Internet Surfer 2.0

home *** CD-ROM | disk | FTP | other *** search

/ Internet Surfer 2.0 / Internet Surfer 2.0 (Wayzata Technology) (1996).iso / pc / text / mac / faqs.094 < prev next >

Wrap

Text File | 1996-02-12 | 27.7 KB | 660 lines

Frequently Asked Questions (FAQS);faqs.094 P. Beckman A history of pi Golem Press, CO, 1971 (fourth edition 1977) J.M. Borwein and P.B. Borwein The arithmetic-geometric mean and fast computation of elementary functions SIAM Review, Vol. 26, 1984, pp. 351-366 J.M. Borwein and P.B. Borwein More quadratically converging algorithms for pi Mathematics of Computation, Vol. 46, 1986, pp. 247-253 J.M. Borwein and P.B. Borwein Pi and the AGM - a study in analytic number theory and computational complexity Wiley, New York, 1987 Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computation of pi Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231-244 Y. Kanada and Y. Tamura Calculation of pi to 10,013,395 decimal places based on the Gauss-Legendre algorithm and Gauss arctangent relation Computer Centre, University of Tokyo, 1983 Morris Newman and Daniel Shanks On a sequence arising in series for pi Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp. 199-217 E. Salamin Computation of pi using arithmetic-geometric mean Mathematics of Computation, Vol. 30, 1976, pp. 565-570 D. Shanks and J.W. Wrench, Jr. Calculation of pi to 100,000 decimals Mathematics of Computation, Vol. 16, 1962, pp. 76-99 Daniel Shanks Dihedral quartic approximations and series for pi J. Number Theory, Vol. 14, 1982, pp.397-423 David Singmaster The legal values of pi The Mathematical Intelligencer, Vol. 7, No. 2, 1985 Stan Wagon Is pi normal? The Mathematical Intelligencer, Vol. 7, No. 3, 1985 J.W. Wrench, Jr. The evolution of extended decimal approximations to pi The Mathematics Teacher, Vol. 53, 1960, pp. 644-650 11Q: There are three doors, and there is a car hidden behind one of them, Master Mind and other games .. A: Read frequently asked questions from rec.puzzles, where the problem is solved and carefully explained. (The Monty Hall problem). MANY OTHER "MATHEMATICAL" GAMES ARE EXPLAINED IN THE REC.PUZZLES FAQ. READ IT BEFORE ASKING IN SCI.MATH. Your chance of winning is 2/3 if you switch and 1/3 if you don't. For a full explanation from the frequently asked questions list for rec.puzzles, send to the address netlib@peregrine.com an email message consisting of the text send switch References American Mathematical Monthly, January 1992. For the game of Master Mind it has been proven that no more than five moves are required in the worst case. For references look at One such algorithm was published in the Journal of Recreational Mathematics; in '70 or '71 (I think), which always solved the 4 peg problem in 5 moves. Knuth later published an algorithm which solves the problem in a shorter # of moves - on average - but can take six guesses on certain combinations. Donald E. Knuth, The Computer as Master Mind, J. Recreational Mathematics 9 (1976-77), 1-6. 12Q: What is the formula for the "Surface Area" of a sphere in Euclidean N-Space. That is, of course, the volume of the N-1 solid which comprises the boundary of an N-Sphere. A: The volume of a ball is the easiest formula to remember: It's r^N times pi^(N/2)/(N/2)!. The only hard part is taking the factorial of a half-integer. The real definition is that x! = Gamma(x+1), but if you want a formula, it's: (1/2+n)! = sqrt(pi)*(2n+2)!/(n+1)!/4^(n+1) To get the surface area, you just differentiate to get N*pi^(N/2)/(N/2)!*r^(N-1). There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of e^(-x^2) is sqrt(pi). Therefore the integral over N-space of e^(-x_1^2-x_2^2-...-x_N^2) is sqrt(pi)^n. Now we change to spherical coordinates. We get the integral from 0 to infinity of V*r^(N-1)*e^(-r^2), where V is the surface volume of a sphere. Integrate by parts repeatedly to get the desired formula. 13Q: Anyone knows a name (or a closed form) for f(x)^f(x)=x Solving for f one finds a "continued fraction"-like answer f(x) = log x ----- log (log x ------ ........... A: This question has been repeated here from time to time over the years, and no one seems to have heard of any published work on it, nor a published name for it (D. Merrit proposes "lx" due to its (very) faint resemblence to log). It's not an analytic function. The "continued fraction" form for its numeric solution is highly unstable in the region of its minimum at 1/e (because the graph is quite flat there yet logarithmic approximation oscillates wildly), although it converges fairly quickly elsewhere. To compute its value near 1/e, I used the bisection method with good results. Bisection in other regions converges much more slowly than the "logarithmic continued fraction" form, so a hybrid of the two seems suitable. Note that it's dual valued for the reals (and many valued complex for negative reals). A similar function is a "built-in" function in MAPLE called W(x). MAPLE considers a solution in terms of W(x) as a closed form (like the erf function). W is defined as W(x)*exp(W(x))=x. If anyone ever runs across something published on the subject, please post. 14Q: The existence of a projective plane of order 10 has long been an outstanding problem in discrete mathematics and finite geometry. A: More precisely, the question is: is it possible to define 111 sets (lines) of 11 points each such that: for any pair of points there is precisely one line containing them both and for any pair of lines there is only one point common to them both. Analogous questions with n^2 + n + 1 and n + 1 instead of 111 and 11 have been positively answered only in case n is a prime power. For n=6 it is not possible. The n=10 case has been settled as not possible either by Clement Lam. See Am. Math. Monthly, recent issue. As the "proof" took several years of computer search (the equivalent of 2000 hours on a Cray-1) it can be called the most time-intensive computer assisted single proof. The final steps were ready in January 1989. 15Q: Is there a formula to determine the day of the week, given the month, day and year? A: Here is the standard method. A. Take the last two digits of the year. B. Divide by 4, discarding any fraction. C. Add the day of the month. D. Add the month's key value: JFM AMJ JAS OND 144 025 036 146 E. Subtract 1 for January or February of a non-leap year. F. For a Gregorian date, add 0 for 1900's, 6 for 2000's, 4 for 1700's, 2 for 1800's; for other years, add or subtract multiples of 400. G. For a Julian date, add 1 for 1700's, and 1 for every additional century you go back. H. Add the year. Now take the remainder when you divide by 7; 0 is Sunday, the first day of the week, 1 is Monday, and so on. Another formula is: W == k + [2.6m - 0.2] - 2C + Y + [Y/4] + [C/4] mod 7 where [] denotes the integer floor function (round down), k is day (1 to 31) m is month (1 = March, ..., 10 = December, 11 = Jan, 12 = Feb) Treat Jan & Feb as months of the preceding year C is century ( 1987 has C = 19) Y is year ( 1987 has Y = 87 except Y = 86 for jan & feb) W is week day (0 = Sunday, ..., 6 = Saturday) This formula is good for the Gregorian calendar (introduced 1582 in parts of Europe, adopted in 1752 in Great Britain and its colonies, and on various dates in other countries). It handles century & 400 year corrections, but there is still a 3 day / 10,000 year error which the Gregorian calendar does not take. into account. At some time such a correction will have to be done but your software will probably not last that long :-) ! References: Winning Ways by Conway, Guy, Berlekamp is supposed to have it. Martin Gardner in "Mathematical Carnaval". Michael Keith and Tom Craver, "The Ultimate Perpetual Calendar?", Journal of Recreational Mathematics, 22:4, pp. 280-282, 1990. K. Rosen, "Elementary Number Theory", p. 156. 16Q: What is the Axiom of Choice? Why is it important? Why some articles say "such and such is provable, if you accept the axiom of choice."? What are the arguments for and against the axiom of choice? A: There are several equivalent formulations: -The Cartesian product of nonempty sets is nonempty, even if the product is of an infinite family of sets. -Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member from each element of S. C can thus be thought of as the result of "choosing" a representative from each set in S. Hence the name. >Why is it important? All kinds of important theorems in analysis require it. Tychonoff's theorem and the Hahn-Banach theorem are examples. AC is equivalent to the thesis that every set can be well-ordered. Zermelo's first proof of this in 1904 I believe was the first proof in which AC was made explicit. AC is especially handy for doing infinite cardinal arithmetic, as without it the most you get is a *partial* ordering on the cardinal numbers. It also enables you to prove such interesting general facts as that n^2 = n for all infinite cardinal numbers. > What are the arguments for and against the axiom of choice? The axiom of choice is independent of the other axioms of set theory and can be assumed or not as one chooses. (For) All ordinary mathematics uses it. There are a number of arguments for AC, ranging from a priori to pragmatic. The pragmatic argument (Zermelo's original approach) is that it allows you to do a lot of interesting mathematics. The more conceptual argument derives from the "iterative" conception of set according to which sets are "built up" in layers, each layer consisting of all possible sets that can be constructed out of elements in the previous layers. (The building up is of course metaphorical, and is suggested only by the idea of sets in some sense consisting of their members; you can't have a set of things without the things it's a set of). If then we consider the first layer containing a given set S of pairwise disjoint nonempty sets, the argument runs, all the elements of all the sets in S must exist at previous levels "below" the level of S. But then since each new level contains *all* the sets that can be formed from stuff in previous levels, it must be that at least by S's level all possible choice sets have already been *formed*. This is more in the spirit of Zermelo's later views (c. 1930). (Against) It has some supposedly counterintuitive consequences, such as the Banach-Tarski paradox. (See next question) Arguments against AC typically target its nonconstructive character: it is a cheat because it conjures up a set without providing any sort of *procedure* for its construction--note that no *method* is assumed for picking out the members of a choice set. It is thus the platonic axiom par excellence, boldly asserting that a given set will always exist under certain circumstances in utter disregard of our ability to conceive or construct it. The axiom thus can be seen as marking a divide between two opposing camps in the philosophy of mathematics: those for whom mathematics is essentially tied to our conceptual capacities, and hence is something we in some sense *create*, and those for whom mathematics is independent of any such capacities and hence is something we *discover*. AC is thus of philosophical as well as mathematical significance. It should be noted that some interesting mathematics has come out of an incompatible axiom, the Axiom of Determinacy (AD). AD asserts that any two-person game without ties has a winning strategy for the first or second player. For finite games, this is an easy theorem; for infinite games with duration less than \omega and move chosen from a countable set, you can prove the existence of a counter-example using AC. Jech's book "The Axiom of Choice" has a discussion. An example of such a game goes as follows. Choose in advance a set of infinite sequences of integers; call it A. Then I pick an integer, then you do, then I do, and so on forever (i.e. length \omega). When we're done, if the sequence of integers we've chosen is in A, I win; otherwise you win. AD says that one of us must have a winning strategy. Of course the strategy, and which of us has it, will depend upon A. From a philosophical/intuitive/pedagogical standpoint, I think Bertrand Russell's shoe/sock analogy has a lot to recommend it. Suppose you have an infinite collection of pairs of shoes. You want to form a set with one shoe from each pair. AC is not necessary, since you can define the set as "the set of all left shoes". (Technically, we're using the axiom of replacement, one of the basic axioms of Zermelo-Fraenkel (ZF) set theory.) If instead you want to form a set containing one sock from each pair of an infinite collection of pairs of socks, you now need AC. References: Maddy, "Believing the Axioms, I", J. Symb. Logic, v. 53, no. 2, June 1988, pp. 490-500, and "Believing the Axioms II" in v.53, no. 3. Gregory H. Moore, Zermelo's Axiom of Choice, New York, Springer-Verlag, 1982. H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice, Amsterdam, North-Holland, 1963. A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, Amsterdam, North-Holland, 1984 (2nd edition, 2nd printing), pp. 53-86. 17Q: Cutting a sphere into pieces of larger volume. Is it possible to cut a sphere into a finite number of pieces and reassemble into a solid of twice the volume? A: This question has many variants and it is best answered explicitly. Given two polygons of the same area, is it always possible to dissect one into a finite number of pieces which can be reassembled into a replica of the other? Dissection theory is extensive. In such questions one needs to specify (A) what a "piece" is, (polygon? Topological disk? Borel-set? Lebesgue-measurable set? Arbitrary?) (B) how many pieces are permitted (finitely many? countably? uncountably?) (C) what motions are allowed in "reassembling" (translations? rotations? orientation-reversing maps? isometries? affine maps? homotheties? arbitrary continuous images? etc.) (D) how the pieces are permitted to be glued together. The simplest notion is that they must be disjoint. If the pieces are polygons [or any piece with a nice boundary] you can permit them to be glued along their boundaries, ie the interiors of the pieces disjoint, and their union is the desired figure. Some dissection results 1) We are permitted to cut into FINITELY MANY polygons, to TRANSLATE and ROTATE the pieces, and to glue ALONG BOUNDARIES; then Yes, any two equal-area polygons are equi-decomposable. This theorem was proven by Bolyai and Gerwien independently, and has undoubtedly been independently rediscovered many times. I would not be surprised if the Greeks knew this. The Hadwiger-Glur theorem implies that any two equal-area polygons are equi-decomposable using only TRANSLATIONS and ROTATIONS BY 180 DEGREES. 2) THM (Hadwiger-Glur, 1951) Two equal-area polygons P,Q are equi-decomposable by TRANSLATIONS only, iff we have equality of these two functions: PHI_P() = PHI_Q() Here, for each direction v (ie, each vector on the unit circle in the plane), let PHI_P(v) be the sum of the lengths of the edges of P which are perpendicular to v, where for such an edge, its length is positive if v is an outward normal to the edge and is negative if v is an inward normal to the edge. 3) In dimension 3, the famous "Hilbert's third problem" is: "If P and Q are two polyhedra of equal volume, are they equi-decomposable by means of translations and rotations, by cutting into finitely many sub-polyhedra, and gluing along boundaries?" The answer is "NO" and was proven by Dehn in 1900, just a few months after the problem was posed. (Ueber raumgleiche polyeder, Goettinger Nachrichten 1900, 345-354). It was the first of Hilbert's problems to be solved. The proof is nontrivial but does *not* use the axiom of choice. "Hilbert's Third Problem", by V.G.Boltianskii, Wiley 1978. 4) Using the axiom of choice on non-countable sets, you can prove that a solid sphere can be dissected into a finite number of pieces that can be reassembled to two solid spheres, each of same volume of the original. No more than nine pieces are needed. This construction is known as the "Banach-Tarski" paradox or the "Banach-Tarski-Hausdorff" paradox (Hausdorff did an early version of it). The "pieces" here are non-measurable sets, and they are assembled *disjointly* (they are not glued together along a boundary, unlike the situation in Bolyai's thm.) An excellent book on Banach-Tarski is: "The Banach-Tarski Paradox", by Stan Wagon, 1985, Cambridge University Press. Also read in the Mathematical Intelligencier an article on the Banach-Tarski Paradox. The pieces are not (Lebesgue) measurable, since measure is preserved by rigid motion. Since the pieces are non-measurable, they do not have reasonable boundaries. For example, it is likely that each piece's topological-boundary is the entire ball. The full Banach-Tarski paradox is stronger than just doubling the ball. It states: 5) Any two bounded subsets (of 3-space) with non-empty interior, are equi-decomposable by translations and rotations. This is usually illustrated by observing that a pea can be cut up into finitely pieces and reassembled into the Earth. The easiest decomposition "paradox" was observed first by Hausdorff: 6) The unit interval can be cut up into COUNTABLY many pieces which, by *translation* only, can be reassembled into the interval of length 2. This result is, nowadays, trivial, and is the standard example of a non-measurable set, taught in a beginning graduate class on measure theory. References: In addition to Wagon's book above, Boltyanskii has written at least two works on this subject. An elementary one is: "Equivalent and equidecomposable figures" in Topics in Mathematics published by D.C. HEATH AND CO., Boston. It is a translation from the 1956 work in Russian. Also, the article "Scissor Congruence" by Dubins, Hirsch and ?, which appeared about 20 years ago in the Math Monthly, has a pretty theorem on decomposition by Jordan arcs. ``Banach and Tarski had hoped that the physical absurdity of this theorem would encourage mathematicians to discard AC. They were dismayed when the response of the math community was `Isn't AC great? How else could we get such unintuitive results?' '' 18Q. Is there a theory of quaternionic analytic functions, that is, a four- dimensional analog to the theory of complex analytic functions? A. Yes. This was developed in the 1930s by the mathematician Fueter. It is based on a generalization of the Cauchy-Riemann equations, since the possible alternatives of power series expansions or quaternion differentiability do not produce useful theories. A number of useful integral theorems follow from the theory. Sudbery provides an excellent review. Deavours covers some of the same material less thoroughly. Brackx discusses a further generalization to arbitrary Clifford algebras. Anthony Sudbery, Quaternionic Analysis, Proc. Camb. Phil. Soc., vol. 85, pp 199-225, 1979. Cipher A. Deavours, The Quaternion Calculus, Am. Math. Monthly, vol. 80, pp 995-1008, 1973. F. Brackx and R. Delanghe and F. Sommen, Clifford analysis, Pitman, 1983. 19Q. What is Erdos Number? Form an undirected graph where the vertices are academics, and an edge connects academic X to academic Y if X has written a paper with Y. The Erdos number of X is the length of the shortest path in this graph connecting X with Erdos. What is the Erdos Number of X ? for a few selected X in {Math,physics} Erdos has Erdos number 0. Co-authors of Erdos have Erdos number 1. Einstein has Erdos number 2, since he wrote a paper with Ernst Straus, and Straus wrote many papers with Erdos. Why people care about it? Nobody seems to have a reasonable answer... Caspar Goffman, And what is your Erdos number?, American Mathematical Monthly v. 76 (1969), p. 791. -------------------------------------------------------------------------- Questions and Answers _Compiled_ by: Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca Deparment of Computer Science University of Waterloo Waterloo, Ontario Canada Xref: bloom-picayune.mit.edu rec.scuba:15643 news.answers:4596 Newsgroups: rec.scuba,news.answers Path: bloom-picayune.mit.edu!enterpoop.mit.edu!news.media.mit.edu!micro-heart-of-gold.mit.edu!news.bbn.com!noc.near.net!uunet!scifi!scifi!njs From: njs@scifi.uucp (Nick Simicich) Subject: [rec.scuba] FAQ: Frequently Asked Questions about Scuba, Monthly Posting Message-ID: <1993.Jan.15.scuba.faq@scifi.uucp> Followup-To: rec.scuba Sender: njs@scifi.uucp (Nicholas J. Simicich) Supersedes: <1993.Dec.15.scuba.faq@scifi.uucp> Organization: N.J. Simicich, Peekskill, NY Date: Tue, 15 Dec 1992 10:01:23 GMT Approved: news-answers-request@MIT.Edu Expires: 28 Jan 1993 Lines: 714 Archive-name: scuba-faq This posting was last modified on 7/14/92 to correct the information on how to get rec-scuba by email. One paragraph was added to the end of the Spare Air discussion regarding bottle transport on airlines. Additionally, a network-wide general disclaimer for all of rec-scuba is included. The FAQ is now being posted twice a month, on the first and 15th. Please feel free to follow-up with comments or email them to uunet!bywater!scifi!njs or scifi!njs@uunet.uu.net. -=-=-=-=- Welcome to rec.scuba. The newsgroup is for discussion of scuba, diving, snorkeling, dive travel, and other underwater activities. Frequent topics are safety, equipment, and certification. We welcome postings from new folks and old hands. Rec.scuba has had a fairly flame-free history. Not that we don't speak out against practices that we disagree with, but we do try to avoid flaming people. Flaming for flaming sake is to be discouraged. Before posting to this group for the first time, please check the FAQ list (this posting), and also read the newsgroup news.announce.newusers, which contains many answers to questions about usenet in general. Summary of rec.scuba FAQ: 1. Differences between certification agencies. (PADI/NAUI/YMCA/SSI etc.) 2. New Diver buying first piece of equipment. 3. Some sources for mail order equipment. 4. rec.scuba archive sites and how to access them. 5. How to find out about dive destinations. 6. Basic discussion of thermal protection (wetsuit, drysuit, darlexx). 7. Liquid breathing in the movie "Abyss". 8. Scuba magazines and periodicals. 9. Diving in contact lenses. 10. What about Spare Air or Pony Bottles? 11. What about Casio Dive watches and the depth ratings thereon? 12. I lost my C-card. What do I do? 13. I need a resort referral, cause I want to do my checkout dives on my upcoming vacation to TinyIsland. Who do I call? 14. I think I got a shoddy course. What can I do? 15. They are cutting off my rec newsfeed. How can I get rec.scuba by email? General Disclaimer: Scuba Diving is a dangerous sport which can only be performed in relative safety if you (a) get training (b) pay attention to that training and apply it (c) recognize that no matter who you are and how trained you are, there are dives which are beyond your personal ability, dives which cannot be safely done with your equipment, and dives that are beyond your training. Finally, some dives are just plain more dangerous. Your certification course should have trained you to recognize your limitations, or, conversely, to recognize the sorts of diving you were trained to do. Various people who post to rec.scuba discuss advanced diving. This stuff is just a discussion. It is not meant to be a replacement for a certification course with an instructor, and it is not meant to be an encouragement to you to go out and engage in similar diving without evaluating your personal skills, and/or getting the appropriate training and equipment, as required. Specifically, Cave or Wreck or Deep diving requires advanced equipment, training, and a careful self examination. Finally, it should be obvious that not everyone who posts their opinions to the net is or can be (a) an expert or (b) correct. It is likely that your instructor, for example, would disagree with a number of the points of view expressed herein, and would probably disagree with part of this FAQ. The fact that someone who identifies themselves as an instructor posts to rec.scuba does not create an instructional situation. Frequently Asked Questions: 1: I'm planning on getting certified. I've been to several shops, and they all offer different certifications. I've heard of PADI, NAUI, YMCA, NASDS and SSI. Which one should I go with? 1a: This question has frequently come up in rec.scuba. One of the discussion threads has been summarized as whosbest.txt in the rec.scuba archives at ames. See the explanation of Peter Yee's archive, below, for how to access the ames archives. The short, widely agreed answer, is that agencies all must follow a minimum standard set by an industry organization, so they differ less than you might expect. However, instructors differ a lot, and you should try to talk to the instructor you will be taking the course from and determine exactly what will be offered, and how you feel about them. Finally, some instructors add significantly to the standard course (and may also charge more). You should ask exactly what you are going to get for your course fees, what else you will have to buy, and where you have to buy it. 2: I'm new to diving, and I want to buy some equipment. Which piece of equipment should be the first? 2a: There are two schools of thought on this. One is that you should consider only purchasing your personal gear until you are sure what type of diving you like. This school believes you should buy only mask, fins, and snorkel, for fit and sanitary reasons. The other school of thought is that the rental gear you can rent, especially in tropical locations, is second rate and poorly maintained, and that gear you purchase will be better and more reliable. Typically, people agree that you should not buy a tank until you believe that you will be doing a significant amount of local diving.