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