Concrete mathematics : a foundation for computer science

More documents

Recommendations

Info

348 GENERATING FUNCTIONS Notice that these are not easy questions. In the ordinary Catalan case (m = 2), we solved (7.68) for G(z) and its coefficients by using the quadratic formula and the binomial theorem; but when m = 3, none of the standard techniques gives any clue about how to solve the cubic equation G = zG3 + 1. So it has turned out to be easier to answer this question before asking it. Now, however, we know enough to ask even harder questions and deduce their answers. How about this one: “What is [z”] G(z)‘, if 1 is a positive integer and if G(z) is the power series defined by (7.68)?” The argument we just gave can be used to show that [PI G(z)’ is the number of sequences of length mn + 1 with the following three properties: . Each element is either $-1 or (1 - m). . The partial sums are all positive. . The total sum is 1. For we get all such sequences in a unique way by putting together 1 sequences that have the m-Raney property. The number of ways to do this is t n, +nr t...+n,=n c’m’c’m’ n, n* t.. CL:) = [znl G(z)'. Raney proved a generalization of his lemma that tells us how to count such sequences: If (XI, x2,. . . , x,) is any sequence of integers with xi 6 1 for all j, and with x1 + x2 + . . . -1 x, = 1 > 0, then exactly 1 of the cyclic shifts (x1,x2,.. .,xm), (X2,...,Xm,Xl), . ..1 (%il,Xl,... ,xTn 1 ) have all positive partial sums. For example, we can check this statement on the sequence (-2,1, -l,O, l,l,-l,l,l,l). The cyclic shifts are (-2,1,-l,O,l,l,-l,l,l,l) (1,~l,l,l,l,-2,1,-l,O,l) (l,-l,O,l,l,-l,l,l,l,--2) (-l,l,l,l,-2,1,-l,O,l,l) (-l,O,l,l,-l,l,l,l,-2,l) (l,l,l,-2,1,-1,0,1,1,-l) J (O,l,l,-1,1,1,1,-2,1,--l) (l,l,-2,1,-l,O,l,l,-1,l) (1,1,-~,1,1,1,-2,1,~1,0) J (l,-2,1,-l,O,l,l,-l,l,l) and only the two examples marked ‘J’ have all partial sums positive. This generalized lemma is proved in exercise 13. A sequence of +1's and (1 - m)‘s that has length mn+ 1 and total sum 1 must have exactly n occurrences of (1 - m). The generalized lemma tells us that L/(mn + 1) of these (,‘“‘t+‘) sequences have all partial sums positive;
hence our tough question has a surprisingly simple answer: [znl G(z)’ = (“I+‘) $1 7.5 CONVOLUTIONS 349 for all integers 1 > 0. Readers who haven’t forgotten Chapter 5 might well be experiencing dkjjh vu: “That formula looks familiar; haven’t we seen it before?” Yes, indeed; equation (5.60) says that [z”]B,(z)’ = -Jr &. ( ) Therefore the generating function G(z) in (7.68) must actually be the generalized binomial series ‘B,(z). Sure enough, equation (5.59) says cBm(z)‘-m - Tim(z)-” = 2) which is the same as T3B(z)-l = zB,(z)" Let’s switch to the notation of Chapter 5, now that we know we’re dealing with generalized binomials. Chapter 5 stated a bunch of identities without proof. We have now closed part of the gap by proving that the power series IBt (z) defined by TQ(z) = x y & n ( 1 has the remarkable property that %(z)’ = x (yr)$&, n whenever t and T ;Ire positive integers. Can we extend these results to arbitrary values oft and I-? Yes; because the coefficients (t:T’) & are polynomials in t and T. The general rth power defined by ‘B,(z)’ = erln’Bt(z) - -9 rln93t(z))n ll20 n! = t $ (- 2 (I-y)nl)‘, ll>O llI>l has coefficients that are polynomials in t and r; and those polynomials are equal to (tnn+‘) &; for infinitely many values oft and r. So the two sequences of polynomials must be identically equal.
Page 1 and 2:
CONCRETE MATHEMATICS Dedicated to L
Page 3 and 4:
CONCRETE MATHEMATICS Ronald L. Grah
Page 5 and 6:
“A odience, level, and treatment
Page 7 and 8:
‘I a concrete life preserver thro
Page 9 and 10:
I’m unaccustomed to this face. De
Page 11 and 12:
n [ n-l 1 n {I m Prestressed concre
Page 13:
CONTENTS xiii 5.3 Tricks of the Tra
Page 16 and 17:
2 RECURRENT PROBLEMS It’s not imm
Page 18 and 19:
4 RECURRENT PROBLEMS Of course the
Page 20 and 21:
6 RECURRENT PROBLEMS through any of
Page 22 and 23:
8 RECURRENT PROBLEMS From these sma
Page 24 and 25:
10 RECURRENT PROBLEMS But what abou
Page 26 and 27:
12 RECURRENT PROBLEMS that is, in t
Page 28 and 29:
14 RECURRENT PROBLEMS by separating
Page 30 and 31:
16 RECURRENT PROBLEMS For example,
Page 32 and 33:
18 RECURRENT PROBLEMS Homework exer
Page 34 and 35:
20 RECURRENT PROBLEMS 21 Suppose th
Page 36 and 37:
22 SUMS The three-dots notation has
Page 38 and 39:
24 SUMS A slightly modified form of
Page 40 and 41:
26 SUMS where A(n), B(n), and C(n)
Page 42 and 43:
28 SUMS Let’s apply these ideas t
Page 44 and 45:
30 SUMS 2.3 MANIPULATION OF SUMS No
Page 46 and 47:
32 SUMS Typically we use rule (2.20
Page 48 and 49:
34 SUMS because the derivative of a
Page 50 and 51:
36 SUMS instead of summing over all
Page 52 and 53:
38 SUMS The second sum here is zero
Page 54 and 55:
40 SUMS The normal way to evaluate
Page 56 and 57:
42 SUMS First, as usual, we look at
Page 58 and 59:
44 SUMS order to get an equation fo
Page 60 and 61:
46 SUMS One way to use this fact is
Page 62 and 63:
48 SUMS definition where the factor
Page 64 and 65:
50 SUMS Let’s try to recap this;
Page 66 and 67:
52 SUMS we need an appropriate defi
Page 68 and 69:
54 SUMS This formula indicates why
Page 70 and 71:
56 SUMS u(x) = x and Av(x) = 2’;
Page 72 and 73:
58 SUMS The definition in the previ
Page 74 and 75:
60 SUMS In fact, our definition of
Page 76 and 77:
62 SUMS tCj,kiEF a. J, k < A for al
Page 78 and 79:
64 SUMS 18 Let 9%~ and Jz be the re
Page 80 and 81:
66 SUMS 34 35 36 Prove that if the
Page 82 and 83:
68 INTEGER FUNCTIONS below the line
Page 84 and 85:
70 INTEGER FUNCTIONS same inequalit
Page 86 and 87:
72 INTEGER FUNCTIONS An important s
Page 88 and 89:
74 INTEGER FUNCTIONS By the way, th
Page 90 and 91:
76 INTEGER FUNCTIONS (Previously K
Page 92 and 93:
78 INTEGER FUNCTIONS This derivatio
Page 94 and 95:
80 INTEGER FUNCTIONS We’ve got mo
Page 96 and 97:
82 INTEGER FUNCTIONS division, and
Page 98 and 99:
84 INTEGER FUNCTIONS more than one
Page 100 and 101:
86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
Page 102 and 103:
88 INTEGER FUNCTIONS For this purpo
Page 104 and 105:
90 INTEGER FUNCTIONS it comes out i
Page 106 and 107:
92 INTEGER FUNCTIONS To summarize,
Page 108 and 109:
94 INTEGER FUNCTIONS Also, as we gu
Page 110 and 111:
96 INTEGER FUNCTIONS Basics 10 Show
Page 112 and 113:
98 INTEGER FUNCTIONS 31 Prove or di
Page 114 and 115:
100 INTEGER FUNCTIONS 43 Find an in
Page 116 and 117:
4 Number Theory INTEGERS ARE CENTRA
Page 118 and 119:
104 NUMBER THEORY The left side can
Page 120 and 121:
106 NUMBER THEORY product of primes
Page 122 and 123:
108 NUMBER THEORY only finitely man
Page 124 and 125:
110 NUMBER THEORY But 2P - 1 isn’
Page 126 and 127:
112 NUMBER THEORY arrangements in a
Page 128 and 129:
114 NUMBER THEORY The binary repres
Page 130 and 131:
116 NUMBER THEORY A fraction m/n is
Page 132 and 133:
118 NUMBER THEORY A mediant fractio
Page 134 and 135:
120 NUMBER THEORY This representati
Page 136 and 137:
122 NUMBER THEORY This means that w
Page 138 and 139:
124 NUMBER THEORY Since x mod m dif
Page 140 and 141:
126 NUMBER THEORY than to say that
Page 142 and 143:
128 NUMBER THEORY require division,
Page 144 and 145:
130 NUMBER THEORY Now we must show
Page 146 and 147:
132 NUMBER THEORY And it’s possib
Page 148 and 149:
134 NUMBER THEORY For example, (~(1
Page 150 and 151:
136 NUMBER THEORY Conversely, if f(
Page 152 and 153:
138 NUMBER THEORY (We must add 1 to
Page 154 and 155:
140 NUMBER THEORY The problem of co
Page 156 and 157:
142 NUMBER THEORY n is odd, n4 and
Page 158 and 159:
144 NUMBER THEORY But the inner sum
Page 160 and 161:
146 NUMBER THEORY 22 The number 111
Page 162 and 163:
148 NUMBER THEORY 40 If the radix p
Page 164 and 165:
150 NUMBER THEORY 57 Let S(m,n) be
Page 166 and 167:
152 NUMBER THEORY 73 If the 0(n) +
Page 168 and 169:
154 BINOMIAL COEFFICIENTS For examp
Page 170 and 171:
156 BINOMIAL COEFFICIENTS And now t
Page 172 and 173:
158 BINOMIAL COEFFICIENTS property
Page 174 and 175:
160 BINOMIAL COEFFICIENTS (3, then
Page 176 and 177:
162 BINOMIAL COEFFICIENTS Binomial
Page 178 and 179:
164 BINOMIAL COEFFICIENTS Table 164
Page 180 and 181:
166 BINOMIAL COEFFICIENTS not rows.
Page 182 and 183:
168 BINOMIAL COEFFICIENTS Equation
Page 184 and 185:
170 BINOMIAL COEFFICIENTS that m =
Page 186 and 187:
172 BINOMIAL COEFFICIENTS ifweuse (
Page 188 and 189:
Page 190 and 191:
176 BINOMIAL COEFFICIENTS sum on th
Page 192 and 193:
178 BINOMIAL COEFFICIENTS some data
Page 194 and 195:
180 BINOMIAL COEFFICIENTS Problem 4
Page 196 and 197:
182 BINOMIAL COEFFICIENTS We’re l
Page 198 and 199:
184 BINOMIAL COEFFICIENTS We can’
Page 200 and 201:
186 BINOMIAL COEFFICIENTS 5.3 TRICK
Page 202 and 203:
188 BINOMIAL COEFFICIENTS if we app
Page 204 and 205:
190 BINOMIAL COEFFICIENTS The Newto
Page 206 and 207:
192 BINOMIAL COEFFICIENTS One examp
Page 208 and 209:
194 BINOMIAL COEFFICIENTS We can de
Page 210 and 211:
196 BINOMIAL COEFFICIENTS This obse
Page 212 and 213:
198 BINOMIAL COEFFICIENTS Generatin
Page 214 and 215:
200 BINOMIAL COEFFICIENTS can be pu
Page 216 and 217:
Page 218 and 219:
204 BINOMIAL COEFFICIENTS This hold
Page 220 and 221:
206 BINOMIAL COEFFICIENTS It’s ou
Page 222 and 223:
208 BINOMIAL COEFFICIENTS into this
Page 224 and 225:
210 BINOMIAL COEFFICIENTS integer.
Page 226 and 227:
212 BINOMIAL COEFFICIENTS into to b
Page 228 and 229:
214 BINOMIAL COEFFICIENTS Now sin(
Page 230 and 231:
216 BINOMIAL COEFFICIENTS But look
Page 232 and 233:
218 BINOMIAL COEFFICIENTS It’s al
Page 234 and 235:
220 BINOMIAL COEFFICIENTS A similar
Page 236 and 237:
222 BINOMIAL COEFFICIENTS One use o
Page 238 and 239:
224 BINOMIAL COEFFICIENTS Therefore
Page 240 and 241:
226 BINOMIAL COEFFICIENTS Now f(-b)
Page 242 and 243:
228 BINOMIAL COEFFICIENTS negative
Page 244 and 245:
230 BINOMIAL COEFFICIENTS valid for
Page 246 and 247:
232 BINOMIAL COEFFICIENTS 18 Find a
Page 248 and 249:
234 BINOMIAL COEFFICIENTS 37 Show t
Page 250 and 251:
236 BINOMIAL COEFFICIENTS 57 Use Go
Page 252 and 253:
238 BINOMIAL COEFFICIENTS 72 Prove
Page 254 and 255:
240 BINOMIAL COEFFICIENTS 86 Let al
Page 256 and 257:
242 BINOMIAL COEFFICIENTS Research
Page 258 and 259:
244 SPECIAL NUMBERS Table 244 Stirl
Page 260 and 261:
246 SPECIAL NUMBERS There are eleve
Page 262 and 263:
248 SPECIAL NUMBERS cycle arrangeme
Page 264 and 265:
250 SPECIAL NUMBERS Table 250 Basic
Page 266 and 267:
252 SPECIAL NUMBERS We can remember
Page 268 and 269:
254 SPECIAL NUMBERS Table 254 Euler
Page 270 and 271:
{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
Page 272 and 273:
258 SPECIAL NUMBERS Table 258 Stirl
Page 274 and 275:
260 SPECIAL NUMBERS ’ ’ ’ (Th
Page 276 and 277:
262 SPECIAL NUMBERS we found that a
Page 278 and 279:
264 SPECIAL NUMBERS Rearranging, we
Page 280 and 281:
266 SPECIAL NUMBERS Thus we have th
Page 282 and 283:
268 SPECIAL NUMBERS (We have used t
Page 284 and 285:
270 SPECIAL NUMBERS Bernoulli numbe
Page 286 and 287:
272 SPECIAL NUMBERS trigonometric f
Page 288 and 289:
274 SPECIAL NUMBERS Recurrence (6.9
Page 290 and 291:
276 SPECIAL NUMBERS Gnethingwecando
Page 292 and 293:
278 SPECIAL NUMBERS cases are: a0 =
Page 294 and 295:
280 SPECIAL NUMBERS The process of
Page 296 and 297:
282 SPECIAL NUMBERS Fk < n < Fk+l;
Page 298 and 299:
284 SPECIAL NUMBERS We have now boi
Page 300 and 301:
286 SPECIAL NUMBERS Before we stop
Page 302 and 303:
288 SPECIAL NUMBERS The continuant
Page 304 and 305:
290 SPECIAL NUMBERS in four steps:
Page 306 and 307:
292 SPECIAL NUMBERS Thus we can con
Page 308 and 309:
294 SPECIAL NUMBERS numbers are 1,
Page 310 and 311:
296 SPECIAL NUMBERS 6 An explorer h
Page 312 and 313: 298 SPECIAL NUMBERS 24 Prove that t
Page 314 and 315: 300 SPECIAL NUMBERS 44 Prove the co
Page 316 and 317: 302 SPECIAL NUMBERS 61 Prove the id
Page 318 and 319: 304 SPECIAL NUMBERS 80 Show that co
Page 320 and 321: 7 Generating Functions THE MOST POW
Page 322 and 323: 308 GENERATING FUNCTIONS are added
Page 324 and 325: 310 GENERATING FUNCTIONS (The last
Page 326 and 327: 312 GENERATING FUNCTIONS Now we hav
Page 328 and 329: 314 GENERATING FUNCTIONS The first
Page 330 and 331: 316 GENERATING FUNCTIONS determine
Page 332 and 333: 318 GENERATING FUNCTIONS operation
Page 334 and 335: 320 GENERATING FUNCTIONS Table 320
Page 336 and 337: 322 GENERATING FUNCTIONS middle of
Page 338 and 339: 324 GENERATING FUNCTIONS Fortunatel
Page 340 and 341: 326 GENERATING FUNCTIONS Once we’
Page 342 and 343: 328 GENERATING FUNCTIONS Step 1 is
Page 344 and 345: 330 GENERATING FUNCTIONS nice prope
Page 346 and 347: 332 GENERATING FUNCTIONS Finally, s
Page 348 and 349: 334 GENERATING FUNCTIONS Step 3 was
Page 350 and 351: 336 GENERATING FUNCTIONS This is a
Page 352 and 353: 338 GENERATING FUNCTIONS Table 337
Page 354 and 355: 340 GENERATING FUNCTIONS F(z)' = i
Page 356 and 357: 342 GENERATING FUNCTIONS We can app
Page 358 and 359: 344 GENERATING FUNCTIONS Step 3 is
Page 360 and 361: 346 GENERATING FUNCTIONS Raney’s
Page 364 and 365: 350 GENERATING FUNCTIONS Chapter 5
Page 366 and 367: 352 GENERATING FUNCTIONS Now let’
Page 368 and 369: 354 GENERATING FUNCTIONS function f
Page 370 and 371: 356 GENERATING FUNCTIONS 7.7 DIRICH
Page 372 and 373: 358 GENERATING FUNCTIONS 5 Find a g
Page 374 and 375: 360 GENERATING FUNCTIONS 20 A power
Page 376 and 377: 362 GENERATING FUNCTIONS 29 What is
Page 378 and 379: 364 GENERATING FUNCTIONS 43 The New
Page 380 and 381: 366 GENERATING FUNCTIONS 53 The seq
Page 382 and 383: 368 DISCRETE PROBABILITY total of 6
Page 384 and 385: 370 DISCRETE PROBABILITY about R if
Page 386 and 387: 372 DISCRETE PROBABILITY The mean o
Page 388 and 389: 374 DISCRETE PROBABILITY more sprea
Page 390 and 391: 376 DISCRETE PROBABILITY hence VS =
Page 392 and 393: 378 DISCRETE PROBABILITY And we can
Page 394 and 395: 380 DISCRETE PROBABILITY variance b
Page 396 and 397: 382 DISCRETE PROBABILITY (1 +z+...
Page 398 and 399: 384 DISCRETE PROBABILITY ~1 and ~2
Page 400 and 401: 386 DISCRETE PROBABILITY where Ug i
Page 402 and 403: 388 DISCRETE PROBABILITY Repeating
Page 404 and 405: 390 DISCRETE PROBABILITY There’s
Page 406 and 407: 392 DISCRETE PROBABILITY But the co
Page 408 and 409: 394 DISCRETE PROBABILITY In the spe
Page 410 and 411: 396 DISCRETE PROBABILITY Here 1 is
Page 412 and 413:
398 DISCRETE PROBABILITY For exampl
Page 414 and 415:
400 DISCRETE PROBABILITY Thus the m
Page 416 and 417:
402 DISCRETE PROBABILITY Let sj be
Page 418 and 419:
404 DISCRETE PROBABILITY Therefore
Page 420 and 421:
406 DISCRETE PROBABILITY The mean v
Page 422 and 423:
408 DISCRETE PROBABILITY Complicate
Page 424 and 425:
410 DISCRETE PROBABILITY all k and
Page 426 and 427:
412 DISCRETE PROBABILITY to be a pg
Page 428 and 429:
414 DISCRETE PROBABILITY 10 What’
Page 430 and 431:
416 DISCRETE PROBABILITY 28 What is
Page 432 and 433:
418 DISCRETE PROBABILITY 38 What is
Page 434 and 435:
420 DISCRETE PROBABIL1T.Y a Once he
Page 436 and 437:
422 DISCRETE PROBABILITY 50 Conside
Page 438 and 439:
424 DISCRETE PROBABILITY 58 Are the
Page 440 and 441:
426 ASYMPTOTICS The word asymptotic
Page 442 and 443:
428 ASYMPTOTIC3 How about the funct
Page 444 and 445:
430 ASYMPTOTICS N. G. de Bruijn beg
Page 446 and 447:
432 ASYMPTOTICS subtle consideratio
Page 448 and 449:
434 ASYMPTOTICS inefficient “beca
Page 450 and 451:
436 ASYMPTOTICS 9.3 0 MANIPULATION
Page 452 and 453:
138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455:
440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457:
442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459:
444 ASYMPTOTICS Problem 4: A sum fr
Page 460 and 461:
446 ASYMPTOTICS Problem 5: An infin
Page 462 and 463:
448 ASYMPTOTICS We might as well do
Page 464 and 465:
450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467:
452 ASYMPTOTICS This last estimate
Page 468 and 469:
454 ASYMPTOTICS Here’s another ex
Page 470 and 471:
456 ASYMPTOTICS On the left is a ty
Page 472 and 473:
458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475:
460 ASYMPTOTICS The graph of B,(x)
Page 476 and 477:
462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
Page 478 and 479:
464 ASYMPTOTICS The integral JT B4(
Page 480 and 481:
466 ASYMPTOTIC3 where 0 < a,,,, < 1
Page 482 and 483:
468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485:
470 ASYMPTOTICS For example, we hav
Page 486 and 487:
472 ASYMPTOTICS The final task seem
Page 488 and 489:
474 ASYMPTOTICS This is our approxi
Page 490 and 491:
476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493:
478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495:
480 ASYMPTOTICS a What is the asymp
Page 496 and 497:
482 ASYMPTOTIC3 Research problems 6
Page 498 and 499:
484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501:
486 ANSWERS TO EXERCISES move the b
Page 502 and 503:
488 ANSWERS TO EXERCISES 2.2 This i
Page 504 and 505:
490 ANSWERS TO EXERCISES 2.24 Summi
Page 506 and 507:
492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509:
494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511:
496 ANSWERS TO EXERCISES the circle
Page 512 and 513:
498 ANSWERS TO EXERCISES 3.40 Let L
Page 514 and 515:
500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517:
502 ANSWERS TO EXERCISES 4.22 (b,
Page 518 and 519:
504 ANSWERS TO EXERCISES 4.40 Let f
Page 520 and 521:
506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523:
508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525:
510 ANSWERS TO EXERCISES In both ca
Page 526 and 527:
512 ANSWERS TO EXERCISES 5.7 Yes, b
Page 528 and 529:
514 ANSWERS TO EXERCISES because al
Page 530 and 531:
516 ANSWERS TO EXERCISES 5.44 Cance
Page 532 and 533:
518 ANSWERS TO EXERCISES and this i
Page 534 and 535:
520 ANSWERS TO EXERCISES The infini
Page 536 and 537:
522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539:
524 ANSWERS TO EXERCISES Since ~06j
Page 540 and 541:
526 ANSWERS TO EXERCISJES 5.94 This
Page 542 and 543:
528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545:
530 ANSWERS TO EXERCISIES 6.40 If 6
Page 546 and 547:
532 ANSWERS TO EXERCISES solutions
Page 548 and 549:
534 ANSWERS TO EXERCISES 6.62 (a) A
Page 550 and 551:
536 ANSWERS TO EXERCISES 6.74 If p(
Page 552 and 553:
538 ANSWERS TO EXERCISIES One of th
Page 554 and 555:
540 ANSWERS TO EXERCISES 6.85 The p
Page 556 and 557:
542 ANSWERS TO EXERCISES Hence gzn+
Page 558 and 559:
544 ANSWERS TO EXERCISES 7.24 ntk,+
Page 560 and 561:
546 ANSWERS TO EXERCISES 7.40 The e
Page 562 and 563:
548 ANSWERS TO EXERCISES 1 + a. Hen
Page 564 and 565:
550 ANSWERS TO EXERCISES applied to
Page 566 and 567:
552 ANSWERS TO EXERCISES 8.13 (Solu
Page 568 and 569:
554 ANSWERS TO EXERCISES 8.24 (a) A
Page 570 and 571:
556 ANSWERS TO EXERCISES 8.32 By sy
Page 572 and 573:
558 ANSWERS TO EXERCISES 8.37 The n
Page 574 and 575:
560 ANSWERS TO EXERCISES probabilit
Page 576 and 577:
562 ANSWERS TO EXERCISES And since
Page 578 and 579:
564 ANSWERS TO EXERCISES 8.56 If m
Page 580 and 581:
566 ANSWERS TO EXERCISES We can now
Page 582 and 583:
568 ANSWERS TO EXERCISES 9.19 Hlo =
Page 584 and 585:
570 ANSWERS TO EXERCIS:ES 9.30 Let
Page 586 and 587:
572 ANSWERS TO EXERCISES 9.42 The h
Page 588 and 589:
574 ANSWERS TO EXERCISIES length of
Page 590 and 591:
576 ANSWERS TO EXERCISES 9.58 Let 0
Page 592 and 593:
B Bibliography HERE ARE THE WORKS c
Page 594 and 595:
580 BIBLIOGRAPHY 24' 25 26 27 28 29
Page 596 and 597:
582 BIBLIOGRAPHY 51 52 53 54 55 56
Page 598 and 599:
584 BIBLIOGRAPHY 78 79 80 81 82 83
Page 600 and 601:
586 BIBLIOGRAPHY 100 R. A. Fisher,
Page 602 and 603:
588 BIBLIOGRAPHY 128 Ronald L. Grah
Page 604 and 605:
590 BIBLIOGRAPHY 160 K. Inkeri, “
Page 606 and 607:
592 BIBLIOGRAPHY 188 E.E. Kummer,
Page 608 and 609:
594 BIBLIOGRAPHY 214 Z.A. Melzak, C
Page 610 and 611:
596 BIBLIOGRAPHY 243 George N. Rane
Page 612 and 613:
598 BIBLIOGRAPHY 271 A. D. Solov’
Page 614 and 615:
600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
Page 618 and 619:
604 CREDITS FOR EXERCISES 6.46 6.47
Page 620 and 621:
Index WHEN AN INDEX ENTRY refers to
Page 622 and 623:
608 INDEX Bois-Reymond, Paul David
Page 624 and 625:
610 INDEX Degenerate hypergeometric
Page 626 and 627:
612 INDEX Feder, Tom&s, 604. Feigen
Page 628 and 629:
614 INDEX Hall, Marshall, Jr., 588.
Page 630 and 631:
616 INDEX Lincoln, Abraham, 387. Li
Page 632 and 633:
618 INDEX Patashnik, Oren, iii, iv,
Page 634 and 635:
620 INDEX Running time, 411-412. Ru
Page 636 and 637:
622 INDEX Sylvester, James Joseph,
Page 638 and 639:
List of Tables Sums and differences
show all

Concrete mathematics : a foundation for computer science

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?