# Mathematics for computer science

## concrete mathematics a foundation for computer science phần 1 pdf

... the concrete ” -Z A Melzak 12 1 41 Concrete Ma the- matics is a bridge to abstract mathematics “The advanced reader who skips parts that appear too elementary may miss more than the less advanced ... Cataloging-in-Publication Data Graham, Ronald Lewis, 19 3 5Concrete mathematics : a foundation for computer science / Ronald L Graham, Donald E Knuth, Oren Patashnik xiii,625 p 24 cm Bibliography: ... time: n 10 11 12 13 14 S, 10 15 21 28 36 45 55 66 78 91 105 These values are also called the triangular numbers, because S, is the number of bowling pins in an n-row triangular array For example,...
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## concrete mathematics a foundation for computer science phần 2 pptx

... generality, we can assume that < a < 1; let us write a = ~ap’J , b = [va-‘l , a- ’ = a+ a’; va-’ = b -v’ Thus a = {a ‘} is the fractional part of a- ‘, and v’ is the mumble-fractional part of va-‘ ... zero, and the first can be evaluated by our usual routine: k,m>O = tm((m+l)‘-m2)[m+16al ll@O = ~m(2m+l)[m...
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## concrete mathematics a foundation for computer science phần 3 ppsx

... BBBB Each of the nm possible patterns appears at least once in this array of mN(m,n) strings, and some patterns appear more than once How many times does a pattern a~ a, ,-, appear? That’s easy: ... mathematical formulas that are easy to deal with? A bit of experimentation suggests that the best way is to maintain a x matrix that holds the four quantities involved in the ancestral fractions ... congruences are almost like equations For example, congruence is an equivalence relation; that is, it satisfies the reflexive law a = a , the symmetric law a b =\$ b E a , and the transitive law a E...
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## concrete mathematics a foundation for computer science phần 4 ppsx

... named A, B, C, D, the 4! = 24 possible ways for hats to land generate the following numbers of rightful owners: ABCD ABDC ACBD ACDB ADBC ADCB 2 1 BACD BADC BCAD BCDA BDAC BDCA 0 CABD CADB CBAD ... the air The hats come back randomly, one hat to each of the n fans How many ways h(n, k) are there for exactly k fans to get their own hats back? For example, if n = and if the hats and fans are ... b;k! We can now combine all these operations and make a mathematical “pun” by expressing the same quantity in two different ways Namely, we have (9 +a, ) (4 +a, )F q = al a, F altl, a, +1 (8 +...
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## concrete mathematics a foundation for computer science phần 5 pps

... drone has one grandfather and one grandmother; he has one greatgrandfather and two great-grandmothers; he has two great-great-grandfathers and three great-great-grandmothers In general, it is easy ... name We have, for example, a0 + 1 a1 + ~ a2 + G = K(ao,al,az ,a3 ) -K(al,az ,a3 ) ' (6.1 35) The same pattern holds for continued fractions of any depth It is easily proved by induction; we have, for ... correctly claims that 3 .5- tS2 = A polynomial formula that involves Fibonacci numbers of the form F,,+k for small values of k can be transformed into a formula that involves only F, and F,+I , because...
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## concrete mathematics a foundation for computer science phần 6 doc

... following problem: How many sequences (al ,a2 , al,,) of +1's and -1's have the property that al + a2 + + azn = and have all their partial sums al, al +a2 , al +a2 + +aZn nonnegative? There must be ... the pattern.) There is no closed form for p(n), but the theory of partitions is a fascinating branch of mathematics in which many remarkable discoveries have been made For example, Ramanujan proved ... for rational functions tells us that the answer can be obtained from a partial fraction representation We can use the general expansion theorem (7.30) and grind away; or we can use the fact that...
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## concrete mathematics a foundation for computer science phần 7 pot

... space, and we want to estimate the mean of a random variable X by sampling its value repeatedly (For exa.mple, we might want to know the average temperature at noon on a January day in San Francisco; ... special case p = i we can interpret these formulas in a particularly simple way Given a pattern A of m heads and tails, let A: A = fIkpl [Ack’ =A( kj] (8 .76 ) k=l We can easily find the binary representation ... information 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, K, and some data D(K) about that...
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## concrete mathematics a foundation for computer science phần 8 doc

... Stirling’s approximation behaves for generalized factorials (and for the Gamma function r( a + 1) = a! ) exactly as for ordinary factorials Summation 4: A bell-shaped summand Let’s turn now to a sum that ... formula does give us O(n mP1 ) for arbitrarily large m, even though we haven’t evaluated any remainders explicitly Summation 1, again: Recapitulation and generalization Before we leave our training ... But in practice, we have no reason to believe that an adversary is trying to trap us, so we can assume that the unknown O-constants are reasonably small With a pocket calculator we find that S4...
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## concrete mathematics a foundation for computer science phần 9 pps

... aa2 -a a+ aa -a2 +a+ a > 0, which is a consequence of aa( a - a) + (1 + a) a ( + a) a > a2 - a Hence we can replace x, and a by a - and (3, repeating this transformation until cases or apply Another ... the easily proved identity a ( a - b ) - (b + II)” (a+ llEemb< = oP (b+l)k bk as well as to the operator formula a - b = (4 + a) - (4 + b) Similarly, we have al ,a2 ,a3 , (al - a2 1 F = alF am ... express F(al , , , a, ,,; bl , , b,; z) as a linear combination of F( a2 + j, a3 , , a, ,,; b2, , b,; z) for j d, thereby eliminating an upper parameter and a lower parameter Thus, for example,...
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## concrete mathematics a foundation for computer science phần 10 docx

... 2982: A double infinite sum for 1x1,” American Mathematical Monthly 96 (1989), 525-526 602 131 Ronald L Graham, Donald E Knuth, and Oren Patashnik, Concrete 102 Mathematics: A Foundation for Computer ... Ramshaw Yossi Shiloach Yossi Shiloach Frank Liang, Chris Tong, Mark Haiman Andrei Broder, Jim McGrath Oren Patashnik Joan Feigenbaum, Dave Helmbold Anna Karlin Oren Patashnik, Alex Schaffer Pang ... 135 Samuel L Greitzer, International Mathematical Olympiads, 1959-1977 Mathematical Association of America, 1978 602 136 Oliver A Gross, “Preferential arrangements,” American Mathematical 604...
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