Mathematics for informatics and computer science:
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| Sprache: | English |
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ISTE [u.a.]
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| Beschreibung: | XXV, 914 S. Ill., graph. Darst. |
| ISBN: | 9781848211964 |
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| 015 | |a GBB056343 |2 dnb | ||
| 020 | |a 9781848211964 |9 978-1-84821-196-4 | ||
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| 084 | |a MAT 023f |2 stub | ||
| 100 | 1 | |a Audibert, Pierre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Mathematics for informatics and computer science |c Pierre Audibert |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a London [u.a.] |b ISTE [u.a.] |c 2010 | |
| 300 | |a XXV, 914 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Computer science / Mathematics | |
| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Mathematik | |
| 650 | 0 | 7 | |a Informatik |0 (DE-588)4026894-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematik |0 (DE-588)4037944-9 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Mathematik |0 (DE-588)4037944-9 |D s |
| 689 | 0 | 1 | |a Informatik |0 (DE-588)4026894-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Mathematik |0 (DE-588)4037944-9 |D s |
| 689 | 1 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bamberg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020569181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-020569181 | ||
Datensatz im Suchindex
| _version_ | 1804143264264093696 |
|---|---|
| adam_text | Table
of
Contents
General Introduction
.................................. xxiii
Chapter
1.
Some Historical Elements
........................ 1
1.1.
Yi King
...................................... 1
1.2.
Flavor combinations in India
......................... 2
1.3.
Sand drawings in Africa
............................ 3
1.4.
Galileo s problem
................................ 4
1.5.
Pascal s triangle
................................. 7
1.6.
The combinatorial explosion: Abu Kamil s problem, the palm grove
problem and the Sudoku grid
............................ 9
1.6.1.
Solution to Abu Kamil s problem
.................... 11
1.6.2.
Palm Grove problem, where N =
4.................... 12
1.6.3.
Complete Sudoku grids
.......................... 14
Part
1.
Combinatorics
................................ 17
Part
1.
Introduction
................................... 19
Chapter
2.
Arrangements and Combinations
................... 21
2.1.
The three formulae
............................... 21
2.2.
Calculation of
СД
Pascal s triangle and binomial formula
........ 25
2.3.
Exercises
..................................... 27
2.3.1.
Demonstrating formulae
.......................... 27
2.3.2.
Placing rooks on a chessboard
...................... 28
2.3.3.
Placing pieces on a chessboard
...................... 29
2.3.4.
Pascal s triangle modulo
k
......................... 30
2.3.5.
Words classified based on their blocks of letters
........... 31
2.3.6.
Diagonals of a polygon
.......................... 33
vi
Mathematics for Informatics and Computer Science
2.3.7.
Number of times a number is present
ina
list of numbers
...... 35
2.3.8.
Words of length
η
based on
0
and
1
without any block
of Is repeated
.................................... 37
2.3.9.
Programming: classification of applications of a set with
η
elements
in itself following the form of their graph
................... 39
2.3.10.
Individuals grouped
2x2......................... 42
Chapter
3.
Enumerations in Alphabetical Order
................. 43
3.1.
Principle of enumeration of words in alphabetical order
......... 43
3.2.
Permutations
................................... 44
3.3.
Writing binary numbers
............................ 46
3.3.1.
Programming
................................ 46
3.3.2.
Generalization to expression in some base
ß.............. 46
3.4.
Words in which each letter is less than or equal to the position
..... 47
3.4.1.
Number of these words
.......................... 47
3.4.2.
Program
................................... 47
3.5.
Enumeration of combinations
......................... 47
3.6.
Combinations with repetitions
......................... 49
3.7.
Purchase of
Ρ
objects out of TV types of objects
............... 49
3.8.
Another enumeration of permutations
.................... 50
3.9.
Complementary exercises
........................... 52
3.9.1.
Exercise
1 :
words with different successive letters
.......... 52
3.9.2.
Exercise
2:
repeated purchases with a given sum of money
..... 56
3.10.
Return to permutations
............................ 58
3.11.
Gray code
.................................... 60
Chapter
4.
Enumeration by Tree Structures
................... 63
4.1.
Words of length n, based on
N
letters
1,2,3,
...,Ν,
where each letter
is followed by a higher or equal letter
....................... 63
4.2.
Permutations enumeration
........................... 66
4.3.
Derangements
.................................. 67
4.4.
The queens problem
............................... 69
4.5.
Filling up containers
.............................. 72
4.6.
Stack of coins
.................................. 76
4.7.
Domino tiling a chessboard
.......................... 79
Chapter
5.
Languages, Generating Functions and Recurrences
....... 85
5.1.
The language of words based on two letters
................. 85
5.2.
Domino tiling a 2xn chessboard
........................ 88
5.3.
Generating function associated with a sequence
.............. 89
Table
of
Contents
vii
5.4.
Rational generating function and
linear
recurrence
............ 91
5.5.
Example: routes in a square grid with rising shapes
without entanglement
................................. 92
5.6.
Exercises on recurrences
............................ 94
5.6.1.
Three types of purchases each day with a sum of
Ν
dollars
..... 94
5.6.2.
Word building
................................ 96
5.7.
Examples of languages
............................. 98
5.7.1.
Language of parts of an element set {a,b,c,d,
...}......... 98
5.7.2.
Language of parts of a multi-set based on
η
elements a, b, c, etc.,
where these elements can be repeated as much as we want
......... 99
5.7.3.
Language of words made from arrangements taken from
η
distinct
and non-repeated letters a, b, c, etc., where these words are shorter than
or equal to
и
..................................... 99
5.7.4.
Language of words based on an alphabet of
и
letters
......... 100
5.8.
The exponential generating function
..................... 101
5.8.1.
Exercise
1:
words based on three letters a, b and c,
with the letter a at least twice
........................... 101
5.8.2.
Exercise
2:
sending
n
people to three countries, with at least
one person per country
.............................. 103
Chapter
6.
Routes in a Square Grid
......................... 105
6.1.
Shortest paths from one point to another
................... 105
6.2.
«-length paths using two (perpendicular) directions of
the square grid
..................................... 108
6.3.
Paths from
Oto B (n, x)
neither touching nor crossing
the horizontal axis and located above it
...................... 109
6.4.
Number of «-length paths that neither touch nor cross the axis
of the adscissae until and including the final point
............... 110
6.5.
Number of
и
-length
paths above the horizontal axis that can touch
but not cross the horizontal axis
..........................
Ill
6.6.
Exercises
..................................... 112
n
6.6.1.
Exercise
1:
show that C 2n
= £
(C*)2
................. 112
P
6.6.2.
Exercise
2:
show that
Y^CkNĄ+k
=
C^+p
.............. 113
6.6.3.
Exercise
3:
show that
¿2¿C¿Í*
=и С^............... ИЗ
6.6.4.
Exercise
4:
a geometrico-linguistic method
.............. 114
6.6.5.
Exercise
5:
paths of a given length that never intersect each other
and where the four directions are allowed in the square grid
........ 115
viii
Mathematics for Informatics and Computer Science
Chapter
7.
Arrangements and Combinations with Repetitions
........ 119
7.1.
Anagrams
..................................... 119
7.2.
Combinations with repetitions
......................... 121
7.2.1.
Routes in a square grid
........................... 121
7.2.2.
Distributing (indiscernible) circulars in personalized letter boxes
. 121
7.2.3.
Choosing /objects out of ./V categories of object
........... 121
7.2.4.
Number of positive or
nul
integer solutions to
the equation xO +
xl
+ ...+ ХП-1
=
Ρ
....................... 122
7.3.
Exercises
..................................... 125
7.3.1.
Exercise
1:
number of ways of choosing six objects out of three
categories, with the corresponding prices
................... 125
7.3.2.
Exercise
2:
word counting
......................... 125
7.3.3.
Exercise
3:
number of words of
Ρ
characters based on an alphabet
of
N
letters and subject to order constraints
.................. 127
7.3.4.
Exercise
4:
choice of objects out of several categories taking
at least one object from each category
..................... 128
7.3.5.
Exercise
5:
choice of
Ρ
objects out of TV categories
when the stock is limited
............................. 128
7.3.6.
Exercise
6:
generating functions associated with the number
of integer solutions to an equation with
и
unknowns
............. 129
7.3.7.
Exercise
7:
number of solutions to the equation*
+
у
+
ζ
=
к,
where
к
is a given natural integer and
0 <
χ
<
у
<
ζ
.............. 130
7.3.8.
Exercise
8:
other applications of the method using
generating functions
................................ 131
7.3.9.
Exercise
9:
integer-sided triangles
.................... 132
7.3.10.
Revision exercise: sending postcards
................. 133
7.4.
Algorithms and programs
........................... 135
7.4.1.
Anagram program
............................. 135
7.4.2.
Combinations with repetition program
................. 136
Chapter
8.
Sieve Formula
............................... 137
8.1.
Sieve formula on sets
.............................. 138
8.2.
Sieve formula in combinatorics
........................ 142
8.3.
Examples
..................................... 142
8.3.1.
Example
1:
filling up boxes with objects, with at least one box
remaining empty
.................................. 142
8.3.2.
Example
2:
derangements
......................... 144
8.3.3.
Example
3:
formula giving the
Euler
number
φ(η)
.......... 145
8.3.4.
Example
4:
houses to be painted
..................... 146
8.3.5.
Example
5:
multiletter words
....................... 148
8.3.6.
Example
6:
coloring the vertices of a graph
.............. 150
Table
of
Contents
ix
8.4.
Exercises
..................................... 153
8.4.1.
Exercise
1 :
sending nine diplomats,
1, 2, 3,
...,
9,
to three countries
Л, В, С
............................. 153
8.4.2.
Exercise
2:
painting
aroom
........................ 153
8.4.3.
Exercise
3:
rooks on a chessboard
.................... 155
8.5.
Extension of sieve formula
........................... 158
8.5.1.
Permutations that have
ł
fixed points
.................. 159
8.5.2.
Permutations wither disjoint cycles that are
к
long
.......... 160
8.5.3.
Terminal nodes of trees with« numbered nodes
............ 161
8.5.4.
Revision exercise about a word: intelligent
............... 163
Chapter
9.
Mountain Ranges or Parenthesis Words: Catalan Numbers
. . 165
9.1.
Number c(n) of mountain ranges 2n long
.................. 166
9.2.
Mountains or primitive words
......................... 167
9.3.
Enumeration of mountain ranges
....................... 168
9.4.
The language of mountain ranges
....................... 169
9.5.
Generating function of the
Сгп
and Catalan numbers
........... 171
9.6.
Left factors of mountain ranges
........................ 173
9.6.1.
Algorithm for obtaining the numbers of these left factors a(N, X)
. 175
9.6.2.
Calculation following the lines of Catalan s triangle
......... 176
9.6.3.
Calculations based on the columns of the Catalan triangle
..... 177
9.6.4.
Average value of the height reached by left factors
.......... 178
9.6.5.
Calculations based on the second bisector of the Catalan triangle
. 180
9.6.6.
Average number of mountains for mountain ranges
......... 183
9.7.
Number of peaks of mountain ranges
.................... 184
9.8.
The Catalan mountain range, its area and height
.............. 187
9.8.1.
Number of mountain ranges 2n long passing through a given point
on the square grid
.................................. 187
9.8.2.
Sum of the elements of lines in triangle OO B of mountain
ranges In long
.................................... 188
9.8.3.
Sum of numbers in triangle OO B
.................... 189
9.8.4.
Average area of a mountain In long
................... 190
9.8.5.
Shape of the average mountain range
.................. 192
9.8.6.
Height of the Catalan mountain range
.................. 194
Chapter
10.
Other Mountain Ranges
........................ 197
10.1.
Mountain ranges based on three lines
/
.
197
10.2.
Words based on three lines
/
N^
______
with as many
rising lines as falling lines
.............................. 198
χ
Mathematics for Informatics and Computer Science
10.2.1.
Explicit formula v(n)
.......................... 199
10.2.2.
Return to u(n) number of mountain ranges based
on three letters
a, b, c
and a link with v(ri)
................... 200
10.3.
Example
1:
domino tiling of an enlarged Aztec diamond
........ 200
10.4.
Example
2:
domino tiling of half an Aztec diamond
........... 204
10.4.1.
Link between
Schröder
numbers and Catalan numbers
....... 207
10.4.2.
Link with Narayana numbers
...................... 207
10.4.3.
Another way of programming three-line mountain ranges
..... 208
10.5.
Mountain ranges based on three types of lines
210
10.6.
Example
3:
movement of the king on a chessboard
........... 213
Chapter
11.
Some Applications of Catalan Numbers and
Parenthesis Words
.................................... 215
11.1.
The number of ways of placing
«
chords not intersecting each other
on a circle with an even number 2n of points
................... 215
11.2.
Murasaki diagrams and partitions
...................... 216
11.3.
Path couples with the same ends in a square grid
............. 218
11.4.
Path couples with same starting point and length
............. 220
11.5.
Decomposition of words based on two letters as a product of words
linked to mountain ranges
.............................. 222
Chapter
12.
Burnside s Formula
........................... 227
12.1.
Example
1:
context in which we obtain the formula
........... 227
12.2.
Burnside s formula
............................... 231
12.2.1.
Complementary exercise: rotation-type colorings of the vertices
of a square
...................................... 232
12.2.2.
Example
2:
pawns on a chessboard
.................. 232
12.2.3.
Example
3:
pearl necklaces
....................... 237
12.2.4.
Example
4:
coloring of a stick
..................... 239
12.3.
Exercises
..................................... 239
12.3.1.
Coloring the vertices of a square
.................... 239
12.3.2.
Necklaces with stones in several colors
................ 241
12.3.3.
Identical balls in identical boxes
.................... 244
12.3.4.
Tiling an Aztec diamond using/-squares
............... 244
12.3.5.
The
4x4
Sudoku: search for fundamentally different
symmetry-type girls
................................ 246
Chapter
13.
Matrices and Circulation on a Graph
................ 253
13.1.
Number of paths of a given length on a complete or a regular graph
. 254
13.2.
Number of paths and matrix powers
.................... 255
Table
of
Contents
xi
13.2.1.
Example
1 :
«-length words in an alphabet of three letters
1, 2, 3,
with prohibition of blocks
11
and
23...................... 257
13.2.2.
Simplification of the calculation
.................... 259
13.2.3.
Example
2:
w-length words based on three letters
1, 2, 3
with blocks
11, 22
and
33
prohibited
...................... 261
13.3.
Link between cyclic words and closed paths in an oriented graph
. . . 262
13.4.
Examples
.................................... 263
13.4.1.
Dominos on a
chessboard
........................ 263
13.4.2.
Words with a dependency link between two successive
Jetters
of words
................................... 265
13.4.3.
Routes on a graded segment
....................... 266
13.4.4.
Molecular chain
.............................. 270
Chapter
14.
Parts and Partitions of a Set
..................... 275
14.1.
Parts of a set
................................... 275
14.1.1.
Program getting all parts of a set
.................... 275
14.1.2.
Exercises
.................................. 277
14.2.
Partitions of a
и
-object
set
.......................... 281
14.2.1.
Definition
.................................. 281
14.2.2.
A second kind of Stirling numbers, and partitions of a n-element
set in
к
parts
..................................... 281
14.2.3.
Number of partitions of a set and Bell numbers
........... 283
14.2.4.
Enumeration algorithm for all partitions of a set
........... 285
14.2.5.
Exercise: Sterling numbers modulo
2................. 286
Chapter
15.
Partitions of a Number
......................... 289
15.1.
Enumeration algorithm
............................ 289
15.2.
Euler
formula
.................................. 290
15.3.
Exercises
..................................... 292
15.3.1.
Exercise
1:
partitions of a number
w
in
£
distinct elements
..... 292
15.3.2.
Exercise
2:
ordered partitions
...................... 296
15.3.3.
Exercise
3:
sum of the products of all the ordered partitions
of a number
..................................... 297
15.3.4.
Exercise
4:
partitions of a number incompletely distinct parts
. . 298
15.3.5.
Exercise
5:
partitions and routes in a square grid
.......... 299
15.3.6.
Exercise
6:
Ferrers graphs
........................ 302
Chapter
16.
Flags
.................................... 305
16.1.
Checkered flags
................................ 305
16.2.
Flags with vertical stripes
........................... 306
xii
Mathematics for Informatics and Computer Science
Chapter
17.
Walls and Stacks
............................. 315
17.1.
Brick walls
................................... 315
17.2.
Walls of bricks made from continuous horizontal rows
......... 316
17.2.1.
Algorithm for classifying various types of walls
........... 317
17.2.2.
Possible positions of one row above another
............. 317
17.2.3.
Coordinates of bricks
........................... 318
17.3.
Heaps
....................................... 319
17.4.
Stacks of disks
................................. 322
17.5.
Stacks of disks with continuous rows
.................... 324
17.6.
Horizontally connected poly
ominös.................... 326
Chapter
18.
Tiling of Rectangular Surfaces using Simple Shapes
...... 331
18.1.
Tiling of a 2xn chessboard using
dominos
................. 331
18.1.1.
First algorithm for constructing tilings
................ 332
18.1.2.
Second construction algorithm
..................... 333
18.2.
Other tilings of a chessboard 2xn squares long
.............. 334
18.2.1.
With squares and horizontal
dominos
................. 334
18.2.2.
With squares and horizontal or vertical
dominos
.......... 335
18.2.3.
With
dominos
and/-squares we can turn and reflect
........ 335
18.2.4.
With squares,/-squares and
dominos
................. 336
18.3.
Tilings of
а Зхи
chessboard using
dominos
................ 337
18.4.
Tilings of
а 4хя
chessboard with
dominos
................. 339
18.5.
Domino tilings of a rectangle
........................ 340
Chapter
19.
Permutations
............................... 345
19.1.
Definition and properties
........................... 345
19.2.
Decomposition of a permutation as a product of disjoint cycles
.... 347
19.2.1.
Particular cases of permutations defined by their decomposition
in cycles
....................................... 349
19.2.2.
Number of permutations of« elements with
k
cycles:
Stirling numbers of the first kind
........................ 352
19.2.3.
Type of permutation
........................... 353
19.3.
Inversions in a permutation
.......................... 354
19.3.1.
Generating function of the number of inversions
.......... 356
19.3.2.
Signature of a permutation: odd and even permutations
...... 357
19.4.
Conjugated permutations
........................... 359
19.5.
Generation of permutations
.......................... 360
19.5.1.
The symmetrical group
S„
is generated by the transpositions (ij)
, 361
19.5.2.
S„
is generated by transpositions of adjacent elements
of the form (i
,
і
+ 1)................................ 362
19.5.3.
Sn is generated by transpositions
(0 1) (0 2)... (0
n
- 1)...... 362
Table
of
Contents xiii
19.5.4. S„
is generated by cycles
(0 1)
and
(0 1 2 3 ...
η
- 1)........ 363
19.6.
Properties of the alternating group An
.................... 363
19.6.
.An is generated by cycles three units long: (ij k)
........... 363
19.6.2. A„
is generated by«
-2
cycles
(0 1
k)
................. 363
19.6.3.
For
η
> 3,
An is generated by the cycle chain three units long,
oftheform(0
1
2)(234)(456)
...
(и-3
w-2
n- )
.......... 364
19.7.
Applications of these properties
....................... 365
19.7.1.
Card shuffling
............................... 365
19.7.2.
Taquin
game in
a
η
by
ρ
(и
and p>
1)
rectangle
........... 368
19.7.3.
Cyclic shifts in a rectangle
........................ 371
19.7.4.
Exchanges of lines and columns in a square
............. 375
19.8.
Exercises on permutations
.......................... 376
19.8.1.
Creating a permutation at random
................... 376
( 0 1 2 ...
И-1
Ì
19.8.2.
Number of permutations
[a(0) a(l) a{2)
...
a(n-l)J
with
η
elements
0, 1, 2, ...,
и
— 1,
such that a(i) ~
і
= 0
or
1....... 377
19.8.3.
Permutations with a(i)~i = ± or
±2................. 379
19.8.4.
Permutations with
η
elements
0, 1, 2,
..., η
- 1
without
two consecutive elements
............................. 379
19.8.5.
Permutations with
η
elements
0, 1,2,
...,η-
1,
made up of a
single cycle in which no two consecutive elements modulo
η
are found
. 381
19.8.6.
Involute permutations
.......................... 383
19.8.7.
Increasing subsequences in a permutation
.............. 384
19.8.8.
Riffle shuffling of type
О
and I for
N
cards when
N
is a power of
2................................... 386
Part
2.
Probability
.................................. 387
Part
2.
Introduction
................................... 389
Chapter
20.
Reminders about Discrete Probabilities
.............. 395
20.1.
And/or in probability theory
......................... 396
20.2.
Examples
.................................... 398
20.2.1.
The Chevalier
de
Mere problem
.................... 398
20.2.2.
From combinatorics to probabilities
.................. 399
20.2.3.
From combinatorics of weighted words to probabilities
...... 400
20.2.4.
Drawing a parcel of objects from a box
................ 401
20.2.5.
Hypergeometric law
........................... 401
20.2.6.
Draws with replacement in a box
.................... 402
20.2.7.
Numbered balls in a box and the smallest number
obtained during draws
............................... 403
xiv
Mathematics for Informatics and Computer Science
20.2.8.
Wait for the first double heads in a repeated game
of heads or tails
................................... 404
20.2.9.
Succession of random cuts made in
agame
of cards
........ 405
20.2.10.
Waiting time for initial success
.................... 407
20.2.11.
Smallest number obtained during successive draws
........ 409
20.2.12.
The pool problem
............................ 411
20.3.
Total probability formula
........................... 412
20.3.1.
Classic example
.............................. 412
20.3.2.
The formula
................................ 413
20.3.3.
Examples
.................................. 413
20.4.
Random variable X, law of X, expectation and variance
......... 418
20.4.1.
Average value
οΐΧ
............................ 418
20.4.2.
Variance and standard deviation
.................... 418
20.4.3.
Example
................................... 419
20.5.
Some classic laws
............................... 420
20.5.1.
Bernoulli s law
.............................. 420
20.5.2.
Geometric law
............................... 420
20.5.3.
Binomial law
................................ 421
20.6.
Exercises
..................................... 422
20.6.1.
Exercise
1 :
throwing balls in boxes
.................. 422
20.6.2.
Exercise
2:
series of repetitive tries
.................. 423
20.6.3.
Exercise
3:
filling two boxes
...................... 425
Chapter
21.
Chance and the Computer
....................... 427
21.1.
Random number generators
......................... 428
21.2.
Dice throwing and the law of large numbers
............... 429
21.3.
Monte Carlo methods for getting the approximate value
of the number
π
.................................... 430
21.4.
Average value of a random variable X, variance
and standard deviation
................................ 432
21.5.
Computer calculation of probabilities, as well as expectation
and variance, in the binomial law example
.................... 433
21.6.
Limits of the computer
............................ 437
21.7.
Exercises
..................................... 439
21.7.1.
Exercise
1 :
throwing balls in boxes
.................. 439
21.7.2.
Exercise
2:
boys and girls
........................ 439
21.7.3.
Exercise
3:
conditional probability
................... 441
21.8.
Appendix: chi-squared law
.......................... 443
21.8.1.
Examples of the test
for uniform
distribution
............. 443
21.8.2.
Chi-squared law and its link with
Poisson
distribution
....... 445
Table
of
Contents
xv
Chapter
22.
Discrete and Continuous
........................ 447
22.1.
Uniform law
................................... 448
22.1.1.
Programming
................................ 448
22.1.2.
Example
1................................. 449
22.1.3.
Example
2:
two people meeting
.................... 450
22.2.
Density function for a continuous random variable
and distribution function
............................... 451
22.3.
Normal law
................................... 452
22.4.
Exponential law and its link with uniform law
.............. 454
22.4.1.
An application: geometric law using exponential law
........ 456
22.4.2.
Program for getting the geometric law with parameter/)
...... 457
22.5.
Normal law as an approximation of binomial law
............ 458
22.6.
Central limit theorem: from uniform law to normal law
......... 460
22.7.
Appendix: the distribution function and its inversion
-
application
to binomial law
B(n,p)
................................ 465
22.7.1.
Program
................................... 465
22.7.2.
The inverse function
........................... 467
22.7.3.
Program causing us to move from distribution function
to probability law
.................................. 468
Chapter
23.
Generating Function Associated with a Discrete Random
Variable in a Game
................................... 469
23.1.
Generating function: definition and properties
.............. 469
23.2.
Generating functions of some classic laws
................. 470
23.2.1.
Bernoulli s law
.............................. 470
23.2.2.
Geometric law
............................... 470
23.2.3.
Binomial law
................................ 473
23.2.4.
Poisson
distribution
............................ 475
23.3.
Exercises
..................................... 476
23.3.1.
Exercise
1 :
waiting time for double heads in a game of heads
or tails
........................................ 476
23.3.2.
Exercise
2:
in a repeated game of heads or tails, what is the parity
of the number of heads?
.............................. 481
23.3.3.
Exercise
3:
draws until a certain threshold is exceeded
....... 482
23.3.4.
Exercise
4:
Pascal s law
......................... 487
23.3.5.
Exercise
5:
balls of two colors in a box
................ 488
23.3.6.
Exercise
6:
throws of
N
dice until each gives the number
1 .... 492
Chapter
24.
Graphs and Matrices for Dealing with Probability Problems.
497
24.1.
First example: counting of words based on three letters
......... 497
24.2.
Generating functions and determinants
................... 499
xvi
Mathematics for Informatics and Computer Science
24.3.
Examples
.................................... 500
24.3.1.
Exercise
1:
waiting time for double heads in a game of heads
or tails
........................................ 500
24.3.2.
Draws from three boxes
......................... 503
24.3.3.
Alternate draws from two boxes
.................... 505
24.3.4.
Successive draws from one box to the next
.............. 506
Chapter
25.
Repeated Games of Heads or Tails
................. 509
25.1.
Paths on a square grid
............................. 509
25.2.
Probability of getting a certain number of wins after
η
equiprobable tosses
.................................. 511
25.2.1.
Probability p(n, x) of getting winnings of
χ
at the end of
η
moves
512
25.2.2.
Standard deviation in relation to a starting point
........... 512
25.2.3.
Probability
/э(2и )
of a return to the origin at stage
η
—
2rí.....
513
25.3.
Probabilities of certain routes over
и
moves
................ 514
25.4.
Complementary exercises
........................... 516
25.4.1.
Last visit to the origin
.......................... 516
25.4.2.
Number of winnings sign changes throughout the game
...... 517
25.4.3.
Probability of staying on the positive winnings side for a certain
amount of time during the N=2n equiprobable tosses
............ 519
25.4.4.
Longest range of winnings with constant sign
............ 520
25.5.
The gambler s ruin problem
......................... 521
25.5.1.
Probability of ruin
............................. 522
25.5.2.
Average duration of the game
...................... 524
25.5.3.
Results and program
........................... 525
25.5.4.
Exercises
.................................. 526
25.5.5.
Temperature equilibrium and random walk
.............. 530
Chapter
26.
Random Routes on a Graph
...................... 535
26.1.
Movement of
apartide
on a polygon or graduated segment
...... 535
26.1.1.
Average duration of routes between two points
........... 535
26.1.2.
Paths of a given length on a polygon
.................. 542
26.1.3.
Particle circulating on a pentagon: time required using one side
or the other to get to the end
........................... 546
26.2.
Movement on a polyhedron
......................... 547
26.2.1.
Case of the regular polyhedron
..................... 547
26.2.2.
Circulation on a cube with any dimensions
.............. 550
26.3.
The robot and the human being
....................... 555
26.4.
Exercises
..................................... 559
26.4.1.
Movement of a particle on a square-based pyramid
......... 559
26.4.2.
Movement of two particles
ona
square-based pyramid
....... 561
26.4.3.
Movement of two particles on a graph with five vertices
...... 563
Table
of
Contents
xvii
Chapter
27.
Repetitive Draws until the Outcome of a Certain Pattern
. . . 565
27.1.
Patterns are arrangements of
К
out of
./Vletters
.............. 566
27.1.1.
Wait for a given arrangement of the
К
letters in the form
of a block
...................................... 566
27.1.2.
Wait for a given cyclic arrangement of AT letters in
the form of a block
................................. 568
27.1.3.
The pattern is a given arrangement of
К
out of jV letters
in scattered form
.................................. 570
27.2.
Patterns are combinations of
К
letters drawn from
N
letters
...... 571
27.2.1.
Wait for the outcome of a part made of
К
numbers in the form
ofablock
...................................... 571
Π
2.2.
Wait for the outcome of any part of AT numbers in the form
of a block, out of
N................................. 574
27.2.3.
Wait for the outcome of a part with AT given numbers out of TV
in scattered form
.................................. 577
27.2.4.
Wait for the outcome of any part of
К
numbers out of N,
in scattered form
.................................. 577
27.2.5.
Some examples of comparative results for waiting times
..... 579
27.3.
Wait for patterns with eventual repetitions of identical letters
..... 580
27.3.1.
For an alphabet of
Л
letters, we wait for a given pattern
in the form of a
и
-length
block
.......................... 580
27.3.2.
Wait for one of two patterns of the same length
L
.......... 581
27.4.
Programming exercises
............................ 586
27.4.1.
Wait for completely different letters
.................. 586
27.4.2.
Waiting time for a certain pattern
.................... 588
27.4.3.
Number of words without two-sided factors
............. 589
Chapter
28.
Probability Exercises
.......................... 597
28.1.
The elevator
................................... 597
28.1.1.
Deal with the case where
P
= 2
floors and the number
ofpeopleNis at least equal to
2......................... 597
28.1.2.
Determine the law ofX, i.e. the probability associated
with each value of X
................................ 598
28.1.3.
Average value E(X)
............................ 599
28.1.4.
Direct calculation of S(K+l,K)
..................... 600
28.1.5.
Another way of dealing with the previous question
......... 601
28.2.
Matches
..................................... 601
28.3.
The tunnel
.................................... 602
28.3.1.
Dealing with the specific case where
N=3.............. 606
28.3.2.
Variation with an absorbing boundary and another method
.... 608
28.3.3.
Complementary exercise: drunken man s walk on a straight line,
with resting time
.................................. 610
xviii
Mathematics for Informatics and Computer Science
28.4.
Repetitive draws from a box
......................... 613
28.4.1.
Probability law for the number of draws
............... 615
28.4.2.
Extra questions
.............................. 616
28.4.3.
Probability of getting ball number
к
during the game
........ 617
28.4.4.
Probability law associated with the number of balls drawn
.... 617
28.4.5.
Complementary exercise: variation of the previous problem
. . . 618
28.5.
The sect
..................................... 620
28.5.1.
Can the group last forever?
....................... 620
28.5.2.
Probability law of the size of the tree
................. 621
28.5.3.
Average tree size
............................. 622
28.5.4.
Variance of the variable size
...................... 624
28.5.5.
Algorithm giving the probability law of
the organization s lifespan
............................ 625
28.6.
Surfing the web (or how Google works)
.................. 627
Part
3.
Graphs
..................................... 637
Part
3.
Introduction
................................... 639
Chapter
29.
Graphs and Routes
........................... 643
29.1.
First notions on graphs
............................ 643
29.1.1.
A few properties of graphs
........................ 645
29.1.2.
Constructing graphs from points
.................... 646
29.2.
Representing a graph in a program
..................... 647
29.2.1.
From vertices to edges
.......................... 649
29.2.2.
From edges to vertices
.......................... 649
29.3.
The tree as a specific graph
.......................... 649
29.3.1.
Definitions and properties
........................ 649
29.3.2.
Programming exercise: network converging on a point
....... 652
29.4.
Paths from one point to another in a graph
................. 654
29.4.1.
Dealing with an example
......................... 654
29.4.2.
Exercise: paths on a complete graph, from one vertex to another
. 656
Chapter
30.
Explorations in Graphs
......................... 661
30.1.
The two ways of visiting all the vertices of a connected graph
..... 661
30.2.
Visit to all graph nodes from one node, following
depth-first traversal
.................................. 662
30.3.
The pedestrian s route
............................. 665
30.4.
Depth-first exploration to determine connected components
of the graph
...................................... 669
30.5.
Breadth-first traversal
............................. 671
30.5.1.
Program
................................... 671
Table
of
Contents
xix
30.5.2.
Example: traversal in a square grid
................... 673
30.6.
Exercises
..................................... 676
30.6.1.
Searching in amaze
............................ 676
30.6.2.
Routes in a square grid, with rising shapes without entangling
. . 680
30.6.3.
Route of a fluid in a graph
........................ 683
30.6.4.
Connected graphs with
и
vertices
.................... 683
30.6.5.
Bipartite graphs
.............................. 685
30.7.
Returning to a depth-first exploration tree
................. 686
30.7.1.
Returning edges in an undirected graph
................ 687
30.7.2.
Isthmuses in an undirected graph
.................... 688
30.8.
Case of directed graphs
............................ 690
30.8.1.
Strongly connected components in a directed graph
......... 690
30.8.2.
Transitive closure of a directed graph
................. 693
30.8.3.
Orientation of a connected undirected graph to become
strongly connected
................................. 696
30.8.4.
The best orientations on a graph
.................... 696
30.9.
Appendix: constructing the maze (simplified version)
.......... 700
Chapter
31.
Trees with Numbered Nodes, Cayley s Theorem
and
Prüfer Code..................................... 705
31.1.
Cayley s theorem
................................ 705
31.2. Prüfer
code
................................... 706
31.2.1.
Passage from a tree to its
Prüfer
code
................. 707
31.2.2.
Reverse process
.............................. 707
31.2.3.
Program
................................... 709
31.3.
Randomly constructed spanning tree
.................... 715
31.3.1.
Wilson s algorithm
............................ 715
31.3.2.
Maze and domino tiling
......................... 718
Chapter
32.
Binary Trees
............................... 723
32.1.
Number of binary trees with
η
nodes
.................... 725
32.2.
The language of binary trees
......................... 725
32.3.
Algorithm for creation of words from the binary tree language
.... 728
32.4. Triangulation
of polygons with numbered vertices and binary trees.
. 729
32.5.
Binary tree sort or quicksort
......................... 733
Chapter
33.
Weighted Graphs: Shortest Paths and Minimum
Spanning Tree
...................................... 737
33.1.
Shortest paths in a graph
........................... 737
33.1.1.
Dijkstra s
algorithm
............................ 738
33.1.2.
Floyd s algorithm
............................. 741
33.2.
Minimum spanning tree
............................ 746
xx
Mathematics for Informatics and Computer Science
33.2.1.
Prim s algorithm
.............................. 747
33.2.2.
Kruskal s algorithm
............................ 749
33.2.3.
Comparison of the two algorithms
................... 754
33.2.4.
Exercises
.................................. 754
Chapter
34.
Eulerian Paths and Cycles, Spanning Trees of a Graph
.... 759
34.1.
Definition of Eulerian cycles and paths
.................. 759
34.2.
Euler
and
Königsberg
bridges
........................ 761
34.2.1.
Returning to
Königsberg
bridges
.................... 763
34.2.2.
Examples
.................................. 764
34.2.3.
Constructing Eulerian cycles by fusing cycles
............ 767
34.3.
Number of Eulerian cycles in a directed graph, link with directed
spanning trees
..................................... 768
34.3.1.
Number of directed spanning trees
................... 771
34.3.2.
Examples
.................................. 774
34.4.
Spanning trees of an undirected graph
................... 776
34.4.1.
Example
1:
complete graph with
ρ
vertices
.............. 777
34.4.2.
Example
2:
tetrahedron
.......................... 778
Chapter
35.
Enumeration of Spanning Trees of an Undirected Graph
. . . 779
35.1.
Spanning trees of the fan graph
....................... 779
35.2.
The ladder graph and its spanning trees
.................. 782
35.3.
Spanning trees in a square network in the form of a grid
........ 784
35.3.1.
Experimental enumeration of spanning trees
of the square network
............................... 785
35.3.2.
Spanning trees program in the case of the square network
..... 786
35.3.3.
Passage to the undirected graph, its dual and formula giving the
number of spanning trees
............................. 788
35.4.
The two essential types of (undirected) graphs based on squares
. . . 789
35.5.
The cyclic square graph
............................ 791
35.6.
Examples of regular graphs
.......................... 792
35.6.1.
Example
1................................. 792
35.6.2.
Example
2:
hypercube with« dimensions
............... 793
35.6.3.
Example
3:
the ladder graph and its variations
............ 793
Chapter
36.
Enumeration of Eulerian Paths in Undirected Graphs
..... 799
36.1.
Polygon graph with
и
vertices with double edges
............. 799
36.2.
Eulerian paths in graph made up of a frieze of triangles
......... 801
36.3.
Algorithm for Eulerian paths and cycles on an undirected graph
. . . 804
36.3.1.
The
arborescence
for the paths
..................... 804
36.3.2.
Program for enumerating Eulerian cycles
............... 805
Table
of
Contents xxi
36.3.3.
Enumeration in the case of
multiple
edges between vertices.
. . . 807
36.3.4.
Another example: square with double diagonals
........... 810
36.4.
The game of
dominos
............................. 813
36.4.1.
Number of domino chains
........................ 813
36.4.2.
Algorithms
................................. 816
36.5.
Congo graphs
.................................. 820
36.5.1.
A simple case: graphs P(2w,
5)..................... 822
36.5.2.
The first type of Congolese drawings, on P(n
+ 1,
ri) graphs,
with their Eulerian paths
............................. 826
36.5.3.
The second type of Congolese drawings, on P{2N,N) graphs.
. . 826
36.5.4.
Case of Eulerian cycles
oпP{2N+ 1,
2N-
1)
graphs
........ 830
36.5.5.
Case of I(2N
+ 1,
27V
+ 1)
graphs with their Eulerian cycles
.... 832
Chapter
37.
Hamiltonian Paths and Circuits
................... 835
37.1.
Presence or absence of Hamiltonian circuits
................ 836
37.1.1.
First examples
............................... 836
37.1.2.
Hamiltonian circuits on a cube
..................... 837
37.1.3.
Complete graph and Hamiltonian circuits
............... 839
37.2.
Hamiltonian circuits covering a complete graph
............. 840
37.2.1.
Case where the number of vertices is a prime number
other than two
.................................... 840
37.2.2.
General case
................................ 841
37.3.
Complete and antisymmetric directed graph
................ 843
37.3.1.
A few theoretical considerations
.................... 843
37.3.2.
Experimental verification and algorithms
............... 848
37.3.3.
Complete treatment of case N=
4................... 851
37.4.
Bipartite graph and Hamiltonian paths
................... 854
37.5.
Knights tour graph on the NxN chessboard
................ 855
37.5.1.
Case where
N
is odd
........................... 855
37.5.2.
Coordinates of the neighbors of a vertex
............... 855
37.5.3.
Hamiltonian cycles program
....................... 856
37.5.4.
Another algorithm
............................. 857
37.6.
de Bruijn
sequences
.............................. 859
37.6.1.
Preparatory example
........................... 859
37.6.2.
Definition
.................................. 860
37.6.3.
de
Bruijn graph
.............................. 862
37.6.4.
Number of Eulerian and Hamiltonian cycles of Gn
......... 865
Appendices
........................................ 867
Appendix
1.
Matrices
.................................. 869
A
1.1.
Notion of linear application
......................... 869
xxii
Mathematics for Informatics and Computer Science
Al.
2.
Bijective linear application
......................... 872
A1.3. Base change
.................................. 873
Al.
4.
Product of two matrices
........................... 874
A
1.5.
Inverse matrix
................................. 875
Al.
6.
Eigenvalues and eigenvectors
........................ 877
Al.
7.
Similar matrices
................................ 879
Al.
8.
Exercise
..................................... 881
A
1.9.
Eigenvalues of
circulant
matrices and circular graphs
.......... 882
Appendix
2.
Determinants and Route Combinatorics
.............. 885
A2.1. Recalling determinants
............................ 885
A2.2. Determinants and tilings
........................... 887
A2.3. Path sets and determinant
.......................... 892
A2.3.1. First example: paths without intersection in a square network
. . 892
A2.3.2. Second example: mountain ranges without intersection,
based on two diagonal lines
............................ 895
A2.3.3. Third example: mountain ranges without intersection based on
diagonal lines and plateaus. Link with Aztec diamond tilings
....... 896
A2.3.4. Diamond tilings
.............................. 899
A2.4. The hamburger graph: disjoint cycles
................... 901
A2.4.1. First example: domino tiling of a rectangular checkerboard
TVlong,
2
wide
.................................... 902
A2.4.
2.
Second example: domino tilings of the Aztec diamond
...... 904
Bibliography
....................................... 907
Index
............................................ 911
|
| any_adam_object | 1 |
| author | Audibert, Pierre |
| author_facet | Audibert, Pierre |
| author_role | aut |
| author_sort | Audibert, Pierre |
| author_variant | p a pa |
| building | Verbundindex |
| bvnumber | BV036649649 |
| classification_rvk | SK 110 SK 130 ST 120 |
| classification_tum | MAT 023f |
| ctrlnum | (OCoLC)699812256 (DE-599)BVBBV036649649 |
| discipline | Informatik Mathematik |
| edition | 1. publ. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV036649649 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:44:54Z |
| institution | BVB |
| isbn | 9781848211964 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020569181 |
| oclc_num | 699812256 |
| open_access_boolean | |
| owner | DE-20 DE-473 DE-BY-UBG DE-573 DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-20 DE-473 DE-BY-UBG DE-573 DE-11 DE-91G DE-BY-TUM |
| physical | XXV, 914 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | ISTE [u.a.] |
| record_format | marc |
| spelling | Audibert, Pierre Verfasser aut Mathematics for informatics and computer science Pierre Audibert 1. publ. London [u.a.] ISTE [u.a.] 2010 XXV, 914 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Computer science / Mathematics Informatik Mathematik Informatik (DE-588)4026894-9 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Mathematik (DE-588)4037944-9 s Informatik (DE-588)4026894-9 s DE-604 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020569181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Audibert, Pierre Mathematics for informatics and computer science Computer science / Mathematics Informatik Mathematik Informatik (DE-588)4026894-9 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4026894-9 (DE-588)4037944-9 (DE-588)4123623-3 |
| title | Mathematics for informatics and computer science |
| title_auth | Mathematics for informatics and computer science |
| title_exact_search | Mathematics for informatics and computer science |
| title_full | Mathematics for informatics and computer science Pierre Audibert |
| title_fullStr | Mathematics for informatics and computer science Pierre Audibert |
| title_full_unstemmed | Mathematics for informatics and computer science Pierre Audibert |
| title_short | Mathematics for informatics and computer science |
| title_sort | mathematics for informatics and computer science |
| topic | Computer science / Mathematics Informatik Mathematik Informatik (DE-588)4026894-9 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Computer science / Mathematics Informatik Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020569181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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