Calculation of the Fourier Transform of a tent map, with a calculus tip….

I’ve been following these excellent lectures by Professor Brad Osgood of Stanford. As an aside: yes, he is dynamite in the classroom, but there is probably a reason that Stanford is featuring him. 🙂

And yes, his style is good for obtaining a feeling of comradery that is absent in my classroom; at least in the lower division “service” classes.

This lecture takes us from Fourier Series to Fourier Transforms. Of course, he admits that the transition here is really a heuristic trick with symbolism; it isn’t a bad way to initiate an intuitive feel for the subject though.

However, the point of this post is to offer a “algebra of calculus trick” for dealing with the sort of calculations that one might encounter.

By the way, if you say “hey, just use a calculator” you will be BANNED from this blog!!!! (just kidding…sort of. 🙂 )

So here is the deal: let represent the tent map: the support of is and it has the following graph:

The formula is:

So, the Fourier Transform is

Now, this is an easy integral to do, conceptually, but there is the issue of carrying constants around and being tempted to make “on the fly” simplifications along the way, thereby leading to irritating algebraic errors.

So my tip: just let and do the integrals:

and substitute and simplify later:

Now the integrals become:
These are easy to do; the first is merely and the next two have the same anti-derivative which can be obtained by a “integration by parts” calculation: ; evaluating the limits yields:

Add the first integral and simplify and we get: . NOW use and we have the integral is by Euler’s formula.

Now we need some trig to get this into a form that is “engineering/scientist” friendly; here we turn to the formula: so so our answer is which is often denoted as as the “normalized” function is given by (as we want the function to have zeros at integers and to “equal” one at (remember that famous limit!)

So, the point is that using made the algebra a whole lot easier.

Now, if you are shaking your head and muttering about how this calculation was crude that that one usually uses “convolution” instead: this post is probably too elementary for you. 🙂

A bit more prior to admin BS

One thing that surprised me about the professor’s job (at a non-research intensive school; we have a modest but real research requirement, but mostly we teach): I never knew how much time I’d spend doing tasks that have nothing to do with teaching and scholarship. Groan….how much of this do I tell our applicants that arrive on campus to interview? 🙂

But there is something mathematical that I want to talk about; it is a follow up to this post. It has to do with what string theorist tell us: . Needless to say, they are using a non-standard definition of “value of a series”.

Where I think the problem is: when we hear “series” we think of something related to the usual process of addition. Clearly, this non-standard assignment doesn’t related to addition in the way we usually think about it.

So, it might make more sense to think of a “generalized series” as a map from the set of sequences of real numbers (or: the infinite dimensional real vector space) to the real numbers; the usual “limit of partial sums” definition has some nice properties with respect to sequence addition, scalar multiplication and with respect to a “shift operation” and addition, provided we restrict ourselves to a suitable collection of sequences (say, those whose traditional sum of components are absolutely convergent).

So, this “non-standard sum” can be thought of as a map where . That is a bit less offensive than calling it a “sum”. 🙂

Nate Silver’s Book: The signal and the noise: why so many predictions fail but some don’t

Reposted from my personal blog and from my Daily Kos Diary:

Quick Review
Excellent book. There are a few tiny technical errors (e. g., “non-linear” functions include exponential functions, but not all non-linear phenomena are exponential (e. g. power, root, logarithmic, etc.).
Also, experts have some (justified) quibbles with the book; you can read some of these concerning his chapter on climate change here and some on his discussion of hypothesis testing here.

But, aside from these, it is right on. Anyone who follows the news closely will benefit from it; I especially recommend it to those who closely follow science and politics and even sports.

It is well written and is designed for adults; it makes some (but reasonable) demands on the reader. The scientist, mathematician or engineer can read this at the end of the day but the less technically inclined will probably have to be wide awake while reading this.

Details
Silver sets you up by showing examples of failed predictions; perhaps the worst of the lot was the economic collapse in the United States prior to the 2008 general elections. Much of this was due to the collapse of the real estate market and falling house/property values. Real estate was badly overvalued, and financial firms made packages of investments whose soundness was based on many mortgages NOT defaulting at the same time; it was determined that the risk of that happening was astronomically small. That was wrong of course; one reason is that the risk of such an event is NOT described by the “normal” (bell shaped) distribution but rather by one that allows for failure with a higher degree of probability.

There were more things going on, of course; and many of these things were difficult to model accurately just due to complexity. Too many factors makes a model unusable; too few means that the model is worthless.

Silver also talks about models providing probabilistic outcomes: example saying that the GDP will be X in year Y is unrealistic; what we really should say that the probability of the GDP being X plus/minus “E” is Z percent.

Next Silver takes on pundits. In general: they don’t predict well; they are more about entertainment than anything else. Example: look at the outcome of the 2012 election; the nerds were right; the pundits (be they NPR or Fox News pundits) were wrong. NPR called the election “razor tight” (it wasn’t); Fox called it for the wrong guy. The data was clear and the sports books new this, but that doesn’t sell well, does it?

Now Silver looks at baseball. Of course there are a ton of statistics here; I am a bit sorry he didn’t introduce Bayesian analysis in this chapter though he may have been setting you up for it.

Topics include: what does raw data tell you about a player’s prospects? What role does a talent scout’s input have toward making the prediction? How does a baseball players hitting vary with age, and why is this hard to measure from the data?

The next two chapters deal with predictions: earthquakes and weather. Bottom line: we have statistical data on weather and on earthquakes, but in terms of making “tomorrow’s prediction”, we are much, much, much further along in weather than we are on earthquakes. In terms of earthquakes, we can say stuff like “region Y has a X percent chance of an earthquake of magnitude Z within the next 35 years” but that is about it. On the other hand, we are much better about, say, making forecasts of the path of a hurricane, though these are probabilistic:

In terms of weather: we have many more measurements.

But there IS the following: weather is a chaotic system; a small change in initial conditions can mean to a large change in long term outcomes. Example: one can measure a temperature at time t, but only to a certain degree of precision. The same holds for pressure, wind vectors, etc. Small perturbations can lead to very different outcomes. Solutions aren’t stable with respect to initial conditions.

You can see this easily: try to balance a pen on its tip. Physics tells us there is a precise position at which the pen is at equilibrium, even on its tip. But that equilibrium is so unstable that a small vibration of the table or even small movement of air in the room is enough to upset it.

In fact, some gambling depends on this. For example, consider a coin toss. A coin toss is governed by Newton’s laws for classical mechanics, and in principle, if you could get precise initial conditions and environmental conditions, the outcome shouldn’t be random. But it is…for practical purposes. The same holds for rolling dice.

Now what about dispensing with models and just predicting based on data alone (not regarding physical laws and relationships)? One big problem: data is noisy and is prone to be “overfitted” by a curve (or surface) that exactly matches prior data but is of no predictive value. Think of it this way: if you have n data points in the plane, there is a polynomial of degree n-1 that will fit the data EXACTLY, but in most cases have a very “wiggly” graph that provides no predictive value.

Of course that is overfitting in the extreme. Hence, most use the science of the situation to posit the type of curve that “should” provide a rough fit and then use some mathematical procedure (e. g. “least squares”) to find the “best” curve that fits.

The book goes into many more examples: example: the flu epidemic. Here one finds the old tug between models that are too simplistic to be useful for forecasting and too complicated to be used.

There are interesting sections on poker and chess and the role of probability is discussed as well as the role of machines. The poker chapter is interesting; Silver describes his experience as a poker player. He made a lot of money when poker drew lots of rookies who had money to spend; he didn’t do as well when those “bad” players left and only the most dedicated ones remained. One saw that really bad players lost more money than the best players won (not that hard to understand). He also talked about how hard it was to tell if someone was really good or merely lucky; sometimes this wasn’t perfectly clear after a few months.

Later, Silver discusses climate change and why the vast majority of scientists see it as being real and caused (or made substantially worse) by human activity. He also talks about terrorism and enemy sneak attacks; sometimes there IS a signal out there but it isn’t detected because we don’t realize that there IS a signal to detect.

However the best part of the book (and it is all pretty good, IMHO), is his discussion of Bayes law and Bayesian versus frequentist statistics. I’ve talked about this.

I’ll demonstrate Bayesian reasoning in a couple of examples, and then talk about Bayesian versus frequentist statistical testing.

Example one: back in 1999, I went to the doctor with chest pains. The doctor, based on my symptoms and my current activity level (I still swam and ran long distances with no difficulty) said it was reflux and prescribed prescription antacids. He told me this about a possible stress test: “I could stress test you but the probability of any positive being a false positive is so high, we’d learn nothing from the test”.

Example two: suppose you are testing for a drug that is not widely used; say 5 percent of the population uses it. You have a test that is 95 percent accurate in the following sense: if the person is really using the drug, it will show positive 95 percent of the time, and if the person is NOT using the drug, it will show positive only 5 percent of the time (false positive).

So now you test 2000 people for the drug. If Bob tests positive, what is the probability that he is a drug user?

Answer: There are 100 actual drug users in this population, so you’d expect 100*.95 = 95 true positives. There are 1900 non-users and 1900*.05 = 95 false positives. So there are as many false positives as true positives! The odds that someone who tests positive is really a user is 50 percent.

Now how does this apply to “hypothesis testing”?

Consider basketball. You know that a given player took 10 free shots and made 4. You wonder: what is the probability that this player is a competent free throw shooter (given competence is defined to be, say, 70 percent).

If you just go by the numbers that you see (true: n = 10 is a pathetically small sample; in real life you’d never infer anything), well, the test would be: given the probability of making a free shot is 70 percent, what is the probability that you’d see 4 (or fewer) made free shots out of 10?

Using a calculator (binomial probability calculator), we’d say there is a 4.7 percent chance we’d see 4 or fewer free shots made if the person shooting the shots was a 70 percent shooter. That is the “frequentist” way.

But suppose you found out one of the following:
1. The shooter was me (I played one season in junior high and some pick up ball many years ago…infrequently) or
2. The shooter was an NBA player.

If 1 was true, you’d believe the result or POSSIBLY say “maybe he had a good day”.
If 2 was true, then you’d say “unless this player was chosen from one of the all time worst NBA free throw shooters, he probably just had a bad day”.

Bayesian hypothesis testing gives us a way to make and informed guess. We’d ask: what is the probability that the hypothesis is true given the data that we see (asking the reverse of what the frequentist asks). But to do this, we’d have to guess: if this person is an NBA player, what is the probability, PRIOR to this 4 for 10 shooting, that this person was 70 percent or better (NBA average is about 75 percent). For the sake of argument, assume that there is a 60 percent chance that this person came from the 70 percent or better category (one could do this by seeing the percentage of NBA players shooing 70 percent of better). Assign a “bad” percentage as 50 percent (based on the worst NBA free throw shooters): (the probability of 4 or fewer made free throws out of 10 given a 50 percent free throw shooter is .377)

Then we’d use Bayes law: (.0473*.6)/(.0473*.6 + .377*.4) = .158. So it IS possible that we are seeing a decent free throw shooter having a bad day.

This has profound implications in science. For example, if one is trying to study genes versus the propensity for a given disease, there are a LOT of genes. Say one tests 1000 genes of those who had a certain type of cancer and run a study. If we accept p = .05 (5 percent) chance of having a false positive, we are likely to have 50 false positives out of this study. So, given a positive correlation between a given allele and this disease, what is the probability that this is a false positive? That is, how many true positives are we likely to have?

This is a case in which we can use the science of the situation and perhaps limit our study to genes that have some reasonable expectation of actually causing this malady. Then if we can “preassign” a probability, we might get a better feel if a positive is a false one.

Of course, this technique might induce a “user bias” into the situation from the very start.

The good news is that, given enough data, the frequentist and the Bayesian techniques converge to “the truth”.

Summary Nate Silver’s book is well written, informative and fun to read. I can recommend it without reservation.

An example to apply Bayes’ Theorem and multivariable calculus

I’ve thought a bit about the breast cancer research results and found a nice “application” exercise that might help teach students about Bayes Theorem, two-variable maximizing, critical points, differentials and the like.

I’ve been interested in the mathematics and statistics of the breast cancer screening issue mostly because it provided a real-life application of statistics and Bayes’ Theorem.

So right now, for women between 40-49, traditional mammograms are about 80 percent accurate in the sense that, if a woman who really has breast cancer gets a mammogram, the test will catch it about 80 percent of the time. The false positive rate is about 8 percent in that: if 100 women who do NOT have breast cancer get a mammogram, 8 of the mammograms will register a “positive”.
Since the breast cancer rate for women in this age group is about 1.4 percent, there will be many more false positives than true positives; in fact a woman in this age group who gets a “positive” first mammogram has about a 16 percent chance of actually having breast cancer. I talk about these issues here.

So, suppose you desire a “more accurate test” for breast cancer. The question is this: what do you mean by “more accurate”?

1. If “more accurate” means “giving the right answer more often”, then that is pretty easy to do.
Current testing is going to be wrong: if C means cancer, N means “doesn’t have cancer”, P means “positive test” and M means “negative test”, then the probability of being wrong is:
. On the other hand, if you just declared EVERYONE to be “cancer free”, you’d be wrong only 1.4 percent of the time! So clearly that does not work; the “false negative” rate is 100 percent, though the “false positive” rate is 0.

On the other hand if you just told everyone “you have it”, then you’d be wrong 98.6 percent of the time, but you’d have zero “false negatives”.

So being right more often isn’t what you want to maximize, and trying to minimize the false positives or the false negatives doesn’t work either.

2. So what about “detecting more of the cancer that is there”? Well, that is where this article comes in. Switching to digital mammograms does increase detection rate but also increases the number of false positives:

The authors note that for every 10,000 women 40 to 49 who are given digital mammograms, two more cases of cancer will be identified for every 170 additional false-positive examinations.

So, what one sees is that if a woman gets a positive reading, she now has an 11 percent of actually having breast cancer, though a few more cancers would be detected.

Is this progress?

My whole point: saying one test is “more accurate” than another test isn’t well defined, especially in a situation where one is trying to detect something that is relatively rare.
Here is one way to look at it: let the probability of breast cancer be , the probability of detection of a cancer be given by and the probability of a false positive be given by . Then the probability of a person actually having breast cancer, given a positive test is given by:
; this gives us something to optimize. The partial derivatives are:
. Note that is positive since is less than 1 (in fact, it is small). We also know that the critical point is a bit of a “duh”: find a single test that gives no false positives and no false negatives. This also shows us that our predictions will be better if goes down (fewer false positives) and if goes up (fewer false negatives). None of that is a surprise.

But of interest is in the amount of change. The denominators of each partial derivative are identical. The coefficients of the numerators are of the same magnitude; there are different signs. So the rate of improvement of the predictive value is dependent on the relative magnitudes of , which is for us, and , which is . Note that is much larger than and occurs in the numerator . Hence an increase in the accuracy of the factor (a decrease in the false positive rate) will have a greater effect on the accuracy of the test than a similar increase in the “false negative” accuracy.
Using the concept of differentials, we expect a change leads to an improvement of about .00136 (substitute into the expression for and multiply by . Similarly an improvement (decrease) of leads to an improvement of .013609.

You can “verify” this by playing with some numbers:

Current () we get . Now let’s change: leads to
Now change: we get

Bottom line: the best way to increase the predictive value of the test is to reduce the number of false positives, while staying the same (or improving) the percentage of “false negatives”. As things sit, the false positive rate is the bigger factor affecting predictive value.

Hypothesis Testing: Frequentist and Bayesian

I was working through Nate Silver’s book The Signal and the Noise and got to his chapter about hypothesis testing. It is interesting reading and I thought I would expand on that by posing a couple of problems.

Problem one: suppose you knew that someone attempted some basketball free throws.
If they made 1 of 4 shots, what would the probability be that they were really, say, a 75 percent free throw shooter?
What if they made 2 of 8 shots instead?
Or, what if they made 5 of 20 shots?

Problem two: Suppose a woman aged 40-49 got a digital mammagram and got a “positive” reading. What is the probability that she indeed has breast cancer, given that the test catches 80 percent of the breast cancers (note: 20 percent is one estimate of the “false negative” rate; and yes, the false positive rate is 7.8 percent. The actual answer, derived from data, might surprise you: it is : 16.3 percent.

I’ll talk about problem two first, as this will limber the mind for problem one.

So, you are a woman between 40-49 years of age and go into the doctor and get a mammogram. The result: positive.

So, what is the probability that you, in fact, have cancer?

Think of it this way: out of 10,000 women in that age bracket, about 143 have breast cancer and 9857 do not.
So, the number of false positives is 9857*.078 = 768.846; we’ll keep the decimal for the sake of calculation;
The number of true positives is: 143*.8 = 114.4.
The total number of positives is therefore 883.246.

The proportion of true positives is So the false positive rate is 83.72 percent.

It turns out that, data has shown the 80-90 percent of positives in women in this age bracket are “false positives”, and our calculation is in line with that.

I want to point out that this example is designed to warm the reader up to Bayesian thinking; the “real life” science/medicine issues are a bit more complicated than this. That is why the recommendations for screening include criteria as to age, symptoms vs. asymptomatic, family histories, etc. All of these factors affect the calculations.

For example: using digital mammograms with this population of 10,000 women in this age bracket adds 2 more “true” detections and adds 170 more false positives. So now our calculation would be , so while the true detections go up, the false positives also goes up!

Our calculation, while specific to this case, generalizes. The formula comes from Bayes Theorem which states:
. Here: is the probability of event A occurring given that B occurs and P(A) is the probability of event A occurring. So in our case, we were answering the question: given a positive mammogram, what is the probability of actually having breast cancer? This is denoted by P(A|B) . We knew: P(B|A) which is the probability of having a positive reading given that one has breast cancer and P(B|not(A)) is the probability of getting a positive reading given that one does NOT have cancer. So for us: and .

The bottom line: If you are testing for a condition that is known to be rare, even a reasonably accurate test will deliver a LOT of false positives.

Here is a warm up (hypothetical) example. Suppose a drug test is 99 percent accurate in that it will detect that a certain drug is there 99 percent of the time (if it is really there) and only yield a false positive 1 percent of the time (gives a positive result even if the person being tested is free of this drug). Suppose the drug use in this population is “known” to be, say 5 percent.

Given a positive test, what is the probability that the person is actually a user of this drug?

Answer: . So, in this population, about 16.1 percent of the positives will be “false positives”, even though the test is 99 percent accurate!

Now that you are warmed up, let’s proceed to the basketball question:

Question: suppose someone (that you don’t actually see) shoots free throws.

Case a) the player makes 1 of 4 shots.
Case b) the player makes 2 of 8 shots.
Case c) the player makes 5 of 20 shots.

Now you’d like to know: what is the probability that the player in question is really a 75 percent free throw shooter? (I picked 75 percent as the NBA average for last season is 75.3 percent).

Now suppose you knew NOTHING else about this situation; you know only that someone attempted free throws and you got the following data.

How this is traditionally handled
The traditional “hypothesis test” uses the “frequentist” model: you would say: if the hypothesis that the person really is a 75 percent free throw shooter is true, what is the probability that we’d see this data?

So one would use the formula for the binomial distribution and use n = 4 for case A, n = 8 for case B and n = 20 for case C and use p = .75 for all cases.

In case A, we’d calculate the probability that the number of “successes” (made free throws) is less than or equal to 1; 2 for case B and 5 for case C.

For you experts: the null hypothesis would be, say for the various cases would be respectively, where the probability mass function is adjusted for the different values of n .

We could do the calculations by hand, or rely on this handy calculator.

The answers are:
Case A: .0508
Case B: .0042
Case C: .0000 ()

By traditional standards: Case A: we would be on the verge of “rejecting the null hypothesis that p = .75 and we’d easily reject the null hypothesis in cases B and C. The usual standard (for life science and political science) is p = .05).

(for a refresher, go here)

So that is that, right?

Well, what if I told you more of the story?

Suppose now, that in each case, the shooter was me? I am not a good athlete and I played one season in junior high, and rarely, some pickup basketball. I am a terrible player. Most anyone would happily reject the null hypothesis without a second thought.

But now: suppose I tell you that I took these performances from NBA box scores? (the first one was taken from one of the Spurs-Heat finals games; the other two are made up for demonstration).

Now, you might not be so quick to reject the null hypothesis. You might reason: “well, he is an NBA player and were he always as bad as the cases show, he wouldn’t be an NBA player. This is probably just a bad game.” In other words, you’d be more open to the possibility that this is a false positive.

Now you don’t know this for sure; this could be an exceptionally bad free throw shooter (Ben Wallace shot 41.5 percent, Shaquille O’Neal shot 52.7 percent) but unless you knew that, you’d be at least reasonably sure that this person, being an NBA player, is probably a 70-75 shooter, at worst.

So “how” sure might you be? You might look at NBA statistics and surmise that, say (I am just making this up), 68 percent of NBA players shoot between 72-78 percent from the line. So, you might say that, prior to this guy shooting at all, the probability of the hypothesis being true is about 70 percent (say). Yes, this is a prior judgement but it is a reasonable one. Now you’d use Bayes law:

Here: A represents the “75 percent shooter” being actually true, and B is the is the probability that we actually get the data. Note the difference in outlook: in the first case (the “frequentist” method), we wondered “if the hypothesis is true, how likely is it that we’d see data like this”. In this case, called the Bayesian method, we are wondering: “if we have this data, what is the probability that the null hypothesis is true”. It is a reverse statement, of sorts.

Of course, we have and we’ve already calculated P(B|A) for the various cases. We need to make a SECOND assumption: what does event not(A) mean? Given what I’ve said, one might say not(A) is someone who shoots, say, 40 percent (to make him among the worst possible in the NBA). Then for the various cases, we calculate respectively.

So, we now calculate using the Bayesian method:

Case A, the shooter made 1 of 4: .1996. The frequentist p-value was .0508
Case B, the shooter made 2 of 8: .0301. The frequentist p-value was .0042
Case C, the shooter made 5 of 20: 7.08 x 10^-5 The frequentist p-value was 3.81 x 10^-6

We see the following:
1. The Bayesian method is less likely to produce a “false positive”.
2. As n, the number of data points, grows, the Bayesian conclusion and the frequentist conclusions tend toward “the truth”; that is, if the shooter shoots enough foul shots and continues to make 25 percent of them, then the shooter really becomes a 25 percent free throw shooter.

So to sum it up:
1. The frequentist approach relies on fewer prior assumptions and is computationally simpler. But it doesn’t include extra information that might make it easier to distinguish false positives from genuine positives.
2. The Bayesian approach takes in more available information. But it is a bit more prone to the user’s preconceived notions and is harder to calculate.

How does this apply to science?
Well, suppose you wanted to do an experiment that tried to find out which human gene alleles correspond so a certain human ailment. So a brute force experiment in which every human gene is examined and is statistically tested for correlation with the given ailment with null hypothesis of “no correlation” would be a LOT of statistical tests; tens of thousands, at least. And at a p-value threshold of .05 (we are willing to risk a false positive rate of 5 percent), we will get a LOT of false positives. On the other hand, if we applied bit of science prior to the experiment and were able to assign higher prior probabilities (called “posterior probability”) to the genes “more likely” to be influential and lower posterior probability to those unlikely to have much influence, our false positive rates will go down.

Of course, none of this eliminates the need for replication, but Bayesian techniques might cut down the number of experiments we need to replicate.

Math in the News (or: here is a nice source of exercises)

I am writing a paper and am through with the mathematics part. Now I have to organize, put in figures and, in general, make it readable. Or, in other words, the “fun” part is over. 🙂

So, I’ll go ahead and post some media articles which demonstrate mathematical or statistical concepts:

Topology (knot theory)

As far as what is going on:

After a century of studying their tangled mathematics, physicists can tie almost anything into knots, including their own shoelaces and invisible underwater whirlpools. At least, they can now thanks to a little help from a 3D printer and some inspiration from the animal kingdom.

Physicists had long believed that a vortex could be twisted into a knot, even though they’d never seen one in nature or the even in the lab. Determined to finally create a knotted vortex loop of their very own, physicists at the University of Chicago designed a wing that resembles a delicately twisted ribbon and brought it to life using a 3D printer.

After submerging their masterpiece in water and using electricity to create tiny bubbles around it, the researchers yanked the wing forward, leaving a similarly shaped vortex in its wake. Centripetal force drew the bubbles into the center of the vortex, revealing its otherwise invisible, knotted structure and allowing the scientists to see how it moved through the fluid—an idea they hit on while watching YouTube videos of dolphins playing with bubble rings.

By sweeping a sheet of laser light across the bubble-illuminated vortex and snapping pictures with a high-speed camera, they were able to create the first 3D animations of how these elusive knots behave, they report today in Nature Physics. It turns out that most of them elegantly unravel within a few hundred milliseconds, like the trefoil-knotted vortex in the video above. […]

Note: the trefoil is the simplest of all of the non-trivial (really knotted) knots in that its projection has the fewest number of crossings, or in that it can be made with the fewest number of straight sticks.

I do have one quibble though: shoelaces are NOT knotted…unless the tips are glued together to make the lace a complete “circuit”. There ARE arcs in space that are knotted:

This arc can never be “straightened out” into a nice simple arc because of its bad behavior near the end points. Note: some arcs which have an “infinite number of stitches” CAN be straightened out. For example if you take an arc and tie an infinite number of shrinking trefoil knots in it and let those trefoil knots shrink toward an endpoint, the resulting arc can be straightened out into a straight one. Seeing this is kind of fun; it involves the use of the “lamp cord trick”

(this is from R. H. Bing’s book The Geometric Topology of 3-Manifolds; the book is chock full of gems like this.)

Social Issues
It is my intent to stay a-political here. But there are such things as numbers and statistics and ways of interpreting such things. So, here are some examples:

Welfare
From here:

My testimony will amplify and support the following points:

A complete picture of time on welfare requires an understanding of two seemingly contradictory facts: the majority of families who ever use welfare do so for relatively short periods of time, but the majority of the current caseload will eventually receive welfare for relatively long periods of time.

It is a good mental exercise to see how this statement could be true (and it is); I invite you to try to figure this out BEFORE clicking on the link. It is a fun exercise though the “answer” will be obvious to some readers.

Speaking of Welfare: there is a debate on whether drug testing welfare recipients is a good idea or not. It turns out that, at least in terms of money saved/spent: it was a money losing proposition for the State of Florida, even when one factors in those who walked away prior to the drug tests. This data might make a good example. Also, there is the idea of a false positive: assuming that the statistic of, say, 3 percent of those on welfare use illegal drugs, how accurate (in terms of false positives) does a test have to be in order to have, say, a 90 percent predictive value? That is, how low does the probability of a false positive have to be for one to be 90 percent sure that someone has used drugs, given that they got a positive drug test?

Lastly: Social Security You sometimes hear: life expectancy was 62 when Social Security started. Well, given that working people pay into it, what are the key data points we need in order to determine what changes should be made? Note: what caused a shorter life expectancy and how does that effect: the percent of workers paying into it and the time that a worker draws from it? Think about these questions and then read what the Social Security office says. There are some interesting “conditional expectation” problems to be generated here.

Mathematics, Statistics, Physics

This is a fun little post about the interplay between physics, mathematics and statistics (Brownian Motion)

Here is a teaser video:

The article itself has a nice animation showing the effects of a Poisson process: one will get some statistical clumping in areas rather than uniform spreading.

Treat yourself to the whole article; it is entertaining.

Quantum Mechanics, Hermitian Operators and Square Integrable Functions

In one dimensional quantum mechanics, the state vectors are taken from the Hilbert space of complex valued “square integrable” functions, and the observables correspond to the so-called “Hermitian operators”. That is, if we let the state vectors be represented by and we say where the overline decoration denotes complex conjugation.

The state vectors are said to be “square integrable” which means, strictly speaking, that is finite.
However, there is another hidden assumption beyond the integral existing and being defined and finite. See if you can spot the assumption in the following remarks:

Suppose we wish to show that the operator is Hermitian. To do that we’d have to show that:
. This doesn’t seem too hard to do at first, if we use integration by parts:
. Now because the functions are square integrable, the term is zero (the functions must go to zero as tends to infinity) and so we have: . Now we use integration by parts again:
which is what we wanted to show.

Now did you catch the “hidden assumption”?

Here it is: it is possible for a function to be square integrable but to be unbounded!

If you wish to work this out for yourself, here is a hint: imagine a rectangle with height and base of width . Let be a function whose graph is a constant function of height for for all positive integers and zero elsewhere. Then has height over all of those intervals which means that the area enclosed by each rectangle (tall, but thin rectangles) is . Hence . is certainly square integrable but is unbounded!

It is easy to make into a continuous function; merely smooth by a bump function whose graph stays in the tall, thin rectangles. Hence can be made to be as smooth as desired.

So, mathematically speaking, to make these sorts of results work, we must make the assumption that and add that to the “square integrable” assumption.

Quantum Mechanics and Undergraduate Mathematics XIV: bras, kets and all that (Dirac notation)

Up to now, I’ve used mathematical notation for state vectors, inner products and operators. However, physicists use something called “Dirac” notation (“bras” and “kets”) which we will now discuss.

Recall: our vectors are integrable functions where converges.

Our inner product is:

Here is the Dirac notation version of this:
A “ket” can be thought of as the vector . Of course, there is an easy vector space isomorphism (Hilbert space isomorphism really) between the vector space of state vectors and kets given by . The kets are denoted by .
Similarly there are the “bra” vectors which are “dual” to the “kets”; these are denoted by and the vector space isomorphism is given by . I chose this isomorphism because in the bra vector space, . Then there is a vector space isomorphism between the bras and the kets given by .

Now is the inner product; that is

By convention: if is a linear operator, and Now if is a Hermitian operator (the ones that correspond to observables are), then there is no ambiguity in writing .

This leads to the following: let be an operator corresponding to an observable with eigenvectors and eigenvalues . Let be a state vector.
Then and if is a random variable corresponding to the observed value of , then and the expectation .

Beware of Randomness…

We teach about p-values in statistics. But rejecting a null hypothesis at a small p-value does not give us immunity from type I error: (via Scientific American)

The p-value puts a number on the effects of randomness. It is the probability of seeing a positive experimental outcome even if your hypothesis is wrong. A long-standing convention in many scientific fields is that any result with a p-value below 0.05 is deemed statistically significant. An arbitrary convention, it is often the wrong one. When you make a comparison of an ineffective drug to a placebo, you will typically get a statistically significant result one time out of 20. And if you make 20 such comparisons in a scientific paper, on average, you will get one signif­icant result with a p-value less than 0.05—even when the drug does not work.

Many scientific papers make 20 or 40 or even hundreds of comparisons. In such cases, researchers who do not adjust the standard p-value threshold of 0.05 are virtually guaranteed to find statistical significance in results that are meaningless statistical flukes. A study that ran in the February issue of the American Journal
of Clinical Nutrition tested dozens of compounds and concluded that those found in blueberries lower the risk of high blood pressure, with a p-value of 0.03. But the researchers looked at so many compounds and made so many comparisons (more than 50), that it was almost a sure thing that some of the p-values in the paper would be less than 0.05 just by chance.

The same applies to a well-publicized study that a team of neuroscientists once conducted on a salmon. When they presented the fish with pictures of people expressing emotions, regions of the salmon’s brain lit up. The result was statistically signif­icant with a p-value of less than 0.001; however, as the researchers argued, there are so many possible patterns that a statistically significant result was virtually guaranteed, so the result was totally worthless. p-value notwithstanding, there was no way that the fish could have reacted to human emotions. The salmon in the fMRI happened to be dead.

Emphasis mine.

Moral: one can run an experiment honestly and competently and analyze the results competently and honestly…and still get a false result. Damn that randomness!