In addition to individual articles, the Electronic Journal of Differential Equations also publishes monographs in the area of differential equations. The first one was a book that I think I remember seeing on library bookshelves: Showalter’s Hilbert Space Methods for Differential Equations.

The Obama administration and Democrats in general are in trouble because they are not urgently and effectively addressing the issue that most Americans want them to: the frightening economic insecurity that has put a chokehold on millions of American families.

Research and Markets has announced the addition of Woodhead Publishing Ltd's new report " A Quick Guide to Fitness Fundamental Engineering Mathematics " to their offering.

A fellow student asked me that question today, and I came up with the following answer: Clearly the top eigenvalue is continuous, as it’s just the norm. Then by writing , the sum of the top eigenvalues of , as the SDP

we see that is convex (since a function of the form , i.e. the supremum of linear functions, is convex). Since a convex function on an open set is continuous, we see that is continuous on the positive semidefinite cone, so is continuous on the same set also.

Another interesting question: are the eigenvalues of a general square matrix continuous w.r.t. to its entries? I believe that’s the case, because we’re talking about roots of polynomials, after all, but I don’t know a simple argument off the top of my head.Possibly relevant posts:

• Semidefinite program from the norm of a PSD matrix (1/21/2010)
• Spectral Mapping Theorem (6/14/2006)
• What good are eigensystems? (9/16/2005)

1. Hard Deadline. PRO- uniform, no favoritism, can post HW Solutions or go over HW in class as soon as it is handed in. CON- there could be legitimate reasons for lateness that are short of a doctors note. CON- you want the student to DO the HW even if it will be late. CON- you need to be TOUGH to say NO.
2. Moral Deadline (what I do, see last post). Same as Hard Deadline, but its a bit easier to say NO.
3. Penalty for lateness. PRO- the students will still do the HW. CON- delay in posting solution. CON- slackers are still slackers. ODDITY- the penalty is supposed to discourage lateness. But it may encourage it (gee, 10% off if I hand it in one day late. OKAY, its a deal)
4. Look at late HW only if they affect the final grade. PRO- less to look at likely, CON- Don't really want to keep track of these things. CON- student may not be discouraged from handing things in late.
5. Only count (say) 10 of the 12 HWs, and have HARD DEADLINES. PRO- same as HARD DEADLINE. CON- students will blow off 2 HW's, possibly the last two which may be important for the final. CAVEAT- raises the much bigger question of whether to treat students like adults or like ...students.
6. Students get x number of late days (this one was new to me). PRO- well defined rule, flexible but no favoritism. CON- delay in posting solutions. CON- keeping track of it.
7. If you miss a HW then the others will count more (up to some limit). PRO- uniform. CON- students may still miss some HW they should do.
8. HW are OPTIONAL. PRO- they sink or swim on their own. CON- they sink or swim on their own.

I was reading some political blogs this morning, when I came across this quote:

Look, there’s an endless list of topics I don’t understand at all. I went through an entire semester of pre-Calculus in high school and was never able to understand what a function is. I still don’t. It’s a complicated subject and I was a lazy student.

I don’t know what to say to that.

HW is due on Tuesday. However, your dog died! Hence you get an extension to Thursday. That is, for all people in the class I assume you have a quasi-legit reason to ask for an extension to Thursday. Hence you can hand it in Thursday for full credit. However, if you want an extension past that you will not get it since I already gave you an extension to Thursday. (There may be some severe exceptions which will have to be documented.)

1. This will save alot of time in terms of students asking permission to hand it in late since I will say I already have you an extension and you are asking for another one?
2. Some students will get into the habit of handing it in Thursday. This is okay so long as they do not ask for an extension past that.
3. Clyde tells me that this is really a cheat- the HW really is due Thursday. I may have a higher moral ground when telling them they can't hand it in later than Thursday, but they will still feel that they deserve an extension if their dog dies on Wednesday. My response: they do not.
4. I do make sure that they have enough knowledge to do the HW by Tuesday.
5. I am teaching one Junior-Senior class and one honors-class so these are already pretty good students. They (I hope) know what I mean when I say that they cannot ask for an extension past Thursday. Also they will likely not need them. I have not tried this in a Freshman class. I would like to but they are usually co-taught and large so it would be harder to manage.

The Oscillating Humans Project, announced here, is searching for a living specimen - an exemplar - of an oscillating human.

DEFINITION: For purposes of the project, an Oscillating Human is someone who consistently, repeatedly, over many years, expresses opinions directly opposite to opinions he or she expressed earlier, always ignoring and/or denying the existence of copious, easily found clear documentation of the earlier opinions.

PURPOSE: The exemplary person, once identified, will serve as an example for teachers to use in logic classes. To minimize the chance of lawsuits, the exemplar must be a "public person", with (as stated above) for whom there is copious, easily found, clear documentation of years and years of oscillation.

If you know of an outstanding specimen, please send:

1. The name and a 20-word biographical sketch of the person.
2. Several URLs pointing to clear, unarguable documentation.

Send to: OSCILLATING HUMANS PROJECT: marca AT improbable.com

NOTE: This is an education project. It is NOT an exercise in naming people you don't like. No screeds, please.

Oscillating Humans Literature Review

Published research about this form of human oscillation may be scarce. One of the few apparently relevant items — judging it by its title, if not by its contents, is this British study:

"Oscillation of Human Performance as a Personality Measure," Michael A. Tainsh, Perceptual and Motor Skills, vol. 35, no. 2, October 1972, pp. 677-8.

Truly pertinent citations will be welcomed.

I’m within of having a new roommate, so naturally I’m planning what I’ll do with the extra $625/month I’ll be saving. Guffaw. The first two months are spoken for already— it’s past time for new glasses, and I haven’t paid off my Caltech dining account in several months—, then I’ll start saving up to get a car, since it looks like I’m going to be here another 3 years or so. After that, I dream of actually adding to my savings account for the first time in over a year.Possibly relevant posts:

• (Not) Getting Paid (11/7/2003)
• Banking and Rolling (10/31/2003)
• The one-month gap (7/27/2006)

Prime numbers have fascinated mathematicians for centuries. A prime number has exactly 2 factors – one and itself. The only even prime is 2, the rest are all odd.

The primes less than 100 are as follows:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

There doesn’t appear to be a pattern in the distribution of primes.

How about the "gap" (spacing) between the primes? Is there a pattern in that?

1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8

There doesn’t appear to be a pattern in the gaps, either.

Spiraling

Stanislaw Ulam was a Polish-American mathematician who was involved in the Manhattan Project during World War II.

One day he was bored in a meeting and began to write numbers in a spiral. He started like this, moving in a clockwise direction.

The next round continued the "spiraling" pattern, as follows.

He kept going (it must have been a long meeting), then highlighted the prime numbers and found something interesting.

Many of the primes appear to line up when arranged in such a sprial.

Let’s go much bigger and see what happens. We observe there are many places where the primes form line segments, mostly at 45°, but sometimes horizontal and vertical.

You can see an even bigger display, courtesy of Prime Curios.

What I found interesting in the large picture is where primes are not – there are distinct blocks and patterns of white space where no primes occur.

This spiral appeared on the cover of Scientific American in March 1964 and continues to generate research interest to this day.

Check out more such prime number information in the book Prime Curios! The Dictionary of Prime Number Trivia  by Chris Caldwell and G. L. Honaker, Jr.

Why do we care about primes?

Apart from many other things, prime numbers are vital in the development of encryption algorithms, used in generating secure Internet transactions.

Related posts:

1. Largest known prime: 2^43,112,609 – 1
2. Friday Math Movie – Math Illuminated
3. Message to G8 Environment Ministers

Qiaochu Yuan recommended Stevenhagen’s mathematical writings in general, so I did some additional searching. I found this page of lecture notes for algebraic number theory courses at Leiden University.

Stevenhagen’s notes on class field theory look particularly interesting. They start with particular examples, and explain what the theory means in those particular examples.

> {-# LANGUAGE RankNTypes #-}

> p1 :: (forall a. [a]) -> [Int]
> p2 :: (forall a. [a]) -> [String]
> p1 x = x
> p2 x = x

π_X:N→F(X)

These notes by Stevenhagen provide an elementary introduction to rings of algebraic numbers.

The Geometry Center at the University of Minnesota was a pioneer in putting mathematics on the web. The Center specialized in visualization of advanced geometric topics.

The Center itself was closed in 1998, but their website is still available. The site is quite old (the pages that note that Netscape 2.0 is required are particularly poignant reminders), and many parts of it no longer work, but much of the content is still there.

I’ve “reduced” a problem to solving this recursion:

This looks nasty, and certainly beyond my experience. Any ideas on how to proceed? I guess I could try Z-transforms, but … just yuck.Possibly relevant posts:

• Linear Recursion Equations (11/24/2005)
• A failed attempt to use an SDP to find a nonorthogonal factorization (1/17/2010)
• Spherical Mean Value (6/6/2005)

Here is a page with individual GIFs/PNGs for each math symbol. This is useful for the occasional inclusion of math formulas on a web page, for example. It also has directions on how to build more complex formulas using just HTML. The page is part of the Metamath project.

Special Issue of the journal Theory of Computing in honor of Rajeev Motwani (1962 - 2009)

Submit contributions by July 30, 2010.

Submissions in all areas of theoretical computer science will be considered, with preference for topics related to Rajeev's work. All papers undergo strict peer review and must meet the standards of Theory of Computing.

See details at http://www.cs.umd.edu/~samir/ToCMotwani.htm.

Daniel Murfat has a nice series of notes on various mathematical topics, mostly algebraic geometry.

Let n be a parameter we pick later. Alice generates a random n x n matrix of elements from {0,...,25} and checks that the Det mod 26 is nonzero. (CORRECTION ADDED LATER: the Det has to have an inverse mod 26, so has to be rel prime to 26.) Alice gives this to Bob. Alice and Bob both compute its inverse. Alice and Bob can exchange messages by encoding every block of n by this matrix. So the first n letters of the text get multiplied by the matrix to get a diff n letters. Then the next n letters after that, etc.

1. n has to be small enough so that Alice and Bob don't mind exchanging n² elements of {0,...,25}
2. n has to be large enough so that going through all possible n x n matrices is not practical for Eve.
3. n has to be large enough so that tables of how often particular n-sized blocks occur are useless.
4. Can combine with other techniques. Perhaps Alice and Bob first encode using the Vigenere Cipher and then apply the Matrix. They would then have to also share the Key for the Vigenere Cipher.
5. QUESTION: If ALL Eve gets is the text then is this a good cipher? Clearly if Eve also somehow gets her hands on a message and what it was coded to she will easily crack the code. But if not then does this work well? This code is not used because Eve might get her hands on such, but I wonder if these are circumstances where it would be reasonable.
6. Is there a value of n that is both big enough and small enough (a Goldilocks n).
7. What else is known about this?

I came across this history of loops, a generalization of groups, where I picked up this interesting tidbit: loops are named after the Chicago Loop, the central business district of Chicago. (The elevated trains tracks form a loop that enclose it.)

The Economist has an article on the question of the hardest language to learn. They suggest that a language called Tuyuca is the answer. What makes Tuyuca unusual is that verbs carry an ending that indicates whether the statement is thought to be true or known to be true with certainty. Imagine a language with one tense for conjectures, and another for theorems.

Update: The dual SDP for can be more simply written as

subject to
,

So it’s surprising that the which solves this seems to be in the 1/2-ball. It’s easy to see that this ball is included in the feasible points, but no reason to think that it is the entire set of feasible points.

I spend mucho time deriving and checking this duality, then implementing it in CVX, so I record it here so hopefully I don’t have to redo this (I realize I could’ve looked this up, but the point is to derive it so I have a better chance of remembering it myself):

The dual of the SDP s.t. , . is the SDP s.t. . ,

In particular, this implies the dual of the SDP I mentioned before gives is the SDP

s.t. , ,

where I decomposed as follows

If , then and its components break down as follows: , , , , , and

As you can see, this is an unwieldy SDP to write out, but it has its benefits as a method for calculating . Specifically, I believe that is a matrix in the unit ball which achieves :

Numerical experiments indicate this is so, but I’d like to establish it. Here’s some CVX code to test this out (mostly this is just here– as is this entire post and all posts relating to my research– for archival purposes):

% solve the dual SDP for a random matrix
A = randn(20,20);
[n,m] = size(A);
cvx_begin
    variable s(n+m);
    variable lambda(n+m+1, n+m+1);
    variable Z(n+m+1, n+m+1) symmetric;
    maximize( trace( lambda * [zeros(n) A zeros(n,1); A.' zeros(m,m+1); zeros(1,n+m+1)] ));
    subject to
        s >= 0;
        Z == semidefinite(n+m+1);
        [diag(s) zeros(n+m,1); zeros(1,n+m) -ones(n+m,1).'*s] - ...
            [zeros(n) lambda(1:n, n+1:n+m+1); ...
             lambda(n+1:n+m, 1:n) zeros(m) lambda(n+1:n+m, n+m+1); ...
             lambda(n+m+1, 1:n+m) 0] + diag([zeros(1,n+m) 1]) - Z == 0; 
cvx_end
B = lambda(1:n, n+1:n+m);
% the solution to the dual SDP, the primal SDP, and  (lambda_2, A) (inner product) should
% all be the same
[cvx_optval gamma2norm(A) trace(2*B.'*A)] 
gamma2dualnorm(2*B) % check 2lambda_2 is in the gamma2dual norm ball

Possibly relevant posts:

• Trace dual of the norm (1/19/2010)
• Equivalence of the norm and the norm (1/27/2010)
• Estimation of the norm using an SDP (1/27/2010)

1. 

   Math Mammoth The Four Operations
   (with a Touch of Algebra)

   The main topics studied in this book are simple equations, expressions that involve a variable, the order of operations, long multiplication, long division, and graphing simple linear functions.
   
   The idea is not to practice each of the four operations separately, but rather to see how they are used together in solving problems and in simple equations. We are trying to develop student's algebraic thinking. Many of the ideas in this chapter are preparing them for algebra in advance.
   
   

2. Math Mammoth Ratios & Proportions & Problem Solving

   This worktext concentrates, first of all, on two important concepts: ratios and proportions, and then on problem solving.
   
   First, we study thoroughly the concept of ratio, including how it connects with fractions and the aspect ratio in rectangles.
   
   Solving proportions is divided into three separate lessons. In the first one, we solve proportions by thinking through equivalent ratios. In the second one, the usual method of cross-multiplying is introduced. Then follows a lesson that explains just why cross-multiplying is allowed. Then there is more practice with solving proportions and word problems. We also study scaling geometric figures and floor plans.
   
   The last lessons concentrate on various kinds of word problems that can be solved with the help of bar or block diagrams (the same kind as in Singapore Math). These involve problems with fractional parts, and then problems involving ratios. The diagrams become a powerful tool to solve such problems without the use of algebra.
   
   
3. 

   Math Mammoth Place Value 5

   Math Mammoth Place Value 5 is a short worktext that covers value up to the trillions—that is, numbers up to 15 digits—rounding them, and using a calculator.
   
   The first lesson covers place value up to billions. Then, we study exponents, and right after that, place value up to trillions, writing the numbers in the expanded form using exponents. After working with addition a little, there are two lessons about calculator usage.
   
   

1. The Guest Speakers look AWESOME!: Khot, Raz, Regev. Also Banquet speaker Hartmanis AWESOME!
2. Based on titles alone (not so reliable) it looks like there are no quantum papers. Someone tell me- is that really true? Even if there are some, I am sure there are not many. Has the field run its course? Doubtful. In fact, it may be the other way around--- the field has grown and their are other places for that work to appear.
3. ADVICE: Try to download some of the papers that (1) interest you AND (2) you have the prereq knowledge OR always wanted to get that knowledge. Either read them or use them to find refs to read.
4. Is there a centralized place to download them? There should be!!!!!! Other conferences manage this (SODA for one).
5. I will be at STOC and CCC. Hope to see you there!

Interest rates are starting to edge up as economic conditions improve.

But even in the traditionally high-interest rate countries (like Australia and New Zealand), banks lend at around 7 to 8% currently, while in much of the world, interest rates are at, or very near, 0%.

So it’s extraordinary (to me) how banks are allowed to charge such whopping amounts on credit card debt.

Here’s an excerpt from Citibank Singapore’s credit card fine print (“RC” stands for “Ready Credit”).

I’ve highlighted some of the charges you’d be up for if you are (heaven forbid) just 1 day late with your payment.

Additional Information Box for Credit Card & RC

The kicker is they are allowed to vary the conditions as they go along.

And did you know if you have loans with other institutions, and you default with them, you can be charged a higher interest rate on this card?

The high interest rates are justified by the banks on the grounds they are lending to people who pose some risk. This is certainly the case in the US where it is still easy to get a credit card. In the Singapore case, there are income limits (normally S$30,000 per year fro locals and $60000 per year for foreigners) before you can apply for a card.

There are thousands of people in the US who are now owing more than they borrowed (using their credit cards) and the interest (and fees) are growing so fast, it’s sending them bankrupt, especially as the Global Financial Crisis results in job losses.

Oh, and what was the main cause of the financial crisis? Lending money to people who couldn’t afford it, and didn’t understand the risks, of course.

It used to be simple. Banks could only lend to people who had the means to pay it off and could secure the loan via some mortgage (this means if you stopped paying, the banks could sell your house.) It reduced excessive and risky lending. And I doubt we’ll see that situation again anytime soon.

To be fair, it’s not just Citibank. All credit card providers do this because they can, giving us a US economy with around 1 trillion in credit card debt.

See Frontline’s The Card Game, from PBS.

Be careful what you sign.

Related posts:

1. Online banking usability
2. Another misleading credit card advertisement
3. Financial literacy for students

The idea of a homomorphism extends neatly to general signatures. A function between two objects with the same signature is a homomorphism if it preserves all function and relation symbols. So φ is a homomorphism if for each n-ary function symbol f

φ( f(x₁, …, x_n) ) = f( φ(x₁, …, φ(x_n) )

and each n-ary relation symbol R

R(x₁, …, x_n) implies R(φ(x₁, …, φ(x_n))

This coincides with the usual definition of homomorphism for groups and rings. For partially-ordered sets, homomorphisms correspond to order-preserving maps.

One of the reasons why I use Matlab more than Mathematica is Matlab supports convenient convex optimization via the open source CVX package, while Mathematica doesn’t seem to have any support even via (open source) third party packages for convex optimization.

Now it seems I’ve found a relatively easy way to rectify this, by interfacing Mathematica with the CVXOPT Python package using Pythonika. At least under Ubuntu, this was a hasslefree process: apt-get python-cvxopt, then download and compile Pythonika. The only modifications that I needed to make to the Pythonika makefile were to add a ‘-lrt’ linker flag and point to the correct python version and the correct Mathematica Linker library. Then I was able to use CVXOPT to solve a linear program from Mathematica!

CVXOPT doesn’t seem to provide the same type of natural mathematical language interface as CVX does, and I’m not sure how flexible/convenient the interface between Python and Mathematica provided by Pyhtonika is, but this is an interesting development.Possibly relevant posts:

• CVX in Mathematica via Mathematica -> Java -> Matlab? (9/1/2009)
• PyPI package basicCAS (7/24/2005)
• User directory python package installation (3/21/2007)

One idea for getting sign decompositions of a matrix, inspired by Grothendieck’s inequality (the dual formulation) is to get a factorization which realizes and then round the entries of to signs somehow, take to be an appropriate scaling matrix, and use as your approximate sign decomposition of .

You can use this as the primitive step in a greedy algorithm, and get pretty good results (if! you use least squares to calculate )– I implemented this method, simply replacing the entries of with their signs–, but it’s not a particularly smart way of getting sign decompositions. For one thing, you could probably equally as well use random sign matrices , because if you use least squares projection, you’re guaranteed that with reasonable probability at each step (take this with a grain of salt: I think this is easy to show, but I've not checked that it's true), which is all that you can hope to say even if you use the from the decomposition. That is to say, there's nothing inherently special or useful about the decomposition.

To drive home the point, first notice that , so . Therefore we can find a wide range of factorizations achieving : take to be any orthonormal matrix. So there's definitely not anything special about the factors, or the signs you'd round them to, in the general case.

Second, and more importantly, it seems difficult to prove anything about the rate of convergence of these types of greedy algorithms. And (this is just a gripe, because even a factorization corresponding to can be far from sparse), although the factorizations I've seen returned with my particular greedy scheme have reasonably many terms, there's no kind of guarantee of sparseness.

Possibly relevant posts:

• Factorization norm details (2/9/2010)
• An observation on the norming functionals of the norm ball (2/24/2010)
• Is uniqueness important? (2/17/2010)

1. Ask you why you were doing that. Seems like an odd use of your time.
2. Point out to you that Automata Theory which usually gets around 8 people got 23 (it competes with crypto as noted a few blog entries ago).

1. I had a prior post on why email is less effective then is used to be (I can't find the post- if you know where it is let me know.) To summarize from memory- people get too much email and some goes to SPAM filters or can be claimed to have.
2. (A colleague of mine suggested this.) If I just EMAIL about a class, I have not spend much effort and the students sense that. If I go out of my way to talk about the class then the students think that I care.
3. There are some other explanations for some of the uptick: Comp Sci enrollment is up (might account for 4 students) and by a fluke we have 2 grad students taking the course (which accounts for 2 students). But going from 12 in Spring 2009 to 24 in Spring 2010 is alot. (It was taught be people who are thought us as good teachers both times.)

Sade is one of those artists I heard about constantly growing up, (you know how black people get about their Saw-day) so it’s kind of odd that I just started listening to her for real last week– growing up, of course I was familiar with “The Sweetest Taboo” and “Smooth Operator”, but I couldn’t say I knew she sung them. The title track from her latest album, “Soldier of Love”, is what caused me to start paying close attention. It must be that I needed to spread my musical wings before I could appreciate her style of music.

Anyhoo, here’re some of my favorite songs by her (if nothing else, check out “By Your Side”):

You is welcome, kinfolk!Possibly relevant posts:

• release the stars? (5/22/2007)
• Brandi Carlile (10/9/2008)
• Canzoni Stonate (5/27/2007)

Here’s a nice observation due to a colleague of mine: if is the sign factorization of corresponding to , and we take , and is an extreme point on the unit ball satisfying , then

so each is either 1 or 0.

Thus, for each rank one sign matrix in the decomposition of , . Inteeeeresting…Possibly relevant posts:

• Equivalence of the norm and the norm (1/27/2010)
• Trace dual of the norm (1/19/2010)
• Factorization norm details (2/9/2010)

1. Regular Languages: DFA's, NFA's, Reg Expressions.
2. PDA's, CFL's.
3. Turing Machines, computable and c.e. sets.
4. NPC

1. Decidability of Weak Second Order with S and ≤. The language has quantifiers that range over finite sets, quantifiers that range over natural numbers, and symbols for Successor and ≤. The proof uses Reg Languages. We do it in the Reg Language section and then revisit it when we do decidability. At that point I will also tell them (but not prove) about some theories that are undecidable. We also do decidability of Presburger arithmetic (quantify over naturals, have + and ≤) which follows from decidability of WS1S easily. Will also talk about decidability of S1S and omega-automta, but not prove anything. This did not take up too much time because I presented alot of it as more examples of regular languages. This material is not in any textbook that I know of, however see pages 8-28 of this PhD thesis.
2. I am NOT going to do the algorithm for MINIMIZING a DFA.
3. I am NOT going to do Context-Sensitive Languages.
4. I am NOT going to have them prove things that are obvious, like that S-->aSb, S-->emptystring generates {aⁿbⁿ}. Generally I am against having students prove things that are obvious.
5. I am NOT going to have them ever program a Turing Machine. I will tell them they can do everything and rarely refer to them ever again. I DO need the definition so that I can prove Cook's theorem.
6. I am NOT going to to Primitive Recursive functions.
7. In the NPC section I WILL DO the protocol for, in our language, NGI (non-Graph-Isom) is in AM.
8. In the NPC section I WILL DO the protocol for, in our language, given bit-commit, 3-COL is in ZK. (I may to other ZK protocols as well.)
9. In the NPC section will do that Vertex Cover with FIXED k is in O(n²) (I know that better is known.) Why? Because this is a very good example of an obvious thing (can't do better than roughly O(n^k)) being WRONG. Shows the NEED to prove things.
10. Might do NSPACE(n) closed under Complementation. Might not- this may be conventionally difficult for this audience. And we really wont' be talking about space anyway.
11. Let SUBSEQ(L) be the set of a subsequence of L. I will, throughout the year, do the following: Show that if L is regular than SUBSEQ(L) is regular, Show that if L is context free than SUBSEQ(L) is context free, Show that if L is c.e. than SUBSEQ(L) is c.e. All of this leads to material I discussed in this old post. I WILL NOT prove that if L is ANY language then SUBSEQ(L) is regular, but I may talk about it.

Inspired by John Allen Paulos, who just now tweeted

Obvious, but NBC hasn’t said: Canada, Norway, Germany way ahead of US in Olympic medals per capita. Many ways to rank: Cf. Arrow’s theorem.

I decided to redo the medals list. Here, the number of medals per capita among the top countries.

(Data taken on February 23, from the current state of olympic medals achieved at that date, and from the Wikipedia page listing populations of the nations of the earth, taken the same date)

It looks like I might soon have some spending money. I haven’t gone on a fiction buying spree in a while. Any suggestions? I’m going to pre-order The Desert Spear, because Warded Man was too awesome for me to miss out on the sequel, but other than that, it’s up in the wind. I’ve been out of the sci-fi/fantasy book loop.

Looking for suggestions… I’d prefer one offs and completed series over incomplete series.Possibly relevant posts:

• Fiction interests (6/13/2005)
• Acquisitions (9/30/2009)
• Twilight zone for book ordering (8/15/2006)

I’ve found a tentative algorithm for sign factorization. I’ve only tested it on random matrices (generated using rand and randn) in Matlab– as opposed to theoretically verifying that it holds, or even testing on real datasets– so it may turn out that it just works with high probability for random matrices generated in those particular ways … but hey, that’d be a result.

The error bound looks to be something like where is the sign matrix factorization and depends on the size of (it may also depend on ), and the integer represents the number of times you do a greedy type recursion.

Fingers crossed! as I’m off to attempt an analysis.

Update: Oops, it seems you can’t get that factor arbitrarily small.Possibly relevant posts:

• A way not to get sign decompositions (2/26/2010)
• Matrix of random signs, redux (6/1/2009)
• Factorization norm details (2/9/2010)

1. The students overwhelmingly take crypto. One year 150 students took Crypto (one section of 50 in the fall, two sections of 50 each in the spring) and 8 students took automata theory. Both courses are always taught by people who are regarded as good teachers, so that is not the issues.
2. I tend to think that Automata Theory is more important, but I may be biased. I also think that Automata Theory can be understood pretty well, whereas to understand crypto you really need to understand some Number Theory and even some security. Hence it is a strange stand-alone course.
3. Some students think that the crypto course will get them a job. A course in security may get them a job, but just crypto I kind of doubt.
4. Since more students choose Crypto we offer it more often. Since we offer it more often more students take it. (I exaggerate the circularity.) Also, its cross listed with Math so some math majors take it. This may account for some of the difference but not even close to all of it.
5. So, how does your school do this? In particular, do you let the students tell you what course is more important, or do you tell them? Is it bad if they tell us? YES if we end up with courses on twitter, NO if the students are more aware of what is important then we old academics are.

• Brand NEW format.
• Be part of setting a world record!
• Designed for all ages and ability levels. Simple to register and participate. All you need is Internet access.
• Great prizes
• And it’s absolutely free!!

I want to know whats bigger e^π or π^e.

Imagine you are stranded on a desert island (without logarithm tables or computers) and--- probably due to an emotional shock---your only concern is to find out which one among numbers π^e and e^π is bigger. The solution is: take h(x)=ln(x)/x, take the derivative twice to prove that x=e is a maximum, and that gives e^π is bigger.

Photo by Caribb

 |  Time  | Distance  | Speed
------------------------------------------
There |   6    |    d      | v + 60           
-----------------------------------------
Back |   8    |    d      | v − 60           

 |  Time  | Distance  | Speed
------------------------------------------
There |   6    |  6(v+60)  | v + 60           
-----------------------------------------
Back |   8    |  8(v-60)  | v − 60      

1. An Elementary Proof is one that does not use Complex Analysis. Basic Calculus is fine. This was the criteria when people asked for an Elementary Proof of The Prime Number Theorem. Such a proof was found by Erdos and Selberg. (See this paper for the history) Later Oliver Sudac (TCS, The Prime Number Theorem is PRA Provable, Vol 257, NOT Online) showed there is a proof in Primitive Recursive Arithmetic which is weaker than Peano Arithmetic. An interesting article on this which IS online is by Jeremy Avigad on all of this is at Number Theory and Elementary Arithmetic. From what I hear, the original proof is still the easiest way to prove it. (NOTE- Oliver Sudac, if you are reading this put your paper on line. If you do not then over time it will be called ``Avigad's theorem''.)
2. An Elementary proof is one that can be taught to a class of bright college students in about two hours. If you hear someone say Shelah's primitive recursive bounds on the VDW bounds is elementary this is what they mean. (See here. for the paper.) )
3. An Elementary proof is one that does not use advanced techniques. The original proof of Szemeredi's Theorem is rather difficult; however, it does not use advanced techniques. You could probably teach it to a class of bright college students in about two months. Even so, I would hesitate to call it elementary. It might be easier to learn the background mathematics for one of the non-elementary proofs rather than follow the elementary one.
4. An Elementary proof is one that some bright high school students can come up with during a High School Math Competition.
   1. The following is elementary: Show that any for any 3-coloring of the natural numbers there exists x,y that are the same color such that x-y is a square.
   2. The following is probably not elementary: Show that any for any 4-coloring of the natural numbers there exists x,y that are the same color such that x-y is a square. I wanted to put it on the Maryland Math Competition to find out if it was elementary, but alas they didn't let me. I do not know of a proof that a bright college student could do on his own. The theorem is true by poly VDW thm; however, I would very much want to see an easier proof.
   3. The following is elementary: Show that if n is a power of 2 then if there are 2n-1 integers then some n of them sum to 0 mod n.
   4. The following is probably not elementary: Show that if n is any integer then if there are 2n-1 integers then some n of them sum to 0 mod n. The proof in here is elementary in that you could explain it to a bright college student in 30 minutes; however, I don't think they could come up with it themselves.
   5. In Elementary Particle Theory it is the particles that are elementary, not the theory.

> {-# LANGUAGE ScopedTypeVariables, UndecidableInstances #-}
> {-# OPTIONS -fglasgow-exts #-}

1. 0+b = b
2. Sa+b = S(a+b)
3. 0.b = 0
4. Sa.b = b+a.b

2+3
= SS0+SSS0 by definition
= S(S0+SSS0) by 2
= S(S(0+SSS0)) by 2
= SSSSS0 by 1
= 5 by definition

> data Void
> instance Show Void where
>     show _ = undefined

> type One   = Maybe Void
> type Two   = Maybe One
> type Three = Maybe Two
> type Four  = Maybe Three
> type Five  = Maybe Four
> type Six   = Maybe Five

> zero  = Nothing
> one   = Just zero
> two   = Just one
> three = Just two
> four  = Just three
> five  = Just four

> class Plussable a b where
>    type Plus a b
>    plus  :: Either a b -> Plus a b
>    plus' :: Plus a b -> Either a b

> instance Plussable Void b where
>    type Plus Void b = b

>    plus (Right b) = b
>    plus'       b  = Right b

> instance Plussable a b => Plussable (Maybe a) b where
>    type Plus (Maybe a) b = Maybe (Plus a b)
>    plus (Left Nothing)  = Nothing
>    plus (Left (Just a)) = Just ((plus :: Either a b -> Plus a b) (Left a))
>    plus (Right b)       = Just ((plus :: Either a b -> Plus a b) (Right b))

>    plus' Nothing = Left Nothing
>    plus' (Just x) =
>        let i' = plus' :: Plus a b -> Either a b
>        in  case i' x of
>            Left  a -> Left (Just a)
>            Right b -> Right b

> class Timesable a b where
>    type Times a b
>    times  :: (a, b) -> Times a b
>    times' :: Times a b -> (a, b)

> instance Timesable Void b where
>    type Times Void b = Void
>    times  _ = undefined
>    times' _ = undefined

> instance (Timesable a b, Plussable b (Times a b)) => Timesable (Maybe a) b where
>     type Times (Maybe a) b = Plus b (Times a b)

>     times (Nothing, b) =
>         let i = plus :: Either b (Times a b) -> Plus b (Times a b)
>         in  i (Left b)

>     times (Just a, b) =
>         let i = plus :: Either b (Times a b) -> Plus b (Times a b)
>         in  i (Right (times ((a, b))))

>     times' b = 
>         let i' = plus' :: Plus b (Times a b) -> Either b (Times a b)
>         in case i' b of
>             Left  b  -> (Nothing, b)
>             Right ab -> let (a, b) = times' ab in (Just a, b)

times :: (Two, Three) -> Six

> class Canonicable a where
>     type Canonical a

>     canonical  :: a -> Canonical a
>     canonical' :: Canonical a -> a

> instance Canonicable Void where
>     type Canonical Void = Void

>     canonical  = id
>     canonical' = id

> instance Canonicable a => Canonicable (Maybe a) where
>     type Canonical (Maybe a) = Maybe (Canonical a)

>     canonical  Nothing  = Nothing
>     canonical  (Just n) = Just (canonical n)
>     canonical' Nothing  = Nothing
>     canonical' (Just n) = Just (canonical' n)

> instance (Canonicable m, Canonicable n, Plussable (Canonical m) (Canonical n)) => Canonicable (Either m n) where
>     type Canonical (Either m n) = Plus (Canonical m) (Canonical n)
>     canonical (Left m) =
>         let i = plus :: Either (Canonical m) (Canonical n) -> Plus (Canonical m) (Canonical n)
>         in  i (Left (canonical m))
>     canonical (Right n) =
>         let i = plus :: Either (Canonical m) (Canonical n) -> Plus (Canonical m) (Canonical n)
>         in  i (Right (canonical n))
>     canonical' x =
>         let i' = plus' :: Plus (Canonical m) (Canonical n) -> Either (Canonical m) (Canonical n)
>         in case i' x of
>             Left  m -> Left  (canonical' m)
>             Right n -> Right (canonical' n)

> instance (Canonicable m, Canonicable n, Timesable (Canonical m) (Canonical n)) => Canonicable (m, n) where
>     type Canonical (m, n) = Times (Canonical m) (Canonical n)
>     canonical (m, n) =
>         let i = times :: (Canonical m, Canonical n) -> Times (Canonical m) (Canonical n)
>         in  i (canonical m, canonical n)
>     canonical' x =
>         let i' = times' :: Times (Canonical m) (Canonical n) -> (Canonical m, Canonical n)
>             (m, n) = times' x
>         in (canonical' m, canonical' n)

> iso :: (Canonical m ~ Canonical n, Canonicable m, Canonicable n) => m -> n
> iso m = canonical' (canonical m)

> test = iso :: Either (Two, Two) Five -> (Three, Three)

> go1 = Left (zero, one)
> go2 = Left (one, zero)
> go3 = Right four