concrete mathematics a foundation for computer science phần 5 pps

... same pattern holds for continued fractions of any depth. It is easilyproved by induction; we have, for example,K(ao,al,az ,a3 +l /a4 ) := K(ao,al ,a2 ,a3 ,a4 )K(al, az, a3 + l /a4 1K(al,az,as,ad) ... arisenaturally. Let’s consider the pedigree of a male bee. Each male (also knownas a drone) is produced asexually from a female (also known as a queen); eachfemale, however, has two parents, a male ... continued fraction can also be obtained directly: Let CQ = a and for k 3 0 letak = Lakj ;1ak = ak+ Kkfl(6.142)The a s are called the “partial quotients” of a. If a is rational, say m/n,this...

concrete mathematics a foundation for computer science phần 3 ppsx

... 492 6 85 878 1071 12641'1'1'2'3' 4' 7'11'18' 25& apos;32' 39' 71'110'181' 252 '323' 394' 4 65& quot;"are the ... a4 ; al = a9 = as; a2 = alo = o6; and a3 = all = a7 .So the values of ao, al, a2 , and as can be chosen in n4 ways, and the remaining a s depend on them. Does this look familiar? ... nm possible patterns appears at least once in this array ofmN(m,n) strings, and some patterns appear more than once. How manytimes does a pattern a~ . . . a, ,-, appear? That’s easy: It’s the...

concrete mathematics a foundation for computer science phần 4 ppsx

... an auxiliary variable z, A( z) = ac +a, z +a2 z2+ = to@“.k>O (5. 52)It’s appropriate to use the letter z as the name of the auxiliary variable, be-cause we’ll often be thinking of z as a ... land generate the following numbers ofrightful owners:ABCD 4 BACD 2 CABD1DABC0ABDC 2 BADC0CADB0DACB1ACBD 2 BCAD1CBAD 2 DBAC1ACDB1BCDA0CBDA1DBCA 2ADBC1BDAC0CDAB0DCAB0ADCB ... n fans. How manyways h(n, k) are there for exactly k fans to get their own hats back? For example, if n = 4 and if the hats and fans are named A, B, C, D,the 4! = 24 possible ways for hats...

concrete mathematics a foundation for computer science phần 9 pps

... LLLRRRRR = & z.3 158 . Incidentally, an average of .334implies at least 287 at bats. 51 8 ANSWERS TO EXERCISESand this is the case x = N of exercise 22. 5. 55 Let Q(k) = (k+Al) (k+AM)Zand R(k) ... resultsabout Fermat’sLast Theorem; seeRibenboim j249j.j A ANSWERS TO EXERCISES 52 7If the harmonicaa and a bb. Thus each bb behaves like a drone in the bee tree and each aabehaves like a queen, ... therebyeliminating an upper parameter and a lower parameter. Thus, for example,we get closed forms for F( a, b; a - 1; z), F( a, b; a - 2; z), etc.Gauss [116, $71 derived analogous relations...

concrete mathematics a foundation for computer science phần 1 pdf

... intended as anantidote to “Abstract Mathematics, ” since concrete classical results were rap-idly being swept out of the modern mathematical curriculum by a new waveof abstract ideas popularly called ... Ronald Lewis, 19 35- Concrete mathematics : a foundation for computer science / Ron-ald L. Graham, Donald E. Knuth, Oren Patashnik.xiii,6 25 p.24 cm.Bibliography: p. 57 8Includes index.ISBN ... hard one to the easy one.time to do warmupLet’s apply these ideas to a useful example. Consider the arrayexercises 4 and 6.)(Or to check outal alal a2 the Snickers bar a2 al a2 a2 languishing...

concrete mathematics a foundation for computer science phần 2 pptx

... name andconquer. a = ~ap’J , a- ’ = a+ a’;The change of vari-able from k to j isb = [va-‘l ,va-’ = b -v’.Thus a = {a ‘} is the fractional part of a- ‘, and v’ is the mumble-fractionalpart ... notations for rising powers as well as falling powers.Mathematicians have long had both sine and cosine, tangent and cotangent,secant and cosecant, max and min; now we also have both floor and ceiling.To ... integers, and in fact to arbitrary real numbers:xmody = x - yLx/yJ, for y # 0.(3.21)This defines ‘mod’ as a binary operation, just as addition and subtraction arebinary operations. Mathematicians...

concrete mathematics a foundation for computer science phần 6 doc

... sequences(al ,a2 . , al,,) of +1's and -1's have the property thatal + a2 +. . . + azn = 0and have all their partial sumsal,al +a2 ,. . . .al +a2 + +aZnnonnegative? There ... many spanning trees f, are in such a graph? A spanning tree is a subgraph containing all the vertices, and containingenough edges to make the subgraph connected yet not so many that it has a ... because 2 - & z0.268. This needs to be taken into account if wetry to use formula (7.38) in numerical calculations. For example, a fairlyexpensive name-brand hand Icalculator comes...

concrete mathematics a foundation for computer science phần 7 pot

... month each person is reassignedto an adjacent state, each adjacent state being equally likely. (Here’s a diagram of the adjacencies:The initial states are circled.) For example, Alice is restationed ... (8.82) that A would probably beat B, that B would probably beat C, and that C wouldprobably beat A; but all three patterns are simultaneously in the game.)What are each player’s chances of ... we can interpret these formulas in a particularlysimple way. Given a pattern A of m heads and tails, let A: A = fIkpl [Ack’ =A( kj] .k=l(8.76)We can easily find the binary representation...

concrete mathematics a foundation for computer science phần 8 doc

... “Eulerian” (eighteenth-century) mathematics. 9.3 0 MANIPULATION 447This may look worse than the sum we began with, but it’s actually a step for- ward, because we have very good approximations for ... the asymptotic expansion(?) = -&(I-&+j&+&-j&g++all have denominators that are powers of 2.61 Exercise 45 proves that the discrepancy D( 01, n) is o(n) for all irrationalnumbers ... is actually 154 858 63. Exercise 21 shows that a still moreaccurate approximation to P, results if we begin with a still more accurateapproximation to n(n) in place of (9.46).9.6 FINAL SUMMATIONS...

concrete mathematics a foundation for computer science phần 10 docx

... sum602. for 1x1,” American Mathematical Monthly 96 (1989), 52 5 -52 6.131 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete 102. Mathematics: A Foundation for Computer Science. Addison-Wesley,1989. ... Warren Weaver, “Lewis Carroll and a geometrical paradox,” AmericanMathematical Monthly 45 (1938), 234-236. 58 2 BIBLIOGRAPHY 51 52 53 54 55 56 57 58 59 6061626364Th. Clausen, “Ueber die Fglle, ... 1990.1 35 Samuel L. Greitzer, International Mathematical Olympiads, 1 959 -1977.602.Mathematical Association of America, 1978.136 Oliver A. Gross, “Preferential arrangements,” American Mathematical...

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