This is a companion web page to the paper
Can anything from Noether's theorem be salvaged
for discrete dynamical systems?
presented at the UC2011 conference in Turku,
and to the subsequent journal paper
Conserved quantities in discrete dynamics:
what can be recovered from Noether's theorem, how, and why?,
submitted to Natural Computing.
It collects the Python experiments devised by the authors,
together with their output.
In addition, some material relevant to the subject is also present.
This is a first example of statistics in a very simple case:
2×2 structure, horizontal bonds all equal, vertical bonds all equal.
For each bonds structure
the corresponding vectorial one-step dynamics is computed:
which means that only two spins are updated.
This is again on a 2×2 structure,
but considers every possible bonds scheme.
Output has two parts.
First one associates each bond structure,
identified with a two-digit hexadecimal,
to the sequence of number of excited bonds per given configuration,
written as a 16-bit array.
Second part associates each possible sequence of number of excited bonds
with the list of all the bonds structure determining it.
This is a revamping of the previous experiment,
with a different output format,
and more information provided.
This time, the output is produced as tables.
The first row of each table is a sequence of energies per state,
as in the previous experiment.
Each next row is indexed by a bonds structure,
and contains the sequence of next states.
As the reader can check,
each bond structure in the table of a given energy sequence
induces the same dynamics.
This experiment considers a 4×4 spins scheme.
For all the bonds layouts in a given list,
it creates a dictionary
where the keys are the sequences of energies
and the values are pairs made of
a bonds layout determining that energy sequence,
plus the list of next states.
Then, for each entry in the dictionary,
it searches whether there are two different lists of next states.
If the latter happens, then the conjecture is disproved.
Otherwise, the conjecture remains plausible.
The current version consider those bonds layouts
which are 2×2 periodic.
The running time on a Pentium D 3.4GHz desktop
with 1GB RAM and Ubuntu 10.04 64-bit,
is slightly more than one hour.
This is a variant of the fourth experiment, where the spin structure is antiperiodic rather than periodic. The 4×4 structure is again partitioned into 2×2 blocks, but blocks on the same horizontal or vertical have corresponding spins opposite in nature.