concrete mathematics a foundation for computer science phần 3 ppsx
English Grammar for ESL Learners phần 3 ppsx
... 5-928 Practice Makes Perfect: English Grammar for ESL Learners 02 (017-0 43) Unit 5 11 /3/ 04 1:41 PM Page 28 24 Practice Makes Perfect: English Grammar for ESL Learners 8. You buy flowers every ... class?exercise 5-14 34 Practice Makes Perfect: English Grammar for ESL Learners 02 (017-0 43) Unit 5 11 /3/ 04 1:41 PM Page 34 Give an emphatic response to each statement.EXAMPLE: He does not speak English. He ... sing?Form the negative by placing not after will:You will sing. ➞You will not sing. 38 Practice Makes Perfect: English Grammar for ESL Learners 02 (017-0 43) Unit 5 11 /3/ 04 1:41 PM Page 38 ...
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Schaum’s Outline Series OF Principles of Computer Science phần 3 ppt
... is a pattern of ones and zeros stored in a word of computer memory. By the way, a “word” of memory is the basic unit of storage for a computer. A 16-bit computer has a word size of 16 bits, or ... *, 6 ) J, ID, AV6 FORMAT( I6, 3X, 'SUBJECT ', I6, 3X, 'AV= ', F9.2 )STOPENDExample input cards:55821 3 7 9 4 7812 1 4 3 3 28 13 3 2 3 1 1824 7 9 9 9 9825 6 9 8 8 ... handling of datais appropriate.CPU/ALUThe CPU is the part of the computer one thinks of first when describing the components of a computer. Therepetitive cycle of the von Neumann computer...
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Schaum’s Outline Series OF Principles of Computer Science phần 4 ppsx
... as part of the state of each individual car (part of the state of each instance). On the other hand, we would maintain acount of the total number of Automobiles as part of the state of the class ... expression/ | \/ | \(9 - 4) - 2 expression add_op termCHAP. 4] SOFTWARE 63 The state of an object is provided by variables. As the values of variables change, the state of the objectchanges. ... ((cdr n)), a list of one element (the element is another list,the cdr of the list of parameters), and the second line of cond would return a list instead of a number.Our version of sum now behaves...
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concrete mathematics a foundation for computer science phần 1 pdf
... recurrence makes it possible to build a table of small values veryquickly. Perhaps we’ll be able to spot a pattern and guess the answer.n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 J(n) 1 1 3 1 3 5 7 1 3 ... index.ISBN o-2 01- 14236-8 1. Mathematics 19 61- 2. Electronic data processing Mathematics. I. Knuth, Donald Ervin, 19 38-.II. Patashnik, Oren, 19 54-.III. Title.QA39.2.C733 19 88 510 dc1988-3779CIPSixth ... J. Hammersley [14 5]THIS BOOK IS BASED on a course of the same name that has been taughtannually at Stanford University since 19 70. About fifty students have taken iteach year-juniors and...
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concrete mathematics a foundation for computer science phần 2 pptx
... as a binary operation, just as addition and subtraction arebinary operations. Mathematicians have used mod this way informally for a long time, taking various quantities mod 10, mod 27 7, and ... 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...
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concrete mathematics a foundation for computer science phần 3 ppsx
... 87 106 1 93 299 492 685 878 1071 12641'1'1'2&apos ;3& apos; 4' 7'11'18'25&apos ;32 ' 39 ' 71'110'181'252&apos ;32 3' 39 4' 465""are ... lo1'10'9'8'7'b'5'4&apos ;3& apos;5'2&apos ;3& apos;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...
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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...
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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...
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concrete mathematics a foundation for computer science phần 6 doc
... 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 ... 0;) as n -+ 00, un:less z = 0. This shows that formal power seriescan be manipulated algebraically without worrying about convergence.Example 6: A recurrence that goes ail the way back.Let’s ... 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...
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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...
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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 ... actually 15 485 863. 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 ... result formally. 9.6 FINAL SUMMATIONS 473Actually 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...
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concrete mathematics a foundation for computer science phần 9 pps
... resultsabout Fermat’sLast Theorem; seeRibenboim j249j.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 ... 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 ... x.(The values of a, b, c must, in fact, be even higher than this resultindicates! Inkeri [160] has proved that A sketch of his proof appears in [2 49, pages 228-2 291 , a book that containsan extensive...
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concrete mathematics a foundation for computer science phần 10 docx
... V. Waugh and Margaret W. Maxfield, “Side-and-diagonal num-bers,” Mathematics Magazine 40 (1967), 74-83.298 Warren Weaver, “Lewis Carroll and a geometrical paradox,” AmericanMathematical Monthly ... Mark HaimanAndrei Broder, Jim McGrathOren PatashnikJoan Feigenbaum, Dave HelmboldAnna KarlinOren Patashnik, Alex SchafferPang Chen, Stefan SharkanskyArif Merchant, Stefan SharkanskyIn addition, ... 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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Algorithms and Networking for Computer Games phần 3 pps
... end for 30 : x ← w 31 : for i ← 0 (c− 1) do Middle vertical. 32 : y ← m 33 : for j ← 0 (c− 2) do 34 : Hx,y← Displacement(H, x, y, −w, 0, +w, 0, 0, −w, 0, +w,d) 35 : y ← y + m 36 : end for 37 : ... information can be used, for instance, in game balancing (e.g.adjusting point rewarding schemes, or testing synthetic players by making them engage in Algorithms and Networking for Computer Games ... of ranks for all players before thetournament can be too demanding a task. To compensate for and to reduce the effect ofseeding, we can introduce a random element into the pairing. For example,...
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