concrete mathematics a foundation for computer science phần 4 ppsx

... 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 ... land generate the following numbers ofrightful owners:ABCD 4 BACD 2 CABD1DABC0ABDC 2 BADC0CADB0DACB1ACBD 2 BCAD1CBAD 2 DBAC1ACDB1BCDA0CBDA1DBCA 2ADBC1BDAC0CDAB0DCAB0ADCB ... 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...

concrete mathematics a foundation for computer science phần 3 ppsx

... 30 49 68 87 106 193 299 49 2 685 878 1071 12 64 1'1'1'2'3' 4& apos; 7'11'18'25'32' 39' 71'110'181'252'323' 3 94& apos; ... lo1'10'9'8'7'b'5&apos ;4& apos;3'5'2'3'1'2'1'2'1'2'1'1'~'1'1'1'1' 1Is it true that m’n - mn’ = 1 whenever m/n immediately ... BBBBBBBBEach of the 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...

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 ... 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 ... mathemat-ics was inferior and no longer worthy of attention. The goal of generalizationhad become so fashionable that a generation of mathematicians had becomeunable to relish beauty in the particular,...

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 5 pps

... 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) ’because of the identityK,(xl,. . .,xn-lrxn+Y)= ... 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 ... 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...

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 ... & 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 up with 41 340 3.0005 ... 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 hasa...

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 ... important algorithms for storing and retrievinginformation inside a computer are based on a technique called “hashing!’The general problem is to maintain a set of records that each contain a “key”value,...

concrete mathematics a foundation for computer science phần 8 doc

... “Eulerian” (eighteenth-century) mathematics. 9.3 0 MANIPULATION 44 7This may look worse than the sum we began with, but it’s actually a step for- ward, because we have very good approximations for ... E S.) 44 4 ASYMPTOTICSProblem 4: A sum from an old final exam.When Concrete Mathematics was first taught at Stanford University dur-ing the 1970-1971 term, students were asked for the asymptotic ... result formally.9.6 FINAL SUMMATIONS 47 3Actually I’m notinto dominance.We have to make D, small enough that we can make a good estimate; for example, we had better not let k get near n, or...

concrete mathematics a foundation for computer science phần 9 pps

... resultsabout Fermat’sLast Theorem; seeRibenboim j 249 j.j A ANSWERS TO EXERCISES 527If 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 ... g(k) = &,,, g(k) X[m/k= 11 = s(m), by (4. 7) and (4. 9). 4. 13 (a) nP 6 1 for all p; (b) p(n) # 0. 4. 14 True when k :> 0. Use (4. 12), (4. 14) , and (4. 15). 4. 15 No. For example,...

concrete mathematics a foundation for computer science phần 10 docx

... Write Mathematics, American Mathematical Society, 1973, 19 -48 . 142 Paul R. Halmos, I Want to Be a Mathematician: An Automathography.v.Springer-Verlag, 1985. Reprinted by Mathematical Association ... lat- 347 erum in polygona m laterum, per diagonales resolvi quzat,” Nova actaacademia: scientiarum Petropolitana: 9 (1791), 243 -251.110 Martin Gardner, “About phi, an irrational number that ... sum602. for 1x1,” American Mathematical Monthly 96 (1989), 525-526.131 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete 102. Mathematics: A Foundation for Computer Science. Addison-Wesley,1989....

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