Last week, my alma mater Yale University announced that the university would work together with New Haven to fund "New Haven Promise," a program which would provide funding to New Haven public school students who attend colleges in Connecticut. The program promises scholarships for New Haven public school students with only a few weak restrictions. For example, students with less than 90% attendance rates in highschool are not eligible.
There are a number of possible criticisms of this program. The most serious criticism to me seems to be the simple one that this program is not Yale's job. Alumni donate money to Yale with certain expectations. They might also donate money to other causes. But there is a basic expectation that money that goes to Yale will be used for Yale purposes such as going to scholarships for poor students at Yale, not to students at random other schools in Connecticut.
There are additional problems with this program. My little brother wrote an op-ed in the Yale Daily News arguing that this program would in fact cover up the real issues in the New Haven public school systems which need to be addressed. He argues that the teacher unions and the lazy and incompetent teaching which they allow are much more of a root cause of the problems. I'm not convinced of his claims. I'm especially unconvinced by his line that "Instead of staying after school to tutor or help run an extracurricular, unionized teachers typically leave as soon as the final bell rings" which seems to underestimate the great difficulty that even hard-working teachers need to put up with daily.
However, I do think that more broadly speaking there's a clear problem with unions in our public schools which prevent the removal of all but the most egregiously bad teachers. For example, consider the case of eighth grade science teacher John Freshwater in Mount Vernon, Ohio. Freshwater taught science so badly that other teachers in later years had to specifically reteach Freshwater's students. Freshwater told students that Catholics were not real Christians. Freshwater burned crosses into students' arms using a tesla coil. Despite all these issues, it has taken more than 2 years to have Freshwater removed. Thus, while I don't have enough detailed experience to personally evaluate whether the unions are a problem in New Haven (although my limited anecdotal evidence suggests that they are a problem), it does fit the general pattern of what is going wrong with American public schools.
Nathaniel's piece generated a variety of responses such this one by a New Haven public school teacher, this one by a New Haven alderman, and this one by another Yale student. Nathaniel has responded to the last piece here. All of these pieces are worth reading.
Sunday, November 14, 2010
Wednesday, November 10, 2010
On Almost Commuting Matrices
When mathematicians encounter a binary operation, one of the first things they ask is "when does the operation commute?" That is, given an operation * when does one have A*B=B*A? Some operations always commute. Addition and multiplication in the real numbers are examples of this. Sometimes they commute under certain restricted circumstances. For example, subtraction rarely commutes (1-2 is not the same as 2-1).
In high school, children are often taught about matrices and matrix multiplication. Matrix multiplication seems to be given in part simply to have an example of an operation which has complicated behavior regarding when it commutes. Of course, we can only meaningfully talk about this when the matrices in question are assumed to be square matrices since otherwise the multiplication won't even be defined. However, matrices have other operations which we can do on them other than just matrix addition and multiplication. In particular, we can also multiply a matrix by a scalar.
This raises the following question: When do matrices almost commute? By almost commute, we mean commute up to multiplication by a non-zero scalar. A general result seems difficult. But there is at least one pretty result which can be proven without too much trouble by looking at the eigenvalues and eigenvectors of a pair of matrices:
If cAB=BA for some constant c, and A is invertible, then c is a dth root of unity for some d such that d divides the number of distinct non-zero eigenvalues of B. It isn't that hard to generalize this result slightly with A not invertible. However, one then needs the slightly technical condition that A sends no non-zero eigenvector of B to zero. Note also that this result is most nicely stated in the slightly more restricted symmetric case when both A and B are invertible.
One pretty corollary of this result is that if A and B are invertible p x p matrices over the real numbers where p is an odd prime, with all distinct eigenvalues, then A and B are almost commuting if and only if they actually commute.
In high school, children are often taught about matrices and matrix multiplication. Matrix multiplication seems to be given in part simply to have an example of an operation which has complicated behavior regarding when it commutes. Of course, we can only meaningfully talk about this when the matrices in question are assumed to be square matrices since otherwise the multiplication won't even be defined. However, matrices have other operations which we can do on them other than just matrix addition and multiplication. In particular, we can also multiply a matrix by a scalar.
This raises the following question: When do matrices almost commute? By almost commute, we mean commute up to multiplication by a non-zero scalar. A general result seems difficult. But there is at least one pretty result which can be proven without too much trouble by looking at the eigenvalues and eigenvectors of a pair of matrices:
If cAB=BA for some constant c, and A is invertible, then c is a dth root of unity for some d such that d divides the number of distinct non-zero eigenvalues of B. It isn't that hard to generalize this result slightly with A not invertible. However, one then needs the slightly technical condition that A sends no non-zero eigenvector of B to zero. Note also that this result is most nicely stated in the slightly more restricted symmetric case when both A and B are invertible.
One pretty corollary of this result is that if A and B are invertible p x p matrices over the real numbers where p is an odd prime, with all distinct eigenvalues, then A and B are almost commuting if and only if they actually commute.
Monday, October 18, 2010
No Free Lunch, Budgetary Constraints, and Gratuitous Promotion of Family Members
My father has a piece up at the Oxford University Press Blog arguing that both major political parties in the US are pretending to the public that there are painless solution to US budgetary problems. I'm not convinced that the current federal budget situation is as dire as he makes it out to be although I do agree that the state and local budgets in many areas are in very bad shape. However, the essential point seems spot on. Neither the Democrats nor the Republicans are willing to be honest with the voters. And this is clearly not a good situation. I don't see any quick resolution to these issues. I will however be slightly partisan in saying that it is clear that some groups are being more unhelpful than others and that the Tea Partiers are clearly one of the most unhelpful groups in question.
Friday, October 8, 2010
The AFA, Youtube, Christine O’Donnell, and Yale: A Rant About The Modern Right Wing
For a long time, I've believed that the anti-intellectualism of the modern right-wing in the United States is a fringe phenomenon.
However, over the last few months, I have become increasingly convinced that anti-intellectualism is not just a fringe phenomenon but a general trend of the modern conservative movement. The leaders of the movement are either ignorant, anti-intellectual buffoons, or they believe that their base is composed of ignorant, anti-intellectual buffoons.
Prior to coming to this conclusion, I had seen much evidence for this claim that did not convince me. The GSS data show that people who self-identify as liberal on average have larger vocabularies than those who self-identify as conservative. However, this did not convince me. Among other problems, the overall trends in that data are complicated. Vocabulary is correlated not just with increased liberalism but with general political extremism. That is, people more likely to identify with extreme political views are more likely to have a large vocabulary. Moreover, having a small vocab does not mean that one is anti-intellectual. It just means one is less likely to be intellectual.
Moreover, as I've discussed before, by some metrics of political knowledge, Democrats perform on average more poorly than Republicans.
However, the evidence for widespread anti-intellectualism among the American right-wing has now reached proportions which are difficult to deny. It is easy to dismiss Sarah Palin's comments about fruit flies as simple ignorance. And it is easy to dismiss Bobby Jindal's remarks about volcano monitoring as an isolated incident. However, these are not isolated incidences and one can point to many similar instances. Two of the most glaring that I've seen recently are remarks by the American Family Association's head Tim Wildmon and remarks made by Delaware senate candidate Christine O'Donnell.
The American Family Association is a right-wing Christian political group most known for organizing the boycotts that the right-wing periodically directs against companies that they have decided are too gay-friendly. I happen to be on their mailing list and received an email recently which contained the following:
The other example was Christine O’Donnell’s recent attack on my alma mater. The Senate candidate, fresh from her prior remarks about scientists engineering ultra-intelligent rats , has now decided that Yale is a bad thing. She tweeted:
Now, in fairness, she included a link to an article in the American Spectator which seemed to prompt her remark. That article didn’t criticize her opponent Chris Coon for going to Yale or for Yale values, but for his statement that he wants to bring the values of the Yale Divinity School to the Senate. That article is an attack piece, but like many attack pieces, it does have some truth to it and points out correctly that the Yale Divinity School is more left-wing than the general American population. That’s not the same thing as complaining about Yale in general. But, apparently to Christine O’Donnell, the problem as a whole is “Yale” values. According to O’Donnell, the values of one of the best universities on the planet are inherently bad values. It is difficult to imagine a more anti-intellectual stance short of book-burning. And yet, O’Donnell won the GOP primary for the U.S. Senate against Mike Castle. Castle is reasonable, well-educated and experienced. He has a law degree from Georgetown. He has demonstrated competence for over 20 years in political offices. And yet, he lost in the primary.
Anti-intellectualism is not at all limited to the GOP. It is becoming increasingly clear that the Obama administration actively interfered with government scientists telling the public how bad the the BP spill really was. Moreover, the Huffington Post, a mainstay of the liberal blogosphere, is filled with proponents of pseudoscience. And they have recently branched out from making absurd medical claims to denying evolution. But even these problems are small compared to the scale and pervasiveness of anti-intellectualism among the current conservative movement.
I don’t know what has happened to the GOP. At this point, the party seems to be engaging in a spiral of anti-intellectualism. There may be a positive feedback loop in that the more intellectuals grow disgusted with the Republican Party, the less incentive Republican candidates have to care about intellectuals. But that cannot be the whole story. Many people who are not intellectuals are not anti-intellectuals. So, I don’t understand what is happening to the Republican Party. But it disturbs me greatly. I’d like to be able to reasonably vote for multiple candidates including moderate Republicans. But as long as these trends continue, I will be forced to keep giving my money, my time, and my vote to Democratic candidates. And when I need to mark down what my politics are in a little box, I’ll answer “liberal” or “progressive” because in the United States right now, putting down anything else is becoming perilously close to writing “I support willful stupidity.”
However, over the last few months, I have become increasingly convinced that anti-intellectualism is not just a fringe phenomenon but a general trend of the modern conservative movement. The leaders of the movement are either ignorant, anti-intellectual buffoons, or they believe that their base is composed of ignorant, anti-intellectual buffoons.
Prior to coming to this conclusion, I had seen much evidence for this claim that did not convince me. The GSS data show that people who self-identify as liberal on average have larger vocabularies than those who self-identify as conservative. However, this did not convince me. Among other problems, the overall trends in that data are complicated. Vocabulary is correlated not just with increased liberalism but with general political extremism. That is, people more likely to identify with extreme political views are more likely to have a large vocabulary. Moreover, having a small vocab does not mean that one is anti-intellectual. It just means one is less likely to be intellectual.
Moreover, as I've discussed before, by some metrics of political knowledge, Democrats perform on average more poorly than Republicans.
However, the evidence for widespread anti-intellectualism among the American right-wing has now reached proportions which are difficult to deny. It is easy to dismiss Sarah Palin's comments about fruit flies as simple ignorance. And it is easy to dismiss Bobby Jindal's remarks about volcano monitoring as an isolated incident. However, these are not isolated incidences and one can point to many similar instances. Two of the most glaring that I've seen recently are remarks by the American Family Association's head Tim Wildmon and remarks made by Delaware senate candidate Christine O'Donnell.
The American Family Association is a right-wing Christian political group most known for organizing the boycotts that the right-wing periodically directs against companies that they have decided are too gay-friendly. I happen to be on their mailing list and received an email recently which contained the following:
A few months ago, AFA commissioned Christian songwriter/singer Eric Horner to write a moving patriotic song to honor our national motto, "In God We Trust."So much about this claim is strange that it is difficult to figure out where to start. The text contains outright lies. Youtube does not in general contact people for making popular videos to “find out what was going on!” It is conceivable that there is some threshold where such contact would occur. But that threshold is surely far beyond 20,000 views. In comparison for example, this video of Christopher Hitchens has around 35,000 hits and it is not the most popular such interview with Hitchens. Or to use a more amusing comparison, this extremely NSFW tribute video to Ray Bradbury has around a million views. 20,000 views is not much on Youtube. And anyone who gave minimal thought would realize this. My conclusion must be that the AFA lied . This is nothing less than political conmen fleecing a mark.
Without any fanfare, we posted it on YouTube. The response was so overwhelming that YouTube called us to find out what was going on!
The fact is, the video is patriotic and inspiring, and it shares the message of faith. People love it!
YouTube has told us that if we can get 20,000 people to watch the video, they will feature it on their front page. That means that the tens of millions of people who visit YouTube's website each day will be offered the opportunity to watch the video - a video with a Christian message!
(fonts and formatting as in in original)
The other example was Christine O’Donnell’s recent attack on my alma mater. The Senate candidate, fresh from her prior remarks about scientists engineering ultra-intelligent rats , has now decided that Yale is a bad thing. She tweeted:
My opponent wants to bring Yale values to US Senate. I want to bring liberty, limited government, fiscal sanity.
Now, in fairness, she included a link to an article in the American Spectator which seemed to prompt her remark. That article didn’t criticize her opponent Chris Coon for going to Yale or for Yale values, but for his statement that he wants to bring the values of the Yale Divinity School to the Senate. That article is an attack piece, but like many attack pieces, it does have some truth to it and points out correctly that the Yale Divinity School is more left-wing than the general American population. That’s not the same thing as complaining about Yale in general. But, apparently to Christine O’Donnell, the problem as a whole is “Yale” values. According to O’Donnell, the values of one of the best universities on the planet are inherently bad values. It is difficult to imagine a more anti-intellectual stance short of book-burning. And yet, O’Donnell won the GOP primary for the U.S. Senate against Mike Castle. Castle is reasonable, well-educated and experienced. He has a law degree from Georgetown. He has demonstrated competence for over 20 years in political offices. And yet, he lost in the primary.
Anti-intellectualism is not at all limited to the GOP. It is becoming increasingly clear that the Obama administration actively interfered with government scientists telling the public how bad the the BP spill really was. Moreover, the Huffington Post, a mainstay of the liberal blogosphere, is filled with proponents of pseudoscience. And they have recently branched out from making absurd medical claims to denying evolution. But even these problems are small compared to the scale and pervasiveness of anti-intellectualism among the current conservative movement.
I don’t know what has happened to the GOP. At this point, the party seems to be engaging in a spiral of anti-intellectualism. There may be a positive feedback loop in that the more intellectuals grow disgusted with the Republican Party, the less incentive Republican candidates have to care about intellectuals. But that cannot be the whole story. Many people who are not intellectuals are not anti-intellectuals. So, I don’t understand what is happening to the Republican Party. But it disturbs me greatly. I’d like to be able to reasonably vote for multiple candidates including moderate Republicans. But as long as these trends continue, I will be forced to keep giving my money, my time, and my vote to Democratic candidates. And when I need to mark down what my politics are in a little box, I’ll answer “liberal” or “progressive” because in the United States right now, putting down anything else is becoming perilously close to writing “I support willful stupidity.”
Thursday, September 2, 2010
Jack Chick and Civil Rights
There's a new Jack Chick tract up. This tract is entitled "Stinky" and is about an eponymous demon assigned to go up to the surface for Halloween. Stinky is supposed to find a gift for Satan so that a higher a ranking demon, #3, hopes to use to get a higher position in the diabolical hierarchy. Unfortunately, school has just resumed, so I don't have the time to go discuss the tract in detail. And honestly, as Chick tracts go, it is pretty mediocre. We don't even have the grand Catholic conspiracy appearing in this one. The plot-line is slightly less coherent than usual which also may lend to the mediocrity.
However, just when I thought that Jack Chick might be losing his did stand out. At one point, Stinky is trying to get past an angel so he can continue to tempt some humans. Stinky cries out "I demand my Civil Rights!" to which the angel responds "That doesn't work here, Stinky!" (eccentric formatting as in in the original). As far as I can tell, Jack Chick is attempting to make some sort of political argument here along the lines of "civil rights are a demonic concept." I don't know what to say to that.
However, just when I thought that Jack Chick might be losing his did stand out. At one point, Stinky is trying to get past an angel so he can continue to tempt some humans. Stinky cries out "I demand my Civil Rights!" to which the angel responds "That doesn't work here, Stinky!" (eccentric formatting as in in the original). As far as I can tell, Jack Chick is attempting to make some sort of political argument here along the lines of "civil rights are a demonic concept." I don't know what to say to that.
Tuesday, August 17, 2010
Conservapedia, P=NP, and the Fields Medal
Conservapedia, the right-wing, Christian alternative to Wikipedia has once again broken new ground. In previous entries we've discussed Conservapedia's founder Andrew Schlafly self-Godwining and Conservapedia's interesting take on Popperian falsifiability. Others have discussed Conservapedia's objections to special and general relativity. Now, Conservapedia is at it again.
Apparently, Conservapedia has taken a recent interest in mathematics. First, they added to their mainpage a take on the recent reports that Deolalikar had proven that P ≠ NP . Conservapedia announced:
Apparently, Aaronson's decision to quote Richard Dawkins meant that math must be evil. I'll leave it as an exercise to the reader to note some of the other more glaring errors. However, this was not the end of Conservapedia's attack on establishment math for being just too liberal. Shortly after this declaration, an addition about the Fields Medal went up. The Fields Medal, awarded every four years, is for mathematics roughly equivalent to a Noble Prize(there is no Nobel in math). Conservapedia announced a "Conservapedia exclusive":
It is not clear what "underachieving woman" Conservapedia is thinking of, but apparently the "communist-trained mathematician" is a reference to Ngo Bao Chau, who made headlines last year for proving the Fundamental Lemma of the Langlands program. Apparently Conservapedia believes that the general media cares so much about mathematics that a failure to speculate on who will win the Fields Medal is a sign of a media conspiracy. Why a "communist-trained" mathematician would be a big deal is not clear given thatabout half of the Fields Medal winners have been either from the USSR or trained in the USSR.
Also, apparently Conservapedia is unhappy that after Terence Tao got the Fields Medal four years ago he then endorsed Barack Obama for President, Finally, we come to the detail that forced me to write this blog entry. To deal with this apparent liberal bias and affirmative action in the Fields Medal, Conservapedia is starting its own award for mathematicians, the "ConservaMath Medal." According to the website this Medal is "the merit-based alternative to the Fields Medal, with winners announced at the same time so that real achievement is recognized." I'll let that speak for itself.
Apparently, Conservapedia has taken a recent interest in mathematics. First, they added to their mainpage a take on the recent reports that Deolalikar had proven that P ≠ NP . Conservapedia announced:
University professors pile on against a non-professor's claim to have solved one of the millennium problems. MIT Assistant Professor Scott Aaronson declares, "Vinay Deolalikar still hasn’t retracted his P≠NP claim, but a clear consensus has emerged that the proof, as it stands, is fatally flawed." He absurdly adds, with a cite to Richard Dawkins, "the only miracle in life is that there are no miracles, neither in mathematics nor in anything else."(internal links and fonts suppressed)
But the best of the public, aided by the internet, will inevitably solve more problems than liberal colleges will - just as Grigori Perelman solved another millennium problem. The future belongs to the conservative public.
Apparently, Aaronson's decision to quote Richard Dawkins meant that math must be evil. I'll leave it as an exercise to the reader to note some of the other more glaring errors. However, this was not the end of Conservapedia's attack on establishment math for being just too liberal. Shortly after this declaration, an addition about the Fields Medal went up. The Fields Medal, awarded every four years, is for mathematics roughly equivalent to a Noble Prize(there is no Nobel in math). Conservapedia announced a "Conservapedia exclusive":
[T]his Thursday liberals will likely give the coveted Fields Medal -- math's highest honor -- to an underachieving woman and to a communist-trained mathematician from Vietnam. Is the lamestream media holding back stories about this to create a bigger splash on Thursday?(internal links and fonts suppressed)
It is not clear what "underachieving woman" Conservapedia is thinking of, but apparently the "communist-trained mathematician" is a reference to Ngo Bao Chau, who made headlines last year for proving the Fundamental Lemma of the Langlands program. Apparently Conservapedia believes that the general media cares so much about mathematics that a failure to speculate on who will win the Fields Medal is a sign of a media conspiracy. Why a "communist-trained" mathematician would be a big deal is not clear given thatabout half of the Fields Medal winners have been either from the USSR or trained in the USSR.
Also, apparently Conservapedia is unhappy that after Terence Tao got the Fields Medal four years ago he then endorsed Barack Obama for President, Finally, we come to the detail that forced me to write this blog entry. To deal with this apparent liberal bias and affirmative action in the Fields Medal, Conservapedia is starting its own award for mathematicians, the "ConservaMath Medal." According to the website this Medal is "the merit-based alternative to the Fields Medal, with winners announced at the same time so that real achievement is recognized." I'll let that speak for itself.
Friday, August 6, 2010
Integer Complexity: An Upper Bound Update
It looks like I may have a final upper bound on the integer complexity problem. Recall we defined f(n) to be the minimum number of 1s needed to represent n as a product or sum of 1s. Thus, f(6)=5 because we can write 6=(1+1)(1+1+1). Harry Altman and I have been working on improving the lower and upper bounds on this function. Prior posts about this problem can be found under the label integer complexity. Harry has a large number of posts on the subject also, but they aren't all tagged so I'll just direct readers to the most recent one.
I'm writing this post to announce that we have a tight upper bound for f(n). Prior to our work, it was known that if n ≥ 2, we have 3log2 n ≥ f(n) ≥ 3log3 n. It is clear that the lower bound is hit infinitely often (whenever n is a power of 3). We (and by we I mean mostly Harry) have worked on understanding what n have close to the lowest possible value, and using this to understand other conjectures about f, such as the conjecture that if n=2^a*3^b then the most efficient representation for n is the obvious one. However, I've been also working on improving the upper bound, and whenever I've thought that I've been done improving the upper bound, I've then realized another way to tweak my proof to reduce the upper bound slightly. However, at this point, I'm confident that cannot happen anymore using the techniques in question, barring substantially new insight. Thus, I'm now announcing the new upper bound. We have for n ≥ 2, f(n) ≤ αlog2 n where α = 70/(log_2 102236160)= 2.630853... Note that this constant is just small enough for us to also state the weaker but prettier result that f(n) ≤ (19/5) log n. Unfortunately, the proof in question is very ugly, involving a large amount of case checking as well as nasty calculation of inequalities. I'm in the process of writing up the last bit. If readers are interested I can email people a draft of the proof when it is fully written.
I'm writing this post to announce that we have a tight upper bound for f(n). Prior to our work, it was known that if n ≥ 2, we have 3log2 n ≥ f(n) ≥ 3log3 n. It is clear that the lower bound is hit infinitely often (whenever n is a power of 3). We (and by we I mean mostly Harry) have worked on understanding what n have close to the lowest possible value, and using this to understand other conjectures about f, such as the conjecture that if n=2^a*3^b then the most efficient representation for n is the obvious one. However, I've been also working on improving the upper bound, and whenever I've thought that I've been done improving the upper bound, I've then realized another way to tweak my proof to reduce the upper bound slightly. However, at this point, I'm confident that cannot happen anymore using the techniques in question, barring substantially new insight. Thus, I'm now announcing the new upper bound. We have for n ≥ 2, f(n) ≤ αlog2 n where α = 70/(log_2 102236160)= 2.630853... Note that this constant is just small enough for us to also state the weaker but prettier result that f(n) ≤ (19/5) log n. Unfortunately, the proof in question is very ugly, involving a large amount of case checking as well as nasty calculation of inequalities. I'm in the process of writing up the last bit. If readers are interested I can email people a draft of the proof when it is fully written.
Subscribe to:
Comments (Atom)
