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\contribution{Grand Unification Schemes:\newline
A Short Introduction}
\author{Max Napf@1, Hans Muller@2}
\address{@1Institut f\"ur Festk\"orperforschung J\"ulich, D-5170
J\"ulich, Fed.\ts Rep. of Germany
@2Institut f\"ur Theoretische Physik, Unversit\"at des Saarlandes,
D-6600 Saarbr\"ucken, Fed.\ts Rep. of Germany }
\abstract{We present a method for constructing Fokker--Planck models,
for which all eigenfunctions \dots}
\titlea{1}{Text}
The group $SU$(5) -- 5 $\times$ 5 unimodular matrices of determimant
1 -- has $5^2 - 1 = 24$ free entries\dots
\lemma{1}{Let $z\in{\rm ex}_t(F)$ and $\varepsilon>0$. Then there
exists a simplex\dots}
Consider as an example the following theorem (Hawking and Ellis 1973;
Meyers 1982, 1983).
 \theorem{1}{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{2}{ For any fixed integer $n \ge1,$ there exists\dots}
\definition{3}{A convex set K is called circumscribed if\dots}
\example{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{All the strict inequalities appearing in the
unconstrained \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}{This is the legend of a figure using half the
pagewidth. It appears below the figure}}
{\figure{2}{This is another figure}}
\enddoublefig
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.

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{2}{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
$$\varphi =-2\pi(\nu_{\rm C}t-n)\ ,\eqno (1)$$
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
$$\int Y_{l^{\prime}}^{m^{\prime}}(\vartheta,\varphi)Y_l^{m\ast}
({mm^{\prime}}) \eqno(2)$$
is used to solve for the coefficients; the
asterisk in upper case 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, we
can rewrite (2) as
$$c_l^m(t)=f_l^m\int_{-1}^1 dx\int_{-\pi}^{\pi}
d\varphi \,B(x,\varphi,t){\rm e}^{-{\rm i}m\varphi}P_l^m(x)\ ,\eqno(3)$$
where $x=\cos\vartheta$, and the factor of proportionality is
$$f_l^m=(-1)^m\, \, \sqrt{{2l +1\over 4\pi}{(l -m)\, !\over (l +m)\,
 !}}\eqno(4)$$
for $m\ge 0$. We have two integrations to perform in something, 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 B.


Now we will code two figures one beside the other with modified
arrangements; this is only if the amount of text in the legends of the
two figures is quite different.
\begfig 6.5 cm
\figure{3}{This is the first small figure legend}
\figure{4}{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
typeset in two columns, one beside the other, 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
 {\it below} the  other.  }
\endfig
\titleb {1.2}{Relaxation from Marginality\newline
in Optical Bistability}
The power spectra $Pc_l^0(\nu)$ for the zonal modes ($m=0$), computed
with our apodized data set using something,
are displayed in Fig.\ts 2.1. 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.
\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
\titleb {1.3}{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.

\begtab 3 cm
\tabcap{3}{This Table will not be produced using \TeX,  but will be pasted in}
\endtab

\titleb{1.4}{New Particles from Supersymmetry or Supergravity}
Let me start with a short answer to the questions:
What is super supersymmetry (SUSY) and what is supergravity (SUGRA)?
SUSY is a (global) symmetry that puts bosons and fermions (i.e. particles
with different spin) into the same multiplet.
SUGRA is gauged SUSY, i.e. SUSY with space-time-dependent parameters.
Equivalently one might say that SUGRA is SUSY in the presence of gravitons
or SUSY on a curved space-time manifold because each one of these three
definitions implies the other ones.

\titlec{1.4.1}{Supersymmetry.}
In the past there have been many attempts to find symmetries which unify
particles with different spins in the sense that they occur in the same
multiplet.
Although this can be achieved in a non-relativistic theory all such attempts
turned out to be incompatible with relativistic QFT.
Later on {\it O'Raifeartaigh} et al. $[4]$ proved ``no-go'' theorems,
which state that the only possible generators of symmetries in a relativistic
local QFT are the generators of the Poincar\'e group plus scalar charges,
and thus all particles in a multiplet must have the same mass {\it  and spin}.
These theorems always (implicitly) assume that the
generators (charges) of the symmetry are bosonic and
therefore transform bosons into bosons and fermions into
fermions.
They do not apply to symmetries such as SUSY with fermionic
generators transforming bosons into fermions and vice versa.
Nevertheless, these theorems impose restrictions on the structure of SUSY
algebras because they do apply to the bosonic part of such algebras.


\titlea{2}{Gauge Theories and the Standard Model}
\titleb{2.1}{Introduction -- The Concept of Gauge Invariance}
The formulation of General Relativity by Einstein, a theory which gave a
new and deep geometrical insight into the force of gravitation, was
followed by attempts to find a unified geometrical description of
electromagnetism and gravitation.


The book consists of two parts: Part I, Basic Theory and Part II,
Applications to Fusion Plasmas. Part I is organized as follows.  After an
introductory description of plasma properties and charged particle motion
in Chaps.\ts ,\ts 3, a concise presentation of the basic formulation of
plasma theory is given in Chap.\ts 4.  In Chap.\ts 5, the two fluid model is
used to describe the fundamental collective responses of plasma.
Chapter\ts 6 deals with the kinetic theory and Chap.\ts 7 with the general
theory of linear response in plasmas.  Examples of nonlinear response are
described in Chap.\ts 8 which contains somewhat advanced topics in nonlinear
plasma theory.


\bigskip\noindent
[{\it Examples of both reference systems follow; you should, of course,
use only one, having consulted the proceedings editor.}]

\begrefchapter{References}
\refno {1.} S. Coleman, F. de Luccia: Phys. Rev.~D {\bf 21}
3305 (1980)
\refno {2.} S.L. Adler, J.N. Bahcall, C.G. Callan,
M.E. Rosenbluth: Phys. Rev. Lett. {\bf 25} 1061 (1970);
S.L. Adler: Ann. Phys. (N.Y.) {\bf 67} 599 (1971)
\refno {3.} H. Euler, B. Kockel:
Naturwissenschaften {\bf 23} 246 (1935);\newline
W. Heisenberg, H. Euler: Z. Phys. {\bf 98} 714 (1936):\newline
V. Wei\ss kopf: Kgl. Danske Vidensk. Selsk.
Mat.-Fys. Medd. {\bf 14} No. 6 (1936).
\refno {4.} S. Coleman, J. Mandula: Phys. Rev. {\bf 159} 1251 (1967)
\refno {5.} R. Haag, J.T. {\L}opuszanski, M. Sohnius:
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\refno {6.} G. 't Hooft: Phys. Rev. Lett. {\bf 37} 8 (1976),
Phys. Rev. D {\bf 14} 3432 (1976)
\refno {7.} R.D. Peccei, H.R. Quinn: Phys. Rev.
Lett. {\bf 38} 1440 (1977)
\refno {8.} S. Weinberg: Phys. Rev. Lett. {\bf 40} 223 (1978);
F. Wilczek: Phys. Rev. Lett. {\bf 40} 279 (1978);
J.E. Kim: Phys. Rev. Lett. {\bf 43} 103 (1979)
\endref
\begrefchapter{References}
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\endref

\byebye