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\titlea{2}{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 shortcommings which must be remedied by\dots
%\titleb{2.1}{$SU$(5) -- GUT}
\titlec{2.1.1}{The Group Structure}
The group $SU$(5) -- 5 $\times$ 5 unimodular matrices of determimant
1 -- has $5^2 - 1 = 24$ free entries\dots
\lemma{2.1}{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 Elles 1973;
Meyers 1982, 1983).
 \theorem{}{The Strong Unconstrained  Convex Function\dots}
Several slightly different forms of such theorems can be found in the
standard textbooks (Hawking and Elles 1973; Wald 1984). In general
the theorems show that the causal Cauchy development of a 3-surface is
incomplete if
 \corollary{2.2}{ For any fixed integer $n \ge1,$ there exists\dots}
\definition{2.3}{A convex set K is called circumscribed if\dots}
\example{2.4}{Let $K\subseteq R^n$ 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{2.5}{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
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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.
\begfigside 1.5 cm 7 cm
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 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
gravitationallens 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
$$\varphi =-2\pi(\nu_{\rm C}t-n)\ ,\eqno (2.1)$$
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 Text in small print: In the evolution of a star usually a point
is reached, just before the core burning is finished, {\it where layers above
the core become thermally unstable}, and large scale mass motions\dots
\endpet
 
The orthonormality condition
$$\int Y_{l^{\prime}}^{m^{\prime}}(\vartheta,\varphi)Y_l^{m\ast}
({mm^{\prime}}) \eqno(2.2)$$
is used to solve for the coefficients:
\framedformula{c_l^m(t)=\int B(\vartheta,\varphi,t)
Y_l^{m\ast}
(\vartheta,\varphi) d\Omega\ .}{}
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 (2.2) as
$$c_l^m(t)=f_l^m\int_{-1}^1 dx\int_{-\pi}^{\pi}
d\varphi \,B(x,\varphi,t){\rm e}^{-{\rm i}m\varphi}P_l^m(x)\ ,\eqno(2.3)$$
where $x=\cos\vartheta$, and the factor of proportionality is
$$f_l^m=(-1)^m\, \, \sqrt{{2l +1\over 4\pi}{(l -m)\, !\over (l +m)\,
 !}}\eqno(2.4)$$
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.
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%\titleb {2.2}{Results for the Zonal Modes}
The power spectra $Pc_l^0(\nu)$ for the zonal modes ($m=0$), computed
with our apodized data set 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. 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-handed coordinate system.
\titlec {2.2.1}{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
\frame{Using (2.1) and (2.4) to eliminate the longitude, the expression
(2.4) for the harmonic coefficients becomes
$$c_l^m(t)=2\pi\nu_{\rm C}\int^\infty_{-\infty}D^m_l(t^{\prime})
w_{\rm C}(t^{\prime}-t)e^{i2\pi m\nu_{\rm C}
t^{\prime}}dt^{\prime}\ ,\eqno(2.5)$$
where we have introduced the notation $w_{\rm C}$ for the ``Carrington
window''}
$$w_{\rm C}(t)=1\, \, \, \, {\rm for} \,\, -{\textstyle{1\over 2}}P_{\rm
C}\le t\le
{\textstyle{1\over 2}}P_{\rm C}\ ,\eqno(2.6)$$ zero otherwise.
coeffifcient, and intermediate values are obtained by interpolation on
this set.
 
To take into account that some of the particles may remain
close together while others will move away, we introduce the notion of
clusters.
\titlea{3}{Gauge Theories and the Standard Model}
%\titleb{3.1}{Introduction -- The Concept of Gauge Invariance}
The formulation of General Relativity by Einstein, a theory which gave a
new an deep geometrical insight into the force of gravitation, was
followed by attempts to find a unified geometrical description of
electromagnetism and gravitation. In General Relativity the
gravitational field is made into a geometrical object by the
introduction of an affine connection, which defines the parallel
transport of vectors, and therefore determines the relative orientation
of local frames in space-time.
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\app{}
%\titleb{A.1}{The Gauge-Invariant Theory of Perturbation}
The discussion here follows [Kodama and Sasaki 1984].
\titlec{A.1.1}{Perturbation of $g_{\mu\nu}$ and $T_{\mu\nu}$}
The background space-time is described by the Robertson-Walker metric
(cf.\ Chap.\ 1)
$$ds^2=-dt^2+R^2(t)d\sigma^2$$
where $d\sigma^2=h_{ij}dx^idx^j$.
 
Scalar quantities are expanded in terms of scalar harmonic functions
$Y(x)$ which satisfy the equation
\framedformula{(\Delta+k^2)Y(x)=0\quad.}{({\rm A}.1)}
Here are two examples of references. The first is numerically ordered
the second is alphabetically ordered.
\begrefbook{References}
\nextchapter{Chapter 1}
\refno {1.1} S. Coleman, F. de Luccia: Phys. Rev.~D {\bf 21}
3305 (1980)
\refno {1.2} 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 {1.3} 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).
\nextchapter{Chapter 2}
\refno {2.1} S. Coleman, J. Mandula: Phys. Rev. {\bf 159} 1251 (1967)
\refno {2.2} R. Haag, J.T. {\L}opuszanski, M. Sohnius:
Nucl. Phys. B {\bf 88} 257 (1975)
\refno {2.3} G. 't Hooft: Phys. Rev. Lett. {\bf 37} 8 (1976).
Phys. Rev. D {\bf 14} 3432 (1976)
\nextchapter{Chapter 3}
\refno {3.1} R.D. Peccei, H.R. Quinn: Phys. Rev.
Lett. {\bf 38} 1440 (1977)
\refno {3.2} 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
\begrefbook{Literature}
\nextchapter{Chapter 1}
\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 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
\nextchapter{Chapter 2}
\ref Ames, W.F. (1969): {\it Numerical Methods for Partial Differential
Equations} (Barnes and Noble, New York)
\ref Belotserkovskii, O.M., Chushkin. P.I. (1965): In {\it Basic
Developements 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)
\nextchapter{Appendix 1}
\ref Gear, C.W. (1971): {\it Numerical Initial Value Problems in
Ordinary Differential Equations} (Prentice-Hall, Englewood Cliffs, N.J.)
\ref Hamming, R.W. (1973): {\it Numerical Methods for Scientists and
Engineers}, 2nd ed. (McGraw-Hill, New York)
\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
\byebye