Wikipedia:Reference desk/Archives/Mathematics/2014 February 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

February 5

Infinite Tetration of a Real Number is an Imaginary Quantity

I was wondering whether or not the following "identity"

$i=e^{i{\frac {\pi }{2}}}=\left(e^{\frac {\pi }{2}}\right)^{i}=\left(e^{\frac {\pi }{2}}\right)^{\left(e^{\frac {\pi }{2}}\right)^{i}}=\ldots =\ ^{^{^{\infty }}}\!\!\!\left(e^{\frac {\pi }{2}}\right)=\infty$

might not be the (exponential) equivalent of various summation methods for divergent series, such as Abel summation, Borel summation, Cesaro summation, Euler summation, Ramanujan summation, and others. Any ideas ? And has this "identity" ever been mentioned before in literature ? Thank you.

(The absurd "identity" above is partially explained by the fact that infinite tetration is only defined for numbers in between $(1/e)^{e}$ and $e^{1/e}$ , but the square root of Gelfond's constant obviously does not belong in that particular interval). — 79.113.217.24 (talk) 00:50, 5 February 2014 (UTC)

I am confused. How did you conclude that

$\left(e^{\frac {\pi }{2}}\right)^{\left(e^{\frac {\pi }{2}}\right)^{i}}=\ldots =\ ^{^{^{\infty }}}\!\!\!\left(e^{\frac {\pi }{2}}\right)$

It does not seems to work.

a=i
a=0+i
a=0+0+i
...
a=0+0+0+...
a=0
I rest my case.
202.177.218.59 (talk) 00:57, 6 February 2014 (UTC)

There's no case to rest, since what you have is $a=0\cdot \infty$ , which is perfectly possible, since the latter product is undetermined. The same goes for $a=1\cdot a$ , which yields $a=1^{\infty }$ , which is also indeterminate. — 79.118.183.38 (talk) 03:31, 6 February 2014 (UTC)

The formula at the top is incorrect, because it mixes up the order of operations. In other word, it mixes up

$a^{(b^{c})}$ with $(a^{b})^{c}$.

Looie496 (talk) 15:35, 6 February 2014 (UTC)

No. It doesn't. It's just that, in our case, ${\color {red}c}=(a^{b})^{\color {red}c}=(a^{b})^{{\big [}(a^{b})^{c}{\big ]}}$ through the repeated replacement of c's value based on its recursive definition. — 79.113.214.241 (talk) 21:02, 6 February 2014 (UTC)

How to measure if a set of sequences converges

I am trying to find ways to measure convergence, but I cannot search for it because I only get billions of pages describing convergence of mathematical sequences. What I have is a data set that has sequences of patient visits. I have patient, date, and location. Grouping by patient and ordering by date, I have about four million sequences of locations. I know that certain locations are far more predominant in the database. In fact, I have identified the distribution of locations, ordered from most frequent to least frequent, to be a logarithmic distribution. So, there is a high possibility that while people begin at many different locations, they converge into a small set. However, it could be that they start in a small set and diverge into a large set. I am looking for methods to measure this idea of converging sequences. I have no interest in identifying if some sine wave function eventually converges. Please don't redirect me to a page about that stuff. 209.149.114.201 (talk) 12:47, 5 February 2014 (UTC)

Stochastic convergence should be enough to get you started. Sławomir Biały (talk) 13:05, 5 February 2014 (UTC)

I'm sorry, but I do not see how that relates. It is, again, convergence of a series to a single value. I am not looking at the convergence of a series. A better way to describe what I have is a set of vectors with origins at many random locations. Hopefully, the total realm of where the vectors end up will be smaller than where they begin. There will be outliers. So, I want a measure that says "Set A (a big set of vectors) has a convergence of 0.2 while Set B (a big set of vectors) has a convergence of 0.6 so Set B converges more." It is silly to think that all four million patients will eventually end up all going to the same hospital. I want to see if they tend to converge on a smaller set of hospitals than where they begin and, more importantly, I need a measure that I can compare to other patient sets. 209.149.114.201 (talk) 14:01, 5 February 2014 (UTC)

You can define the distance of a point to a subset easily enough. I don't see how adding more than one point to the set of limit points would be a big issue. Sławomir Biały (talk) 14:40, 5 February 2014 (UTC)

Can you explain what you mean by "point" and "subset" in this example. This is what my sequences look like: {3,15,3,584,34,34,485,3}, {5,65,3,676,4,3,66,4,7}, {52,6,66,76,7,2,44}, {5}, {52,75,12,467,3,675,87,5,4,25}, {15,4}, etc... What is the "point"? What is the "subset"? 209.149.114.201 (talk) 14:56, 5 February 2014 (UTC)

Presumably a point is just a number and a set is a set of numbers. But without you telling me what these number are, it's hard to be sure if that's the appropriate response. Sławomir Biały (talk) 16:41, 5 February 2014 (UTC)

The numbers are identifiers of locations. Each location has an identifier - just an integer. The sequence is the order in which the patient went to each location. So, a sequence {5,1,10} means the patient went to 5 first, then to 1, and then to 10. Given a few million sequences (with varying length), I want a function that spits out a measure of how much the sequences tend to converge from a large population of locations to a smaller set of locations, but I want it to be continuous. Just looking at the first and last location in each sequence is not acceptable. 209.149.114.201 (talk) 16:46, 5 February 2014 (UTC)

You could just count the number of initial hospitals (the first place visited in each location sequence) and the number of final hospitals. If there are fewer final hospitals than initial hospitals then there is some "convergence" taking place, and the ratio of the two counts is a simple measure of the amount of convergence that is independent of the sample size. Gandalf61 (talk) 14:19, 5 February 2014 (UTC)

That is similar to what I've done. Outliers mess that up. I weighted each count by the number of patients at the hospital. I get a start count and an end count. I identified convergence. But, it is not a continuous measure. This could simply be that everyone goes to the big hospital to die. I want to see if there is a standard measure for continuous convergence as opposed to completely random sequences that suddenly collapse on a small set in the end. Then, I need it to be a measure or a metric because I'm comparing 51 different sets of patient visit sequences. 209.149.114.201 (talk) 14:23, 5 February 2014 (UTC)

How about if you divide the data up differently, to look at how many people visit a given hospital on their first visit, how many people visit that hospital on their second visit, etc. Then, by comparing the data for each hospital, you can conclude if people move towards or away from that hospital on subsequent visits, and the degree of the shift. You might also want to split off everyone's last visit, to isolate those who "go to a big hospital to die". StuRat (talk) 17:10, 5 February 2014 (UTC)

Thank you. I've asked this question multiple times on many different forums for the last two years and this is the first answer I have received that I feel understands the question. The problem with first, second, third visit is that some sequences have 1 visit. Some have 500 visits. I need to work with all varying lengths in the same way to avoid having someone claim that I am over/under weighting short/long sequences. But, based on your suggestion, there is a scheme that I have proposed (it has not been refused yet). For every location, I make an incoming value that is the count of how many unique locations feed into that location. I can calculated that by counting how many unique locations precede a given location in all sequences. Then, I can replace each sequence with a sequence of incoming values. If the overall set converges, the sequences of incoming values should generally go from smaller to larger numbers. If I plot all of them, stretching the short sequences to be the same width on the graph as the longer sequences, I expect to get a bunch of plot points that generally increase. Then, I can put a trendline on that scatter plot and measure the slope. The more positive the slope, the more convergent it is. The problem is that I haven't proven that this is a measure of convergence and math is all about proofs. 209.149.114.201 (talk) 17:36, 5 February 2014 (UTC)

You could also break your data up by total number of known visits per patient. Let's use patients with 3 known visits as an example. So, for each hospital, you could have total number of first visits from those for which you have 3 known visits, the total number of 2nd visits from those for which you have 3 known visits, and the total number of 3rd visits for those which you have 3 known visits. Then repeat this process for each number of known visits. StuRat (talk) 19:31, 5 February 2014 (UTC)

This isn't so much a math question as one in the more murky area of data analysis. The first issue to deal with is the need for operational definitions. The word "converge" is being used here in a way that does not correspond to its usual mathematical meaning, which may be may be the source of some of the confusion here. What it seems to mean is that if a patient visits visit clinic/hospital A, is referred to hospital B, and is referred again to specialist C, and so on to Z, the sequence is said to "converge" to Z. This may seem like a useable definition but it's not operational in terms of the data, since that only lists the sequence of visits. The patient may have been referred to another hospital but not followed up yet; in this case the data only has the last visit, not the planned visit so it would give the incorrect result for convergence. Or, a patient could visit a clinic for a toenail fungus then go to a hospital a week later for chest pains. If the dataset only records to the two visits then it would imply that the toenail visit converged to the chest pain hospitalization. When such issues arise it's often due to a deficiency in the data design, for example if the dataset also recorded a status for each visit (patient cured, referred, followup required, etc.) then the final visit for a given case could be determined without guesswork, so which hospital is being converged to, or whether this is still undetermined, can be accurately reported. The second question to consider is what kind of information you're trying to get out of the dataset, and ultimately how this information will be used. It looks like the kind of statements you want to make are something like "27% of patient cases in the Baltimore area are eventually referred to Maryland Mercy." You need operational definitions to make this statement precise enough so that the statement makes sense in terms of the dataset. For example "in the Baltimore area" might be defined as "Patient Zip code in (list of Zip code)" and "eventually referred to" might mean "status of last visit in sequence in (cured, deceased) and date of last visit < current date - 3 months".

There is an anecdote, possibly false but this is for illustration, that back when the U.S. speed limit was reduced to 55, it was afterwards claimed that the traffic fatality rate went down. Eventually someone noticed that at the same time the speed limit was changed, the NTSB changed their definition of "traffic fatality", and this change accounted for most of the reported difference. If the question had been been asked at the start, "was there a significant decrease in traffic fatalities from 1973 to 1974?" with carefully made operational definitions for all terms involved, then perhaps the existing data could have been queried correctly. Even so the result could still be questioned due to the correlation/causation fallacy. Perhaps a perceived decrease was due to a drop in LSD usage and nothing to do with the speed limit. --RDBury (talk) 10:29, 6 February 2014 (UTC)

• The spirit of this question seems more about finding a basin of attraction than more general notions of convergence (as RDBury says above). Also, whether the time series "concentrates" to a few locations or "spreads out" to many can be measured by the lyapunov exponent. Your set of data is a large collection of time series, methods here go under the heading of time series analysis. These pages won't tell you exactly how to do what you want, but it might give you some better terminology and search terms. I'd start by using google and google scholar to search for things like /estimate "lyapunov exponent" time series data analysis/ SemanticMantis (talk) 16:00, 6 February 2014 (UTC)

I have read the articles you linked. I see the relation, but I want to ensure that I understand this correctly. A "time series" is a time-ordered set of measurements. It is not a time-ordered set of categorical values. For example, {"ford","chevy","toyota","ford","audi"} would not qualify as a time series because there is no numerical relationship between the values. Is that correct? I ask because my values are integers representing healthcare locations. There is no implied relationship between the values. For example, location 4 is not "less than" location 8. 209.149.114.201 (talk) 19:46, 6 February 2014 (UTC)

I think that's ok. I guess usually time series are number-valued, but for the purposes I'm alluding to, it doesn't matter. Imagine a simpler problem (or a mathematical model of your problem).

• I have 100 tokens, labeled 1-100. Let those represent the patients.
• I have 10 bowls, labeled A...J, which represent hospitals.
• At time t=0, I roll a (fair) 10-sided die, and place each token in the appropriate bowl.
• Now, at each successive time t=1,2,..., I flip a coin for each token. If tails, that token has "ended its run" I record its position history, then remove it from play. If the coin is heads, the token remains "in play," and I roll another 10-sided die to see where it goes next time.

See how this is very similar to your example? After 100 turns, you have 100 sequences, of length 1-100 ([A,C], [J,B,D], [A,E,B,B,A], ...), and we can analyze the dynamics of the system. Now, they way I set up my game, we'd have no notion of "spread" or "concentration," because any token can move to any bowl at each time step, and all are equally likely. But, if instead, I had an unfair die , we could see "concentration" where, regardless of initial positions, just a few bowls end up with most of the tokens over time. I have modeled this problem naively as a Markov chain, and, though that assumption is probably wrong for real-life patients and hospitals, it is probably a pretty good place to start. What you need is a way to estimate the transition matrix, e.g. what are the odds of moving from bowl A to B, B to A, A to C and so on. I am sure there are methods out there for estimating transition matrices from data, but I don't know how to do it off the top of my head. Hopefully these additional links and terms will help you on your way. If you can justify that you have a good estimate of the transition matrix, then checking for "concentration" will be fairly easy, using the properties of the resulting Markov chain. SemanticMantis (talk) 22:51, 6 February 2014 (UTC)

After re-reading and thinking a bit, I think that StuRat and Gandalf's responses above are both getting at similar ideas too, with Markov chains as the key modeling framework. Here is a set of lecture notes specifically about estimating transition matrices from data [1], and here is a stackoverflow discussion of the same problem, specifically if, (like your case) the observations are of differing length [2]. That's definitely the way I would proceed. But, are you working for a large healthcare company or something? If so, recall we are all just random strangers on the internet, and do your own due diligence :) SemanticMantis (talk) 23:41, 6 February 2014 (UTC)

Thanks. I am happy to start seeing similar work. I knew someone was doing this, but Googling and Googling didn't help. I think I can use these references to make a method of measurement that will be accepted by the statisticians. And, I don't work for a company. I work for the government. Unlike most in my position, I'm trying to do a good job. 209.149.114.201 (talk) 13:36, 7 February 2014 (UTC)

Glad to help, feel free to post follow-ups here once you get going. Good luck! SemanticMantis (talk) 15:38, 7 February 2014 (UTC)

I ran this over the weekend on all 50 sets. I did this per patient. I loaded the patient's visit history, time ordered. I replaced each visit's location ID with the popularity of the location. Popularity is measured by counting how many unique locations immediately precede the given location in all histories for the data set. Then, I placed the popularities on an X/Y graph such that visit 1 was always plotted at X=1 and the last visit was always plotted at X=500 (stretching the visit history across 1 to 500). With all of the patients plotted, I placed a best-fit trendline on the graph. Finally, I have a slope of the line that I can use. I also have very clear patterns in some of the data. For example, some data sets have a clear linear increase or decrease. Some are cup (upside down U) shaped. Two of them are double-humps. So, I think I can report on more than just convergence. 209.149.114.201 (talk) 14:07, 10 February 2014 (UTC)

Accuracy of a straight edge and compass construction

I have discovered a method for trisecting an angle (almost). I am under no delusion that the construction is a solution to this problem, but I am interested in determining exactly how accurate it is. Is there some mathematical calculation or a computer application that can calculate the accuracy of a construction? — Preceding unsigned comment added by 97.65.82.66 (talk) 18:45, 5 February 2014 (UTC)

If you want to determine the accuracy of your approximation, there are two ways:

A) For the theoretical accuracy, the math approach you asked for is correct.

What is that math approach? — Preceding unsigned comment added by 97.65.82.66 (talk) 21:19, 5 February 2014 (UTC)

B) For the practical accuracy, including the compass not creating a perfect circle, etc., measure some results and then do a statistical analysis.

Of course, in both cases, the accuracy may vary as a function of how large of an angle you trisect. StuRat (talk) 19:22, 5 February 2014 (UTC)

For initial angles between 1.0 and 2.0 radians there is no measurable error. That is why I was looking for a method of mathematical calculation or a computer application. — Preceding unsigned comment added by 97.65.82.66 (talk) 19:54, 5 February 2014 (UTC)

Use complex numbers. The angle is 1,0,e^i3x. Perform your procedure to find the point e^iw. Compare x to w. Bo Jacoby (talk) 21:47, 5 February 2014 (UTC).

How do I represent the accuracy of the width of a pencil lead or the accuracy of placing a compass point on a drawn line? How do I represent the accuracy of placing a straight edge near two points and drawing a line? etc.? How do I propagate the error term to a point created by an intersection of two lines that each have been created by a straight edge that introduced errors that change with distance from the original points used for construction? 97.65.82.66 (talk) 00:04, 6 February 2014 (UTC)

There are two kinds of error. First it is impossible to trisect an angle exactly using only compass and straight edge. Any such construction is approximate, and the error is the one I referred to above. Secondly any actual geometrical construction has finite accuracy. The propagation of inaccuracy may be estimated by repeating the construction and comparing results. Bo Jacoby (talk) 07:37, 6 February 2014 (UTC).

There are many methods to trisect an angle using compass and straight edge (note that I omitted Bo's "only" so I am not disagreeing with the statement above). The one that the OP has discovered might be a good approximation, in which case the error can be calculated by trigonometry if the exact method is known; or it might be a Neusis construction which is, in theory, perfectly accurate if objects can be lined up with perfect accuracy by eye, but in practice this proves humanly impossible. Of course, perfect accuracy with "only compass and straight edge" is also humanly impossible, but the Greek mathematicians saw that the inaccuracy here was only limited by the quality of the instruments used (assuming no user error). Dbfirs 08:34, 6 February 2014 (UTC)

• You might be interested in the info at heptagon. A heptagon is also not constructable with compass-and-straight-edge-only, but there is a nice animation of an approximation, and also some (brief) info on how you can mathematically define the error. If you can do your own work to get the answers there, you should also be able to do your own work to quantify the error in your approximation. SemanticMantis (talk) 15:52, 6 February 2014 (UTC)