Improving quality of tiles generated from Ordnance Survey Raster Data

The cp command now takes the option -c to create copy on write clones of files.

From the cp man file:

-c    copy files using clonefile(2) 

For example: cp -c my-file my-file-clone

Can anyone solve this question using law of conservation of energy?

Of course. Conservation of energy requires that the sum of potential and kinetic energy remain constant. The potential energy is the height of the center of mass relative some arbitrary point. Choosing the rod being horizontal as that arbitrary point, the potential energy as a function of $\theta$ is $P = \frac 1 2 mgl \cos \theta$. The kinetic energy is purely rotational from the perspective of a frame with origin at the pivot point: $T = \frac12 \mathrm{I}\, {\dot\theta}^2$ where $\mathrm I = \frac13 ml^2$ is the moment of inertia of a slender rod about one end. Given that the rod is not moving initially when $\theta = 0$, conservation of energy dictates that $$ \frac 1 2 mgl \cos \theta + \frac12 \frac13 ml^2\,{\dot\theta}^2 = \frac12 mgl $$ Solving for $\dot\theta$ yields $${\dot\theta}^2 = 3\frac g l (1-\cos\theta)$$ Differentiating with respect to time yields $$2{\dot\theta}{\ddot\theta} = 3 \frac g l \sin\theta \dot\theta$$ or $$\ddot \theta = \frac 3 2 \frac g l \sin\theta$$

In general, it is possible for an object to have every mono into it an iso, but not be initial, weakly initial, or quasi-initial. The pre-condition is too easily satisfied just by not having very many morphisms into a given object, so to give it some traction, I'll need an existence-of-morphisms condition.

Theorem: In a category with equalisers, if $I$ is an object such that any mono $A \rightarrowtail I$ is an iso, then $I$ is quasi-initial.

Proof: Let $f, g : I \rightrightarrows X$, and take $A \xrightarrow{e} I$ the equaliser of $f$ and $g$. An equaliser is a (regular) mono, so by hypothesis on $I$ it is an iso, but if the equaliser of two morphisms is an iso then they are equal. So $f = g$, so there is at most one morphism from $I$ to any other object.

In the particular case of categories of functor algebras and homomorphisms, the forgetful functor to the underlying category creates limits, so in particular it is enough for the underlying category to have equalisers (thanks Zhen Lin for reminding me of this).

I've not yet come up with any interesting word on when there is at least one morphism, so that $I$ really is initial instead of quasi-so.