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Sample slidesFranco Vivaldi
Hamiltonian stabilityover discrete spaces
Franco VivaldiQueen Mary, University of London
Smooth area-preserving maps
integrable
Smooth area-preserving maps
integrable
foliation by invariant curves
Smooth area-preserving maps
near-integrable: stableintegrable
foliation by invariant curves
Smooth area-preserving maps
KAM curves
near-integrable: stableintegrable
foliation by invariant curves
Smooth area-preserving maps
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
Smooth area-preserving maps
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
What happens if the space is discrete?
Smooth area-preserving maps
KAM curves
near-integrable: stableintegrable strong perturbation: unstable
foliation by invariant curves
What happens if the space is discrete?
?
Smooth area-preserving maps
Some methods for discretizing space
Some methods for discretizing space
Truncation (computer arithmetic).
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
Geometric discretization.
Some methods for discretizing space
Truncation (computer arithmetic).
Restricting coordinates to a discrete field (ring, module).
Geometric discretization.
Reduction to a finite field.
Early investigations:
F. Rannou (1974)
Early investigations:
F. Rannou (1974)
Early investigations:
First study of an invertible lattice map on the torus
F. Rannou (1974)
Early investigations:
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
F. Rannou (1974)
Early investigations:
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
F. Rannou (1974)
lattice size
Early investigations:
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
Early investigations:
All orbits are periodic.
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
Early investigations:
All orbits are periodic.
Orbit representing curves develop some“thickness”, but remain stable.
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
Early investigations:
All orbits are periodic.
Orbit representing curves develop some“thickness”, but remain stable.
In the chaotic regions the map behaves like a random permutation.
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
Early investigations:
All orbits are periodic.
Orbit representing curves develop some“thickness”, but remain stable.
In the chaotic regions the map behaves like a random permutation.
x
t+1
= x
t
+ y
t
+m
2p
✓1� cos
2pm
y
t
◆(mod m)
y
t+1
= y
t
� lm
2p
✓sin
2pm
x
t+1
+1� cos
2pm
x
t+1
◆(mod m)
1
First study of an invertible lattice map on the torus
rounding to nearest integer
F. Rannou (1974)
lattice size
why?
Renormalizationin parametrised families
of polygon-exchange transformations
Franco VivaldiQueen Mary, University of London
with J H Lowenstein (New York)
Piecewise isometries
a finite collection of pairwise disjoint open polytopes (intersection of open half-spaces),called the atoms.
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
the space:
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
Piecewise isometries
a finite collection of pairwise disjoint open polytopes (intersection of open half-spaces),called the atoms.
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
is an isometrythe dynamics:
the space:
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
Piecewise isometries
a finite collection of pairwise disjoint open polytopes (intersection of open half-spaces),called the atoms.
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
is an isometrythe dynamics:
the space:
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
If F is invertible, then F is volume-preserving.
Piecewise isometries
a finite collection of pairwise disjoint open polytopes (intersection of open half-spaces),called the atoms.
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
is an isometrythe dynamics:
the space:
Theorem (Gutkin & Haydin 1997, Buzzi 2001)The topological entropy of a piecewise isometry is zero.
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
�� �11 0
⇥|�| < 2 R2 ⇤ R2 T2 ⇤ T2
� ⇥ Rn � =⇤
�i F : �⇤ � F |�i
1
If F is invertible, then F is volume-preserving.
Higher dimensions: topology
type I
Higher dimensions: topologyIterate the boundary of the atoms:
type I
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
type I
discontinuity set
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
type I
discontinuity set
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
type I
discontinuity set
not dense, typically
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
type I
discontinuity set
not dense, typically
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
type I
discontinuity set
(union of cells of positive measure)
not dense, typically
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
exceptional set
type I
discontinuity set
(union of cells of positive measure)
not dense, typically
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D \D
l = 2cos(2p p/q)
1
Higher dimensions: topologyIterate the boundary of the atoms:
periodic set
exceptional set
type I
discontinuity set
(union of cells of positive measure)
(asymptotic phenomena)
not dense, typically
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D\D
l = 2cos(2p p/q)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1
W2
W3
W4
|W1
|, |W2
|, |W3
|, |W4
| K =Q(|W1
|, . . . , |Wn|)∂W =
[∂Wi D =
[
t2ZFt(∂W) P = W\D E = D \D
l = 2cos(2p p/q)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O
parameter: position of O along diagonal
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O O
rotate about O
parameter: position of O along diagonal
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
O OO
rotate about O translate back to (parameter-dependent)
parameter: position of O along diagonal
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
quadratic rotation fields:
O OO
rotate about O translate back to (parameter-dependent)
parameter: position of O along diagonal
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
One-parameter families of polygon-exchange transformations
type I
Hooper (2013)Schwartz (2014)Lowenstein & fv (2016)
quadratic rotation fields:
translation module:
O OO
rotate about O translate back to (parameter-dependent)
parameter: position of O along diagonal
s: parameter
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
Wi W ⇢ Rn W =[
Wi F : W ! W F |Wi(1234) 7! (2431)
W1 W2 W3 W4 |W1|, |W2|, |W3|, |W4| K =Q(|W1, . . . , |Wn|)
1
Geometric discretization: strip maps
Geometric discretization: strip maps
A flow with a piecewise-constant vector field is diffracted by a line
Geometric discretization: strip maps
A flow with a piecewise-constant vector field is diffracted by a line
replace the flow by a map, using
the same vectors
Geometric discretization: strip maps
strip
A flow with a piecewise-constant vector field is diffracted by a line
replace the flow by a map, using
the same vectors
flow and map differ within a strip
Geometric discretization: strip maps
strip
A flow with a piecewise-constant vector field is diffracted by a line
replace the flow by a map, using
the same vectors
flow and map differ within a strip
The lattice generated by the
vectors is invariant under the map
Linked strip maps: outer billiards of polygons
Linked strip maps: outer billiards of polygons
Linked strip maps: outer billiards of polygons
Linked strip maps: outer billiards of polygons
Linked strip maps: outer billiards of polygons
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
strip
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
strip
closed polygon (integrable orbit)
Linked strip maps: outer billiards of polygonspiecewise-constant
vector field
strip
closed polygon (integrable orbit)
Linked strip maps: outer billiards of polygons
perturbed orbit
piecewise-constant vector field
The arithmetic of chaos
Franco Vivaldi Queen Mary, University of London
God
‘‘God gave us the integers, the rest is the work of man”L Kronecker
‘‘God gave us the integers, the rest is the work of man”L Kronecker
2
0 1 2 3 4
the work of man
0 1 2 3 4
the work of man
0 1 2 3 4
the work of man
0 1 2 3 4
the work of man
0 1 2 3 4
the work of man
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
print(3846264338327950288419716939937510582097494)
dumb
computer programs to build numbers
print(123123123123123123123123123123123123123123123)
print(123) 15 times
smart
dumb
print(3846264338327950288419716939937510582097494)
dumb
smart ?
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
randomdumb program
output
random numbers
A number is random if the shortest program that can build its digits is the dumb program
Kolmogorov
non-random
output
smart program
randomdumb program
output
do numbers have mass?
0 1
do numbers have mass?
0 1Lebesgue
do numbers have mass?
1 Kg
0 1Lebesgue
do numbers have mass?
1 Kg
0 1
145 g
Lebesgue
do numbers have mass?
1 Kg
0 1
145 g
Lebesgue
The total mass of all fractions is zero
DARK MATTER
DARK MATTER
00
25
50
75
100
universe
visible dark
DARK MATTER
00
25
50
75
100
universe
visible dark
00
25
50
75
100
real numbers
visible dark
Theorem:
Theorem: the random numbers account for thetotal mass of the real number system.
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :dark
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
dark
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
dark
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
they can’t be proved to be random
dark
Theorem: the random numbers account for thetotal mass of the real number system.
Random numbers are :
they can’t be defined individually, or talked about
they can’t be computed, or stored in computers
they can’t be proved to be random
Random numbers are not meant for humans.
dark
Thank you for your attention
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