorem is that it shows that the Riemann sum and the anti-derivative are equivalent concepts, under certain special conditions. This association is not at all natural.

Yes, I know the proof.

He talked about Cauchy's attempt to provide some rigor to calculus, and how even good contemporary mathematicians found some of his stuff difficult to understand. And, he shows one of Cauchy's big mistakes; he thought that he proved that the infinite sum of a convergent series of continuous functions converges to a continuous function; Abel showed that this was false (think: Fourier Series) unless one had the notion of uniform convergence.

He then talked about the so-called Heinie-Borel theorem (ironically, Shoenflies was the first to refer to this theorem by this name) and how this theorem is really only needed at the start of the discussion of measure theory. My take: that might well be true, but H-B sure makes life easier.

For those who don't know what the H-B theorem says, it says that closed bounded sets in the real line are compact under the standard metric topology. That is, if one covers any closed interval by a collection of open intervals, then a finite subset of the collection of open intervals will also cover that given closed interval.

By the way, Heinie had very little to do with this theorem, though he did talk about uniform convergence.
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I attended two afternoon sessions of contributed talks. One was a session on the teaching of numerical methods. There were a few talks of "here is what I did and this is what happened" variety, one talk was about an ambitious plan to talk about discrete methods of solutions to partial differential equations, one was about automatic differentiation, and one talk was about the use of some cool applets for exploring various curve approximation schemes (splines, Lagrange, etc.)

The automatic differentiation talk by Richard Neidinger of Davidson was especially interesting. It basically views differential functions as 2-dimensional vectors: the first coordinate is the value of the function at a point, and the second coordinate is the value of the derivative. Then one does a weird sort of algebra with these vectors; one scalar multiplies and adds in the usual way (component wise). But multiplication of these vectors becomes interesting; one multiplies the first two components in the usual way, but one uses a bizarre dot product to multiply the second component (to respect the chain rule): (a , b) * (c, d) = (ac, ad + bc) under this "algebra".

The value is that differentiation can be done purely numerically once one knows the formula for the functions; for example, one doesn't need to enter the formula for the derivative of a function into, say, a Newton's root finding program.

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I then ended the conference by catching the tail end of the last general contributed paper session. There were a few good talks here as well.

Elena Constantin gave a nice talk about how one can generalize the Lagrange multiplier method/second derivative test even when these methods don't work (e. g., the Hessian matrix is singular). She showed how her method worked, though she didn't have time to give an outline of the proof. My guess is that she used the multi-variable power series method, but that is just a guess.

David Austin gave a nice talk about how to visualize the Cauchy-Riemann equations in complex analysis.

Basically, the theorem is this: if f(x + i y) = u(x,y) + i v(x,y) is a complex function, f is differentiable on an open set if and only if the so-called Cauchy-Riemann equations are satisfied:

u_x = v_y and u_y = - v_x

The usual proof is to use the limit definition of derivative and approach the point in question from the real axis direction and also from the y-axis direction, realize that one must get the same limit, and equate real and imaginary parts.

Austin gave a nice visual way of seeing this that uses the concept of how the complex derivative acts on sets in the domain (stretches by the magnitude of the derivative and rotates by the argument of the derivative evaluated at the given point).

His talk can be found here: http://merganser.math.gvsu.edu/david/mathfest06/

Jay Shiffman gave a talk about Collatz k-tuples. The Collatz conjecture says that any positive integer can be reduced to 1 by the following rules: if the integer is odd, multiply by three and add 1; if it is even, divide by two.

For example: start with 7. 7 * 3 + 1 = 22
22/2 = 11
11* 3 + 1 = 34
34/2 = 17
17*3 + 1 = 52
52 / 2 = 26
26 / 2 = 13
13*3 +1 + 40
40/2 = 20
20/2 = 10
10/2 = 5
5*3 + 1 = 16

16 is a power of 2: 16/ 2 = 8, 8 /2 = 4, 4/2 = 2, 2/2 = 1.
So, we say that 7 has "Collatz length" 16. (my note: to me, it was over when we reached 16 since 16 is a power of 2, so to me a better number to focus on would be 12, but I digress)

Now if, say, there are numbers n, n+ 1 that have the same Collatz length, these are called a Collatz pair. If there are numbers n, n+1, n+2 that have the same Collatz length, these are called a Collatz triple, and so on.

Anyway, Schiffman's talk was about this. Note: it is still unknown if any positive integer can be reduced to 1 by this process, though this has been verified for all numbers <= 3 x 2^(53). Then Mahmoud Almanassra closed the session with a nice talk about finding estimators for parameters of "quality adjusted lifetime hazard functions.

My reliability engineering course is a long way in my past, so here is my best recollection of the issues: a hazard function is a probability distribution which models the lifetime of something (human, electronic component, etc.) Typically, the density function follows a bathtub curve; models have initial defects, then go through ok, then start dying off near the end of their lives. If one wants to measure only for a certain length of time and then stop counting, one is "censoring" the data.

There are estimators for the parameters of such functions, but unfortunately two of the most common ones are not monotonic. This is bad because a non-monotonic estimator might well provide for "births after deaths." Almanassra showed how to correct for this.

Then, there is something called the quality adjusted hazard function; one takes into account the "quality of life" (obvious meaning in humans, say, usefulness in components?) He then develops consistent, monotone estimators for this setting.

Illinois Governor's Race: Misreading the polls.

2006-07-14T11:25:00.000-07:00

On a political website, I read the following:
http://www.dailykos.com/story/2006/7/14/12642/2045

IL-Gov: Is Blago pulling away?

by kos

Fri Jul 14, 2006 at 09:06:42 AM PDT
Rasmussen. 7/6. Likely voters. MoE 4.5% (4/8 results)

Blagojevich (D) 45 (38)
Topinka (R) 34 (44)

Huge turnaround. While Blago may be unliked and corrupt, even that might not be enough to hand the governor's seat to Republicans, not matter how much voters might flirt with Topinka (the only statewide elected Republican in the state). That's how far in the dumps Illinois Republicans are in.

The Blagojevich campaign has run an advertising campaign raising the question "What is She Thinking?" about Topinka and touching on issues ranging from the state budget to Iraq. The Blagojevich campaign has also raised complaints about lease payments by Topinka to a developer who is also a contributor. A better job performance in the state may also have something to do with the turnabout.

So, is this a huge turn-around? Well, maybe, maybe not.

Let's go to the poll itself:

http://www.rasmussenreports.com/2006/State%20Polls/July%202006/ilGovernor.htm

Survey of 500 Likely Voters
July 6, 2006

Election 2006: Illinois Governor
Rod Blagojevich (D)	45%
Judy Baar Topinka (R)	34%

July 13, 2006 After trailing earlier in the campaign season, Illinois Governor Rod
Blagojevich (D) now leads Republican State Treasurer Judy
Baar Topinka by 11-points in his bid for re-election. The latest Rasmussen
Reports poll shows Blagojevich with 45% of the vote to 34% for Topinka.

The Blagojevich campaign has run an advertising campaign raising the
question "What is She Thinking?" about Topinka and touching on issues
ranging from the state budget to Iraq. The Blagojevich campaign has also
raised complaints about lease payments by Topinka to a developer who is
also a contributor. A better job performance in the state may also have
something to do with the turnabout.

The governor, too, has had to contend with charges of impropriety, but
that's old news and he has more money to spend on campaign ads. Many
Republicans in the state concede that he has managed to put Topinka on the
defensive lately.

What is the key? The key is this: the poll was of 500 likely voters. This type of poll is what is known as a proportion. The theory is something like this: when you poll 500 voters, you are interested in, say, the number that say that they will vote for Blagojevich. That number (out of 500) is denoted by the letter "p" and the appropriate distribution is the binomial distribution. Typically, one then divides by the number polled to get a proportion.

The variance turns out to be p*q/n where n = number of voters and p is the assumed proportion (often set to .5 as that is the "null hypothesis"; that is, the initial assumption).

The standard deviation is the square root of the variance, so here we have "sd" = (.5*.5/500)^.5 = .0224. Since we are 95% confident that the true proportion will be within 2 standard deviations of the one that we actually measured, we have that Blagojevich's support is between .45 +- 2*(.0224) = (.4052, .4948).

In the previous poll, it was .38 +-2*(.0224) = (.3352, .4248). Note that these two confidence intervals overlap.

Now that doesn't mean that we can say that there is no change but it does point out that the change might not be as dramatic as it seems at first.

If we now measure the difference between the two proportions, and use the fact that the new variance is the sum of the absolute values of the two variances:

"variance" = 2*.5*5./500, so the standard deviation is: .0316, and so the difference in Blagojevich's support is (.45-.38) +- 2*.0316 = (.007, .133), which means that this difference in support for Blagojevich (between polls) might be less than 1%.

We can say, with confidence, that Blagojevich did not lose support.

What I am up to

2006-07-10T13:52:00.000-07:00

First a couple of notes: I found some sources that help with math symbols in HTML:

http://barzilai.org/math_sym.htm

http://www.utoronto.ca/ian/books/xhtml1/entity/en_symbol.html

and this is interesting too:

http://www.cs.tut.fi/~jkorpela/math/

I am kind of ashamed that I didn't know this earlier.

Next, while looking for good figures to put in this post (I didn't find them yet), I came across a fun little article which relates elementary knot theory with "impossible diagrams" (Escher type stuff)

http://www.mi.sanu.ac.yu/vismath/cerf/

It is a quick, enjoyable read.

Now back to the subject at hand: Imagine a knot inside of a torus:

(figure: http://www.rwgrayprojects.com/Lynn/DoubleTorus/dt01.html)

The yellow string of beads is the knot (trefoil). Now suppose that torus (the red solid object) is itself knotted up:

(figure: http://www.imasters.com.br/artigo/1772)

The knot inside the solid torus is called a satellite knot of the knot formed by the solid torus, and the knot formed by the solid torus is called a companion knot of the knot inside the torus. Of course, there is more to the definition than I am telling; for example, the knot inside of the torus (first diagram) must "really" lie in the torus and not in some ball inside of the torus (think of it this way: at the beginning of this process, the knot inside of the torus to be knotted must hit every meridional disk in the beginning torus (the red torus of the first figure). And the example I gave is of a cable knot.

Anyway, at the moment, I am interested in the following kind of object. Consider a knot inside of a companion. Now thicken that knot so that it is a solid torus. Now put another knot inside that. Thicken that knot so that it is a solid torus as well (inside of the two solid tori). Put a knot inside that one, thicken it again, and repeat the process infinitely.

Now take the intersection of this infinite collection of nested tori; kind of like these dolls:
except that the tori are tied up in more and more complicated fashion.

Now this infinite intersection is an example of what is known as a solenoid.

If we are careful about how we nest these tori; that is, if a longitude of each inner torus just goes once around the torus that contains it, we can get a simple closed curve in the limit.

Arnold Shilepsky came up with criteria that guarantees that we get a simple closed curve in the limit; see the references in his selected publication list. (These types of knots were first studied by a real giant of geometric topology: R. H. Bing), and Shilepsky was one of Bing's students).

These knots are very pathological. They are examples of "wild" knots; we'll say a knot is wild if either it is not the image of any differentiable embedding of the circle into 3-space (as in the sense of ordinary calculus) or if the knot cannot be realized by a finite number of straight line segments placed end to end.

But these knots are not just wild; they are so wild that they fail to "pierce a disk" at any point. Think of it this way: if you think of these knots as pieces of thread, there is no way to run this particular thread through an eye of a needle without running an infinite number of "strands" of this thread though the eye at the same time. That is, if one strand of this type of knot goes through the eye, then an infinite number of strands must go through at the same time as well.

Now the question I am interested in is this: how can one tell if two of these types of wild knots are really different; that is, if space can't be deformed to take one of these wild knots onto the other one?

What symmetries do they have? What about their groups?

A couple of mathematically flavored diaries from the Daily Kos

2006-07-01T09:07:00.000-07:00

A couple of mathematically related diaries taken from the Daily Kos: the first is about ecology and discusses some of the mathematical models. The second is about Bayes' law.
I will warn the reader that there are some politics in these diaries; I thought about trying to edit it out but then decided to leave them "as is."

First, a diary from Jmknapp from the Daily Kos.

http://www.dailykos.com/story/2006/6/29/22011/9126

Mathematical Challenges of Global Ecology

by jmknapp [Unsubscribe]

Thu Jun 29, 2006 at 07:00:11 PM PDT

All this week a group of climate scientists have been meeting for a Global Ecology workshop at Ohio State University. Abstracts of some of the talks can be found at OSU Mathematical Biosciences Institute

* Marine Carbon Cycle Biogeochemistry from the Bottom Up
* Quantifying the Predictability of Noisy Nonlinear Biogeochemical Systems
* Ecosystem-climate dynamics: What are the phenomena that pose mathematical challenges?
* Stoichiometry of wild plants and crops in the high CO2 world
...etc.

jmknapp's diary :: ::

I attended a lot of the talks, to get a feel for current thinking on the global warming crisis and to watch these scientists in action, on the global warming front line.

The conference generally steered clear of political issues, being concerned with more heady topics like transcritical bifurcations in systems of ordinary differential equations, oscillating relationships in predator-prey populations, hysteresis and catastrophic jumps in system behavior, maximum entropy production models, Navier-Stokes equations, ecological stoichiometry, Redfield ratios of C:N, eigenvalues of Jacobian matrices--fun stuff like that.

Still, it was interesting that Bush would come up from time to time and it was evident that the Bush administration is regarded with total disdain by these scientists. Not surprising, given the all-out assault on science that Bush has led, from evolution to the Big Bang to global warming.

One presenter included a Bush photo in one of his slides:

Underneath the photo was the relationship:

plant -> herbivore + waste

That got quite a few chuckles. Also below the photo was the statement: organisms are just abstract molecules

That thought, the speaker said, "is the only thing getting me through the last administration and a half." Another quipped, "It looks like he's saying 'you know where you can stick it!'"

During a talk on stability (or instability) of the global climate over Earth's history, the speaker noted that elements of the climatic system develop "without foresight, much like Republicans."

So this ire is definitely directed at Republicans and not all politicians. This is cutting the Democrats some slack, as I recall it was Congressional Democrats who scuttled Gore's carbon tax early on in the Clinton Administration. But still, Republicans are just so egregious in these areas that even an adminstration that "offered a friendly ear" is infinitely preferable. Several pointed out that however that a carbon tax, which could be designed to be revenue neutral, would be a great step towards mitigation of CO2 emissions. So Democrats need to pick up the ball there.

Gore's movie was seen by most. One esteemed professor from the University of Minnesota said, "I learned a few things. Gore was very careful with the science." Also, "The idea that someone could make a movie of a lecture and people would go to see it!" As for Gore's personal stories in the film, "If you want to present an interesting lecture you should include some personal anecdotes." An Oak Ridge scientist organized a group of his colleagues from the Carbon Dioxide Information Center to see the movie & it was very well received.

Dr. John Pastor gave a talk illustrating how there are several time constants involved in CO2 dynamics, and how perturbations in the climate can persist long after the stimulus is removed, so-called reactive transient responses. It is not known whether increased CO2 currently will produce such a response, even if emissions are curtailed.

Dr. John Harte (UC-Berkeley) points out that Milankovich (orbital) cycles explain the timing of the shifts in the CO2/temperature record over the last 400,000 years or more, but those orbital cycles don't explain the magnitude of the temperature changes, which involve various positive feedback factors making the swings larger. He wonders if perhaps the current trend of making models more and more complex, requiring supercomputers to crunch Navier-Stokes equations and the like, involving hundreds of thousands of variables (the GCM model) might need to be rethought, in favor of simpler mathematical techniques, e.g., current explorations of maximum entropy models.

Dr. Rob Armstrong (SUNY/Stonybrook) says that "the patient is sicker than we thought." He points out that acidification of the oceans (as they absorb CO2 and form carbonic acid) may ultimately dwarf global warming effects in terms of ecological damage (coral reef destruction, dead zones). This acidification is a positive-feedback mechanism for atmospheric CO2 levels, as acidified water is less able to absorb CO2, meaning the oceans become less effective as a buffer. Current pH levels are lower (more acid) than any time in the Vostok ice core record (400,000 years) and perhaps several million years.

Dr. Irakli Loladze (Ohio State) explored the effect of increased CO2 on plant growth and nutrient concentrations. Increased CO2 leads to non-uniform changes in N, P, K, S, Ca, Fe, Zn, Cu and many other trace nutrients. Studies have shown that doubling CO2 can lead to 30-40% boost in biomass, and reduce transpiration of water by 23%. However, much of the world suffers from "hidden hunger," that is, bellies are full but nutrient deficiencies exist, leading to anemia, cretinism, infant and maternal mortality, low birth weight, goiter and other maladies. In the US we have supplements to take up the slack, but in other countries like Bangladesh, 97% of calories come from plants (rice primarily) and natural nutrient levels are critical. Most of the trace elements in plants are lower today than heretofore. As a rule of thumb, doubling CO2 causes trace elements (other than C, O and N) to drop by 10%.

Loladze says that cities like Phoenix currently have up to 600 ppm CO2 levels due to the abundance of cars and the formation of CO2 domes over the city.

Corn productivity is up ten-fold, leading to a glut of nutritionally deficient corn on the market. Harte notes that agricultural interests are pushing for ethanol production to reduce this corn surplus, even though it not a long-term solution.

Dr. Raymond Pierrehumbert (University of Chicago) gave an interesting talk on the idea of "Snowball Earth"--that is, based on calculations that the earth through much, if not all of its history, should have been frozen over, since the sun in the past was dimmer than it is now. Once the earth freezes over, the white surface would reflect solar energy away, tending to lock in the frozen regime. It could take many millions of years for the sitation to reverse, if ever (so-called hysteresis). This paradox was popularized by Carl Sagan. Somehow methane and CO2 concentrations have hit just the right levels over time to keep this from happening. One theory is that CO2 is stabilized by a feedback mechanism where CO2 is removed from the atmosphere by sedimentation into carbonate rock, but gets added again when that rock travels into subduction zones and gets vaporized in volcanic plumes. If CO2 levels rise, rainfall and thus sedimentation rates increase due to the increased temperature, removing CO2--completing a stable negative feedback loop. Of course, lately CO2 levels have been artifically raised dramatically.

There was quite a buzz about a current New York Review of Books article by Jim Hansen, titled "The Threat to the Planet." The author blurb states:

Director of the NASA Goddard Institute for Space Studies and Adjunct Professor of Earth and Environmental Sciences at Columbia University's Earth Institute. His opinions are expressed here, he writes, "as personal views under the protection of the First Amendment of the United States Constitution."

John Pastor said that the article represents "Prometheus unbound." Scientists are seething from an earlier scandal where Hansen was muzzled by the Bush administration, being assigned a 25-year-old political handler censoring his talks. Now the handler is fired (he falsified his resume) the anti-science Bush forces beat a retreat and Hansen is coming out with both barrels blazing.

Much homage was paid to a oceanographic pioneer named Alfred Redfield:

Will have to look him up...

---------------------------------

Now, the Bayes' Law Diary:

http://www.dailykos.com/story/2006/7/1/82621/72221

Statistics 101 Part 11: Why spying is stupid

by plf515 [Unsubscribe]

Sat Jul 01, 2006 at 05:26:21 AM PDT

Spying on people.

We folks oppose it. The Bushites say it is necessary. They say that they only spy on terrorists. They say the spying helps them catch terrorists. They are wrong. We can prove it.

See how, below the fold

plf515's diary :: ::

Thanks to a Kossack who sent me this offline; since he didn't say to cite him, I won't. If he wants to chime in, that's fine too.

He pointed me to an article by Rudmin which is here
Rudmin article. I've adapted their argument, changing things round a bit, and adding some detail (after all, this is a statistics diary).

OK. On to meat of the diary.

Bayes' theorem is a part of elementary probability. A formal statement, with formula, is on the wikipedia page.

prob of A, given B, equals prob. of B, given A, times prob of A, divided by prob of B.

Here's the gist, as applied to the spying situation:

We want to know the probability that a person Bushco spies on is a terrorist. In the formula, then,

A is "person is a terrrorist"
B is "Bushco spies on them"

We need to know
Pr(B|A) - that is, the probability that a person is spied on, given that he is a terrorist
Pr(A) Proabbility that a person is a terrorist
Pr(B) Probability that Bushco spies on a person

Let's make some guesses.

There are about 300 million people in the USA. (That's not a guess). Let's leave out little kids, and make it 250 million

How many terrorists? 1,000? 10,000? Let's be generous to the Bushite and say 10,000 terrorists.

So, Pr(A) = 10,000/250,000,000
= 1/25,000

Now, how many people does Bush spy on?
well, he's looking at all phone records, and now banking records, so that let's out very few people. Say he's spying on 90%. But let's define spying more narrowly. Suppose he is spying on a million people. Then Pr(B) = 1/250.
Next, probability that a person is spied on, given that he is a terrorist. Hmmmm. Terrorists are doing their best to avoid attention. But the NSA is doing their best to find them. Hard to know, so I will plug in various numbers

If NSA is right 10% of the time, then the odds that a person being spied on is a terrorist are 1 in 1,000. Ouch

let's say NSA is right 50% of the time, then odds that a person being spied on is a terrorist is 1 in 200.

Could they be right 90% of the time? I doubt it. But, even if they could, it would mean that the odds are 1 in 111.

Finding terrorists by broad based spying is a violation of our rights. It's also stupid. It can't work well.

Bayes theorem is not esoteric; anyone who's had statistics knows it. NSA is not staffed by idiots. They know Bayes' theorem. They know the spying system can't work.

So, why is Bushco doing it?

Could it be another power grab by a paranoid megalomaniac?

I leave that for you to decide.

Interesting application of Baye's Rule

2006-06-29T12:30:00.000-07:00

This post isn't so much to talk about the politics of the NSA mass surveillance program as it is to demonstrate a well known statistical concept.

Basically, what his article argues is this: suppose you want to determine if someone is a terrorist threat. Suppose you have some algorithm that is reasonably accurate when you apply it (say, if a person meets criteria X, Y, Z, W,.... then it follows that there is a, say, 90% probability that this person is a terrorist threat). Now you randomly apply this process to the population at large and it turns up names, say, including Ollie. So, now that Ollie has been identified, what is the probability that Ollie is indeed a terrorist threat?

Answer: pretty low; probably less than 50%. 50% is what you would get if you were to merely flip a coin with heads: "he is a terrorist" and tails: "he isn't".

Of course, one could argue is that this first mass process is really some weeding out thing so as one has a collection of people, each with a slightly higher than normal chance of being a terrorist threat, and then one could apply more expensive tests to this group, knowing that the "false alarm" rate is going to be very high.

Anyway, here goes with the article:

http://www.counterpunch.org/rudmin05242006.html

Why Does the NSA Engage in Mass Surveillance of Americans When It's Statistically Impossible for Such Spying to Detect Terrorists?

By FLOYD RUDMIN

The Bush administration and the National Security Agency (NSA) have been secretly monitoring the email messages and phone calls of all Americans. They are doing this, they say, for our own good. To find terrorists. Many people have criticized NSA's domestic spying as unlawful invasion of privacy, as search without search warrant, as abuse of power, as misuse of the NSA's resources, as unConstitutional, as something the communists would do, something very unAmerican.

In addition, however, mass surveillance of an entire population cannot find terrorists. It is a probabilistic impossibility. It cannot work.

What is the probability that people are terrorists given that NSA's mass surveillance identifies them as terrorists? If the probability is zero (p=0.00), then they certainly are not terrorists, and NSA was wasting resources and damaging the lives of innocent citizens. If the probability is one (p=1.00), then they definitely are terrorists, and NSA has saved the day. If the probability is fifty-fifty (p=0.50), that is the same as guessing the flip of a coin. The conditional probability that people are terrorists given that the NSA surveillance system says they are, that had better be very near to one (p_1.00) and very far from zero (p=0.00).

The mathematics of conditional probability were figured out by the Scottish logician Thomas Bayes. If you Google "Bayes' Theorem", you will get more than a million hits. Bayes' Theorem is taught in all elementary statistics classes. Everyone at NSA certainly knows Bayes' Theorem.

To know if mass surveillance will work, Bayes' theorem requires three estimations:

1) The base-rate for terrorists, i.e. what proportion of the population are terrorists.

2) The accuracy rate, i.e., the probability that real terrorists will be identified by NSA;

3) The misidentification rate, i.e., the probability that innocent citizens will be misidentified by NSA as terrorists.

No matter how sophisticated and super-duper are NSA's methods for identifying terrorists, no matter how big and fast are NSA's computers, NSA's accuracy rate will never be 100% and their misidentification rate will never be 0%. That fact, plus the extremely low base-rate for terrorists, means it is logically impossible for mass surveillance to be an effective way to find terrorists.

I will not put Bayes' computational formula here. It is available in all elementary statistics books and is on the web should any readers be interested. But I will compute some conditional probabilities that people are terrorists given that NSA's system of mass surveillance identifies them to be terrorists.

The US Census shows that there are about 300 million people living in the USA.

Suppose that there are 1,000 terrorists there as well, which is probably a high estimate. The base-rate would be 1 terrorist per 300,000 people. In percentages, that is .00033% which is way less than 1%. Suppose that NSA surveillance has an accuracy rate of .40, which means that 40% of real terrorists in the USA will be identified by NSA's monitoring of everyone's email and phone calls. This is probably a high estimate, considering that terrorists are doing their best to avoid detection. There is no evidence thus far that NSA has been so successful at finding terrorists. And suppose NSA's misidentification rate is .0001, which means that .01% of innocent people will be misidentified as terrorists, at least until they are investigated, detained and interrogated. Note that .01% of the US population is 30,000 people. With these suppositions, then the probability that people are terrorists given that NSA's system of surveillance identifies them as terrorists is only p=0.0132, which is near zero, very far from one. Ergo, NSA's surveillance system is useless for finding terrorists.

Suppose that NSA's system is more accurate than .40, let's say, .70, which means that 70% of terrorists in the USA will be found by mass monitoring of phone calls and email messages. Then, by Bayes' Theorem, the probability that a person is a terrorist if targeted by NSA is still only p=0.0228, which is near zero, far from one, and useless.

Suppose that NSA's system is really, really, really good, really, really good, with an accuracy rate of .90, and a misidentification rate of .00001, which means that only 3,000 innocent people are misidentified as terrorists. With these suppositions, then the probability that people are terrorists given that NSA's system of surveillance identifies them as terrorists is only p=0.2308, which is far from one and well below flipping a coin. NSA's domestic monitoring of everyone's email and phone calls is useless for finding terrorists. [...]

Note: This article goes on to discuss some politics, which I don't want to get into on this particular blog. My other blog: http://blueollie.blogspot.com doesn't shy away from this discussion.

Some fun; global and local optima

2006-06-23T07:21:00.000-07:00

Today I had a nice swim (4000 yards, via 40 by 100) and decided to blog about some technical stuff that has been on my mind recently.

First: there is a debate going on about the minimum wage. Some wish to raise the federal minimum wage. The question, of course, is "does this do the poor workers any good?" The debate seems to be: "if you raise the minimum wage, the poorest of the workers will get more money, be better off, and have more to spend" vs. "raise the wage, and businesses won't be able to hire as many workers."

I don't have any economic credentials and therefore won't be talking about that issue specifically, though I do know from history that those old "company store" days didn't benefit anyone.

But what did strike me as odd is the following type of argument that I've heard (on a recent PBS Newshour program):

http://www.pbs.org/newshour/bb/business/jan-june06/minwage_06-21.html

GWEN IFILL: June O'Neill, what about the argument that raising the minimum wage would basically be the tide that lifts all boats, that even though people aren't necessarily making $5.15 an hour now, they could be making more?
JUNE O'NEILL: Well, for one thing, the vast bulk of people are nowhere near the minimum wage. They're earning much more than the minimum wage, so I don't think that they could speak too well. They're not talking about themselves.
But if it's such a magic wand, why stop at $5.15? Why stop at $7.25? Why not say $15 an hour? And then, voila, everyone will be earning $15 an hour and we would do all kinds of good, because you know that employers are not going to employ people. There are other places that they can go.

My note: June O'Neill is an economics professor at Baruch College and served in the Office of Management and Budget in the mid 1990's.

Anyway, it is this argument that I want to pick up on. Suppose you wanted to optimize some economic measure for the lowest income people. Dr. O'Neill seems to be saying that if you think that a small raise in the minimum wage will benefit people, then it follows that a larger raise ought to benefit them more, and of course raising the minimum wage to an absurd level doesn't make sense. Therefore, raising it at all (or even having one at all) makes no sense.

But this argument simply doesn't make sense to me, and here is why: sometimes, an optimum can occur at a local optimum point. It depends on the model.

Bear with me while I give an example:

take, for instance, gasoline mileage. Everyone who has driven a car knows that you get fewer miles per gallon when driving 70 miles per hour than when you drive, say, 55 miles per hour. (of course, I am not talking about race cars). So, driving slower always means getting better mileage right? Well, wrong. What happens when, say, you do lots of driving, say at 20 miles per hour? You get bad gas mileage!

So you see, the best gas mileage comes at a certain speed. Here is a real life efficiency curve:
http://www.fsec.ucf.edu/pubs/energynotes/en-19.htm

Click to see a larger version. So, if you are driving, say, at 35 miles per hour, you should speed up to get better gas mileage. If you are driving say, at 65 miles per hour, then you should slow down to get better gas mileage.

The speed corresponding to the peak of the graph (about 45 miles per hour) is what is known as a local maximum; if you are at that speed and change speed in either direction, your gas mileage goes down. It is also a "global" maximum, in that this is the best speed to drive in terms of gas mileage.

The same thing could be true of the minimum wage; we could well be at a point where raising the minimum wage will help things, though if it is raised too much it might end up hurting the poorest workers.

But either way, what Dr. O'Neill said (the part in bold) is not necessarily a valid argument.

Now for some fun stuff:

Darksyde of the Daily Kos has a good science diary; he writes one every Friday.

Today's can be found here:
http://www.dailykos.com/storyonly/2006/6/23/74340/6263
Today he interviews cosmologist Sean Carroll.

Sean Carroll has a cool blog worth checking out: http://cosmicvariance.com/

Finally, reading a comment on another blog reminded me of the film 2001: A Space Odyssey. I have to admit that I never "got" some of the weirder stuff in that film. But there is a website that takes a credible stab at some of the symbolism and makes some sense to me:

http://www.kubrick2001.com/

Anyway, I found that a fun site to visit.

Whew!

2006-06-19T14:06:00.000-07:00

It has been a good long while. This past semester has been a busy one; though my course load wasn't that heavy (the usual three courses) I had mathematical statistics, linear algebra and numerical methods.

Those courses, while enjoyable to teach, require quite a bit of preparation.

During this time, I mostly focused on finishing a paper for publication; I am one more review of the rough draft away from sending it out.

When I send it out, I'll post a link to a preprint and discuss it on this blog. Don't worry; I doubt that this paper will be foundational!

So what happens when one sends a paper out?

The editor gets it, and decides if it is worth sending to a referee. The referee checks to see if it is correct and if it is interesting enough for that journal.

What have been my results? In the past, I've had the following:

Accepted with very minor revisions (typo correction and the like)
Accepted with moderate revisions: explain this, delete that, modify this proof in this way, put in a diagram, etc.
Accepted pending revisions: there is a problem but a fixable one
Rejected with a recommendation to resubmit if a problem is fixed
Rejected with a recommendation that I send the paper elsewhere (paper isn't appropriate for the given journal; i. e., too specialized, not high enough quality for the target journal, etc.)
Rejected with a recommendation that I expand the result to make it more interesting
Rejected because of a math error (a couple of times; I was grateful that something not obviously false was published)
Rejected because my paper just wasn't interesting.

Fortunately, every rejection has lead directly to another publication; most of the time another paper or, in one instance, a published "problem."

To be honest, my research is far from stellar, but my stuff has appeared in the following journals: College Mathematic Journal, American Mathematical Monthly, Proceedings of the American Mathematical Society, Houston Journal of Mathematics, Journal of Knot Theory and its Ramifications (most of my stuff appears here), Bulletin of the Mexican Mathematical Society, Missouri Journal of the Mathematical Sciences and one refereed conference Proceedings (Low-Dimensional Topology, Knoxville 1992).

On another note: I got funding to attend the Math Fest in Knoxville from August 10-12 this year.

Dummies don't know that they are dumb...

2006-01-03T19:42:00.000-08:00

Over this break, I've done some reading and an excessive amount of blogging. I've also fallen prey to the sales at book stores; one "fun" book that I picked off of the $1.99 rack was Marc Abrams book: The Ig Noble Prizes: Annals of Improbable Research. The Ig Noble Prizes have made the mainstream news.

One of the articles they talked about was the one given in psychology in 2000:

PSYCHOLOGY David Dunning of Cornell University and Justin Kreuger of the University of Illinois, for their modest report, "Unskilled and Unaware of It: How Difficulties in Recognizing One's Own Incompetence Lead to Inflated Self-Assessments." [Published in the Journal of Personality and Social Psychology, vol. 77, no. 6, December 1999, pp. 1121-34.]

Basically, the article demonstrates that the more ignorant one is on something, the less that one recognizes their ignorance. This sure rings true for me in my experiences in the class room, both as a professor and as a student.
For example, some of the hardest people to teach are relatively untalented freshmen who did well (grade wise) in high school. And, as for me, the higher I went in mathematics, the more ignorant I felt. For example, when I waltzed into graduate school for the first time, I took a trip to the university co-op book store. I went to the mathematics section and was astonished that I understood the titles of at most 10% of the books on the shelf! It was to only get worse...I've heard many a graduate student lament "I used to be good at math; now I am a moron!"

And, I feel my dumbest when I am at a mathematics research conference!

Are my football predictions accurate?

2006-01-03T14:03:00.000-08:00

First, I'll mention one problem that I am working on over this break. Consider the standard plane in two-space. Then in, say, the first quadrant, consider the "wedge" shaped region W bounded by a line through (0,0) of slope m, where m > 0, and the x-axis. Let X denote some infinite set of points in W which has (0,0) as a limit point. Clearly it is possible to find a single variable function f whose graph runs though an infinite subset of W such that

f is continuous on the entire real line
f is differentiable except for possibly at x = 0.

Here is how one can obtain such an f : Select from W some infinite subset {xi} with the following property: r(i+1)where {r(j)} is some strictly decreasing sequence which converges to 0. Next, construct an polygonal infinite arc by connecting the points xi to each other via line segments, and then setting f(0)=0, and then choosing a real number sequence ti where ti converges to 0 and then setting xi = (ti, f(ti)). Clearly such an f is continuous. To make f differentiable: let yi denote the midpoint of the segment connecting xi to x(i+1). Then do a three point spline smoothing process to obtain a smooth segment through y(i-1), xi, y(i) such that the smooth arc has derivative equal to the slope of the segment containing yi for each i. This modification makes f smooth, except for possibly at t = 0
So the question is: how does one make f differentiable from the right at t = 0 ; that is, I want {limit as dt goes to zero) of (f(0+dt)-0)/dt to exist. The type of thing I am worried about would be things like f(t) = tsin(1/t) which has a limit as t goes to zero, but cannot be made to be differentiable there. I have some rather cumbersome ideas to try out over the next 4-5 days.

Now, about football predictions: I made some bowl game predictions here:

http://blueollie.blogspot.com/2006/01/bowls-and-my-prediction-nds-season.html

and my record was 17-9 straight up (on correctly picking the winner) and 15-10-1 (on correctly picking the winner vs. the published point spread). Example: at the Insight Bowl, Arizona State was favored to be Rutgers by 11.5 points. I picked Arizona State to win (which they did) but I picked Rutgers to cover ( that is, either win or lose by no more than 11 points). The final score was Arizona State 45, Rutgers 40, so I won "straight up" and also won "against the spread". On the other hand, Michigan was an 11 point favorite against Nebraska in the Alamo Bowl. I picked Michigan to win, but Nebraska to cover. Nebraska won 32-28, so I missed the "straight up" pick, but won the "spread" pick.

So, the question obtains: am I a good at making football predictions? I define "good" as being "better than chance". In statistics, one can safely say that they are better than random chance if the probability of, say, a coin flip doing as well is less than or equal to 5%.

So I decided to do a hypothesis test:

Ho: my predictions are no better than a random coin flip

Ha: my predictions are better than a random coin flip.

I used the binomial distribution (26 bowl games, a coin flip has .5 probability of being correct each time, so how many times will a coin be right?). In the days of old, it was usual to use the normal approximation to the binomial or to use tables; but now-a-days there are handy calculators such as this one: http://www.stat.sc.edu/~west/applets/binomialdemo.html

So if X is the random variable that denotes the number of times a coinflip would be correct out of 26 trials, we see that P(X>= 17) = .0843 and P(X>=18) = .0378. Now we see that x = 15 for the spread betting and x = 17 for the straight up betting (note: my point spread picks were correct 15 times and P(X>=15) = .2786. ) So that tells us that in neither case (straight up or by the spread) gives convincing statistical evidence that I can do better than a coin flip, though I was "close" to being better than a coin flip with my "straight up" predictions.

U. S. Math Education: do we really lag behind the rest of the world?

2005-12-24T14:31:00.000-08:00

We’ve often heard how poorly our grade school students (high school and down) stack up to those students in other countries, especially when we compare test scores. A careful look at the data paints a nuanced picture. Yes, our schools (or our schools plus parents plus students) could do better.
But, the current situation, while far from optimal, might not be as dismal as it seems. It appears that our relatively poor showing on the international exams may be influenced by several factors which include:

Sometimes students from our regular comprehensive high schools end up being compared with those in other countries who are attending more specialized academies.

Other countries don’t necessarily have the same socio-economic stratification within their citizens that we do. That is, there are some societies where the lower social classes are not citizens of their respective countries.

Our lack of homogeneity is also a factor. For example, our white highs school students score comparably with white students from Europe. This might indicate that our lower scores may well be reflecting problems with our society rather than problems with our academic education.

Again, this is not to say that all is well. And, frankly, as someone who teaches incoming science, and engineering freshmen, I am appalled at how poorly prepared most of them are, especially those who enter with sterling grade point averages.
I also see big problems with parental attitudes; I’ll go into this a bit later. Nevertheless, the following article gives a balanced view of those distressing test results.

Source: http://www.thenewatlantis.com/archive/9/soa/education.htm

How We Measure Up

Is American Math and Science Education in Decline?

As if coordinated to provoke headlines, top executives at three of the nation’s leading technology firms recently issued bleak appraisals of the American education system, criticizing especially how American students are taught science and mathematics. Microsoft Chairman Bill Gates minced no words at a summit of the nation’s governors: until high schools are redesigned, he declared, “we will keep limiting, even ruining, the lives of millions of Americans every year.” The chief executives of Intel and Cisco Systems shortly followed suit, suggesting that America’s lackluster schools will increasingly force companies to look overseas for talent.
Of course, these concerns are hardly new. But the somber prognoses from the heights of high-tech have added high-profile urgency to recent press reports about the declining performance of U.S. students in science and math compared to other nations, and the potential rise of China as a technological and economic superpower. Leading U.S. media outlets have featured major stories on the consequences of China’s rise for America’s future, like the recent Newsweek cover story by Fareed Zakaria appealing for a “massive new focus” on science and technology in the U.S., lest America “find itself unable to produce the core of scientists, engineers and technicians who make up the base of an advanced industrial economy.” In such a media atmosphere, one could be forgiven for having concluded that the United States is drifting unawares into an educational backwater while the rest of the world paddles furiously past it.
The truth is more complex. Cross-national studies of scientific and mathematical ability, interpreted rightly, tell a complicated story, giving reason to question how well the tests measure America’s real educational standing in the world. The two tests cited most frequently in press reports are the Program for International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMSS). PISA, undertaken by the Organization for Economic Cooperation and Development (OECD), most recently spanned 41 countries and tested 15-year-olds on mathematical word-problems. The latest TIMSS, in 2003, comprised more traditional, textbook-style math and science problems and was administered to fourth- and eighth-graders in 25 countries by an international team of researchers based in Boston and Amsterdam. The Department of Education funded both studies in the U.S., with help from the National Science Foundation.
Both tests have repeatedly been invoked by sensationalists seeking to cast the United States as unprepared for the high-tech, global economy. When the latest PISA results were released toward the end of last year, for instance, the Christian Science Monitor ran with the headline “Math + Test = Trouble for U.S. Economy,” and concluded that the study’s emphasis on “real-life” math skills makes it an accurate and “sobering” predictor of students’ performance in “the kind of life-skills that employers care about.” Federal officials expressed concern about the test results, too. “If we want to be competitive, we have some mountains to climb,” said Deputy Education Secretary Eugene Hickok.
To be sure, the results of neither TIMSS nor PISA should send American educators and policymakers rushing to the champagne. In most math areas tested by PISA, the gap between the average U.S. student and the average student in the highest-scoring countries—often Finland, the Netherlands, Singapore, Japan and Hong Kong—was roughly equivalent to the gap between the United States and low-scoring countries like Uruguay or Mexico. Where 44 percent of Singapore’s students reached the TIMSS “advanced international benchmark,” only 7 percent of U.S. students did. And, in general, the longer students had remained in the U.S. school system, the worse they performed relative to their peers abroad.
The first question that must be asked of such broad results, however, is whether the tests themselves accurately represent the countries’ student populations. International surveys such as these are not given to every student in each participating country; the tests’ organizers pick out statistical samples that are supposed to represent each country’s entire student population. Even so, schools—especially in the United States—sometimes decline to participate in the tests, potentially skewing the sample. As far as accurate sampling is concerned, early incarnations of the tests were not encouraging. In the first TIMSS general achievement test, conducted in 1995, only 5 of 21 participating countries met the study’s guidelines for conducting representative samples. While most countries participating in the latest studies have dramatically improved their overall sampling, the United States remains a notable exception. Only 73 percent of U.S. students chosen to be sampled were actually tested, a figure below the “minimum acceptable” rate of 75 percent. In most other countries, that number was well over 90 percent. If the omitted 27 percent of U.S. students were even slightly above or below average, their exclusion casts serious doubts on the accuracy of the U.S. sample.
The studies also inevitably confront large differences between countries’ school systems. “In Cyprus, students taking the advanced mathematics test were in their final year of the mathematics and science program; in France, the final year of the scientific track; in Lithuania, the final year of the mathematics and science gymnasia; in Sweden, the final year of the natural science or technology lines; and in Switzerland, the final year of the scientific track of gymnasium,” Professor Iris Rotberg of George Washington University wrote in Science concerning the 1995 TIMSS assessment, which tested high-schoolers. “In contrast, students in several countries, including the United States, attended comprehensive secondary schools. The major differences in student selectivity and school specialization across countries make it virtually impossible to interpret the rankings.” In TIMSS, especially, tests are conducted by grade-level rather than by age. In elementary and middle school, where topics are often covered and learned over the course of a few weeks, the risk of comparing students at incommensurate stages of their education is great.
Broad curricular differences have probably had a role in deflating U.S. scores. TIMSS and PISA use the same test in every participating country, and the material that makes it onto the test is selected through a winnowing process that leaves the tests considerably narrower than any single country’s general curriculum. Countries that include large amounts of material in their typical curricula are therefore at a disadvantage compared to those countries that focus their curricula more intensely on fewer subject areas. Regardless of its other merits or failings, the American strategy of repeated exposure to a broad range of subjects—American textbooks are the bulkiest in the world—is likely to lend itself to unduly poor performance on standardized tests, as full understanding of any single concept takes longer to develop.
Demographics and culture are also thought to confound the results of cross-national comparisons. In the United States, the tested students come from every socioeconomic rung, while other countries sometimes lack some rungs because of cross-border employment. For example, much of the labor force in Hong Kong (which is treated on the tests as an independent entity) is made up of tens of thousands of low-paid Filipino household workers whose children live and are educated in the Philippines; in light of the extensive literature tying socioeconomic indicators to educational achievement, this cross-border employment surely affects both countries’ scores. A similar situation obtains in other places with significant immigration and cross-border commerce, as Gerald Bracey points out in a 1997 article in the journal Educational Researcher. “Each morning thousands of Malaysians enter Singapore to sweep streets, pick up garbage, and do other low-level jobs. They return to their homes at night, relieving Singapore of having to educate the children of poor laborers,” Bracey writes. “If one reads the [domestic] educational research literature, one is struck by the lengths—the extreme lengths—that researchers go to to ensure that samples in their studies are comparable....The research community would never accept test results in this country that simply compared scores in an inner-city slum and an affluent suburb as if they were comparable,” he writes. The opposite circumstance holds in the United States: Students from all socioeconomic rungs are educated and scored on these tests.
Amid this deluge of confounding factors, the inference that the U.S. education system is going down the tubes is an unjustified logical leap. The United States is still pumping out tremendous numbers of new Ph.D.s in the sciences—more, in fact, than our economy can presently absorb, as there is a well-reported dearth of jobs for newly-minted science Ph.D.s. The same is true in engineering: According to a recent National Science Foundation report, the number of engineers graduating from U.S. schools will continue to grow into the foreseeable future, outstripping the number of available jobs. Of these new engineers and Ph.D.s, an increasing number are foreign-born—but increasing even faster is the percentage of those who decide to stay in the United States. Federal research funding for scientific research and development has consistently risen in absolute terms and as a fraction of discretionary spending—and industry research dollars have risen dramatically on top of that, to the tune of 7 percent per year in real terms—according to calculations by the Consortium for Science, Policy and Outcomes at Arizona State University. (Alarmist media reports often use GDP, against which research spending has fallen, as a comparative baseline.) And countries that have “outperformed” the United States in educational studies for many years—a number of European countries top this list—still fail to rival the U.S. in any measure of research productivity. When Bill Gates and others seem to appeal for school reform in the U.S., perhaps they are merely providing their companies with political cover and a post hoc justification for employing foreign engineers who, while not better educated than U.S. workers, are often significantly cheaper.
Nevertheless, there remains good reason to worry about what the global economy portends for those American students who really are badly educated. In only one other OECD country (New Zealand) are internal educational inequalities worse than in the United States, according to a recent analysis by researchers in England and Italy. Where these inequalities lie is no mystery. The gap in test scores between white and ethnically Asian students on the one hand and black and Hispanic students on the other is a well-known attribute of U.S. schools and is noted ruefully in nearly all cross-national studies. Two University of Pennsylvania researchers recently aggregated scores from a number of cross-national studies and found that white students in the United States, taken alone, consistently outperform the predominantly white student populations of several other leading industrial nations. “There is compelling evidence,” they write,” that the low scores of [black and Hispanic students] were major factors in reducing the comparative standing of the U.S. in international surveys of achievement. If these minority students were to perform at the same level as white students, the U.S....would lead the Western G5 nations in mathematics and science, though it would still trail Japan.” In PISA, for instance, white students performed above most European countries, whereas black students performed on par with students in Thailand. So while the performance of minority groups in the U.S. does refute the alarmist assertion regarding an across-the-board decline in U.S. schools, it does so in a particularly unfortunate way—namely, it suggests that some American minority groups will be shut out of high-paying jobs as companies look for better-educated workers overseas. Although the most recent TIMSS saw the white-black score gap close slightly, it is almost certain to remain shockingly large in the near future.
None of this is to say that other countries are not catching up technologically, nor that the United States is safe from competition in even a single technological sector. China is without doubt the most aggressive challenger. In the mid-twentieth century, Japan’s economy grew 55-fold over the course of thirty years through stringent government control; observers of Japan’s rise will remember the key role of its Ministry of International Trade and Industry, which employed many of the nation’s brightest stars and guided the economy on a carefully directed path of technological growth. China’s strategy has been similar, though its tremendous size has necessitated delegation of heavy-handed economic control to regional governments in what scholars have termed “local state corporatism.” It has simultaneously harnessed the power of markets in a way Japan did not. Regional governments lavish tax breaks on high-tech industries (many of them funded from overseas) and pump millions into China’s new universities—which are poised to graduate more Ph.D.s than the United States by 2010, according to some projections. Nearly all of China’s top leaders are scientists and engineers by training: President Hu Jintao is a hydroelectric engineer, Premier Wen Jiabao is a geological engineer. Their predecessors, Jiang Zemin and Zhu Rongji, were both electrical engineers. The technocrats steering China’s ship of state are working hard to modernize scientific education in their country.
But the United States need not worry—not yet. The U.S. is by no means in technological decline, though China and India will inevitably pose challenges in years to come. Although not a crisis, this competition should motivate the U.S. to improve its science and math education, especially for poor and minority students who might lose out in a globalized, high-tech economy. If sensationalists must take up a cause, it should be the plight of those students and not a hyped-up “threat” of China or the “impending decline” of technological innovation here at home.
The Editors of The New Atlantis, "How We Measure Up," The New Atlantis, Number 9, Summer 2005, pp. 111-116.

Embedding a Klein Bottle in 4-space

2005-12-12T12:59:00.000-08:00

My wife is leaving for a 3 week trip to India and decided to give me my Christmas present early. What I got was a glass model of an immersed Klien bottle. On the left, you see some models and on the right, a drawing made by Mathematica software. She got the idea for the gift from Dus7's blog and went to Classy Glass to get it.
Ok, so what is a Klien bottle? And, why is it important, and why did I use the word "immersed"?
First, a Klien bottle is an example of what is known as a "2-manifold"; that is, a very nearsighted ant that lived on the surface of the bottle could not distinguish the area immediately around it from what it would see if it lived on an ordinary plane (the kind of plane you did geometry on in high school). Furthermore, for those who have had some calculus, there is a way of "parametrizing" the Klien bottle so we could do calculus on it.
To obtain the Klien bottle from a section of a plane, consider the following figure:

Take this rectangle and glue the upper edge to the lower edge as shown. One obtains a long open "tube". Now if one were to glue the remaining "circle" edges to each other in a normal way, would would obtain a surface that looks like the skin of a donut; this is called a "torus".

But if we glue the circles in the reverse way (as indicated by the arrows; this is similar to what one does when one makes a Mobius Strip) one obtains the Klien bottle. Go ahead and click on the link as it gives better drawings than I give here.

Now, back to the glass model and the drawing. Notice how the bottle seems to "intersect itself" in a circle (where the glass tube goes from the "outside" into the "inside"? This is unavoidable if one tries to put this bottle in 3-space.

But, if one goes up one dimension, one can avoid this self intersection.

Consider the following two figures.

Here, we pretend that we are in 4 dimensional space, with coordinates (x, y, z, w). In the first figure (the shorter tube) we assume that the two circle boundaries of the tube lie in the plane (x, y, 0,0). The rest of the tube lies in the hyperplane (x, y, 0,w). That is, for all points in the first figure, the z coordinate is "0". Label the first circle boundary (closest to you) by C3 and the second one by C1. Now the second tube lies in the hyperplane (x, y, z, 0); that is, all of the points in this tube have "w" coordinate "0". The three circle that I've shown lie in the plane (x, y, 0, 0). The circle closest to you is called C3, and the one just slightly to the right of that is called C1. Now the circles labeled C1 and C3 are the same circle as they have the same coordinates in the (x, y, 0, 0) plane. So, we can join these two tubes in 4 space to obtain a surface that has no "exposed" edges; we call such a thing a "closed" surface. Notice that these tubes do not intersect each other except for these two circles as all of their points differ in their z and w coordinates. And, when you join these two tubes up, you get a Klien bottle! This is because of the way the tubes connect at circle C1.

Why is math so hard?

2005-12-11T16:53:00.000-08:00

I am near the end of the semester and hope to make a few more entries in this blog over break. One of the things I hope to do is to talk a bit about the Nash embedding theorem.

I am teaching a numerical methods class and gave a take-home exam. One student included this photo in his work, though he crossed out the word "you" and wrote "I". This reminded me of a quote from the mathematician Ronald Graham:, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (source: http://www.time.com/time/reports/v21/science/discover.html )
This link goes to an interesting discussion which I've reproduced here:
Will There Be Anything Left To Discover?
Is the great era of scientific inquiry over? Have all the big theories been formulated and important discoveries made—leaving future scientists nothing but fine tuning? Or is the real fun about to begin?By JOHN HORGAN and PAUL HOFFMAN

A spirited debate, conducted via e-mail, between two acclaimed science journalists: John Horgan, author of the controversial book The End of Science, and Paul Hoffman, former editor of Discover magazine and past president of the Encyclopaedia Britannica.
HOFFMAN: The past decade has brought a spate of books sounding the death knell for a host of subjects. Francis Fukuyama served up The End of History and David Lindley The End of Physics. But your more sweeping work The End of Science (1997) attracted a lot more attention and controversy—and with good reason. The idea that science may have had its run—that we've discovered all we can realistically expect to discover and that anything we come up with in the future will be pretty much small-bore stuff—left people either intrigued or outraged. With today's seemingly frenetic pace of scientific discovery, however, how can you say that the whole enterprise is coming to an end? The scientists I know, far from preparing for the undertaker, are ebullient about the future of their field.
HORGAN: Sure, scientists are keeping busy, but what are they actually accomplishing? My argument is that science in its grandest sense—the attempt to comprehend the universe and our place in it—has entered an era of diminishing returns. Scientists will continue making incremental advances, but they will never achieve their most ambitious goals, such as understanding the origin of the universe, of life and of human consciousness. Most people find this prediction hard to believe, because scientists and journalists breathlessly hype each new breakthrough, whether genuine or spurious, and ignore all the areas in which science makes little or no progress. The human mind, in particular, remains as mysterious as ever. Some prominent mind scientists, including [Time Visions contributor] Steven Pinker, have reluctantly conceded that consciousness might be scientifically intractable. Paul, you should jump on the end-of-science bandwagon before it gets too crowded.
HOFFMAN: Don't save a seat for me quite yet, John. Take the human mind. I agree that we are not close to an understanding of consciousness, despite the efforts of some of the best minds in science. And perhaps you're even right that we may never understand it. But what is the evidence for your position? You've criticized scientists for having faith—a dirty word in the scientific lexicon—that our era of explosive progress will continue unabated. Isn't it at least as much a leap to think that the progress will abruptly end—particularly since the trajectory of discoveries so far suggests just the opposite, that supposedly unanswerable questions eventually do get answered?
HORGAN: My faith is based on common sense, Paul, and on science itself. As science advances, it imposes limits on its own power. Relativity theory prohibits faster-than-light travel or communication. Quantum mechanics and chaos theory constrain our predictive abilities. Science's limits are glaringly obvious in particle physics, which, as Steven Weinberg describes [in the Visions issue], seeks a "theory of everything" that will explain the origin of matter, energy and even space and time. The leading theory postulates that reality arises from infinitesimal "strings" wriggling in a hyperspace of 10 (or more) dimensions. Unfortunately, these hypothetical strings are so small that it would take a particle accelerator the size of the Milky Way to detect them! I am not alone in fearing that string theorists are not really practicing science anymore; one leading physicist has derided string theory as "medieval theology." Paul, here is persuasive evidence of science's plight.
HOFFMAN: Yes, but who is to say that all these scientific theories won't ultimately be replaced by ones with greater explanatory power? Galileo and Newton thought their laws of motion were the cat's pajamas, explaining everything under the sun and many things beyond, but 2 1/2 centuries later a Swiss patent clerk toppled their notions of space and time. Obviously, Galileo and Newton did not foresee what Einstein found. I think it's ahistorical to assert that in the future there will never be an Einstein of, say, the mind who will be able to pull together a theory of consciousness. And even if it's true that some of the big unanswered questions of science may never be answered, a lot of new and exciting science could still come from overturning truths that we now take for granted. Robert Gallo, the AIDS researcher, once told me that at the end of the 1970s, he was at a conference where a prominent scientist confidently summed up the truths of biomedicine— such truths as: epidemic diseases are things of the past, at least in so-called developed nations; a widespread outbreak of infectious disease is impossible unless the microbe is casually transmitted; the kind of virus found in animals known as the retrovirus doesn't exist in man; and no virus causes cancer in humans. By the end of the 1980s, these four truisms had hit the dustbin. Or take a more recent example: the newfound plasticity of the human brain. Until a year and half ago, it was a dogma taught in every medical school in the country that the adult human brain is rigid, that its nerve cells can never regenerate. Now we know our brains do have the ability to generate new cells—a discovery that may not only open up a new understanding of the brain but also lead to novel treatments for a host of brain disorders. HORGAN: Here's the big question we're dancing around: Can we keep discovering profound new truths about reality forever, or is the process finite? You seem to assume that because science has advanced so rapidly over the past few centuries, it will continue to do so, possibly forever. But this view is, to use your word, ahistorical, based on faulty inductive logic. In fact, inductive logic suggests that the modern era of explosive scientific progress might be an anomaly, a product of a singular convergence of social, intellectual and political factors. If you accept this, then the only question is when, not if, science will reach its limits. The American historian Henry Adams observed almost a century ago that science accelerates through a positive-feedback effect. Knowledge begets more knowledge; power begets more power. This so-called acceleration principle has an intriguing corollary: If science has limits, then it might be moving at maximum speed just before it hits the wall.
HOFFMAN: Of course, I accept that science has limits—and may even be up against them in some fields. But I believe there is still room for science, even on its grandest scale, that awe-inspiring discoveries will continue to be made over this millennium. The mathematician Ronald Graham once said, "Our brains have evolved to get us out of the rain, find where the berries are and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." Sounds reasonable, except when you consider that it could be similarly said that our brains didn't evolve to invent computers, design spaceships, play chess and compose symphonies. John, I think we'll continue to be surprised by what the brains of scientists turn up.
HORGAN: I hope you're right, Paul. I became a science writer because I believe science is humanity's most meaningful creation. We are here to figure out why we are here. The thought that this grand adventure of discovery might end haunts me. What would it be like to live in a world without the possibility of further revelations as profound as evolution or quantum mechanics? Not everyone finds this prospect disturbing. The science editor of the Economist once pointed out to me that if science does end, we will still have sex and beer. Maybe that's the right attitude, but there aren't any Nobels in it. No matter how far science does or doesn't advance, however, there's one wild card in even the most pessimistic scenario. If we encounter extraterrestrial life—and especially life intelligent enough to have developed its own science—then all bets are off.

Relativity from the Daily Kos

2005-12-09T04:57:00.000-08:00

I doubt that regular readers of this page will learn anything here, but this would be a good place to send your not so scientifically inclined friends to if they want to get a brief glimpse of the basics of Relativity Theory and of the Theory of Quantum Mechanics:

http://www.dailykos.com/storyonly/2005/12/9/64239/6747

And speaking of basics, I can recommend the following books to those who want to learn more about the topological aspects of the mathematics of space-time (not at the research level):

http://www.amazon.com/gp/product/0824707095/104-3534335-8829559?v=glance&n=283155

And if you care to play some of the games that the book recommends, you can go here (play tick-tack-toe on a "flat torus" or a "flat klien bottle"

http://geometrygames.org/

Harsh Reality: Many students are set up to underachieve by their parents

2005-11-29T05:48:00.000-08:00

I've posted before about how many students are showing up on campus without realizing that they will need to work hard. In my mind, I blamed the "self-esteem" centered culture that seems to be prevalent in our school system. I've also talked about the "helicopter parents" in a previous post.

Then in today's Peoria Journal Star, I read the following column by Leonard Pitts:

http://www.tmsfeatures.com/tmsfeatures/subcategory.jsp?custid=67&catid=1050
WHITES FLEEING IN THE WRONG DIRECTION

By Leonard Pitts Jr.

Tribune Media Services

Perhaps you remember white flight.
That is, of course, the term for what happened in the '60s when African Americans, newly liberated from legal segregation, began fanning out from the neighborhoods to which they'd once been restricted. Traumatized at the thought of living in proximity to their perceived inferiors, white people put their houses on the market at fire sale prices and took flight.
Well, something similar is happening now in Northern California. Similar in the sense of being completely different.
Where whites once ran because they felt they were superior to their new neighbors, they are apparently running now because they feel they are not quite as good.
I refer you to a Nov. 19 story in the Wall Street Journal. Reporter Suein Hwang interviewed white parents who are pulling their kids out of elite public high schools, schools known for sending graduates to the nation's top colleges. They are doing this, writes Hwang, because the schools are too academically rigorous, too narrowly focused on subjects like math and science.
Too Asian.
Yes, you read right. Hwang reports that since 1995, the number of white students at Lynbrook High in San Jose has fallen by almost half. At Monta Vista High in Cupertino, white students now make up less than a third of the population.
White parents are putting their kids into private schools or moving to areas where the public schools are whiter, less Asian and less demanding. Where sports and music are also emphasized and educators value, as one parent put it, "the whole child."
One white woman told Hwang how she dissuaded a young white couple from moving to town, telling them their child might be "the only Caucasian kid in the class." Another said, "It does help to have a lower Asian population."
Which plays, of course, into the old stereotype of the hyper-competitive Asian. But the new white flight has also given rise to a new stereotype one educator calls "the white boy syndrome." It says that white kids just don't have it between the ears.
The irony speaks for itself.
I have no idea why Asian kids tend to lap the field, academically speaking. I do know it has nothing to do with the simple fact of being Asian, any more than the fact of being black makes you a great basketball player. To attain proficiency in any field, it helps to want that proficiency and to belong to a culture that rewards it. We strive for the things we deem important.
I make no argument for punishing, joyless education. Sports and music are important, too. On the other hand, most kids are hardly in danger of studying too hard or being insufficiently entertained.
Consider the National Assessment of Educational Progress, a federal study released last month. It found that, despite some improvement, American kids remain academically underwhelming. Only 31 percent of fourth graders, for instance, were rated "proficient'' or better in reading. Just 30 percent of eighth graders managed to hit that mark in math.
In recent years, I've taught writing at an elite public high school and three universities. And I've been appalled how often I've encountered students who simply could not put a sentence together, had no conception of basic grammar and punctuation. They tell me I'm a tough grader, and the funny thing is, I think of myself as a soft touch. "I've always gotten A's before," sniffed one girl to whom I thought I was being generous in awarding a C plus.
It occurs to me that this is the fruit of our dumbing down education in the name of "self-esteem." This is what we get for making the work easier instead of demanding the students work harder - and the parents be more involved.
So this new white flight is less a surprise than a fresh disappointment. And I've got news for those white parents:
They should be running in the opposite direction.

So there you have it. What parents want these days is for their kids to make good grades; whether they learn anything or not seems to be irrelevant. After all, it is "that piece of paper" that is important, right?

What many don't understand is that the value in education is NOT the degree itself, but how it changes one's mind for the better.

This reminds me of something that happened this semseter: I gave an exam and this one student failed yet again; this time with a grade in the 40's (out of 100). she came up to me with tears in her eyes exclaiming in anguish that she "had taken a two semester class in calculus out of a college book and had gotten A's". Well, on this exam, she couldn't even differentiate a simple function like "cos(x)". I told her that she could drop the class, settle for a D or F, or do what I told her and to learn the material properly.

Happily, on the next quiz, she made a low "A" (on more difficult material) and did everything I told her to do.

I've noticed similar stuff even when I was back in grad school (1985-1991). I remember several times, Asian students would see me in my office late a night, working away and say "hey, what are YOU doing here; aren't you an American?" And, many times when I took the 11 pm bus home, I'd be the only "round eyes" on the bus.

Judging a book by its cover

2005-11-29T05:47:00.000-08:00

Judging a book by its cover.

First, I'd like to thank Mawk for altering me to what follows. Ok take a look at the following photo. This person plays the part of a physicist in a popular television series. Do you have a reaction?

I know that I do. What a joke! This model probably couldn't integrate e^x if you spotted her with a computer, right? Why can't they cast someone who at least looks like a scientist, right?

Well, to find out who this lady is, go here: http://www.edge.org/3rd_culture/bios/randall.html

Yes, she is a world famous particle physicist in real life, with professorships at Harvard, numerous awards as well as some popular books. She speaks a bit about having some traditional female interests here: http://www.msnbc.msn.com/id/7374458/page/3/

Teaching Remedial Students

2005-11-10T18:58:00.000-08:00

Cross posted at blueollie:

At my university, we have a special class called "Calculus with review". What we do is we take our standard first semester science/engineering major calculus class and split it into two semesters and cover the material at half speed. The students who take this are those who don't do well on our placement tests or who have low (for us) entrance exam scores.
We really hit the algebra hard during this course. Now, some students are doing well; I have an unusual number of A's so far (3 out of 28 starters). Others have had to drop. Many have told me that they have grossly understudied so far.
The real trouble that we have in this class are with the students who claim to have taken a calculus course in high school (and have gotten an A or B!). I had the following conversation with a student this morning (who asked me if she could still pass; she got a 38 out of 100 on the last exam): "but I understand this stuff; I just can't answer your questions." I laughed and said "that is like saying, 'I know how to swim, but when I get in the pool I drown.' If you understand it, you can do it." She exclaimed "that's not true!"
This person just hasn't grasped that they don't know what they are doing; evidently their high school grade gave this student an overinflated opinion of where they are, in terms of mathematical ability.
I suppose it is a fine line between trying to build a student's confidence and being realistic; what this student doesn't want to accept is that he/she is going to have to work very very hard just to achieve at an average level; the current attitude seems to be "hey, I am bright so if I have to work hard just to get a C, then something is wrong with the course or my professor."
So, I have to find a nice way of saying: "this course is designed for people who are better prepared and more talented than you are. Therefore if you don't work your rear end off, you are doomed to fail. Forget what you were told; you have an inflated opinion about your current abilities."
I had another interesting conversation with a student; this person wanted to know why they got no credit on a problem. The problem said something like:
"if y = (x^2 + 3x + 1/x)/x, then y' = He wrote: y = x + 3 + 1/x^2 and then circled his answer.
I said: ok, the problem told you to differentiate, right? He read the problem and nodded his head. Then I said: "ok, where did you differentiate?" He looked open mouthed and slinked away.
The problem was is that he was so sure that I had shorted him; this kid has had an attitude for the whole semester. He is another one who has an inflated self image.
I think that what we have to work toward is an attitude of "yes, you are smart enough to be successful, IF you are humble enough to do what you are told and IF you work hard. Your success is far from guaranteed; this is a very conditional sort of thing!""

----
I posted the above at http://blueollie.blogspot.com. I recieved the following comment, which is hysterical:

"When I was a graduate student in physics, we were required to spend 1 hr per week in the "study hall" for the physics undergraduate students. I was trying to help a student who was in the honors program with a simple electrostatics problem. It was a line charge, so he had to integrate dx/x. He couldn't do the integral! But he didn't think that was a big deal. He told me that his high school calculus class they had used Maple for everything, so he was used to just typing stuff in.
But, my best story was from tutoring. I had this student, who by the way she dressed, came from some money. She wasn't great at physics, but she wa a hard working student. I always forced them to do the algebra 1st, then substitute in numbers at the end. So we get to the "numbers part", and she reaches for the calculator. I say: "you can do that may w/o a calculator."
She says: "I always use a calculator for math."
I say: "But what do you do when your shopping & the tag says 25% off and you have to figure out the price?"
She deadpans: "I never look at prices, I just buy what I want."
Silence. I deadpan: "I'm not charging enough am I" "

Those Pesky Error terms...

2005-11-08T19:18:00.000-08:00

I am teaching the "numerical methods" class for engineers this year. I've learned many things while preparing the lessons. But one of the most fascinating things I've learned is how one can use the error terms of a series (say, a Taylor series) to obtain a weighted average of an approximating scheme to improve the accuracy of the approximation! Here is but one example: suppose one wants to approximate the derivative of a function, and one has only the function itself to work with, along with the fact that it is known (or hypothesized) that a certain number of higher order derivatives exist. Assume that we are working about x = 0.
Then, from the theory of Taylor Series:
f(x) = f(0) + f'(0)x + f''(0)(1/2)x^2 + f'''(0)(1/6)x^3+ terms where x has order 4 or more.
So, to approximate f'(0) one does some algebra to obtain
f'(0) = (f(x)-f(0))/x -f''(0)(1/2)x -f'''(0)(1/6)x^2 - terms where x has order 3 or more
Which is not a surprise as we are approximating the derivative by a difference quotient. Note that the error terms have order 2 or more. Call this approximating scheme F(x). Now, if we want to get our error terms to have order 2 or more:
Let F(x/2)= 2(f(x/2)-f(0))/x -f''(0)(1/4)x -f'''(0)(1/24)x^2 - terms where x has order 3 or more. So now, F(x)-2F(x/2) = -f'(0) + f'''(0) (-1/6 +1/12)x^2 - terms where x has order 3 or more
Then 2F(x/2)-F(x) = (4f(x/2)-f(x)-3f(0))/x is an approximation scheme which reduces the error of estimating f'(0) to a second order (roughly 4 times as good) at the expense of using one closer value of f in the scheme.
Similar ideas are used in many other approximation schemes, such as those to approximate integrals (e. g., Romberg integration) and those to approximate solutions to differential equations (Runge-Kutta schemes, for example). If you always wondered why those funny weighted averages were the way that they were, it was to reduce the order of the error terms as stated above.
Anyway, I never realized any of that when I previously taught calculus. Those pesky error terms in the series have a practical use!

More things you can do with a math degree

2005-10-11T18:42:00.000-07:00

Of course, the top prize for mathematics is the Field's Medal. But, you could also win a Nobel prize in another field, as John Nash (the subject of "A Beautiful Mind") did.

Or more recently:

Game theory economists share Nobel prize Israeli-American and American win for work on political, economic conflict

The Associated PressUpdated: 2:49 p.m. ET Oct. 10, 2005
STOCKHOLM, Sweden - A pair of game theorists who defined chess-like strategies in politics and business that can be applied to arms races, price wars and actual warfare won the Nobel Prize in Economic Sciences on Monday.
Israeli-American Robert J. Aumann and U.S. citizen Thomas C. Schelling won the award for research on game theory, a branch of applied mathematics that uses models to study interactions between countries, businesses or people.
The theory, devised in 1944 by John von Neumann and Oskar Morgenstern, is often used in a political or military context to explain conflicts between countries. More recently it has been used to map trends in the business world, ranging from how cartels set prices to how companies can better sell their goods and services in new markets.
“The understanding of game theory helps explain economic conflicts like price competition and trade wars,” said Jorgen Weibull, chairman of the prize committee. “I think the main impact is on economics, but it also applies to other social sciences.”
[...]

It said the pair’s work, which built on research by the 1994 winners of the same prize, could be applied to understand how merchant guilds, international trade treaties and even organized crime groups are formed and operate.[...]
Schelling, who teaches at the University of Maryland, used game theory in his 1960 book “The Strategy of Conflict” to focus on how the U.S. and the former Soviet Union maintained credible threats that were not likely to be used, given the threat of nuclear annihilation.
“If you have second-strike capacity, then it makes your opponent think twice,” said Carl-Gustaf Lofgren, a member of the prize committee.
In an interview with The Associated Press, Schelling said: “I use game theory to help myself understand conflict situations and opportunities.”
He said the prize committee linked the two laureates on the virtue of their respective research.
“They linked us together because he is a producer of game theory and I am a user of game theory,” said Schelling, who worked with the U.S. Marshall plan to rebuild Europe after World War II.

[...]
In the economic world, Aumann (pronounced OW-man) and Schelling’s work is used to prevent illegal cartels between rival companies, Lofgren said.
Two competitors can use game theory to agree on joint price structures that benefit both parties, thereby eliminating fair competition, he explained. But the theory also lets regulators pick up on signs of collusion and, ultimately, break up illegal cartels.
Monday’s award also highlighted developments in game theory that were lauded with the 1994 economics prize to Americans John Harsanyi and John Nash and German Reinhard Selten. Nash was portrayed in the 2001 Academy Award-winning film “A Beautiful Mind,” starring Russell Crowe.
Lofgren said Aumann’s theories differed from Nash’s by introducing an infinite repetition of the same game so as to find the best solution in long-term relationships instead of in a single encounter.
The difference is illustrated in the so-called “prisoner’s dilemma,” one of game theory’s best-known situations in which two partners in crime are put in separate cells and given an ultimatum: If one implicates the other, he may go free while his partner faces a firing squad.
Facing that situation once, both prisoners are likely to talk, meaning both would be executed, Lofgren said. However, if they could repeat the situation an infinite number of times and add the results of each action, they would realize that the best option is for both to keep mum, he said.
That supposition formed the basis for a TV game show, “Friend or Foe?” that aired on the Game Show Network in the U.S.
Aumann is a professor at the Center for Rationality at the Hebrew University of Jerusalem."

For the full article: URL: http://www.msnbc.msn.com/id/9649575/

One job you can get with a math major

2005-10-04T18:39:00.000-07:00

One can get a nomination to be on the Supreme Court of the United States!

Hat tip to Harriet Miers, a mathematics major at SMU (prior to getting her law degree at SMU).

Yeah, I am a Democrat, but it is good to see a math major make good.

I haven't posted much here as math blog posts take time, and most of my math time has been devoted to teaching a "numerical methods" course.

This course is time consuming as

My last numerical analysis course was a long time ago.
I wasn't that good at it then.
I am learning the software packages as we go along.

I am not sure as to how I feel about the software packages; in my day we wrote our own code which helped us learn how the algorithms worked. On the other hand, we can cover more topics since coding isn't slowing us down.

What I can say is that I finally learned where the error formulas in the Simpson's rule approximation formula comes from.

What is Simpson's rule? It is a way of approximating a definite integral by approximating the function being integrated in a piecewise fashion by parabollas. Basically, you chop up the interval that you are integrating over into "n" equal intervals, subdivide each interval into two pieces, and over each piece you fit a parabola (three points determine a unique parabola (a line is considered a degenerate parabola). And it is easy to integrate a quadratic function.

So, over each interval, there is a maximum error; if the function being integrated has four continuous derivatives (can differentiate the function 4 times and still get a continuous function) then the error over each piece is bounded by the absolute value of

(M/2880)(b-a)^5 where the interval runs from "a" to "b" and M is the absolute value of the maximum value of the 4'th derivative of the function being integrated, taken over the interval from a to b.

Where in the world does the 2880 come from?

Well, if one uses the 3'rd degree Lagrange polynomial to approximate f over the interval from a to b, the maximum error is (M/4!)(where M is as before and the points x0, x1, x2, x3 run from a to b. But if you integrate the Lagrange polynomial over the interval, the cubic term vanishes (integrates to zero) and what you have left is the integral of the Lagrange quadratic polynomial through x0, x1, x2, which can be set so x0 =a, x1 = (1/2)(a+b) and x2 = b. that is, this is the unique quadratic that one integrates in Simpson's rule!

So, to find the error, we integrate (M/4!)(x-x0)(x-x1)(x-x2)(x-x3) from a to b, setting x0, x1 and x2 as before. we set x3=x2 which you really can't do if you are doing an approximation, but remember we are obtaining an upper bound for the error.

Set h = b-a and then make the substitution x = a + th, then dx = hdt and

x-x0 becomes th, x-x1 becomes h(t-1/2), (x-x2)(x-x3) becomes (h(t-1))^2

and the limits of the integral become 0 to 1. Pull all of the "h's" out of the integral sign to get the h^5 term, and integrate (t)(t-1/2)(t-1)^2 from 0 to 1 to obtain 1/120

and then 1/120 * (1/24) =2400 + 480 = 2880.

It has been a while; a philosophical question:

2005-09-17T14:43:00.000-07:00

The start of the semester has gotten in the way of my blogging. :-) But I am in between watching football and grading exams.

For the first part of the semester, I had a "refereeing" job to do for a math journal. Basically, when someone writes an article for a math journal, the journal sends the paper to a mathematician to be checked for accuracy and to be checked for quality (i. e., is what is being offered really a good contribution to the literature?).

In this referee job, the authors submitted a result that was worth publishing, and their proof was, on technical grounds, basically correct. But their proof was quite dreary; there were far more elegant and elementary ways to reach their conclusions and any specialist who read their article would note that immediately.

So, what was my duty in this case? What I ended up doing is telling the editor what I just said and made the necessary suggestions.

Now, as far as grading exams: I am teaching a "calculus with precalculus" class. Basically, we go through the first semester of a standard engineering/science/math calculus class at half speed; the two semesters of this class put together counts as one semester of calculus.

On this exam, several (20-25% of the class) presented the following answer: "9 < x < 3".

Ok, perhaps a few made a blunder during the pressure of an exam; we've all done that. But, there are students in this class who simply don't see why this statement is complete nonsense. And frankly, they won't be able to unless it is explained to them "very slowly".

Folks, these students simply don't have a chance of success in any major that requires mathematical ability. I actually have stronger thought than this (in that there are majors that these students could succeed in) but I probably better keep these thoughts to myself.

Sigh....I suppose we sometimes have to find things out the hard way.

small brain teaser

2005-08-31T05:13:00.000-07:00

Yesterday, I turned 46 years old. Next July, my wife, Barbara, turns 64. (No, you fans of Ben Franklin, they are not "so grateful").

Barbara noticed that, for a couple of months, the digits of our ages will be transposed. She said "that won't happen again."

Well, I pointed out: 75 - 57 = 18, 86 - 68 = 18, and being very optimistic, 97 - 69 = 18.
So the question:

(blindingly easy): what is the pattern?
(more challenging): what is going on here? Why does this work?

---------------------------------------------------------

Reference to Ben Franklin for those who don't want to follow the source:

Franklin had a friend who did not want to get married, but was battling with urges and lustful inclinations for the opposite sex. In the letter which follows, Franklin first advises the friend that the best solution for his urges is marriage. However, since he knows the friend will not take that advice, Franklin goes on to suggest that his friend have sexual affairs with old women. We know from the context that he is suggesting women over 45 years of age (see #3). His words about putting a basket over her head or turning out the light (see #5) illustrate an aspect of Franklin's character which is seldom exposed.

June 25, 1745
My dear Friend,
I know of no Medicine fit to diminish the violent natural Inclinations you mention; and if I did, I think I should not communicate it to you. Marriage is the proper Remedy. It is the most natural State of Man, and therefore the State in which you are most likely to find solid Happiness. You Reasons against entering into it at present, appear to me not well-founded. The circumstantial Advantages you have in View by postponing it, are not only uncertain, by they are small in comparison with that of the Thing itself, the being married and settled. It is the Man and Woman united that make the compleat human Being. Separate, she wants his Force of Body and Strength of Reason; he, her Softness, Sensibility and acute Discernment. Together they are more likely to succeed in the World. A single Man has not nearly the Value he would have in that State of Union. He is an incomplete Animal. He resembles the odd half of a Pair of Scissars. If you get a prudent healthy Wife, your Industry in your Profession, with her good Oeconomy, will be a Fortune sufficient.
But if you will not take this Counsel, and persist in thinking a Commerce with the Sex inevitable, then I repeat my former Advice, that in all your Amours you should prefer old Women to young ones. You call this a Paradox, and demand my Reasons. They are these:
1. Because as they have more Knowledge of the World and their Minds are better stor'd with Observations, their Conversation is more improving and more lastingly agreable.
2. Because when Women cease to be handsome, the study to be good. To maintain their Influence over Men, they supply the Dimunition of Beauty by the Augmentation of Utility. They learn to do 1000 Services small and great, and are the most tender and useful of all Friends when you are sick. Thus they continue amiable. And hence there is hardly such thing to be found as an old Woman who is not a good Woman.
3. Because there is no hazard of Children, which irregularly produc'd may be attended with much Inconvenience.
4. Because thro' more Experience, they are more prudent and discreet in conducting and Intrigue to prevent Suspicion. The Commerce with them is therefore safer with regard to your Reputation. And with regard to theirs, if the Affair should happen to be known, considerate People might be rather inclin'd to excuse an old Woman who would kindly take care of a young Man, form his Manners by her good Counsels, and prevent his ruining his Health and Fortune among mercenary Prostitutes.
5. Because in every animal that walks upright, the Deficiency of the Fluids that fill the Muscles appears first in the highest Part: the Face first grows lank and wrinkled; then the Neck; then the Breast and Arms; the lower Parts continuing to the last as plump as ever: So that covering all above with a Basket, and regarding only what is below the Girdle, it is impossible of two Women to know an old from a young one. And as in the dark all Cats are grey, the Pleasure of corporal Enjoyment with an Old Woman is at least equal, and frequently superior, every Knack being by Practice capable of Improvement.
6. Because the Sin is less. The debauching of a Virgin may be her Ruin, and make her for Life unhappy.
7. Because the Compunction is less. The having made a young Girl miserable may give you frequent bitter Reflections; none of which can attend the making an old Woman happy.
8. They are so grateful!!
Thus much for my Paradox. But still I advise you to marry directly; being sincerely Your affectionate Friend,
Benjamin Franklin.

"Numbers" and "largest primes"

2005-08-16T13:33:00.000-07:00

While visiting one of my favorite political websites (commongroundcommonsense.org) I noticed a post that mentioned the TV series "numbers". The episode in question dealt with codes; in particular one can create a difficult to crack code by using a "key" which consists of the product of two very large prime numbers. The idea is that this key would take years and years of computer time to factor.

So, knowing various large prime numbers has some value.

But, the poster on this site (a poet) said something to the effect of finding the "largest prime number". Well, one thing that is easy to prove is that there is no largest prime number!

Here goes the proof:
suppose there is a largest prime number. Let it be "q".

Now consider the number p = q! + 1 (where q! means 1*2*3...*(q-1)*q, e. g., 6! = 1*2*3*4*5*6 = 720)
Now p cannot be prime as p > q and q was the largest prime. So p has at least one prime factor, call it r. r<= q hence r divides q!. r divides p.
So r divides (p - q!) = 1 which is impossible since no prime number divides 1.
Therefore there can be no largest prime number.

Another way of putting this is that if you know that q is a prime number, you can always find a prime number that is larger than q by finding the prime factors of q! + 1.

Of course, it can be very difficult to perform this factorization, but that is another matter.

A good teaser

2005-08-09T18:48:00.000-07:00

Courtsey of a common ground, common sense member:

http://www.commongroundcommonsense.org/forums/index.php?act=ST&f=16&t=35269

There are five houses in a row in different colors. In each house lives a person with a different nationality. The five owners drink a different drink, smoke a different brand of cigar and keep a different pet, one of which is a Walleye Pike.

The question is-- who owns the fish?

Hints:
1. The Brit lives in the red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house owner drinks coffee.
6. The person who smokes Pall Malls keeps birds.
7. The owner of the yellow house smokes Dunhills.
8. The man living in the house right in the center drinks milk.
9. The man who smokes Blends lives next to the one who keeps cats.
10. The Norwegian lives in the first house.
11. The man who keeps horses lives next to the one who smokes Dunhills.
12. The owner who smokes Bluemasters drinks beer.
13. The German smokes Princes.
14. The Norwegian lives next to the blue house.
15. The man who smokes Blends has a neighbor who drinks water.

Solution: (don't cheat! Hey, you don't get stronger by watching someone else lift weights, do you?)

http://www.geocities.com/onanyes/teaser.htm

Weird Functions; why continuity matters so much

2005-07-13T14:19:00.000-07:00

This summer I have been working through a delightful book by Gelbaum and Olmsted called Counterexamples in Analysis. No, I don't own stock in the Dover book company, nor do I know either of the authors.

I can recommend this book to anyone who is either learning elementary analysis, interested in learning the "whys" behind calculus, or is teaching calculus, real analysis or complex variables.

I'll share a few cool examples that I was not familiar with prior to reading this book.

A function that is continuous precisely at the irrational numbers whose discontinuities are removable.

A reminder: if the limit as "x" approaches "a" of a function "f" exists but is not equal to the value of the function at "a", we say that the discontinuity is "removable". For example, the function f(x) = (x^2 -1)/(x + 1) has a removable discontinuity at x = 1 because f(1) doesn't exist (zero in the denominator) but the limit of f as x approaches 1 is clearly zero.

When we bring this concept up in calculus, we show a graph with a single "hole" where (a, f(a)) should be.

Now recall that a rational number is a number that can be put in the form p/q where p and q are integers and p, q have no common factors other than 1 or -1. If we insist that, say, q be positive, then this form is unique.

Now define f(x) to be "1/q" if x = p/q (i.e., x is rational) and let f(x) = 0 if x is irrational (e. g. f(pi) = 0).

Then note that f is continuous at every irrational number; it is a wonderful delta-epsilon argument to see this. Also note that if r is any rational number and we redefine f(r)=0 and leave f alone elsewhere, f becomes continuous at r. Hence each discontinuity is removable.

A function that is bounded at every point but is unbounded on any interval (closed or open; we don't allow for empty or single point intervals)

Now let g(x) = q if x is rational and x = p/q and let g(x) = 0 if x is irrational.

Note that any interval whatsoever contains rational numbers of arbitrarily high magnitude (to see this, divide the real number line into, say, 1/10'ths, 1/100'ths, 1/1000's and so on, the way you would a ruler). Hence g is unbounded on any given interval, even though g(x) is finite for any given x.

Cameron Gordon

2005-07-13T14:08:00.000-07:00

Photo: Cameron Gordon (my Ph. D. advisor) with about half of his students at a conference for his 60'th birthday. I am the one with long hair, a gray beard, and a black t-shirt with planets on it.

Cameron Gordon is one of knot theory's top all time researchers. He is probably best known the Gordon-Lueke result (John Lueke was one of his top students and is the blonde guy behind me and to the left) which states that two knots are equivalent if and only if their complements are homeomorphic.

To the non-specialist: a knot will be considered as a smooth (smooth as meaning "differentiable") closed loop in the three sphere (which we consider as the standard 3 space with "infinity" considered to be a point; to visualize this, go one dimension down. Look at the plane and imagine what would happen if "infinity" was declared to be a point. That point could be, say, the north pole). Because of the differentiability condition, one can think of a knot as something like a shoestring whose ends have been fused.

A knot complement is what is left when the shoestring is removed from three space. So the Gordon-Lueke result says that if two knot complements have a one to one continuous function between them that has a continuous inverse, then the knots themselves must have been the "same knot".

Note: this result is well known to be FALSE for "links" (systems of more than one closed loop).

Hyperbolic Space and Crochet?

2005-07-11T09:33:00.000-07:00

From the New York Times:

Professor Lets Her Fingers Do the Talking

By MICHELLE YORKITHACA, N.Y. - Some people looking at the crocheted objects on Daina Taimina's kitchen table would see funky modern art. Others would see advanced geometry.
The curvy creations, made of yarn, are actually both. And they are helping two very different groups - artists and mathematicians - learn more about each other. Increasingly, they are also making a quirky celebrity out of the woman who created them.

"The forms are amazing," said Binnie B. Fry, the gallery director of the Eleven Eleven Sculpture Space, an art gallery in Washington, where Dr. Taimina's creations are part of a summer exhibition called "Not the Knitting You Know."

Dr. Taimina, a math researcher at Cornell University, started crocheting the objects so her students could visualize something called hyperbolic space, which is an advanced geometric shape with constant negative curvature. Say what?

Well, balls and oranges, for example, have constant positive curvature. A flat table has zero curvature. And some things, like ruffled lettuce leaves, sea slugs and cancer cells, have negative curvatures.

This is not some abstract - or frightening - math lesson. Hyperbolic space is useful to many professionals - engineers, architects and landscapers, among others. So Dr. Taimina expected some attention for her yarn work, especially from math students destined for those professions. But her work has recently drawn interest from crocheting enthusiasts.

Math professors have been teaching about hyperbolic space for decades, but did not think it was possible to create an exact physical model. In the 1970's, some educators, including Dr. Taimina's husband, David Henderson, a math professor at Cornell, created hyperbolic models, but the first ones, made from paper and cellophane tape, were too fragile to be of much use.
Though she did not realize it at the time, Dr. Taimina was a good candidate to create a better model. As a precocious child in her native Latvia, she tried her elementary school teacher's patience. When her fellow second graders did not understand a math lesson, she recalled, she would jump up and yell, "I can't stand these idiots," prompting her teacher to send notes home.
By high school she had settled down, and was most impressed by a teacher who was known to keep his advanced students laughing and engaged. When she became an educator, she decided that no student, regardless of aptitude level, would feel out of place in her classroom. One way she assured that was by using everyday objects to explain theories. (She was known for peering so intently at the oranges at her local grocery to see if she could find perfectly round ones to use in her geometry class that she scared the clerks.)

But it was her crocheting hobby that would prove really useful and make her something of a star - at least to knitters and math lovers.

In 1997, while on a camping trip with her husband, she started crocheting a simple chain, believing that it might yield a hyperbolic model that could be handled without losing its original shape. She added stitches in a precise formula, keeping the yarn tight and the stitches small. After many flicks of her crocheting needle, out came a model.

One professor who had taught hyperbolic space for years saw one and said, "Oh, so that's how they look," Dr. Taimina recalled in an interview at her home here, not far from the Cornell campus. A year after she created the models, she and her husband gave a talk about them to mathematicians at a workshop at Cornell. "The second day, everyone had gone to Jo-Ann fabrics, and had yarn and crochet hooks," said Dr. Taimina. "And these are math professors."
The crossover to the art world began last year. An official of the Institute for Figuring, an educational organization based in Los Angeles, spotted an article about Dr. Taimina's models in New Scientist magazine and invited her and her husband to California to speak about them. An audience that included artists and movie producers was there. "They told us this helps with their imagination," Dr. Henderson said.

In February, the two spoke in New York City. To their surprise, the talk, at the Kitchen, a performance space in Chelsea, sold out. Some enthusiasts asked if they were going on tour.
Gwen Blakley Kinsler, the director of the Crochet Guild of America, believes Dr. Taimina's objects will be of interest to free-form crocheters. "It's always nice to be validated," she said. "People think only grannies do this and it's just doilies."

She plans to publish an article about Dr. Taimina and her hyperbolic creations in Crochet Fantasy magazine later this year.

That would be interesting notoriety for someone who, as a child, was told by her teachers not to waste time in art classes. As an adult, when terrified artists started showing up in her math classes to fulfill their degree requirements, she signed up for a watercolor class, thinking, "Then I will know how they feel."

Now when students tell her they simply cannot understand math, she pulls out one of her paintings and says, "I learned that in three months." Then she might pull out one of her crochet models.

The order of addition matters!

2005-06-14T13:26:00.000-07:00

Now we are ready for our demonstration that the infinite sum:

1 - 1/2 + 1/3 - 1/4 + 1/5....... - 1/(2n) + 1/(2n +1)......
can be rearranged to equal any number that we chose. In other words, the order of addition matters!

The idea is this: choose any number in the real number system that you please. For example, choose, say "2".

Now since we picked a positive number, start with the positive terms.

Start with 1 + 1/3 + 1/5 + 1/7 and keep adding until you first get a number greater than 2.
How do you know that you can do this? Because 1 + 1/3 + ....+ 1/(2k+1) + 1/(2k+3) +....
is an infinite sum by the integral test. Exercise: find how many terms it takes to get bigger than 2.

Next start with the negative terms -1/2 - 1/4 - 1/6....and keep going until you first get a number less than 2. Why can this always be done?

Next start with the first positive term you didn't use in your first step and work your way past 2 again, and then use the negative terms to get less than 2 again, and keep repeating this process.

To see why this process yields a sum of 2, let s1, s2, s3,...denote the following partial sums:

s1 = sum of the first set of positives, s2 = sum of the first set of negative terms, s3 = sum of the second set of positive terms, and so on.

Notice that s1 can exceed 2 by at most some 1/n1. Then when we use s2, note that we only go at most 1/m1 past 2 in the negative direction. Then using s3, we exceed 2 by at most some 1/n2, where 1/n2 < 1/n1 (why does this inequality hold?) and then using s4, we go at most 1/m2 below 2 where 1/m2 < 1/m1.

So, given any interval about our chosen number 2, the partial sums must eventually lie within this interval (this is an "epsilon-delta" argument).

In other words, we have shown that our series has been rearragned to have partial sums which go "back and forth" about 2 and get arbitrarily close to 2.

Infinite Series: integral test

2005-06-14T13:09:00.000-07:00

Ok, let's develop an important tool: this is called the integral test.

First of all, remember what a Riemann Sum is; in particular recall the left and right endpoint sums. (Recall all of those little rectangles that you drew when you were first learning about definite integrals).

Next, recall what an improper integral is. This is where you take a limit as one of the limits of the definite integral goes to infinity.

Now suppose f is a function which, on the interval [1, infinity) is decreasing and positive (examples: f = 1/x, f = exp(-x) ). We then have the following:

the sum f(1)+ f(2) +.....+ f(k) + f(k+1) +..... converges if and only if the improper integral
integral(f(x), 1, infinity) converges.

Why? Note that f(1)+ f(2) + .....+f(k) + f(k+1)+...... is a left hand endpoint sum for the integral and therefore is larger than the integral. (why? draw a picture and draw the rectangles for the left hand sum). Note that f(2)+...+f(k) + f(k+1)+.... is a right hand endpoint sum for the improper integral and is smaller than the integral.

Also remember that both sums have a sequence of partial sums which are increasing.

So, if the sum is finite, so is the integral. If the sum is infinite, then so is the sum starting with
f(2) and so is the integral.

That is the integral test, in a nutshell.

As a consequence: sums whose terms look like 1/(ak+b) where a and b are constants are infinite and sums whose terms look like 1/p(k) where p(k) is a polynomial of degree 2 or larger converge.

We have our math-studs too!

2005-06-13T07:45:00.000-07:00

We are used to reading about those who excel in many areas, including those in sports, business, acting, politics, as well as the various popular art areas.

But we in mathematics have our stars too; here is one who has achieved the ultimate in excellence in my research area (topology): Michael Freedman.

He was one of our Field's Medal winners and is currently empolyed by Microsoft.

Infinite Series Part III

2005-06-12T11:25:00.000-07:00

So, we see that in order for an infinite sum to make sense, we must use limits.

So, that is how an infinite sum is defined: if Sn is the "n'th partial sum" (what you get when you sum the first "n" terms with ordinary addition, then the infinite sum is the limit of Sn as n goes to infinity, provided the limit exists.

So that is why we need limits and sequences to talk about infinite sums.

Now what about that strange infinite sum 1 - 1/2 + 1/3 - 1/4 + 1/5......

This is an example of an alternating sum because the signs alternate. To find what this sums to, one must study infinite series. But to see that this sum exists:

Let S2n+1 be the odd partial sums. Since each S2n+1 is a finite sum, we can group and sum the terms in any order; we get (1-1/2) + (1/3-1/4).....+ 1/(2n+1) =

1/2 + 1/12 + .....(1/(2n-1) - 1/2n) + 1/(2n+1) =

1/2 + 1/12 + ....((2n - (2n-1))/(2n(2n-1)) + 1/(2n+1) =

1/2 + 1/12 + 1/(4n^2 -2n) + 1/(2n+1)

We need some results about infinite series here, but as n goes to infinity in we get a convergent

series (p test and comparison test). That is why we have all of those tests.

Now here is what is bizarre: if we change the order of the original series, we change the result of the sum! This is not obvious, and to see this we need some infinite series results.

Infinite Series Part II

2005-06-10T09:09:00.000-07:00

We've seen that an infinite sum can sometimes fail to exist for "obvious" reasons. Examples 2, 4, and 5 are more subtle.

Let's look at examples 2 and 4: can these sums exist? Note that one does have an infinite number of terms in each sum. But also note that each term is getting progressively smaller. So, there might be a chance that these sums exist!

It turns out that the sum of example 4 indeed exists, but the sum of example 2 does not. Here is how we can see this:

In example 1, recall from calculus the idea of an indefinite integral. Integrate the function "1/x" from x = 1 to x = "3". The value of this integral is "ln(3)". Notice what "1 + 1/2" represents: it represents the left hand Riemann sum with n = 2 and delta x = 1 for this integral and remember that because the function 1/x is decreasing, the left hand sum is greater than the integral.

Now keep doing this; note that 1+ 1/2 + ....+1/n represents the left hand sum with delta x = 1, n = n for the function 1/x. Notice that this partial sum exceeds the value of the integral of 1/x as x goes from 1 to n; that is, 1+1/2 +....1/n exceeds ln(n).

So we see that the n'th partial sum exceeds ln(n) and we know from calculus that ln(n) increases without bound as n increases. Hence these partial sums will eventually exceed and finite number. This sum cannot make sense.

On the other hand, if 1 + 1/2 + ....+(1/2)^n = Sn
Then we can multiply both sides by "1/2" to get
1/2 + 1/4 + ....(1/2)^(n+1) = (1/2)*Sn
Now subtract: Sn - (1/2)Sn = 1 - (1/2)^(n+1) (subtract the two sums and note that the leading 1 doesn't get subtracted off and the last term (1/2)^(n+1) isn't cancelled either.)

Now solve for Sn: Sn = (1 - (1/2)^(n+1))/(1 - (1/2))

Now notice that as n gets bigger (as we take a limit as n goes to infinity) the right hand side becomes 1/(1-(1/2))= 1/(1/2) = 2 because (1/2)^(n+1) goes to zero as n goes to infinity.

So, you see that a sequence of partial sums is determined by the limit as n goes to infinity! Here, that limit is 2 and that is the value of this infinite sum.

Example 5 is trickier. It turns out that this sum exists, but we need some theory of sequences to see this. Also example 5 is just downright weird.

Remember back in algebra where you were told that addition is commutative; that is, you could add things in different order and still get the same answer? This is false with infinite sums! To see why, keep reading.

What is an infinite series?

2005-06-10T08:48:00.000-07:00

This entry is designed to help a student understand what an infinite series is and why we need to be fussy.

Regular Addition

Not a problem; we "know" what, say, "3 + 4" means, or even "4 + 7 + 5" means, etc. In a similar way, we have an intuitive idea of what "x + 2x^2" means. A sum with any finite number of terms is always well defined.

Infinite Addition: where the problem begins.

So, now what does it mean to add an infinite number of terms together? Let's look at the following examples:

1 + 2 + 3 + ....+ n + (n + 1) +....
1 + 1/2 + 1/3 +.....+ 1/n + 1/(n+1) +......
1 - 1 + 1 - 1 + 1 - 1....... + (-1)^(n)+ (-1)^(n+1) +....
1 + 1/2 + 1/4 + 1/8 + .....+ 1/2^(n) + 1/2^(n+1) + .....
1 - 1/2 + 1/3 - 1/4 + 1/5 +.....+ (-1)^n)*(1/(n+1))+ (-1)^(n+1)*(1/(n+2))

The first thing to notice is that an infinite sum need not make sense at all! First note that it should be clear that the first sum not make sense. After all, it is clear that as one is adding an infinite number of terms which are getting infinitely large.

To see this we introduce the concept of the partial sum; the first partial sum is 1, the second is 1+2 = 3, the third is 1+2+3 = 6. Exercise: what is the fourth partial sum?

So, we look at the partial sums as a sequence: 1, 3, 6, 10, 15, 21....

and we see that the partial sums are getting bigger without bound. Clearly this infinite sum can't make sense.

Now look at example three: the first partial sum is 1, the second is zero, the third is 1, the fourth is zero, and so on. Exercise: what is the (2k+1)'st partial sum? The 2k'th?

Here the sequence of partial sums (1, 0, 1, 0, 1, 0...) are never going to "steady out" on anything. So this sum can't make sense either.

Ok, what about examples 2, 4, and 5? That is the topic of the next blog entry.

Math Journal

2005-06-07T18:31:00.000-07:00

Here I am hoping to record my progress in mathmetics research.

My summer projects are to learn about hyperbolic geometry, review some knot theory and to get current, and to

1) work on how classes of curves in the plane detect discontinuities of two variable functions
2) work on polynomial invariants of proper knots (proper embeddings of the real line into 3 space)
3) explore wild knots that have symmetries of all orders (are set wise fixed under peroidic homeomorphisms of S^3 which have S^1 as a fixed point set.