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The book by Ross and Wright is quite good.

The mediocre reviews at Amazon are likely at anomaly.

The book by Ross and Wright is quite good.

The mediocre reviews at Amazon are likely at anomaly.

However, unlike validation, testcases cannot definitively say that an application is conformant

True, you can't prove a negative (impossibility of non-conforming output) without proving correctness of the whole program, which is generally considered intractable. But if you've produced a wide variety of test cases that when put through a program result in conforming or non-conforming documents, you can state with a high statistical confidence level how much work needs to be done to reduce the probability of non-conforming output from the program to an acceptable minimum. You might find essay "When is a proof?" by Keith Devlin interesting.

I would just tell you to Google yourself, but you're being annoying, so here:

It was actually smallpox.

Whenever a large population of non-immunes exists, epidemics happen.

The model does not aim to predict the emergence of new strains of influenza, but it does suggest that a short-lived general immunity to the virus might affect the virus's evolution.

The model takes into account the effects of specific immunization against viral strains, but also infectivity randomness and the presence of a short lived strain-transcending immunity recently suggested in the literature.

A pandemic is possible when an influenza A virus makes a dramatic change (i.e., "shift") and acquires a new H or H+N. This shift results in a new or "novel" virus to which the general population has no immunity... Since, by definition, a novel virus is a virus that has never previously infected humans, or hasn't infected humans for a long time, it's likely that almost no one will have immunity, or antibody to protect them against the novel virus. If the novel virus is related to a virus that circulated long ago, older people might have some level of immunity.

Most of this seems really obvious to me, but what the hell do I know...

I already posted this problem, here:

http://ask.slashdot.org/comments.pl?sid=165444&thr eshold=1&commentsort=0&tid=228&tid=4&mode=thread&c id=13802487

Nobody has replied yet. :-( Too many other puzzles...

However, traditionally you are allowed to open the envelope before deciding whether to switch. (See, e.g., http://www.maa.org/devlin/devlin_0708_04.html ).

Also, here's where it really gets interesting. It turns out that both the arguments above about whether to switch are wrong! You can do better than to always keep the one you drew, but not by always switching. Since you know this puzzle already, I'll bet you're very surprised by this, and likely do not believe it. You can email me if you want me to prove it, though.

/PEDANTIC

Google is a mispelling of googol which, as a word, has it's its own interesting history:

Its very name is a derivation of the word googol, a term invented in 1938 by nine-year-old Milton Sirotta to denote a 1 followed by 100 zeros.

the use of computers to generate proofs is causing mathematicians to 're-examine the foundations of their discipline.'

They should be concerned about the use of computers to generate new formulas and conjectures, which is where the creativity is.

critics of computer-aided proofs say that the proofs are hard to verify due to the large number of steps and hence, may be inherently flawed.

Bullshit. Computer-aided proofs are also computer-verified. And only computers generate useful computer-verified proofs, because most humans are too proud to submit themselves to such unforgiving examination. (And because it takes an awful lot of boring work, too).

What you loose is the elegance of a short proof. But a nice-looking proof can be incorrect too (e.g. a geometric figure which doesn't cover all cases.)

The article sounds a bit like Microsoft PR saying they are investing in state-of-the art techniques to produce bug-free software. Which is good. But be sure to read the full credits.

However, critics of computer-aided proofs say that the proofs are hard to verify due to the large number of steps and hence, may be inherently flawed.

However, critics of critics of computer-aided proofs say that critics of computer-aided proofs should get around to solving Hilbert's 24th problem.

Hilbert prepared 24 open problems to present at the second International Congress of Mathematicians in 1900. For reasons that are not entirely clear, he only presented the first 23.

The 24th problem is a metamathematical problem in the field of proof theory. In brief, it asks for criteria of simplicity, i.e. guidelines by which you can prove that a particular proof is of maximal simplicity.

Until Hilbert's 24th problem is solved, our only hope for proving that particular proofs are maximally simple is through computational means.

• R, including several add-on packages (such as tseries, RODBC, coda, mcmcpack, gtkdevice, rgtk, rquantlib, qtl, dbi, rmysql), out-of-the box support for the powerful ESS modes for XEmacs as well as the Ggobi visualisation program;
• A complete teTeX, TeX, and LaTeX setup for scientific publishing, along with TeXmacs and LyX for wysiwyg editing;
• Perl and Python with loads of add-ons, plus ruby, tcl, Lua, and Scientific and Numeric Python;
• The Emacs and Vim editors, as well as Gnumeric, kate, Koffice, jed, joe, nedit and zile;
• Octave, with add-on packages octave-forge, octave-sp, octave-epstk, and matwrap;
• Computer-algebra systems Maxima, Pari/GP, GAP, GiNaC and YaCaS;
• the QuantLib quantitative finance library including its Python interface;
• GSL, the Gnu Scientific Library (GSL) including example binaries;
• The GNU compiler suite comprising gcc, g77, g++ compilers;
• the OpenDX, Plotmtv, and Mayavi data visualisation systems;
• it includes apcalc,aribas,autoclass,

Full disclosure: I work for Wolfram Research. But oh -- the irony! I am also a columnist for Math Games at maa.org, and I wrote an article about the Quantian Distribution. I didn't want a spammer to start using quantian.org just as the distro was getting popular, so I bought it, and provided a redirect to the main Quantian site. So now, I'm getting doubly Slashdotted. Huzzah. A student should definitely be getting Mathematica for Students -- but check with the college first. They might be on a Mathematica Campus, and can get it for free.

Full disclosure: I work for Wolfram Research. But oh -- the irony! I am also a columnist for Math Games at maa.org, and I wrote an article about the Quantian Distribution. I didn't want a spammer to start using quantian.org just as the distro was getting popular, so I bought it, and provided a redirect to the main Quantian site. So now, I'm getting doubly Slashdotted. Huzzah. A student should definitely be getting Mathematica for Students -- but check with the college first. They might be on a Mathematica Campus, and can get it for free.

Or is anyone else coming to the conclusion that any organisation named *IA or *AA is, in fact, corrupt and evil?

AAA? MAA?

He's also enjoyed the reputation of being someone who capitalizes on other people's hard work and clever ideas.

And he's not even the best at it.

If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture.

You can follow discussions on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's along with almost-proofs like Castro-Mahecha's and Dunwoody's.

Linkified goodness for the lazy (like me):

http://www.public.asu.edu/~mmcbeath/mcbeath.resear ch/CatchFly/CatchFly.html

http://www.maa.org/mathland/mathtrek_10_14_02.html

Here is an Keith Devlin article "Do software engineers need mathematics that addresses a number of these issues.

The overall point is that math courses often develop abstract thinking skills, which may be more important for developing efficient, correct code than learning a specific toolset which may not age well. I know Keith Devlin has written other articles about this but this was the only one I could find online.

IMHO, the discovery of a real-world application of the idempotent law that was Boole's greatest accomplishment. One could argue that Lebnitz and Boole had independently discovered this. This is not unlike Hamilton's discovery of an application for non-commutative algebra.

Boole's contribution to logic was profound. First, a real world model for any mathematical property ensures the consistency of that model. Boole's work provided an abstraction for elementary set theory. The key to this abstraction is idempotency. The aggregate of set A and itself is the set A (i.e. A+A=A). Thus, Boolean algebra formalizes the basic set theoretic operations of union and intersection, which in turn is almost trivially isomorphic to a Boolean ring. I could create all kinds of stupid rules [insert your favorite slam on mathematics here] that have no meaning in the real world. Most importantly, Boole seemed to be the first to attempt to bridge the gap between abstract thought and mathematics. Admittedly there was some previous work in attempting to formalize|classify all syllogistic reasoning. It was the first step towards a unified theory of logic and ultimately what is hope to be a universal theory of symbolism (see Chomsky's mathematical linguistics).

The irony about mathematics is that often the best ideas are childishly simple. It's not the proof of deep theorems (although that has it's place) that often has the greatest impact. It's the fresh applications of mathematical rigour to some real world scenario. Thus, mathematics is often at it's weakest when done in isolation. Incidentally, Knuth's work in algorithm analysis was revolutionary. In a world described by (K-Complexity (AIT)|cellular automata|simple computer programs) algorithm analysis and ultimately a proof of P not= NP may be to hold the key to the fundamental laws of nature (i.e. physics, biology, and chemistry).

Incidentally, the Martin Davis' The Universal Computer is a great popular science book on this topic. A free copy of the introduction is here. This book manages to introduce the ideas of Turing (Turing-Post?) Machines and the Diagonal Method to the lay reader. The author is a respected logician and computer scientist who studied under Church and Post.

Corrected link: http://www.maa.org/editorial/mathgames/mathgames_1 2_22_03.html

Whether it's something obvious like the Pentium off by 1+1=1.9999943 error

The Pentium math bug was with division, not addition, and it only occurred in very specific circumstances. So while it supports your general point that complicated systems are more difficult to debug, that wasn't a very good example of an "obvious" bug. Careless, yes.

One thing that was good for the industry was to move away from the complex instruction set (CISC) towards a reduced set of instructions (RISC), and we have seen the speed improvements as well as a general reduction in hardware bugs since that time.

You do realize that Intel x86 processors are still CISC, right? (OK, actually internally they do execute things very much like a RISC chip, but the instruction set is still CISC, and modern x86 processors are certainly not any _simpler_ for having some RISC-like elements to them.

Besides, RISC chips don't actually have fewer instructions. Most of them these days have more. The difference between CISC and RISC is that RISC chips don't have certain complicated, slow instructions, but rather break these up into smaller pieces. For example, CISC processors usually have an instruction to move memory-to-memory while RISC only moves memory-to-register and register-to-memory. Also, CISC processors often have a division instruction while many RISC processors instead just have a multiplicitive inverse instruction (so to compute a/b you instead compute a*inv(b)).

But to add Hyperthreading, an untested and unproven technology which can guarantee no more than a 12% speed improvement, is folly. Better to amp the CPU clock and deal with a known like heat than to risk your company's livelihood on letting the CPU figure out which thread is which. That is something an OS is much more reliable in handling.

Now that's just ridiculous. Hyperthreading is not untested or unproven. Similar ideas have been discussed in academic papers for years; Intel was just the first to put it into a modern CPU. It's hardly untested, either - Intel started seeding the first Hyperthreading-capable processors what, two years ago now? At that point I wouldn't have suggested running a mission-critical application on a machine with Hyperthreading enabled, but now? You'd be crazy not to if it actually speeds up the application you need to run.

The reality is that in order to advance the speed of computer processors, it's necessary to make them more complicated.

The main problem in the USA isn't how we gather votes, although there are problems in some states (Florida). There is a more fundamental problem in that we aren't using the right voting mechanism. In the US, we use plurality voting -- a.k.a "first across the line" -- to determine who wins an election. This means that a candidate for whom only 30% of the people voted can win an election simply because there was no other single candidate with more votes.

This has a number of problems, but they can all be summed up by saying that plurality is one of the least fair, if not the least fair, way of determining the winner of a democratic election that you can get. Consider:

• Say 40% of the people vote for candidate A
• 35% of the people want candidate B
• 25% want candidate C

This situation encourages strategic voting; that is to say, voters for C have to decide whether they want to vote honestly, for C, or whether they should vote for B just to make that they don't get their least favorite candidate, A.

This is why we only have two parties in the US, and why -- despite the large number of Greens and Libertarians, neither party has a chance of winning. We don't even know what percentage of the US population is Green or Libertarian (or anything else, for that matter) because they aren't voting honestly. They're voting for the lesser of two evils. This system practically guarantees alienation of the largest number of people -- the majority ends up with a candidate they don't want, unless they lie when voting and vote for the candidate that they dislike the least who also has the best chance of winning.

There are voting mechanisms which allow people to vote their true opinion without being alienated. The most popular are Condorcet -- complex, but the most fair; Approval Voting -- not as fair as Condorcet, but much simpler, and can be implemented with existing voting technology; and Instant Runoff -- less fair than approval, no more simple -- but better than plurality.

Many democratic countries do not use plurality voting, although plurality is the most common. For example, Australia, Northern Ireland, and the Irish Republic (among others) use single transferable vote[1]. In fact, 68 countries (~2b ppl) use plurality, 31 countries (~400m ppl) use single transferable vote, and two countries (~18m ppl) use IRV (instant runoff) -- this is according to International IDEA Handbook.

There is a huge amount of information about Condorcet and Approval Voting available on the web. The Citizens for Approval Voting page is a good start, if you're at all interested in improving voting in the US. If you're interested in the mechanics and mathematics of the systems, start with Condorcet -- most sites that talk about Condorcet are less about how to get it implemented politically, and are more about how it works, fairness tests, and how it compares to other systems. The Wikipedia entry for "voting system" is particularly useful.

I was trying to take the first possible move available (e.g. for 2 1 0 4, try 2, then 1, then 0, then 4) but it's apparently not what I should be doing...

Thanks!

Rating: 9.9 - Cannot find a fault

Obviously, the review was calculated using an early Pentium.

Especially if you are looking to -suppliment- an education system that is based on "here is a TYPE X problem ... you solve it this way with this algorithm ..."

I suggest books such as Proofs from THE BOOK. Firstly it deals with topics outside the -regular curriculum- but whose problems are easily explained to (and understood by) just about anyone. "Graph Theory" ... simply put "connect-the-dots". "Combinatorics", well that's just "counting things".

This book is particularly good as it offers, in many cases, more than one proof for a given problem, looking at problems in different creative ways to find elegant solutions. Feynman was a huge advocate of this (see the "Cargo Cult Science" chapter in his book Surely you're joking Mr. Feynman!

They are the following.

1. Maurer, Stephen B. and Ralston, Anthony. Discrete Algorithmic Mathematics Reading, MA: Addison-Wesley, 1991.

2. Ross, Kenneth A. and Wright, Charles R.B. Discrete Mathematics, Englewood Cliffs, NJ: Prentice Hall, 1985, 1988, 1999. Third Edition.

In particular, the second textbook has plenty of examples. Answers to many of the odd-numbered problems are also included in the back of the book.

The book by Ross and Wright is essentially the best book on discrete mathematics if you are pursuing a course of self study. The best book also costs plenty of money but is worth it. You will find it to be a useful reference long after you have graduated with your degree in computer science. Discrete mathematics is, after all, the foundation of modern computer science.

I think he's already stated the point:

I'd like to educate my peers on the alternatives to Windows (Linux and Open Source), how hardware works and fits together, job offerings in computer-related fields, and anything else that may be of interest.

When I was in college, our student chapter of the MAA met for no other reason than to eat pizza, paid for by the math department book sales, and to listen to guest speakers give colloquia.

However, I think, given his reasons, the club will fail. Here's why:

1. I'd like to educate my peers on the alternatives to Windows (Linux and Open Source)
   noble cause, but the school probably won't let you use their hardware or your hardware on their net for demonstrations (however, since GNU/Linux and BSD can run on commodity stuff, not such a big deal)
2. how hardware works and fits together
   this assumes hardware works and fits together
3. job offerings in computer-related fields
   What are these so-called "job-offerings"?

There is actually a scientific explanation for the bunching phenomenon. It is explained thoroughly here

You can find the nth digit of Pi in hexadecimal, but not decimal.

See: http://www.maa.org/mathland/mathtrek_3_2_98.html

I once found a bug in my Pentium computer.

Godel, Escher, Bach

"The Advent of the Algorithm" - David Berlinksi
A quirky, very accessible treatment of the link between mathematics and computation.

"Concrete Mathematics" - Ronald Graham, Donald Knuth, and Oren Patashnik
If you're hard-core about the mathematical aspect.

"Object-Oriented Analysis and Design with Applications" - Grady Booch
If you're interested in OO.

Books by or about:
alan turing
godel
noam chomsky
claude shannon

fields 2002

-Kevin

Why would Putnam want to sponsor a literature course?

...they'd blame this on the Pentium floating-point bug.

See here for a page that links to about 15 reviews of ANKOS. My favorite is this review for the Mathematical Association of America.

See here for a page that links to about 15 reviews of ANKOS. My favorite is this review for the Mathematical Association of America.

OK, I may be biased (I work for one of them) but they can be quite helpful. Most if not all have reduced rates for students. I know that SIAM has activity groups focusing on different areas of Mathematics. (Anyone care to guess who I work for?).

I am a 2nd year undergrad at a local university studying Computer Science, and I work tech support here. I can't form a real accurate opinion of what we do for our members, as I'm not all that interested in it, or do I work with members much. But I know we have conferences all over the country and have Job listing and such. Plus it's a good way to network with other geeks... Umm... I mean Mathematicians.

I suggest you check out some websites, see what you like, and perhaps pose the question towards some of them.

Society for Industrial and Applied Mathematics
American Mathematical Society
Association for Women in Mathematics
Mathematical Association of America

Also, here is a direct link to the AMS's link page

Good Luck, and feel free to email me with questions.

for my .02, almost every AMD system i've seen has been rock-solid stable. Better than the intel's i've used/seen. The (AMD) ones that weren't stable were flawed not in processors, but in via/misc chipsets. bad motherboards. Intel has had trouble with bad mobo's, too - but what about that pentium bug? Oh well, give them time - I just like the competition. As much fun as i've had being on the AMD side of things, i'd switch if Intel's products were faster, cheaper, and as easily overclocked.

If anyone can go to the American Mathematical MOnthly website and find an article by Klehr or Bliss or both I'd appreciate you letting me know.

The Monthly article in question appears to be "Morley Related Triangles on the Nine-Point Circle", by Floor van Lamoen, Amer. Math. Monthly vol. 107, no. 10, Dec. 2000, pages 941-945. The introduction says: "We identify two points M and H on Euler's nine point circle CN, found as intersections of three reflected lines. M and H each depend on the direction of a set of parallel lines. Posing the condition that M and H coincide for a certain direction, or that MH is a diameter of CN, we find two equilateral triangles in CN homothetic to Morley's famous trisector triangles."

The fastest sprinter at the Sydney Olympics will not surpass 37km/h (23 mph)

World's Fastest Man
The 1998 edition of The Guinness Book of World Records features an article about Donovan Bailey, billed as the fastest man alive.
The article begins: "Canadian Donovan Bailey rocketed into the record books when he set a new world mark of 9.84 seconds for the 100-meter dash at the Atlanta Olympics." It briefly recounts Bailey's career and, toward the end, quotes Bailey: "No one has ever run as fast as I have, running 27 mph."

When Roy D. North, a computer programmer and mathematical gadfly now based in Connecticut, came across that passage, something bothered him. When he calculated Bailey's speed from the given time and distance, he obtained 22.7 miles per hour.

"What went wrong here?" North wondered. "Was Bailey misquoted? Was there a typo?"

North had to find out how that apparent mismatch had come about, and his search turned up an article in the Aug. 5, 1996, Sports Illustrated. The account mentioned that Bailey's speed at the 60-meter mark of the race was 27.1 miles per hour.

Mystery solved! One speed was the average over the entire race, and the other was the instantaneous velocity at a particular point in the race.

I really don't care for their choices at all. A lot of them are more like general approaches than algorthms, and I'm not at all sure they are the most influential. I think they are supposed to be "the cleverest of the common fancy methods"

Simple algorithms for common problems are much more widely used, and have far more impact and influence, but try telling *them* that!

I hope these links help. (Warning: many are technical) If anyone has personal favorites that are less dry than many of these, please post!.

10. 1987: Fast Multipole Method. A breakthrough in dealing with the complexity of n-body calculations, applied in problems ranging from celestial mechanics to protein folding. [Overview] [A math/visual approach]

9. 1977: Integer Relation Detection. A fast method for spotting simple equations satisfied by collections of seemingly unrelated numbers. [Nice article with links]

8. 1965: Fast Fourier Transform. Perhaps the most ubiquitous algorithm in use today, it breaks down waveforms (like sound) into periodic components. Everyone knows this one (or should) [Part II of my personal favorite FFT and wavelet tutorial]

7. 1962: Quicksort Algorithms for Sorting. For the efficient handling of large databases. [Definition][Basic Method][Mathworld][More technical explanation][A lecture with animations and simulations]

6. 1959: QR Algorithm for Computing Eigenvalues. Another crucial matrix operation made swift and practical. [Math] [Algorithm

5. 1957: The Fortran Optimizing Compiler. Turns high-level code into efficient computer-readable code. (pretty much self-explanatory) [History and lots of info]

4. 1951: The Decompositional Approach to Matrix Computations. A suite of techniques for numerical linear algebra. [matrix decomposition theorem] [Strategies]

3. 1950: Krylov Subspace Iteration Method. A technique for rapidly solving the linear equations that abound in scientific computation. [History] [various Krylov subspace iterative methods]

2. 1947: Simplex Method for Linear Programming. An elegant solution to a common problem in planning and decision-making. [English} [Explanation with Java simulator] [An interactive teaching tool

1. 1946: The Metropolis Algorithm for Monte Carlo. Through the use of random processes, this algorithm offers an efficient way to stumble toward answers to problems that are too complicated to solve exactly. [English] [Code and Math] [Math explained]

Here is a couple of really good plots of some elliptic curves.

The mathematical association of america has some nice information on this:
http://www.maa.org/mathland/math trek_11_22_99.html

I have been told by an applied math geek friend of mine that STW is another one of those "it's all connected, maaaan..."-type theories along the line of "e^(pi * i) + 1 = 0", although a good deal messier. I've also been informed that STW was used heavily in Wiles' proof, not unlike a load-bearing block in Jenga.

(Never mind "First Post!" I hereby start the new tradition of "Most Links!" After all, it's more productive, and more importantly, it's all connected, maaaaaan....)