mathematics for computer science eric lehman and tom leighton 2012

... Eric Lehman and Tom Leighton Mathematics for Computer Science 2004 Contents What is a Proof? 15 1.1 Propositions ... one for each natural value of n: • If student gets candy, then student also gets candy • If student gets candy, then student also gets candy • If student gets candy, then student also gets candy, ... divisibility hold If a | b, then a | bc for all c If a | b and b | c, then a | c If a | b and a | c, then a | sb + tc for all s and t For all c = 0, a | b if and only if ca | cb Proof We’ll only...

Ngày tải lên: 05/03/2014, 23:20

... Stueben, Michael and Diane Sandford Association of America, ©1998 p ab Twenty Years Before the Blackboard, Mathematical “mcs” — 2013/1/10 — 0:28 — page 21 — #29 1.10 References 21 for all nonnegative ... of numbers of the form rn, where r is a positive real number and n N Well ordering commonly comes up in Computer Science as a method for proving that computations won’t run forever The idea is ... proposition for each possible set of truth values for the variables For example, the truth table for the proposition “P AND Q” has four lines, since there are four settings of truth values for the...

Ngày tải lên: 23/03/2014, 23:20

... (International Standards Organization) This means that now your C++ program can be written and compiled on any computer platform (PCs, minicomputers, mainframe computers) that has a standard C++ compiler ... the computer, it can then process that data, often performing some calculations on that information When the calculations are finished, the computer outputs that set of data and the results For ... following information: the employee’s id, their age and the years they have been employed at Acme To be considered a candidate, the employee must have worked for Acme for ten years and be at least...

Ngày tải lên: 19/03/2014, 14:14

... papers by Roy Anderson and Robert May, summarized in their book (Anderson and May 1991) Anderson and May have developed population dynamic models of the host engaged in reproduction and migration ... stochastic models and direct computer simulations References Anderson, R M and R M May (1991), Infectious diseases of humans Oxford University Press, Oxford UK Kermack, W O and A G McKendrick ... Environmental Sciences”, held at Hamamatsu, Japan, March 14th-17th, 2004, under the chairmanship of one of the editors (Y.T.), gave the editors the idea for the book Mathematics for Life Science and Medicine...

Ngày tải lên: 27/06/2014, 10:20

... SUMS 31 “What’s one and one and one and one and one and one and one and one and one and one?” “1 don’t know,” said Alice ‘7 lost count.” “She can’t Addition.” -Lewis Carroll [44] And we can now apply ... 6, and y and trying to find a closed form for the more general recurrence f ( ) = cc; f(2n) = 2f(n) + fi, f(2n+1)=2f(n)+y, for n 1; for n (1.11) (Our original recurrence had a = 1, fi = -1, and ... “Discrete Mathematics! ’ Therefore the subject needs a distinctive name, and “Concrete Mathematics has proved to be as suitable as any other The original textbook for Stanford’s course on concrete mathematics...

Ngày tải lên: 14/08/2014, 04:21

... These formulas for even and odd n slightly resemble those for n = and 1, but no clear pattern has emerged yet; so we had better continue exploring some more small cases For m = the sum is and we ... arbitrary positive integer n in the form n = 2m + 1, where < < 2” Give explicit formulas for and m as functions of n, using floor and/ or ceiling brackets What is a formula for the nearest integer to a ... 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...

Ngày tải lên: 14/08/2014, 04:21

... 4) QED for the case m = 12 So far we’ve proved our congruence for prime m, for m = 4, and for m = 12 Now let’s try to prove it for prime powers For concreteness we may suppose that m = p3 for some ... whenever m I n and < m n we can find integers a and b such that m a - n b = Fdrey ‘nough (4.32) (Actually we said m’m + n’n = gcd( m, n), but we can write for gcd( m, n), a for m’, and b for -n’.) ... 19 For every positive integer n there’s a prime p such that n < p 2n (This is essentially “Bertrand’s postulate,” which Joseph Bertrand verified for n < 3000000 in 1845 and Chebyshev proved for...

Ngày tải lên: 14/08/2014, 04:21

... a hypergeometric and plug into the general identity for Vandermonde’s convolution Problem in Section 5.2 asked for the value of This problem is a natural for hypergeometrics, and after a bit of ... right-hand side of (5.107) always expands into a sum of terms of the form c%kzkFiki (z), where Flk’(z) is the kth derivative DkF(k); the left-hand side always expands into a sum of terms of the form ... 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 named...

Ngày tải lên: 14/08/2014, 04:21

... male and a female Here are the first few levels of the tree: The drone has one grandfather and one grandmother; he has one greatgrandfather and two great-grandmothers; he has two great-great-grandfathers ... assuming an infinite longevity for K and W, an infinite elasticity of the band, and an infinitely tiny worm.) Let’s write down some formulas When K stretches the rubber band, the fraction of it that ... a few things For one, kim’ isn’t a polynomial if j = 0; so we will need to split off that term and handle it separately For another, we’re missing the term k = from the formula for nth difference;...

Ngày tải lên: 14/08/2014, 04:21

... expressed in the form R(z) = S(z) t T(z), (7.28) where S(z) has the form (7.26) and T(z) is a polynomial Therefore there is a closed form for the coefficients [z”] R(z) Finding S(z) and T(z) is equivalent ... exist for negative n Two kinds of “closed forms” come up when we work with generating functions We might have a closed form for G(z), expressed in terms of z; or we might have a closed form for ... The first term stands for the way to leave no pennies, the second term stands for one penny, then two pennies, three pennies, and so on Now if we’re allowed to use both pennies and nickels, the...

Ngày tải lên: 14/08/2014, 04:21

... when X and Y are independent Let F(z) and G(z) be the pgf’s for X and Y, and let H(z) be the pgf for X + Y Then H(z) = F(z)G(z), and our formulas (8.28) through must have (8.31) for mean and variance ... element of n has the form (T + HT)"HH for some n 0, and each term of T has the form (0 + E)n Therefore by (7.4) we have s = (I-T-HT)-'HH, and the probability generatin.g function for our problem is ... Prove (8.74) and (8.75), the general formulas for mean and variance of the time needed to wait for a given pattern of heads and tails 21 What does the value of N represent, if H and T are both...

Ngày tải lên: 14/08/2014, 04:21

... stands for “‘real part.” valid for !.Xz > 9%~ The asymptotic formula for Bernoulli numbers B, in Table 438 illustrates this principle On the other hand, the asymptotic formulas for H,, n!, and ... negative for < x < i and positive for i < x < Therefore its integral, Bdk+~ (x)/(4k+2), decreases for < x < and increases for i < x < Moreover, we have bk+l(l - X) = -&+I (X) , for < x < 1, and it ... appear in Sloane’s Handbook [270]; therefore a closed form for gn seems out of the question, and asymptotic information is probably the best we can hope to derive Our first handle on this problem...

Ngày tải lên: 14/08/2014, 04:21

... the hint, we get andasimilarformulafor&,(z) Thustheformulas (ztB;‘(z)‘B[(z)+l)Bt(z)r and (ztE;‘(z)&:(z) + l)&,(z)’ give the respective right-hand sides of (5.61) We must therefore prove that (zwwJm ... holds for n = only Case 1, n > and n is prime Then (n - l)!/(n + 1) is an integer and it can’t be a multiple of n Case 2, n 5, n is composite, and n + is composite Then n and n+l divide (n-l)!,andnIn+l; ... where a, = /n! and Xk = k3 5.86 (a) First expand with n(n- 1) index variables Lij for all i # j Setting kii = li’ -Lji for :< i < j < n and using the constraints tifi (lij -iii) = for all i < n...

Ngày tải lên: 14/08/2014, 04:21

... by (n) requires a different C for each k; but each stands for a single C In fact, the context of this requires it to stand for a set of functions of two variables k and n It would be correct to ... constants Q, B, no, C such that la(n)/ Blf(n)[ for n mc and [b(n)1 Clg(n)l for n no Therefore the left-handfunctionisatmostmax(B,C)(lf(n)l+Ig(n)l),forn3max(~,no), so it is a member of the right ... E Knuth, and Oren Patashnik, Concrete 102 Mathematics: A Foundation for Computer Science Addison-Wesley, 1989 (The first printing had a different Iversonian notation.) 132 R L Graham and H Pollak,...

Ngày tải lên: 14/08/2014, 04:21

... Teaching and learning reading English for Computer Science at Nghe An JTTC is aimed at helping students get acquainted with various vocabulary, terms of computer and the computer information and events ... English for Science and Technology (EST), English for Business and Economics (EBE), and English for Social Studies (ESS) Each of these subject areas is further divided into two branches: English for ... learned for years, others for years and the rest has learnt for years In the department of information technology at our college, they engage in ESP course, it means English for Computer Science, ...

Ngày tải lên: 18/12/2013, 15:37

... elements of S and the elements of T that relates exactly one element of T to each element of S We use a letter like f to stand for a function and use f (x) to stand for the one and only one element ... S1 = S2 ∪ S3 and S2 and S3 are disjoint, and so, by the sum principle, Equation 1.7 must hold We now give a proof for general n and k Theorem 1.3 If n and k are integers with n > and < k < n, ... B, and D precede C and E Since there are 3! ways to arrange A, B, and D, and 2! ways to arrange C and E, by the product principal, there are 3!2! = 12 lists in which A, B, and D precede C and...

Ngày tải lên: 21/02/2014, 09:20

... emphasis on proof theory and procedures for constructing formal proofs of formulae algorithmically This book is designed primarily for computer scientists, and more generally, for mathematically inclined ... string v, and the symbol “)”, and similarly for the string (u ∗ v) Also, note that H+ and H∗ are defined for all strings in A, and not just legal arithmetic expressions For example, if u = and v = ... Function and Constant Symbols, 339 7.5.3 Substitutions, 342 7.5.4 Herbrand’s Theorem for Prenex Formulae, 344 xx TABLE OF CONTENTS PROBLEMS, 353 7.6 ∗ 7.7 Skolem-Herbrand-G¨del’s Theorem for Formulae...

Ngày tải lên: 29/03/2014, 09:20

... impossible It takes time and effort to decipher the mathematics and understand the meaning It is a little like programming, it takes time to understand a lot of code and you never understand how to write ... Gareth J Janacek & Mark Lemmon Close Mathematics for Computer Scientists Download free eBooks at bookboon.com Mathematics for Computer Scientists © 2011 Gareth J Janacek, Mark Lemmon ... bookboon.com Mathematics for Computer Scientists Numbers The Euclidean algorithm Algorithms which are schemes for computing and we cannot resist putting one in at this point The Euclidean algorithm for...

Ngày tải lên: 24/10/2014, 17:32

... and learning ESP reading, and ESP reading difficulties for foreign language learners 13 CHAPTER 2: AN INVESTIGATION INTO THE CURRENT SITUATION OF TEACHING AND LEARNING ENGLISH FOR COMPUTER SCIENCE ... the computer science topics such as: History of computer, characteristics and applications of computers, the CPU, memory, hardware and software, the operating systems, computer graphics and design, ... 120–period GE and 60period ESP 15 Being responsible for teaching English for Computer Science (ECS) – a type of ESP at VTTTU, we realize that teaching and learning ESP in general and ECS in particular...

Ngày tải lên: 28/03/2015, 10:36