| GS: Gilbert Strang | IAM: Introduction to Applied Mathematics |
| JPB: John P. Boyd | CFSM: Chebyshev and Fourier Spectral Methods |
| AI: Arieh Iserles | AFCINADE: A First Course in Numerical Analysis of Differential Equations |
| NJH: Nicholas J. Higham | ASNA: Accuracy and Stability of Numerical Algorithms |
| LNT: Llyod Nick Trefethen | NLA: Numerical Linear Algebra (with D. Bau, III) |
| TG: Timothy Gowers | PCM: The Princeton Companion to Mathematics |
| AMM: American Mathematical Monthly | |
| RPF: Richard P. Feynman | SYJMF: Surely You’re Joking Mr. Feynman! (Adventures of a Curious Character) |
| AM: Arthur Mattuck |
Caveat Lector: Tip 74 of Desmond J. Higham’s 101 Writing Tips is not followed in this page.
Considerable computational grief arises from inappropriate or even perverse formulations of otherwise tractable problems. The man who insists on solving a boundary-value problem in ordinary differential equations via initial-value techniques may get away with it for a while, but sooner or later he will reap the trouble he has sown. At the very least he pays through inefficiency; at worst, through instability. [Acton, Numerical Methods that (Usually) Work, p. 248]
Since the first electronic computers were developed in the 1940s, comments along the following lines have often been made: “The enormous speed of current machines means that in a typical problem many millions of floating point operations are performed. This in turn means that rounding errors can potentially accumulate in a disastrous way.” This sentiment is true, but misleading. Most often, instability is caused not by the accumulation of millions of rounding errors, but by the insidious growth of just a few rounding errors. [Higham, ASNA, p. 14]
Rather than think of 
The two main classes of rounding error analysis are not, as my audience might imagine, `backwards’ and `forwards’, but rather `one’s own’ and `other people’s’. One’s own is, of course, a model of lucidity; that of others serves only to obscure the essential simplicity of the matter in hand. [J.H. Wilkinson, The State of the Art in Error Analysis]
Fingers or fists? (The choice of decimal or binary representation). [Title of Buchholz’s paper, CACM-2-3]
Nobody should ever have to know that much about floating-point arithmetic. But I’m afraid sometimes you might. [Hamming of Sterbenz’s book “Floating-Point Computation”]
It is rather conventional to obtain a “realistic” estimate of the possible overall error due to 
On November 15, I posted a summary of what I knew then [about the Pentium FDIV bug] to both the Intel and the MATLAB newsgroups, using Nicely’s prime and Coe’s ratio as examples. I also pointed out that the divisors in both cases are a little less than three times a power of two: 
Considering how much time we spend thinking about what the computer will do for us, we should be surprised if its ways did not alter our ways of thought a little. But who would expect the computer’s treatment of the sign of zero to influence our thinking? In fact, the ways computers perform arithmetic can affect the way we think profoundly, much though we may wish it were the other way around. [Kahan, Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing’s Sign Bit]
The cancellation in the subtraction only gives an indication of the unhappy consequence of a loss of information in previous steps, due to rounding of [at least] one of the operands, and is not the cause of the inaccuracy. [Dahlquist and Bjork, Numerical Methods in Scientific Computing, Volume 1, p. 17]
The universe is probably littered with the one-planet graves of cultures which made the sensible economic decision that there’s no good reason to go into space—each discovered, studied, and remembered by the ones who made the irrational decision. [xkcd, 65 Years]
Numerical analysis is often considered neither beautiful nor, indeed, profound. Pure mathematics is beautiful if your heart goes after the joy of abstraction, applied mathematics is beautiful if you are excited by mathematics as a means to explain the mystery of the world around us. But numerical analysis? Surely, we compute only when everything else fails, when mathematical theory cannot deliver an answer in a comprehensive, pristine form and thus we are compelled to throw a problem onto a number-crunching computer and produce boring numbers by boring calculations. This, I believe, is nonsense. A mathematical problem does not cease being mathematical just because we have discretized it. The purpose of discretization is to render mathematical problems, often approximately, in a form accessible to efficient calculation by computers. This, in particular, means rephrasing and approximating analytic statements as a finite sequence of algebraic steps. Algorithms and numerical methods are, by their very design, suitable for computation but it makes them neither simple nor easy as mathematical constructs. Replacing derivatives by finite differences or an infinite-dimensional space by a hierarchy of finite-dimensional spaces does not necessarily lead to a more fuzzy form of reasoning. We can still ask proper mathematical questions with uncompromising rigour and seek answers with the full mathematical etiquette of precise definitions, statements and proofs. The rules of the game do not change at all. Actually, it is almost inevitable that a discretized mathematical problem is, as a mathematical problem, more difficult and more demanding of our mathematical ingenuity. To give just one example, it is usual to approximate a partial differential equation of evolution, an infinite-dimensional animal, in a finite-dimensional space (using, for example, finite differences, finite elements or a spectral method). This finite-dimensional approximation makes the problem tractable on a computer, a machine that can execute a finite number of algebraic operations in finite time. However, once we wish to answer the big mathematical question underlying our discourse, how well does the finite-dimensional model approximate the original equation, we are compelled to consider not one finite-dimensional system but an infinite progression of such systems, of increasing (and unbounded) dimension. In effect, we are not just approximating a single equation but an entire infinite-dimensional function space. Of course, if all you want is numbers, you can get away with hand-waving arguments or use the expertise and experience of others. But once you wish to understand honestly the term `analysis’ in `numerical analysis’, prepare yourself for real mathematical experience. [AI, AFCINADE, Preface to 2nd edition.]
Finite difference approximations have a more complicated “physics” than the equations they are designed to simulate. This irony is no paradox, however, for finite differences are used not because the numbers they generate have simple properties, but because those numbers are simple to compute. [LNT, Finite Difference and Spectral Methods for O/PDEs, p.~192]
Gauss and Clenshaw–Curtis formulas should perhaps be regarded as equally valuable and fundamental, with the former having an edge in elegance and the latter in simplicity. [LNT, Is Gauss Quadrature Better than Clenshaw–Curtis?]
The days when it made sense to do your own memory management in a new program are long over, outside of a few specialty areas like kernel hacking, scientific computing and 3-D graphics—places where you absolutely must get maximum speed and tight control of memory usage, because you need to push the hardware as hard as possible. For most other situations, accepting the debugging overhead of buffer overruns, pointer-aliasing problems, malloc/free memory leaks and all the other associated ills is just crazy on today’s machines. Far better to trade a few cycles and a few kilobytes of memory for the overhead of a scripting language’s memory manager and economize on far more valuable human time. [Eric Raymond, Why Python?]
Ugly programs are like ugly suspension bridges: they’re much more liable to collapse than pretty ones, because the way humans (especially engineer-humans) perceive beauty is intimately related to our ability to process and understand complexity. A language that makes it hard to write elegant code makes it hard to write good code. [Eric Raymond, Why Python?]
Suppose you are given a power series and are asked to determine whether it is an asymptotic series as 
When a measure becomes a target, it ceases to be a good measure. [Goodhart’s Law]
That brings me to Dennis Ritchie. Our collaboration has been a thing of beauty. In the ten years that we have worked together, I can recall only one case of miscoordination of work. On that occasion, I discovered that we both had written the same 20-line assembly language program. I compared the sources and was astounded to find that they matched character-for-character. [Ken Thompson, Reflections on Trusting Trust]
Qui cupit, capit omnia. [Komensky, Janua Linquarum Reserata, quoted by Bella Bollobas, Linear Analysis, preamble] [From the chapter De Statu Regio of Janua: Author: “He who has learned the nomenclature of all things of Nature and Art has laid the foundation of all erudition.” Pupil: “But that must surely be very difficult.” Author: “It certainly is so if you attempt it unwillingly, and if you allow your prejudiced imagination to frighten you. Besides, if there is any difficulty, it will be at the beginning. Do not the shapes and characters of letters also appear to children who first see them singular, wonderful, and monstrous? But when they have taken some trouble and pains, they understand that they (the letters) are but a play and a recreation. The same applies to all things; they appear superficially more difficult than they are. But if you not only begin a work but also persevere, there is nothing that will not yield and submit itself to your intellect. Who wishes to do so can understand everything [= Qui cupit, capit omnia]. Therefore, whoever you are, I order you to hope; I forbid you to despair. See this small work (the Janua). Here—I say this without boasting—I shall place the whole world before your eyes and show you the Latin, French, Spanish, Italian, and German languages as in a summary or handbook.”]
To tell the truth, this whole discussion [of Cauchy’s theorem] is really about Kirchhoff’s voltage law. It had three forms in the continuous case and we have used them all. Either the integral around every closed loop is zero, or the curl 
The Fundamental Theorem of Algebra asserts that every polynomial equation over the complex field has a root. It is almost beneath the dignity of such a majestic theorem to mention that in fact it has precisely 
The analytic functions are indeed the aristocrats of the complex plane, but provided they only mate with their own kind, and only in ways sanctioned by the rules (which allow many forms of incest!), their offspring will also be aristocrats. [Needham, Visual Complex Analysis, p. 226]
The less sharp [a bound], the less informative. For instance, to say that the author owns less than a million neckties is less informative than saying that he owns less than seven, which is still less informative than saying that owns exactly five. [M.D. Greenberg, Advanced Engineering Mathematics, p. 1187, footnote]
A generating function is a clothesline on which we hang up a sequence of numbers for display. [Herbert Wilf]
According to formalist philosophy, all of mathematics is tautology. Chapter 2 [Duality] might strike the reader—as it does the author—as quintessential tautology. Yet even this trivial looking material has some interesting consequences. [Lax, Linear Algebra, p. 12] [The master then goes on to “prove” his point by deriving the quadrature formula by proving that if a set of 
Theorem 1 [rank+nullity theorem] and its corollaries have many applications, possibly more than any other theorem of mathematics. [Lax, Linear Algebra, p. 16]
Theories of the known, which are described by different physical ideas, may be equivalent in all their predictions and hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest differentkinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in one’s attempt to understand what is not yet understood. [RPF, quoted in Tristen Needham, Visual Complex Analysis]
Idleness is a disease which must be combated; but I would not advise a rigid adherence to a particular plan of study. I myself have never persisted in any plan for two days together. A man ought to read just as inclination leads him: for what he reads as a task will do him little good. A young man should read five hours in a day, and so may acquire a great deal of knowledge. [Samuel Johnson, quoted in James Boswell, The Life of Samuel Johnson, p. 365, and Spivak, Calculus, SUGGESTED READING, p. 597 (via The Endeavour)]
Colleges aren’t really in the education business. Colleges are in the credentialing business. [Josh Kaufman]
To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour. [William Blake, “Auguries of Innocence”]
It was 1949 in Southern California. Our computer was a very new CPC (model 1, number 1)—a 1-second-per-arithmetic-operation clunker that was holding the computational fort while an early electronic monster was being coaxed to life in an adjacent room. From a nearby aircraft company there arrived one day a 
A matrix with non-positive off-diagonal entries is an 
Very often we come across people who declare that some classes of mathematics is “ugly” or “boring”. When I was a young student, such a class was non-existent—there was only mathematics that I could understand and that which I could not. That which I could not understand was of course more challenging and required greater effort. Of course, some mathematics becomes boring because we understand it well and do not see any challenges in pursuing it. However, a more common context in which people call some subject ugly is when they do not understand the point of the exercise. In this case it probably needs more study and a clearer perspective to see the worth of what one’s colleagues are doing. Ugliness, like beauty, is often in the eye of the beholder. [V.I. Arnold]
I think students should do research, in their subject, at their level. They can’t invent the Calculus for themselves but you can give them a definition; you can give them a direction; you can ask them a question and you say “Find out; tell me the answer; tell me the answer right here now. Go up to the blackboard and start talking; start thinking. And if you cannot do it, then go home and think; but for heaven’s sake don’t look it up in a book. Looking it up in a book means giving up.” [P. Halmos]
Two students came forward after class with another idea. They knew that the intersection of two sets is the complement of the union of complements. So if “union” is replaced by “sum” and the complement of a set changes to the orthogonal complement of a subspace, they had what they wanted. And they knew the Fundamental Theorem: nulbasis(A’) produces a complement to colbasis(A). So the set identity [mentioned above] converted into a basis formula for the intersection of subspaces: intbasis = nulbasis([nulbasis(A’) nulbasis(B’)]’). Their names were Yan and Dianne. They got A’s on the spot. [GS, Row Reduction and 
Someone calls the Yale Mathematics Department, and asks for Serge Lang. The assistant who answers says, “He can’t talk now; he is writing a book. I will put you on hold.” [Notices of the AMS, May 2006, via The Captain]
I hope that this section has convinced you that eigenvectors are useful tools, and not just bizarre torture devices inflicted on you by your professors for the pleasure of watching you suffer (although the latter is a nice fringe benefit). [J.R. Shewchuck, CG without the Agonizing Pain, p. 19]
There are two kinds of generalizations. One is cheap and the other is valuable. It is easy to generalize by diluting a little idea with a big terminology. It is much more difficult to prepare a refined and condensed extract from several good ingredients. [George Pólya, via The Endeavor]
It is an established fact that the best algorithms for most problems do no better, in general, than to compute exact solutions for slightly perturbed data. Backward error analysis is a method of reasoning fitted neatly to this backward reality. [LNT, NLA, p. 112]
A language that doesn’t affect the way you think about programming is not worth knowing. [Alan Perlis, Epigram’s in Programming]
Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretical emphases. Our mathematical language continues to improve, just as “the \d-ism of Leibniz overtook the dotage of Newton” in past centuries. [Donald Knuth, AMM-99-403, Two notes on notation]\footnote{The quoted part is from Charles Babbage, “Passages from the Life of a Philosopher” (London, 1864). Reprinted in Charles Babbage and his Calculating Engines, edited by Philip Morrison and Emily Morrison 19 (New York: Dover, 1961)}
Tim Gowers once mentioned to me the nice concept of the “width” of a mathematical lecture or talk. This is defined as the supremum, over all the times 
Ideally, the width of a talk should always be kept less than or equal to the amount of space available to the lecturer. For instance, if the lecturer has three blackboards to work with, the talk should always be at most three blackboards wide. This is one reason why computer or transparency presentations can be more difficult to follow than blackboard talks, as the “bandwidth” provided by the spatial resources is significantly smaller.
Encapsulation is a good way to keep the width down to manageable levels. If a fact or technique is only needed to prove one key statement in the talk, then phrasing that statement as a lemma, and encapsulating the special facts or techniques inside the proof of that lemma, means that that fact or technique will no longer occupy precious bandwidth outside of that lemma. In a similar spirit, concluding one story before starting an unrelated story requires less width than telling both stories in parallel. (But if it is important to highlight the relationship between the two stories, then some parallelism may be needed, though perhaps only the “moral” of each story has to be retained going forward.) [Terence Tao]
As Don Braben so aptly put it, funding the technology but not the basic research on which it depends is “living off the seedcorn”. [Leslie Ann Goldberg]
Q: When did Bourbaki stop writing books?
A: When they realized that Serge Lang was a single person….\footnote{Serge Lang was himself a member of Bourbaki group.} [via The Captain]
[It is not] possible to distinguish between “numerical” and “nonnumerical” algorithms, as if numbers were somehow different from other kinds of precise information. All data are numbers, and all numbers are data. Mathematicians work much more with symbolic entities than with numbers. [D. Knuth, Letter to the Patent Office]
It is curious to track the path by which the word `argument’ came to have two different meanings, one in mathematics and the other in everyday English. According to the Oxford English Dictionary, the word derives from the Latin for ‘to make clear, prove’; thus it came to mean, by one thread of derivation, `the evidence offered as proof’, which is to say, `the information offered’, which led to its meaning in Lisp. But in the other thread of derivation, it came to mean `to assert in a manner against which others may make counter assertions’, which led to the meaning of the word as a disputation. (Note here that the English word has two different definitions attached to it at the same time. By contrast, in Emacs Lisp, a symbol cannot have two different function definitions at the same time.) [Robert J. Chassell, Programming in Emacs Lisp]
Procrastinate later. [Bumper sticker]
If you can’t convince them, confuse them. [Harry S. Truman]
Money is better than poverty, if only for financial reasons. [Woody Allen]
One can measure the importance of a scientific work by the number of erlier publications rendered superfluous by it. [David Hilbert]
In the highly nonnormal case, vivid though the image may be, the location of the eigenvalues may be as fragile an indicator of underlying character [of an operator] as the hair color of a Hollywood actor. [LNT and M. Embree, Spectra and Pseudospectra, p. 11]
An affine space is a vector space that’s forgotten its origin. [John Baez]
If you find the concept of Hilbert space hard to grasp, then you are not alone. Hilbert was a professor at Göttingen University which was a center of mathematics research from the days of Gauss and Riemann up until the mid-1930s. The Hungarian–American mathematician John von Neumann (1903–1957), one of the founders of the modern theory of function spaces, visited Göttingen in the mid-1920s to give a lecture on his work. The legend is that shortly into the lecture Hilbert raised his hand and asked, “Herr von Neumann, could you explain to us again what a Hilbert space is?” [Y. Pinchover and J. Rubinstein, Introduction to Partial Differential Equations, p. 298]
Think of this [Banach–Tarski Paradox] from an alchemist’s point of view. A one inch gold ball can be cut into five disjoint pieces and the pieces rigidly re-assembled to make two one inch gold balls. Repeating the process would make you very rich. [C.C. Puch, Real Mathematical Analysis, p. 409]
In mathematics one cannot stop at drawing with a big, wide brush; all the details have to be filled in at some time. [S. Ulam]
Some mathematicians are always careful to distinguish the function 
People who search for elementary proofs in analytic number theory are like the algebraists I once heard Herman Weyl talk about at the Institute for Advanced Study. He was lecturing on Lie algebras or something like that. “Of course,” he said, “this was the problem before the algebraists got hold of it. You know what an algebraist is? An algebraist is a person who comes to a river and says, `I wonder if I can jump across that river!’ And then he takes a big leap and he jumps across that river. And then he swims back and says, `NOW, I wonder if I can jump across that river with my hands tied behind my back!’{”} [AM in J. Segel, Recountings: Conversations with MIT Mathematicians, p. 99]
The reasonable man adapts himself to the world; the unreasonable persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable. [George Bernard Shaw]
Problem. Show that through every phase point there is one and only one phase curve. Hint Refer to a textbook on ordinary differential equations. [V.I. Arnold, Mathematical Methods of Classical Mechanics, p. 16]
The first incredible success for that [model theory] was in 1964—a thirty-year-old problem in pure math first formulated by Emil Artin, halfway between algebra and number theory, which was solved by logic. It was a collaboration between an algebraist and a logician, but when it was written up, all the logic was removed from their proof, because the algebraist felt that otherwise nobody would trust it! [AM in J. Segel, Recountings: Conversations with MIT Mathematicians, p. 85]
I find television very educating. Every time somebody turns on the set, I go into the other room and read a book. [Groucho Marx]
I guess I regard the books quite personally. They’re like children that are part of you: you take a few years to give birth to them, and they’re pretty special. So I don’t see writing about mathematics as just putting things on the record. I regard textbooks as speaking to the student and persuading him or her that this is worth learning and making it clearer and clearer and clearer, as far as possible. I don’t think you can take the attention of students for granted. [GS in J. Segel, Recountings: Conversations with MIT Mathematicians, p. 174]
Located somewhere between behaviorism and introspection, the school of gestalt psychology teaches that thinking starts with a problem, a difficulty, a contradiction. It sounds like a truism, yet is widely ignored in practice. Teachers say their aim is to get their students to think, yet in classroom after classroom they violate this psychological principle by giving the solution before there is any problem. [AM, The Torch or Firehose]
MIT is known as an excellent place to get an education in mathematics, both at the undergraduate and graduate levels. How does a department earn such a reputation? Even senior department faculty commonly teach basic courses at MIT, and the faculty has written well over a hundred textbooks in the last half-century or so. But […] excellence in teaching is more than just a matter of books and blackboards. It is Arthur Mattuck’s decades of devotion to undergraduate education, including his wry booklet on the art of recitation teaching, “The Torch or the Firehose,” […] . It is the Research Science Institute (RSI) and Summer Program for Undergrad Research (SPUR), programs that pair high school students and undergraduates with graduate students as mentors in a summer research project. It is Michael Artin’s continued search, even in his third decade of teaching undergraduate algebra, for the perfect exercise. It is Steven Kleiman’s focus on imparting the craft of setting out one’s research findings in a coherent paper and Gilbert Strang striving for that trademark conversational tone in his latest textbook. It is Daniel Kleitman dreaming up Java applets to illustrate the concepts of calculus more viscerally. It is videoed lectures on MIT’s OpenCourseWare, downloaded and viewed free of charge by students all over the world. [J. Segel, Recountings: Conversations with MIT Mathemticians, p. xii]
When I get a little money I buy books; and if any is left I buy food and clothes. [Desiderius Erasmus]
To Joanna,
My brilliant and beautiful wife without whom I would be nothing. She always comforts and consoles, never complains or interferes, asks nothing and endures all, and writes my dedications. [Albert Paul Malvino, “Electronic Principles” (via Tal Cohen’s Bookshelf)]
Code is poetry. [WordPress.org]
One can roughly divide mathematical education into three stages:
1. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.
2. The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory. This stage usually occupies the later undergraduate and early graduate years.
3. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond. [Terence Tao, There’s more to mathematics than rigour and proofs]
The chair of the department of a Big Ten university once observed, probably after a bad day, that it was possible for a student to graduate with a mathematics major without ever having solved a single problem correctly. Partial credit can go a long way. This was in the 1950s, looked on by many as a golden age of mathematics education. [Underwood Dudley-NAMS-57-608-What is Mathematics For?]
When once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been and there you will always long to return. [Leonardo da Vinci]
No gnus is bad news. [Gnus]
“git” can mean anything, depending on your mood.
– random three-letter combination that is pronounceable, and not actually used by any common UNIX command. The fact that it is a mispronounciation of “get” may or may not be relevant.
– stupid. contemptible and despicable. simple. Take your pick from the dictionary of slang.
– “global information tracker”: you’re in a good mood, and it actually works for you. Angels sing, and a light suddenly fills the room.
– “goddamn idiotic truckload of sh*t”: when it breaks [Linus Torvalds, Git README]
For the first 10 years of kernel maintenance, we literally used tarballs and patches, which is a much superior source control management system than CVS is, but I did end up using CVS for 7 years at a commercial company [Transmeta] and I hate it with a passion. When I say I hate CVS with a passion, I have to also say that if there are any SVN (Subversion) users in the audience, you might want to leave. Because my hatred of CVS has meant that I see Subversion as being the most pointless project ever started. The slogan of Subversion for a while was “CVS done right,” or something like that, and if you start with that kind of slogan, there’s nowhere you can go. There is no way to do CVS right. [Linus Torvalds]
Nobody actually creates perfect code the first time around, except me. But there’s only one of me. [Linus Torvalds, Google Tech talk on Git]
There is a species of mathematician that denotes the `positive integers’ by 
I should say something about tenure, I guess. About that time there was a story in the “Los Angeles Times,” that students at UCLA were unhappy because a professor at UCLA hadn’t gotten tenure, and I’d never heard of the word tenure before, so I want to Marshall Hall, I said, “Marshall, what is this word, tenure? What does it mean, that she didn’t get tenure?” And he said, “well Don, remember when you got your appointment form from Caltech last year, where it said you’re appointed Associate professor, you know?” And I said, “yeah?” “Well there’s a line on there that says, `the ending date for your appointment,’ remember that that line was blank?” And I said “yes.” “Well,” he says, “that’s tenure. You know, your job goes on.” Well I’d never heard of the concept before, so I had tenure before I knew it existed, and I’m glad, because a lot of students now are so worried about not getting tenure, that it interferes with their professional development, you know, they’re spending more time strategizing about how to get tenure than about how to do good science. [Donald Knuth, The emergence of computer science as an academic subject]
Lisppaste pastes can be made by anyone at any time. Imagine a fearsomely comprehensive disclaimer of liability. Now fear, comprehensively. [Lisppaste disclaimer]
My phone is an old blackberry curve that I found used on craigslist. I’m cheap. It was in decent shape, but there was one scratch on the screen that kept bothering me. The internet said I might be able to polish it out with brasso metal polish. It wasn’t too expensive, so I ordered some and gave it a try. After polishing until my hand was cramped, the one scratch had become lost amongst a million new scratches. My eyes watered as I tried to focus on the screen through the damaged plastic. Dismayed, but still hopeful, I rested a bit and then polished some more, thinking that sometimes things have to get worse before they get better. Well, I’m still wondering if better was just around the corner, because I finally gave up. [Bryan Murdock, Don’t Use Brasso On Your Blackberry]
“Don’t Reinvent The Wheel” should be used as a call to arms for deeply educating yourself about all the existing solutions—not as a bludgeoning tool to undermine those who legitimately want to build something better or improve on what’s already out there. In my experience, sadly, it’s much more the latter than the former. [Jeff Atwood, Don’t Reinvent The Wheel, Unless You Plan on Learning More About Wheels]
In common parlance, what we must check is that the operations are `well defined.’ Really this is over-polite: what we are checking is that they are `defined’ at all! A fraudulent definition is no true definition whatsoever. [Ian Stewart and David Tall, The Foundations of Mathematics, p. 78]
`Well-defined’ is perfectly well-defined. It means `single-valued’ and carries with it the suggestion that the function that is single-valued might easily not have been. [TG, Is the phrase `well-defined’ well-defined?]
Abstraction of meaning does not imply loss of meaning. [Frank DuSua, AMM-63-295-Consistency and Completeness–A Resume]
Gödel’s theorem has been a battleground for philosophers arguing about whether the human brain is a deterministic machine (in which case, presumably, we would not be able to prove any formally unprovable statement). Fortunately, there is not enough space in this article for more details! [Peter J. Cameron, Gödel’s Theorem, PCM]
There is an obvious way of showing that a mathematical statement has a proof: you just find one. [Joan Bagaria, Set Theory, PCM]
Some failure in life is inevitable. It is impossible to live without failing at something, unless you live so cautiously that you might as well not have lived at all—in which case, you fail by default.
…
Many [humans] prefer not to exercise their imaginations at all. They choose to remain comfortably within the bounds of their own experience, never troubling to wonder how it would feel to have been born other than they are. They can refuse to hear screams or to peer inside cages; they can close their minds and hearts to any suffering that does not touch them personally; they can refuse to know.
…
The way you vote, the way you live, the way you protest, the pressure you bring to bear on your government, has an impact way beyond your borders. That is your privilege, and your burden.
If you choose to use your status and influence to raise your voice on behalf of those who have no voice; if you choose to identify not only with the powerful, but with the powerless; if you retain the ability to imagine yourself into the lives of those who do not have your advantages, then it will not only be your proud families who celebrate your existence, but thousands and millions of people whose reality you have helped change. We do not need magic to change the world, we carry all the power we need inside ourselves already: we have the power to imagine better. [J.K. Rowling in her Harvard University Commencement Speech The Fringe Benefits of Failure, and the Importance of Imagination]
He who knows not mathematics cannot know the other sciences nor the things of this world. And those who have no knowledge of mathematics do not perceive their own ignorance and so do not look for a cure. [Roger Bacon, quoted in Robin Wilson’s talk Who invented algebra?]
One night after a deep carouse, when on his way from Carfax to Merton [one of the first three Oxford colleges], he found it advisable to take his bearings. Whipping out his astrolabe he observed the altitude of the stars, but, on getting the view of the firmament through the sights, he fancied that sky and stars were rushing down upon him. Stepping quickly aside he quietly fell into a large pond. `Ah, ah,’ says he, `now I’m in a nice soft bed I will rest in the Lord.’ Recalled to his senses when the cold struck through, he rose from the watery couch and proceeded to his room where he retired to bed fully clothed. On the morrow, in answer to kind inquiries, he denied all knowledge of the pond. [Story of a Merton student named Robert Dobbys and his Merton astrolabe recounted in Robin Wilson’s talk Who invented algebra?]
Mathematics reveals every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters. Whoever then has the effrontery to study physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom. [Thomas Bradwardine (the greatest English mathematician of the fourteenth century whose discourses were so learned that he was known as Dr Profundus), quoted in Robin Wilson’s talk Who invented algebra?]
We’ve come a long way from the geometry of Euclid and his contemporaries, and I hope that you have considered the journey worthwhile. If not, I conclude by recounting the story that someone who had begun to read geometry with Euclid asked him, “What advantage shall I get by learning these things?” Euclid called his slave and said, “Give him three pence, for he must make profit out of what he learns.” I do hope that you don’t all want to claim three pence from me. [Robin Wilson in his talk Euclid]
Read Euler, read Euler; he is the master of us all…. [Pierre-Simon Laplace]
He [Euler] had full possession of his faculties and apparently all of his strength and absolutely no change announced itself that the Sciences were in fear of losing him. On 7 September 1783, after having enjoyed some calculations on his blackboard concerning the laws of ascending motion for aerostatic machines for which the recent discovery was the rage of Europe, he dined with Mr. Lexell and his family, spoke of Herschel’s planet (Uranus) and the mathematics concerning its orbit and a little while later he had his grandson come and play with him and took a few cups of tea, when all of a sudden the pipe that he was smoking slipped from his hand and he ceased to calculate and live.
Such was the end of one of the greatest and most extraordinary men that Nature ever produced, whose genius was equally capable of great effort and continuous work which multiplied his productive work span to beyond what one dares to expect from a human. And each of which was original, during which his mind was always occupied and his soul always calm. He had finally achieved what he had always deserved, a happiness that was cloudless coupled to a glory that has never been questioned. [the Marquis de Condorcet, Eulogy to Euler, quoted (with a somewhat different translation) in Robin Wilson’s talk Euler]
There were a lot of fools at that conference [on Ethics of Equality]—pompous fools—and pompous fools drive me up the wall. Ordinary fools are all right; you can talk to them, and try to help them out. But pompous fools—guys who are fools and are covering it all over and impressing people as to how wonderful they are with all this hocus pocus—THAT, I CANNOT STAND! An ordinary fool isn’t a faker; an honest fool is all right. But a dishonest fool is terrible! And that’s what I got at the conference, a bunch of pompous fools, and I got very upset. I’m not going to get upset like that again, so I won’t participate in interdisciplinary conferences any more. [RPF et al., SYJMF-Is Electricity Fire?]
R & S: Occasionally one can hear within the mathematical community statements that the theory of nonlinear partial differential equations, though profound and often very important for applications, is fraught with ugly theorems and awkward arguments. In pure mathematics, on the other hand, beauty and aesthetics rule. The English mathematician G.H. Hardy is an extreme example of such an attitude, but it can be encountered also today. How do you respond to this? Does it make you angry?
Lax: I don’t get angry very easily. I got angry once at a dean we had, terrible son of a bitch, destructive liar, and I got very angry at the mob that occupied the Courant Institute and tried to burn down our computer. Scientific disagreements do not arouse my anger. But I think this opinion is definitely wrong. I think Paul Halmos once claimed that applied mathematics was, if not bad mathematics, at least ugly mathematics, but I think I can point to those citations of the Abel Committee dwelling on the elegance of my works!
Now about Hardy: When Hardy wrote A Mathematician’s Apology he was at the end of his life, he was old, I think he had suffered a debilitating heart attack, he was very depressed. So that should be taken into account. About the book itself: There was a very harsh criticism by the chemist Frederick Soddy, who was one of the co-discoverers of the isotopes—he shared the Nobel Prize with Rutherford. He looked at the pride that Hardy took in the uselessness of his mathematics and wrote: “From such cloistral clowning the world sickens.” It was very harsh because Hardy was a very nice person.
My friend Joe Keller, a most distinguished applied mathematician, was once asked to define applied mathematics and he came up with this: “Pure mathematics is a branch of applied mathematics.” Which is true if you think a bit about it. Mathematics originally, say after Newton, was designed to solve very concrete problems that arose in physics. Later on, these subjects developed on their own and became branches of pure mathematics, but they all came from applied background. As von Neumann pointed out, after a while these pure branches that develop on their own need invigoration by new empirical material, like some scientific questions, experimental facts, and, in particular, some numerical evidence. [Martin Raussen and Christian Skau, Interview with Peter D. Lax, Notices of the American Mathematical Society-53-p. 225]
Repeat in Unison
OH-WA
TAH-GOO
SIAM
Repeat until you attain enlightenment! [Charles Leiserson at the end of his talk in SIAM Annual Meeting 2009, Design and Analysis of Multithreaded Algorithms]
[The Euclidean algorithm] is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. [Donald Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd edition (1981), p. 318]
If you want to build a ship, don’t drum up people together to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea. [Antoine de Saint-Exupery]
This [PCM] is one of those books that makes you wish you had a desert island to be marooned on. [Brian Hayes, bit-player.org]
Since a number of years I am familiar with the observation that the quality of programmers is a decreasing function of the density of go to statements in the programs they produce. [E. Dijkstra, Communications of ACM-11-147-Go To Statement Considered Harmful]
… it [sentimentalism] may be used quite legitimately to indicate certain type of unbalanced emotion. Many people, of course, use `sentimentalism’ as a term of abuse for other people’s decent feelings, and `realism’ as a disguise for their own brutality. [G.H. Hardy, A Mathematician’s Apology-p. 45-footnote]
Read the masters! [H.M. Edwards in L.A. Steen, ed., Mathematics Tomorrow]
If you were plowing a field, which would you rather use? Two strong oxen or 1024 chickens? [Seymore Cray]
Barrier synchronization is a popular method of coordinating parallel activities. In real life, we often use barrier synchronization to coordinate parallel activities of multiple persons. For example, assume that four friends go to a shopping mall in a car. They can all go to different stores to buy their own clothes. This is a parallel activity and is much more efficient than if they all remain as a group and sequentially visit the stores for their clothes. However, barrier synchronization is needed before they leave the mall. They have to wait until all four friends have returned to the car before they can leave. Without the barrier synchronization, one or more persons can be left in the mall when the car leaves, which can seriously damage the friendship! [David Kirk and Wen-mei Hwu, Programming Massively Parallel Processors]
A day or two before the talk I saw Wigner in the hall. “Feynman,” he said, “I think that work you’re doing with Wheeler is very interesting, so I’ve invited Russell to the seminar.” Henry Norris Russell, the famous, great astronomer of the day, was coming to the lecture!
Wigner went on. “I think Professor von Neumann would also he interested.” Johnny von Neumann was the greatest mathematician around. “And Professor Pauli is visiting from Switzerland, it so happens, so I’ve invited Professor Pauli to come”—Pauli was a very famous physicist—and by this time, I’m turning yellow. Finally, Wigner said, “Professor Einstein only rarely comes to our weekly seminars, but your work is so interesting that I’ve invited him specially, so he’s coming, too.”
By this time I must have turned green, because Wigner said, “No, no! Don’t worry! I’ll just warn you, though: If Professor Russell falls asleep—and he will undoubtedly fall asleep—it doesn’t mean that the seminar is bad; he falls asleep in all the seminars. On the other hand, if Professor Pauli is nodding all the time, and seems to be in agreement as the seminar goes along, pay no attention. Professor Pauli has palsy.”
…
Then the time came to give the talk, and here are these monster minds in front of me, waiting! My first technical talk—and I have this audience! I mean they would put me through the wringer! I remember very clearly seeing my hands shaking as they were pulling out my notes from a brown envelope. [RPF et al., SYJMF-Monster Minds]
The next paper selected for me was by Adrian and Bronk. They demonstrated that nerve impulses were sharp, single-pulse phenomena. They had done experiments with cats in which they had measured voltages on nerves.
I began to read the paper. It kept talking about extensors and flexors, the gastrocnemius muscle, and so on. This and that muscle were named, but I hadn’t the foggiest idea of where they were located in relation to the nerves or to the cat. So I went to the librarian in the biology section [in Princeton] and asked her if she could find me a map of the cat.
“A map of the cat, sir?” she asked, horrified. “You mean a zoological chart!” From then on there were rumors about some dumb biology graduate student who was looking for a “map of the cat.”
When it came time for me to give my talk on the subject, I started off by drawing an outline of the cat and began to name the various muscles.
The other students in the class interrupt me: “We know all that!”
“Oh,” I say, “you do? Then no wonder I can catch up with you so fast after you’ve had four years of biology.” They had wasted all their time memorizing stuff like that, when it could be looked up in fifteen minutes. [RPF et al., SYJMF-A Map of the Cat?]
I often listened to my roommates—they were both seniors—studying for their theoretical physics course. One day they were working pretty hard on something that seemed pretty clear to me, so I said, “Why don’t you use the Baronallai’s equation?” “What’s that!” they exclaimed. “What are you talking about!” I explained to them what I meant and how it worked in this case, and it solved the problem. It turned out it was Bernoulli’s equation that I meant, but I had read all this stuff in the encyclopedia without talking to anybody about it, so I didn’t know how to pronounce anything. [RPF et al., SYJMF-Who Stole the Door?]
Just think of 
Chaos is an unavoidable part of reality. It’s also a slap in the face. Scientists, like everyone else, had always known that complicated things like human relationships or wars or history could be unpredictable. But at least we had our pendulums for comfort. Now chaos was taking those away too. [Steven Strogatz, The Calculus of Friendship-p. 108]
We all worked so hard and selflessly because we believed—we knew—it was happening here and at a few other places right then, and we were lucky to be in on it. We were sure because von Neumann cleared the cobwebs from our minds as nobody else could have done. A tidal wave of computational power was about to break and inundate everything in science and much elsewhere, and things would never be the same… [Julian Bigelow, shown in George Dyson’s talk on The Birth of the Computer]
I have now duplicated both results. How will I know which is right assuming one result is correct?
…
This now is the 3rd different output. I know when I’m licked. [Electronic Computer Project Engineers’ Log, shown in George Dyson’s talk on The Birth of the Computer]
A mouse has climbed into the blower behind the regulating rack, set blower to vibrating. Result: no more mouse.
Here lies mouse. Born ?, Died 4:50am, May 1953. [Electronic Computer Project Engineers’ Log, shown in George Dyson’s talk on The Birth of the Computer]
Code error—machine not guilty. [Electronic Computer Project Engineers’ Log, shown in George Dyson’s talk on The Birth of the Computer]
Professor John von Neumann
Institute for Advanced Study
Princeton, New Jersey
Dear John:
I am a little troubled about the tea service in the electronic computer building. Apparently the members of your staff consume several times as much supplies as the same number of people do in Fuld Hall and they have been especially unfair in the matter of sugar. Sugar is rationed and for a member of your staff to come up here as Thompson did and carry down a large quantity of sugar in excess of your rations is not cricket. I understand, furthermore, that the tea is served in several different places. We have never undertaken in the Institute to provide tea service in a large number of private offices and I should like to raise the question whether it would not be better for the computer people to come up to Fuld Hall at the end of the day at 5 o’clock and have their tea here under proper supervision. [Letter from the director of the Institute to John von Neumann shown in George Dyson’s talk on The Birth of the Computer—“This is hackers getting in trouble for the first time,” as George Dyson so wittily puts it.]
The heart of the system is a central clock, carrying an enormous load…
This [modular] sort of design is favorable for mass production…
`Words’ coding the orders are handled in the memory just like numbers… [First meeting of the IAS Electronic Computer Project (November 12, 1945), shown in George Dyson’s talk on The Birth of the Computer]
“[…] the idea of fractional integro–differentiation […] occurred to Leibniz, as soon as he has developed his version of calculus and invented the notations 
Johann Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way that successive differentials are in geometric progression. One can ask what would be a differential having as its exponent a fraction. […] Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility.” [B. Mandelbrot, The fractal geometry of nature-p. 405] I read the first part of this quotation as a reference to what became later known as Taylor series: in the formula 
…although analysis often involves limiting processes and algebra usually does not, a more significant distinction is that algebraists like to work with exact formulas and analysts use estimates. Or, to put it even more succinctly, algebraists like equalities and analysts like inequalities. [TG, Algebra versus Analysis, PCM]
At the supper table, my young daughter once said, “Put your hand up if you are a girl.” One of my sons, to tease her, put his hand up on the grounds that, since she had not added, “and keep it down if you are a boy,” his doing so was compatible with her command. [TG, Logical Connectives, PCM]
Integration is a pairing of two objects: one geometric, the domain of integration, and one algebraic, the differential form that we are integrating. On each type of objects there is a natural operation: taking the boundary of the domain on the geometric side, and taking the de Rham differential on the algebraic side. Both operations are nilpotent: for the de Rham differential 
Calculus is the mathematical study of change. Its essence is best captured by its original name, “fluxions,” coined by its inventor, Isaac Newton. The name calls to mind systems that are ever in motion, always unfolding.
…
Calculus thrives on continuity. At its core is the assumption that things change smoothly, that everything is only infinitesimally different from what it was a moment before. Like a movie, calculus reimagines reality as a series of snapshots, and then recombines them, instant by instant, frame by frame, the succession of imperceptible changes creating an illusion of seamless flow. [Steven Strogatz, The Calculus of Friendship-p. xii, 1]
This concludes our tour, now that we have arrived back at your first real analysis class, on that special day when you first saw the Heine–Borel theorem proven. For those of you out there that have yet to take real analysis, but are advanced and motivated enough to be reading this article, pay attention. When you find yourself in an analysis class, and the professor draws that little box at the end of the proof of Heine–Borel, raise your hand, and inquire:
“Doesn’t that just follow from König’s infinity lemma, and the standard ultrametric on the space of binary sequences?” [Mattew Macauley, Brian Rabern, and Landon Rabern, A novel proof of the Heine–Borel theorem]
Grades will be based entirely on homework. Homework problems will be graded right/wrong, but you may re-submit the problems you get wrong within two weeks of getting them back to convert them to “right.” (If you turn in a homework late, you forfeit this possibility.) [Jon Wilkening]
Computer scientists, or at least some computer scientists, deal with algorithms that you might call “practical in theory” but not “practical in practice.” [TG about the notion of Polynomial Complexity, The Importance of Mathematics at The Millennium Meeting (2000)]
Most unfortunately, the habit in the numerical analysis literature is to speak not of the convergence of these magnificently efficient methods [Adams multistep integration methods, or other numerical methods for that matter], but of their error, or more precisely their discretization or truncation error as distinct from rounding error. This ubiquitous language of error analysis is dismal in tone, but seems ineradicable. [LNT, Numerical Analysis, PCM]
On one occasion he [von Neumann] and his colleague Jacob Bronowski had been arguing late into the night about the correct solution to a problem. “When I called his hotel in London,” Bronowski recalled, “he answered the phone in bed, and I said `Johnny, you’re quite right.’ And he said to me, `You wake me up early in the morning to tell me that I’m right? Please wait until I’m wrong.’ ” [Jacob Bronowski about John von Neumann, in Johnny Jiggles the Planet, The New York Times (November 8, 1992)]
In the 1950s and 1960s, the founding fathers of the field [Numerical Analysis] discovered that inexact arithmetic can be a source of danger, causing errors in results that “ought” to be right. The source of such problems is numerical instability: that is, the amplification of rounding errors from microscopic to macroscopic scale by certain modes of computation. These men, including von Neumann, Wilkinson, Forsythe, and Henrici, took great pains to publicize the risks of careless reliance on machine arithmetic. These risks are very real, but the message was communicated all too successfully, leading to the current widespread impression that the main business of numerical analysis is coping with rounding errors. In fact, the main business of numerical analysis is designing algorithms that converge quickly; rounding-error analysis, while often a part of the discussion, is rarely the central issue. If rounding errors vanished, 90% of numerical analysis would remain. [LNT, Numerical Analysis, PCM]
The LaTeX user is recommended to define macros with the “\newcommand” command rather than “\def”. With “\newcommand” an attempt to define an existing command produces an informative error message, while if “\def” is used, an error can result that is hard to track down. I tend to use “\def” anyway! [NJH, Handbook of Writing for Mathematical Sciences-p. 190-footnote]
I prefer to use the accents “\widehat” and “\widetilde” (
Calvin: I think we’ve got enough information now, don’t you?
Hobbes: All we have is one “fact” you made up.
Calvin: That’s plenty. By the time we add an introduction, a few illustrations, and a conclusion, it will look like a graduate thesis. [Calvin and Hobbes by Bill Watterson (1991) in NJH, Handbook of Writing for Mathematical Sciences]
I could tell about the time that he [Richard P. Feynman] learned Spanish before he went to give a series of lectures in Brazil, but I won’t. (They speak Portuguese in Brazil.) [Cornell Provost Dale R. Corson, introducing RPF before his lecture “Law of Gravitation: An Example of Physical Law” (November 9, 1964)]
Another of his [RPF’s] specialties is safecracking. One legend says that he once opened a locked safe in a secret establishment, removed a secret document, and left a note saying GUESS WHO. [Cornell Provost Dale R. Corson, introducing RPF before his lecture “Law of Gravitation: An Example of Physical Law” (November 9, 1964)]
I thought it might be interesting to see what was said about him [RPF] when he was appointed at Cornell, so I searched the minutes of our board of trustees—and there’s absolutely no record of his appointment. There are, however, some twenty references to leaves of absence, salary, and promotions. One reference interested me especially: on July 31 1945 the chairman of the Physics department wrote the dean of the Arts College stating that “Dr. Feynman is an outstanding teacher and investigator, the equal of whom develops infrequently. The chairman suggested that an annual salary of $3,000 was a bit too low for a distinguished faculty member and recommended that Professor Feynman’s salary be increased $900. The dean, in an act of unusual generosity and with complete disregard for the solvency of the university, crossed out the $900 and made it an even $1,000. You can see that we thought highly of Professor Feynman, even then. [Cornell Provost Dale R. Corson, introducing RPF before his lecture “Law of Gravitation: An Example of Physical Law” (November 9, 1964)]
The following table shows the results of applying various operations to the number 
When a physicist shows data you often hear, “these are some typical data.” The word “typical” is a code word for “these are the best data I ever obtained in my whole career.” [Eric Mazur, Confessions of a Converted Lecturer]
The plural of “anecdote” is not “data.” [Eric Mazur quoting Lee Shulman (president of Carnegie Foundation for the Advancement of Teaching) on importance of basing educational research on data, Confessions of a Converted Lecturer]
Each truth that I discovered became a rule that served me afterwards in the discovery of others. [Rene Descartes, Le Discours de la Methode]
The merely Difficult, we do immediately; the Impossible, will take slightly longer. [Old British naval maxim, beginning the section “Why Is Equation Solving Provably Impossible?” in Hewlett-Packard Journal-p. 23 (December 1974)]
-----BEGIN GS-----
The editors [David Y. Gao and Hanif D. Sherali, Advances in Applied Mathematics and Global Optimization] have kindly invited me to write a few words of introduction to this volume. They even expressed the hope that I would go beyond mathematics, to say something about my own life experiences. I think every reader will recognize how hard it is (meaning impossible) to do that properly. If I choose a single word to describe an approach to the complications of life (and of mathematics too), it would be “optimism.” Eventually I realized that, if you allow that word in its mathematical sense too, this whole book is for optimists.
If I may give one instance of my own optimism, it has come from writing textbooks. I enjoy the hopeless effort to express simple ideas clearly. Beyond that, I have come to expect (without knowing any reason, perhaps this defines an optimist) that the connections between all the pieces of the book will somehow appear. Suddenly a topic fits into its right place. This irrational certainty may also be the experience of a hopeful novelist who doesn’t know how the characters will interact and how the plot will turn out.
Looking seriously at this approach, to applied mathematics or to life, an unconstrained optimism is hard to justify. Mathematically, an immediate constraint on all of us is that we are “not Gauss.” Far wiser to accept constraints, and continue to optimize. The connection that did finally bring order to my own thinking and writing about applied mathematics and computational engineering was constrained optimization. I now call that the “Fundamental Problem of Scientific Computing.”
Examples are everywhere, or those words would not be justified. So many problems involve three steps, and flows in networks are a good model. The potentials at the nodes, and the currents on the edges, are the unknowns (somehow dual). A first step goes from potentials to potential differences (by an edge-node matrix 
The dual role of 
![\displaystyle \text{\bf Optimization with constraint:} \left[ \begin{matrix} C^{-1} & A\\ A' & 0 \end{matrix} \right] \left[ \begin{matrix} w\\ u \end{matrix} \right] = \left[ \begin{matrix} b\\ f \end{matrix} \right]](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Ctext%7B%5Cbf+Optimization+with+constraint%3A%7D+%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09%09C%5E%7B-1%7D%09%26%09A%5C%5C+%09%09A%27%09%26%090+%09%5Cend%7Bmatrix%7D+%5Cright%5D+%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09%09%09%09w%5C%5C+%09%09%09%09u+%09%5Cend%7Bmatrix%7D+%5Cright%5D+%3D+%5Cleft%5B+%5Cbegin%7Bmatrix%7D+%09%09%09%09%09b%5C%5C+%09%09%09%09%09f+%09%5Cend%7Bmatrix%7D+%5Cright%5D+&bg=ffffff&fg=000000&s=0 "\displaystyle \text{\bf Optimization with constraint:} \left[ \begin{matrix} C^{-1} & A\\ A' & 0 \end{matrix} \right] \left[ \begin{matrix} w\\ u \end{matrix} \right] = \left[ \begin{matrix} b\\ f \end{matrix} \right]")
One way to solve this fundamental problem is to eliminate 
May I thank the editors and authors and readers of the present book.
I hope you will accept constraints as inevitable, and go forward. [GS, Constrained Optimism, in the introduction of Advances in Applied Mathematics and Global Optimization (@ Google Books)]
-----END GS-----
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors. [James Caballero]
There’s only one idea here; if Gram had the idea, I don’t know what Schmidt did! [GS, in his lecture on GS (Gram–Schmidt)]
…Cramer’s rule is a disastrous way to go because to compute these determinants takes like, approximately, forever! [GS]
Knowledge is of two kinds. We know a subject ourselves or we know where we can find information upon it [Samuel Johnson, Boswell’s Life of Johnson quoted at the beginning of the Subject Index of NJH’s ASNA]
Cogito, Ergo Sum. [Descartes]
When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false. [Carl Friedrich Gauss]
The total number of Dirichlet’s publications is not large; jewels are not weighed on a grocery scale. [Carl Friedrich Gauss]
-----BEGIN AM-----
By geometry, [pause] well, just by looking at it actually.
Trivial: go home and stare at it for 3 hours until you’re convinced.
After working at it long enough to understand it, you should agree that it’s trivial.
The margin of this blackboard is too small for the proof of this theorem.
Pure math is divided into analysis, algebra, and topology. Never mind what they are, they’re not important.
I have a feeling I must be a windbag, since my notes for this lecture consist of 4 lines, and [in 44 minutes] I’ve only covered 2.
How many people don’t know 
You’ll use it in the next problem set. If that isn’t applied mathematics I don’t know what is.
Some of you said 
I didn’t do it last time, but I stated that you could read it in your book, and that’s the same thing.
Aha! Your have Mattuck syndrome! You fall asleep at unpredictable moments.
So, so, so…what am I trying to say?
It looks like the hypothesis has nothing to do with the solution. That makes a good theorem.
You can’t argue with me there. Well, you could, but it would be pointless, because this is the right answer.
I erased the theorem. It makes it easier to prove.
Well, let’s just defer that. Maybe we’ll defer that off to infinity.
So, 
At certain critical values of 
This is one of those things which you probably already understand but won’t after I’m finished explaining it.
[After scribbling all over one board, and mumbling a few disjointed, incoherent, sentences] That was a proof, by the way.
The sum of the heights of eight Canadians is close enough to infinity.
So far, I have prefaced every lecture by saying that this one is trivial. You may be asking yourself, “is every lecture trivial?” Fortunately, or unfortunately, no.
[While discussing multivariable max/min problems] It’s obvious that there’s a maximum. It’s obvious to me, anyway, and I’m giving the lecture.
There is a point I want to make. [pause] That was my point.
[Looking at a stack of problem sets being returned, each set about 50 pages long] I’m glad I’m not taking this class. It looks like a lot of work!
No hardcore math people here? Then I can get away with this.
Don’t worry; I can’t pass the exams either.
-----END AM-----
Think freely, think deeply, think differently…. [Ehssan]
One may wish mournfully for faster silicon, but the only absolutely fatal disease to a scientist is a deficiency of thinking. [JPB, CFSM-p. 41]
We all owe it. It is our duty to repay it. And repaying it is the way we pass the debt on to our descendants. [W. Kahan, in his honorary doctoral speech, U of Waterloo]
The analysis was very complicated—a member of the National Academy of Sciences once described it to me, laughing, as “the most complicated damn thing I’ve ever seen”—but the final answer fits on one line. [JPB, CFSM-p. xii]
-----BEGIN AI-----
This is not a review of undergraduate mathematics or a distillation of the wisdom of many lecture courses into a few pages. Certainly, nobody should use it to understand new material. Mathematics is not learnt from crib-sheets and brief compendia but by careful study of definitions, theorems and—most importantly, perhaps—proofs, by elucidating the intuition behind ideas and grasping the interconnectedness between what might seem disparate concepts at first glance. There are no shortcuts and no cherry-tasting knowledge capsules to help you along your path….
A conscious attempt has been made throughout the volume not to take for granted any knowledge that an advanced mathematics undergraduate is unlikely to possess. If we need it, we explain it. However, every book has to start from somewhere.
Unless you have basic knowledge of the first two years of university or college mathematics, this appendix will not help you and, indeed, this is the wrong book for you. However, it is not unusual for students to attend a lecture course, study material, absorb it, pass an exam with flying colours—and yet, a year or two later, a concept is perhaps not entirely forgotten but resides so deep in the recesses of memory that it cannot be used here and now. In these circumstances a virtuous reader consults another textbook or perhaps her lecture notes. A less virtuous reader usually means to do so—not just yet—in the meantime plunges ahead with a decreased level of comprehension. This appendix has been written in recognition of poverty and scarcity of virtue.
While trying to read a mathematical textbook, nothing can be worse than gradually losing the thread, progressively understanding less and less. This can happen either because the reader fails to understand the actual material—and the fault may well rest with the author—or when she encounters unfamiliar mathematical constructs.
If in this volume you occasionally come across a mathematical concept and, for the life of you, simply cannot recall exactly what it means (or perhaps are not sure of the finer details of its definition), do glance in this appendix—you may find it here! However, if these glances become a habit, rather than an exception, perhaps you had better use a proper textbook!
There are two sections to this appendix, one on linear algebra and the second on analysis. Neither is complete—they both endeavour to answer possible queries arising from this book, rather than providing a potted summary of a subject.
There is nothing on basic calculus. Unless you are familiar with calculus then, I am afraid, you are trying to dance the samba before you can walk. [AI, AFCINADE, Appendix A: Bluffer’s guide to useful mathematics]
-----END AI-----
Mathematics is replete with diverse concepts bearing the identical sobriquet “stability” and a careful mathematician should always verify whether a casual reference to “stability” has to do with stable ultrafilters in logic, with stable fluid flow or…. [AI, AFCINADE]
Divide et impera—Divide and conquer. [Louis XI]
A lemma is a helping proposition, or subsidiary proposition, as well as a bract in a grass spikelet just below the pistil and stamens. [M.D. Greenberg, Foundations of Applied Mathematics-p. 81]
Lewin is known at MIT for his clear and often dazzling presentations. His legendary lectures are not only popular in Cambridge and on iTunes U: On May 9, five of his lectures were in Google video’s top 100 listing. Lecture 32 from Electricity and Magnetism was number one in the list for that day, ahead of all sex videos. [Elizabeth Knox, MIT News Office Correspondent July 25, 2007]
What counts is not what you cover, but what you uncover. [Walter G.H. Lewin]
A tourist stops someone on the street in New York City and asks how to get to Carnegie Hall. The response is “Practice! Practice! Practice!” [Anonymous]
We could be led, by following this structure of nullspaces and boundary conditions, to homology groups and elliptic complexes and Hodge theory—somewhere near the center of pure mathematics. It is beautiful, but to go that way seems unwise. We want the framework to be general, but not that general. [GS, AMM-30-283-A Framework for Equilibrium Equations]
We emphasize that this pattern is not new (revolutionary inventions are doubtful models). [GS, AMM-30-283-A Framework for Equilibrium Equations]
What makes it possible to study such a sequence of different applications is that mathematically they fit into a single framework. The theories are separate but parallel.\footnote{The resistance of such a long series of problems would normally be too great; but their resistance in parallel….} [GS, IAM-p. 87]
Mixed elements lead to mixed results. [GS’s famous dictum in FEM]
[For positive pivots] it is the sign of the diagonal and the size of the off-diagonal that are jointly decisive. [GS, IAM-p. 17]
The system looks for a point of equilibrium, and an equilibrium is successful only if it is stable. It should require the input of energy to move away from equilibrium; otherwise, if a movement releases rather than consumes energy, that movement will grow.\footnote{I think this also applies to us.} [GS, IAM-p. 32]
We could check (11) by computing 
Less is more, more or less. [Ludwig Mies van der Rohe]
That is the Euler equation.\footnote{This differential equation for the minimizing 
Then the key to 
This is the boundary condition that comes NATURALLY from the minimum principle. [GS, IAM-p. 169]
FEM can be summarized in the following sentence: Project the weak form of the differential equation onto a finite-dimensional function space. The rest of this section deals with explaining the above statement. [MATLAB Help]
What properties are desirable in the trial functions?…These requirements are a mixture of engineering mathematics and computer science and numerical analysis. So is the finite element method. [GS, IAM-p. 433]
The concept of trial functions was old, but the CHOICE of trial functions (as piecewise polynomials) was the right idea at the right time. [GS, IAM-p. 445]
The principles are clear—but if you are going to use it [FEM] tomorrow there is more I should say. [GS, IAM-p. 439]
FAQ: If I miss a lecture or recitation should I find the professor or TA and ask “Did I miss anything important?” Answer: Be aware that this question, phrased in this manner, has been known to provoke sarcastic answers from faculty. (“No, we saw you were not there and realized that we could not discuss anything of substance today.”) [J. Wolfe, MIT Intro to Psychology]
We feel sorry for people who come in with sad stories about how they came to plagiarize. We feel sorry, but they still fail. [Wolfe, MIT Intro to Psychology]
The gradient is called “del”—I think it is a nickname for delta—and the Laplacian is “del squared.” [GS, IAM-p. 183]
It is almost true that the battle between the Boeing 767 and the Airbus depends on the solution to this equation.\footnote{It also depends on politics.} [GS, IAM-p. 184]
For fluids, this [KVL] corresponds to irrotational flow. A vortex is not allowed; you cannot mix a drink or pour cream into coffee. [GS, IAM-p. 185]
In economics the potential becomes the “price,” and money flows like water. [GS, IAM-p. 199]
And we had better not close without emphasizing that div grad u = 0 is not an identity! It is Laplace’s equation, and it has to be solved. [GS, IAM-p. 207]
One point about this framework. The author hoped, and he hopes the reader will hope, that the two side-by-side systems fit into one. The time derivative has to join with the space derivatives. By some miracle this does happen. [GS, IAM-p. 208]
At one time so many units were defined by independent experiments that by official Act of Congress, Ohm’s law was repealed. [GS, IAM-p. 209]
At the end we discuss the invariance under coordinate transformations—the intrinsic geometrical meaning—that makes a tensor a tensor [GS, IAM-p. 220]
The underlying principle is expressed beautifully but almost too concisely by 
Half the world reserves theta and phi. The situation is unbelievable. [GS, IAM-p. 216]
advice to graduate applicants:
I am accepting no new graduate students [F.E.C. Culick, Caltech]
The strain might have been 
That is Laplace’s equation for w, after discarding the zeros. To be honorable we have to show it has a solution. [GS, IAM-p. 226]
The gradient and divergence look different for different systems; but in themselves, they do not change. The laws of mechanics are coordinate-free (or we are absolutely lost). [GS, IAM-p. 227]
Einstein needed curvilinear coordinates and the Schwarz–Christoffel symbols that come with derivatives. That would carry us pretty far, and not with the speed of light…. [GS, IAM-p. 227]
The Eulerian description gives the velocity at each point. The Lagrangian description gives the velocity of each particle. The fluid is flowing past Euler, who sits at point and watches Lagrange go by. [GS, IAM-p. 228]
Pattern recognition is an inverse problem—to recover the coloring book from the finished picture—like recovering the coefficients of a differential equation from its solutions. [GS, IAM-p. 198]
The gods may throw a dice
Their minds as cold as ice
And someone way down here
Loses someone dear. [Abba, The Winner Takes It All]
People often wonder where the name “Chain Rule” comes from. I was just wondering about that myself. [Pulls a chain out of his briefcase.] Is it because it chains you down? Is it like a chain fence? I think I’ve decided what it is. It’s because by using it you BURST the chains of differentiation and you can differentiate many more functions using it. It lets you burst free. [Haynes Miller (substituting for David Jerrison), MIT 18.01-Calculus (Lect. 4, 00h:29m)]
After a day of discussion of the various challenges facing the industry, an audience member from Fogra tried to put things in perspective with a little color science humor: In the last 10-15 years, he said, we’ve gone from “different colors to color differences.” [from Steve Eddins’ blog, http://blogs.mathworks.com/steve/2008/11/13/icc-devcon-2008]
Let me just introduce you to a little bit about matrices just enough for what we’ll need later in this class. I should say if you want to know everything about the secret life of matrices, then you should take 18.06 some day. [Denis Auroux, MIT 18.02-Multivariable Calculus (Lect. 3, 00h:14m)]
Just your luck. We have them in your size. Over the limit. Under arrest. [Drunk driving infomercial showing a pair of handcuffs]
Those who speak two languages? Bilingual. Three languages? Trilingual. One language? American [Yale N. Patt, The microprocessor ten years from now, Talk given at CMU]
[In response to an introduction given before his talk] Thank you and there’s even 2 minutes left for the talk. [Yale N. Patt, The microprocessor ten years from now, Talk given at CMU]
What is Moore’s law a law of? Physics, Psychology, [Somebody in the audience answers: a business plan] If the only thing you have is a hammer, everything looks like a nail to you. [Yale N. Patt, The microprocessor ten years from now, Talk given at CMU]
I actually have a data slide! If you have too many data slides, then you’re a new PhD looking for a job. If you have no data slides, then you are a department head. Say, you have to have a data slide. [Yale N. Patt, The microprocessor ten years from now, Talk given at CMU]
[Somebody from the audience] But who’s gonna fund it?
—I don’t care about who’s gonna fund it; I’m a college professor. [Yale N. Patt, The microprocessor ten years from now, Talk given at CMU]
[Denis Auroux, MIT 18.02-Multivariable Calculus, (Lect. 4, 00h:43m), erases two sliding blackboards as one is moving down the other!] The class claps.
Warning: this comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors). [xkcd]
In summary, I claim that some of the reasons why so many people who have greatness within their grasp don’t succeed are: they don’t work on important problems, they don’t become emotionally involved, they don’t try and change what is difficult to some other situation which is easily done but is still important, and they keep giving themselves alibis why they don’t. They keep saying that it is a matter of luck. I’ve told you how easy it is; furthermore I’ve told you how to reform. Therefore, go forth and become great scientists! [Hamming, You and Your Research]
There are wavelengths that people cannot see, there are sounds that people cannot hear, and maybe computers have thoughts that people cannot think. [Hamming, You and Your Research]
A pointer might be handy. [Somebody in the audience, perhaps the one who introduced him, says, “There’s [a] laser pointer just right to the viewgraph.”] I’m always worried about these things. I point them in the wrong direction and zap a member of the audience. Ah! Got it the right way! [Sidney Coleman, Quantum Mechanics in Your Face]
My passion has become my profession and I’m a very lucky person. [Sidney Coleman to his brother Robert]
A lot of memory companies claim to be memory manufacturers even though they’re just memory module assemblers. By putting two pre-manufactured memory parts together to build a memory module, they feel they’ve manufactured memory. [Crucial, Truth about Memory manufacturers]
Linux never pats you on the back when you do something right but it slaps you on the head when you do something wrong [Mike, Linux+]
We find you guilty of closing your torrents as soon as they finish. Your sentence is unremovable Hungarian subtitles on everything. [xkcd, Pirate Bay]
(Charlie) So, first plane crash?
(Claire) What gave it away?
(Charlie) I can always spot the newbies. [Lost, S03E21]
I hope you like these topics in geometrical probability; linear algebra is a much bigger world. Mathematics is even bigger than linear algebra, slightly. And we allow calculus, every once in a while. And just enjoy. [GS, BLOSSOMS, Are Random Triangles Acute or Obtuse?]
@humphd (with a “d”, like “Didn’t sign-up soon enough to get humph”) [David Humphrey, http://vocamus.net/dave/?p=554]
This is the “ballerina effect.” She becomes tall and thin, like the ellipse, and she spins faster and faster like an ice-skater. I am a little sorry she is spinning around the [horizontal]
One theme of this book is the relation of equations to minimum principles. To minimize 
At each step the examples will be as familiar (and famous) as possible. In two dimensions that means Laplace’s equation, and minimal surfaces in the nonlinear case. In time-dependent problems it means F=ma, and relativity in the nonlinear case. I hope you reach that part. In one dimension we rediscover the circle. This section is also the opening to control theory—the modern form of the calculus of variations. Its constraints are differential equations, and Pontryagin’s maximum principle yields solutions. But something has to be left for the next book. [GS, IAM-p. 242]
That diagram is worth a chapter of words. It is the shortest path to nonlinear equilibrium equations [GS, IAM-p. 251]
Since these nonlinear things are in front of us, why not take the last step? It is never seen in advanced calculus, but there is nothing incredibly difficult. It is the direct link between 
The whole nonlinear theory is there, provided the material laws are conservative—the energy in the system should be constant. This conservation law seems to be destroyed by dissipation, or more spectacularly by fission, but in some ultimate picture of the universe it must remain true. [GS, IAM-p. 253]
Fortunately or unfortunately, the world is not in equilibrium. The energy stored in springs and beams and nuclei and people is waiting to be released. [GS, IAM-p. 253]
Anyone who views aliasing solely as a source of errors is missing out on the most important (and intriguing) strengths of PS methods. [Fornberg, A Practical Guide to Pseudospectral Methods-p. 43]
It is all too easy to equate multiple windows with hard work, and multiple contour plots with progress. [JPB, CFSM-p. xii]
The worst sin of a thesis adviser or a textbook writer is to have no opinions. [JPB, CFSM-p. xii]
The Heart of Africa has lost its mystery; the planets of Tau Ceti are currently unknown and unreachable. Nevertheless, the rise of digital computers has given this generation its galleons and astrolabes. The undiscovered lands exist, in one sense, only as intermittent electric rivers in dendritic networks of copper and silicon, invisible as the soul. And yet the mystery of scientific computing is that its new worlds over the water, wrought only of numbers and video images, are as real as the furrowed brow of the first Cro-Magnon who was mystified by the stars, and looked for a story. [JPB, CFSM-p. xiii]
What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school…. It is my task to convince you not to turn away because you don’t understand it. You see my physics students don’t understand it…. That is because I don’t understand it. Nobody does. [RPF, QED, The Strange Theory of Light and Matter]
If calculation is the first thing that enters your mind when thinking of mathematics, I can assure you that your knowledge of math does not exceed simple addition and multiplication. [Ehssan in his blog, http://www.artofproblemsolving.com/Forum/weblog.php?w=509]
…low order methods are like the chrysalis of a butterfly…inside every low order program is a high order algorithm waiting to burst free. [JPB, CFSM-p. 18]
All happy family resemble one another but each unhappy family is unhappy in its own way. [The opening line of “Anna Karenina” quoted as the opening line of AI, AFCINADE-p. 105]
The reason that a multi-dimensional BVP solver may take weeks or months to develop is not because of the intricacies of Sobolev spaces, but because of the care and fussing that is needed to correctly use the “!*#S%&” [expletive deleted] matrix indices. [JPB, CFSM-p. 120]
It is essential, however, that the reader be prepared for these problems; the alternative is despair and cursing when a Gaussian elimination subroutine bombs out with the error message: SINGULAR MATRIX. [JPB, CFSM-p. 120]
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different. [Johann Wolfgang von Goethe]
In order not to get fooled by the irrelevant information it is necessary to meet the data with a prejudice of some sort. A typical prejudice is of the form “Nature is Simple.” [Ljung, IFAC Conference]
Our starting point is a nonlinear example from geometry. I am tempted to call it simple, and it is, even though it contains more mathematics that we can do. [GS, IAM-p. 587]
Do you ever want to kick the computer? Does it iterate endlessly on your newest algorithm that should have converged in three iterations? And does it finally come to a crashing halt with the insulting message that you divided by zero? These minor trauma are, in fact, the ways the computer manages to kick you and, unfortunately, you almost always deserve it! For it is a sad fact that most of us can more easily compute than think—which might have given rise to that famous definition, “Research is when you don’t know what you’re doing.” [Forman S. Acton, Numerical Methods That Work]
Before embarking on 9 hours of lectures on Monte Carlo methods, let me offer a warning: Monte Carlo is an extremely bad method; it should be used only when all alternative methods are worse. [A. Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms]
And if you don’t enjoy the book, it might at least save be cheaper than a prescription sleep aid. [Don M. Chance on his book “Essays in Derivatives”]
Some say the glass is half-empty, others say it’s half-full, and some say the glass is just too large. [?]
Life is what happens to you while you’re busy making other plans. [John Lennon, Beautiful Boy (Darling Boy)]
Stephen Belloti: “Myron, what do you have more of—money or brains?”
Myron Scholes: “Brains, but it’s getting close.” [?]
the web site address has a blank space between uk and aas instead of a “~”. So the correct address is
http://www.maths.strath.ac.uk/~aas96106/option_book.html
This error arose at the final copyediting stage, and LaTeX user’s will appreciate that it was caused by somebody overriding my use of the verbatim environment. [Desmond J. Higham, An Introduction to Financial Option Valuation, errata]
A first-rate theory predicts, a second-rate theory forbids and a third-rate theory explains after the event. [Alexander Kitaigoroski]
Using a large sheet of paper and a pen with plenty of ink, show that for 
Where one path zig-zags, the other path zag-zigs. [Desmond J. Higham, An Introduction to Financial Option Valuation-p. 223]
In order to solve this differential equation you look at it till a solution occurs to you. [George Polya]
One way or another I’m gonna find ya
I’m gonna getcha getcha getcha getcha
One way or another I’m gonna win ya [Fermat’s Last Theorem Documentary, the song played when the bridge (Taniyama–Shimura Elliptic-Modular conjecture) is shown]
There are a number of great texts that do measure theory justice. This is not one of them. [Maya Gupta, A Measure Theory Tutorial]
>>why
He wanted it that way. [An output of MATLAB’s why]
At one point Visio told me that my drawing was too complicated and I should simplify it. I kid you not. That is when my love for Xfig fully blossomed. [http://www.xfig.org intro]
The first step turns away from 
For vectors the picture is similar but harder to draw. [GS, IAM-p. 374]
Normally Newton’s \footnote{Also known as Newton-Raphson. We take the unjustified liberty of burying Raphson.} method converges quickly. [GS, IAM-p. 374]
If it [Newton’s method] starts near the unstable point at (0,0), it will slowly accelerate toward a minimum (I don’t know which one). [GS, IAM-p. 379]
A person with one watch always knows what time it is. A person with two watches is never sure. Avoid making copies. [Nick Parlante, Stanford CSE, Pointers and Memory]
What’s Your Mental Model for MATLAB Memory Management? Say that three times fast! [Loren Shure, http://blogs.mathworks.com/loren/2006/05/10/memory-management-for-functions-and-variables]
When a task cannot be partitioned because of sequential constraints, the application of more effort has no effect on the schedule. The bearing of a child takes nine months, no matter how many women are assigned. [Frederick P. Brooks Jr., The Mythical Man-Month: Essays on Software Engineering. Chapter 2傍he Mythical Man Month]
Parity is for farmers. [Seymour Cray, when he was told that the Cray-1 memory system, which did not have parity checking, was malfunctioning at Los Alamos due to the altitude.]
Forget UNIX—it will be gone in 5 years. [Tom Jermoluk, SGI (1990s)]
Real men don’t use backups, they post their stuff on a public ftp server and let the rest of the world make copies. [Linus Torvalds]
After being referred to as the father of the multi-computer, I will be very pleased to contribute to its demise. [Chuck Seitz]
…before developing the model, I make a brief digression to summarize the relevant findings in option pricing theory. (About a full page later, after he’s given the Black–Scholes formula) This completes the digression. [Metron, JBF-1-3-An analytic derivation of the cost of deposit insurance and loan guarantees: An application of modern option pricing theory]
PETER LAX was born in Hungary in 1926; he came to the U.S. in December, 1941 on the last boat. He is a fixture at the Courant Institute of New York University; his mathematical interests are too numerous to mention. He has always liked to teach at all levels, hence this paper. [AMM-106-497-Change of Variables in Multiple Integrals]
Transposing a matrix is too easy; the underlying rule is mostly forgotten. That rule is to keep the inner product 
There is an astonishing imagination, even in the science of mathematics…. We repeat, there was far more imagination in the head of Archimedes than in that of Homer. [Voltaire. A Philosophical Dictionary, Boston, 1881, vol. 3, p40]
It [QR] is also available in MATLAB, a useful code for matrix computations on the IBM PC. [GS, IAM-p. 391]
Go!
I’m going nowhere! [Notepad++, Button labels in Go to… window]
If the code and the comments disagree, then both are probably wrong. [N. Schryer]
Documentation is like sex; when it’s good, it’s very, very good, and when it’s bad, it’s better than nothing. [Dick Brandon]
Any fool can write code that a computer can understand. Good programmers write code that humans can understand. [Martin Fowler]
Programming can be fun, so can cryptography; however they should not be combined. [Kreitzberg and Shneiderman]
If only God would give me some clear sign! Like making a large deposit in my name at a Swiss bank. [Woody Allen]
I don’t know the question, but sex is definitely the answer. [Woody Allen]
I have bad reflexes. I was once run over by a car being pushed by two guys. [Woody Allen]
How big is the big picture? [Andrew Mullhaupt, Berkeley]
The beginner is often confused between the model and reality. When tossing a coin, the model will generally not include the result that the coin ends up on its edge, nor that it is lost when it rolls off the table, nor that the experimenter drops dead in the middle of the trial. [Hamming, The Art of Probability for Scientists and Engineers-p. 8]
A remark on “A note on constructing a symmetric matrix with specified diagonal entries and eigenvalues” [Title of Ikramov’s paper, BIT-38-p. 807]
I also wish to acknowledge the creators of an excellent software package, MATLAB, which made running experiments as easy as thinking of them. [A. Edelman, SJMAA-9-543]
I was done with finals. Matt was done with finals. Liz was done with finals. Angie wasn’t done with finals. So we went to the beach! [Andrew Duchi, Stanford undergrad]
reducing the value of TolFun need not reduce the error of the fit. If an optimizer has converged to its global optimum, reducing these tolerances cannot produce a better fit. Blood cannot be obtained from a rock, no matter how hard one squeezes. The rock may become bloody, but the blood came from your own hand. [John D’Errico, Optimization Tips]
The numbers in Pascal’s triangle satisfy, practically speaking, infinitely many identities. [Graham, Knuth, Patashnik, Concrete Mathematics]
Looking at the algorithms that are most successful [in solving matrix problems] the key seems to be an intelligent use of orthogonal matrices. They are certainly stable; what we need is a fast way to produce them. There is always Gram–Schmidt, but Householder had a better idea. [GS, IAM-p. 392]
Because of the Householder matrices, it [
[This is probably the shortest preface ever]
Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician. [Charles Chapman Pugh, Real Analysis]
Like an exiled monarch, it [supremum] rules from without. [AM, Introduction to Analysis]
Airplanes land on automatic control while a Kalman filter is fitting speed and position to the laws of motion. [GS, IAM-p. 148]
It is too easy after many late nights spent in front of a computer screen and/or laboratory bench to convince yourself that your work is the best invention since sliced bread. More than likely it is not…. [Philip E. Bourne, Ten Simple Rules for Getting Published]
Will my computer go up in smoke if I do not buy a license after 30 days of use? [TexPoint Licensing FAQ’s]
You go to the Greek islands for three weeks; you come back and computers are faster. [Stephen Boyd, Convex Optimization I (Lect. 1), Stanford]
Are you from Statistics? When I’m making fun of a department, I like to have a face to look at. [Stephen Boyd, Convex Optimization I (Lect. 1), Stanford]
I said this was my favorite course. Maybe I elaborate a little. My life is here to teach you, NOT to grade you. I’m not gonna spend the semester worrying about grades, and please don’t; they come out fine. We’ve got lots to learn and I’ll do my very best to explain it clearly. And I know you’ll do your best. I know from experience this class goes for it and does it right. That’s what makes it so good. [GS, MIT CSE (Lect. 1)]
…because if I’ve brought fun to academia, it was about damn time. [Randy Pausch, Time Management]
I put Physics here [under Calculus], to give them a head start. They’ve got differential equation and they integrate stuff. I’m a little shaky on putting that [Economics] there [under Linear Algebra] ’cause you know we’re in the middle of this financial meltdown and some people blame it on mathematicians for creating these very sophisticated instruments. You know what those instruments are called; they’re often called “derivatives.” So I blame it on Calculus. [GS, Life of a Mathematician, (00h:11m)]
The astronomers are terrific at that. I don’t know if you realize that, there aren’t too many of them but they have the best pictures. [GS, Random Triangle]
Can I call them [the angles] 
There is at present a strong movement called Bayesianism. It is much more connected with statistics than with probability and hence strictly falls outside the field of this book. Still, because of its prominence it is necessary to look briefly at it—though one of its practitioners, I. J. Good, claims that there are 46656 varieties of Bayesians! [Richard Hamming, The Art of Probability, p. 297]
What’s right isn’t always popular. What’s popular isn’t always right. [Howard Cosell]
The solution is to encapsulate the symmetric key inside a message encrypted with an asymmetric algorithm. You have never transmitted your private key to anybody, then the message encrypted with the public key is secure (relatively secure, nothing is certain except death and taxes). [Franck Martin, GAtech]
Major mode are called major because there are also minor modes. [Emacs tutorial]
This tutorial is meant to be understandable to all users, so if you found something unclear, don’t sit and blame yourself—complain! [Emacs tutorial]
What starts here will change the world. [The University of Texas at Austin]
What is to give light must endure burning. [Viktor Frankl]
from intelligent people (me) for intelligent people (you) [K3DSurf]
If you have problems with this page it probably means you are using a junk browser. Especially the one from the monopolist. Get something better. Use Firefox or Mozilla. [Ulrich Drepper, http://people.redhat.com/drepper]
Question: How many TI execs does it take to change a light bulb? Answer: Three. One to turn the bulb, one to paint it orange and yellow, and one to add an extra filament and call it the lightbulb silver edition. [Michael, TICalc of quoting a joke written by Ray Kremer in a TI discussion board post]
Converting numbers into Roman numerals as easy as I, II, III. [Google Docs]
Artificial intelligence is no match for stupidity. [Anonymous]
The shortest path between two truths in the real domain passes through the complex domain. [Jacques Hadamard]
Customizing Emacs is a sport pursued world-wide, at least. [Charles Curly, http://www.charlescurley.com/emacs.init.html]
Unless you have used some very dirty tricks, the compiler will know that the thing you are trying to modify is constant, so it can warn you. [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/chapter5/pointers.html]
In fact the only restriction is that a structure cannot contain an example of itself as a member—in which case its size would be an interesting concept for philosophers to debate, but hardly useful to a C programmer. [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/chapter6/structures.html]
It is the job of a good language to do more than just allow you to do something; it must actively help as well. [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/chapter6/history.html]
You won’t ever go back later and document your code. You just won’t. Don’t lie to yourself, the world, and your mother by saying that you will. [Todd Hoff, C++ Coding Standard]
When all is said and done, more is said than done. [Anonymous]
To not decide, is to decide. [Martin Luther]
The previous chapters have introduced the fundamentals of the language and have covered nearly all of the language that the Standard defines. There are a number of murky and convoluted backwaters left unexplored on grounds of sympathy and compassion for the sufferer, and some without any better home. This chapter gathers them together—it’s the toxic waste dump for the nasty bits of C. Pull on your rubber gloves, read the following sections and make notes where you think the material is important to you; re-read them from time to time as well. What seemed uninteresting and painful the first time round may change as your experience grows, or your natural immunity improves. What we cover here is not an exhumation of all the pathogenic elements—we leave that for another book—but it does serve to round up most of the commonly encountered difficult or extraordinary material. [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/chapter8/health_warning.html]
It is impossible to read C when it is laced with these things every few lines. The urge to maim the author of a piece of code becomes very strong when you suddenly come across
“#else } #endif”
with no “#if” or whatever immediately visible above. They should be treated like chilli sauce; essential at times, but more than a tiny sprinkle is too much. [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/chapter7/directives.html]
Permission is hereby granted for anyone to do anything that they want with this material—you may freely reprint it, redistribute it, amend it or do whatever you like with it. In doing so you must accept that you do so strictly on your own liability and that you accept any consequences with no liability whatsoever remaining with the original authors. If you find the material useful and happen to encounter one of the authors, it is unlikely that they will refuse offers to buy them a drink. You may therefore like to consider this material “drinkware”. (Offer void where prohibited by law, in which case fawning and flattery may be substituted.) [Mike Banahan, Declan Brady and Mark Doran, The C Book, http://publications.gbdirect.co.uk/c_book/copyright.html]
A random sequence is a vague notion…in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians… [Cleve Moler quoting D.H. Lehmer, Random Thoughts]
“Giving up smoking is easy,” he quipped. “I’ve done it hundreds of times.” [?]
Law of Total Probability—“How to put Humpty Dumpty back together again.” [W.H. Press, Lect. 1, Computational Statistics with Application to Bioinformatics]
Watch your steps kids, ’cause I’m about to drop some knowledge. [Barney, How I met your mother (S02E012)]
The difference between the right word and the almost right word is really a large matter—it’s the difference between a lightning bug and the lightning. [Mark Twain]
All the world’s a stage,
And all the men and women merely players;
They have their exits and their entrances;
And one man in his time plays many parts,
His acts being seven ages. At first the infant,
Mewling and puking in the nurse’s arms;
And then the whining school-boy, with his satchel
And shining morning face, creeping like snail
Unwillingly to school. And then the lover,
Sighing like furnace, with a woeful ballad
Made to his mistress’ [eye]brow. Then a soldier,
Full of strange oaths, and bearded like the pard[oner],
Jealous in honour, sudden and quick in quarrel,
Seeking the bubble reputation
Even in the cannon’s mouth. And then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part. The sixth age shifts
Into the lean and slippered pantaloon,
With spectacles on nose and pouch on side;
His youthful hose, well saved, a world too wide
For his shrunk shank; and his big manly voice,
Turning again toward childish treble, pipes
And whistles in his sound. Last scene of all,
That ends this strange eventful history,
Is second childishness and mere oblivion;
Sans teeth, sans eyes, sans taste, sans everything. [William Shakespeare, Jaques (Act II, Scene VII)]
It is not true that we have only one life to live; if we can read, we can live as many lives and as many kinds of lives as we wish. [S. I. Hayakawa]
The numbers you entered aren’t what we thought they should be. We could be wrong, but just in case, please try again. [Mozy Registration, incorrect CAPTCHA error message]
TeX capacity exceeded, sorry.
If you really absolutely need more capacity, you can ask a wizard to enlarge me. [LaTeX error message]
Sorry, I already gave what help I could….
Maybe you should try asking a human?
An error might have occurred before I noticed any problems.
“If all else fails, read the instructions.” [LaTeX error message]
Other go to-less languages for system programming have similarly introduced other statements which provide “equally powerful” alternative ways to jump. In other words, it seems that there is widespread agreement that go to statements are harmful, yet programmers and language designers still feel the need for some euphemism that “goes to” with saying go to. [Donald Knuth, Structure Programming with go to Statements]
Next time you are rude on the mailing list remember…
You are in range! [Eric Ludlam, the caption of a picture of himself and a catapult throwing a projectile]
RSA’s success bears two important lessons for readers: If Ron Rivest comes to you with an engineering problem, listen, and if he wants to put your name first on the resulting paper, let him. “ARS sounds better and better to me now,” Adleman quips. [SIAM News, Vol. 36, No. 5, June 2003, http://www.siam.org/news/news.php?id=326]
MathWorks’ first Massachusetts office phone number was 653-1415 (ignoring country and area codes). The astute reader will notice that the last 5 digits are an approximation for 
It’s easy to crash the package using styles. Write “\lstdefinestyle{crash}{style=crash}” and “\lstset{style=crash}”. TeX’s capacity will exceed, sorry [parameter stack size]. Only bad boys use such recursive calls, but only good girls use this package. Thus the problem is of minor interest. [Brooks Moses, The Listings Package]
To be able to use integration by part, perhaps it’s better LIATE than never. [Herbert E. Kasube, AMM-90-210-A Technique for Integration by Parts]
Links to other sites are other sites. [Howstuffworks disclaimer]
…when a man is tired of London, he is tired of life. [Samuel Johnson, September 20, 1777]
What is an Epigram? A dwarfish whole;
Its body brevity, and wit its soul. [Samuel Taylor Coleridge]
Little strokes
Fell great oaks. [Benjamin Franklin]
Microsoft English: You can forward your mail to one other e-mail address that ends in hotmail.com, msn.com, live.com, or is part of Windows Live Custom Domains. [Hotmail Forwarding Option]
English: You cannot set up hotmail to forward emails. Nice Try. [MATLABician]
There [in Federal Polytechnic Institute in Zurich] he [Ahlfors] produced his first major work, a study of asymptotic values of an entire function, based on his own new approach to conformal mapping. Self-effacingly, Ahlfors credited Nevanlinna and another teacher, George Polya, for their “considerable help.” They, in turn, insisted that he publish the results solely in his own name. Thereafter, as he expressed it, “I have tried to repay my debt by never accepting to appear as coauthor with a student.” [Harvard University Gazette, http://www.news.harvard.edu/gazette/1996/10.17/MathematicianLa.html]
CAUTION
THIS SIGN HAS
SHARP EDGES
DO NOT TOUCH THE EDGES OF THIS SIGN
(In super small font) Also, the bridge is out ahead.
Wir müssen wissen, wir werden wissen. (We must know, we will know.) [David Hilbert]
If you think C++ is not overly complicated, just what is a protected abstract virtual base pure virtual private destructor and when was the last time you needed one? [Tom Cargill]
Ex falso quodlibet. [From falsity, whatever you like.]
The Perl language…bridges the gap between shell and C programming (or between doing it on the command line and pulling your hair out). [Ted Timar, http://www.faqs.org/faqs/unix-faq/faq/part1/section-3.html]
I have something to show you. It may even be new. [Ralph Byers]
The present paper came from trying to move a chair onto a hotel balcony (it went through but wouldn’t go back). [GS, AMM-89-529-The Width of a Chair]
Bruce Reznick
Professor
“The John and Harriet J. Absentminded Professor of Mathematics” [Prof. Bruce Reznick, UIUC, at the beginning of his homepage, http://www.math.uiuc.edu/~reznick]
If you order a bowl of soup and there’s a fly in it, well just remember the fly is not having a great day either. [Larry King]
This release was brought to you as a result of weekend’s worth of work. Please donate, even you have already donated. I’ve already finished all the beer! [Alex Gorbatchev, SyntaxHighlighter, http://alexgorbatchev.com/wiki/SyntaxHighlighter]
Books serve to show a man that those original thoughts of his aren’t very new after all. [Abraham Lincoln]
There is a wonder in reading Braille that the sighted will never know: to touch words and have them touch you back. [Jim Fiebig]
For friends…do but look upon good Books: they are true friends, that will neither flatter nor dissemble. [Francis Bacon]
You may have tangible wealth untold;
Caskets of jewels and coffers of gold.
Richer than I you can never be
I had a mother who read to me. [Strickland Gillilan]
A book is the only place in which you can examine a fragile thought without breaking it, or explore an explosive idea without fear it will go off in your face. It is one of the few havens remaining where a man’s mind can get both provocation and privacy. [Edward P. Morgan]
It is what you read when you don’t have to that determines what you will be when you can’t help it. [Oscar Wilde]
