Mathematics for computer science

... 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 xii...

Ngày tải lên: 14/08/2014, 04:21

... 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

Ngày tải lên: 14/08/2014, 04:21

... 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 fra...

Ngày tải lên: 14/08/2014, 04:21

... 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 fan...

Ngày tải lên: 14/08/2014, 04:21

... 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;...

Ngày tải lên: 14/08/2014, 04:21

... 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, Ram...

Ngày tải lên: 14/08/2014, 04:21

... 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 f...

Ngày tải lên: 14/08/2014, 04:21

... 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 Bef...

Ngày tải lên: 14/08/2014, 04:21

... 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 , (...

Ngày tải lên: 14/08/2014, 04:21

... 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,...

Ngày tải lên: 14/08/2014, 04:21

... are reading, we can read again or slow down for intensive reading For the teachers, silent reading is helpful for controlling the class In silent reading, students are in fact concentrating on ... included: definition of reading and reading comprehension, classification of reading according to the purposes of reading; reading in ESP teaching and learning such as d...

Ngày tải lên: 18/12/2013, 15:37