mathematics for computer science course outline

... 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 ... c This seems very reasonable! But, of course, there is room for disagreement about what consitutes a reasonable axiom For example, one of Euclid’s axioms for geometry is equivalent to the following: ... 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...

Ngày tải lên: 05/03/2014, 23:20

... 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 ... 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 to assign a value to the successive steps ... 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

... undergraduate courses entitled “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 ... 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 ... 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), ... 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...

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, ... 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 that ... currently used for the second year students in the department of information technology at Nghe An JTTC belongs to the latter type The course book named “English for Computer Science is a collection...

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

... 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

... 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

... RESOURCES 7.1 Course Resources 7.2 Other Resources, Support and Information CONTINUAL COURSE IMPROVEMENT COURSE SCHEDULE MFIN6205 – Financial Risk Management for Financial Institutions Page COURSE STAFF ... 1 COURSE STAFF 1.1 Communication with Staff INFORMATION ABOUT THE COURSE 2.1 Teaching times and Locations 2.2 Units of Credit 2.3 Parallel teaching in the course 2.4 Relationship of this course ... in this course 2.4 Relationship of this course to other course offerings This course aims to give an overview of the risks faced by financial institutions and how they can be managed The course...

Ngày tải lên: 20/06/2014, 14:20

... RESOURCES 7.1 Course Resources 7.2 Other Resources, Support and Information 6 CONTINUAL COURSE IMPROVEMENT COURSE SCHEDULE MFIN6205 – Financial Risk Management for Financial Institutions Page COURSE ... Management for Financial Institutions Page 7.1 Course Resources Prescribed text for this course is: Fundamentals of Risk Management, C Marrison, McGrawHill, 2002 Recommended text for this course ... in this course 2.4 Relationship of this course to other course offerings This course aims to give an overview of the risks faced by financial institutions and how they can be managed The course...

Ngày tải lên: 20/06/2014, 14:20

... RESOURCES 7.1 Course Resources 7.2 Other Resources, Support and Information 6 CONTINUAL COURSE IMPROVEMENT COURSE SCHEDULE MFIN6205 – Financial Risk Management for Financial Institutions Page COURSE ... Management for Financial Institutions Page 7.1 Course Resources Prescribed text for this course is: Fundamentals of Risk Management, C Marrison, McGrawHill, 2002 Recommended text for this course ... in this course 2.4 Relationship of this course to other course offerings This course aims to give an overview of the risks faced by financial institutions and how they can be managed The course...

Ngày tải lên: 20/06/2014, 14:20