Additions to the book
Industrial Mathematics
C.R. MacCluer
Chapter 1
Typo: equation (1.8) should be an expectedvalue. In Latex source it should be
E[g(X)] = \int_{-\infty}^{\infty} g(x) dF(x) = \int_{-\infty}^{\infty} g(x) f(x) dx
Likewise, equation (1.9) should be
E[g(X)] = \int_{-\infty}^{\infty} g(x) dF(x) = \sum_{i} g(x_{i}) m_{i}
Typo: Routine 1.1, page 8: lines 3 and 4should be interchanged to read
pk = (1-p)^n;
sum = pk;
Not serious --- will not change the result.
Typo: page 14, in both (1.36) and (1.37) thebottom subscript on L should be m, not n+1.
Exercise 1.33. This is known as Benford'sLaw--- the digit 1 leads more often in tables of
physical constants (since the table isinvariant under change of units).
Typo: Exercise 1.35 should read p n =\lambda T not p/n.
Typo: Exercise 1.24 should read"Normalize appropriately and graph against (1.15). Use hist."
(Hint: The area under the histogram should alsoequal 1.)
Contributed by John Ferden (ajunior at Ovid-Elsie HS): What is theprobability of
guessing at least 6 correct answers on atrue-or-false exam?
using simple, inexpensive, low-pixel-count CCD cameras by taking
multiple (n>50) exposures, then combining them with Adobe PhotoShop.
Their results are comparable to older photos taken with the Mt. Palomar
200 inch. See the most notable of these amateurs, Thierry Legault:
Why does this work?
Outline: Show that a random variable X withstandard deviation sigma, when (independently) sampled
Chapter 2
Contributed by Will Brockman:
by Poisson --- see
Exercise: Show that when the number ofarrivals are governed by the Poisson distribution (1.26),
then interarrival times are given by Poissonflow (2.7) and conversely.
Exercise. Modify slightly the previous derivation of Poisson flow by assumingthat the rate of
failure (hazard rate, force of mortality) increases with time (as withautomobile tires) according
to some power rule h = at^b. Deduce that the time before first failure is givenby
the Weibull cumulative distribution
F(t) = 1 - exp( -(t/alpha)^beta).
Project: Simulate the double slitexperiment via
photons grow, the impacts on the screen(behind the double slit) build up a histogram similar to interference
patterns, as if each photon were a waveinterfering with itself.
Project: Experimentally verify L\'{e}vi's theorem: Suppose $\zeta(x,y)$ is specified to be either 0 or 1 ateach
point on the boundary of the bounded domain$\Omega$ in the plane. Let $u(x,y)$ denote the probability that
a particle released at $(x,y)$ in $\Omega,$thereafter undergoing Browian motion, will exit through the boundary
at a point with $\zeta$-value 1. Then $u(x,y)$is a harmonic function that agrees with $\zeta(x,y)$ on the boundary
of $\Omega.$
Chapter 3
Typo: Equation (3.30) should be (in Latexsource) F \{ u_{k-k_{0}} \}_{k=k_{0}}^{\infty}
= \{ y_{k-k_{0}} \}_{k=k_{0}}^{\infty}
Typo: page 40, line 9: should be u_{k} =\alpha^{k}, not u_{k} = \alpha.
Exercise. Find a linear, causal filter thatis not time-invariant. (Answer: y_{k} = u_{[k/2]})
Exercise. Find
Exercise3.14. The simulated noisy sinusoidal should be pulsed. The matched filter
will correlate the nearby noise onto the center frequency and thereby misleadthe
student into believing the filter is performing over-well.
Chapter 4
Project. Contributed by George Stockman. Whatis the least number of pixels and greyscale levels
needed to recognize familiar faces?
web. Reduce the number ofpixels and greyscale level, until the image is just recognizable, keeping the
size of the image constant. Ithelps to blur the sharp edges as pixels become coarser. Recognition seems
possible down to 32 X 32pixels, and 3 bit greyscale, i.e., W = 7. For example, if we startwith a 256 x 256
image of 256 grey levels, thenwe can next try a "128 x 128" image of 256 grey levels formed asfollows.
Each 2x2 block of the inputimage is averaged and the average is repeated 4 times in the output image:
the output still has 256 x 256pixels, but each is really a 2x2 block of identical grey levels.
Typo: Should be an " i " inthe exponent in (4.1) and exercise 4.8. Should be "filter(4.14)" in exercise 4.6.
Chapter 5
Chapter6
Chapter 7
Typo: Exercise 7.4. Assume that inflationrate i = 3%. Inflate savings at 3%, subtract costs, then discount
at 5%. The actual answer is about $11, 174.08.
Project: Can a solar cell ever pay foritself? Assuming present and future electric rates and projected
manufacturing costs of photovoltaics, will thelife-cycle savings ever exceed 0?
Chapter 8
Typo: In the right hand side of (8.16), both D's should be differentiated, i.e.,
.... = - b_{n}D'(b_{n}) (8.16a)
.... = - a_{n-1}D'(a_{n-1}) (8.16b)
and the consequent changes
a_{n} =F(b_{n}) (8.17a)
a_{n} = F(F(a_{n-1})} (8.18a)
Exercise. Prove that maximum profitand maximum revenue necessarily occur at different
prices. Which price is lower?
Project. Contributed by JimCase. E.H. Chamberlin in 1933 proposed an alternative model for
duopolistic competition. For each firm, graphprofit against price. Call the price where profit
maximizes the optimal price for thatfirm. Then the lowest optimal price determines the price
for the commodity. Is this a well-definedconcept?
Exercise: (V. P. Sreedharan) For the Leontiev input/output model in 8.6, prove the following converse.
If some positive bill of goods b > 0 isrealizable by some nonnegative production output x, then every
nonnegative bill of goods b is realizable, and,in fact, all eigenvalues of A are less than 1 in modulus.
Proof outline: x \geq Ax \geq A^{2}x \geq \cdots and A^{n} x= A^{n+1}x + A^{n} b. Thus
x = b + Ax = b + Ab + A^{2}x = b + Ab + A^{2}b+ A^{3}x = \cdots giving that \sum_{0}^{\infty} A^{n}b < \infty.
But then, because b > 0, \sum_{0}^{\infty}A^{n} < \infty. Apply to v where Av = \lambda v.
Exercise: Prove Result C (page 120) byshowing the geometic series \sum_{n=0}^{\infty} J^{n} converges
if and only if | \lambda | < 1,where J is a
diagonal, 1 down the subdiagonal).
Chapter 9
Chapter 10
Exercise 10.57 Hint. Contributed byBrigit Jacob and Jonathan Partington. Try p(t) = sin (t^2/2). See
T.W. Koerner's book "Fourier Analysis," Cambridge University Press, 1988, page 406.
Chapter 11
Typo: Exercise 11.3. Replace the u(r,z) byu(r,z) = 0 + 2 \sum etc. The sentence following
should be "Can v = 1 -u beextended oddly to solve .... ?"
Typo: Exercise 11.42. Should be u_{tt} -c^{2} u_{xx} = 0
Typo: Exercise 11.49. \alpha =\kappa/\rho c
Typo: Exercise 11.53. The answer should bep(t) = - 2 \pi \sum etc
Chapter 12
Typo: Runge-Kutta (12.18a)should be k_1 + 2 k_2 + 2 k_3 + k_4.
Chapter 13
Typo:
Typo: page 253: line 2should read (Exercise 13.11)
Typo: In Exercise 13.2,
Typo: In Exercise 13.8, thesubscripts of c should be mn.
Typo: In Exercise 13.23,line -3, should be, “ orespecially, the square root of the sum of the squares …”
Chapter 14
Chapter 15
Index