Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics

268 Pages·2001·1.69 MB·English

by  John Casti

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title: Five More Golden Rules : Knots, Codes,
Chaos, and Other Great Theories of 20th
Century Mathematics
author: Casti, J. L.
publisher: John Wiley & Sons, Inc. (US)
isbn10 | asin: 0471395285
print isbn13: 9780471395287
ebook isbn13: 9780471437895
language: English
subject Mathematics--Popular works.
publication date: 2000
lcc: QA93.C383 2000eb
ddc: 510
subject: Mathematics--Popular works.
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Page 1
Chapter 1
The Alexander Polynomial:
Knot Theory

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Page 10
The crossing number is thus an invariant of the knot, in the sense that it
depends only on the intrinsic "knottiness" of the closed curve and not on the
way it happens to be projected onto the plane. Therefore, we can use it as a
crude filter to separate some knots from others. For instance, the trefoil with
crossing number 3 cannot possibly be deformed into the Figure-8 knot
shown in Figure 1.5, which has crossing number 4. Unfortunately, the
crossing number is a pretty crude invariant because there are lots of
inequivalent knots having the same crossing number. For example, it has
been shown that there are exactly 9,988 different knots having at most 13
crossings. Consequently, we need much more subtle invariants if we hope
to be able to separate this set of nearly ten thousand knots into distinct
classes.
To illustrate the concept of the crossing number, Figure 1.6 shows a
tabulation of all inequivalent knots have eight or fewer crossings. It is of
some aesthetic interest to note that the famed German Renaissance painter
Albrecht Dürer created six ornamental knots stimulated by knot designs of
Leonardo da Vinci. These six Dürer knots are shown in Figure 1.7. But we
still need better ways than the crossing number to separate knots.
A knot is a simple, closed curve, so it makes sense to think about starting at
some point on the knot, and then traversing it and returning to your starting
point. The direction one moves in such a traversal of the knot is called the
knot's orientation. This is a basic aspect of a knot that is of great value in
finding invariants. We can also attach a sign to each doublepoint of a knot.
We do this by using the French traffic rule that cars on the right
approaching an intersection have right-of-way. This means that a
doublepoint is "positive" if the strand passing under the
Figure 1.5
The oriented
Figure-8 knot.

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Page 101
Chapter 3
The Kalman Filter:
Control Theory

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Page 103
Looking for Life
In summer 1975, two Viking spacecraft were launched from Cape
Canaveral, Florida, on their way to a rendevous with Mars one year later,
where their mission was to search for signs of life on the Martian surface.
Upon arrival, both spacecraft were put into a predetermined orbit around
Mars, from which the search for suitable landing places began. On July 20,
1976, the Viking I lander came to rest in the Chryse Planitia region of Mars,
some 23 degrees north of the equator, and six weeks later Viking II settled
down onto the Utopia Planitia plain, which is almost exactly 180 degrees of
longitude away from the first landing site, thus putting the two landers on
opposite sides of the planet.
There were six different instruments on the landers used for the detection of
signs of life: two cameras for photographing the landscape, a combined gas
chromatograph and mass spectrometer for analyzing the surface for organic
material, and three instruments designed to detect the metabolic activities of
any microorganisms present in the Martian soil. Sad to say, none of the
tests produced unequivocal evidence of life at either of the landing sites.
Nevertheless, much data of great scientific value was obtained, including
signs of extraordinary chemical reactivity of the Martian soil. But for our
story here, what is notable about the entire Viking mission is that it took
place at all! Think for a moment what was involved in just getting the two
landers to their antipodal locations on the Martian surface.
Leaving aside the mechanical, electrical, and chemical engineering aspects
of the Viking spacecraft, the sine qua non of the mission is the obvious
point that NASA had to be able to get the landers from Cape Canaveral to
their resting points on Mars. This task involved several aspects of guidance
and control: determining what kind of propulsion was necessary to transfer
a certain kind of vehicle from Earth to a Martian orbit; calculating the
various trajectories possible; identifying the trajectories of minimal cost in
some combination of time, energy, and other resources; developing the
capability to measure where the spacecraft was located at any time during
its flight; and being able to exert midcourse corrections if and when the
spacecraft drifted off its design trajectory. At the center of all these tasks
sits what has come to be called the Kalman filter, a mathematical theory
that, along with the simplex method of optimization theory (see Chapter 5
of Five Golden Rules), is

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Page 104
probably the single most useful piece of mathematics developed in this
century. Our story in this chapter is an account of what this gadget is and
how it came to occupy its pivotal role in the mathematical theory of control
systems.
Can You Get There from Here?
To see what's involved in the problem of guidance and control, let's look at
a simple toy version of the problem. Suppose we have a spacecraft whose
trajectory is constrained to move along the real line, as shown in Figure 3.1.
If the vehicle starts at the origin with an initial velocity in the positive
direction, then it's clear that by simple inertia it will just sail along,
eventually visiting every location on the positive real axis. But if we want
to bring the spacecraft to a soft landing at, say, the point marked X in Figure
3.1, then we would have to turn it around somehow, and then exert a
retrothrust to neutralize the inertia from the rocket's initial thrust, thereby
reducing the vehicle's velocity to zero at the point X. Under what
circumstances can this be done? In other words, when can we transfer the
spacecraft from the origin with a given initial velocity to the point X with
zero velocity? This is a question that control theorists call a problem of
reachability.
To see what's at issue in solving the reachability problem, consider a linear,
discrete-time control system moving in the plane. To make things as
transparent as possible, suppose the control input at each moment is a
simple scalar, and that it enters into the dynamics in a linear fashion. Under
these conditions, the system dynamics can be written in the form
Figure 3.1
A spacecraft moving along the real line.
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Page 105
For ease of writing, let's write the above system in the vector-matrix form
where F and g are given by
Here the quantities fij, gi(i, j = 1, 2) are just real numbers. Furthermore, it
turns out to involve no loss of generality to assume that the initial state of
the system is x1(0) = x2(0) = 0, that is, the system starts at the origin. The
reachability problem can now be stated as: What points (x1, x2) in the plane
can be "reached" within, say, T time steps by a suitable choice of the
control inputs u(0), u(l), . . . , u(T 1)?
First of all, let's dispose of the two extreme cases, (1) when there is no
control at all (u(t) = 0 for all t), and (2) when the control input cannot
influence the system state (if g1 = g2 = 0). In both these cases, the system
state remains at the origin forever; hence, no state other than the origin is
ever reachable. Trivial as these two cases are, they illustrate the twin points
that (1) a control system isn't a control system if you can't exert some
influence on the behavior of the system's state, and (2) the only way you
can influence the state is when the channel between the control and the
system state remains open. So we will henceforth assume without further
comment that these conditions are satisfied.
Suppose, then, that the initial control input u(0) is nonzero. Then the system
state at time t = 1 is x1(l) = u(0)g1 and x2(1) = u(0)g2. Thus, as long as the
quantities g1 and g2 are not both zero, the system moves away from the
origin after one time step to the point (g1, g2)u(0). If we introduce the state
vector x(t) = (x1 (t), x2 (t))' and the input vector g = (g1, g2)', this can be
written more compactly as x(l) = gu(0). (Note: Here the notation v' means
simply the transpose of the vector v.) This means that in one time step the
system can be transferred to any point on the line from the origin through
the point g. By a similar line of argument, the state at the second time step
is