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Wednesday November 8, 2006 7:20 am Lethbridge Sunrise 7:31 Sunset 17:59 Hours of daylight: 9:28
A. Morning Musings
7:20 am It is +1 C at the moment with a high of +9 C forecast.
I am up late but feeling very refreshed and eager to begin something.
The model trains activities that I am most interested in completing all require some additional materials. The quad hopper cars need Kadee #5 couplers (I would like to buy a bulk package of 20 which are not available in Lethbridge) and the nailing down of track requires more foam under bed (I will try to buy that today) as well as more under the track uncoupling magnets (not available in Lethbridge). Diesel locomotive 5930 requires a Kadee #37 coupler (not available in Lethbridge).
The mathematics will require that I purchase the preview copy of MathType as my 30-day free trial is over. That should only take a few minutes so I will likely do that shortly. The chapter on quadratic reciprocity contains a number of new terms and ideas - the actual math is not that difficult but the notation is quite new and takes some getting used to.
B. Plan
Immediate Health Walk & exercise 1 hr Model Trains Begin assembly of 6 quad hopper cars 1 hr Purchase 2 turnouts & complete track layout for The Channon 1 hr Mathematics Read & make notes for "Fearless Symmetry" chap 7: quadratic reciprocity 2 hr History Continue reading "Citizens" 1 hr Literature Begin reading "Empire of the Sun" by J. G. Ballard 1 hr Later Chores Investigate water softeners for home Technology Read manual for cell phone Make notes for chap. 4 of "Switching to the Mac" Begin reading "iPhoto" digital photography - learn about using the various manual settings
Philosophy Read "The Art of Living" by Epictetus Mathematics Larson "Calculus" Read "The Computational Beauty of Nature" Chap 3 Gardner "The Colossal Book of Short Puzzles" History Watson "Ideas" Model Trains Build oil refinery diorama: add ground cover Assemble second oil platform kit Assembly of CN 5930, an SD40-2 with a NAFTA logo Puzzles The Orange Puzzle Cube: puzzle #9
C. Actual/Notes
8:10 am
I am back to chapter 7 of "Fearless Symmetry" which is on the idea of quadratic reciprocity.
This is an introduction to the general idea of looking at groups of Z-equations (i.e. equations with integral coefficients). The chapter restricts itself to the simplest types of interesting Z-equations, namely quadratic equations. Furthermore it restricts the solutions to those in the field
, for various prime numbers p.
Fearless Symmetry (2006). Avner Ash & Robert Gross.
Chapter 7 Quadratic Reciprocity [p. 67 - 83]
- "... quadratic reciprocity is the tip of the gigantic iceberg of 'reciprocity laws' ... [p. 68]
- One might begin with equations of degree 1, that is, equations of the form ax + b = 0. For any field R, the solution set is simply the element -b/a which is always defined since we are working within a field.
- The next most difficult case is that of the quadratic. The general form of a quadratic equation is
. But rather than start with this case, we will deal with the simpler situation
.
- If a is 0, then the solution set will be x = 0, that is, it will contain just one number, 0.
- If a is 1, then the solution set S(Z) = {1, -1}. Also S(Q), S(R) and S(C) are the same set {1, -1}.
- If a is -1, then the solutions sets S(Z), S(Q) and S(R) are all the empty set, but S(C) = {i, -i}.
- In general the solution set for the equation
- Staying with the same equation
Pedagogically, the authors have done a superb job of introducing this idea beginning with the simplest possible case and then slowly and meticulously extending it to more complex and interesting cases. This is also a great example of how new mathematics is actually done!
- Begin with a = 1. Consider the polynomial
, or equivalently, the equation
. Now examine various values of p, beginning with p = 3.
- if x = 0, we have
which is not congruent to 0 modulo 3 (Note: being congruent to 0 modulo p is equivalent to saying that this value is a root of the equation).
- if x = 1, we have
which is not congruent to 0 modulo 3.
- if x = 2, we have
which is not congruent to 0 modulo 3.
- Therefore there are no solutions to the equation. Notationally,
.
- Consider next the various values of the polynomial when p = 5. It is easy to verify that
. Similarly one can work out values for other values of p. After a while one may notice that when
, it is also true that n + m = p.
- Thus we have shown that the solution set
may have 0, 1 or 2 elements modulo p.
- Legendre created the notation
to represent the number of solutions to
where:
At first this may seem a bit strange, if not simply arbitrary. However this notation allows one to "see" certain patterns that would otherwise be very difficult to notice. [p. 73]
2:10 PM More progress to report. I visited the local hobby shop this morning and bought a couple of packages of foam road bed that will allow me to continue working on nailing down the track.
I also bought 2 more turnouts which I have already placed on the layout. This allowed me to complete the track layout for The Channon, although it still needs to have the road bed installed and then to have everything nailed down.
I was able to buy 3 packages of Kadee couplers (each package contains 2 pair of couplers) which will allow me to proceed assembling all 6 of the 100 ton quad hopper cars for my Black Diamond operation.
I spotted a new boxcar on the shelf in the store (CN 11059) and bought it for my layout. I have often seen photos of short passenger trains with one baggage boxcar and a couple of passenger cars. I can now run trains that look like this.
Here are a few more photos:
D. Reflection