October 24, 2008
I concluded my series of talks by showing the following theorem of Viale:
Theorem (Viale). Assume and let be an inner model where is regular and such that . Then .
This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .
An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .
Under an additional assumption, Viale has shown this: If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be . The new assumption on allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and , Fund. Math. 148 (1995), 165-198, in place of the structure imposed by . It is still open if the corresponding covering statement follows from , which would eliminate the need for this the strong limit requirement.
- Go to the intermezzo for a discussion of consistency strengths.
October 23, 2008
The text we are using for Calculus III introduces the notion of unit tangent vector, principal unit normal vector, and curvature, for smooth curves , and it also mentions that circles and lines are planar curves of constant curvature. (A curve is planar if its image is contained in a plane, i.e., if it describes a two-dimensional trajectory.)
Surprisingly, though, the text does not explain (not even in the exercises) that circles and lines are the only smooth planar curves of constant curvature. The argument for this is simple enough, so I will show it here:
Read the rest of this entry »
October 21, 2008
Remember that on October 31 is the second midterm. There will not be a homework set due on November 4. However, we will cover new material during the final week of October. If you want to read ahead, in 175 we will be covering the beginning of Chapter 8, probably up to section 8.3. In 275, we will be covering the beginning of Chapter 13; I doubt we will get to section 13.4, but you may want to start reading the first 4 sections of this chapter.
The exam will concentrate on material covered after the first midterm, but it is cumulative. For 175, it will include the basics of Taylor’s theorem (and it may be a good idea to take a look at Talman’s paper; see here); but it won’t include the proof that the method of partial fractions decomposition works. For 275, you may want to review the polar expression of the Laplacian and how to derive it. For both courses, a safe assumption is that if something was covered in lecture (up to October 24), then it may be included.
October 21, 2008
Homework 8 is due Tuesday, October 28, at the beginning of lecture. Same remarks as before apply.
175: Section 7.7, exercises 11, 20, 32, 42, 50, 65, 74.
Section 8.1, exercises 2, 8, 11, 12, 16, 42, 49, 81.
Solve as many problems as you want; of those, up to 12 will be chosen randomly and graded. Each exercise is worth 1 point, so you may obtain two extra credit points.
275: Section 12.7, exercises 2, 20, 36, 40, 44, 46.
Section 12.9, exercises 4, 10, 12.
Section 13.1, exercises 6, 24.
This homework will be graded out of 10 points. Each exercise is worth 1 point. You may obtain one extra credit point.
October 20, 2008
Here is a .pdf file of a Maple worksheet showing how Simpson’s method can be used to compute , following the exercise I suggested in lecture. The accuracy I obtained is larger than what I asked for in lecture, at the cost of a large number of intervals being required.
There are of course much faster methods to compute , based on other ideas.
October 17, 2008
I presented a sketch of a nice proof due to Todorcevic that implies the P-ideal dichotomy . I then introduced Viale’s covering property and showed that it follows from . Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:
Theorem (Viale). Assume is an inner model.
- If holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .
- If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be .
It follows from this result and the last theorem from last time that if is a model of and a forcing extension of an inner model by a cardinal preserving forcing, then .
In fact, the argument from last time shows that we can weaken the assumption that is a forcing extension to the assumption that for all there is a regular cardinal such that in we have a partition where each is stationary in .
It is possible that this assumption actually follows from in . However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of that preserves cardinals, does not add reals, and (for some cardinal ) the set of points of countable -cofinality in is nonstationary for every regular . Obviously, this situation is incompatible with in , by Viale’s result.