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theorems-ams-extended
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\begin_body

\begin_layout Chapter
Methods
\end_layout

\begin_layout Section
Algorithms
\end_layout

\begin_layout Standard
Lyx doesn't have a nice pseudocode editor for algorithms.
 To write an algorithm in pseudocode, insert a box for \SpecialChar TeX
 code and use a package
 such as algorithmic.
 Here's the conjugate gradients (
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "CG"
symbol "CG"
description "Conjugate gradients"

\end_inset

CG) algorithm
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Trefethen,Kelley,Shewchuk"

\end_inset


\begin_inset space ~
\end_inset

for solving 
\begin_inset Formula $Ax=b$
\end_inset

.
 
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{algorithmic}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{Given: matrix $A$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{Given: right-hand side $b$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{Given: initial guess $x_0$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{Given: stopping condition $
\backslash
tau$ and maximum iterations $N_{max}$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$r_0 
\backslash
gets b-Ax_0$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$p 
\backslash
gets r_0$}
\end_layout

\begin_layout Plain Layout


\backslash
FOR{$n=1 
\backslash
to N_{max}$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$z 
\backslash
gets A*p$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$v 
\backslash
gets 
\backslash
frac{(r_{n-1},r_{n-1})}{(p,z)}$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$x_{n} 
\backslash
gets x_{n-1}+v*p$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$r_n 
\backslash
gets r_{n-1}-v*z$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$
\backslash
mu 
\backslash
gets 
\backslash
frac{(r_n,r_n)}{(r_{n-1},r_{n-1})}$}
\end_layout

\begin_layout Plain Layout


\backslash
STATE{$p 
\backslash
gets r_n +
\backslash
mu*p$}
\end_layout

\begin_layout Plain Layout


\backslash
IF{converged$(r_n,A,b,
\backslash
tau)$} 
\end_layout

\begin_layout Plain Layout


\backslash
RETURN{$x_n$}
\end_layout

\begin_layout Plain Layout


\backslash
ENDIF
\end_layout

\begin_layout Plain Layout


\backslash
ENDFOR
\end_layout

\begin_layout Plain Layout


\backslash
STATE{Failure: maximum iterations exceeded}
\end_layout

\begin_layout Plain Layout


\backslash
end{algorithmic}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
The method of Conjugate Gradients
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Some good scientific prose
\end_layout

\begin_layout Standard
Authors of the highest eminence seem to be fully satisfied with the view
 that each species has been independently created.
 To my mind it accords better with what we know of the laws impressed on
 matter by the Creator, that the production and extinction of the past and
 present inhabitants of the world should have been due to secondary causes,
 like those determining the birth and death of the individual.
 When I view all beings not as special creations, but as the lineal descendants
 of some few beings which lived long before the first bed of the Silurian
 system was deposited, they seem to me to become ennobled.
 Judging from the past, we may safely infer that not one living species
 will transmit its unaltered likeness to a distant futurity.
 And of the species now living very few will transmit progeny of any kind
 to a far distant futurity; for the manner in which all organic beings are
 grouped, shows that the greater number of species of each genus, and all
 the species of many genera, have left no descendants, but have become utterly
 extinct.
 We can so far take a prophetic glance into futurity as to foretel that
 it will be the common and widely-spread species, belonging to the larger
 and dominant groups, which will ultimately prevail and procreate new and
 dominant species.
 As all the living forms of life are the lineal descendants of those which
 lived long before the Silurian epoch, we may feel certain that the ordinary
 succession by generation has never once been broken, and that no cataclysm
 has desolated the whole world.
 Hence we may look with some confidence to a secure future of equally inapprecia
ble length.
 And as natural selection works solely by and for the good of each being,
 all corporeal and mental endowments will tend to progress towards perfection.
\end_layout

\begin_layout Standard
It is interesting to contemplate an entangled bank, clothed with many plants
 of many kinds, with birds singing on the bushes, with various insects flitting
 about, and with worms crawling through the damp earth, and to reflect that
 these elaborately constructed forms, so different from each other, and
 dependent on each other in so complex a manner, have all been produced
 by laws acting around us.
 These laws, taken in the largest sense, being Growth with Reproduction;
 inheritance which is almost implied by reproduction; Variability from the
 indirect and direct action of the external conditions of life, and from
 use and disuse; a Ratio of Increase so high as to lead to a Struggle for
 Life, and as a consequence to Natural Selection, entailing Divergence of
 Character and the Extinction of less-improved forms.
 Thus, from the war of nature, from famine and death, the most exalted object
 which we are capable of conceiving, namely, the production of the higher
 animals, directly follows.
 There is grandeur in this view of life, with its several powers, having
 been originally breathed into a few forms or into one; and that, whilst
 this planet has gone cycling on according to the fixed law of gravity,
 from so simple a beginning endless forms most beautiful and most wonderful
 have been, and are being, evolved.
\end_layout

\begin_layout Subsection
Some pretty pictures
\end_layout

\begin_layout Standard
Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:MyFig"

\end_inset


\begin_inset space ~
\end_inset

shows the result of a PDE simulation.
 Its role in this document is to show how to create a labeled float that
 is referenced by label within the document.
 Notice that it's also added automatically to the list of figures in the
 front matter.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
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	filename nonlinFwd.jpg
	scale 60

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:MyFig"

\end_inset

If you can't convince them with logic, then use a color picture.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Yet more good scientific prose
\end_layout

\begin_layout Standard
It is known that Maxwell's electrodynamics—as usually understood at the
 present time—when applied to moving bodies, leads to asymmetries which
 do not appear to be inherent in the phenomena.
 Take, for example, the reciprocal electrodynamic action of a magnet and
 a conductor.
 The observable phenomenon here depends only on the relative motion of the
 conductor and the magnet, whereas the customary view draws a sharp distinction
 between the two cases in which either the one or the other of these bodies
 is in motion.
 For if the magnet is in motion and the conductor at rest, there arises
 in the neighbourhood of the magnet an electric field with a certain definite
 energy, producing a current at the places where parts of the conductor
 are situated.
 But if the magnet is stationary and the conductor in motion, no electric
 field arises in the neighbourhood of the magnet.
 In the conductor, however, we find an electromotive force, to which in
 itself there is no corresponding energy, but which gives rise—assuming
 equality of relative motion in the two cases discussed—to electric currents
 of the same path and intensity as those produced by the electric forces
 in the former case.
\end_layout

\begin_layout Standard
Examples of this sort, together with the unsuccessful attempts to discover
 any motion of the earth relatively to the “light medium,” suggest that
 the phenomena of electrodynamics as well as of mechanics possess no properties
 corresponding to the idea of absolute rest.
 They suggest rather that, as has already been shown to the first order
 of small quantities, the same laws of electrodynamics and optics will be
 valid for all frames of reference for which the equations of mechanics
 hold good.1 We will raise this conjecture (the purport of which will hereafter
 be called the “Principle of Relativity”) to the status of a postulate,
 and also introduce another postulate, which is only apparently irreconcilable
 with the former, namely, that light is always propagated in empty space
 with a definite velocity c which is independent of the state of motion
 of the emitting body.
 These two postulates suffice for the attainment of a simple and consistent
 theory of the electrodynamics of moving bodies based on Maxwell's theory
 for stationary bodies.
 The introduction of a “luminiferous ether” will prove to be superfluous
 inasmuch as the view here to be developed will not require an “absolutely
 stationary space” provided with special properties, nor assign a velocity-vecto
r to a point of the empty space in which electromagnetic processes take
 place.
\end_layout

\begin_layout Standard
The theory to be developed is based—like all electrodynamics—on the kinematics
 of the rigid body, since the assertions of any such theory have to do with
 the relationships between rigid bodies (systems of co-ordinates), clocks,
 and electromagnetic processes.
 Insufficient consideration of this circumstance lies at the root of the
 difficulties which the electrodynamics of moving bodies at present encounters.
\end_layout

\end_body
\end_document
