i. On the youtube channel of the Institute for Advanced Studies there has been a lot of activity over the last week or two (far more than 100 new lectures have been uploaded, and it seems new uploads are still being added at this point), and I’ve been watching a few of the recently uploaded astrophysics lectures. They’re quite technical, but you can watch them and follow enough of the content to have an enjoyable time despite not understanding everything:
This is a good lecture, very interesting. One major point made early on: “the take-away message is that the most common planet in the galaxy, at least at shorter periods, are planets for which there is no analogue in the solar system. The most common kind of planet in the galaxy is a planet with a radius of two Earth radii.” Another big take-away message is that small planets seem to be quite common (as noted in the conclusions, “16% of Sun-like stars have an Earth-sized planet”).
Of the lectures included in this post this was the one I liked the least; there are too many (‘obstructive’) questions/interactions between lecturer and attendants along the way, and the interactions/questions are difficult to hear/understand. If you consider watching both this lecture and the lecture below, I would say that it would probably be wise to watch the lecture below this one before you watch this one; I concluded that in retrospect some of the observations made early on in the lecture below would have been useful to know about before watching this lecture. (The first half of the lecture below was incidentally to me somewhat easier to follow than was the second half, but especially the first half hour of it is really quite good, despite the bad start (which one can always blame on Microsoft…)).
ii. Words I’ve encountered recently (…or ‘recently’ – it’s been a while since I last posted one of these lists): Divagations, periphrasis, reedy, architrave, sett, pedipalp, tout, togs, edentulous, moue, tatty, tearaway, prorogue, piscine, fillip, sop, panniers, auxology, roister, prepossessing, cantle, catamite, couth, ordure, biddy, recrudescence, parvenu, scupper, husting, hackle, expatiate, affray, tatterdemalion, eructation, coppice, dekko, scull, fulmination, pollarding, grotty, secateurs, bumf (I must admit that I like this word – it seems fitting, somehow, to use that word for this concept…), durophagy, randy, (brief note to self: Advise people having children who ask me about suggestions for how to name them against using this name (or variants such as Randi), it does not seem like a great idea), effete, apricity, sororal, bint, coition, abaft, eaves, gadabout, lugubriously, retroussé, landlubber, deliquescence, antimacassar, inanition.
iii. “The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.” (Terry Tao, There’s more to mathematics than rigour and proofs)
iv. A century of trends in adult human height. A figure from the paper (Figure 3 – Change in adult height between the 1896 and 1996 birth cohorts):
(Click to view full size. WordPress seems to have changed the way you add images to a blog post – if this one is even so annoyingly large, I apologize, I have tried to minimize it while still retaining detail, but the original file is huge). An observation from the paper:
“Men were taller than women in every country, on average by ~11 cm in the 1896 birth cohort and ~12 cm in the 1996 birth cohort […]. In the 1896 birth cohort, the male-female height gap in countries where average height was low was slightly larger than in taller nations. In other words, at the turn of the 20th century, men seem to have had a relative advantage over women in undernourished compared to better-nourished populations.”
v. I found this paper, on Exercise and Glucose Metabolism in Persons with Diabetes Mellitus, interesting in part because I’ve been very surprised a few times by offhand online statements made by diabetic athletes, who had observed that their blood glucose really didn’t drop all that fast during exercise. Rapid and annoyingly large drops in blood glucose during exercise have been a really consistent feature of my own life with diabetes during adulthood. It seems that there may be big inter-individual differences in terms of the effects of exercise on glucose in diabetics. From the paper:
“Typically, prolonged moderate-intensity aerobic exercise (i.e., 30–70% of one’s VO2max) causes a reduction in glucose concentrations because of a failure in circulating insulin levels to decrease at the onset of exercise.12 During this type of physical activity, glucose utilization may be as high as 1.5 g/min in adolescents with type 1 diabetes13 and exceed 2.0 g/min in adults with type 1 diabetes,14 an amount that quickly lowers circulating glucose levels. Persons with type 1 diabetes have large interindividual differences in blood glucose responses to exercise, although some intraindividual reproducibility exists.15 The wide ranging glycemic responses among individuals appears to be related to differences in pre-exercise blood glucose concentrations, the level of circulating counterregulatory hormones and the type/duration of the activity.2“
I find it difficult to find the motivation to finish the half-finished drafts I have lying around, so this will have to do. Some random stuff below.
(15.000 views… In some sense that seems really ‘unfair’ to me, but on the other hand I doubt neither Beethoven nor Gilels care; they’re both long dead, after all…)
ii. New/newish words I’ve encountered in books, on vocabulary.com or elsewhere:
Agley, peripeteia, dissever, halidom, replevin, socage, organdie, pouffe, dyarchy, tauricide, temerarious, acharnement, cadger, gravamen, aspersion, marronage, adumbrate, succotash, deuteragonist, declivity, marquetry, machicolation, recusal.
iii. A lecture:
It’s been a long time since I watched it so I don’t have anything intelligent to say about it now, but I figured it might be of interest to one or two of the people who still subscribe to the blog despite the infrequent updates.
iv. A few wikipedia articles (I won’t comment much on the contents or quote extensively from the articles the way I’ve done in previous wikipedia posts – the links shall have to suffice for now):
Russian political jokes. Some of those made me laugh (e.g. this one: “A judge walks out of his chambers laughing his head off. A colleague approaches him and asks why he is laughing. “I just heard the funniest joke in the world!” “Well, go ahead, tell me!” says the other judge. “I can’t – I just gave someone ten years for it!”).
v. World War 2, if you think of it as a movie, has a highly unrealistic and implausible plot, according to this amusing post by Scott Alexander. Having recently read a rather long book about these topics, one aspect I’d have added had I written the piece myself would be that an additional factor making the setting seem even more implausible is how so many presumably quite smart people were so – what at least in retrospect seems – unbelievably stupid when it came to Hitler’s ideas and intentions before the war. Going back to Churchill’s own life I’d also add that if you were to make a movie about Churchill’s life during the war, which you could probably relatively easily do if you were to just base it upon his own copious and widely shared notes, then it could probably be made into a quite decent movie. His own comments, remarks, and observations certainly made for a great book.
Below are three new lectures from the Institute of Advanced Study. As far as I’ve gathered they’re all from an IAS symposium called ‘Lens of Computation on the Sciences’ – all three lecturers are computer scientists, but you don’t have to be a computer scientist to watch these lectures.
Should computer scientists and economists band together more and try to use the insights from one field to help solve problems in the other field? Roughgarden thinks so, and provides examples of how this might be done/has been done. Applications discussed in the lecture include traffic management and auction design. I’m not sure how much of this lecture is easy to follow for people who don’t know anything about either topic (i.e., computer science and economics), but I found it not too difficult to follow – it probably helped that I’ve actually done work on a few of the things he touches upon in the lecture, such as basic auction theory, the fixed point theorems and related proofs, basic queueing theory and basic discrete maths/graph theory. Either way there are certainly much more technical lectures than this one available at the IAS channel.
I don’t have Facebook and I’m not planning on ever getting a FB account, so I’m not really sure I care about the things this guy is trying to do, but the lecturer does touch upon some interesting topics in network theory. Not a great lecture in my opinion and occasionally I think the lecturer ‘drifts’ a bit, talking without saying very much, but it’s also not a terrible lecture. A few times I was really annoyed that you can’t see where he’s pointing that damn laser pointer, but this issue should not stop you from watching the video, especially not if you have an interest in analytical aspects of how to approach and make sense of ‘Big Data’.
I’ve noticed that Scott Alexander has said some nice things about Scott Aaronson a few times, but until now I’ve never actually read any of the latter guy’s stuff or watched any lectures by him. I agree with Scott (Alexander) that Scott (Aaronson) is definitely a smart guy. This is an interesting lecture; I won’t pretend I understood all of it, but it has some thought-provoking ideas and important points in the context of quantum computing and it’s actually a quite entertaining lecture; I was close to laughing a couple of times.
ii. “The man who knows everyone’s job isn’t much good at his own.” (-ll-)
iii. “It is amazing what little harm doctors do when one considers all the opportunities they have” (Mark Twain, as quoted in the Oxford Handbook of Clinical Medicine, p.595).
iv. “A first-rate theory predicts; a second-rate theory forbids and a third-rate theory explains after the event.” (Aleksander Isaakovich Kitaigorodski)
v. “[S]ome of the most terrible things in the world are done by people who think, genuinely think, that they’re doing it for the best” (Terry Pratchett, Snuff).
vi. “That was excellently observ’d, say I, when I read a Passage in an Author, where his Opinion agrees with mine. When we differ, there I pronounce him to be mistaken.” (Jonathan Swift)
vii. “Death is nature’s master stroke, albeit a cruel one, because it allows genotypes space to try on new phenotypes.” (Quote from the Oxford Handbook of Clinical Medicine, p.6)
viii. “The purpose of models is not to fit the data but to sharpen the questions.” (Samuel Karlin)
ix. “We may […] view set theory, and mathematics generally, in much the way in which we view theoretical portions of the natural sciences themselves; as comprising truths or hypotheses which are to be vindicated less by the pure light of reason than by the indirect systematic contribution which they make to the organizing of empirical data in the natural sciences.” (Quine)
x. “At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too.” (-ll-)
xi. “A statistician carefully assembles facts and figures for others who carefully misinterpret them.” (Quote from Mathematically Speaking – A Dictionary of Quotations, p.329. Only source given in the book is: “Quoted in Evan Esar, 20,000 Quips and Quotes“)
xii. “A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances.” (Quote from Mathematically Speaking – A Dictionary of Quotations, p. 328. The source provided is: “Elements of Statistics, Part I, Chapter I (p.4)”).
xiii. “We own to small faults to persuade others that we have not great ones.” (Rochefoucauld)
xiv. “There is more self-love than love in jealousy.” (-ll-)
xv. “We should not judge of a man’s merit by his great abilities, but by the use he makes of them.” (-ll-)
xvi. “We should gain more by letting the world see what we are than by trying to seem what we are not.” (-ll-)
xvii. “Put succinctly, a prospective study looks for the effects of causes whereas a retrospective study examines the causes of effects.” (Quote from p.49 of Principles of Applied Statistics, by Cox & Donnelly)
xviii. “… he who seeks for methods without having a definite problem in mind seeks for the most part in vain.” (David Hilbert)
xix. “Give every man thy ear, but few thy voice” (Shakespeare).
xx. “Often the fear of one evil leads us into a worse.” (Nicolas Boileau-Despréaux)
As I’ve observed many times before, a wordpress blog like mine is not a particularly nice place to cover mathematical topics involving equations and lots of Greek letters, so the coverage below will be more or less purely conceptual; don’t take this to mean that the book doesn’t contain formulas. Some parts of the book look like this:
That of course makes the book hard to blog, also for other reasons than just the fact that it’s typographically hard to deal with the equations. In general it’s hard to talk about the content of a book like this one without going into a lot of details outlining how you get from A to B to C – usually you’re only really interested in C, but you need A and B to make sense of C. At this point I’ve sort of concluded that when covering books like this one I’ll only cover some of the main themes which are easy to discuss in a blog post, and I’ve concluded that I should skip coverage of (potentially important) points which might also be of interest if they’re difficult to discuss in a small amount of space, which is unfortunately often the case. I should perhaps observe that although I noted in my goodreads review that in a way there was a bit too much philosophy and a bit too little statistics in the coverage for my taste, you should definitely not take that objection to mean that this book is full of fluff; a lot of that philosophical stuff is ‘formal logic’ type stuff and related comments, and the book in general is quite dense. As I also noted in the goodreads review I didn’t read this book as carefully as I might have done – for example I skipped a couple of the technical proofs because they didn’t seem to be worth the effort – and I’d probably need to read it again to fully understand some of the minor points made throughout the more technical parts of the coverage; so that’s of course a related reason why I don’t cover the book in a great amount of detail here – it’s hard work just to read the damn thing, to talk about the technical stuff in detail here as well would definitely be overkill even if it would surely make me understand the material better.
I have added some observations from the coverage below. I’ve tried to clarify beforehand which question/topic the quote in question deals with, to ease reading/understanding of the topics covered.
On how statistical methods are related to experimental science:
“statistical methods have aims similar to the process of experimental science. But statistics is not itself an experimental science, it consists of models of how to do experimental science. Statistical theory is a logical — mostly mathematical — discipline; its findings are not subject to experimental test. […] The primary sense in which statistical theory is a science is that it guides and explains statistical methods. A sharpened statement of the purpose of this book is to provide explanations of the senses in which some statistical methods provide scientific evidence.”
On mathematics and axiomatic systems (the book goes into much more detail than this):
“It is not sufficiently appreciated that a link is needed between mathematics and methods. Mathematics is not about the world until it is interpreted and then it is only about models of the world […]. No contradiction is introduced by either interpreting the same theory in different ways or by modeling the same concept by different theories. […] In general, a primitive undefined term is said to be interpreted when a meaning is assigned to it and when all such terms are interpreted we have an interpretation of the axiomatic system. It makes no sense to ask which is the correct interpretation of an axiom system. This is a primary strength of the axiomatic method; we can use it to organize and structure our thoughts and knowledge by simultaneously and economically treating all interpretations of an axiom system. It is also a weakness in that failure to define or interpret terms leads to much confusion about the implications of theory for application.”
It’s all about models:
“The scientific method of theory checking is to compare predictions deduced from a theoretical model with observations on nature. Thus science must predict what happens in nature but it need not explain why. […] whether experiment is consistent with theory is relative to accuracy and purpose. All theories are simplifications of reality and hence no theory will be expected to be a perfect predictor. Theories of statistical inference become relevant to scientific process at precisely this point. […] Scientific method is a practice developed to deal with experiments on nature. Probability theory is a deductive study of the properties of models of such experiments. All of the theorems of probability are results about models of experiments.”
But given a frequentist interpretation you can test your statistical theories with the real world, right? Right? Well…
“How might we check the long run stability of relative frequency? If we are to compare mathematical theory with experiment then only finite sequences can be observed. But for the Bernoulli case, the event that frequency approaches probability is stochastically independent of any sequence of finite length. […] Long-run stability of relative frequency cannot be checked experimentally. There are neither theoretical nor empirical guarantees that, a priori, one can recognize experiments performed under uniform conditions and that under these circumstances one will obtain stable frequencies.” [related link]
What should we expect to get out of mathematical and statistical theories of inference?
“What can we expect of a theory of statistical inference? We can expect an internally consistent explanation of why certain conclusions follow from certain data. The theory will not be about inductive rationality but about a model of inductive rationality. Statisticians are used to thinking that they apply their logic to models of the physical world; less common is the realization that their logic itself is only a model. Explanation will be in terms of introduced concepts which do not exist in nature. Properties of the concepts will be derived from assumptions which merely seem reasonable. This is the only sense in which the axioms of any mathematical theory are true […] We can expect these concepts, assumptions, and properties to be intuitive but, unlike natural science, they cannot be checked by experiment. Different people have different ideas about what “seems reasonable,” so we can expect different explanations and different properties. We should not be surprised if the theorems of two different theories of statistical evidence differ. If two models had no different properties then they would be different versions of the same model […] We should not expect to achieve, by mathematics alone, a single coherent theory of inference, for mathematical truth is conditional and the assumptions are not “self-evident.” Faith in a set of assumptions would be needed to achieve a single coherent theory.”
On disagreements about the nature of statistical evidence:
“The context of this section is that there is disagreement among experts about the nature of statistical evidence and consequently much use of one formulation to criticize another. Neyman (1950) maintains that, from his behavioral hypothesis testing point of view, Fisherian significance tests do not express evidence. Royall (1997) employs the “law” of likelihood to criticize hypothesis as well as significance testing. Pratt (1965), Berger and Selke (1987), Berger and Berry (1988), and Casella and Berger (1987) employ Bayesian theory to criticize sampling theory. […] Critics assume that their findings are about evidence, but they are at most about models of evidence. Many theoretical statistical criticisms, when stated in terms of evidence, have the following outline: According to model A, evidence satisfies proposition P. But according to model B, which is correct since it is derived from “self-evident truths,” P is not true. Now evidence can’t be two different ways so, since B is right, A must be wrong. Note that the argument is symmetric: since A appears “self-evident” (to adherents of A) B must be wrong. But both conclusions are invalid since evidence can be modeled in different ways, perhaps useful in different contexts and for different purposes. From the observation that P is a theorem of A but not of B, all we can properly conclude is that A and B are different models of evidence. […] The common practice of using one theory of inference to critique another is a misleading activity.”
Is mathematics a science?
“Is mathematics a science? It is certainly systematized knowledge much concerned with structure, but then so is history. Does it employ the scientific method? Well, partly; hypothesis and deduction are the essence of mathematics and the search for counter examples is a mathematical counterpart of experimentation; but the question is not put to nature. Is mathematics about nature? In part. The hypotheses of most mathematics are suggested by some natural primitive concept, for it is difficult to think of interesting hypotheses concerning nonsense syllables and to check their consistency. However, it often happens that as a mathematical subject matures it tends to evolve away from the original concept which motivated it. Mathematics in its purest form is probably not natural science since it lacks the experimental aspect. Art is sometimes defined to be creative work displaying form, beauty and unusual perception. By this definition pure mathematics is clearly an art. On the other hand, applied mathematics, taking its hypotheses from real world concepts, is an attempt to describe nature. Applied mathematics, without regard to experimental verification, is in fact largely the “conditional truth” portion of science. If a body of applied mathematics has survived experimental test to become trustworthy belief then it is the essence of natural science.”
Then what about statistics – is statistics a science?
“Statisticians can and do make contributions to subject matter fields such as physics, and demography but statistical theory and methods proper, distinguished from their findings, are not like physics in that they are not about nature. […] Applied statistics is natural science but the findings are about the subject matter field not statistical theory or method. […] Statistical theory helps with how to do natural science but it is not itself a natural science.”
I should note that I am, and have for a long time been, in broad agreement with the author’s remarks on the nature of science and mathematics above. Popper, among many others, discussed this topic a long time ago e.g. in The Logic of Scientific Discovery and I’ve basically been of the opinion that (‘pure’) mathematics is not science (‘but rather ‘something else’ … and that doesn’t mean it’s not useful’) for probably a decade. I’ve had a harder time coming to terms with how precisely to deal with statistics in terms of these things, and in that context the book has been conceptually helpful.
Below I’ve added a few links to other stuff also covered in the book:
Radon-Nikodyn theorem. (not covered in the book, but the necessity of using ‘a Radon-Nikodyn derivative’ to obtain an answer to a question being asked was remarked upon at one point, and I had no clue what he was talking about – it seems that the stuff in the link was what he was talking about).
A very specific and relevant link: Berger and Wolpert (1984). The stuff about Birnbaum’s argument covered from p.24 (p.40) and forward is covered in some detail in the book. The author is critical of the model and explains in the book in some detail why that is. See also: On the foundations of statistical inference (Birnbaum, 1962).
This one was mostly review for me, but there was also some new stuff and it was a ‘sort of okay’ lecture even if I was highly skeptical about a few points covered. I was debating whether to even post the lecture on account of those points of contention, but I figured that by adding a few remarks below I could justify doing it. So below a few skeptical comments relating to content covered in the lecture:
a) 28-29 minutes in he mentions that the cutoff for hypertension in diabetics is a systolic pressure above 130. Here opinions definitely differ, and opinions about treatment cutoffs differ; in the annual report from the Danish Diabetes Database they follow up on whether hospitals and other medical decision-making units are following guidelines (I’ve talked about the data on the blog, e.g. here), and the BP goal of involved decision-making units evaluated is currently whether diabetics with systolic BP above 140 receive antihypertensive treatment. This recent Cochrane review concluded that: “At the present time, evidence from randomized trials does not support blood pressure targets lower than the standard targets in people with elevated blood pressure and diabetes” and noted that: “The effect of SBP targets on mortality was compatible with both a reduction and increase in risk […] Trying to achieve the ‘lower’ SBP target was associated with a significant increase in the number of other serious adverse events”.
b) Whether retinopathy screenings should be conducted yearly or biennially is also contested, and opinions differ – this is not mentioned in the lecture, but I sort of figure maybe it should have been. There’s some evidence that annual screening is better (see e.g. this recent review), but the evidence base is not great and clinical outcomes do not seem to differ much in general; as noted in the review, “Observational and economic modelling studies in low-risk patients show little difference in clinical outcomes between screening intervals of 1 year or 2 years”. To stratify based on risk seems desirable from a cost-effectiveness standpoint, but how to stratify optimally seems to not be completely clear at the present point in time.
c) The Somogyi phenomenon is highly contested, and I was very surprised about his coverage of this topic – ‘he’s a doctor lecturing on this topic, he should know better’. As the wiki notes: “Although this theory is well known among clinicians and individuals with diabetes, there is little scientific evidence to support it.” I’m highly skeptical, and I seriously question the advice of lowering insulin in the context of morning hyperglycemia. As observed in Cryer’s text: “there is now considerable evidence against the Somogyi hypothesis (Guillod et al. 2007); morning hyperglycemia is the result of insulin lack, not post-hypoglycemic insulin resistance (Havlin and Cryer 1987; Tordjman et al. 1987; Hirsch et al. 1990). There is a dawn phenomenon—a growth hormone–mediated increase in the nighttime to morning plasma glucose concentration (Campbell et al. 1985)—but its magnitude is small (Periello et al. 1991).”
I decided not to embed this lecture in the post mainly because the resolution is unsatisfactorily low so that a substantial proportion of the visual content is frankly unintelligible; I figured this would bother others more than it did me and that a semi-satisfactory compromise solution in terms of coverage would be to link to the lecture, but not embed it here. You can hear what the lecturer is saying, which was enough for me, but you can’t make out stuff like effect differences, p-values, or many of the details in the graphic illustrations included. Despite the title of the lecture on youtube, the lecture actually mainly consists of a brief overview of pharmacological treatment options for diabetes.
If you want to skip the introduction, the first talk/lecture starts around 5 minutes and 30 seconds into the video. Note that despite the long running time of this video the lectures themselves only take about 50 minutes in total; the rest of it is post-lecture Q&A and discussion.
This is a book full of quotes on the topic of mathematics. As is always the case for books full of quotations, most of the quotes in this book aren’t very good, but occasionally you come across a quote or two that enable you to justify reading on. I’ll likely include some of the good/interesting quotes in the book in future ‘quotes’ posts. Below I’ve added some sample quotes from the book. I’ve read roughly three-fifths of the book so far and I’m currently hovering around a two-star rating on goodreads.
“Since authors seldom, if ever, say what they mean, the following glossary is offered to neophytes in mathematical research to help them understand the language that surrounds the formulas …
ANALOGUE. This is an a. of: I have to have some excuse for publishing it.
APPLICATIONS. This is of interest in a.: I have to have some excuse for publishing it.
COMPLETE. The proof is now c.: I can’t finish it. […]
DIFFICULT. This problem is d.: I don’t know the answer. (Cf. Trivial)
GENERALITY. Without loss of g.: I have done an easy special case. […]
INTERESTING. X’s paper is I.: I don’t understand it.
KNOWN. This is a k. result but I reproduce the proof for convenience of the reader: My paper isn’t long enough. […]
NEW. This was proved by X but the following n. proof may present points of interest: I can’t understand X.
NOTATION. To simplify the n.: It is too much trouble to change now.
OBSERVED. It will be o. that: I hope you have not noticed that.
OBVIOUS. It is o.: I can’t prove it.
READER. The details may be left to the r.: I can’t do it. […]
STRAIGHTFORWARD. By a s. computation: I lost my notes.
TRIVIAL. This problem is t.: I know the answer (Cf. Difficult).
WELL-KNOWN. The result is w.: I can’t find the reference.” (Pétard, H. [Pondiczery, E.S.]).
Here are a few quotes similar to the ones above, provided by a different, unknown source:
“BRIEFLY: I’m running out of time, so I’ll just write and talk faster. […]
HE’S ONE OF THE GREAT LIVING MATHEMATICIANS: He’s written 5 papers and I’ve read 2 of them. […]
I’VE HEARD SO MUCH ABOUT YOU: Stalling a minute may give me time to recall who you are. […]
QUANTIFY: I can’t find anything wrong with your proof except that it won’t work if x is a moon of Jupiter (popular in applied math courses). […]
SKETCH OF A PROOF: I couldn’t verify all the details, so I’ll break it down into the parts I couldn’t prove.
YOUR TALK WAS VERY INTERESTING: I can’t think of anything to say about your talk.” (‘Unknown’)
“Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines which conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view.” (Carl Boyer)
“It’s the nature of mathematics to pose more problems than it can solve.” (Ivars Peterson)
“the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed […] The chief merit of mathematization is that it compels us to become conscious of what we are assuming.” (Bertrand de Jouvenel)
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” (Albert Einstein)
“[Mathematics] includes much that will neither hurt one who does not know it nor help one who does.” (J. B. Mencke)
“Pure mathematics consists entirely of asseverations to the extent that, if such and such a proposition is true of anything, then such and such another proposition is true of anything. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” (Bertrand Russell)
“Mathematical rigor is like clothing; in its style it ought to suit the occasion, and it diminishes comfort and restricts freedom of movement if it is either too loose or too tight.” (G. F. Simmons).
“at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up … In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.” (John von Neumann)
i. A lecture on mathematical proofs:
ii. “In the fall of 1944, only seven percent of all bombs dropped by the Eighth Air Force hit within 1,000 feet of their aim point.”
From wikipedia’s article on Strategic bombing during WW2. The article has a lot of stuff. The ‘RAF estimates of destruction of “built up areas” of major German cities’ numbers in the article made my head spin – they didn’t bomb the Germans back to the stone age, but they sure tried. Here’s another observation from the article:
“After the war, the U.S. Strategic Bombing Survey reviewed the available casualty records in Germany, and concluded that official German statistics of casualties from air attack had been too low. The survey estimated that at a minimum 305,000 were killed in German cities due to bombing and estimated a minimum of 780,000 wounded. Roughly 7,500,000 German civilians were also rendered homeless.” (The German population at the time was roughly 70 million).
iii. Also war-related: Eddie Slovik:
“Edward Donald “Eddie” Slovik (February 18, 1920 – January 31, 1945) was a United States Army soldier during World War II and the only American soldier to be court-martialled and executed for desertion since the American Civil War.
Although over 21,000 American soldiers were given varying sentences for desertion during World War II, including 49 death sentences, Slovik’s was the only death sentence that was actually carried out.
During World War II, 1.7 million courts-martial were held, representing one third of all criminal cases tried in the United States during the same period. Most of the cases were minor, as were the sentences. Nevertheless, a clemency board, appointed by the Secretary of War in the summer of 1945, reviewed all general courts-martial where the accused was still in confinement. That Board remitted or reduced the sentence in 85 percent of the 27,000 serious cases reviewed. The death penalty was rarely imposed, and those cases typically were for rapes or murders. […] In France during World War I from 1917 to 1918, the United States Army executed 35 of its own soldiers, but all were convicted of rape and/or unprovoked murder of civilians and not for military offenses. During World War II in all theaters of the war, the United States military executed 102 of its own soldiers for rape and/or unprovoked murder of civilians, but only Slovik was executed for the military offense of desertion. […] of the 2,864 army personnel tried for desertion for the period January 1942 through June 1948, 49 were convicted and sentenced to death, and 48 of those sentences were voided by higher authority.”
What motivated me to read the article was mostly curiosity about how many people were actually executed for deserting during the war, a question I’d never encountered any answers to previously. The US number turned out to be, well, let’s just say it’s lower than I’d expected it would be. American soldiers who chose to desert during the war seem to have had much, much better chances of surviving the war than had soldiers who did not. Slovik was not a lucky man. On a related note, given numbers like these I’m really surprised desertion rates were not much higher than they were; presumably community norms (”desertion = disgrace’, which would probably rub off on other family members…’) played a key role here.
iv. Chess and infinity. I haven’t posted this link before even though the thread is a few months old, and I figured that given that I just had a conversation on related matters in the comment section of SCC (here’s a link) I might as well repost some of this stuff here. Some key points from the thread (I had to make slight formatting changes to the quotes because wordpress had trouble displaying some of the numbers, but the content is unchanged):
“Shannon has estimated the number of possible legal positions to be about 1043. The number of legal games is quite a bit higher, estimated by Littlewood and Hardy to be around 1010^5 (commonly cited as 1010^50 perhaps due to a misprint). This number is so large that it can’t really be compared with anything that is not combinatorial in nature. It is far larger than the number of subatomic particles in the observable universe, let alone stars in the Milky Way galaxy.
As for your bonus question, a typical chess game today lasts about 40 to 60 moves (let’s say 50). Let us say that there are 4 reasonable candidate moves in any given position. I suspect this is probably an underestimate if anything, but let’s roll with it. That gives us about 42×50 ≈ 1060 games that might reasonably be played by good human players. If there are 6 candidate moves, we get around 1077, which is in the neighbourhood of the number of particles in the observable universe.”
“To put 1010^5 into perspective:
There are 1080 protons in the Universe. Now imagine inside each proton, we had a whole entire Universe. Now imagine again that inside each proton inside each Universe inside each proton, you had another Universe. If you count up all the protons, you get (1080 )3 = 10240, which is nowhere near the number we’re looking for.
You have to have Universes inside protons all the way down to 1250 steps to get the number of legal chess games that are estimated to exist. […]
Imagine that every single subatomic particle in the entire observable universe was a supercomputer that analysed a possible game in a single Planck unit of time (10-43 seconds, the time it takes light in a vacuum to travel 10-20 times the width of a proton), and that every single subatomic particle computer was running from the beginning of time up until the heat death of the Universe, 101000 years ≈ 1011 × 101000 seconds from now.
Even in these ridiculously favorable conditions, we’d only be able to calculate
1080 × 1043 × 1011 × 101000 = 101134
possible games. Again, this doesn’t even come close to 1010^5 = 10100000 .
Basically, if we ever solve the game of chess, it definitely won’t be through brute force.”
v. An interesting resource which a friend of mine recently shared with me and which I thought I should share here as well: Nature Reviews – Disease Primers.
vi. Here are some words I’ve recently encountered on vocabulary.com: augury, spangle, imprimatur, apperception, contrition, ensconce, impuissance, acquisitive, emendation, tintinnabulation, abalone, dissemble, pellucid, traduce, objurgation, lummox, exegesis, probity, recondite, impugn, viscid, truculence, appurtenance, declivity, adumbrate, euphony, educe, titivate, cerulean, ardour, vulpine.
i. Lock (water transport). Zumerchik and Danver’s book covered this kind of stuff as well, sort of, and I figured that since I’m not going to blog the book – for reasons provided in my goodreads review here – I might as well add a link or two here instead. The words ‘sort of’ above are in my opinion justified because the book coverage is so horrid you’d never even know what a lock is used for from reading that book; you’d need to look that up elsewhere.
On a related note there’s a lot of stuff in that book about the history of water transport etc. which you probably won’t get from these articles, but having a look here will give you some idea about which sort of topics many of the chapters of the book are dealing with. Also, stuff like this and this. The book coverage of the latter topic is incidentally much, much more detailed than is that wiki article, and the article – as well as many other articles about related topics (economic history, etc.) on the wiki, to the extent that they even exist – could clearly be improved greatly by adding content from books like this one. However I’m not going to be the guy doing that.
ii. Congruence (geometry).
I’d note that this is a topic which seems to be reasonably well covered on wikipedia; there’s for example also a ‘good article’ on the Everglades and a featured article about the Everglades National Park. A few quotes and observations from the article:
“The geography and ecology of the Everglades involve the complex elements affecting the natural environment throughout the southern region of the U.S. state of Florida. Before drainage, the Everglades were an interwoven mesh of marshes and prairies covering 4,000 square miles (10,000 km2). […] Although sawgrass and sloughs are the enduring geographical icons of the Everglades, other ecosystems are just as vital, and the borders marking them are subtle or nonexistent. Pinelands and tropical hardwood hammocks are located throughout the sloughs; the trees, rooted in soil inches above the peat, marl, or water, support a variety of wildlife. The oldest and tallest trees are cypresses, whose roots are specially adapted to grow underwater for months at a time.”
“A vast marshland could only have been formed due to the underlying rock formations in southern Florida. The floor of the Everglades formed between 25 million and 2 million years ago when the Florida peninsula was a shallow sea floor. The peninsula has been covered by sea water at least seven times since the earliest bedrock formation. […] At only 5,000 years of age, the Everglades is a young region in geological terms. Its ecosystems are in constant flux as a result of the interplay of three factors: the type and amount of water present, the geology of the region, and the frequency and severity of fires. […] Water is the dominant element in the Everglades, and it shapes the land, vegetation, and animal life of South Florida. The South Florida climate was once arid and semi-arid, interspersed with wet periods. Between 10,000 and 20,000 years ago, sea levels rose, submerging portions of the Florida peninsula and causing the water table to rise. Fresh water saturated the limestone, eroding some of it and creating springs and sinkholes. The abundance of fresh water allowed new vegetation to take root, and through evaporation formed thunderstorms. Limestone was dissolved by the slightly acidic rainwater. The limestone wore away, and groundwater came into contact with the surface, creating a massive wetland ecosystem. […] Only two seasons exist in the Everglades: wet (May to November) and dry (December to April). […] The Everglades are unique; no other wetland system in the world is nourished primarily from the atmosphere. […] Average annual rainfall in the Everglades is approximately 62 inches (160 cm), though fluctuations of precipitation are normal.”
“Between 1871 and 2003, 40 tropical cyclones struck the Everglades, usually every one to three years.”
“Islands of trees featuring dense temperate or tropical trees are called tropical hardwood hammocks. They may rise between 1 and 3 feet (0.30 and 0.91 m) above water level in freshwater sloughs, sawgrass prairies, or pineland. These islands illustrate the difficulty of characterizing the climate of the Everglades as tropical or subtropical. Hammocks in the northern portion of the Everglades consist of more temperate plant species, but closer to Florida Bay the trees are tropical and smaller shrubs are more prevalent. […] Islands vary in size, but most range between 1 and 10 acres (0.40 and 4.05 ha); the water slowly flowing around them limits their size and gives them a teardrop appearance from above. The height of the trees is limited by factors such as frost, lightning, and wind: the majority of trees in hammocks grow no higher than 55 feet (17 m). […] There are more than 50 varieties of tree snails in the Everglades; the color patterns and designs unique to single islands may be a result of the isolation of certain hammocks. […] An estimated 11,000 species of seed-bearing plants and 400 species of land or water vertebrates live in the Everglades, but slight variations in water levels affect many organisms and reshape land formations.”
“Because much of the coast and inner estuaries are built by mangroves—and there is no border between the coastal marshes and the bay—the ecosystems in Florida Bay are considered part of the Everglades. […] Sea grasses stabilize sea beds and protect shorelines from erosion by absorbing energy from waves. […] Sea floor patterns of Florida Bay are formed by currents and winds. However, since 1932, sea levels have been rising at a rate of 1 foot (0.30 m) per 100 years. Though mangroves serve to build and stabilize the coastline, seas may be rising more rapidly than the trees are able to build.”
iv. Chang and Eng Bunker. Not a long article, but interesting:
“Chang (Chinese: 昌; pinyin: Chāng; Thai: จัน, Jan, rtgs: Chan) and Eng (Chinese: 恩; pinyin: Ēn; Thai: อิน In) Bunker (May 11, 1811 – January 17, 1874) were Thai-American conjoined twin brothers whose condition and birthplace became the basis for the term “Siamese twins”.”
I loved some of the implicit assumptions in this article: “Determined to live as normal a life they could, Chang and Eng settled on their small plantation and bought slaves to do the work they could not do themselves. […] Chang and Adelaide [his wife] would become the parents of eleven children. Eng and Sarah [‘the other wife’] had ten.”
A ‘normal life’ indeed… The women the twins married were incidentally sisters who ended up disliking each other (I can’t imagine why…).
v. Genie (feral child). This is a very long article, and you should be warned that many parts of it may not be pleasant to read. From the article:
“Genie (born 1957) is the pseudonym of a feral child who was the victim of extraordinarily severe abuse, neglect and social isolation. Her circumstances are prominently recorded in the annals of abnormal child psychology. When Genie was a baby her father decided that she was severely mentally retarded, causing him to dislike her and withhold as much care and attention as possible. Around the time she reached the age of 20 months Genie’s father decided to keep her as socially isolated as possible, so from that point until she reached 13 years, 7 months, he kept her locked alone in a room. During this time he almost always strapped her to a child’s toilet or bound her in a crib with her arms and legs completely immobilized, forbade anyone from interacting with her, and left her severely malnourished. The extent of Genie’s isolation prevented her from being exposed to any significant amount of speech, and as a result she did not acquire language during childhood. Her abuse came to the attention of Los Angeles child welfare authorities on November 4, 1970.
In the first several years after Genie’s early life and circumstances came to light, psychologists, linguists and other scientists focused a great deal of attention on Genie’s case, seeing in her near-total isolation an opportunity to study many aspects of human development. […] In early January 1978 Genie’s mother suddenly decided to forbid all of the scientists except for one from having any contact with Genie, and all testing and scientific observations of her immediately ceased. Most of the scientists who studied and worked with Genie have not seen her since this time. The only post-1977 updates on Genie and her whereabouts are personal observations or secondary accounts of them, and all are spaced several years apart. […]
Genie’s father had an extremely low tolerance for noise, to the point of refusing to have a working television or radio in the house. Due to this, the only sounds Genie ever heard from her parents or brother on a regular basis were noises when they used the bathroom. Although Genie’s mother claimed that Genie had been able to hear other people talking in the house, her father almost never allowed his wife or son to speak and viciously beat them if he heard them talking without permission. They were particularly forbidden to speak to or around Genie, so what conversations they had were therefore always very quiet and out of Genie’s earshot, preventing her from being exposed to any meaningful language besides her father’s occasional swearing. […] Genie’s father fed Genie as little as possible and refused to give her solid food […]
In late October 1970, Genie’s mother and father had a violent argument in which she threatened to leave if she could not call her parents. He eventually relented, and later that day Genie’s mother was able to get herself and Genie away from her husband while he was out of the house […] She and Genie went to live with her parents in Monterey Park. Around three weeks later, on November 4, after being told to seek disability benefits for the blind, Genie’s mother decided to do so in nearby Temple City, California and brought Genie along with her.
On account of her near-blindness, instead of the disabilities benefits office Genie’s mother accidentally entered the general social services office next door. The social worker who greeted them instantly sensed something was not right when she first saw Genie and was shocked to learn Genie’s true age was 13, having estimated from her appearance and demeanor that she was around 6 or 7 and possibly autistic. She notified her supervisor, and after questioning Genie’s mother and confirming Genie’s age they immediately contacted the police. […]
Upon admission to Children’s Hospital, Genie was extremely pale and grossly malnourished. She was severely undersized and underweight for her age, standing 4 ft 6 in (1.37 m) and weighing only 59 pounds (27 kg) […] Genie’s gross motor skills were extremely weak; she could not stand up straight nor fully straighten any of her limbs. Her movements were very hesitant and unsteady, and her characteristic “bunny walk”, in which she held her hands in front of her like claws, suggested extreme difficulty with sensory processing and an inability to integrate visual and tactile information. She had very little endurance, only able to engage in any physical activity for brief periods of time. […]
Despite tests conducted shortly after her admission which determined Genie had normal vision in both eyes she could not focus them on anything more than 10 feet (3 m) away, which corresponded to the dimensions of the room she was kept in. She was also completely incontinent, and gave no response whatsoever to extreme temperatures. As Genie never ate solid food as a child she was completely unable to chew and had very severe dysphagia, completely unable to swallow any solid or even soft food and barely able to swallow liquids. Because of this she would hold anything which she could not swallow in her mouth until her saliva broke it down, and if this took too long she would spit it out and mash it with her fingers. She constantly salivated and spat, and continually sniffed and blew her nose on anything that happened to be nearby.
Genie’s behavior was typically highly anti-social, and proved extremely difficult for others to control. She had no sense of personal property, frequently pointing to or simply taking something she wanted from someone else, and did not have any situational awareness whatsoever, acting on any of her impulses regardless of the setting. […] Doctors found it extremely difficult to test Genie’s mental age, but on two attempts they found Genie scored at the level of a 13-month-old. […] When upset Genie would wildly spit, blow her nose into her clothing, rub mucus all over her body, frequently urinate, and scratch and strike herself. These tantrums were usually the only times Genie was at all demonstrative in her behavior. […] Genie clearly distinguished speaking from other environmental sounds, but she remained almost completely silent and was almost entirely unresponsive to speech. When she did vocalize, it was always extremely soft and devoid of tone. Hospital staff initially thought that the responsiveness she did show to them meant she understood what they were saying, but later determined that she was instead responding to nonverbal signals that accompanied their speaking. […] Linguists later determined that in January 1971, two months after her admission, Genie only showed understanding of a few names and about 15–20 words. Upon hearing any of these, she invariably responded to them as if they had been spoken in isolation. Hospital staff concluded that her active vocabulary at that time consisted of just two short phrases, “stop it” and “no more”. Beyond negative commands, and possibly intonation indicating a question, she showed no understanding of any grammar whatsoever. […] Genie had a great deal of difficulty learning to count in sequential order. During Genie’s stay with the Riglers, the scientists spent a great deal of time attempting to teach her to count. She did not start to do so at all until late 1972, and when she did her efforts were extremely deliberate and laborious. By 1975 she could only count up to 7, which even then remained very difficult for her.”
“From January 1978 until 1993, Genie moved through a series of at least four additional foster homes and institutions. In some of these locations she was further physically abused and harassed to extreme degrees, and her development continued to regress. […] Genie is a ward of the state of California, and is living in an undisclosed location in the Los Angeles area. In May 2008, ABC News reported that someone who spoke under condition of anonymity had hired a private investigator who located Genie in 2000. She was reportedly living a relatively simple lifestyle in a small private facility for mentally underdeveloped adults, and appeared to be happy. Although she only spoke a few words, she could still communicate fairly well in sign language.“
(This was a review lecture for me as I read a textbook on these topics a few months back going into quite a lot more detail – the post I link to has some relevant links if you’re curious to explore this topic further).
A few relevant links: Group (featured), symmetry group, Cayley table, Abelian group, Symmetry groups of Platonic solids, dual polyhedron, Lagrange’s theorem (group theory), Fermat’s little theorem. I think he was perhaps trying to cover a little bit too much ground in too little time by bringing up the RSA algorithm towards the end, but I’m sort of surprised how many people disliked the video; I don’t think it’s that bad.
The beginning of the lecture has a lot of remarks about Fourier‘s life which are in some sense not ‘directly related’ to the mathematics, and so if this is what you’re most interested in knowing more about you can probably skip the first 11 minutes or so of the lecture without missing out on much. The lecture is very non-technical compared to coverage like this, this, and this (…or this).
I think one thing worth mentioning here is that the lecturer is the author of a rather amazing book on the topic he talks about in the lecture.
i. Invasion of Poland. I recently realized I had no idea e.g. how long it took for the Germans and Soviets to defeat Poland during WW2 (the answer is 1 month and five days). The Germans attacked more than two weeks before the Soviets did. The article has lots of links, like most articles about such topics on wikipedia. Incidentally the question of why France and Britain applied a double standard and only declared war on Germany, and not the Soviet Union, is discussed in much detail in the links provided by u/OldWorldGlory here.
ii. Huaynaputina. From the article:
“A few days before the eruption, someone reported booming noise from the volcano and fog-like gas being emitted from its crater. The locals scrambled to appease the volcano, preparing girls, pets, and flowers for sacrifice.”
This makes sense – what else would one do in a situation like that? Finding a few virgins, dogs and flowers seems like the sensible approach – yes, you have to love humans and how they always react in sensible ways to such crises.
I’m not really sure the rest of the article is really all that interesting, but I found the above sentence both amusing and depressing enough to link to it here.
iii. Albert Pierrepoint. This guy killed hundreds of people.
On the other hand people were fine with it – it was his job. Well, sort of, this is actually slightly complicated. (“Pierrepoint was often dubbed the Official Executioner, despite there being no such job or title”).
Anyway this article is clearly the story of a guy who achieved his childhood dream – though unlike other children, he did not dream of becoming a fireman or a pilot, but rather of becoming the Official Executioner of the country. I’m currently thinking of using Pierrepoint as the main character in the motivational story I plan to tell my nephew when he’s a bit older.
iv. Second Crusade (featured). Considering how many different ‘states’ and ‘kingdoms’ were involved, a surprisingly small amount of people were actually fighting; the article notes that “[t]here were perhaps 50,000 troops in total” on the Christian side when the attack on Damascus was initiated. It wasn’t enough, as the outcome of the crusade was a decisive Muslim victory in the ‘Holy Land’ (Middle East).
v. 0.999… (featured). This thing is equal to one, but it can sometimes be really hard to get even very smart people to accept this fact. Lots of details and some proofs presented in the article.
vi. Shapley–Folkman lemma (‘good article’ – but also a somewhat technical article).
vii. Multituberculata. This article is not that special, but I add it here also because I think it ought to be and I’m actually sort of angry that it’s not; sometimes the coverage provided on wikipedia simply strikes me as grossly unfair, even if this is perhaps a slightly odd way to think about stuff. As pointed out in the article (Agustí points this out in his book as well), “The multituberculates existed for about 120 million years, and are often considered the most successful, diversified, and long-lasting mammals in natural history.” Yet notice how much (/little) coverage the article provides. Now compare the article with this article, or this.
“This book was originally developed alongside the lecture Systems Analysis at the Swiss Federal Institute of Technology (ETH) Zürich, on the basis of lecture notes developed over 12 years. The lecture, together with others on analysis, differential equations and linear algebra, belongs to the basic mathematical knowledge imparted on students of environmental sciences and other related areas at ETH Zürich. […] The book aims to be more than a mathematical treatise on the analysis and modeling of natural systems, yet a certain set of basic mathematical skills are still necessary. We will use linear differential equations, vector and matrix calculus, linear algebra, and even take a glimpse at nonlinear and partial differential equations. Most of the mathematical methods used are covered in the appendices. Their treatment there is brief however, and without proofs. Therefore it will not replace a good mathematics textbook for someone who has not encountered this level of math before. […] The book is firmly rooted in the algebraic formulation of mathematical models, their analytical solution, or — if solutions are too complex or do not exist — in a thorough discussion of the anticipated model properties.”
I finished the book yesterday – here’s my goodreads review (note that the first link in this post was not to the goodreads profile of the book for the reason that goodreads has listed the book under the wrong title). I’ve never read a book about ‘systems analysis’ before, but as I also mention in the goodreads review it turned out that much of this stuff was stuff I’d seen before. There are 8 chapters in the book. Chapter one is a brief introductory chapter, the second chapter contains a short overview of mathematical models (static models, dynamic models, discrete and continuous time models, stochastic models…), the third chapter is a brief chapter about static models (the rest of the book is about dynamic models, but they want you to at least know the difference), the fourth chapter deals with linear (differential equation) models with one variable, chapter 5 extends the analysis to linear models with several variables, chapter 6 is about non-linear models (covers e.g. the Lotka-Volterra model (of course) and the Holling-Tanner model (both were covered in Ecological Dynamics, in much more detail)), chapter 7 deals briefly with time-discrete models and how they are different from continuous-time models (I liked Gurney and Nisbet’s coverage of this stuff a lot better, as that book had a lot more details about these things) and chapter 8 concludes with models including both a time- and a space-dimension, which leads to coverage of concepts such as mixing and transformation, advection, diffusion and exchange in a model context.
How to derive solutions to various types of differential equations, how to calculate eigenvalues and what these tell you about the model dynamics (and how to deal with them when they’re imaginary), phase diagrams/phase planes and topographical maps of system dynamics, fixed points/steady states and their properties, what’s an attractor?, what’s hysteresis and in which model contexts might this phenomenon be present?, the difference between homogeneous and non-homogeneous differential equations and between first order- and higher-order differential equations, which role do the initial conditions play in various contexts?, etc. – it’s this kind of book. Applications included in the book are varied; some of the examples are (as already mentioned) derived from the field of ecology/mathematical biology (there are also e.g. models of phosphate distribution/dynamics in lakes and models of fish population dynamics), others are from chemistry (e.g. models dealing with gas exchange – Fick’s laws of diffusion are e.g. covered in the book, and they also talk about e.g. Henry’s law), physics (e.g. the harmonic oscillator, the Lorenz model) – there are even a few examples from economics (e.g. dealing with interest rates). As they put it in the introduction, “Although most of the examples used here are drawn from the environmental sciences, this book is not an introduction to the theory of aquatic or terrestrial environmental systems. Rather, a key goal of the book is to demonstrate the virtually limitless practical potential of the methods presented.” I’m not sure if they succeeded, but it’s certainly clear from the coverage that you can use the tools they cover in a lot of different contexts.
I’m not quite sure how much mathematics you’ll need to know in order to read and understand this book on your own. In the coverage they seem to me to assume some familiarity with linear algebra, multi-variable calculus, complex analysis (/related trigonometry) (perhaps also basic combinatorics – for example factorials are included without comments about how they work). You should probably take the authors at their words when they say above that the book “will not replace a good mathematics textbook for someone who has not encountered this level of math before”. A related observation is also that regardless of whether you’ve seen this sort of stuff before or not, this is probably not the sort of book you’ll be able to read in a day or two.
I think I’ll try to cover the book in more detail (with much more specific coverage of some main points) tomorrow.
It’s been quite a while since the last time I posted a ‘here’s some interesting stuff I’ve found online’-post, so I’ll do that now even though I actually don’t spend much time randomly looking around for interesting stuff online these days. I added some wikipedia links I’d saved for a ‘wikipedia articles of interest’-post because it usually takes quite a bit of time to write a standard wikipedia post (as it takes time to figure out what to include and what not to include in the coverage) and I figured that if I didn’t add those links here I’d never get around to blogging them.
iii. I found this article about the so-called “Einstellung” effect in chess interesting. I’m however not sure how important this stuff really is. I don’t think it’s sub-optimal for a player to spend a significant amount of time in positions like the ones they analyzed on ideas that don’t work, because usually you’ll only have to spot one idea that does to win the game. It’s obvious that one can argue people spend ‘too much’ time looking for a winning combination in positions where by design no winning combinations exist, but the fact of the matter is that in positions where ‘familiar patterns’ pop up winning resources often do exist, and you don’t win games by overlooking those or by failing to spend time looking for them; occasional suboptimal moves in some contexts may be a reasonable price to pay for increasing your likelihood of finding/playing the best/winning moves when those do exist. Here’s a slightly related link dealing with the question of the potential number of games/moves in chess. Here’s a good wiki article about pawn structures, and here’s one about swindles in chess. I incidentally very recently became a member of the ICC, and I’m frankly impressed with the player pool – which is huge and includes some really strong players (players like Morozevich and Tomashevsky seem to play there regularly). Since I started out on the site I’ve already beaten 3 IMs in bullet and lost a game against Islandic GM Henrik Danielsen. The IMs I’ve beaten were far from the strongest players in the player pool, but in my experience you don’t get to play titled players nearly as often as that on other sites if you’re at my level.
v. You may already have seen this one, but in case you have not: A Philosopher Walks Into A Coffee Shop. More than one of these made me laugh out loud. If you like the post you should take a look at the comments as well, there are some brilliant ones there as well.
vi. Amdahl’s law.
vii. Eigendecomposition of a matrix. On a related note I’m currently reading Imboden and Pfenninger’s Introduction to Systems Analysis (which goodreads for some reason has listed under a wrong title, as the goodreads book title is really the subtitle of the book), and today I had a look at the wiki article on Jacobian matrices and determinants for that reason (the book is about as technical as you’d expect from a book with a title like that).
This post will be brief but I thought that since it’s been a while since I last posted anything and since I just finished reading this book, I wanted to add a few remarks about it here while it was still ‘fresh in my mind’. I’m gradually coming to the conclusion that if I’m to blog all the books I’m reading in the amount of detail I’d ideally like to, I’ll have to read a lot less. This option does not appeal to me; I’d rather provide limited coverage of a book I’ve actually read than not read a book in order to provide more extensive coverage of another book.
Anyway, the book is a rather nice collection of interviews with mathematicians from MIT’s ‘early days’ (in some sense at least – MIT is a rather old institution, but at least some of the people interviewed in this book came along during the days before MIT was what it is today), who talk about the history of the mathematics department of MIT, and other stuff – the people interviewed include an Abel Prize winner and a few people who’ve been members of the Institute for Advanced Study, a former MacArthur Fellow, as well as a guy who used to be on the selection committee for the MacArthur Foundation. All of them are really, really smart, and some of them have lived quite interesting lives. To the extent that these guys aren’t impressive enough on their own, some of them also knew some people most non-mathematicians have probably heard about – this book includes contributions from people who were friends of people like John Nash, Grothendieck, Shannon, Minsky, and Chomsky, and they are people who’ve met and talked to people like John von Neumann, Oppenheimer, Weyl, Heisenberg, and Albert Einstein. They talk a little bit about their work and the history of the mathematics department, but they also talk about other stuff as well; there are various amusing anecdotes along the way (for example one interviewee tells the story about the time he lectured in a gorilla suit at MIT), there are stories about the private parties and social lives of the MIT staff during the fifties (and later), we get some personal stories about mathematicians who fled Europe when the Nazis started to cause trouble, and there are stories about student protests in the late sixties and how they were dealt with – the books spans widely. There was some repetition across the interviews (various people answering similar questions in similar ways), and there was more talk about ‘administrative matters’ than I’d have liked – probably a natural consequence of the fact that a few of them (3? At least three of the contributors..) were former department heads – which is part of why I didn’t give it five stars, but it’s really a quite nice book. I may or may not blog it later in more detail.
“Most elementary statistics books discuss inference for proportions and probabilities, and the primary readership for this monograph is the student of statistics, either at an advanced undergraduate or graduate level. As some of the recommended so-called ‘‘large-sample’’ rules in textbooks have been found to be inappropriate, this monograph endeavors to provide more up-to-date information on these topics. I have also included a number of related topics not generally found in textbooks. The emphasis is on model building and the estimation of parameters from the models.
It is assumed that the reader has a background in statistical theory and inference and is familiar with standard univariate and multivariate distributions, including conditional distributions.”
The above quote is from the the book‘s preface. The book is highly technical – here’s a screencap of a page roughly in the middle:
I think the above picture provides some background as to why I do not think it’s a good idea to provide detailed coverage of the book here. Not all pages are that bad, but this is a book on mathematical statistics. The technical nature of the book made it difficult for me to know how to rate it – I like to ask myself when reading books like this one if I would be able to spot an error in the coverage. In some contexts here I clearly would not be able to do that (given the time I was willing to spend on the book), and when that’s the case I always feel hesitant about rating(/’judging’) books of this nature. I should note that there are pretty much no spelling/formatting errors, and the language is easy to understand (‘if you know enough about statistics…’). I did have one major problem with part of the coverage towards the end of the book, but it didn’t much alter my general impression of the book. The problem was that the author seems to apply (/recommend?) a hypothesis-testing framework for model selection, a practice which although widely used is frankly considered bad statistics by Burnham and Anderson in their book on model selection. In the relevant section of the book Seber discusses an approach to modelling which starts out with a ‘full model’ including both primary effects and various (potentially multi-level) interaction terms (he deals specifically with data derived from multiple (independent?) multinomial distributions, but where the data comes from is not really important here), and then he proceeds to use hypothesis tests of whether interaction terms are zero to determine whether or not interactions should be included in the model or not. For people who don’t know, this model selection method is both very commonly used and a very wrong way to do things; using hypothesis testing as a model selection mechanism is a methodologically invalid approach to model selection, something Burnham and Anderson talks a lot about in their book. I assume I’ll be covering Burnham and Anderson’s book in more detail later on here on the blog, so for now I’ll just make this key point here and then return to that stuff later – if you did not understand the comments above you shouldn’t worry too much about it, I’ll go into much more detail when talking about that stuff later. This problem was the only real problem I had with Seber’s book.
Although I’ll not talk a lot about what the book was about (not only because it might be hard for some readers to follow, I should point out, but also because detailed coverage would take a lot more time than I’d be willing to spend on this stuff), I decided to add a few links to relevant stuff he talks about in the book. Quite a few pages in the book are spent on talking about the properties of various distributions, how to estimate key parameters of interest, and how to construct confidence intervals to be used for hypothesis testing in those specific contexts.
Some of the links below deal with stuff covered in the book, a few others however just deal with stuff I had to look up in order to understand what was going on in the coverage:
Binomial proportion confidence interval. (Coverage of the Wilson score interval, Jeffreys interval, and the Clopper-Pearson interval included in the book).
Fisher’s exact test.
Factorial moment-generating function.
Multidimensional central limit theorem (the book applies this, but doesn’t really talk about it).
It’s been a long time since I had one of these. Questions? Comments? Random observations?
I hate posting posts devoid of content, so here’s some random stuff:
If you think the stuff above is all fun and games I should note that the topic of chiralty, which is one of the things talked about in the lecture above, was actually covered in some detail in Gale’s book, which hardly is a book which spends a great deal of time talking about esoteric mathematical concepts. On a related note, the main reason why I have not blogged that book is incidentally that I lost all notes and highlights I’d made in the first 200 pages of the book when my computer broke down, and I just can’t face reading that book again simply in order to blog it. It’s a good book, with interesting stuff, and I may decide to blog it later, but I don’t feel like doing it at the moment; without highlights and notes it’s a real pain to blog a book, and right now it’s just not worth it to reread the book. Rereading books can be fun – I’ve incidentally been rereading Darwin lately and I may decide to blog this book soon; I imagine I might also choose to reread some of Asimov’s books before long – but it’s not much fun if you’re finding yourself having to do it simply because the computer deleted your work.
Here’s the abstract:
“Statistical power analysis provides the conventional approach to assess error rates when designing a research study. However, power analysis is flawed in that a narrow emphasis on statistical significance is placed as the primary focus of study design. In noisy, small-sample settings, statistically significant results can often be misleading. To help researchers address this problem in the context of their own studies, we recommend design calculations in which (a) the probability of an estimate being in the wrong direction (Type S [sign] error) and (b) the factor by which the magnitude of an effect might be overestimated (Type M [magnitude] error or exaggeration ratio) are estimated. We illustrate with examples from recent published research and discuss the largest challenge in a design calculation: coming up with reasonable estimates of plausible effect sizes based on external information.”
If a study has low power, you can get into a lot of trouble. Some problems are well known, others probably aren’t. A bit more from the paper:
“design calculations can reveal three problems:
1. Most obvious, a study with low power is unlikely to “succeed” in the sense of yielding a statistically significant result.
2. It is quite possible for a result to be significant at the 5% level — with a 95% confidence interval that entirely excludes zero — and for there to be a high chance, sometimes 40% or more, that this interval is on the wrong side of zero. Even sophisticated users of statistics can be unaware of this point — that the probability of a Type S error is not the same as the p value or significance level.
3. Using statistical significance as a screener can lead researchers to drastically overestimate the magnitude of an effect (Button et al., 2013).
Design analysis can provide a clue about the importance of these problems in any particular case.”
“Statistics textbooks commonly give the advice that statistical significance is not the same as practical significance, often with examples in which an effect is clearly demonstrated but is very small […]. In many studies in psychology and medicine, however, the problem is the opposite: an estimate that is statistically significant but with such a large uncertainty that it provides essentially no information about the phenomenon of interest. […] There is a range of evidence to demonstrate that it remains the case that too many small studies are done and preferentially published when “significant.” We suggest that one reason for the continuing lack of real movement on this problem is the historic focus on power as a lever for ensuring statistical significance, with inadequate attention being paid to the difficulties of interpreting statistical significance in underpowered studies. Because insufficient attention has been paid to these issues, we believe that too many small studies are done and preferentially published when “significant.” There is a common misconception that if you happen to obtain statistical significance with low power, then you have achieved a particularly impressive feat, obtaining scientific success under difficult conditions.
However, that is incorrect if the goal is scientific understanding rather than (say) publication in a top journal. In fact, statistically significant results in a noisy setting are highly likely to be in the wrong direction and invariably overestimate the absolute values of any actual effect sizes, often by a substantial factor.”
iii. I’m sure most people who might be interested in following the match are already well aware that Anand and Carlsen are currently competing for the world chess championship, and I’m not going to talk about that match here. However I do want to mention to people interested in improving their chess that I recently came across this site, and that I quite like it. It only deals with endgames, but endgames are really important. If you don’t know much about endgames you may find the videos available here, here and here to be helpful.
iv. A link: Crosss Validated: “Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.”
A friend recently told me about this resource. I knew about the existence of StackExchange, but I haven’t really spent much time there. These days I mostly stick to books and a few sites I already know about; I rarely look for new interesting stuff online. This also means you should not automatically assume I surely already know about X when you’re considering whether to tell me about X in an Open Thread.
I finished this book earlier today. This is a really hard book to blog, and I ended up concluding that I should just cover the material here by adding some links to relevant wiki articles dealing with stuff also covered in the book. It’s presumably far from obvious to someone who has not read the book how the different links are related to each other, but adding details about this as well would mean rewriting the book. Consider the articles samples of the kind of stuff covered in the book.
It should be noted that the book has very few illustrations and a lot of (/nothing but) formulas, theorems, definitions, and examples. Lots of proofs, naturally. I must admit I found the second half of the book unnecessarily hard to follow because it unlike the first half included no images and illustrations at all (like the ones in the various wiki articles below) – you don’t need an illustration on every second page to follow this stuff, but in my opinion you do need some occasionally to help you imagine what’s actually going on; this stuff is quite abstract enough to begin with without removing all ‘visual aids’. Some of the stuff covered in the book was review, but a lot of the stuff covered was new to me and I did have a few ‘huh, I never really realized you could think about [X] that way‘-experiences along the way. One problem making me want to read the book was also that although I’ve encountered some of the concepts introduced along the way – to give an example, the axiom of choice and various properties of sets, such as countability, came up during a micro course last year – I’ve never really read a book about that stuff so it’s sort of felt disconnected and confusing to some extent because I’ve been missing the big picture. It was of course too much to ask for to assume that this book would give me the big picture, but it’s certainly helped a bit to read this book. I incidentally did not find it too surprising that the stuff in this book sort of overlaps a bit with some stuff I’ve previously encountered in micro; in a book like this one there are a lot of thoughts and ideas about how to set up a mathematical system, and you start out at a very simple level and then gradually add more details along the way when you encounter problems or situations which cannot be dealt with with the tools/theorems/axioms already at your disposal – this to me seems to be a conceptually similar approach to numbers and mathematics as the approach to modelling preferences and behaviour which is applied in e.g. Mas-Colell et al.‘s microeconomics textbook, so it makes sense that some similar questions and considerations pop up. A related point is that some of the problems the two analytical approaches are trying to solve are really quite similar; a microeconomist wants to know how best to define preferences and then use the relevant definitions to compare people’s preferences, e.g. by trying to order them in various ways, and he wants to understand which ways of mathematically manipulating the terms used are permissible and which are not – activities which, at least from a certain point of view, are not that different from those of set theorists such as those who wrote this book.
I decided not to rate the book in part because I don’t think it’s really fair to rate a math book when you’re not going to have a go at the exercises. I wanted an overview, I don’t care enough about this stuff to dive in deep and start proving stuff on my own. I was as mentioned annoyed by the lack of illustrations in the second part and occasionally it was really infuriating to encounter the standard ‘…and so it’s obvious that…’ comments which you can find in most math textbooks, but I don’t know how much these things should really matter. Another reason why I decided not to rate the book is that I’ve been suffering from some pretty bad noise pollution over the last days, which has meant that I’ve occasionally had a really hard time concentrating on the mathematics (‘if I’d just had some peace and quiet I might have liked the book much better than I did, and I might have done some of the exercises as well’, is my working hypothesis).
Okay, as mentioned I’ve added some relevant links below – check them out if you want to know more about what this book’s about and what kind of mathematics is covered in there. I haven’t read all of those links, but I’ve read about those things by reading the book. I should note, in case it’s not clear from the links, that some of this stuff is quite weird.
Bijection, injection and surjection.
Family of sets.
Axiom of choice.
Axiom of infinity.
Cantor’s diagonal argument.
Cantor’s continuum hypothesis.
(A minor note: These days when I’m randomly browsing wikipedia and not just looking up concepts or terms found in the books I read, I’m mostly browsing the featured content on wikipedia. There’s a lot of featured stuff, and on average such articles more interesting than random articles. As a result of this approach, all articles covered in the post below are featured articles. A related consequence of this shift may be that I may cover fewer articles in future wikipedia posts than I have in the past; this post only contains five articles, which I believe is slightly less than usual for these posts – a big reason for this being that it sometimes takes a lot of time to read a featured article.)
i. Woolly mammoth.
“The woolly mammoth (Mammuthus primigenius) was a species of mammoth, the common name for the extinct elephant genus Mammuthus. The woolly mammoth was one of the last in a line of mammoth species, beginning with Mammuthus subplanifrons in the early Pliocene. M. primigenius diverged from the steppe mammoth, M. trogontherii, about 200,000 years ago in eastern Asia. Its closest extant relative is the Asian elephant. […] The earliest known proboscideans, the clade which contains elephants, existed about 55 million years ago around the Tethys Sea. […] The family Elephantidae existed six million years ago in Africa and includes the modern elephants and the mammoths. Among many now extinct clades, the mastodon is only a distant relative of the mammoths, and part of the separate Mammutidae family, which diverged 25 million years before the mammoths evolved. […] The woolly mammoth coexisted with early humans, who used its bones and tusks for making art, tools, and dwellings, and the species was also hunted for food. It disappeared from its mainland range at the end of the Pleistocene 10,000 years ago, most likely through a combination of climate change, consequent disappearance of its habitat, and hunting by humans, though the significance of these factors is disputed. Isolated populations survived on Wrangel Island until 4,000 years ago, and on St. Paul Island until 6,400 years ago.”
“The appearance and behaviour of this species are among the best studied of any prehistoric animal due to the discovery of frozen carcasses in Siberia and Alaska, as well as skeletons, teeth, stomach contents, dung, and depiction from life in prehistoric cave paintings. […] Fully grown males reached shoulder heights between 2.7 and 3.4 m (9 and 11 ft) and weighed up to 6 tonnes (6.6 short tons). This is almost as large as extant male African elephants, which commonly reach 3–3.4 m (9.8–11.2 ft), and is less than the size of the earlier mammoth species M. meridionalis and M. trogontherii, and the contemporary M. columbi. […] Woolly mammoths had several adaptations to the cold, most noticeably the layer of fur covering all parts of the body. Other adaptations to cold weather include ears that are far smaller than those of modern elephants […] The small ears reduced heat loss and frostbite, and the tail was short for the same reason […] They had a layer of fat up to 10 cm (3.9 in) thick under the skin, which helped to keep them warm. […] The coat consisted of an outer layer of long, coarse “guard hair”, which was 30 cm (12 in) on the upper part of the body, up to 90 cm (35 in) in length on the flanks and underside, and 0.5 mm (0.020 in) in diameter, and a denser inner layer of shorter, slightly curly under-wool, up to 8 cm (3.1 in) long and 0.05 mm (0.0020 in) in diameter. The hairs on the upper leg were up to 38 cm (15 in) long, and those of the feet were 15 cm (5.9 in) long, reaching the toes. The hairs on the head were relatively short, but longer on the underside and the sides of the trunk. The tail was extended by coarse hairs up to 60 cm (24 in) long, which were thicker than the guard hairs. It is likely that the woolly mammoth moulted seasonally, and that the heaviest fur was shed during spring.”
“Woolly mammoths had very long tusks, which were more curved than those of modern elephants. The largest known male tusk is 4.2 m (14 ft) long and weighs 91 kg (201 lb), but 2.4–2.7 m (7.9–8.9 ft) and 45 kg (99 lb) was a more typical size. Female tusks averaged at 1.5–1.8 m (4.9–5.9 ft) and weighed 9 kg (20 lb). About a quarter of the length was inside the sockets. The tusks grew spirally in opposite directions from the base and continued in a curve until the tips pointed towards each other. In this way, most of the weight would have been close to the skull, and there would be less torque than with straight tusks. The tusks were usually asymmetrical and showed considerable variation, with some tusks curving down instead of outwards and some being shorter due to breakage.”
“Woolly mammoths needed a varied diet to support their growth, like modern elephants. An adult of six tonnes would need to eat 180 kg (397 lb) daily, and may have foraged as long as twenty hours every day. […] Woolly mammoths continued growing past adulthood, like other elephants. Unfused limb bones show that males grew until they reached the age of 40, and females grew until they were 25. The frozen calf “Dima” was 90 cm (35 in) tall when it died at the age of 6–12 months. At this age, the second set of molars would be in the process of erupting, and the first set would be worn out at 18 months of age. The third set of molars lasted for ten years, and this process was repeated until the final, sixth set emerged when the animal was 30 years old. A woolly mammoth could probably reach the age of 60, like modern elephants of the same size. By then the last set of molars would be worn out, the animal would be unable to chew and feed, and it would die of starvation.”
“The habitat of the woolly mammoth is known as “mammoth steppe” or “tundra steppe”. This environment stretched across northern Asia, many parts of Europe, and the northern part of North America during the last ice age. It was similar to the grassy steppes of modern Russia, but the flora was more diverse, abundant, and grew faster. Grasses, sedges, shrubs, and herbaceous plants were present, and scattered trees were mainly found in southern regions. This habitat was not dominated by ice and snow, as is popularly believed, since these regions are thought to have been high-pressure areas at the time. The habitat of the woolly mammoth also supported other grazing herbivores such as the woolly rhinoceros, wild horses and bison. […] A 2008 study estimated that changes in climate shrank suitable mammoth habitat from 7,700,000 km2 (3,000,000 sq mi) 42,000 years ago to 800,000 km2 (310,000 sq mi) 6,000 years ago. Woolly mammoths survived an even greater loss of habitat at the end of the Saale glaciation 125,000 years ago, and it is likely that humans hunted the remaining populations to extinction at the end of the last glacial period. […] Several woolly mammoth specimens show evidence of being butchered by humans, which is indicated by breaks, cut-marks, and associated stone tools. It is not known how much prehistoric humans relied on woolly mammoth meat, since there were many other large herbivores available. Many mammoth carcasses may have been scavenged by humans rather than hunted. Some cave paintings show woolly mammoths in structures interpreted as pitfall traps. Few specimens show direct, unambiguous evidence of having been hunted by humans.”
“While frozen woolly mammoth carcasses had been excavated by Europeans as early as 1728, the first fully documented specimen was discovered near the delta of the Lena River in 1799 by Ossip Schumachov, a Siberian hunter. Schumachov let it thaw until he could retrieve the tusks for sale to the ivory trade. [Aargh!] […] The 1901 excavation of the “Berezovka mammoth” is the best documented of the early finds. It was discovered by the Berezovka River, and the Russian authorities financed its excavation. Its head was exposed, and the flesh had been scavenged. The animal still had grass between its teeth and on the tongue, showing that it had died suddenly. […] By 1929, the remains of 34 mammoths with frozen soft tissues (skin, flesh, or organs) had been documented. Only four of them were relatively complete. Since then, about that many more have been found.”
ii. Daniel Lambert.
“Daniel Lambert (13 March 1770 – 21 June 1809) was a gaol keeper[n 1] and animal breeder from Leicester, England, famous for his unusually large size. After serving four years as an apprentice at an engraving and die casting works in Birmingham, he returned to Leicester around 1788 and succeeded his father as keeper of Leicester’s gaol. […] At the time of Lambert’s return to Leicester, his weight began to increase steadily, even though he was athletically active and, by his own account, abstained from drinking alcohol and did not eat unusual amounts of food. In 1805, Lambert’s gaol closed. By this time, he weighed 50 stone (700 lb; 318 kg), and had become the heaviest authenticated person up to that point in recorded history. Unemployable and sensitive about his bulk, Lambert became a recluse.
In 1806, poverty forced Lambert to put himself on exhibition to raise money. In April 1806, he took up residence in London, charging spectators to enter his apartments to meet him. Visitors were impressed by his intelligence and personality, and visiting him became highly fashionable. After some months on public display, Lambert grew tired of exhibiting himself, and in September 1806, he returned, wealthy, to Leicester, where he bred sporting dogs and regularly attended sporting events. Between 1806 and 1809, he made a further series of short fundraising tours.
In June 1809, he died suddenly in Stamford. At the time of his death, he weighed 52 stone 11 lb (739 lb; 335 kg), and his coffin required 112 square feet (10.4 m2) of wood. Despite the coffin being built with wheels to allow easy transport, and a sloping approach being dug to the grave, it took 20 men almost half an hour to drag his casket into the trench, in a newly opened burial ground to the rear of St Martin’s Church.”
“Sensitive about his weight, Daniel Lambert refused to allow himself to be weighed, but sometime around 1805, some friends persuaded him to come with them to a cock fight in Loughborough. Once he had squeezed his way into their carriage, the rest of the party drove the carriage onto a large scale and jumped out. After deducting the weight of the (previously weighed) empty carriage, they calculated that Lambert’s weight was now 50 stone (700 lb; 318 kg), and that he had thus overtaken Edward Bright, the 616-pound (279 kg) “Fat Man of Maldon”, as the heaviest authenticated person in recorded history.
Despite his shyness, Lambert badly needed to earn money, and saw no alternative to putting himself on display, and charging his spectators. On 4 April 1806, he boarded a specially built carriage and travelled from Leicester to his new home at 53 Piccadilly, then near the western edge of London. For five hours each day, he welcomed visitors into his home, charging each a shilling (about £3.5 as of 2014). […] Lambert shared his interests and knowledge of sports, dogs and animal husbandry with London’s middle and upper classes, and it soon became highly fashionable to visit him, or become his friend. Many called repeatedly; one banker made 20 visits, paying the admission fee on each occasion. […] His business venture was immediately successful, drawing around 400 paying visitors per day. […] People would travel long distances to see him (on one occasion, a party of 14 travelled to London from Guernsey),[n 5] and many would spend hours speaking with him on animal breeding.”
“After some months in London, Lambert was visited by Józef Boruwłaski, a 3-foot 3-inch (99 cm) dwarf then in his seventies. Born in 1739 to a poor family in rural Pokuttya, Boruwłaski was generally considered to be the last of Europe’s court dwarfs. He was introduced to the Empress Maria Theresa in 1754, and after a short time residing with deposed Polish king Stanisław Leszczyński, he exhibited himself around Europe, thus becoming a wealthy man. At age 60, he retired to Durham, where he became such a popular figure that the City of Durham paid him to live there and he became one of its most prominent citizens […] The meeting of Lambert and Boruwłaski, the largest and smallest men in the country, was the subject of enormous public interest”
“There was no autopsy, and the cause of Lambert’s death is unknown. While many sources say that he died of a fatty degeneration of the heart or of stress on his heart caused by his bulk, his behaviour in the period leading to his death does not match that of someone suffering from cardiac insufficiency; witnesses agree that on the morning of his death he appeared well, before he became short of breath and collapsed. Bondeson (2006) speculates that the most consistent explanation of his death, given his symptoms and medical history, is that he had a sudden pulmonary embolism.”
“The exposed geology of the Capitol Reef area presents a record of mostly Mesozoic-aged sedimentation in an area of North America in and around Capitol Reef National Park, on the Colorado Plateau in southeastern Utah.
Nearly 10,000 feet (3,000 m) of sedimentary strata are found in the Capitol Reef area, representing nearly 200 million years of geologic history of the south-central part of the U.S. state of Utah. These rocks range in age from Permian (as old as 270 million years old) to Cretaceous (as young as 80 million years old.) Rock layers in the area reveal ancient climates as varied as rivers and swamps (Chinle Formation), Sahara-like deserts (Navajo Sandstone), and shallow ocean (Mancos Shale).
The area’s first known sediments were laid down as a shallow sea invaded the land in the Permian. At first sandstone was deposited but limestone followed as the sea deepened. After the sea retreated in the Triassic, streams deposited silt before the area was uplifted and underwent erosion. Conglomerate followed by logs, sand, mud and wind-transported volcanic ash were later added. Mid to Late Triassic time saw increasing aridity, during which vast amounts of sandstone were laid down along with some deposits from slow-moving streams. As another sea started to return it periodically flooded the area and left evaporite deposits. Barrier islands, sand bars and later, tidal flats, contributed sand for sandstone, followed by cobbles for conglomerate and mud for shale. The sea retreated, leaving streams, lakes and swampy plains to become the resting place for sediments. Another sea, the Western Interior Seaway, returned in the Cretaceous and left more sandstone and shale only to disappear in the early Cenozoic.”
“The Laramide orogeny compacted the region from about 70 million to 50 million years ago and in the process created the Rocky Mountains. Many monoclines (a type of gentle upward fold in rock strata) were also formed by the deep compressive forces of the Laramide. One of those monoclines, called the Waterpocket Fold, is the major geographic feature of the park. The 100 mile (160 km) long fold has a north-south alignment with a steeply east-dipping side. The rock layers on the west side of the Waterpocket Fold have been lifted more than 7,000 feet (2,100 m) higher than the layers on the east. Thus older rocks are exposed on the western part of the fold and younger rocks on the eastern part. This particular fold may have been created due to movement along a fault in the Precambrian basement rocks hidden well below any exposed formations. Small earthquakes centered below the fold in 1979 may be from such a fault. […] Ten to fifteen million years ago the entire region was uplifted several thousand feet (well over a kilometer) by the creation of the Colorado Plateaus. This time the uplift was more even, leaving the overall orientation of the formations mostly intact. Most of the erosion that carved today’s landscape occurred after the uplift of the Colorado Plateau with much of the major canyon cutting probably occurring between 1 and 6 million years ago.”
Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, “Tangencies”); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them […] and each solution circle encloses or excludes the three given circles in a different way […] The general statement of Apollonius’ problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. Solutions to Apollonius’ problem are sometimes called Apollonius circles, although the term is also used for other types of circles associated with Apollonius. […] A rich repertoire of geometrical and algebraic methods have been developed to solve Apollonius’ problem, which has been called “the most famous of all” geometry problems.”
v. Globular cluster.
“A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is derived from the Latin globulus—a small sphere. A globular cluster is sometimes known more simply as a globular.
Globular clusters, which are found in the halo of a galaxy, contain considerably more stars and are much older than the less dense galactic, or open clusters, which are found in the disk. Globular clusters are fairly common; there are about 150 to 158 currently known globular clusters in the Milky Way, with perhaps 10 to 20 more still undiscovered. Large galaxies can have more: Andromeda, for instance, may have as many as 500. […]
Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters. The Sagittarius Dwarf galaxy and the disputed Canis Major Dwarf galaxy appear to be in the process of donating their associated globular clusters (such as Palomar 12) to the Milky Way. This demonstrates how many of this galaxy’s globular clusters might have been acquired in the past.
Although it appears that globular clusters contain some of the first stars to be produced in the galaxy, their origins and their role in galactic evolution are still unclear.”
“Mathematical models underpin much ecological theory, […] [y]et most students of ecology and environmental science receive much less formal training in mathematics than their counterparts in other scientific disciplines. Motivating both graduate and undergraduate students to study ecological dynamics thus requires an introduction which is initially accessible with limited mathematical and computational skill, and yet offers glimpses of the state of the art in at least some areas. This volume represents our attempt to reconcile these conflicting demands […] Ecology is the branch of biology that deals with the interaction of living organisms with their environment. […] The primary aim of this book is to develop general theory for describing ecological dynamics. Given this aspiration, it is useful to identify questions that will be relevant to a wide range of organisms and/or habitats. We shall distinguish questions relating to individuals, populations, communities, and ecosystems. A population is all the organisms of a particular species in a given region. A community is all the populations in a given region. An ecosystem is a community related to its physical and chemical environment. […] Just as the physical and chemical properties of materials are the result of interactions involving individual atoms and molecules, so the dynamics of populations and communities can be interpreted as the combined effects of properties of many individuals […] All models are (at best) approximations to the truth so, given data of sufficient quality and diversity, all models will turn out to be false. The key to understanding the role of models in most ecological applications is to recognise that models exist to answer questions. A model may provide a good description of nature in one context but be woefully inadequate in another. […] Ecology is no different from other disciplines in its reliance on simple models to underpin understanding of complex phenomena. […] the real world, with all its complexity, is initially interpreted through comparison with the simplistic situations described by the models. The inevitable deviations from the model predictions [then] become the starting point for the development of more specific theory.”
I haven’t blogged this book yet even if it’s been a while since I finished it, and I figured I ought to talk a little bit about it now. As pointed out on goodreads, I really liked the book. It’s basically a math textbook for biologists which deals with how to set up models in a specific context, that dealing with questions pertaining to ecological dynamics; having read the above quote you should at this point at least have some idea which kind of stuff this field deals with. Here are a few links to examples of applications mentioned/covered in the book which may give you a better idea of the kinds of things covered.
There are 9 chapters in the book, and only the introductory chapter has fewer than 50 ‘named’ equations – most have around 70-80 equations, and 3 of them have more than 100. I have tried to avoid equations in this post in part because it’s hell to deal with them in wordpress, so I’ll be leaving out a lot of stuff in my coverage. Large chunks of the coverage was to some extent review but there was also some new stuff in there. The book covers material both intended for undergraduates and graduates, and even if the book is presumably intended for biology majors many of the ideas also can be ‘transferred’ to other contexts where the same types of specific modelling frameworks might be applied; for example there are some differences between discrete-time models and continuous-time models, and those differences apply regardless of whether you’re modelling animal behaviour or, say, human behaviour. A local stability analysis looks quite similar in the contexts of an economic model and an ecological model. Etc. I’ve tried to mostly talk about rather ‘general stuff’ in this coverage, i.e. model concepts and key ideas covered in the book which might also be applicable in other fields of research as well. I’ve tried to keep things reasonably simple in this post, and I’ve only talked about stuff from the first three chapters.
“The simplest ecological models, called deterministic models, make the assumption that if we know the present condition of a system, we can predict its future. Before we can begin to formulate such a model, we must decide what quantities, known as state variables, we shall use to describe the current condition of the system. This choice always involves a subtle balance of biological realism (or at least plausibility) against mathematical complexity. […] The first requirement in formulating a usable model is […] to decide which characteristics are dynamically important in the context of the questions the model seeks to answer. […] The diversity of individual characteristics and behaviours implies that without considerable effort at simplification, a change of focus towards communities will be accompanied by an explosive increase in model complexity. […] A dynamical model is a mathematical statement of the rules governing change. The majority of models express these rules either as an update rule, specifying the relationship between the current and future state of the system, or as a differential equation, specifying the rate of change of the state variables. […] A system with [the] property [that the update rule does not depend on time] is said to be autonomous. […] [If the update rule depends on time, the models are called non-autonomous].”
“Formulation of a dynamic model always starts by identifying the fundamental processes in the system under investigation and then setting out, in mathematical language, the statement that changes in system state can only result from the operation of these processes. The “bookkeeping” framework which expresses this insight is often called a conservation equation or a balance equation. […] Writing down balance equations is just the first step in formulating an ecological model, since only in the most restrictive circumstances do balance equations on their own contain enough information to allow prediction of future values of state variables. In general, [deterministic] model formulation involves three distinct steps: *choose state variables, *derive balance equations, *make model-specific assumptions.
Selection of state variables involves biological or ecological judgment […] Deriving balance equations involves both ecological choices (what processes to include) and mathematical reasoning. The final step, the selection of assumptions particular to any one model, is left to last in order to facilitate model refinement. For example, if a model makes predictions that are at variance with observation, we may wish to change one of the model assumptions, while still retaining the same state variables and processes in the balance equations. […] a remarkably good approximation to […] stochastic dynamics is often obtained by regarding the dynamics as ‘perturbations’ of a non-autonomous, deterministic system. […] although randomness is ubiquitous, deterministic models are an appropriate starting point for much ecological modelling. […] even where deterministic models are inadequate, an essential prerequisite to the formulation and analysis of many complex, stochastic models is a good understanding of a deterministic representation of the system under investigation.”
“Faced with an update rule or a balance equation describing an ecological system, what do we do? The most obvious line of attack is to attempt to find an analytical solution […] However, except for the simplest models, analytical solutions tend to be impossible to derive or to involve formulae so complex as to be completely unhelpful. In other situations, an explicit solution can be calculated numerically. A numerical solution of a difference equation is a table of values of the state variable (or variables) at successive time steps, obtained by repeated application of the update rule […] Numerical solutions of differential equations are more tricky [but sophisticated methods for finding them do exist] […] for simple systems it is possible to obtain considerable insight by ‘numerical experiments’ involving solutions for a number of parameter values and/or initial conditions. For more complex models, numerical analysis is typically the only approach available. But the unpleasant reality is that in the vast majority of investigations it proves impossible to obtain complete or near-complete information about a dynamical system, either by deriving analytical solutions or by numerical experimentation. It is therefore reassuring that over the past century or so, mathematicians have developed methods of determining the qualitative properties of the solutions of dynamic equations, and thus answering many questions […] without explicitly solving the equations concerned.”
“[If] the long-term behaviour of the state variable is independent of the initial condition […] the ‘end state’ […] is known as an attractor. […] Equilibrium states need not be attractors; they can be repellers [as well] […] if a dynamical system has an equilibrium state, any initial condition other than the exact equilibrium value may lead to the state variable converging towards the equilibrium or diverging away from it. We characterize such equilibria as stable and unstable respectively. In some models all initial conditions result in the state variable eventually converging towards a single equilibrium value. We characterize such equilibria as globally stable. An equilibrium that is approached only from a subset of all possible initial conditions (often those close to the equilibrium itself) is said to be locally stable. […] The combination of non-periodic solutions and sensitive dependence on initial conditions is the signature of the pattern of behaviour known to mathematicians as chaos.
“Most variables and parameters in models have units. […] However, the behaviour of a natural system cannot be affected by the units in which we chose to measure the quantities we use to describe it. This implies that it should be possible to write down the defining equations of a model in a form independent of the units we use. For any dynamical equation to be valid, the quantities being equated must be measured in the same units. How then do we restate such an equation in a form which is unaffected by our choice of units? The answer lies in identifying a natural scale or base unit for each quantity in the equations and then using the ratio of each variable to its natural scale in our dynamic description. Since such ratios are pure numbers, we say that they are dimensionless. If a dynamic equation couched in terms of dimensionless variables is to be valid, then both sides of any equality must likewise be dimensionless. […] the process of non-dimensionalisation, which we call dimensional analysis, can […] yield information on system dynamics. […] Since there is no unique dimensionless form for any set of dynamical equations, it is tempting to cut short the scaling process by ‘setting some parameter(s) equal to one’. Even experienced modellers make embarrasing blunders doing this, and we strongly recommend a systematic […] approach […] The key element in the scaling process is the selection of appropriate base units – the optimal choice being dependent on the questions motivating our study.”
“The starting point for selecting the appropriate formalism [in the context of the time dimension] must […] be recognition that real ecological processes operate in continuous time. Discrete-time models make some approximation to the outcome of these processes over a finite time interval, and should thus be interpreted with care. This caution is particularly important as difference equations are intuitively appealing and computationally simple. […] incautious empirical modelling with difference equations can have surprising (adverse) consequences. […] where the time increment of a discrete-time model is an arbitrary modelling choice, model predictions should be shown to be robust against changes in the value chosen.”
“Of the almost limitless range of relations between population flux and local density, we shall discuss only two extreme possibilities. Advection occurs when an external physical flow (such as an ocean current) transports all the members of the population past the point, x [in a spatially one-dimensional model], with essentially the same velocity, v. […] Diffusion occurs when the members of the population move at random. […] This leads to a net flow rate which is proportional to the spatial gradient of population density, with a constant of proportionality D, which we call the diffusion constant. […] the net flow [in this case] takes individuals from regions of high density to regions of low density” […] […some remarks about reaction-diffusion models, which I’d initially thought I’d cover here but which turned out to be too much work to deal with (the coverage is highly context-dependent)].
It’s been a long time since I had one of these.
Some random stuff I’ve come across:
i. Reviews of Anything. Some pretty funny stuff there. Examples include: Our solar system: 1 star. Reviews of this review. The 5 star Rating System: 9/10. Obese Americans, 1 out of 4. Spell Checker: 1 satr.
iii. The Bad Writing Contest. A quote from the link:
“The move from a structuralist account in which capital is understood to structure social relations in relatively homologous ways to a view of hegemony in which power relations are subject to repetition, convergence, and rearticulation brought the question of temporality into the thinking of structure, and marked a shift from a form of Althusserian theory that takes structural totalities as theoretical objects to one in which the insights into the contingent possibility of structure inaugurate a renewed conception of hegemony as bound up with the contingent sites and strategies of the rearticulation of power.”
In contexts where you socialize with people who write that way, dumbpiphanies may happen.
v. I’m not actually sure I liked this lecture very much (I was very much annoyed by the word ‘cristal’ in the slides in the last part of the lecture; he repeatedly misspells the word crystal in the slides. I find that kind of sloppiness irritating, because I tend to use the existence of spelling errors in lecture notes/slides in mathematical lectures as what might be termed a caution heuristic; if the lecturer did not bother to correct spelling errors, I figure he probably also didn’t bother to correct other errors in the slides – and if you start to think along the way that there might be errors in the slides, a lecture to me becomes less enjoyable to watch, especially when the lecture deals with complicated stuff which is hard enough to follow as it is), but I figured I might as well share it anyway:
vi. arXiv vs snarXiv.