A function f(n) with range in the integers is said to be multiplicative if f(ab)=f(a)f(b) whenever a and b are relatively prime. A function is said to be completely multiplicative if this applies for all a and b whether or not a and b are relatively prime. Some of my prior posts have discussed multiplicative functions (e.g. this post on Mersenne primes and perfect numbers). An important thing to note is that a multiplicative function is determined by its values at prime powers, while a completely multiplicative function is determined by its values at primes.
Let f and g be two functions from the positive integers to the integers. Let K(f,g)(x) be the number of n less than x such that f(n) does not divide g(n). We say that f almost always divides g if f(n) divides g(n) for almost all n if K(f,g)(x) = o(x) (or to put it another way, the set of exceptions has density zero). Obviously, almost divisibility is weaker than divisibility. Almost divisibility shares many of the properties of divisibility; it is a transitive relation. One important difference between divisibility and almost divisibility is that among positive functions, divisibility is an anti-symmetric relation (that is, if fg, and gf then f=g). This is not true for almost divisibility, but is true for almost divisibility if we restrict our attention to multiplicative functions.
Are there interesting examples of non-trivial almost divisibility where one doesn't have divisibility? Yes. Let σ(n) be the sum of the positive divisors of n, and let τ(n) be the number of positive divisors of n. Erdos showed that τ almost divides σ and showed through similar logic that τ almost divides the Euler φ function.
Why am I thinking about these functions? I noticed that some multiplicative functions divide a positive completely multiplicative function. Thus, for example φ divides the completely multiplicative f such that for any prime p, f(p)=p(p-1). However, other cases don't have such examples. It isn't too hard to show that there's no such completely multiplicative function for τ or σ. But, the situation changes drastically in the case of almost divisibility. We in fact have that for any function f from the natural number to the natural numbers, there is a completely multiplicative function g such that f almost divides g. We shall call such a g, a "completely multiplicative almost multiple" (or CMAM) Moreover, we can construct such a function explicitly.
Proposition: Let f(n) be a function from N to N, and let g(n) be the completely multiplicative function defined by g(p) = is the least common multiple of f(1),f(2),f(3)...f(2^p). Then g is a completely multiplicative almost multiple of f.
In order to prove this result, we need another notion, that of normal order. Often, to describe the behavior of a poorly behaved function, number theorists like to show that it asymptotic to a well behaved function in the sense that the limit of their ratios is 1. Thus, for example, the prime number theorem says that the poorly behaved function π(n) that counts the number of primes that are at most n is asymptotic to n/log n. But for many functions we care about this sort of statement is much too strong. Let ω(n) be the number of distinct prime divisors of n, and let Ω(n) be the total number of prime factors. Then these functions hit 1 infinitely often and so clearly are not asymptotic to any nice functions. However, ω and Ω possess a normal order, in the sense that for any ε > 0, we have that excepting a set of density 0, (1-ε) log log n < ω(n) < (1+ε) log log n and the same holds for Ω(n). (More refined estimates exist but we don't need them for these remarks.) An important heuristic that stems from this result is that most numbers look very close to square free in the sense that they have a lot of prime factors and very few of the prime factors occur more than once.
Now, to prove our result, we observe that excepting a set of density zero, we have Ω(n) < 2 log log n . So ignoring a set of density zero, the largest prime factor of n, call it p, is at least n^(1/(2 log log n)), and so f(n) appears as a term in g(p).
Obviously, for any f, the constructed CMAM grows very fast. However, this doesn't need to be the case, and it might be that in specific contexts one can substantially reduce this growth rate. So, this leaves me with three questions, one broad and two specific:
1) How much can we improve on reducing the growth rate of the CMAM for a function f? There are obvious steps to take in the construction but they still make it grow very quickly.
2) Let g(n) be a CMAM for σ. Does it follow that g(n) is not O(n^k) for any k? I strongly suspect this is true but don't see how to prove it.
3) One can through more careful work show that there is a CMAM for τ that is 0(x). What is the best possible growth rate for such a function?
While I've seen individual papers that have discussed specific functions being almost divisible by each other, I'm not aware any substantial work on their general structure. I don't know if this is due to my ignorance of the literature, the fault of almost divisibility being too weak a property to be interesting, or some other cause.
Tuesday, May 17, 2011
Monday, May 2, 2011
Osama bin Laden's death, Barack Obama, and the Judeo-Christian heritage of the United States: A response to my brother
In the last 24 hours, everyone has been talking about Osama bin Laden's death. Last night, after the President's speech, people around the country celebrated. Here in Boston, this apparently became another excuse for college students to get drunk. I heard well into the night screams of "USA! USA!" and at one point an attempt to sing "America, Fuck Yeah!" from Team America.
There were more measured responses including a piece in the Huffington Post by my brother Nathaniel. Unfortunately, many of the more measured responses, including this one, are misguided.
Nathaniel listed five aspects of the President's remarks which in his view stood out:
I don't have any significant problems with the second or fifth points, but the other three are problematic.
In his first point, Nathaniel portrays as a positive that which isn't. He also confuses a variety of different issues. Targeting high ranking terrorists and killing them is distinct from whether or not people such as Khalid Shaikh Mohammed should get civilian trials once they are in our custody. But we can direct our military against targets at the same time that we use civllian trials for those who are captured. The first World Trade Center bombers and the Oklahoma City bombers were both tried successfully in civil courts. Being at war does not mean we need to ignore due process.
Nathaniel's third point is deeply wrong. The Judeo-Christian heritage of this country is deeply exaggerated. The US Constitution bears almost no signs of Judeo-Christian values. There are only three signs of such religious influence in the Constitution, and all are comparatively minor. First, laws presented to the President become law in 10 days after presentation to the President, excepting Sundays. Second, the treason clause requires two witnesses of an overt act to convict or a confession in court. This requirement echos the Old Testament rule for conviction of severe crimes which requires testimony of two witnesses. Third, the Constitution is dated "in the year of our Lord," a conventional phrase at the time.
Moreover, Nathaniel's statement portrays the Judeo-Christian heritage in the worst light possible. I'm proud of my Jewish heritage. But there's something deeply wrong when that heritage's primary lesson is an endorsement of retributive justice. It is noteworthy that the most substantial impact of the Bible on the text of the Constitution is to make conviction and punishment more difficult, not to endorse retribution. Indeed, it is a common theme in that heritage that we understand that even our enemies are people who can suffer. At the Passover Seder, even as the deliverance of the Israelites is celebrated, we remove a drop of wine from the cup for each of the Ten Plagues, remembering the Egyptian suffering.
Most troubling of all is the notion that mentioning God constitutes a " welcome shift from the president who went out of his way to give a nod to "non-believers" in his inaugural address." Approximately 10% of the people in United States self-identify as having no religion, and about 2% of the U.S. population identifies as either atheist or agnostic. Non-believers in the US have ranged from Carl Sagan to George Clooney, from Neil deGrasse Tyson to Bill Gates. Non-believers are an important part of the United States, intertwined with everything that makes America a great nation. All citizens deserve the same respect, whether they are differ by skin color, politics, or religious beliefs.
Nathaniel's fourth point is also misguided. The building of the transcontinental railroad was a triumph of the American spirit. And yes, America is really in a decline when our example of "we can do everything" is to kill our enemies. We are the only nation that has ever sent people to the moon. Yet, no one has walked on the moon in forty years. The shuttle will soon no longer be operational, and the US will need to rely on Russia for space flight. We are in a decline. No speech can hide that. And pretending otherwise is not a good thing. We must fight that decline. But we cannot fight it if we do not acknowledge the threat.
Nathaniel ended his piece by saying that he was proud of the troops and proud of the President. I can understand being proud of the troops. They risked their lives. We should be thankful to those soldiers who put their lives on the line to protect what we hold dear. This is not a good reason to be proud of the Ppresident. Nothing he did substantially impacted this result. It is possible that actual policy changes by Obama somehow lead to these events by making it easier to track down Osama. But I've seen no indication of that. Let's not give him credit that he isn't due. It is likely that in the 2012 election I will vote to reelect Obama, but that has almost nothing to do with these events, and it shouldn't. Instead of responding to these recent events, we should all vote for whichever candidate we think will be the most competent President with the policies that are best.
There were more measured responses including a piece in the Huffington Post by my brother Nathaniel. Unfortunately, many of the more measured responses, including this one, are misguided.
Nathaniel listed five aspects of the President's remarks which in his view stood out:
1) Obama emphasized that America and al Qaeda are "at war." This is an important shift from the president who wanted to try Khalid Shaikh Mohammed in a civilian court in New York, and who vowed to shut down Gitmo during the 2008 campaign. The tone here tonight was clear: The terrorists who plot against the United States are (illegal) combatants who deserve the full front of our military fury, not our legal rights.
2) The president gave a nod to his predecessor, in an acknowledgment that America has never been, nor ever will be, at war with Islam. This took class and grace, and Obama merits credit for it.
3) This speech was traditional. From the inclusion of "under God" in his closing remarks, to the references to retributive "justice," Obama channeled the Judeo-Christian values that still define our nation -- again, a welcome shift from the president who went out of his way to give a nod to "non-believers" in his inaugural address.
4) Somehow, Obama managed to take this moment to combat feelings of American declinism. The memo: We can do anything we set out to do. Compare this simple yet effective message to his recent flop of a State of the Union speech, in which the example of our greatness was the fact that "America is the nation that built the transcontinental railroad." This moment disproves those who sing the song of the "fall of American Empire," resolve, and spirit.
5) Most importantly, Obama tonight reaffirmed America's role as a force for good in the world, a force that extends beyond our borders. After U.S. troops took a backseat in NATO operations against Muammar Gaddafi, many (including me) worried that our will to "oppose any foe" in the defense of liberty played second fiddle to the whims of the UN, EU, and the Arab League. Thankfully and surprisingly, Obama reaffirmed our commitment to be a "shining beacon on a hill" to light the world.
I don't have any significant problems with the second or fifth points, but the other three are problematic.
In his first point, Nathaniel portrays as a positive that which isn't. He also confuses a variety of different issues. Targeting high ranking terrorists and killing them is distinct from whether or not people such as Khalid Shaikh Mohammed should get civilian trials once they are in our custody. But we can direct our military against targets at the same time that we use civllian trials for those who are captured. The first World Trade Center bombers and the Oklahoma City bombers were both tried successfully in civil courts. Being at war does not mean we need to ignore due process.
Nathaniel's third point is deeply wrong. The Judeo-Christian heritage of this country is deeply exaggerated. The US Constitution bears almost no signs of Judeo-Christian values. There are only three signs of such religious influence in the Constitution, and all are comparatively minor. First, laws presented to the President become law in 10 days after presentation to the President, excepting Sundays. Second, the treason clause requires two witnesses of an overt act to convict or a confession in court. This requirement echos the Old Testament rule for conviction of severe crimes which requires testimony of two witnesses. Third, the Constitution is dated "in the year of our Lord," a conventional phrase at the time.
Moreover, Nathaniel's statement portrays the Judeo-Christian heritage in the worst light possible. I'm proud of my Jewish heritage. But there's something deeply wrong when that heritage's primary lesson is an endorsement of retributive justice. It is noteworthy that the most substantial impact of the Bible on the text of the Constitution is to make conviction and punishment more difficult, not to endorse retribution. Indeed, it is a common theme in that heritage that we understand that even our enemies are people who can suffer. At the Passover Seder, even as the deliverance of the Israelites is celebrated, we remove a drop of wine from the cup for each of the Ten Plagues, remembering the Egyptian suffering.
Most troubling of all is the notion that mentioning God constitutes a " welcome shift from the president who went out of his way to give a nod to "non-believers" in his inaugural address." Approximately 10% of the people in United States self-identify as having no religion, and about 2% of the U.S. population identifies as either atheist or agnostic. Non-believers in the US have ranged from Carl Sagan to George Clooney, from Neil deGrasse Tyson to Bill Gates. Non-believers are an important part of the United States, intertwined with everything that makes America a great nation. All citizens deserve the same respect, whether they are differ by skin color, politics, or religious beliefs.
Nathaniel's fourth point is also misguided. The building of the transcontinental railroad was a triumph of the American spirit. And yes, America is really in a decline when our example of "we can do everything" is to kill our enemies. We are the only nation that has ever sent people to the moon. Yet, no one has walked on the moon in forty years. The shuttle will soon no longer be operational, and the US will need to rely on Russia for space flight. We are in a decline. No speech can hide that. And pretending otherwise is not a good thing. We must fight that decline. But we cannot fight it if we do not acknowledge the threat.
Nathaniel ended his piece by saying that he was proud of the troops and proud of the President. I can understand being proud of the troops. They risked their lives. We should be thankful to those soldiers who put their lives on the line to protect what we hold dear. This is not a good reason to be proud of the Ppresident. Nothing he did substantially impacted this result. It is possible that actual policy changes by Obama somehow lead to these events by making it easier to track down Osama. But I've seen no indication of that. Let's not give him credit that he isn't due. It is likely that in the 2012 election I will vote to reelect Obama, but that has almost nothing to do with these events, and it shouldn't. Instead of responding to these recent events, we should all vote for whichever candidate we think will be the most competent President with the policies that are best.
Saturday, April 2, 2011
Political Affiliation and Scientific Knowledge Levels
Previously on this blog, I've talked about differing intelligence levels and knowledge levels of different political groups. I've also decried the extreme anti-science and anti-intellectual views that have been articulated by conservative spokespersons in the United States. Thus, I was interested in some recent work which suggests that by some metrics the political right is more pro-science than the left. Audacious Epigone showed that GSS data demonstrated that, on average, Republicans are more pro-science and scientifically literate than Democrats. Epigone made no effort to control for variables, such as income, education and race.
Epigone's statement prompted Razib Kahn to do a similar, more detailed analysis focusing on science knowledge and attitudes. Kahn organized the data by political self-identification along the conservative-liberal continuum rather than by an individual's party affiliation. Razib's analysis suggests shows that conservatives and liberals are almost indistinguishable in overall knowledge level. However, when one removes the questions that discuss specific pet issues of the modern right-wing (i.e. those related to evolution and the age of the Earth), conservatives arguably pull ahead slightly in terms of scientific knowledge. At the same time, the data shows that moderates are less scientifically literate and less science-friendly than both conservatives and liberals. On all of the 19 variables that Razib examined, political moderates never come out on top. That is, for each question, sometimes conservatives perform best, and sometimes liberals, but never moderates. This is consistent with other results that show that in general moderates are less intelligent and less educated than other groups. For example, moderates have lower vocabulary scores than the general population. Razib performed additional analysis to try to control for other variables and his piece is worth reading.
This data suggests that there is a significant and underappreciated disconnect between right-wing leaders and self-identifying conservatives. If the individuals on the right right aren't statistically distinguishable from the left when it comes to science issues why do so many conservative politicans go out of their way to make anti-science remarks? There are a variety of possible explanations, but none of them are satisfactory.
First, many of these anti-science comments have been directed towards biology and matters related to biology (e.g. John McCain's remarks about bear DNA and Sarah Palin's remark about fruit fly research.) It is possible that the religious right's negative attitude towards evolution is carrying over to biology as a whole. However, this doesn't explain the remarks about other scientific areas (such as Bobby Jindal's remark about volcano monitoring). Moreover, although the human evolution question is by far the one with the most extreme difference between liberals and conservatives, (The percentages that accept human evolution according to the GSS are 69% for liberals, 52% for moderates and 39% for conservatives), the other questions suggest that conservative attitudes about evolution have not spread to other areas of biology. For example, when asked if the statement "Antibiotics kill viruses as well as bacteria" is true or false such as understanding that antibiotics cannot harm viruses 60% of liberals answered correctly while 63% of conservatives answered correctly. (The statement is false.)
Second, right-wing leaders may understand that most conservatives are not anti-science but think that the more active conservatve base is heavily anti-science. Without more data, it is hard to test if the active conservative base is substantially more anti-science than rank-and-file conservatives. Eeven if this is the case, it seems there are more effective ways of energizing the conservative base than anti-science rhetoric.
Third, right-wing politicians mayhave erroneously bought into the false stereotypes about their own constituents. Given the prevalence of such stereotypes, this seems most likely. This hypothesis is also difficult to test because politicians aren't going to admit that they've been pandering to rubes. Unfortunately, this belief among the right-wing leaders that conservatives are anti-science could easily act as a self-fulfilling prophecy if it causes pro-science conservatives to either stop being conservative, or causes some conservatives to become more anti-science to fit their tribal allegiance. However, this possibility does have a bright side: If this explanation is correct, then right-wing politicians are likely to be more pro-science in practice than they appear to be in public. Moreover, conservative politicians may act pro-science if they can be convinced that their constituents really aren't as anti-science as the politicians them to be.
Epigone's statement prompted Razib Kahn to do a similar, more detailed analysis focusing on science knowledge and attitudes. Kahn organized the data by political self-identification along the conservative-liberal continuum rather than by an individual's party affiliation. Razib's analysis suggests shows that conservatives and liberals are almost indistinguishable in overall knowledge level. However, when one removes the questions that discuss specific pet issues of the modern right-wing (i.e. those related to evolution and the age of the Earth), conservatives arguably pull ahead slightly in terms of scientific knowledge. At the same time, the data shows that moderates are less scientifically literate and less science-friendly than both conservatives and liberals. On all of the 19 variables that Razib examined, political moderates never come out on top. That is, for each question, sometimes conservatives perform best, and sometimes liberals, but never moderates. This is consistent with other results that show that in general moderates are less intelligent and less educated than other groups. For example, moderates have lower vocabulary scores than the general population. Razib performed additional analysis to try to control for other variables and his piece is worth reading.
This data suggests that there is a significant and underappreciated disconnect between right-wing leaders and self-identifying conservatives. If the individuals on the right right aren't statistically distinguishable from the left when it comes to science issues why do so many conservative politicans go out of their way to make anti-science remarks? There are a variety of possible explanations, but none of them are satisfactory.
First, many of these anti-science comments have been directed towards biology and matters related to biology (e.g. John McCain's remarks about bear DNA and Sarah Palin's remark about fruit fly research.) It is possible that the religious right's negative attitude towards evolution is carrying over to biology as a whole. However, this doesn't explain the remarks about other scientific areas (such as Bobby Jindal's remark about volcano monitoring). Moreover, although the human evolution question is by far the one with the most extreme difference between liberals and conservatives, (The percentages that accept human evolution according to the GSS are 69% for liberals, 52% for moderates and 39% for conservatives), the other questions suggest that conservative attitudes about evolution have not spread to other areas of biology. For example, when asked if the statement "Antibiotics kill viruses as well as bacteria" is true or false such as understanding that antibiotics cannot harm viruses 60% of liberals answered correctly while 63% of conservatives answered correctly. (The statement is false.)
Second, right-wing leaders may understand that most conservatives are not anti-science but think that the more active conservatve base is heavily anti-science. Without more data, it is hard to test if the active conservative base is substantially more anti-science than rank-and-file conservatives. Eeven if this is the case, it seems there are more effective ways of energizing the conservative base than anti-science rhetoric.
Third, right-wing politicians mayhave erroneously bought into the false stereotypes about their own constituents. Given the prevalence of such stereotypes, this seems most likely. This hypothesis is also difficult to test because politicians aren't going to admit that they've been pandering to rubes. Unfortunately, this belief among the right-wing leaders that conservatives are anti-science could easily act as a self-fulfilling prophecy if it causes pro-science conservatives to either stop being conservative, or causes some conservatives to become more anti-science to fit their tribal allegiance. However, this possibility does have a bright side: If this explanation is correct, then right-wing politicians are likely to be more pro-science in practice than they appear to be in public. Moreover, conservative politicians may act pro-science if they can be convinced that their constituents really aren't as anti-science as the politicians them to be.
Wednesday, March 9, 2011
Illinois and the Death Penalty
Governor Pat Quinn of Illinois just signed a law which abolishes the death penalty in that state. This legislation is to some extent symbolic, in that Illinois has had a moratorium on the death penalty for almost a decade. With this legislation, Illinois joins New Mexico and New Jersey as states that have recently abolished the death penalty.
I have discussed here before my attitude towards the death penalty, especially in regards to specific cases such as the ongoing case of Hank Skinner . I'm not intrinsically against the death penalty: societies have the right in general to execute those who have violated the social contract in particularly heinous fashions if that will assist the public good. However, the death penalty as practiced in the United States is capricious and disorderly. Prosecutors push for death sentences when it is politically convenient, and their are huge racial disparities in who is executed.
It is also clear that innocent people have been wrongly convicted, and that in at least some cases, innocent people have been executed such as Cameron Todd Willinngham. Evidentiary standards that allow junk science and superficially persuasive eyewitness testimony are leaving blood on all our hands. For a long time, I have found striking a certain section in the Biblical book of my namesake, Joshua. the Israelites are called thieves when one man steals at Jericho. How much worse are we as a society that we as a democracy repeatedly elect and reelect officials who kill innocent people in our name?
In the particular case of the state of Illinois, I have additional personal reason to be interested in this legislation. My uncle Seymour Simon was a justice on the Illinois Supreme Court. He was a staunch opponent of the death penalty and argued forcefully and unceasingly that the death penalty as implemented in Illinois was unconsciousable and unconstitutionally capricious. He died in 2006, and so did not live to see this legislation. I suspect I know how he would have responded if he had seen this: Illinois is down. Only thirty-five more states to go.
I have discussed here before my attitude towards the death penalty, especially in regards to specific cases such as the ongoing case of Hank Skinner . I'm not intrinsically against the death penalty: societies have the right in general to execute those who have violated the social contract in particularly heinous fashions if that will assist the public good. However, the death penalty as practiced in the United States is capricious and disorderly. Prosecutors push for death sentences when it is politically convenient, and their are huge racial disparities in who is executed.
It is also clear that innocent people have been wrongly convicted, and that in at least some cases, innocent people have been executed such as Cameron Todd Willinngham. Evidentiary standards that allow junk science and superficially persuasive eyewitness testimony are leaving blood on all our hands. For a long time, I have found striking a certain section in the Biblical book of my namesake, Joshua. the Israelites are called thieves when one man steals at Jericho. How much worse are we as a society that we as a democracy repeatedly elect and reelect officials who kill innocent people in our name?
In the particular case of the state of Illinois, I have additional personal reason to be interested in this legislation. My uncle Seymour Simon was a justice on the Illinois Supreme Court. He was a staunch opponent of the death penalty and argued forcefully and unceasingly that the death penalty as implemented in Illinois was unconsciousable and unconstitutionally capricious. He died in 2006, and so did not live to see this legislation. I suspect I know how he would have responded if he had seen this: Illinois is down. Only thirty-five more states to go.
Thursday, February 24, 2011
Space Shuttle Discovery launch
I'm currently in Florida with friends watching the launch of Discovery. This will be the second to last shuttle launch (there's a small chance that this will be the third to last launch if they get enough funding for a final launch). I and Tyrol5 wrote an article for Wikinews about the launch which includes a picture taken by me of the launch. There are additional photos on my Facebook page.
Monday, February 14, 2011
On rotations of spheres
Consider a sphere living in some number of dimensions. The sphere we are used to is the 2-sphere, denoted by S2. Mathematicians call it this because even though it lives in 3-dimensions, it is itself a 2-dimensional object (in the sense that small sections of it look like the 2-dimensional plane). One can similarly talk about the n-sphere, which lives in n+1 dimensions. Thus for example, the 1-sphere, S1, can be thought of all all points on a plane that are of distance one from the origin.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
Wednesday, February 2, 2011
Obama's State of the Union Address: A Response to Nathaniel
President Obama delivered his State of the Union address last week. My younger brother has a piece in the Yale Daily News which extensively criticizes the speech. Nathaniel complains that the speech started with an inspirational message but "soon fell into a quagmire of policy." Nathaniel criticizes the speech for focusing too much on policy minutia and argues that, while referencing Kennedy, Obama fails to understand that inspiration requires large goals, not policy details. Overall, Nathaniel is correct that this was a disappointing speech. However, it was not disappointing because it went into detail: it was disappointing because the details were unimpressive.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
Subscribe to:
Posts (Atom)
