In this essay, I calculate the cost of a leaking
hot water tap. I link hot water loss to heat energy
loss to extra electricity consumption to resultant
monetary cost.
The DNA molecules making up the chromosomes of
eukaryotic cells are very long. So to fit inside
the nucleus of the cell, the molecules usually
undergo four levels of compact helical winding.
While minding that winding, let's have helix fun!
For a rooftop solar power installation, two
important questions are: 1. How many solar panels
are required? And 2. how many many batteries are
required? In this study, a physical model of
solar insolation is developed to help answer. In
the physical model, a novel concept dubbed the
Effective Global Horizontal Irradiation is
introduced to tractably account for the effect of
tilting the solar panels towards the sun.
Mathematics is beautiful. Its beauty comes from its
utility and from its surprise. And if there is a god (or
gods), then mathematics would surely be its language.
Both. Languages are invented, and mathematics is a
language. But in the physical reality in which we
are embedded, objects and relations between objects
seem to exist independently of us. And we use
mathematical symbols to describe our discovery of them.
A new form of functional interpolation is introduced,
dubbed plateau function interpolation.
As per usual, an approximating function is
synthesised as a linear combination of simple
well-known functions. But under the new interpolation
scheme introduced here, the simple functions need not
be taken from a single class of functions. For
example, trigonometrics, exponentials and rationals
may all be summoned at once.
This study offers a systematic approach for addressing
the difficulty in parametrising arbitrary paths on
surfaces embedded in three-dimensional space, ℝ3.
Specifically, attempts at finding global
parametrisations of paths on such surfaces will give
way to finding a connected set of locally-derived
parametrisations, all of which are regular and
tractable.
A most basic notion of reality is contemplated: the
notion of dimension.
I motivate for the existence of a reality much
richer than what we may easily intuit, but no less
real. I offer an epistemologically rigorous pathway
to help discover such possible metaphysical realities.
Rendering a three-dimensional landscape requires that
a view of the landscape be computed as a two-dimensional
projection of the landscape onto a viewing plane which
lies perpendicular to some line of sight
vector. I present a mathematical formalism for
specifying such a vector and for computing the
resultant projection.
I take a tour of rotational transformations in
three-dimensions, beginning with the well-known
formulae for a rotation in two dimensions. I then
generalise to three dimensions, deriving a useful
coordinate-free formula.
A systematic procedure is introduced for calculating
both contour and gradient paths on surfaces embedded
in ℝ3. For any such surface, both
paths admit coordinates which seem natural for the
surface. This is studied. Also, the commutivity of
contour path and gradient path traversal is
investigated.
Why are some soccer players able to move more
quickly than others? It turns out that for two
players of equal mass, the shorter player has
a mobility advantage over the taller player,
according to the square root of the ratio of their
respective heights.
A formal analysis of circular variables is given.
Sensible distance measures are derived both for
when a circular variable is continous and for
when it is discrete.
Confronted by a randomly selected group of people, one
would ask, What is the probability that at least one
other person in the group shares my birthday?
While this may be easy to ask, answering it is
less easy.
