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Qualitative computing : a computational journey into nonlinearity
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Description
Rating
Title
Qualitative
computing
: a
computational
journey
into
nonlinearity
Creator
ChaitinChatelin, Francoise.
Contributors
World Scientific (Firm)
DescriptionAbstract
High
technology
industries
are in
desperate
need
for
adequate
tools
to
assess
the
validity
of
simulations
produced
by
ever
faster
computers
for
perennial
unstable
problems
. In
order
to
meet
these
industrial
expectations
,
applied
mathematicians
are
facing
a
formidable
challenge
summarized
by these
words

nonlinearity
and
coupling
. This
book
is
unique
as
it
proposes
truly
original
solutions
:
(1)
Using
hypercomputation
in
quadratic
algebras
, as
opposed
to the
traditional
use
of
linear
vector
spaces
in the
20th
century
;
(2)
complementing
the
classical
linear
logic
by the
complex
logic
which
expresses
the
creative
potential
of the
complex
plane
. The
book
illustrates
how
qualitative
computing
has been the
driving
force
behind
the
evolution
of
mathematics
since
Pythagoras
presented
the
first
incompleteness
result
about
the
irrationality
of
[symbol]2
. The
celebrated
results
of
Godel
and
Turing
are but
modern
versions
of the
same
idea
: the
classical
logic
of
Aristotle
is
too
limited
to
capture
the
dynamics
of
nonlinear
computation
.
Mathematics
provides
us with the
missing
tool
, the
organic
logic
,
which
is
aptly
tailored
to
model
the
dynamics
of
nonlinearity
. This
logic
will be the
core
of the "
Mathematics
for
Life
" to be
developed
during
this
century
.
DescriptionTable Of Contents
1
.
Introduction
to
qualitative
computing
.
1.1
. The
art
of
computing
before
the
20th
century
.
1.2
. The
unending
evolution
of
logic
due
to
complexification
.
1.3
. The
20th
century
.
1.4
.
Back
to the
art
of
computing

2
.
Hypercomputation
in
Dickson
algebras
.
2.1
.
Associativity
in
algebra
.
2.2
.
Dickson
algebras
over
the
real
field
.
2.3
.
Properties
of the
multiplication
.
2.4
.
Left
and
right
multiplication
maps
.
2.5
. The
partition
A[symbol]
.
2.6
.
Alternative
vectors
in
A[symbol]
for
K
[symbol]
4
.
2.7
.
Coalternativity
in
A[symbol]
for
K
[symbol]
4
.
2.8
. The
power
map
in
A[symbol]\{lcub}0{rcub}
.
2.9
. The
exponential
function
in
A[symbol]
,
k[symbol]0
.
2.10
.
Some
extensions
of the
Fundamental
Theorem
of
Algebra
, from A
[symbol]
.
2.11
.
Normwise
qualification
mod
2
[symbol]
.
2.12
.
Bibliographical
notes

3
.
Variable
complexity
within
noncommutative
Dickson
algebras
.
3.1
. The
multiplication
tables
in
A[symbol]
,
n
[symbol]
0.
3.2
. The
algorithmic
computation
of the
standard
multiplication
table
M[symbol]
.
3.3
.
Another
algorithmic
derivation
of
M[symbol]
,
n
[symbol]
0.
3.4
. The
right
and
left
multiplication
maps
.
3.5
.
Representations
of
A[symbol]
,
k
[symbol]
2
with
variable
complexity
.
3.6
.
Multiplication
in
[symbol]
.
3.7
. The
algebra
Der(A[symbol])
of
derivations
for
A[symbol]
,
k
[symbol]
0.
3.8
.
Beyond
linear
derivation
.
3.9
. The
nature
of
hypercomputation
in
A[symbol]
,
k
[symbol]
0.
3.10
.
Bibliographical
notes

4
.
Singular
values
for the
multiplication
maps
.
4.1
.
Multiplication
by a
vector
x
in
A[symbol]
,
k
[symbol]
0.
4.2
. a
is
not
alternative
in
[symbol]
,
k
[symbol]
4
.
4.3
.
x
=
[symbol]
.
4.4
.
Complexification
of the
algebra
A[symbol]
,
k
[symbol]
3
.
4.5
.
Zerodivisors
with
two
alternative
parts
in
[symbol]
,
k
[symbol]
3
.
4.6
.
[symbol]
=
(a,b)
has
alternative
,
orthogonal
parts
with
equal
length
in
[symbol]
,
k
[symbol]
3
.
4.7
. The
SVD
for
L[symbol]
in
A[symbol]
.
4.8
.
Other
types
of
zerodivisors
in
[symbol]
,
k
[symbol]
4
.
4.9
.
Bibliographical
notes

5
.
Computation
beyond
classical
logic
.
5.1
.
Local
SVD
derivation
.
5.2
.
Pseudozerodivisors
associated
with
[symbol]
.
5.3
.
Local
and
global
SVD
analyzed
in
[symbol]
for
k
[symbol]
3
.
5.4
. The
measure
of a
vector
[symbol]
in
A[symbol]
evolves
with
k
[symbol]
.
5.5
.
Complexification
of
A[symbol]
into
A[symbol]
,
k
[symbol]
2
.
5.6
.
Local
SVD
for
L[symbol]
=
0,
2
,
5
,
7
.
5.7
.
About
the
inductive
computation
of
[symbol]
from
[symbol]
tino
[symbol]
,
k
[symbol]
4
.
5.8
. An
epistemological
conclusion
.
5.9
.
Bibliographical
notes
. ;
8
6
.
Complexification
of the
arithmetic
.
6.1
. The
resurgence
of
[symbol]
in
A[symbol]
,
k
[symbol]
3
.
6.2
.
Selfinduction
in
[symbol]
by
[symbol]
,
k
[symbol]
2
.
6.3
.
Complex
selfinduction
by
[symbol]
in
[symbol]
,
k
[symbol]
3
.
6.4
.
Spectral
analysis
of
[symbol]
for
s
=
(a,a
,
x
h)
,
[symbol]
,
k
[symbol]
3
.
6.5
. A
geometric
viewpoint
based
on
+
.
6.6
.
Monomorphisms
from
A[symbol]
.
6.7
.
Inductive
construction
of
Der
.
6.8
. An
algorithmic
evolution
of the
field
[symbol]
into
[symbol]
by the
logistic
iteration
.
6.9
.
Other
algorithmic
evolutions
of
t
from
[symbol]
to
[symbol]
.
6.10
.
Evolution
of
u
without
divergence
at
[symbol]
.
6.11
. An
application
: the
isophasic
exponentiation
of
z
in
[symbol]
as a
function
of the
parameter
[symbol]
.
6.12
.
Bibliographical
notes

7
.
Homotopic
deviation
in
linear
algebra
.
7.1
. An
introduction
to
complex
homotopic
deviation
.
7.2
. The
algebraic
tools
.
7.3
. The
resolvent
R(t,z)
for
z
[symbol]
.
7.4
. The
spectral
field
[symbol]
.
7.5
.
Study
of the
limit
set
Lim
under
(7.4.1)
.
7.6
.
About
the
limit
and
frontier
points
in
re(A)
.
7.7
. The
mutation
matrix
B[symbol]
at
[symbol]
.
7.8
. The
observation
point
is
the
eigenvalue
[symbol](A)
.
7.9
.
Algorithmic
complexification
of the
homotopy
parameter
t
,
[symbol]
<
1
.
7.10
. The
family
of
pencils
P[symbol](t)
=
[symbol]
,
where
the
parameter
z
varies
in
[symbol]
.
7.11
.
About
contextual
algebraic
computation
.
7.12
.
Visualization
tools
.
7.13
.
Bibliographical
notes

8
. The
discrete
and the
continuous
.
8.1
. The
selfconjugate
binary
algebras
B[symbol]
,
k
[symbol]
0.
8.2
. The
multiplication
tables
for
k
=
1
,
2
.
8.3
.
Partial
emergence
of
multiplication
mod
2[symbol]
in
B[symbol]
,
k
[symbol]
3
.
8.4
. The
linear
space
C[symbol]
of
binary
sequences
of
length
n
[symbol]
1
.
8.5
.
n
=
2[symbol]
: an
alternative
complex
order
.
8.6
. The
base
bexpansion
of
n
,
b
[symbol]
2
.
8.7
.
Mechanical
uncomputability
.
8.8
. The
arithmetic
triangle
.
8.9
. The
arithmetic
triangle
mod
2
.
8.10
. The
triangle
mod
3
.
8.11
.
Connections
between
2
and
3
.
8.12
.
Two
digital
representations
of
real
numbers
.
8.13
. The
BorelNewcomb
paradox
for
real
numbers
.
8.14
.
Sum
of
random
variables
computed
modulo
1
.
8.15
.
Finite
precision
computation
over
[symbol]
.
8.16
. A
dynamical
perspective
on the
natural
integers
.
8.17
.
Bibliographical
notes

9
.
Arithmetic
in the
four
Dickson
division
algebras
.
9.1
. A
review
of the
three
theorems
of
squares
.
9.2
. The
rings
R[symbol]
of
hypercomplex
integers
,
k
[symbol]
3
.
9.3
.
Isometries
in
3
and
4
dimensions
.
9.4
. The
rate
of
association
in
G
.
9.5
. The
first
cycle
(f1,f2,f3)
.
9.6
. A
second
epistemological
pause
.
9.7
. The
last
three
canonical
vectors
f5
to
f7
.
9.8
.
Conclusion
.
9.9
.
Bibliographical
notes
. ;
8
10
. The
real
and the
complex
.
10.1
.
About
the
relativity
stemming
from an
algorithmic
quantification
of a
quality
.
10.2
.
Setwise
inclusion
in
[symbol]
.
10.3
.
Isophasic
inclusion
inside
[symbol]
by
exponentiation
.
10.4
.
Metric
inclusion
inside
[symbol]
under
exponentiation
.
10.5
. The
Cantor
space
{lcub}0,1{rcub}[symbol]
.
10.6
.
Doubly
infinite
sequences
.
10.7
.
Evolution
from
R[symbol]
to
C[symbol]
.
10.8
. The
continuous
Fourier
transform
as a
cognitive
tool
.
10.9
. The
scalar
product
[symbol]
.
10.10
.
Bibliographical
notes

11
. The
organic
logic
of
hypercomputation
.
11.1
.
About
the
representations
of
complex
integers
.
11.2
. The
inductive
points
of
[symbol]
with
norm
n
[symbol]
2
.
11.3
. An
algorithm
for
organic
arithmetic
in
[symbol][b]
.
11.4
.
Comparison
between
z
and
vis(z)
for
z
[symbol]
.
11.5
. The
rings
[symbol]
=
n
[symbol]
2
for
hyperarithmetic
.
11.6
. The
synthetic
power
of
[symbol]
stemming
from
[symbol]
.
11.7
. The
organic
logic
for
hypercomputation
.
11.8
. The
organic
measure
set
for the
source
vector
a
[symbol]
, for
k
[symbol]
3
.
11.9
. The
angles
[symbol]
=
[symbol]
for
j
=
1
to
4
.
11.10
.
About
the
coincidence
of a with
one
of the
a[symbol]
when
[symbol]
=
[symbol]
[symbol]
0.
11.11
. The
autonomous
evolution
of
[symbol]
as a
function
of
r
=
N(a)
=
1
+
N(h)
[symbol]
1
.
11.12
.
Computational
evolution
of
t
out
of
[symbol]
,
k
[symbol]
3
.
11.13
.
Autonomous
evolution
based
on the
spectral
information
in
[symbol]
.
11.14
.
Bibliographical
notes

12
. The
organic
intelligence
in
numbers
.
12.1
.
About
the
zeros
of the
[symbol]
function
.
12.2
.
Algebraic
depth
and
p
=
[symbol]
.
12.3
. The
two
families
of
complex
zeros
for
[symbol]
in the
light
of
hypercomputation
.
12.4
. The
algebras
with
d[symbol]
[symbol]
2
are
sources
of
common
sense
.
12.5
. The
algebraic
reductions
with
p
=
1/2
.
12.6
.
Thinking
in
1
or
2
dimensions
:
thought
or
intuition
.
12.7
. A
review
of
hypercomputation
.
Publisher
World Scientific Pub. Co.
Subject
Mathematical optimization.
Nonlinear theories.
Coupled problems (Complex systems)
Identifier
9789814322935
(electronic
bk.)
;
981432292X
;
9789814322928
;
http://www.worldscientific.com/worldscibooks/10.1142/7904#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xv, 582 p. : ill. (some col.), ports.
Date
c2012
.
OCLC number
874501140
CONTENTdm number
305
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