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\input lecmono.cmm
\pageno=7
%%% ----------------- extra line for explanation --------------
\topinsert\vbox{\centerline{\vbox{\hrule\smallskip\hbox{\quad
Example for layout only:  text is
not intended to make scientific sense.\quad}\smallskip\hrule}}}\endinsert
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\titlea{1.}{Grand Unification Schemes:\newline
A Short Introduction}
\titlearunning{Grand Unification Schemes}
\motto{``To the inventor the products of his imagination appear
necessary and natural, such that he does not look at them as objects of
the mind, but rather as things belonging to reality, and thus he wishes
them to be seen like that.''}{A. Einstein\fonote{``Wer da n\"amlich
erfindet, dem erscheinen die Erzeugnisse seiner Phantasie so notwendig
und naturgegeben, da\ss{} er sie nicht f\"ur Gebilde des Denkens,
sondern f\"ur gegebene
Realit\"aten ansieht und angesehen wissen m\"ochte.''}    }
\motto{``I hate reality, but it is still the only place where I can get
a decent steak''}{Woody Allen}
There is no experimental evidence so far that could not be
accommodated within the standard model. There are, however, several
theoretical shortcomings which must be remedied by\dots
\titleb{1.1}{$SU$(5) -- GUT}
The group $SU$(5) -- 5 $\times$ 5 unimodular matrices of determimant
1 -- has $5^2 - 1 = 24$ free entries\dots
\lemma{1.1}{Let $z\in{\rm ex}_t(F)$ and $\varepsilon>0$. Then there
exists a simplex\dots}
Consider as an example the following theorem (Ames 1969).
 \theorem{}{The Strong Unconstrained  Convex Function\dots}
Several slightly different forms of such theorems can be found in the
standard textbooks (Hawking and Ellis 1973; Wald 1984). In general,
the theorems show that the causal Cauchy development of a 3-surface is
incomplete if \dots
 \corollary{1.2}{ For any fixed integer $n \ge1,$ there exists\dots}
\definition{1.3}{A convex set K is called circumscribed if\dots}
\example{1.4}{Let $K\subseteq \bbbr^n$ be a polytope defined
as the convex hull of a given finite set\dots }
\proof {Since ${\rm ex}_t(F)\subseteq F$, it is trivial that
${\rm hull}_t\bigl ({\rm ex}_t(F)\bigr )\subseteq {\rm
hull}_t(F)$\qed}
  \remark{1.5}{All the strict inequalities appearing in the strong
versions\dots}
Somewhat larger masses are possible in particular models. One can, in a
general, qualitative way, argue as follows\dots
\begdoublefig 2.8 cm
{\figure{1.1}{This is the first figure legend in this section.
 The width of this legend
is about the same as for the second figure}}
{\figure{1.2}{This is another figure}}
\enddoublefig
{\it [The space for the figure and its legend may appear where it is
necessary for the page layout and not where it is coded in the input
file.]}

These were taken during August 1959 -- December 1976
and at the National Solar Observatory
(Kitt Peak) during December 1976 -- July 1984. As the sun rotates, the
longitude of the central meridian decreases.
\begfig 5 cm
\figure{1.3}{This is the legend of a figure. The space for the figure is
5 cm}
\endfig
 From each daily magnetogram,
the magnetic fields around the central meridian have been used to cover
all latitudes within this particular longitude band. During the course
of one solar rotation, all longitudes get covered, resulting in a synoptic
map representing the observed longitudinal (line-of-sight) magnetic field
as a function of latitude and longitude.
\smallskip
\item{a.} Supernovae of Type I (SNI) are believed to occur when a white
dwarf, consisting mainly of carbon and oxygen, is accreting matter in a
close binary system, and is pushed by the mass transfer beyond the
Chandrasekahar limit of stability\dots
\item{b.} The ``double quasar" was discovered in 1979 as a pair of
quasars, 6 arcseconds apart, with identical redshifts and spectra. The
initial guess that these were two images of one quasar produced by the
gravitational lens effect\dots
\item{c.} The extragalactic distance scale still depends on the use of
standard candles and standard reference distances (such as the distances
of the Hyades star cluster)\dots
\smallskip
Now we start with
$$\varphi =-2\pi(\nu_{\rm C}t-n)\ ,\eqno (1.1)$$
where $n$ is an integer ensuring that $\varphi$ stays in the interval
$-\pi<\varphi \le \pi$, and where the zero point of the time scale $t$
has been chosen such that $\varphi =0$ when $t=0$.

\begpet During the evolution of a star, a point is usually
 reached, just before the core burning is finished, {\it when layers above
the core become thermally unstable}, and large scale mass motions\dots
\endpet

The orthonormality condition
$$\int Y_{l^{\prime}}^{m^{\prime}}(\vartheta,\varphi)Y_l^{m\ast}
({mm^{\prime}}) \eqno(1.2)$$
is used to solve for the coefficients; the
asterisk in superscript denotes complex conjugation.

Using the explicit expressions for the spherical harmonics and the definition
of the associated Legendre functions $P_l^m(x)$ in Appendix A.1, we
can rewrite (1.2) as
$$c_l^m(t)=f_l^m\int_{-1}^1 dx\int_{-\pi}^{\pi}
d\varphi \,B(x,\varphi,t){\rm e}^{-{\rm i}m\varphi}P_l^m(x)\ ,\eqno(1.3)$$
where $x=\cos\vartheta$, and the factor of proportionality is
$$f_l^m=(-1)^m\, \, \sqrt{{2l +1\over 4\pi}{(l -m)\, !\over (l +m)\,
 !}}\eqno(1.4)$$
for $m\ge 0$. We have two integrations to perform, one Fourier transform
over the longitude window, which is done numerically using a fast
Fourier transform, and one Legendre transform, for which we apply an
improved (in relation to Stenflo and Vogel 1986) numerical method appropriate
when the spatial resolution of the data is limited, as described in Appendix
A.2.


{\it [Now we will code two figures
next to one another, with a modified
arrangement; this is only to be used
if the amount of text in the legends of the
two figures is quite different.]}
\begfig 6.5 cm
\figure{1.4}{This is the first small figure legend}
\figure{1.5}{This is the extremely long legend of the second figure
and should therefore be typeset using  the full width of the type area. For
typographical and aesthetic reasons it would be unacceptable if these
legends were  set next to one another
in two columns, with a legend
of  2 lines for the first figure and a legend of 16 lines for the
second. Therefore  we suggest that  the legends for the
two figures should be placed one
 below the  other. It is not necessary to place
the figure numbers below or beside  the two figures because it should
be clear
that the left one is the first figure and the right one the second   }
\endfig
\titleb {1.2}{Results for the Zonal Modes}
The power spectra $Pc_l^0(\nu)$ for the zonal modes ($m=0$), computed
with our apodized data set,
are displayed in Fig.\ts 1.1 As the modes of odd and even parity
behave\fonote{The spherical harmonics are given by
$$Y_l^m(\vartheta ,\varphi)=(-1)^m\, \, \sqrt{{2l +1\over 4\pi}
{(l -m)\, !\over (l +m)\, !}}\, \, \,
{\rm e}^{im\varphi}P_l^m(\cos\vartheta)$$
for $m\ge 0$. When $m$ is negative, a common definition\dots}
so differently, they are plotted separately. Odd parity (for odd values of
$l\, $) corresponds to modes that are anti-symmetric with respect to
reflections in the equatorial plane, even parity to symmetric modes.
The vectors $\vec {x, y, z}$ create a right-hand coordinate system.
\titlec {1.2.1}{Interpolation}
In practice the continuous capacity distribution function is replaced by
a discrete set of pairs ($w,x$) covering the full range of the
absorption\dots

To take into account that some of the particles may remain
close together while others will move away, we introduce the notion of
clusters.
\begtab 2.5 cm
\tabcap{1.1}{This Table will not be produced using \TeX, but will be
pasted in}
\endtab
\vfill\eject
\null\vskip0pt plus20fill
\noindent [{\it This page is left blank because the next section of }
\verb|\titlea| {\it must begin on a right-hand page (here}
\verb|\reference|{\it )}]
\begrefbook{References}
\ref Ames, W.F. (1969): {\it Numerical Methods for Partial Differential
Equations} (Barnes and Noble, New York)
\ref Arlinger, B.G. (1986): ``Computation of Supersonic Flow Including
Leading-Edge Vortex Flows using Marching Euler Techniques'', in Proc.
Int. Symp. Comp. Fluid Dynamics, ed. by K. Oshima (Japan Computational
Fluid Dynamics Society, Tokyo) Vol. 2, pp.~1--12
\ref Bailey, F.R. (1986): ``Overview of NASA's Numerical Aerodynamic
Simulation Program", in Proc. Int. Symp. Comp. Fluid Dynamics, ed.
by K. Oshima (Japan Computational Fluid Dynamics Society,
Tokyo) Vol. 1, pp. 21--32
\ref Belotserkovskii, O.M., Chushkin, P.I. (1965): In {\it Basic
Developments in Fluid Dynamics}, ed. by M. Holt (Academic, New York)
pp. 1-126
\ref Chester, C.R. (1971): {\it Techniques in Partial Differential
Equations} (McGraw-Hill, New York)
\ref Courant, R., Hilbert, D. (1962): {\it Methods of Mathematical
Physics, Vol II} (Intersience, New York)
\ref Gear, C.W. (1971): {\it Numerical Initial Value Problems in
Ordinary Differential Equations} (Pren\-tice-Hall, Englewood Cliffs, N.J.)
\ref Hamming, R.W. (1973): {\it Numerical Methods for Scientists and
Engineers}, 2nd ed. (McGraw-Hill, New York)
\ref Holst, T.L., Thomas, S.D., Kaynak, U., Grundy, K.L., Flores, J.,
Chaderjian, N.M. (1986): ``Computational Aspects of Zonal Algorithms for
Solving the Compressible Navier-Stokes Equation in Three Dimensions", in Proc.
Int. Symp. Comp. Fluid Dynamics, ed. by K. Oshima (Japan Computational
Fluid Dynamics Society, Tokyo) Vol. 1, pp.~113--122
\ref de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H.G., Jr. (1976):
{\it Second Reference Catalogue of Bright Galaxies}, (Univ. of Texas
Press, Austin)
\endref
\null\vskip0pt plus20fill
\noindent [{\it Please note that the author--date reference system has
been
used throughout this example; you could of course
use the number-only system instead.}]
%\verb|\reference|)]
\byebye