CAN ONE DO SERIOUS MATHEMATICS
USING PICTURES AND CALCULUS?
Raimond A. Struble, PhD.
Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
Special Seminar, 201 Harrelson Hall,
Department of Mathematics, Undergraduate Division,
North Carolina State University at Raleigh, Raleigh, NC,
4:00 PM, Tuesday, September 28, 2004.
Send comments and correspondence to:
George.Moore4@med.va.gov
See also:
http://www.medparse.com/struifpr.htm .............
http://www.medparse.com/struitgr.htm .............
http://www.medparse.com/strucomm.htm .............
http://www.medparse.com/infnpapl.htm
ABSTRACT.
A visual-type problem will be discussed, using geometrical notions
and calculus, leading easily to the Lebesgue integral. The main point,
however, is to illustrate some serious mathematics in a context which
can be appreciated by those with limited mathematical experiences.
All that is required to completely understand this work
is to employ the logic exhibited by simple pictures,
and to follow simple manipulations of primitive expressions and concepts
(nothing more advanced than convergence of infinite series
and infinite products) that belong to calculus.
REFERENCES.
1. Mikusiński J.
The Bochner integral.
New York, San Francisco: Academic Press.
Harcourt Brace Jovanovich, Publishers. 1978;:.
Pure and Applied Mathematics.
Basel: Birkhäuser. Lehrbücher und Monographien
aus dem Gebiete der exakten Wissenschaften: Mathematische Reihe.
[Textbooks and monographs from the area of exact sciences:
mathematical series.] 1978;55:.
ISBN: 3764308656, 233 pages.
2. Mikusiński J, Mikusiński P.
An Introduction to Analysis.
From Number to Integral.
New York: John Wiley and Sons Ltd. 1993 Apr 8;:.
ISBN: 0471599778.
3. Derbyshire J.
Prime Obsession: Bernhard Riemann
and the Greatest Unsolved Problem in Mathematics.
New York: Plume Books. 2004 May 25;:.
ISBN: 0452285259, 448 pages.
CHAPTER 2. TABLE OF CONTENTS.
Chapter 1. Abstract.
Chapter 2. Table of Contents.
Chapter 3. The Problem.
Chapter 4. The Method.
Chapter 5. The Painting Lemma.
Chapter 6. Being Specific.
Chapter 7. Change of Perspective.
Chapter 8. Painted Triangles as Ordinary Functions.
Chapter 9. Pictures vs. Tent Functions.
Chapter 10. The Big Picture.
Chapter 11. Advanced Theorem.
Chapter 12. Elementary Example from Calculus.
Chapter 13. Riemann and Picasso-type Painting.
Chapter 14. Acknowledgements.
Chapter 15. References.
Chapter 16. Appendix A. Continuous Painting.
Chapter 17. Appendix B. Overpainting the Barn.
Chapter 18. Appendix C. More on the Riemann Zeta Function.
Chapter 19. Appendix D. Imaginative painting and school spirit: North Carolina State University Wolfpack.
Chapter 20. Appendix E. Sample Calculations.
Chapter 21. Appendix F. Additional Reading.
Chapter 22. Appendix G. Historical Notes.
Chapter 23. Appendix H. Glossary.
Chapter 24. Appendix I. Skin Cancer Model.
CHAPTER 3. THE PROBLEM.
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To paint a barn a little at-a-time, one isosceles triangle at-a-time.
CHAPTER 4. THE METHOD.
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Suppose that we successively select isosceles triangles
pointing upward, to fill unpainted portion of our barn.
If we do this in a sequence of infinitely many steps,
the question to ask is: "does the whole barn really get painted?".
A precise display of the steps follows:
STEP 1:
31.
STEP 2:
32.
STEP 3:
33.
STEP 4:
34.
STEP 5:
28.
The sensible procedure, of course, is to maximize the area
of each triangle to be painted, realizing it is to be placed
within a skewed triangle pointing downward.
FAT:
36.
MAX:
73.
SLIM:
38.
This maximum (1/4) occurs for a triangle with (1/2)
the dimensions of the unpainted triangle. Returning to the central question
concerning the success of the painting, we should do a little analysis
of the process.
PAINTING ANALYSIS, STEPS: 1 2 3 4 5
1 1 1 1 3 1 1 3 3 1 1 3 3 3 1 ....
Fraction painted: 2 2 4 2 4 4 2 4 4 4 2 4 4 4 4
1 1 3 1 3 3 1 3 3 3 1 3 3 3 3 ....
Fraction not yet painted: 2 2 4 2 4 4 2 4 4 4 2 4 4 4 4
TOTAL fraction painted: ½ +⅛[1 +¾ +¾2 +¾3 + .... ]
½ +⅛[(1/(1-¾))] = ½ + ½ = 1.
Sweet success: the barn has been successfully painted.
Just lucky. Painting the maximum amount at each step? What if we choose
some smaller fraction, say, r<(1/4), at each step? What then?
PAINTING ANALYSIS, STEPS: 1 2 3 4
Fraction painted: ½ ½r ½(1-r)r ½(1-r)(1-r)r ....
Fraction not yet painted: ½ ½(1-r) ½(1-r)(1-r) ½(1-r)(1-r)(1-r) ....
TOTAL fraction painted: ½ +½r +½(1-r)r +½(1-r)(1-r)r + ....
½ +(r/2)[1+(1-r) +(1-r)2 + .... ]
½ +(r/2)[1/(1-(1-r))] = ½ + ½ = 1.
WOW! For even small values of r, where the painting
is done inefficiently, we nonetheless always succeed in painting the barn;
and this with the same number of triangles at each step! (See figure).
91.
Note: the above illustration more accurately represents
the case r = 1/8.
Can we ever fail? Suppose we choose smaller and smaller fractions
at each stage.
PAINTING ANALYSIS, STEPS: 1 2 3 4
Fraction painted: ½ ½r1 ½(1-r1)r2 ½(1-r1)(1-r2)r3 ....
Fraction not yet painted: ½ ½(1-r1) ½(1-r1)(1-r2) ½(1-r1)(1-r2)(1-r3) ....
TOTAL fraction painted:
F = ½ + ½r1 + ½(1-r1)r2
+ ½(1-r1)(1-r2)r3
+ ½(1-r1)(1-r2)(1-r3)r4
+ ....
? (MYSTERY)
FINAL fraction not yet painted:
P = ½(1-r1)(1-r2)(1-r3)(1-r4) ....
Infinite product.
This infinite product should be zero (nothing left to paint),
if the painting of the whole barn has been a success. So maybe
one CAN fail. In fact, we now know exactly when this will happen.
CHAPTER 5. THE PAINTING LEMMA.
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Suppose at the nth stage of painting, one paints an
rn fraction of the unpainted area,
then the barn gets painted if and only if the infinite product:
P = ½(1- r1)(1- r2)(1- r3)(1- r4) ... (1- rn) ....
diverges to zero as n ⇒ ∞.
We can put this all in a more useful form,
employing a familiar result from calculus.
A CALCULUS LEMMA.
The infinite product, P, is zero
if and only if the infinite series:
r1 + r2 + r3
+ ... + rn + ....
diverges. We can now state our fundamental painting theorem.
THE PAINTING THEOREM.
Suppose that at the nth stage of painting, one paints an
rn fraction of the unpainted area, then the barn
gets painted if and only if the series
r1 + r2 + r3 +... diverges.
So the convergence or non-convergence of an infinite series
tells us whether or not the barn gets painted.
Example:
r = r1 = r2 = r3
= r4 = ... rn ...; r + r + r+ r + r + ... = ∞
Example:
rn = (1/n); (1/4) + (1/5) + (1/6) + (1/7) + ... (1/n)
+ ... = ∞; the harmonic series.
These two examples result in successful paintings. An unsuccessful
painting results for rn = (1/4)n,
since the infinite series, (1/4) + (1/16) + (1/64) + ...,
converges, to (1/3). Then the infinite product, P (>0),
gives the fraction of the barn never painted. Of course, a (1-P)=F
fraction of the barn does get painted, where F is that mystery
series, labeled TOTAL fraction painted. Note that this series
sums to 1, when the barn gets completely painted, yet the
rn's can be selected randomly
(rn<(1/4), in order to satisfy
∑rn = ∞. Strange?
This situation contrasts sharply with that of the traditional interest
in infinite products and infinite series, where success is associated
only with convergence. Here, divergence creates success
in painting.
CHAPTER 6. BEING SPECIFIC.
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For clarity (and trickery), let us now specify dimensions
for our barn. Let the barn be 1 barn unit tall
and 2 barn units wide. The barn thus has an area equal
to 2 barn units2. At step 1, ½
the barn gets painted (big triangle, area 1 barn unit2).
and 1 barn unit2 remains. With this modification,
the
TOTAL fraction painted
(after step 1)
becomes:
F = r1 + (1-r1)r2
+ (1-r1)(1-r2)r3
+ (1-r1)(1-r2)(1-r3)r4 + ....
The final fraction not yet painted
(after step 1)
becomes:
FINAL fraction not yet painted:
P = (1-r1)(1-r2)(1-r3)(1-r4) ....
The barn is successfully painted if and only if P=0.
When this happens, F=1.
The fundamental PAINTING THEOREM is unchanged except
that the rn fractions now account for
the painting process after the initial (big) triangle has been painted.
The fundamental identity:
1 = F+P = [r1 + (1-r1)r2 +
(1-r1)(1-r2)r3 + ...] +
[(1-r1)(1-r2)(1-r3) ....]
yields an interesting (and useful) formula for an infinite product:
P = (1-r1)(1-r2)(1-r3) ...
= 1 - [r1 + (1-r1)r2 +
(1-r1)(1-r2)r3 + ... ]
which can also be gotten from just a formal expansion of the factors
on the left, collecting the terms as one moves through that product.
(Note: (1-r1)(1-r2) =
1-r1 - (1-r1)r2;
(1-r1)(1-r2)(1-r3) =
1-r1 - (1-r1)r2
- (1-r1)(1-r2)r3, etc.)
It is instructive to examine why (analytically)
it is that the series (total fraction painted)
F = r1 + (1-r1)r2 +
(1-r1)(1-r2)r3 +
(1-r1)(1-r2)(1-r3)r4 + ....
always converges (it is obvious pictorially from the painting problem).
When the series, r1 + r2 + r3
+ ... is convergent, it clearly dominates F. But when
the series r1 + r2 + r3 + ...
diverges, the infinite product P=0, so that the convergence
of F results from the rapid decay of the product factors,
(1-rk) of each term.
Example (Euler-Riemann):
Let us now make a somewhat strange choice of painting fractions:
r1=0, r2=1/22,
r3=1/32, r4=0,
r5=1/52, r6=0,
r7=1/72,
r8=r9=r10=0,
r11=1/112, r12=? r13=?
78.
AN EULER-RIEMANN PAINTING RULE:
rp=1/p2 for primes p;
rn=0, otherwise.
Then:
r1 + r2 + r3 + r4 +
r5 + r6 + r7 + ...
= 1/22 + 1/32 + 1/52 +
1/72 + ...
(converges), and the final fraction not yet painted:
P = (1 - 1/22)(1 - 1/32)(1 - 1/52)(1 - 1/72) ...
exceeds zero, and tells us then this particular rule results in
a FAILED PAINTING.
The final fraction painted becomes
F = 1/22 +
(1 - 1/22)1/32 +
(1 - 1/22)(1 - 1/32)1/52 +
(1 - 1/22)(1 - 1/32)(1 - 1/52)1/72 +
...
The painting fractions employed are required to be (1/4), (1/9),
(1/25), (1/49), (1/121), ... , and the first two steps are routine,
and appear nearly as in our initial painting of the barn.
The next ones present something of a challenge. To meet those
decreasing fractions, one must use either very slim or very fat triangles.
To illustrate, let us choose slim ones throughout. The results are
(very incompletely) exhibited on the accompanying picture.
169.
Note: the above illustration more accurately represents
the case of s = 1.5.
All the triangles to be painted during the first three steps
are shown.
We here exhibit only some of the more visible ones for the next
three steps. Beyond these, all triangles to be painted, if exhibited,
would be indistinguishable from vertical spiked segments.
The picture (beyond the initial triangles), if completed, would appear
throughout mostly as a dense forest of vertical segments. Notice how even
the visible triangles shown climb up the big triangle in ever-decreasing
increments. The last ones exhibited here correspond to the fraction,
(1/121). We have chosen not to show the forest throughout,
but suggest what will happen in a portion of the barn.
This dense forest pattern is omnipresent in the upper regions of
all areas between the visible, or nearly visible triangles.
It's a startling picture contemplated here, particularly realizing that
approximately 60% of this area of the barn receives no paint
at all! Just what does this have to do with Euler and Riemann?
In 1859, Riemann defined a function (of s>1) as
an infinite series:
ζ(s) = 1 + 1/2s + 1/3s + 1/4s +
1/5s + 1/6s + 1/7s + ...
(all integers)
and chose to call it the zeta function.
One hundred twenty-two years earlier (1737), Euler had shown
that the reciprocal of this infinite series could be written
as an infinite product:
1/ζ(s) = (1 - 1/2s)(1 - 1/3s)(1 - 1/5s)(1 - 1/7s) ...
(all primes)
Of course, he did not know that Riemann was going to abscond with
his infinite series and call it zeta. Actually Riemann gave Euler
full credit for it all. For s=2, this infinite product
is just our final not yet painted expression for P above.
So in general (for any s>1), if we paint by the rule
rp = 1/ps for primes p and
rn = 0 otherwise, our final fraction not yet painted,
P, is just 1/ζ(s). Our final fraction painted
is given by F = 1-P = 1 - 1/ζ(s) for
s>1.
But we can do barn painting even if s<1. (One may have to
drop some early terms greater than 1/4). In which case, the series
∑rn = ∑1/ps diverges, the final
fraction not yet painted P=0, and the total fraction painted is
then F=1, i.e., the barn is successfully painted. This somewhat
puzzling situation is illustrated by the plot of F as shown
in the picture:
79.
We give the plot of F, the fraction painted, for all
s>0. The result is F=1 for
0<s<1 and F=1 - 1/ζ(s)
for s>1. Hmmmm.
For a slightly different (and more detailed) treatment of this example,
and for some other idle chit-chat concerning our painting process
with the Riemann zeta function, see Appendix C.
If you are reading this manuscript on the internet, you may perform
sample calculations from Appendix E.
Concerning the restriction, rn<(1/4), employed here,
this constraint arises merely because we are very disciplined, and
restrict the painting process to isosceles triangles. If we were more
liberal, and allowed painting in any form (maybe just paint globs tossed
at the barn), this restriction could be relaxed to simply
rn<1.
(See picture.)
82.
Nonetheless, the painting theorem still yields the maxim:
the barn gets painted if and only if
r1 + r2 + r3 + ... = ∞,
no matter how we paint (and so long as we don't waste any paint,
i.e., paint only in unpainted regions.) Another interesting variation
in the painting process is elaborated on in
Appendix A.
This results in a painting theorem for a continuous painting process
(as contrasted to the step-by-step process employed here), completely
analogous to the discrete case. In Appendix B,
we indicate what happens when one is a little careless in the
painting process, and chooses fractions to paint which lead to triangles
too big to be accomodated in unpainted regions.
CHAPTER 7. CHANGE OF PERSPECTIVE.
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Suppose that at each position, x, along the barn, we examine
the vertical segments within the painted triangles, and attempt to add up
their lengths. (See Picture.)
83.
Then as x varies from 0 to 2, these combined
lengths L(x), encountering infinitely many painted triangles,
will vary between 0 and 1. Clearly L(1)=1
where we encounter but a single triangle (the first one painted),
and also L(0)=L(2)=0, since, there, no segments
ever lie within any painted triangles. If the painting job has been
successful, we might expect (perhaps) to find the total length
L(x) to be constantly 1 for 0<x<2.
In order to investigate this question further, we revisit the original
step-by-step procedure used for painting the barn.
As each triangle is chosen for painting, suppose that we move
the vertical segments (within) downward on the barn, until we reach
a previously approximating graph on the barn (see figure).
Starting at STEP 2 with two triangles, then adding
three new triangles at STEP 3, and then numerous triangles at
STEP 5,
STEP 2:
174.
STEP 3:
175.
STEP 5:
66.
(where not all painted triangles are shown), one obtains graphs
of the increased area which become improved approximations
to the (ultimate) total sum L(x) (various x) we seek.
The new estimates move upward. Continuing in this fashion, we can obtain
better and better approximations of L(x), as a graph
along the barn. For the case, r=(1/4), as one adds
more and more of the painted segments and then plots these additions
on the barn, the resulting pictures strongly suggest that the barn graphs
eventually climb to the roof. Note the new locations of the vertices
of the transformed, non-isosceles triangles formed, particularly
the steady upward movement of their peaks. Can the rest of the painted
segments be far behind? Anyway, this is something of a
picture proof that indeed L(x)=1 for 0<x<2,
when r=(1/4). Analogous pictures are not so convincing
when very slim and very fat triangles are substituted, successively,
with r<(1/4). Imagine what the pictures might look like
when the barn is completely painted in a mode prescribed
by the harmonic series, where rn = (1/n).
No simple pictures at all, as one has to select at each step
either fat or slim triangles dictated by the number (1/n).
(Not nearly as wierd as the Riemann zeta picture,
where the barn-painting fails.)
Let us now change perspective again, and re-interpret the painted
triangles as giving us ordinary graphs of ordinary functions.
To this end, we push all the painted triangles down to the x-axis
(see pictures).
CHAPTER 8. PAINTED TRIANGLES AS ORDINARY FUNCTIONS.
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ON THE BARN:
39.
GRAPHS OF THE LOWERED TRIANGLES:
40.
COMBINED PLOT OF THE LOWERED TRIANGLES:
41.
Along the x-axis, their graphs look like tents,
and we could give very simple formulas for them. (Zero mostly,
and linear segments otherwise). Mostly they look like
lonely tents. It might be convenient to enumerate them,
say, t1, t2, t3,...,
more-or-less in the order that we paint them. When we combine
their graphs into one plot, the picture gets messed up, but we can
symbolically add up the lengths of the painted segments anyway
as the sum of an infinite series of tent functions:
L(x) = t1(x) + t2(x) + t3(x) + ....
This gives us L(x) in the usual form. Its graph may
(or may not) be the straight line function with value 1 for
0<x<2?
84.
CHAPTER 9. PICTURES VS. TENT FUNCTIONS.
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LET US SUMMARIZE.... We can either obtain the sum
L(x) of lengths of the painted segments pictorially by moving
all segments downward on the barn, to obtain its graph; or else we can sum
an infinite series of simple tent functions, defined for
0<x<2, to obtain a graph in the ordinary way. (See picture.)
24.
Similarly, we can symbolically add up the areas of the painted triangles,
as an infinite series:
∫L(x)dx = ∫t1(x)dx + ∫t2(x)dx
+ ∫t3(x)dx + ...
(here, ∫ stands for 0∫2)
of integrals which then gives the area under the L-curve.
Our painting procedure summed this series for us rather easily.
Here it looks like a rather difficult job, finding the areas
of those triangles and summing their areas.
Just finding the triangles seems like a big task.
Of course, one knows that the barn has been successfully painted
whenever this series adds up to the number 2. We know it
from the painting lemma, when P=0.
If on the other hand, ∫L(x)dx<2, then certainly
L(x)<1 for some x's, and we know what the painting
theorem tells about the fractions rn used to paint
(to fail to paint) the barn. (Their sum converges.) The integral signs
above on the right are just window dressing, since these numbers
are merely the areas of triangles (=(1/2)bh).
More interesting in some ways are these cases wherein the painting
of the barn IS a failure. In such cases, using almost exclusively
VERY FAT or VERY SLIM triangles, one might expect the
L(x) function to look more like the Dow-Jones averages (or worse);
and the barn itself like some bizarre Picasso-like painting.
It is important to emphasize that these graphs of L(x)
and of the barn painting occur when the painting process actually fails.
Recall the failed Riemann zeta painting case and try (just try!)
to imagine the appearance of the resulting L(x) function!
77.
We must move on, and so we're led inexplicably to the domain
of the "REALLY BIG PICTURE". This picture is, however,
of necessity more analytical than visual,
although in describing it,
we'll strive to keep it as visual as possible.
CHAPTER 10. THE BIG PICTURE.
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Consider an ARBITRARY sequence, t1,
t2, t3,..., of tent functions.
Some of these tents will be permitted to be upside-down,
and their function values, together with their areas,
count as negative numbers. Moreover, to cover all situations,
we extend the tent functions from -∞ to +∞,
and they become really lonely.
20.
Also they can be very big or not:
21.
They can bip up or down anywhere along the x-axis,
and if we are only interested in the interval 0 to 2,
then the triangles will bip up or down only in that interval.
Of the sequence of tents, we impose only one restriction,
namely, that if they are all turned right-side-up,
then the total area must be finite. Simply stated,
the infinite series of positive and negative numbers,
∫t1(x)dx + ∫t2(x)dx +
∫t3(x)dx + ....
must converge absolutely, i.e.,
|∫t1(x)dx| + |∫t2(x)dx| +
|∫t3(x)dx| + .... < ∞
Recall that these integral signs are just window-dressing
for the ± areas of triangles. Nothing fancy here.
For the sum of the segments within the tents, we have
L(x) = t1(x) + t2(x) + t3(x) + ....
where some of these numbers may be negative, but,
this sum is to be accepted only if the infinite series
converges absolutely. With other x's, we just ignore them,
and consider L(x) to be undefined. (Seems fishy?)
We then define the number ∫L(x)dx to be the sum
of the series of integrals:
∫L(x)dx = ∫t1(x)dx + ∫t2(x)dx
+ ∫t3(x)dx +....
All the advanced concepts lie with the integral on the left.
CHAPTER 11. ADVANCED THEOREM.
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If t1, t2, t3, ...
is a set of tent functions (positive and/or negative), and if the
infinite series of numbers,
∫t1(x)dx + ∫t2(x)dx
+ ∫t3(x)dx +....
converges absolutely, then this sum is the Lebesgue integral
of a Lebesgue-integrable function, L(x), defined
as the sum of the infinite series,
L(x) = t1(x) + t2(x) + t3(x) +....
whenever this series is absolutely convergent.
We then write
∫L(x)dx = ∫t1(x)dx + ∫t2(x)dx
+ ∫t3(x)dx +....
and call it the Lebesgue-integral of L(x). Moreover,
every Lebesgue-integrable function possesses such a
(many, of course) representation.
Finding such a representation for a given Lebesgue-integrable function,
L(x), may, however, be a very difficult task. One can get
some idea of how this might be done in simple cases, for example,
we actually did this when we painted the barn using those
initial pictures. The triangles, while yielding the area of the barn,
inadvertently defined a Lebesgue-integrable-function, L(x),
that we sought pictorially, and speculated was a constant function,
L(x)=1 for 0<x<2. The advanced theorem above
assures us that this equality does indeed hold whenever the barn
is successfully painted, since absolute convergence of the series
is guaranteed (the terms are bounded and positive).
Any Dow-Jones L(x) resulting from a failed painting case is always
just another such Lebesgue-integrable-function, given by
the advanced theorem. One wonders just what this special class
of functions could be. Recall the example using the Riemann zeta numbers,
1/ps, and the picture illustrated for s=2,
with those dense forests and all. Some Lebesgue function, indeed!
Before venturing further along this bumpy road, it is useful to change
the setting slightly, and consider an equivalent advanced theorem.
What we do is replace the tent functions throughout with
brick functions, that look like bricks. These have graphs
of the type:
85.
They assume a positive or negative constant value on some
finite interval, anywhere along the x-axis, and are zero
otherwise. They can be very big or not.
EQUIVALENT ADVANCED THEOREM.
Let b1, b2, b3,...
be a sequence of brick functions. If the infinite series
(of positive or negative numbers),
∫b1(x)dx + ∫b2(x)dx
+ ∫b3(x)dx +....
is absolutely convergent, then this sum is the Lebesgue-integral of a
Lebesgue-integrable function, L(x), given by the sum of the infinite series,
L(x) = b1(x) + b2(x) + b3(x) +....
wherever it converges absolutely. We write:
∫L(x)dx = ∫b1(x)dx + ∫b2(x)dx
+ ∫b3(x)dx +....
Moreover, every Lebesgue-integrable-function possesses such a
(many, of course) representation. Note again that the integral signs
on the right are merely window-dressings for the ± areas
of the bricks (hb). The advanced concepts lie with the integral
on the left.
CHAPTER 12. ELEMENTARY EXAMPLE FROM CALCULUS.
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Consider a continuous function, y = f(x), on some interval,
a<x<b:
23.
Clearly, we can choose the bricks, bn (i.e.,
brick functions, bn(x)) arbitrarily but
judiciously, to fill up the total area under the y = f(x)
curve, as illustrated. Then:
∫f(x)dx = ∫b1(x)dx + ∫b2(x)dx
+ ∫b3(x)dx +....
à la the EQUIVALENT ADVANCED THEOREM. This is not the
customary way to compute such an integral, but does reflect a very
appropriate scheme for numerically evaluating the integral. Note that
we arbitrarily pick the brick functions once and for all as a definition
of the integral. Then we simply sum the series to evaluate it.
This contrasts sharply with the traditional definition and evaluation
of a Riemann integral in calculus. In this situation, of course:
f(x) = b1(x) + b2(x) + b3(x) + ....
holds for a<x<b since the series converges
absolutely (terms are bounded and positive). Looking at the picture,
this is perhaps not surprising, though the context is a bit unusual:
deriving an expansion of a function on an interval from its own
Riemann integral over that interval. On the other hand, this is but a
discrete version of the fundamental theorem of calculus,
sans differentiation.
One can employ the traditional (continuous) version of an anti-derivative
using the bricks, where, however, the fundamental theorem of calculus,
namely:
(d/dt)(∫tf(x)dx) = f(t),
holds only almost everywhere, since the termwise differentiation
(d/dt)(∫tb(x)dx) = b(t), is violated at the edges
of each brick. "Almost everywhere" means: although there are
many exceptions to the formula, the exceptions do not account for much
in the real world. In this scheme, repeated anti-derivatives,
for computing:
∫t...∫...∫f(x)dx ... dx
introduce no great difficulties, and require only the power rules:
∫xndx = (xn+1/n+1)
to evaluate. No other anti-derivatives are needed. The cost for this
simplification is that, in each case, one must sum an infinite series.
Finally to come full circle, so to speak, one might start to fill up
the countably many bricks, in turn, with countably many triangles
each (as we did at the beginning). Doing this and re-ordering
the double-sum, we construct a tent expansion for the function,
y=f(x), à la the ADVANCED THEOREM. Of course,
this compounding trick works in EVERY CASE, not just for
continuous functions, and so we can now move back to where we started,
using our tent expansions, throughout.
CHAPTER 13. RIEMANN AND PICASSO-TYPE PAINTING
OF THE BARN.
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As a last word (picture), let us return to the initial,
pure barn-painting problem, with some very imaginative painting,
here, a highly-abstracted frog, based upon a
1929 Picasso painting, using Riemman and his zeta function numbers,
(1/ps). We abandon those pesky tents entirely.
As step one, we now paint (1/2) of the barn (with area = 2)
anyway whatsoever: perhaps one glob with some interesting shape,
or many globs scattered about the barn, perhaps some speckles
scattered all around, just whatever suits one's fancy.
Also, we'll color the globs, etc., with many different colors,
à la Picasso. The unpainted area of the barn is now
1 barn unit2.
(See picture.)
261.
Having finished step one, we repeat a Picasso-type painting job as step
two, painting a 1/2s fraction of the unpainted portion
of the barn. The area covered will be (1/2s)
barn units2, leaving 1-(1/2s)
barn units2, unpainted. (See picture:)
262.
As step three, we will again repeat a Picasso-type painting job, painting
a (1/3s) fraction of the unpainted portion of the barn.
The area covered will be
(1-(1/2s))(1/3s) barn units2,
leaving (1-(1/2s))(1-(1/3s))
barn units2 unpainted. (See picture:)
263.
Note, these expressions now are not just some "abstract algebraic
symbols", but (for a fixed s) down-to-earth specific
arithmetical numbers. Repeat Picasso-type painting processes for each
prime, p. At each step, one knows exactly how much area to paint
(compute a partial Euler product, multiplied by (1/ps),
but how and with which colors is left entirely to one's imagination.
When the painting job is finished and s is some number between
0 and 1, we will discover that, lo and behold,
the whole barn is colorfully and imaginatively painted,
and there are no gaps--no holes. (See picture:)
264.
The painting job is made easiest for small values of s,
with more generous areas to paint early on.
If, however, s is a number exceeding 1, then we will
discover that the barn is not completely painted, and we will
be left with, what is perhaps, a marvelous Picasso-type painting
(we've made it that way by clever choices) with holes in it.
Actually, had we noted beforehand that s>1 would cause holes,
we could have done some designing of them along the way.
The total unpainted area is 1/ζ(s) (the complete Euler
product) barn units2, and one can select portions
of this arbitrarily along the way (or all at once), choosing
where and how to exclude select patches from our paintbrush.
This way, one more actively participates in the final appearance
of the picture. Should we concentrate entirely on the painting aspect
of the process, the holes will just show up automatically anyway.
It is easiest to design the picture if s is close to 0.
(See picture.)
411.
FINAL COMMENT. CAUTION: Though rather innocent-looking, the
admission of positive and negative tents or bricks without bounds in these
advanced theorems is the essential step leading from calculus to the
big picture, and makes all the difference in the world for the subject.
So, can one do serious mathematics using pictures and calculus?
CHAPTER 14. ACKNOWLEDGMENTS.
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The EQUIVALENT ADVANCED THEOREM (Mikusiński's Theorem) using bricks,
and all other serious mathematics mentioned here, is due entirely
to the work and inspiration of the late Prof. Jan Mikusiński.
The total theory of Lebesgue integration, of course, consists of
much more than a definition. The textbooks of Prof. Jan Mikusiński
(and son Dr. Piotr Mikusiński) cover the subject thoroughly,
and also very much more.
The use of isosceles triangles and tents is the speaker's amusing little
innovation, so as to create a visual painting game with a link to
integration. It was done originally to show that the tent expansions
in the ADVANCED THEOREM suffice for the same sweeping claims as given
in Mikusiński's Theorem, and subsequently proved useful in the
investigation of Fourier transforms.
I acknowledge the assistance of G. William Moore, MD, PhD,
Geoffrey W. Moore, G. Vincent Moore, and Lawrence A. Brown, MD,
in reviewing and formatting the manuscript.
I further acknowlege the inspiration supplied by my wife of fifty-eight
years, Marilyn Struble, who rekindled in me a dormant enthusiasm
in mathematics, after a sixteen-year hiatus. She had the insight
to give me a copy of John Derbyshire's stimulating book,
Prime Obsession. Prime Obsession, of course, mainly
concerns itself with the Riemann Hypothesis. Having read that,
things began to bubble up a little, and this presentation is one
of those little bubbles.
CHAPTER 15. REFERENCES.
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1. Mikusiński J.
The Bochner integral.
Basel: Birkhäuser. Lehrbücher und Monographien
aus dem Gebiete der exakten Wissenschaften: Mathematische Reihe.
[Textbooks and monographs from the area of exact sciences:
mathematical series.] 1978;55:.
ISBN: 3764308656, 233 pages.
2. Mikusiński J, Mikusiński P.
An Introduction to Analysis.
From Number to Integral.
New York: John Wiley and Sons Ltd. 1993 Apr 8;:.
ISBN: 0471599778.
3. Derbyshire J.
Prime Obsession: Bernhard Riemann
and the Greatest Unsolved Problem in Mathematics.
New York: Plume Books. 2004 May 25;:.
ISBN: 0452285259, 448 pages.
A fun book about this incredible unsolved problem in mathematics,
for which a $1,000,000 prize has been offered to the first person
with a bona-fide proof or counterexample. The book includes a lot
of interesting gossip about the great mathematicians of nineteenth-century
and early twentieth-century Europe and North America.
CHAPTER 16. APPENDIX A. CONTINUOUS PAINTING.
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As a sideline venture, we might consider a continuous
painting process (spray painting, where the paint specks
never land on one another.) To this end, we let:
F(t) = the instantaneous fraction of barn painted.
P(t) = the instantaneous fraction of barn not yet painted.
Then the identity 1 = F(t)+P(t) is demanded
of our painting process. The process itself is given by
(dF/dt) = P(t)r(t) = (1-F(t))r(t), where r(t)
is the instantaneous fraction being painted. (See picture).
165.
Separating variables in this differential equation yields
[1/(1-F(t)]dF = r(t)dt, so that, upon integration, we obtain
log(1-F(t)) = −
0∫tr(s)ds,
where we have imposed the initial condition, F(0)=0. Thus:
1-F(t) = P(t) = e −
0∫tr(s)ds,
which gives us now:
THE CONTINUOUS PAINTING THEOREM.
The barn gets painted if and only if:
0∫tr(s)ds ⇒ ∞
as t ⇒ ∞. Thus, a divergent accumulation of fractions,
r(t) gets the barn painted; otherwise, no go!. In fact, if
0∫∞r(s)ds < ∞
then a total fraction,
P(∞) = e − 0
∫∞ r(s)ds,
of the barn never gets painted, and an F(∞) = 1 - P(∞)
fraction of the barn does get painted.
Note: The intermediary infinite product of the painting lemma,
traversing as it does through the calculus lemma in order to establish
the discrete painting theorem, morphs into the
limt⇒∞P(t) in this case. Thence we obtain
the continuous painting theorem directly in terms of a
convergent or divergent accumulation of the painting fractions,
r(t). (Like the convergence or divergence of the series,
r1 + r2 + r3 + r4 + ...
for discrete painting.)
If we change our perspective (as we did in the discrete case)
and seek the continuous analogs of the segment summing functions
along the barn, it becomes simply a search for a family,
L(x,t), of such functions requiring only that
the instantaneous total fraction painted satisfy the condition:
F(t)= 1 - e
− 0
∫tr(s)ds
= 0∫2L(x,t)dx.
The simplest selection scheme is to prescribe
a continuously rising family of constant functions (of x),
with values L(x,t) = F(t)/2, so that
0∫2L(x,t)dx = F(t)
holds.
The next simplest is to prescribe a continuously rising family
of step functions, such as we encountered in the
ELEMENTARY EXAMPLE. More generally, one could even invoke the
ALTERNATE ADVANCED THEOREM, and prescribe families
of continuously rising brick functions, bn(x,t),
accordingly requiring that F(t) =
∑0∞∫bn(x,t)
hold for each t (absolute convergence assured here), and that
L(x,t) = ∑0∞bn(x,t)
hold for each t (absolute convergence assured here).
These latter, of course, would be (for each fixed value of t)
Lebesgue-integrable-functions of x (Dow-Jones averages, perhaps,
or whatever), satisfying:
F(t) = ∫L(x,t)dx =
∑0∞
∫bn(x,t)dx.
(See picture.)
168.
One could similarly treat the Riemann zeta case, s>1:
1 - 1/ζ(s)= (1/2s) +
(1-(1/2s))(1/3s) +
(1-(1/2s))(1-(1/3s))(1/5s) + ...
= F(s)
and select sequences of positive brick functions,
bp(x,s) accordingly, requiring
F(s) = ∑primes∫bp(x,s)dx
to hold for each s (with absolute convergence assured here); and
L(x,s) = ∑primesbp(x,s)
to hold for each s (with absolute convergence assured here).
In this discrete case, we guarantee the above specific expansion
by requiring ∫bp(x,s)dx = (1/ps).
Perhaps choose bricks, say, with base 1 and height
(1/ps), scattered far and yon (anywhere)
along the x-axis (maybe non-overlapping, for simplicity),
so that then L(x,s) = ∑primesbp(x,s)
holds for each s and every x, while
F(s) = ∑primes∫bp(x,s)dx
= (1/2s) + (1-(1/2s))(1/3s) +
(1-(1/2s))(1-(1/3s))(1/5s) + ...
= 1 - 1/ζ(s)
holds for each s. Again, for each fixed value of s,
L(x,s) will be a Lebesgue-integrable function of x,
(Dow Jones?) (See picture.) Of course, one could just as well select
tent functions, tp(x,s) (scattered about the
x-axis), satisfying ∫tp(x,s)dx
= (1/ps), and thus obtain other Lebesgue-integrable
functions, L(x,s) = ∑primestp(x,s).
Why not select tents and bricks together in some way,
anywhere along the x-axis), with areas (1/ps).
Sure, that will work!
CHAPTER 17. APPENDIX B. OVERPAINTING THE BARN.
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In order to interpret the following, modified painting procedure,
we need to install a billboard above our barn. Following this,
we will now introduce a modified painting process, using
isosceles triangles with painting fractions restricted
only by rn<1. If rn<1/4,
then the process is as before: at each step an attempt to paint
the unpainted triangular portions of the barn. When, however,
some rn exceeds 1/4, we are to select
the unpainted triangular portions of the barn, just as if
rn did not exceed 1/4, but then paint
isosceles triangles there with increased altitudes to accommodate
the fraction, rn. These means that the painted
triangles will not fit into those particular unpainted portions,
and will violate the original requirement that only unpainted portions
of the barn are allowed to be painted. (See picture).
173.
Also as another violation of an original requirement,
some portions of the barn will receive repeated painting, while some
regions above the barn will have to be painted (repeatedly also).
Hence the need for a billboard to received the paint. This modified
process we will label overpainting the barn.
As always, one has the fundamental identity:
P =(1-r1)(1-r2)(1-r3) ...
= 1 - [r1 +
(1-r1)r2 +
(1-r1)(1-r2)r3 + ... ]
= 1-F,
which governs the infinite painting process. If we were to modify the
shapes of the oversized triangles, so as to accommodate all
the fractions rn within unpainted portions
(such, of course, is possible, since rn<1),
then P and F could be interpreted as resulting from
the original barn painting process exactly. We conclude that P=0
(i.e., r1 + r2 + r3+... = ∞)
again insures that the barn is completely painted.
Should numerous rn be close to 1,
the resulting paint job will result in tall painted spikes
on the billboard, as illustrated.
(See picture).
171.
This appearance suggests that of a cathedral or, in medicine
(see Appendix I), papillomatosis.
Should one choose to base the overpainting scheme by employing
slim triangles corresponding to fractions rn
even less than 1/4, the spires of the cathedral become
even more pronounced. When particular fractions
rn approach 1, then the spires will grow
so as to suggest the painting of a tower
(shown subsequently for overpainting, in excess of 100%).
Of course, the illustration given neglects
to show that the barn itself may be completely covered
via other painted triangles.
Should P be positive (r1 + r2 +
r3+... < ∞) the picture would be very similar,
but the barn does not get compeltely painted. There is a residue
of unpainted barn, despite the overpainting. If one imposes an upper limit
on the spikes of the cathedral, this puts another additional restraint
on the rn, generally forcing the
rn
away from 1.
So long as we are considering the overpainting of the barn, it is
interesting to ask: what interpretation can be given to the process
if the fractions rm<1 contribute to overpainting
unpainted portions by more than 100%? In such cases, we paint
an isosceles triangle with greatly increased altitude, in order to
accommodate a number (1+rm), which depicts
the excess of the unpainted triangle.
(See picture).
172.
For purposes of illustration, let us
first consider a situation where all steps of the overpainting
process require a (1+rm) "part" of the
unpainted portions encountered.
In this situation, the infinite product formula now reads:
P0 =(1+r1)(1+r2)(1+r3) ...
= 1 + [r1 + (1+r1)r2 +
(1+r1)(1+r2)r3 + ... ]
= 1 + F0,
and P0 becomes an indicator of the final (accumulative)
effects of the overpainting of the barn. It is greater than 1,
and P0 - 1 = F0 equals the final fractional
overpainting.
If r1 + r2 + r3 +... < ∞,
then P0 is finite (converges), and one anticipates
a picture depicting the painting of a tower, if P0
is large, as illustrated. (See picture). This is because the area
of the painted traingle must exceed that of the whole
unpainted region first selected for rm.
170.
Again in the tower illustration, we have omitted the painted triangles
showing the partial painting of the barn. If we were to impose
an upper limit to the tower, this becomes an additional restriction
on the fractions, rm, generally bounding them further
from above.
When r1 + r2 + r3+... = ∞,
and
P0 = F0 = ∞, then some
of the spires of the tower must stretch to infinity.
Yet the overpainting process itself conveys a real meaning to the limit
situation, differing only in the final details from the finite case,
P0 < ∞
:
a visible, unbounded tower being painted. The corresponding
mathematical limiting process:
P0 = limm⇒∞
[(1+r1)(1+r2)...(1+rm) ... ]
= 1 + limm⇒∞
[r1 + (1+r1)r2 +
(1+r1)(1+r2)r3 + ...
(1+r1)(1+r2)...(1+rm-1)rm]
= 1 + F0,
is void of any content ∞
= 1+ ∞.
The more general situation consists of combinations of over 100%
overpainting steps, and under 100% overpainting steps.
The controlling infinite product then consists of (1+rm)
factors and (1-rn) factors in various and sundry orders.
In this situation, the governing (finite) product tends to P, the
infinite product, which is zero, a positive number, infinite,
or nothing at all.
If the fractions, rm, rn,
together are summable, than ordinary (mathematical) convergence
must occur, which rules out zero, infinity, and "nothing at all".
The overpainting process accommodates all these simple outcomes,
but also allows for a real interpretation of the "nonconvergent" limits
zero and infinity as well. It is only the "nothing at all" situation
which escapes a painting interpretation.
Briefly the nearly complete story is as follows: the barn is
completely painted if P=0. The exceptions occur when 0<P,
r1 + r2 + r3+... < ∞,
and rn<1, which includes the original painting
scheme and the mildly overpainting situation discussed earlier.
Except when using the original painting scheme, we expect the picture
on the billboard to resemble a cathedral or tower, (i.e., employing
a combination of (1+rm) and (1-rn)
factors to paint by).
When P=0, the painting process always covers the barn. Of course,
if all factors are (1-rn) with
rn<1/4, then the barn alone gets fully painted
(no billboard painting); otherwise one expects to end up with cathedrals.
When P=∞, one certainly ends up with unbounded towers.
We can rephrase the general painting case (using ±
and ㅜ signs) as follows: (No negative factors
are permitted, i.e., 1-rn>0,
rn>0.)
P = (1ㅜr1)(1ㅜr2)
(1ㅜr3)...
= 1 ㅜ [(±r1) ±
(1ㅜr1)(±r2)
±(1ㅜr1)
(1ㅜr2)(±r3)
± ...)]
= 1ㅜ F.
P will be finite (greater than zero) if and only if the series,
r1 + r2 + r3+..., converges.
The barn is completely painted if and only if P=0, F=1,
and a tower stretches to ∞
if P = F = ∞.
CHAPTER 18. APPENDIX C. MORE ON THE
RIEMANN ZETA FUNCTION.
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First: barn painting again. We again consider some
fixed s>1, and let rp = (1/ps)
whenever p is a prime; and rn = 0 otherwise.
Since the series ∑(1/ps) converges, we recall
that we deal here with an unsuccessful painting case, which necessitates
using increasingly fatter and fatter triangles, or slimmer and slimmer
ones. Recall the intriguing picture for s=2.
169.
Note: the above illustration more accurately represents
the case of s = 1.5.
In this particular example, the infinite product P is positive, and
is always equal to the reciprocal, [1/ζ(s)], of the
RIEMANN ZETA FUNCTION (according to Euler's celebrated
product formula, where:
[1/ζ(s)] = (1-(1/2s)(1-(1/3s)(1-(1/5s)(1-(1/7s) ....
= Πprimes(1-(1/ps)
= Πprimes(1-p-s).
Recall, the zeta function originated as the infinite series:
ζ(s) = 1 + 1/(2s) + 1/(3s) +
1/(4s) + ... = ∑(1/ns) = ∑n-s
(all integers, n). Our alternative expression
P = 1 - [(1/(2s) + (1-(1/(2s)(1/(3s)
+ (1-(1/(2s)(1-(1/(3s)(1/(5s) +....]
for the infinite product, P = (1 - (1/2s)(1 -
(1/3s)(1 - (1/5s) ... in this special case,
is a simple (?) formula for the reciprocal [1/ζ(s)]
of the infinite series. There is so much stuff evolving
around the Riemann hypothesis that this is probably out there somewhere.
Who knows? The well-known series ∑µ(n)/ns,
using the Möbius function µ would appear to be
a far simpler formula for the reciprocal [1/ζ(s)].
COROLLARY: Since 0<P<1, then
ζ(s)>1 for s>1.
Clearly old stuff, but of interest coming as it does from our painting
problem. Now in our barn-painting game, we can let s
be less than 1. Then the series ∑1/ps
diverges, the Euler infinite product P is zero (diverges),
and so according to our painting theorem (and only then),
the barn gets painted! Also from the above expression for P
one obtains a really surprising result: the equality,
1 = (1/(2s)) + (1-(1/(2s))(1/(3s))
+ (1-(1/(2s))(1-(1/(3s)))(1/(5s)) +...
holds IDENTICALLY for 0<s<1.
For s=0, trivial. For s=1, definitely not trivial:
1 = 1/2 + (1/2)(1/3) + (1/2)(2/3)(1/5) + (1/2)(2/3)(4/5)(1/7) + ...
Familiar? Thus the sum
[(1/2s) + (1-(1/2s))(1/3s) +
(1-(1/2s))(1-(1/3s))(1/5s) + ...]
= 1-P = F
is constantly equal to 1 for 0<s<1,
but is equal to (1-(1/ζ(s)) for s>1.
It is well-known, of course, that the zeta function has a simple pole at
s=1, and we have just moved in on it from above and
strangely from below. (See picture).
79.
Second: prime counting and the
Riemann hypothesis. The Riemann zeta function, ζ(s),
connects with the prime number counting function, π(x)
(i.e., the number of primes less than x), through the equation
(called the Golden Key in Derbyshire's book,
Prime Obsession):
K: (1/s)logζ(s) = 0∫∞
J(x) x-s-1dx
where
J(x) = π(x) + π(2√x)(1/2) +
π(3√x)(1/3) + π(4√x)(1/4)
+ ....
(a finite sum for every x). As derived in Prime Obsession,
this equation holds for s>1 (real part of s>1).
In this range, our painting process furnishes an alternate expression,
namely:
P = 1 - [(1/2s) + (1-(1/2s))(1/3s)
+ (1-(1/2s))(1/3s)(1/5s) + ...]
= 1 - F(s)
for 1/ζ(s). Thus we can write K in the form:
-logP(s) = -log(1 - F(s)) =
s 0∫∞J(x)x-s-1dx,
a new formulation for of the Golden Key. Using the
well-known series expansion for log(1 - F(s)), one can obtain
a second alternate form:
K: F(s) + F2(s)/2 + F3(s)/3 +
F5(s)/5 + ... =
s 0∫∞J(x)x-s-1dx.
with
F(s) = [(1/2s) + (1-(1/2s)(1/3s)
+ (1-(1/2s)(1-1/3s)(1/5s) + ...]
Why one should care, is another question. It is known, of course,
that F(s) analytically extends (as does ζ(s))
into the remainder of the complex plane, sans 1, where the zeros
of ζ(s) are located.
The Riemann Hypothesis claims that the real parts of the
non-trivial zeros of ζ(s) are always ½, where
s=½<1! Why ½?
Because the (extended) zeta function satisfies the relationship
ζ(1-s)
∝
ζ(s) and 1-s = s if and only if
s=½. When we paint the barn for s=½,
our total fraction formula expresses the number one in the
special series,
1 = 1/√2 + (1-1/√2)1/√3 +
(1-1/√2)(1-1/√3)1/√5 + ... .
Why should this be any more of interest than having the number one
similarly expressed for any s between 0 and 1?
Beats me! Somehow, these formulas (the Golden Key) must sashay
into the critical strip, 0<s<1 (0<real part s<1)
intact in order to accomodate these zeros. For zeta it's done
via the equation, ζ(s) = η(s)/(1-21-s),
where η(s) = 1 - 2-s + 3-s - 4-s
+ ..., which converges for all s>0. But the right-hand
member of K clearly converges for s>0, and so both
members make it there, intact.
What does our painting theorem have to say about the zeros? For example,
their (positive) imaginary parts, t1, t2,
t3, ... satisfy an equation of the (usual) form:
F1(s) =
1/t1s
+ (1 - 1/t1s)1/t2s
+ (1 - 1/t1s)(1 - 1/t2s)1/t3s
+ ...,
= 1 - [(1-1/t1s)(1-1/t2s)(1-1/t3s)....]
= 1 - P1(s)
= 1 - (1-1/t1s)(1-1/t2s)(1-1/t3s)...
for any s>1, where it is known that
∑1/tns < ∞,
since tn ∼ n. On the other hand, the identity
1 = 1/t1s
+ (1 - 1/t1s)1/t2s
+ (1 - 1/t1s)(1 - 1/t2s)1/t3s
+ ...,
(P1(s) = 0).
holds for 0<s<1, where
∑1/tns = ∞.
This P1(s)
for s>1, is not the reciprocal of the Riemann
zeta function, of course, though it might be of interest to somehow
discover a direct connection between the two. But what if some zeros
are masquerading as complex numbers? (Zeros off the critical line).
Does the F1(s)
series still converge for some s? (Probably for
s>1 at least.) If so, then any complex values
exhibited by F1(s)
would negate the Riemann hypothesis. This is just fantasy,
of course, since we don't really know all the tn's
in the first place.
One might speculate about an analogous situation for the zeta function,
where all the primes are "sort of known". Here, if some or all
of the primes wander off the real line, i.e., some or all
pn are replaced by qn
= pn + iε, say, then the series
F(s) = 1/q1s +
(1- 1/q1s)1/q2s +
(1- 1/q1s)(1-1/q2s)1/q3s ...
converges for s>1, since
∑1/|pn+iε|s < ∞,
and F(s) becomes complex-valued. Interestingly enough,
our pseudo-painting scheme (complex painting?) still comes into play
and guarantees the identity (when 0<s<1):
1 = 1/(2+iε)s
+ (1 - 1/(2+iε)s)1/(3+iε)s
+ (1 - 1/(2+iε)s)(1 - 1/(3+iε)s)1/(5+iε)s + ...
in the complex domain. Indeed,
∑1/(pn+iε)s diverges if
0<s<1, and so F(s)≡1. A similar
identity holds if only some of the primes are moved off the real axis.
What fun one can have painting (or pretending to paint) a barn!
CHAPTER 19. APPENDIX D.
IMAGINATIVE PAINTING AND SCHOOL SPIRIT:
NORTH CAROLINA STATE UNIVERSITY WOLFPACK.
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As a last word (picture), let us return to the initial,
pure barn-painting problem, with some very imaginative painting,
using Riemman, his zeta function numbers, (1/ps).
and the profile for the North Carolina State University Wolfpack.
We abandon those pesky tents entirely. As step one, we now paint
(1/2) of the barn (with area = 2) anyway whatsoever:
perhaps one glob with some interesting shape, or many globs scattered
about the barn, perhaps some specs scattered all around, just whatever
suits one's fancy. Also, we'll color the globs, etc., with many different
colors, à la Picasso. The unpainted area of the barn is now
1 barn unit2. (See picture.)
241.
Having finished step one, we repeat a Picasso-type painting job
as step two, painting a 1/2s fraction
of the unpainted portion of the barn. The area covered
will be (1/2s) barn units2,
leaving 1-(1/2s) barn units2, unpainted.
(See picture.)
249.
As step three, we will again repeat a Picasso-type painting job,
painting a (1/3s) fraction of the unpainted portion
of the barn. The area covered will be
(1-(1/2s))(1/3s) barn units2,
leaving (1-(1/2s))(1-(1/3s))
barn units2 unpainted.
(See picture.)
250.
Note, these expressions now are not just some
"abstract algebraic symbols", but (for a fixed s)
down-to-earth specific arithmetical numbers. Repeat Picasso-type painting
processes for each prime, p. At each step, one knows exactly
how much area to paint (compute a partial Euler product, multiplied by
(1/ps), but how and with which colors
is left entirely to one's imagination. When the painting job is finished
and s is some number between 0 and 1,
we will discover that, lo and behold, the whole barn is colorfully
and imaginatively painted and there are no gaps--no holes.
(See picture.)
251.
The painting job is made easiest for small values of s
with more generous areas to paint early on.
If, however, s is a number exceeding 1, then we will
discover that the barn is not completely painted, and we will
be left with, what is perhaps, a marvelous Picasso-type painting
(we've made it that way by clever choices) with holes in it.
Actually, had we noted beforehand that s>1 would cause holes,
we could have done some designing of them along the way.
The total unpainted area is 1/ζ(s) (the complete Euler
product) barn units2, and one can select portions
of this arbitrarily along the way (or all at once), choosing
where and how to exclude select patches from our paintbrush.
This way, one more actively participates in the final appearance
of the picture. Should we concentrate entirely on the painting aspect
of the process, the holes will just show up automatically anyway.
It is easiest to design the picture if s is close to 0.
CHAPTER 20. APPENDIX E. SAMPLE CALCULATION
OF PAINTING FUNCTION.
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Here are three examples of the F-calculation for r1,
r2, r3, r4, r5, r6,
r7. To calculate the value for F, the fraction
of triangle painted, click on SUBMIT. Three examples
are presented: a convergent (geometric) series,
that never paints the barn; a divergent (constant) series,
that paints the barn; and a divergent (harmonic) series,
that paints the barn. You may wish to alter the numbers,
and examine the behavior of the series.
CONVERGENT SERIES (GEOMETRIC).
DIVERGENT SERIES (CONSTANT=1/10).
DIVERGENT SERIES (HARMONIC).
NOW TRY YOUR OWN, by changing the values of
rn in the boxes, and clicking
on SUBMIT.
CHAPTER 21. APPENDIX F: ADDITIONAL READING.
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1. Struble RA.
Curriculum Vitae.
http://www.medparse.com/strublcv.htm
2. Struble RA.
Series expansions of Fourier transforms and Lebesgue functions.
Studia Math. 1984;77:479-484.
3. Burniston EE.
Raimond Struble.
Raleigh, NC: Math History Home Page. 1988;:.
1987-1988 Harrelson News.
Rai Struble retired in December of 1987 ...
http://www4.ncsu.edu/~njrose/Special/Bios/Struble.html
4. Lauwerier H.
Fractals. Endlessly Repeated Geometrical Figures.
Trnsl by Gill-Hoffstaedt S.
Princeton, NJ: Princeton Science Library.
Princeton University Press. 1991;:.
ISBN 0-691-02445-6, 209 pages.
A review of the history of fractals, of which the triangles
in this manuscript and Sierpinski's Triangle are two examples.
Much of the history of calculus leading up to the understanding
of fractals is covered in this book.
5. Sierpinski's Triangle. 1916;:.
http://serendip.brynmawr.edu/playground/sierpinski.html
A concise explanation of Sierpinski's Triangle.
6. Sierpinski's Triangle. 1916;:.
http://www.shodor.org/interactivate/activities/gasket/
A live demonstration of the construction of Sierpinski's Triangle.
7. Edwards HM.
Riemann's Zeta Function.
New York: Dover Publications. 1 Jun 2001;:.
ISBN: 0486417409, 315 pages.
8. Pesic P.
Abel's Proof: An Essay on the Sources and Meaning
of Mathematical Unsolvability.
Boston: The MIT Press. 1 Apr 2004;:.
ISBN: 0262661829, 221 pages.
9. Sabbagh K.
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics.
New York: Farrar, Straus and Giroux. 26 May 2004;:.
ISBN: 0374529353, 352 pages.
10. Dunham W.
Journey Through Genius: The Great Theorems of Mathematics.
New York: Penguin Books. 1 August 1991;:.
ISBN: 014014739X, 320 pages.
11. Halpern P.
The Great Beyond: Higher Dimensions, Parallel Universes
and the Extraordinary Search for a Theory of Everything.
New York: John Wiley & Sons, Inc. Jun 2004;:.
ISBN: B0002IW1IO, File Size: 1933K (Adobe Reader).
12. Berman JJ, Moore GW.
Spontaneous regression of residual tumour burden:
prediction by Monte Carlo simulation.
Anal Cell Pathol. 1992 Sep;4(5):359-368.
13. Moore GW, Berman JJ.
Cell growth simulations predicting polyclonal origins
for 'monoclonal' tumors.
Cancer Lett. 1991 Nov;60(2):113-119.
Discussion of a Monte-Carlo simulation of tumor growth,
in which it is hypothesized that all tumor growth rates
may be predicted by the rate of programmed cell death.
14. Maor E.
e: Story of a Number.
New Haven: Princeton University Press. 4 May 1998;:.
ISBN: 0691058547, 232 pages
15. Agnew RP.
Calculus.
New York: McGraw-Hill. 1962;:.
16. Maor E.
To Infinity and Beyond.
New Haven: Princeton University Press. 9 Jul 1991;:.
ISBN: 0691025118, 304 pages.
17. Beckmann P.
A history of Pi.
New York: St. Martin's Griffin. 15 Jul 1976;:.
ISBN: 0312381859, 208 pages.
18. Livio M.
The Golden Ratio. The Story of Phi,
the World's Most Astonishing Number.
New York: Broadway Books. 2003.
ISBN 0-7679-0816-3, 290 pages.
Phi, Φ, is (1+√5)/2, the limit
of the ratio of consecutive Fibonacci numbers,
1, 1, 2, 3, 5, 8, 13,.... (Each Fibonacci number is the sum
of the previous two Fibonacci numbers.) Leonardo Bigollo Pisano Fibonacci
was the son of a merchant in Pisa, Italy, who was tutored
by Arabic mathematicians in North Africa
as a teenager in the late twelfth century. Fibonacci later
introduced the Arabic numeral system to Europe in his 1202 classic book,
Liber Abaci (Book of the Abacus).
19. Huntley HE.
The Divine Proportion. A Study of Mathematical Beauty.
New York: Dover Publications, Inc. 1970.
ISBN 486-22254-3, 186 pages.
20. Brown D.
The Da Vinci Code.
New York: Doubleday. 2003.
ISBN 0-385-50420-9, 454 pages.
A best-seller murder mystery.
p. 91, ch 20.
Nice discussion of the Golden Ratio (1.618....)
and the Fibonacci Sequence. Repeats the legend
that the ratio of the height to the umbilicus-to-ground-height
of a beautiful person is the Golden Ratio, phi.
p. 199, ch. 45.
Mention of cryptographers Bruce Schneier and Philip K. Zimmerman.
(See [Schneier, 1996]).
21. Singh S.
Fermat's Enigma.
The Epic Quest to solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Divsion of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.
p. 62.
"Cuius rei demonstrationem mirabilem sane detexi hanc marginis
exiguitas non caperet."
[Latin: I have detected this truly miraculous demonstration,
of which thing the tightness of the margin might not capture."]
p. 52.
Photograph, Frontispiece of Claude Gaspar Bachet's French translation
of Diophantus's Arithemetica (originally in Latin). Published 1621.
Found in Fermat's literary estate.
"Diophanti Alexandrini Arithmeticorum Libri Sex.
Et de Numeris Multangulis Liber Unus."
[Latin: Six Books of Arithemetic by Diophantus of Alexandria.
Book One of multangular numbers.] Six books extant from a total
of thirteen books. Other seven books lost in the tragic burning
of the Library of Alexandria in 389 CE, by order of Emperor Theodosius,
certainly one of the least distinguished Roman Emperors in an altogether
very undistinguished line of rulers.
22. Bühler WK.
Gauss : A Biographical Study .
Berlin: Springer Verlag; ISBN: 0387106626.
Hardcover (April, 1981) .
An excellent biography of the Prince of Mathematics.
23. Pascal B.
Traité du Triangle Arithmétique. 1653.
As cited in: Huntley HE.
The Divine Proportion. A Study of Mathematical Beauty.
Discussion of Pascal's Triangle, first discovered
by the 13th century Chinese. chap 10, pp. 131-140.
24. Bernstein PL.
Against the Gods. The Remarkable Story of Risk.
New York: John Wiley & Sons, Inc. 1996.
ISBN 0-471-29563-9, 383 pages.
A fantastic excursion through the history of probability and chance,
starting with the ancient Egyptians and ending with modern
worldwide business practices. Probability was originally studied
in order to INCREASE BENEFITS, as in winning at gambling
or staying alive longer. Now, probability has its most important
applications in AVOIDING DAMAGE.
25. Lemay L, Tyler D.
SAMS Teach Yourself Web Publishing with HTML4 in 21 Days.
Indianapolis, IN: SAMS. A division of Macmillan Computer Publishing.
1998;:.
ISBN 0-672-31345-6, 795 pages.
The present manuscript is written in the universal formatting language
of the internet, Hypertext Markup Language, HTML. The advantage
of HTML over its competitors is that HTML syntax is public-domain,
and anybody with access to a webpage-server and either a PC or Macintosh
computer can create an HTML file with available text-writers
on these computers. Therefore, anybody with these tools becomes
an instant publisher to the world. The simplest HTML file is:
<html></html>
Another simple HTML file is:
<html><body>
Another simple HTML file.
</body></html>
For more details, read the book.
26. von Neumann J.
The Computer and the Brain.
New Haven: Yale University Press. 1958;:.
ISBN not stated, 82 pages.
A book written by John von Neumann, one of the primary inventors
of the modern digital computer. Von Neumann's argument that
computers cannot equal the brain because the brain has more
components in a smaller space is somewhat dated by the computers
of his day, but I still think that his conclusion is correct.
27. Kleene SC.
Mathematical Logic.
Mineola, NY: Dover Publications, Inc. 1967;:.
ISBN 0-486-42533-9, 398 pages.
One of the classics of symbolic logic.
28. Schneier B.
Applied Cryptography, Second Edition.
Protocols, Algorithms, and Source Code in C.
New York: John Wiley & Sons. 1996;:249-250.
ISBN 0-471-12845-7, 758 pages.
An excellent overview of the history and theory of cryptography,
for beginners and experienced practitioners alike. One of the
main cryptography methods, so-called PUBLIC/PRIVATE CRYPTOGRAPHY,
involves the distribution of a PUBLIC KEY to the world at large,
say on the internet, and absolute secrecy of a PRIVATE KEY,
owned solely by the receiver of the encrypted message. The general public,
or sender, can encrypt a message using the public key,
but only the receiver can decrypt it, using the private key.
The public key is the product of two large prime numbers, and
private key is the two large prime numbers themselves.
The security of the private key depends upon the (mathematically unproved,
but widely accepted) assumption that it is much easier and faster
to multiply two large prime numbers, in order to generate the private key,
than it is to factor the product of two prime numbers, the public key,
using the sieve of Eratosthenes. If, somehow, it became possible
to factor the product of two large prime numbers as efficiently as it
is to multiply them, then the world banking and financial system
would quiver for a few days, while new cryptography methods (such as
the so-called one-time-pad method used in medical privacy regulations)
were programmed into computer systems.
29. Euclid.
Euclid. Greek Mathematics.
Goold GP, ed. Thomas I, translator.
Loeb Classical Library. #335.
Cambridge, MA: Harvard University Press. 1939.
ISBN 0-674-99369-1, 511 pages.
Euclid produced, in addition to your high-school geometry book,
the first published proof that there is an infinity of prime numbers.
Read the book. You'll be amazed how little has changed in plane geometry
over the past two millennia.
30. Seife C.
Zero. The Biography of a Dangerous Idea.
London: Penguin Books. 2000.
ISBN: 0-670-88457-X, 248 pages.
A charming book on the history of the number zero,
which was abhorred by the ancient Greeks, but eventually
embraced by Indian mathematicians in the eighth century.
31. Courant, Robbins, Stewart.
What is Mathematics. Second Edition.
Discusses the origins of calculus, including the regrettable
priority disputes between
Sir Isaac Newton
and Gottfried Leibniz.
which hobbled the world of mathematics in the eighteenth century.
Essentially all the famous names in European mathematics in that century
were non-British, due to those disputes. See also:
Hawking S. A Brief History of Time.
32. Hawking S.
A Brief History of Time.
New York: Bantam Books. 1988;:.
ISBN....
There is an uncomplimentary mini-biography of
Sir Isaac Newton on the last page
of this popular book, by Newton's successor in the
Chair of Natural Philosophy at Cambridge, UK, that begins:
"Newton was not a pleasant man..."
33. Leslie R.
Pablo Picasso. A Modern Master.
New York: Todtri Productions, Ltd. 1996;:.
ISBN 1-880908-73-5, 128 pages.
The frog-painting is based upon Picasso's Nude in an Armchair,
1929, oil on canvas. Musée Picasso. Paris. pp. 80-82.
"The 1929 Nude in an Armchair, has an interior with the
linear, flat, and interlocking Cubist planes of color.
The nude woman is a crudely drawn and roughly painted form
of distorted body parts, who tip back a tooth-filled face in what
is either a grin or scream...".
CHAPTER 22. APPENDIX G: HISTORICAL NOTES.
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1. Archimedes
(Αρχιμεδης)
[287? BCE-212 BCE]. Ancient Greek mathematician,
one of the three greatest mathematicians of all time
(Archimedes, Newton, Gauss). Archimedes' The Sand Reckoner
is the first serious effort to deal with
large-number problems, in this case, the number of grains of sand
on the entire Earth. The elements of calculus are present
in this document. Believe it or not, Archimedes was pretty close
to the right number of grains! Statistics is the mathematical study
of repeated sampling from a larger, possibly infinite, population.
Archimedes laid the foundation for such large-number studies.
2. Zeno
(Ζενο).
...........
3. Pythagoras
(Ρυθαγορας).
..........
4. Euclid(Ευκλιδης)
[?-?300 BCE].
............
Ancient Greek mathematician, who summarized the rules of geometry,
or Elements, which had been known as an empirical science
by the ancient Egyptians for at least a millennium previously.
Euclid's main contribution is that he collected the known truths
of geometry and derived all geometric theorems
from two undefined concepts (point, line) and five postulates,
using the idea of proof by deduction. The Pythagorean theorem,
namely that in a right triangle with legs a and b,
and hypoteneuse c:
a2 + b2 = c2
is the basis for the Theory of Least Squares,
one of the bedrock methods of statistics.
5.
Sun-Tse. First Century Chinese mathematician.
Inventor of the Chinese Remainder Theorem.
(See [Schneier, 1996]).
6. Brahmagupta. Eighth Century Indian Mathematician.
Inventor of Zero.
(See [Seife, 2000].)
7.
Abu Abdullah Muhammad bin Musa al-Khwarizmi
(أبو عبد
الله
محمد
بن موسى
الخوارزمي)
Seventh
Eighth Century Arab Mathematician.
Inventor of the Algorithm. Note the similarity of name.
(See [Seife, 2000].)
8.
Leonardo Bigollo Pisano Fibonacci [1170?-1240?].
Leonard of Pisa, pre-renaissance Italian mathematician. Fibonacci's
Liber Abaci (1202) [Latin: Book of the Abacus]
introduces Arabic numerals to European mathematics, namely:
٠١٢٣٤٥٦٧٨٩
Yes, the real Arab numerals are: 0=٠, 1=١, 2=٢,
3=٣, 4=٤, 5=٥, 6=٦, 7=٧, 8=٨,
9=٩. Do you see the resemblance? These numerals are used today
in the Middle-East, and are used to enumerate verses in the Holy Quran
(written in 7th century, Classical Arabic). Europe had been
in an intellectual dark ages between the sacking of Rome in 476 A.D.
until the Renaissance (13th-15th centuries).
Fibonacci imported Arabic numerals to 13th century Italy.
Arabic numerals, including zero, are far superior to Roman numerals
(or the even more primitive numerals of Ancient Greek, Hebrew, and Chinese
cultures) for both accounting and mathematics.
Fibonacci was the son of a merchant from Pisa, Italy, living in Algeria,
where he was educated by Arab teachers, who were familiar
with the Arab numeral system, and the great Arab masters of mathematics,
such as Al-Khwárizmí and Abu-Kamil.
The Fibonacci sequence is:
1, 1, 2, 3, 5, 8, 13, 21, 34, ...
where each number in the sequence is the sum of the two previous numbers.
The ratio of a given Fibonacci number to its immediate predecessor
approaches the so-called Golden Ratio or Divine Proportion,
(1+√5)/2 = 1.6181..., for large Fibonacci numbers
(in the language of calculus, Fibonacci numbers approaching infinity).
The Golden Ratio is observed in many areas of nature and esthetics.
For example, the petals of a rose and a sunflower are arranged
in this ratio; allegedly the ratio of body-height to the height
of the umbilicus (belly-button) of a beautiful woman
satisfy this ratio; and the dimensions of nearly all national flags
(including the U.S. flag, of course) are in the Golden Ratio.
(See [Livio, 2003].)
9. John Napier, Laird of Merchiston [1550-1617].
Sixteenth century Scottish nobleman, gentleman of leisure,
and the inventor of logarithms, which transformed multiplication problems
into addition problems, and division problems into subtraction problems.
This advance in calculation methods led the way to increased accuracy
of navigation and astronomy. Essentially there would never have been
a Galileo, a Kepler, or a Newton, without Napier's incredibly
powerful advance in calculation, the 16th century equivalent
of the digital computer.
Napier's discovery is a simple extension of the idea that:
bi×bj = bi+j
which had been known since Euclid and Archimedes. For example:
8×16 = 23×24 = 23+4
= 27 = 128.
That is, if you want to multiply the two numbers, x and y, where
x = bi and y = bj, all you have to do is look up
the value of i for which x = bi
and the value of j for which y = bj.
Then add i+j (a lot easier than multiplication), and look up
the value z = bi+j. Napier spent twenty years
developing a table of LOGARITHMS, where
the value of i for which x = bi is called the
logarithm of x to the base b, or i = logbx.
A reversal of this same process converts long division into subtraction,
which is worlds of difficulty easier.
Napier used 10,000,000 as the base for his logarithm table,
and a contemporary mathematician, Wolfgang Burgi, converted Napier's
tables into base-10, which is to be found in most high-school algebra
texts. The so-called BASE OF NAPIERIAN LOGARITHMS, e, is the value
2.718281828..., and has a deep significance in calculus,
is named in honor of Napier, but was not known during Napier's lifetime.
10. Johannes Kepler. [1571-1630]. German astronomer..
Discoverer of the three laws of planetary motion. Kepler gave
the first mathematical treatment of close packing of equal spheres;
gave the first proof of how logarithms worked; devised a method
of finding the volumes of solids of revolution; and calculated
the most exact astronomical tables then known (Rudolphine Tables,
Ulm, Germany, 1627). Kepler entered Prague on January 1, 1600,
to begin a fruitful collaboration with Tycho Brahe,
a Danish-born astronomer who had collected extensive astronomical data
on the orbit of Mars.
11. Galileo Galilei [1564-1642].
Seventeenth century Italian physicist and astronomer,
and beneficiary of the patronage of the Medici family,
who discovered four of the moons of Jupiter,
and provided a mathematical basis for Polish physicist
Copernicus' hypothesis that the Sun, not the Earth,
lay at the center of the universe. For making the latter assertion,
Galileo was placed under house arrest by the Papal Court in 1630
(a better fate than that of Italian physicist Giordano Bruno,
who was burnt at the stake in 1630 for a similar crime).
In the late 17th century, Catholic university physics departments
began to teach Galileo's doctrines. In 1993, more than
three-and-a-half centuries after Galileo's arrest, Pope John Paul II
exonerated him [Beckmann, 1960].
12. René Descartes [1596-1650].
Inventor of Analytic Geometry, an algebraic mirror of Euclidean geometry.
In an instant, all the true statements of geometry were true for algebra,
and all the true statements of algebra were true for geometry,
thus doubling the body of provable statements (theorems) in mathematics.
This exact relationship between the statements of algebra and
the statements of geometry is called a ONE-TO-ONE MAPPING.
Dr. Lawrence A. Brown has suggested a Biblical prophesy foreshadowing
Descartes's Theory is Matt 16:18 (known to all educated Roman Catholics):
"Thou art Peter, and upon this rock shall I build my church,
and I shall give thee the Keys to the Kingdom of Heaven,
and whatsoever is bound on earth will be bound in heaven;
and whatsoever is loosed on earth will be loosed in heaven."
St. Peter, the first Roman Catholic pope, serves as a one-to-one mapping
between heaven and earth.
13. Sir Isaac Newton [1642-1727].
Seventeenth century British physicist and mathematician,
one of the three greatest mathematicians of all time
(Archimedes, Newton, Gauss). Inventor of Classical Physics,
in his classic work: Principia Mathematica Philosophiae Naturalis
[Latin: Mathematical Principles of Natural Philosophy]. Co-inventor, with
Blaise Pascal, of the Binomial Formula, which is used to build
the Method of Least Squares. Co-inventor, with Gottfried Leibniz,
of differential and integral calculus, which are used in proving
the Methods of Least Squares. 1642 was a big year in physics
and mathematics: the death of Galileo and the birth of Newton.
14. Gottfried Leibniz .
Seventeenth century German philosopher and mathematician.
Co-inventor, with Sir Isaac Newton, of differential and integral calculus.
15. Leonhard Euler [1707-1783]. Swiss mathematician.
Inventor of topology (the famous Königsberg bridge problem).
It is said that every calculus textbook is either Euler,
a copy of Euler, or a copy of a copy of Euler
[Agnew, 1960]. Worked extensively
with Pascal's triangle and the binomial distribution.
The Swiss government has honored Prof. Euler on their ten-franc note.
16. Pierre de Fermat [1601-1665].
Seventeenth century French civil servant and amateur mathematician.
In Fermat's literary estate, there is an assertion written in one
of the books in Fermat's library (the works of Diophantus, the ancient
Greek mathematician, famed for his work with integer arithmetic), that
for integers x, y, z, and k, the equation:
xk = yk + zk
is only true for k<2, so-called Fermat's Last Theorem.
When k=2, there are numerous examples of this equation,
the most being the famous 3-4-5 triangle, for which
32 = 42 + 52,
where 3 and 4 are legs of a right triangle
with hypoteneuse equal to 5. Fermat wrote in the margin
that he knew a proof, but didn't have room to write it
into the book margins. No proof and no counterexample
were found for the next 350 years.
Sir Andrew Wiles, mathematician from Cambridge and Princeton,
and his student, Dr. Richard Taylor, finally
proved this theorem in the early 1990s, three-and-a-half centuries,
after it had tormented the minds of virtually every great mathematician
living during those years. The Wiles/Taylor proof was not simple,
and we may never know whether Fermat had a correct proof
of the assertion. Leonhard Euler subsequently paid to have
Fermat's effects searched in a vain effort to find Fermat's proof.
17. Karl F. Gauss [1777-1855].
Nineteenth century German mathematician, one of the three greatest
mathematicians of all time (Archimedes, Newton, Gauss). Gauss invented
the method of least squares, which forms the backbone of
modern statistical theory. (Actually, Gauss always claimed that he copied
the method from a mathematical colleague [Bühler, 1980].)
Gauss also created a mathematical framework for arithmetic
and classical physics.
Gauss attempted to verify from actual physical measurements whether
Euclid's controversial theorem, that the angles of a triangle
always add up to 180o, was actually true.
18. Jerzy Neyman [1894-1981].
Polish-American statistician, who developed a method for assessing
the robustness of a statistical formula, the Neyman-Pearson Theorem.
Neyman was a playful genius, who called the null hypothesis, "the devil".
Prof. Neyman gave a lecture at North Carolina State University
Department of Statistics in the early 1970s.
19. Albert Einstein [1879-1955].
Swiss-American Physicist, and winner of the 1921 Nobel Prize in Physics.
Einstein's theory of relativity revolutionized our concepts
of space and time in physics, but Einstein was always suspicious
of non-deterministic, i.e., probabilistic, methods
to describe physical observations. His famous statement,
in a letter to Max Born: Der Herrgott würfelt nicht.
[German: The Lord God does not play dice.]
There are at least four interpretations of this statement.
First, that Einstein deeply believed that every move made by the
Lord God was planned and pre-determined. Not a swallow falls
from the sky without knowledge of the Lord God. Second, that Einstein
did not understand probability theory very well, and was trying
to rescue physics from the evils of quantum mechanics,
a probabilistic-statistical theory of physics invented by Erwin
Schrödinger in the 1930s. (Einstein was an extremely smart man;
this is an unlikely interpretation.) Third, that Einstein lined up
with the 19th-20th century anti-gambling activists in Europe
and North America, who believe that games of chance are morally wrong.
Fourth, that Einstein, not a man with a small ego, was giving
marching orders to the Lord God.
20. Jan Mikusiński [1913-1987].
See:
Kaminski's Life and Work of Jan
Mikusiński.
See:
Antosik P, Kaminski A, eds.
Generalized Functions and Convergence:
Memorial Volume for Professor Jan Mikusiński.
13-18 June, 1988, Katowice, Poland.
New York: World Scientific Pub Co Inc (December 1, 1990)
ISBN: 9810201834
CHAPTER 23. APPENDIX G: GLOSSARY.
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MATHEMATICS. MATHEMATICS is the study of proof.
The five major divisions of mathematics are:
1. Arithmetic, including number theory.
2. Geometry, including topology.
3. Algebra.
4. Analysis, including calculus.
5. Foundations of mathematics and formal logic.
ARITHMETIC.
Mathematical study of the fundamental numbering system,
including natural numbers (1, 2, 3, ...),
integers (..., -3, -2, -1, 0, 1, 2, 3, ...),
rational numbers, real numbers, and complex numbers.
Number theory is devoted largely to the study of prime numbers (q.v.).
GEOMETRY. Mathematical study of spatial relations.
ALGEBRA. Mathematical study of the operations
of addition and multiplication, and related operations.
ANALYSIS. Mathematical study of limits and infinitesimals.
FOUNDATIONS OF MATHEMATICS.
Mathematical study of sets and logic.
NUMBERS/NUMERALS. A NUMBER is an abstract concept,
that includes all instances of anything that can be counted.
That is, the number one, or one-ness, includes every occurrence
of things that are one; number two, or two-ness, includes every
occurrence of things that are two or paired; number three,
or three-ness, includes every occurrence of things that are three,
etc.
r1 + r2 + r3 +... diverges.
A NUMERAL is the physical representation of a number, which
is different in different cultures. All modern cultures use so-called
ARABIC NUMERALS, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,..., although
these numerals actually appear as 1=١, 2=٢, 3=٣,
4=٤, 5=٥, 6=٦, 7=٧, 8=٨, 9=٩,
in Arabic script. Ancient Chinese/Far-Eastern numerals are:
1=一, 2=二, 3=三, 4=..., 5=五, 10=十.
The ancient Phoenicians, Hebrews, and Greeks reused their alphabets
as numerals. Thus, the ancient Greek numerals, seen in the works
of Euclid and Archimedes, are: 1=α, 2=β, 3=γ,
4=δ, 5=ε, 6=ζ, 7=η, 8=θ, 9=ι, 10=κ.
The ancient Roman numerals, of course, are: 1=Ⅰ, 2=Ⅱ,
3=Ⅲ, 4=Ⅳ, 5=Ⅴ, 6=Ⅵ, 7=Ⅶ, 8=Ⅷ,
9=Ⅸ, 10=Ⅹ. The Hindu/Sanskrit numerals are:
0=०, 1=१, 2=२, 3=३, 4=४,
5=५, 6=६, 7=७, 8=८, 9=९.
The Gujarati numerals are: 0=૦, 1=૧, 2=૨, 3=૩,
4=૪, 5=૫, 6=૬, 7=૭, 8=૮, 9=૯.
However it is represented, whether numeral 3, ٣, 三,
३, ૩, or Ⅲ, all represent the same
number, namely three. The classic philosophical work,
Principia Mathematica, by Alfred North Whitehead
and Bertrand Russell, devotes 345 pages to the number three.
If you think of mathematical numbers as being merely
an academic abstraction, consider this: EVERYTHING
in the universe that occurs as one (e.g., one partridge, one pear tree)
belongs to the number one; everything that occurs as two
(e.g., two turtle doves, two eyes, two hands) belongs to the number two,
etc. With this philosophical definition, mathematicians have seized
ownership of everything in the universe that can be counted.
Concepts such as number and numeral, which are so obvious to
those of us living in the twenty-first century, were great mysteries
to the ancient peoples. The ancient Greeks had no zero, and had
a great deal of difficulty with one. After that, they were OK
up to about a million.... No ancient culture (Mediterranean, Indian,
Chinese) had a concept or name for zero, and the Greeks abhorred zero.
(See [Seife, 2000].)
Zero was introduced in eighth century India by Brahmagupta.
NATURAL NUMBERS.
The counting numbers, 1, 2, 3, 4, 5, 6, ..., known to
ancient cultures.
ORDINAL NUMBERS. [Latin: ordo = order.] Numbers in order:
1st, 2nd, 3rd, 4th, 5th, 6th, .... There is nothing between
a pair of consecutive ordinal numbers, e.g., there is no one-and-a-halfth.
CARDINAL NUMBER. [Latin: cardo = hinge or linchpin.
Signifying great importance, since a door without its hinge
will not function as a door. A cardinal in the Roman Catholic Church
is a priest of great importance.] Numbers of magnitude:
1, 2, 3, 4, 5, 6, .... Between any two cardinal numbers,
there is always another cardinal number.
PRIME NUMBERS. A PRIME NUMBER is a whole number
(i.e., natural number) divisible without remainder only
by itself and 1. The first few prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, ...,
INTEGERS.
The natural numbers, the negative natural numbers,
and zero: ..., -3, -2, -1, 0, 1, 2, 3, ....
RATIONAL NUMBERS.
The values of the ratios of two integers,
i.e., integers divided by integers.
REAL NUMBERS.
Cuts formed from the ordering of rational numbers.
COMPLEX NUMBERS.
Symbolized as (x+iy), where x and y are real,
and are formally added and multiplied, with i2=-1.
TRIANGLE. A three-sided, planar figure, consisting
of three lines (EDGES), connected at three points
(VERTICES). Each vertex lies opposite of an edge; and
each edge lies opposite of an vertex. In this report, one edge
of the triangle is placed horizontally, called the BASE, b.
The vertex opposite to the base is the APEX. A line that
is dropped from the apex to the line through the base is the
HEIGHT, h. The rules of plane geometry were known
on an intuitive basis by the ancient Egyptians (second millennium B.C.),
and were systematized and summarized by Euclid (230-275 B.C.)
ISOSCELES TRIANGLE.
A triangle with two equal edges. By convention, the unequal edge
is placed at the base. Proof that the base angles of an isosceles triangle
are equal is called PONS ASINORUM [Latin: bridge of asses],
and was one of the first successes attributed to computer programs
for discovering mathematical proofs.
FRACTAL.
Fractals are endlessly repeated geometric figures,
as for example, Sierpinski's triangle.
(See [Lauwerier, 1995].)
The barn-painting problem is an example of a fractal.
SIERPINSKI'S TRIANGLE.
"...Sierpinski's Triangle (or gasket), after the
Polish mathematician, Waclaw Sierpinski, who described some of its
interesting properties in 1916. Among these is its fractal
or self-similar character. The large blue triangle consists
of three smaller blue triangles, each of which itself consists
of three smaller blue triangles, each of which ..., a process
of subdivision which could, with adequate screen resolution,
be seen to continue indefinitely. Fractals and self-similarity
are of considerable interest in their own right, but our interest here
is in how to construct Sierpinski's triangle. One way to do so
is to inscribe a second triangle inside the original one,
by joining the midpoints of the three sides, and then repeat
the process for the resulting three outer triangles,
for the three outer triangles that result from that, and so forth...."
(See [Sierpinski, 1916].)
FAT ISOSCELES TRIANGLE. An isosceles triangle in which
the base, b, is much greater than the height, h.
SLIM ISOSCELES TRIANGLE. An isosceles triangle in which
the height, h, is much greater than the base, b.
MAXIMUM ISOSCELES TRIANGLE. An isosceles triangle in which
the height, h, and the base, b are balanced,
to give maximum area, when placed within an inverted triangle.
LIMIT, L.
Traditionally, that number, L, on the real line, to which
a sequence, x1, x2, x3,
... xn, ..., approaches, as n ⇒ ∞.
For a finite limit, xn ⇒ L, the
xn get "ever so close to" L. For an infinite
limit, the xn ⇒ ∞, i.e., the
xn "gallop off to" ∞.
Weierstrass interpretation. For a finite limit,
xn ⇒ L, for every ε>0,
there exists an N>0 such that for every n>N,
|xn-L|<ε.
Weierstrass interpretation.
For an infinite limit, xn ⇒ ∞
and for every n>N, xn > ∞.
CALCULUS.
[Latin: stone, used in an abacus, for calculation.]
Any method of computation. Calculus has come to mean
differential and integral calculus.
INFINITY, ∞. The entity greater than any real number.
That is, for every N, N<∞.
INTEGRAL, ∫. The calculation of area under a function,
f(x), between the range where x=a and x=b, denoted
a∫b f(x)dx.
71.
Integration is performed by making smaller and smaller
areas under the function, that get closer and closer to
the actual curvature of the function.
RIEMANN INTEGRAL, ∫.
An integral where the limiting areas are vertical columns.
LEBESGUE INTEGRAL, ∫.
An integral where the limiting areas are horizontal slabs.
BOCHNER INTEGRAL, ∫.
An integral where the limiting areas are bricks.
PAINTING INTEGRAL, ∫.
An integral where the limiting areas are triangles or arbitrary
SUMMATION, ∑. For a sequence of values,
x1, x2, x3, ..., xn,
the notation ∑i=1n xi
denotes
x1+x2+x3+...+xn,
that is, the sum of xi from i=1 to i=n.
For example, the notation
∑i=15 1/ni denotes
1 + 1/2 + 1/3 + 1/4 + 1/5 = 1 + 0.5 + 0.33333 + 0.25 + 0.2 =
2.28333.... These are the first five terms
of the harmonic series.
PRODUCT, Π. For a sequence of values,
x1, x2, x3, ..., xn,
the notation Πi=1n xi
denotes x1×x2×x3×
...×xn, that is, the product of xi
from i=1 to i=n. For example, the notation
Πi=15 1/ni denotes
1 × 1/2 × 1/3 × 1/4 × 1/5 =
1 × 0.5 × 0.33333 × 0.25 × 0.2 =
0.0083332. These are the first five terms
of the harmonic product.
WEIERSTRASS PARADIGM.
Statements about differential and integral calculus of the form:
"for every ε>0, however small, there exists
a δ>0, however small, or an N>0, however large, such
that...". This paradigm replaces the otherwise sloppy concepts of
"getting ever so close to a limit" and
"galloping off to infinity".
See