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\begin{document} 
%
\title*{Searching for Supersymmetric Dark Matter-
\protect\newline The Directional Rate for Caustic Rings.
}
\titlerunning{ Searching for Supersymmetric Dark Matter}
% allows abbreviation of title, if the full title is too long
% to fit in the running head
%
%
\author{J. D. Vergados} 
%
\authorrunning{J. D. Vergados} 
%
\institute{ Theoretical Physics Division, University of Ioannina, 
GR-45110, 
Greece\\E-mail:Vergados@cc.uoi.gr} 
\maketitle              % typesets the title of the contribution
\begin{abstract} 
The detection of the theoretically expected dark matter
\index{Dark Matter Detection} 
is central to particle physics and cosmology. Current fashionable supersymmetric
models provide a natural dark matter candidate which is the lightest
supersymmetric particle (LSP). 
The theoretically obtained event rates
are usually  very low or even undetectable.
 So the experimentalists would like to exploit special signatures
like the directional rates and the modulation effect. 
In the
present paper we study these signatures focusing on a specific class of
 non-isothermal models
involving flows of caustic rings.
\end{abstract}

\section{Introduction}
In recent years the consideration of exotic dark matter has become necessary
in order to close the Universe  \cite{Jungm}.  Recent data from the 
 High-z Supernova Search Team \cite{HSST} and the
Supernova Cosmology Project  ~\cite {SPF}
$^,$~\cite {SCP} suggest the presence of a cosmological connstanta
 $\Lambda$. In fact 
 the situation can be adequately described by 
 a barionic component $\Omega_B=0.1$ along with 
%the exotic components 
$\Omega _{CDM}= 0.3$ and $\Omega _{\Lambda}= 0.6$ (see also
Turner, these proceedings). 

 Since this particle is expected to be very massive, $m_{\chi} \geq 30 GeV$, and
extremely non relativistic with average kinetic energy $T \leq 100 KeV$,
it can be directly detected ~\cite{JDV}$^-$\cite{KVprd} mainly via the
recoiling nucleus.

 Using an effective supersymmetric Lagrangian at the  
quark level, see e.g. Jungman et al 
~\cite{Jungm} and references therein , a quark model for the nucleon
 ~\cite{Dree,Chen}
and nuclear  wave functions ~\cite{KVprd} one can obtain the needed
detection rates.  They are typically very low. So experimentally one 
would like to exploit the modulation of the event rates due to the 
earth's revolution around the sun. In our previous work
\cite{JDV99}$^,$ \cite{JDV99b} we found
enhanced modulation, if one uses appropriate asymmetric velocity
distribution.  
 The isolated galaxies are, however, surrounded by cold dark matter
, which,  due to gravity, keeps falling continuously on 
them from all directions \cite{SIKIVIE}.
It is the purpose of our present paper to exploit the results of
such a scenario.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Basic Ingredients for LSP Nucleus Scattering}

The differential cross section can be cast in the form 
\cite{JDV99b}:
\begin{equation}
d\sigma (u,\upsilon)= \frac{du}{2 (\mu _r b\upsilon )^2} [(\bar{\Sigma} _{S} 
                   +\bar{\Sigma} _{V}~ \frac{\upsilon^2}{c^2})~F^2(u)
%\frac{u}{\rangle~u~\rangle})\frac{\upsilon^2}{c^2}\frac{u}{\rangle~u~\rangle}) 
                       +\bar{\Sigma} _{spin} F_{11}(u)]
\label{2.9}
\end{equation}
\begin{equation}
\bar{\Sigma} _{S} = \sigma_0 (\frac{\mu_r}{m_N})^2  \,
 \{ A^2 \, [ (f^0_S - f^1_S \frac{A-2 Z}{A})^2 \, ] \simeq \sigma^S_{p,\chi^0}
        A^2 (\frac{\mu_r}{m_N})^2 
\label{2.10}
\end{equation}
\begin{equation}
\bar{\Sigma} _{spin}  =  \sigma^{spin}_{p,\chi^0}~\zeta_{spin}
\label{2.10a}
\end{equation}
\begin{equation}
\zeta_{spin}= \frac{(\mu_r/m_N)^2}{3(1+\frac{f^0_A}{f^1_A})^2}
[(\frac{f^0_A}{f^1_A} \Omega_0(0))^2 \frac{F_{00}(u)}{F_{11}(u)}
  +  2\frac{f^0_A}{ f^1_A} \Omega_0(0) \Omega_1(0)
\frac{F_{01}(u)}{F_{11}(u)}+  \Omega_1(0))^2  \, ] 
\label{2.10b}
\end{equation}
\begin{equation}
\bar{\Sigma} _{V}  =  \sigma^V_{p,\chi^0}~\zeta_V 
\label{2.10c}
\end{equation}
\begin{equation}
\zeta_V = \frac{(\mu_r/m_N)^2} {(1+\frac{f^1_V}{f^0_V})^2} A^2 \, 
(1-\frac{f^1_V}{f^0_V}~\frac{A-2 Z}{A})^2 [ (\frac{\upsilon_0} {c})^2  
[ 1  -\frac{1}{(2 \mu _r b)^2} \frac{2\eta +1}{(1+\eta)^2} 
\frac{\langle~2u~ \rangle}{\langle~\upsilon ^2~\rangle}] 
\label{2.10d}
\end{equation}
\\
$\sigma^i_{p,\chi^0}=$ proton cross-section with $i=S,spin,V$ given by:\\
$\sigma^S_{p,\chi^0}= \sigma_0 ~(f^0_S)^2$   (scalar) , 
(the isovector scalar is negligible, i.e. $\sigma_p^S=\sigma_n^S)$\\
$\sigma^{spin}_{p,\chi^0}= \sigma_0~~3~(f^0_A+f^1_A)^2$
  (spin) ,
$\sigma^{V}_{p,\chi^0}= \sigma_0~(f^0_V+f^1_V)^2$  
(vector)   \\
where $m_p$ is the proton mass,
 $\eta = m_x/m_N A$, and
 $\mu_r$ is the reduced mass and  
\begin{equation}
\sigma_0 = \frac{1}{2\pi} (G_F m_N)^2 \simeq 0.77 \times 10^{-38}cm^2 
\label{2.7} 
\end{equation}
\begin{equation}
u = q^2b^2/2~~or~~
Q=Q_{0}u, \qquad Q_{0} = \frac{1}{A m_{N} b^2} 
\label{2.15} 
\end{equation}
where
b is (the harmonic oscillator) size parameter, 
q (Q) is the momentum (energy) transfer to the nucleus.
In the above expressions $F(u)$ is the nuclear form factor and
$F_{\rho \rho^{\prime}}(u)$
are the spin form factors \cite{KVprd} ($\rho , \rho^{'}$ are isospin indices)
The differential non-directional  rate can be written as
\begin{equation}
dR=dR_{non-dir} = \frac{\rho (0)}{m_{\chi}} \frac{m}{A m_N} 
d\sigma (u,\upsilon) | {\boldmath \upsilon}|
\label{2.18}  
\end{equation}
 where
 $\rho (0) = 0.3 GeV/cm^3$ is the LSP density in our vicinity and 
 m is the detector mass 

 The directional differential rate \cite{Copi99} in the direction
$\hat{e}$ is 
given by :
\begin{equation}
dR_{dir} = \frac{\rho (0)}{m_{\chi}} \frac{m}{A m_N} 
{\boldmath \upsilon}.\hat{e} H({\boldmath \upsilon}.\hat{e})
 ~\frac{1}{2 \pi}~  
d\sigma (u,\upsilon)
\label{2.20}  
\end{equation}
where H the Heaviside step function. The factor of $1/2 \pi$ is 
introduced, since we have chosen to normalize our results to the
usual differential rate.
 
 We will now  examine the consequences of the earth's
revolution around the sun (the effect of its rotation around its axis is
expected to be negligible) i.e. the modulation effect. 

 Following Sikivie we will consider $2 \times N$ caustic rings, (i,n)
, i=(+.-) and n=1,2,...N (N=20 in the model of Sikivie et al),
each of which
contributes to the local density a fraction $\bar{\rho}_n$ of the
the summed density $\bar{\rho}$ of each type $i=+,-$. and has
velocity ${\bf y}_n=(y_{nx},y_{ny},y_{nz})$
, in units of $\upsilon_0=220~Km/s$, with respect to the
galactic center.

We find it convenient to choose the z-axis 
in the direction of the motion of the
the sun, the y-axis is normal to the plane of the galaxy and 
the x-axis is in the radial direction. 
The needed quantities are taken from the 
work of Sikivie (table 1 of last Ref. \cite{SIKIVIE}) by the 
definitions
$y_n=\upsilon_n/\upsilon_0
,y_{nz}=\upsilon_{n\phi}/\upsilon_0
,y_{nx}=\upsilon_{nr}/\upsilon_0
,y_{ny}=\upsilon_{nz}/\upsilon_0$
. This leads to a
velocity distribution of the form:
\begin{equation}
f(\upsilon^{\prime}) = \sum_{n=1}^N~\delta({\bf \upsilon} ^{'}
    -\upsilon_0~{\bf y}_n)
\label{3.1} 
\end{equation}
The velocity of the earth around the
sun is given by \cite{KVprd}. 
\begin{equation}
{\bf \upsilon}_E \, = \, {\bf \upsilon}_0 \, + \, {\bf \upsilon}_1 \, 
= \, {\bf \upsilon}_0 + \upsilon_1(\, sin{\alpha} \, {\bf \hat x}
-cos {\alpha} \, cos{\gamma} \, {\bf \hat y}
+cos {\alpha} \, sin{\gamma} \, {\bf \hat z} \,)
\label{3.6}  
\end{equation}
where $\alpha$ is the phase of the earth's orbital motion, $\alpha =0$
around second of June. In the laboratory frame we have
\cite{JDV99b}
$ {\bf \upsilon}={\bf \upsilon}^{'} \, - \, {\bf \upsilon}_E \,$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Event Rates}


Integrating Eq. (\ref {2.18}) we obtain for the total
non-directional rate
\begin{equation}
R =  \bar{R}\, t \, \frac{2 \bar{\rho}}{\rho(0)}
          [1 - h(a,Q_{min})cos{\alpha}] 
\label{3.55}  
\end{equation}
The integration was performed from$u=m_{min}$ to $u=u_{max}$, where
\begin{eqnarray}
u_{min}= \frac{Q_{min}}{Q_0},
u_{max}=min(\frac{y^2_{esc}}{a^2},max(\frac{y_{n} ^2}{a^2})~,~ n=1,2,...,N)
\label{3.30c}  
\end{eqnarray}
Here $y_{esc}=\frac{\upsilon_{esc}}{\upsilon_0}$, with 
$\upsilon_{escape}=625 Km/s$
is the escape velocity from the galaxy.
$Q_{min}$ is the energy transfer cutoff imposed by the detector
and $a =[\sqrt{2} \mu _rb\upsilon _0]^{-1}.  $ Also
$\rho_{n}=d_n/\bar{\rho}
,\bar{\rho}=\sum_{n=1}^N~d_n$ (for each flow +,-).
In the Sikivie model
\cite {SIKIVIE} $2\bar{\rho}/\rho(0)=1.25$. 
$\bar{R}$ is obtained  
\cite {JDV} by neglecting the folding with the LSP velocity and the
momentum transfer dependence, i.e. by
\begin{eqnarray}
\bar{R}& =&\frac{\rho (0)}{m_{\chi}} \frac{m}{Am_N} \sqrt{\langle
v^2\rangle } [\bar{\Sigma}_{S}+ \bar{\Sigma} _{spin} + 
\frac{\langle \upsilon ^2 \rangle}{c^2} \bar{\Sigma} _{V}]
\label{3.39b}  
\end{eqnarray}
and it contains all SUSY parameters except $m_{\chi}$
 The modulation is described in terms of the parameter $h$. 
 The effect of folding
with LSP velocity and the nuclear form factor is taken into account by 
$t$ (see table 1) 
%%%%%%%%%%%%    TABLE 1  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}
\caption{The quantities $t$ and $h$ entering the total non-directional
rate in the case of the
target $_{53}I^{127}$ for various LSP masses and $Q_{min}$ in KeV. 
Also shown are the quantities $r^i_j,h^i_j$
 $i=u,d$ and $j=x,y,z,c,s$, entering the directional rate for no energy
cutoff. For definitions see text. 
}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{5pt}
\begin{tabular}{|c|c|rrrrrrr|}
\hline
& & & & & & & &     \\
&  & \multicolumn{7}{|c|}{LSP \hspace {.2cm} mass \hspace {.2cm} in GeV}  \\ 
\hline 
& & & & & & & &     \\
Quantity &  $Q_{min}$  & 10  & 30  & 50  & 80 & 100 & 125 & 250   \\
\hline 
& & & & & & & &     \\
t      &0.0&1.451& 1.072& 0.751& 0.477& 0.379& 0.303& 0.173\\
h      &0.0&0.022& 0.023& 0.024& 0.025& 0.026& 0.026& 0.026\\
$r^u_z$&0.0&0.726& 0.737& 0.747& 0.757& 0.760& 0.761& 0.761\\
$r^u_y$&0.0&0.246& 0.231& 0.219& 0.211& 0.209& 0.208& 0.208\\
$r^u_x$&0.0&0.335& 0.351& 0.366& 0.377& 0.380& 0.381& 0.381\\
$h^u_z$&0.0&0.026& 0.027& 0.028& 0.029& 0.029& 0.030& 0.030\\
$h^u_y$&0.0&0.021& 0.021& 0.020& 0.020& 0.019& 0.019& 0.019\\
$h^u_x$&0.0&0.041& 0.044& 0.046& 0.048& 0.048& 0.049& 0.049\\
$h^u_c$&0.0&0.036& 0.038& 0.040& 0.041& 0.042& 0.042& 0.042\\
$h^u_s$&0.0&0.036& 0.024& 0.024& 0.023& 0.023& 0.022& 0.022\\
$r^d_z$&0.0&0.274& 0.263& 0.253& 0.243& 0.240& 0.239& 0.239\\
$r^d_y$&0.0&0.019& 0.011& 0.008& 0.007& 0.007& 0.007& 0.007\\
$r^d_x$&0.0&0.245& 0.243& 0.236& 0.227& 0.225& 0.223& 0.223\\
$h^d_z$&0.0&0.004& 0.004& 0.004& 0.004& 0.004& 0.004& 0.004\\
$h^d_y$&0.0&0.001& 0.000& 0.000& 0.000& 0.000& 0.000& 0.000\\
$h^d_x$&0.0&0.022& 0.021& 0.021& 0.020& 0.020& 0.020& 0.020\\
$h^d_c$&0.0&0.019& 0.018& 0.018& 0.017& 0.017& 0.017& 0.017\\
$h^d_s$&0.0&0.001& 0.001& 0.000& 0.000& 0.000& 0.000& 0.000\\
\hline 
& & & & & & & &     \\
t    &10.0&0.000& 0.226& 0.356& 0.265& 0.224& 0.172& 0.098\\
h    &10.0&0.000& 0.013& 0.023& 0.025& 0.025& 0.026& 0.026\\
\hline 
& & & & & & & &     \\
t    &20.0&0.000& 0.013& 0.126& 0.139& 0.116& 0.095& 0.054\\
h    &20.0&0.000& 0.005& 0.017& 0.024& 0.025& 0.026& 0.026\\
\hline
\end{tabular}
\end{center}
\label{apptab1.1b}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 There are now experiments under way aiming at measuring directional rates
, i.e. the case in which the nucleus is observed in a certain direction.
The rate will depend on the direction of observation, showing a strong
correlation with the direction of both the sun's and the earth's motion. In 
the  favorable 
situation the rate will merely be suppressed by about a factor of $2 \pi$
relative to the non-directional rate. This is due to the fact that one 
does not now integrate over the
azimuthal angle of the nuclear recoiling momentum. 

 We need  distinguish the following cases: a) $\hat{e}$ has a
component in the sun's direction of
motion, i.e. $0<\theta < \pi /2$, labeled by i=u (up). b) Detection
in the opposite direction, $\pi /2 <\theta < \pi $, labeled by 
i=d (down).  
Thus we find :

1. In the first quadrant (azimuthal angle $0 \leq \phi \leq \pi/2)$.
\begin{eqnarray}
R^i_{dir} & = &\bar{R} \frac{2 \bar{\rho}}{\rho(0)}
    \frac{t}{2 \pi} [(r^i_z  - \cos \alpha~ h^i_1) {\bf e}_z.{\bf e}  
\nonumber \\ 
&+& (r^i_y +cos \alpha h^i_2 +\frac{h^i_c }{2}(|cos\alpha|+cos\alpha))
     |{\bf e}_y.{\bf e} | 
\nonumber \\ 
&+& (r^i_x -sin \alpha h^i_3 +\frac{h^i_s }{2}(|sin\alpha|-sin\alpha))
     |{\bf e}_x.{\bf e} | ]
\label{3.56}  
\end{eqnarray}
2. In the second quadrant (azimuthal angle $\pi/2 \leq \phi \leq \pi)$
\begin{eqnarray}
R^i_{dir} & = &\bar{R} \frac{2 \bar{\rho}}{\rho(0)}
    \frac{t}{2 \pi}  [(r^i_z  - \cos \alpha~ h^i_1) {\bf e}_z.{\bf e}  
\nonumber \\ 
&+& (r^i_y +cos \alpha h^i_2 (u)+\frac{h^i_c }{2}(|cos\alpha|-cos\alpha))
     |{\bf e}_y.{\bf e} | 
\nonumber \\ 
&+& (r^i_x +sin \alpha h^i_3 +\frac{h^i_s }{2}(|sin\alpha|+sin\alpha))
     |{\bf e}_x.{\bf e} | ]
\label{3.57}  
\end{eqnarray}
3. In the third quadrant (azimuthal angle $\pi \leq \phi \leq 3 \pi/2)$.
\begin{eqnarray}
R^i_{dir} & = &\bar{R} \frac{2 \bar{\rho}}{\rho(0)}
    \frac{t}{2 \pi}  [(r^i_z  - \cos \alpha~ h^i_1) {\bf e}_z.{\bf e}  
\nonumber \\ 
&+& (r^i_y -cos \alpha h^i_2 (u)+\frac{h^i_c (u)}{2}(|cos\alpha|-cos\alpha))
     |{\bf e}_y.{\bf e} | 
\nonumber \\ 
&+& (r^i_x +sin \alpha H^i_3 +\frac{h^i_s }{2}(|sin\alpha|+sin\alpha))
     |{\bf e}_x.{\bf e} | ]
\label{3.58}  
\end{eqnarray}
4. In the fourth quadrant (azimuthal angle $3 \pi/2 \leq \phi \leq 2 \pi)$
\begin{eqnarray}
R^i_{dir} & = &\bar{R} \frac{2 \bar{\rho}}{\rho(0)}
    \frac{t}{2 \pi}  [(r^i_z  - \cos \alpha~ h^i_1) {\bf e}_z.{\bf e}  
\nonumber \\ 
&+& (r^i_y -cos \alpha h^i_2 +\frac{h^i_c }{2}(|cos\alpha|-cos\alpha))
     |{\bf e}_y.{\bf e} | 
\nonumber \\ 
&+& (r^i_x -sin \alpha h^i_3 +\frac{h^i_s }{2}(|sin\alpha|-sin\alpha))
     |{\bf e}_x.{\bf e} | ]
\label{3.59 }  
\end{eqnarray}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
 We have calculated the parameters describing
characteristic signatures needed 
to reduce the formidable backgrounds
in the direct detection of SUSY dark matter,
 such as : a) The modulation effect, 
correlating the rates with the motion of the Earth  and b)
 The directional  rates, correlated with both with the velocity of the sun
and that of the Earth (see table 1).

 We have focused on the LSP density and velocity spectrum 
obtained from a recently proposed non-isothermal model, involving caustic
rings 
\cite{SIKIVIE}. Our results for isothermal models have appeared
elsewhere 
\cite{JDV99,JDV99b}. 

 The quantities  $t$ and $h$ are given in table 1. 
We see that the maximum in this model does not  
occur around June 2nd, but about six months later. The difference
between the maximum and the minimum is about $4\%$, i.e. smaller
than that predicted by the asymmetric isothermal models 
\cite{JDV99,JDV99b}. 

 For the directional experiments we found that 
the biggest rates are obtained close to the direction of the sun's motion.
They are suppressed compared to the usual 
non-directional rates by the factor 
$f_{red}=\kappa/(2 \pi)$, $\kappa=u^i_z$. We find $\kappa \simeq 0.7$, 
$i=up$ (observation in the sun's direction of motion) while
$\kappa\simeq 0.3$, $i=down$ ( in the opposite direction).
The modulation is a bit larger than in the non-directional case. The
largest difference between the maximum and the minimum, 
 $8\%$, occurs  not the sun's direction of motion, but in the x-direction
 (galactocentric direction).

 In the case of the isothermal models the reduction factor 
along the sun's direction of motion
is now given $f_{red}=t_0/(4 \pi~t)=\kappa/(2 \pi)$. 
Using the values of $t_0$ obtained previously
\cite{JDV99b},
we find
that $\kappa$ is around 0.6 for the symmetric case  and around 0.7 for maximum
asymmetry ($\lambda=1.0$). 
The modulation of the directional rate depends on the direction of 
observation. It is generally larger and 
increases with the asymmetry parameter $\lambda$. 
 For $Q_{min}=0$ it can reach values up 
to $23\%$. Values up to $35\%$ are possible for large
 $Q_{min}$, but at the expense of the total number of counts
\cite{JDV99b}.

 Finally in  all cases $t$
deviates from unity for large reduced mass.
Thus when extracting the LSP-nucleon cross section from
the data one must divide by $t$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\it Acknowledgments:} The author happily 
acknowledges partial support of this work by
TMR No ERB FMAX-CT96-0090 of the European Union. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
\begin{document}
%
\title*{Focusing of a Parallel Beam to Form\protect\newline a Point
in the Particle Deflection Plane}
%
%
\toctitle{Focusing of a Parallel Beam to Form a Point
\protect\newline in the Particle Deflection Plane}
% allows explicit linebreak for the table of content
%
%
\titlerunning{Focusing of a Parallel Beam}
% allows abbreviation of title, if the full title is too long
% to fit in the running head
%
\author{Ivar Ekeland\inst{1}
\and Roger Temam\inst{2}
\and Jeffrey Dean\inst{2}
\and David Grove\inst{1}
\and Craig Chambers\inst{2}
\and Kim~B.~Bruce\inst{2}
\and Elsa Bertino\inst{1}}
%
\authorrunning{Ivar Ekeland et al.}
% if there are more than two authors,
% please abbreviate author list for running head
%
%
\institute{Princeton University, Princeton NJ 08544, USA
\and Universit\'{e} de Paris-Sud,
     Laboratoire d'Analyse Num\'{e}rique,
     B\^{a}timent 425,\\
     F-91405 Orsay Cedex, France}

\maketitle              % typesets the title of the contribution

\begin{abstract}
The abstract\index{abstract} should summarize the contents of the paper
in at least 70 and at most 150 words; neither too long
nor too short but to the point!
\end{abstract}

\section{Text}
%
Please consult the file or printout of \emph{1readme}.$^*$ to find detailed
instructions on the layout and style of text elements.

\subsection{Fixed-Period Problems: The Sublinear Case}
%
Of course this text has no scientific relevance but should be used
as a practical example of the Springer layout specifications for
\emph{physics books}~\cite{fhjl1.1}.

We begin now the search for periodic solutions to Hamiltonian systems\index{Hamiltonian
systems}. All this will be done
in the convex case; that is, we shall study the
boundary-value\index{boundary-value problem} problem
\begin{eqnarray}
  \dot{x}(0)&=&JH' (t,x)\\
  x(0) &=& x(T)
\end{eqnarray}
with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
$\left\|x\right\| \to \infty$.


\section{Equations}
%
Please consult the file or printout of \emph{1readme}.$^*$ to find detailed
instructions on the layout and style of mathematical elements.

The following equation is the compiled result of using the command
\verb|\vec| and using the newly defined command \verb|\umu|:
\begin{equation}
\vec{A}=\mu y = 50\; \umu \mathrm{m}\; .
\end{equation}

\subsection{Sub-equations}
As you have seen above, equations are numbered automatically
when the equations are defined with \verb|\begin{equation}|
(but not when \verb|$$| is used). Here you will find an example
for the automatic sub-numbering of equation arrays, using the style \verb|subeqnar.sty|:
\begin{subeqnarray}
a & = & c + d \; , \label{f1.1a}\\
e & = & f - d \;  .\label{f1.1b}
\end{subeqnarray}

In order to refer to the equations within the main text you must simply label your
equations (within the equation's environment) and quote the labels within the text
environment. Upon running \TeX\ you should receive the corresponding
subnumbers, e.g.
(\ref{f1.1a}) and (\ref{f1.1b}).

\subsubsection*{No Sub-numbering.}
If you want to suppress the sub-numbering of an equation
array you may use the original \verb|\begin{subeqnarray}| environment, put
\verb|\nonumber|
at the end of the line that is to have no number, and set the sub-counter to zero:
\begin{subeqnarray}
a & = & c + d \; , \nonumber \\
e & = & f - d \; .
\setcounter{eqsubcnt}{0}
\end{subeqnarray}



\section{Figures}
\begin{figure}[b]
\begin{center}
\includegraphics[width=.3\textwidth]{figure.eps}
\end{center}
\caption[]{Example of an electronically included eps-figure}
\label{eps1.1}
\end{figure}

Please consult the file or printout of \emph{1readme}.$^*$ to find
detailed instructions on how to treat figures.

If your figures are available as electronic data it is advisable to
convert them to eps-format and include them directly into the text with
the help of the graphicx package (see the following example in
Fig.~\ref{eps1.1})


In order to mark the desired amount of space for a (centered!) figure
which has to be pasted into the manuscript manually please provide a
vertical line on the lefthand side of the figure. This
is reached by using the commands
\verb|\mpicplace{width in cm}{height in cm}|.


For further instructions
e.g. on the structure and layout of the caption see
the captions of Figs.~\ref{partfig1.1} and~\ref{labelfig1.1}.

\begin{figure}
\begin{center}
\leavevmode
\mpicplace{9 cm}{3 cm}
\end{center}
\caption{General description. (\textbf{a}) The `name' of the sub-figure should be set
boldface and in round brackets. (\textbf{b}) In general the last
sentence of a figure caption should not
end with a fullstop}
\label{partfig1.1}
\end{figure}

\begin{figure}
\begin{center}
\leavevmode
\mpicplace{5 cm}{3 cm}
\end{center}
\caption{This could show a figure consisting of different types of
lines to describe individual aspects. These descriptive lines ({\it
dotted line\,}) should be set italic and put into round brackets
as shown in this sample figure caption. In general the last
sentence of a figure caption ({\it straight line\,}) should not
end with a fullstop}
\label{labelfig1.1}
\end{figure}

\section{Tables}
If equations and figures are centered then tables should be centered as
well. Table captions should be treated in the same way as figure
legends, except that the
table captions appear {\it above} the tables. Overwide tables should
be reduced to the page width, if possible, or exceed the type area by a
maximum of 5\,mm. Please check the \TeX\ files of the following
tables
(Table~\ref{Tab1.1a} and ~\ref{Tab1.1b}) and use them as an example for
setting your own tables.

\begin{table}
\caption{Critical $N$ values}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{5pt}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
${\mathrm M}_\odot$ & $\beta_{0}$ & $T_{\mathrm c6}$ & $\gamma$
  & $N_{\mathrm{crit}}^{\mathrm L}$
  & $N_{\mathrm{crit}}^{\mathrm{Te}}$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\
 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\
120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\
\hline
\end{tabular}
\end{center}
\label{Tab1.1a}
\end{table}


\begin{table}
\caption{Please write your table caption here. Multi-line captions as
well as single-line captions automatically will be set flushleft}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{15pt}
\begin{tabular}{@{}llp{1.8cm}l}
\hline\noalign{\smallskip}
Nominal dimension & Angle & Tolerance & Evaluation factor \\
(mm) & & (\umu {m}) & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$10^1$ & $1^\circ$ & Untoleranced dimension& $c_1 =0$ \\
$10^{-1}$ & $1^{\prime\, {\mathrm a}}$ & 101--200 & $c_1 = 1$ \\
$10^{-2}$ & $1^{\prime\prime\, {\mathrm b}}$ & 40--100 & $c_1 = 2$ \\
$10^{-3}$ & & $< 50 $ & $c_1 = 3$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
$^{\mathrm a}$ One minute of arc. \\
$^{\mathrm b}$ One second of arc.
\label{Tab1.1b}
\end{table}


\section{Lists}
We have redefined the {\it itemize} environment (labelitemi) so that you
will
receive bullets instead of dashes to introduce the individual items.
We think that this way the list
\begin{itemize}
\item
is clearer
\item
looks better
\item
is more noticeable
\end{itemize}

\appendix

\section*{Appendix}
%
Of course you may use the standard \LaTeX\ command for your
appendix. But please be aware of the fact that we have modified so
that the numbering
of equations, figures and tables remains arabic, see examples below.
\begin{eqnarray}
a & = & c + d \; , \nonumber \\
e & = & f - d \; .
\setcounter{eqsubcnt}{0}
\end{eqnarray}

\begin{table}
\caption{Critical $N$ values}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{5pt}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
${\mathrm M}_\odot$ & $\beta_{0}$ & $T_{\mathrm c6}$ & $\gamma$
  & $N_{\mathrm{crit}}^{\mathrm{L}}$
  & $N_{\mathrm{crit}}^{\mathrm{Te}}$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\
 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\
120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\
\hline
\end{tabular}
\end{center}
\label{apptab1.1b}
\end{table}


\begin{thebibliography}{8.}
\addcontentsline{toc}{section}{References}

\bibitem{journ1.1} W. Frank, A. Seeger: Appl. Phys. A \textbf{3}, 66
(1988)

\bibitem{proc1.1} W. Greiner, D.N. Poenaru: `Cluster Preformation in
Closed- and Mid-shell Nuclei'. In:
\emph{Atomic and Nuclear Clusters, 2nd International Conference at Santorini, Greece,
June 28--July 2, 1993}, ed. by G.S. Anagnostatos, W. von Oertzen (Springer, Heidelberg 1994)
pp. 264--266

\bibitem{fhjl1.1} F. Holzwarth, J. Lenz  et al.: \emph{1readme.
Further Details on Layout and \LaTeX\, code}. (Springer, Berlin
Heidelberg 1999)

\bibitem{mono1.1} B. Jirgensons: \emph{Optical Activity for Proteins
and Other Macro-Molecules}, 2nd edn.
(Springer, New York 1984)

\bibitem{contr1.1} D.M. MacKay: `Visual Stability and Voluntary Eye
Movements'. In: \emph{Handbook of
Sensory Physiology VII/3}. ed. by R. Jung (Springer, Berlin, Heidelberg 1973) pp. 307--331

\bibitem{mono1.2} M. M\"uller, F.J. Becker: \emph{On Generalized
Hamiltonian Dynamics} (Cambridge University Press, Cambridge 1930)

\bibitem{journ1.2} S. Nakamura, M. Senoh, N. Iwasa, S. Nagahama: Jpn. J.
Appl. Phys. \textbf{34}, L797 (1995)
W. Frank, A. Seeger: Appl. Phys. A \textbf{3}, 66 (1988)

\bibitem{thesis1.1} D.W. Ross: Lysosomes and Storage Diseases. MA
Thesis,
Columbia University, New York
(1977)



\end{thebibliography}


%INDEX%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Please check with the editor of your book whether he plans to
% include a "mutual" subject index - if so, please code your entries
% in the standard syntax. For your own purposes you may print your
% "personal" index by using the following commands:
%
%\clearpage
%\addcontentsline{toc}{section}{Index}
%\flushbottom
%\printindex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


