basic mathematics for computer science 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 ... “simple” closed form For example, the product of the first n integers, n!, has proved to be so important that we now consider it a basic operation The formula ‘n!’ is therefore in closed form, although ... 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 ... Therefore q(n) is the number of reduced basic fractions with denominator n; and the Farey series 3,, contains all the reduced basic fractions with denominator n or less, as well as the non -basic ... 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),...

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 ... 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 ... 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 tell us that we...

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 ... 386-387 Forcadel, Pierre, 586, 603 Formal power series, 206, 317, 517 FORTRAN, 432 Fourier, Jean Baptiste Joseph, 22, 586 series, 481 Fractional part, 70, 83, 87, 456 Fractions, 116-123, 151 basic, ...

Ngày tải lên: 14/08/2014, 04:21

... reduced form equation for one dependent variable has been derived then the reduced form equation for the other dependent variable can be found This is done by substituting the reduced form for the ... the reduced form equation for equilibrium q is not a simple linear function of t Lastly, we can derive the reduced form equation for equilibrium p by substituting the reduced form for q that we ... formula should always appear in the formula bar just above the spreadsheet When a formula is copied down a column any cell’s numbers that the formula contains should also change As the main formulae...

Ngày tải lên: 14/08/2014, 22:20

... (1 + i)n The formula for the final value F of an investment of £A for n time periods at interest rate i is therefore F = A(1 + i)n Let us rework Examples 7.3 and 7.4 using this formula just to ... requirements are 10 units of K plus 30 units of L for each unit of A 25 units of K plus 15 units of L for each unit of B If the per-unit profit is £80 for A and £30 for B, what combination of A and B should ... intersection of the constraints for W and X This means that the minimum requirements for W and X are only just met and so 20A + 90B = 7,200 for W (1) 5A + 7B = 1,400 for X (2) Multiplying (2) by...

Ngày tải lên: 14/08/2014, 22:20

... 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

... Calculators and computers Some students may ask, ‘what’s the point in spending a great deal of time and effort studying mathematics when nowadays everyone uses calculators and computers for calculations?’ ... answer that the computer gives you Also, many economic problems have to be set up in the form of a mathematical problem before they can be fed into a computer program package for solution Most ... Solution 1p = £0.01 Therefore 0.01p = £0.0001 In mathematics a decimal format is often required for a value that is usually specified as a percentage in everyday usage For example, interest rates...

Ngày tải lên: 15/03/2014, 01:12

... Introduction to Programming 16 Basic Computer Architecture In order to effectively write programs on the computer, some basic knowledge of computer architecture is required The computer can be viewed ... on any computer platform (PCs, minicomputers, mainframe computers) that has a standard C++ compiler Introduction to Programming 19 In this chapter, we are going to examine the basic format of ... 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...

Ngày tải lên: 19/03/2014, 14:14

... 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