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In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured with high accuracy, while distances to these bodies are much more difficult to find and can be determined only indirectly.
Continue reading ‘Embedding: Reconstructing Solutions from a Delay Map’
and 
having the same underlying set 
but different families of open sets, a function may be continuous in one but discontinuous in the other.
[Image from Wikimedia Commons].
.
is the group of isometries of 
-dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane 
, we have the group 
, comprising all combinations of translations, rotations and reflections of the plane.
to diameter 
the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that 
is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post 
. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.
. More generally, for an N-gon the ratio is easily shown to be 
. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum 
is the angle through which the disk has rotated. The centre of the disk is at 
.
be the radius of the circle and 
the distance from the axis to the centre of the circle, with 
.
as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result
. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,
can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.
. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:
are the
, and ratios of these, the positive rational numbers 
. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.
, which include rationals, irrationals like 
and transcendental numbers like 
.
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