mathematics for computer science 2013 pdf

... simplification Therefore, P (n) is true for all natural n by induction, and the theorem is proved Induction was helpful for proving the correctness of this summation formula, but not helpful for discovering ... core truth of mathematics 2.3 Using Induction Induction is by far the most important proof technique in computer science Generally, induction is used to prove that some statement holds for all natural ... Eric Lehman and Tom Leighton Mathematics for Computer Science 2004 Contents What is a Proof? 15 1.1 Propositions ...

Ngày tải lên: 05/03/2014, 23:20

... 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 ... before we start into mathematics, we need to investigate the problem of how to talk about mathematics To get around the ambiguity of English, mathematicians have devised a special language for...

Ngày tải lên: 23/03/2014, 23:20

... “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 ... foundation for computer science / Ronald L Graham, Donald E Knuth, Oren Patashnik xiii,625 p 24 cm Bibliography: p 578 Includes index ISBN o-201-14236-8 Mathematics 19612 Electronic data processing Mathematics ... expression for the quantity of interest For the Tower of Hanoi, this is the recurrence (1.1) that allows us, given the inclination, to compute T,, for any n Find and prove a closed form for our...

Ngày tải lên: 14/08/2014, 04:21

... closed form for f(n) , when n L Provethatf(n)=n-l+f([n/2~)+f(~n/Z])foralln~l 35 Simplify the formula \(n + )‘n! e] mod n 36 Assuming that n is a nonnegative integer, find a closed form for the ... 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 ... v(x+l) Au(x) (2.54) This formula can be put into a convenient form using the shij?! operator E, defined by Ef(x) = f(x+l) Substituting this for v(x+l) yields a compact rule for the difference of...

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 ... cl(P) = -1; p(pk) = for k > Therefore by (4.52), we have the general formula ifm=pjpz p,; if m is divisible by some p2 (4.57) That’s F If we regard (4.54) as a recurrence for the function q(m), ... write for gcd( m, n), a for m’, and b for -n’.) The Farey series gives us another proof of (4.32), because we can let b/a be the fraction that precedes m/n in 3,, Thus (4.5) is just (4.31) again For...

Ngày tải lên: 14/08/2014, 04:21

... summation formula of the form al, a,, z kbk = CF h, b, 1) AI, AM, '5, , BN (5.127) k’ then we also have al, a,, bl, bn k+l ’ for any integer There’s a general formula for shifting ... ignore convergence if we are using z simply as a formal symbol It is not difficult to verify that formal infinite sums of the form tk3,, (Xkzk form a field, if the coefficients ak lie in a field ... on such formal sums without worrying about convergence; any identities we derive will still be formally true For example, the hypergeometric F( “i ,’ /z) = tkZO k! zk doesn’t converge for any...

Ngày tải lên: 14/08/2014, 04:21

... a closed formula for them We haven’t found closed forms for Stirling numbers, Eulerian numbers, or Bernoulli numbers either; but we were able to discover the closed form H, = [“:‘]/n! for harmonic ... -F; = (-l).", for n > (6.103) When n = 6, for example, Cassini’s identity correctly claims that 3.5-tS2 = A polynomial formula that involves Fibonacci numbers of the form F,,+k for small values ... 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

... 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 ... that R(0) # 00 can be 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 ... (7.10) and get a closed form for the coefficients, but it’s bett,er to save that for later in the chapter after we’ve gotten more experience So let’s divest ourselves of dominoes for the moment and...

Ngày tải lên: 14/08/2014, 04:21

... 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 ... PROBABILITY For example, suppose ,the keys are names, and suppose that there are m = lists based on the first letter of a name: 1, for ,4-F; 2, for G-L; h(name) = 3, for M-R; 4, for !3-Z We start ... form for P~,~ Generalizing your a.nswer to part (a), find a closed form for the probability that exactly k matches are in the other box when an empty one is first th.rown away Find a closed form...

Ngày tải lên: 14/08/2014, 04:21

... that R 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!, ... is 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, ... 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 is the observation that < gn for...

Ngày tải lên: 14/08/2014, 04:21

... 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 allows us to carry out the sums on li, for j < n and then on iii for < i ... 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 ... of F( a2 + j, a3, , a,,,; b2, , b,; z) for j d, thereby eliminating 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),...

Ngày tải lên: 14/08/2014, 04:21

... 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 ... 525-526 602 131 Ronald L Graham, Donald E Knuth, and Oren Patashnik, Concrete 102 Mathematics: A Foundation for Computer Science Addison-Wesley, 1989 (The first printing had a different Iversonian ... 196-204, 283-285, 306-366 for Bernoulli numbers, 271, 337, 351 for convolutions, 339-350, 355, 407 Dirichlet, 356-357, 359, 418, 437 for Eulerian numbers, 337 exponential, 350-355 for Fibonacci numbers,...

Ngày tải lên: 14/08/2014, 04:21

... 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, ... challenges in learning reading ESP for computer science, the questionnaires were designed for students Then data were analyzed and the information was displayed in the form of tables III.4 Summary ... examine the areas of challenges in reading comprehension of English for Computer Science for second-year students in the department of information technology at Nghe An JTTC To be more specific, the...

Ngày tải lên: 18/12/2013, 15:37

... Our algorithm performs a certain set of multiplications For any given i, the set of multiplications performed in lines through can be divided into the set S1 of multiplications performed when j ... multiplications performed when j = 2, and, in general, the set Sj of multiplications performed for any given j value Each set Sj consists of those multiplications the inner loop carries out for a particular ... using the formula for n is is straightforward to show that n n−1 = n (n − 2) However this proof just uses blind substitution and simplification Find a more conceptual explanation of why this formula...

Ngày tải lên: 21/02/2014, 09:20

... meet the Form PF filing thresholds (but that were Form ADV (Parts and 2A) within 90 days of the end of its not required to file the Form PF in 2012) must make their ini- fiscal year In 2013 (a ... plinary information) for- certain-registered investment-advisers-to-privatefunds-form-pf-11-04-2011/ TIP: You should ensure that your IARD account is adequately funded (generally $225 for registered ... Delivery of Form ADV Part type of borrowing and short selling (See the Form PF at A registered investment adviser must, within 120 days of http://www.sec.gov/rules/final/2011/ia-3308-formpf .pdf (page...

Ngày tải lên: 16/03/2014, 05:20

... substantially longer to perform.) Thus, it is the ability of the modern computer to perform reliably and to perform at great speed that has made it so powerful Introduction to Programming Computers have ... was formally standardized by ISO (International Standards Organization) This means that now your C++ program can be written and compiled on any computer platform (PCs, minicomputers, mainframe computers) ... written as a computer program.) Problem Cs01-4 — Your First Actual Real Computer Program Write a program that outputs your personal information in a card-like format Your card information should...

Ngày tải lên: 19/03/2014, 14:14

... Overview of the platform for computeraided terminology preceding step through a self-indexing procedure followed by a graph-based classification This task is basically performed by FASTR, a term ... reordering For example, the index library catalogs is transformed into catalog librar through such simplification techniques In the platform presented in this paper, term normalization is performed ... used in automatic indexing for conflating various term occurrences into unique canonical forms More or less linguistically-oriented techniques are used in the literature for this task Basic procedures...

Ngày tải lên: 24/03/2014, 03:20

... for constructing formal proofs of formulae algorithmically This book is designed primarily for computer scientists, and more generally, for mathematically inclined readers interested in the formalization ... decision procedures for testing the validity of quantifier-free formulae are presented This book grew out of a number of class notes written for a graduate course in logic for computer scientists, ... Theorem for Prenex Formulae, 344 xx TABLE OF CONTENTS PROBLEMS, 353 7.6 ∗ 7.7 Skolem-Herbrand-G¨del’s Theorem for Formulae in NNF, 355 o 7.6.1 Skolem-Herbrand-G¨del’s Theorem in Unsatisfiability o Form,...

Ngày tải lên: 29/03/2014, 09:20

... store them, etc4 For an example of an application form see Supplement No 13 IAPSS Manual Address for all sorts of information Write the address where the participants can get other information Also ... who is the organizer; - deadline for applications; - how many applicants from one organization; - costs; - official application form; - address for all sorts of information; - special requirements ... application form Official application form is the form through which you’ll accept applications Usually you put it on the web site as most of applications are done via internet Application form is...

Ngày tải lên: 30/03/2014, 16:20