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% Author Package file for use with AMS-LaTeX 1.2
\controldates{19-MAR-1998,19-MAR-1998,19-MAR-1998,19-MAR-1998}
 
\documentclass{era-l}

%\begin{document}
%\def
\newcommand\ds{\displaystyle }
\newtheorem{theorem}{Theorem}
%\bibliographystyle{plain} 

\begin{document}

\title[EIGENVALUE FORMULAS FOR THE UNIFORM TIMOSHENKO BEAM]
{Eigenvalue formulas for the uniform
Timoshenko beam: the free-free problem}
%\markboth{BRUCE GEIST AND JOYCE R. MCLAUGHLIN}{

\author[BRUCE GEIST AND JOYCE R. MCLAUGHLIN]{Bruce Geist}
\address{ Unisys Corporation, 41100 Plymouth Road, Plymouth, MI 48170 }
\email{Bruce.Geist@unisys.com}

\thanks{The work of both authors
was completed at Rensselaer Polytechnic Institute, and was partially
supported by funding from the Office of Naval Research, grant number
N00014-91J-1166. The work of the first author was also partially
supported by the Department of Education fellowship grant number
6-28069. The work of the second author was also partially supported by
the National Science Foundation, grant number DMS-9410700.}

%\addtocounter{footnote}{-1}

\author{Joyce R. McLaughlin}
\address{Department of Mathematical Sciences, Rensselaer Polytechnic Institute,
Troy, NY 12180}
\email{mclauj@rpi.edu}

%\maketitle

\keywords{Timoshenko beam, asymptotic distribution of
eigenvalues, boundary value problems}
\subjclass{Primary 34Lxx; Secondary 73Dxx}

\date{January 5, 1998}

\commby{Michael Taylor}

\issueinfo{4}{03}{}{1998}
\dateposted{March 20, 1998}
\pagespan{12}{17}
\PII{S 1079-6762(98)00041-9}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\begin{abstract}
This announcement presents asymptotic formulas for
the eigenvalues of a free-free uniform Timoshenko beam.
\end{abstract}

\maketitle



\section*{}Suppose a structural beam is driven by a laterally
oscillating
sinusoidal force. As the frequency of this applied force is varied, the
response varies. Experimental frequencies for which the response is
maximized are called natural frequencies of the beam. Our goal is to
address the question: if a beam's natural frequencies are known, what
can be inferred about its bending stiffnesses or its mass density? To
answer this question we need to know asymptotic formulas for the
frequencies. Here we establish these formulas for a uniform beam.


     
One widely used mathematical model for describing the transverse
vibration of beams was developed by Stephen Timoshenko in the 1920s (see
\cite{Timoshenko1}, \cite{Timoshenko2}). In this model, two coupled
partial differential equations arise,
\begin{displaymath} 
(EI \psi_x )_x + kAG(w_x - \psi) -\rho I \psi _{tt} = 0,
\end{displaymath}
%
\begin{displaymath} 
(kAG(w_x-\psi))_x - \rho A w_{tt} = P(x,t).
\end{displaymath}
The dependent variable $w = w(x, t)$ represents the lateral
displacement
at time $t$ of a cross-section located $x$ units from one end of the
beam. $\psi = \psi(x, t)$ is the cross-sectional rotation due to
bending. $E$ is Young's modulus, i.e., the modulus of elasticity in
tension and compression, and $G$ is the modulus of elasticity in shear.
The nonuniform distribution of shear stress over a cross-section
depends on cross-sectional shape. The coefficient $k$ is introduced to
account for this geometry dependent distribution of shearing stress. $I$
and $A$ represent cross-sectional inertia and area, $\rho$ is the mass
density of the beam per unit length, and $P(x, t)$ is an applied force.
If we suppose the beam is anchored so that the so-called ``free-free''
boundary conditions hold (i.e., shearing forces and moments are assumed
to be zero at each end of the beam), then $w$ and $\psi$ must satisfy
the following four boundary conditions:
\begin{equation} \label{tbc}
w_x - \psi |_{x=0,L} = 0, \qquad \psi_x |_{x=0,L} = 0.
\end{equation}

After making a standard separation of variables argument, one finds that
the Timoshenko differential equations for $w$ and $\psi$ lead to a
coupled system
of two second order ordinary differential equations for $y(x)$ and
$\Psi(x)$,
\begin{equation} \label{tde1}
(EI \Psi_x)_x + kAG(y_x - \Psi) + p^2 I \rho \Psi = 0,
\end{equation}
\begin{equation} \label{tde2}
(kAG(y_x - \Psi))_x + p^2 A \rho y = 0.
\end{equation}
%\noindent 
Here, $p^2$ is an eigenvalue parameter. The conditions on $w$
and $\psi$
in (\ref{tbc}) imply $y$ and $\Psi$ must satisfy the same free-free
boundary conditions: 
\begin{equation} \label{tbcsp}
y_x - \Psi|_{x=0,L}=0,~~~\Psi_x|_{x=0,L} = 0.
\end{equation}

This boundary value problem for $y$ and $\Psi$ is self-adjoint, which
implies
that the values of $p^2$ for which nontrivial solutions to this problem
exist, the eigenvalues for this model, are real. Furthermore, it is not
difficult to show that the collection of all eigenvalues for this
problem forms a discrete, countable, unbounded set of real nonnegative
numbers. Moreover, it can be shown that if $\sigma$ is a natural
frequency for
a beam, then $p^2 = (2 \pi \sigma)^2$ is one of the beam's eigenvalues.
Therefore, it
is possible to determine eigenvalues from natural frequency data
obtained in an experiment like the one indicated in the opening
paragraph. 

Suppose from vibration experiments we have determined a set of natural
frequen\-cies for a beam with unknown elastic moduli and mass density, and
have constructed a sequence of eigenvalues from this data. What
information can the eigenvalues provide about these unknown material
parameters? To address this question, we must determine how eigenvalues
depend on $E$, $I$, $kG$, $A$ and $\rho$. This determination is not
easy, since the dependence of eigenvalues on these coefficients is
highly nonlinear. The first step towards general eigenvalue formulas for
beams with nonconstant material and geometric parameters is to derive
eigenvalue formulas for the uniform Timoshenko beam. (A beam is uniform
when $E$, $kG$, $A$, $I$, and $\rho$ are constants.) In a forthcoming
paper (and in
Geist \cite{Geist}), the formulas for the uniform free-free beam given
below are
crucial in establishing asymptotic formulas for the eigenvalues of
free-ended beams which have variable density and constant material
parameters otherwise.

This announcement presents asymptotic formulas for the eigenvalues of
the free-ended, uniform Timoshenko beam. 
\section{A frequency equation}In this section a frequency equation is
derived. The notation follows that given in Huang \cite{Huang}. When
$E$, $kG$, $A$, $I$, and $\rho$ are constant, the boundary value problem
defined by (\ref{tde1}), (\ref{tde2}), and (\ref{tbcsp}) may be written
in the following simplified notation. Let $\xi = x/L$, $b^2 = \rho A L^4
p^2/ (EI)$, $r^2 = I/(AL^2)$, and $s^2 = EI/(kAGL^2)$. Interpreting a
prime $(')$ as differentiation with respect to $\xi$, equations
(\ref{tde1}) and (\ref{tde2}) can be proved equivalent to the
differential equations
\begin{equation} \label{5}
s^2 \Psi'' - (1-b^2 r^2 s^2) \Psi + y'/L = 0,
\end{equation}
%\noindent 
and
\begin{equation} \label{6}
y'' + b^2 s^2 y - L \Psi' = 0;
\end{equation}
%\noindent 
similarly, the four boundary conditions given in (\ref{tbcsp})
may be written as
\begin{equation} \label{7}
\Psi' |_{\xi=0,1} = 0, \ \  \left [ \frac{y'}{L} - \Psi  \right
]_{\xi=0,1} = 0.
\end{equation}

By eliminating $y$ or $\Psi$ from (\ref{5}) or (\ref{6}), we find that
these two second order equations imply that $y$ and $\Psi$ must satisfy
two
decoupled fourth order ordinary differential equations
\begin{equation} \label{8}
y^{(iv)} + b^2(r^2+s^2)y'' - b^2(1-b^2 r^2 s^2)y=0,
\end{equation}
\begin{equation} \label{9}
\Psi^{(iv)} + b^2(r^2+s^2)\Psi'' - b^2(1-b^2r^2s^2)\Psi = 0.
\end{equation}
%\noindent 
Coupling between $y$ and $\Psi$ still occurs through the
boundary conditions (\ref{7}). Define $a$ and $B$ as
\begin{eqnarray}
a = \left [ \frac{r^2+s^2}{2} - \sqrt{\left( \frac{r^2 - s^2}{2}\right
)^2 + \frac{1}{b^2}} \right ] ^{1/2}, \label{10a}\\
B = \left [ \frac{r^2+s^2}{2} + \sqrt{\left( \frac{r^2 - s^2}{2}\right
)^2 + \frac{1}{b^2}} \right ] ^{1/2}. \label{10b}
\end{eqnarray}
%\noindent 
In \cite{Huang}, Huang derives general solutions to equations
\eqref{8} and
\eqref{9}, valid when $b^2 r^2 s^2$ is not 1 or 0. These solutions are 
\begin{eqnarray*}
y = c_1 \cos ba\xi + c_2 \sin ba\xi + c_3 \cos bB\xi + c_4 \sin bB\xi ,
\\
\Psi = d_1 \sin ba\xi + d_2 \cos ba\xi + d_3 \sin bB\xi + d_4 \cos bB\xi
.
\end{eqnarray*}
%\noindent 
The $c_i$ may be determined in terms of the $d_i$ by
substituting the general solutions for $y$ and $\Psi$ into the second
order differential equations (\ref{5}) and (\ref{6}); $y$ can then be
expressed in terms of the $d_i$. For $b \ne 0$ or $1/rs$, one can then
show that solutions to the boundary value problem given in (\ref{5}),
(\ref{6}), and (\ref{7}) exist if and only if $\hat A \vec d = \vec 0$,
where $\vec d = (d_1, d_2, d_3, d_4)^T$ and 
\begin{equation} \label{11} \hat A = \left (
\begin{array}{cccc}
0                              &      \frac{\ds s^2}{\ds a^2 - s^2}       &
0               &        \frac{\ds s^2}{\ds -s^2 + B^2} \\
\noalign{\vskip 5pt}
ba                             &            0                     &
bB              &        0                      \\
\noalign{\vskip 5pt}
\frac{\ds s^2 \sin ba}{\ds a^2 - s^2}  & \frac{\ds s^2 \cos ba}{\ds a^2 - s^2}    &
\frac{\ds s^2 \sin bB}{\ds B^2 - s^2} & \frac{\ds s^2 \cos bB}{\ds B^2 - s^2} \\
\noalign{\vskip 5pt}
ba \cos ba                     & -ba \sin ba                     & bB
\cos bB                    & -bB \sin bB  \\
\end{array} \right ).
\end{equation}

The matrix equation $\hat A \vec d = \vec 0$ has nontrivial solutions
if and only if the determi\-nant of the matrix $\hat A$ vanishes. Let
$\tilde A(b)
= | \hat A |$. After carrying out some lengthy but routine calculations,
it can
be shown that 
\begin{equation}
\begin{split}
\tilde A(b)&= 2(1- \cos ba \cos bB )\\
&\quad + \frac{\ds b}{\ds (b^2r^2s^2 - 1)^{1/2}}
\cdot (b^2r^2(r^2-s^2)^2 + 3r^2 - s^2)\sin ba \sin bB.
\end{split}
\end{equation}
The matrix equation $\hat A \vec d = \vec 0$ defines solutions
to
(\ref{5})--(\ref{7}) whenever the (nonnegative) frequency parameter
$b$ is not equal to $0$ or $1/rs$.\footnote{For a discussion of when
$b=1/rs$ defines an eigenvalue, see \cite[pp. 26--29] {Geist}. For any
choice of the beam parameters $E$, $kG$, $\rho$, $I$ and $A$, the
equation $b=0$ defines an eigenvalue. In this case, if $k_1$ and $k_2$
are arbitrary constants, then $\left ( \begin{array}{c} Y \\ \Psi
\end{array} \right ) = k_1 \left ( \begin{array}{c} L \xi \\ 1
\end{array} \right ) + k_2 \left ( \begin{array}{c} 0 \\ 1 \end{array}
\right )$ solves the boundary value problem (\ref{5})--(\ref{7}). Since
the two vectors on the right of the last equation are linearly
independent, two linearly independent solutions to (\ref{5})--(\ref{7})
always exist when $b = 0$. Hence $b=0$ always defines the ``double''
eigenvalue $p^2 = 0$ for the free-free Timoshenko beam. Solutions to the
boundary value problem when $b=0$ correspond to rigid body rotation and
displacement of the beam. Nonzero double eigenvalues are also possible.
See Geist and McLaughlin \cite{Geist0} for a discussion and examples of
nonzero double eigenvalues.} This implies that when $b > 0$ and $b \ne
1/rs$, the free-free Timoshenko boundary value problem will have
nontrivial solutions if and only if $\tilde A(b) = 0$. Recall that the
nonnegative eigenvalue parameter $p^2$ appearing in equations
(\ref{tde1}) and (\ref{tde2}) satisfies $b = \sqrt{\rho A / (EI)} L^2
p$. It follows that with the possible exception of $p = (rs L^2)^{-1}
\sqrt{EI/(\rho A)}$, $p > 0$ is the square root of a nonzero eigenvalue
if and only if $\tilde A ( \sqrt{\rho A / (EI)} L^2 p ) = 0$. The
eigenvalue formulas documented in the next section are established by
estimating the roots of the frequency function $\tilde A$.

\section{Asymptotic formulas} The asymptotic formulas for the
eigenvalues of the free-free uniform Timoshenko beam are presented in
the theorem below. The discussion following the theorem is intended to
provide an overview of the theorem's proof. For a comprehensive and
thorough demonstration of the following result, the interested reader
should consult \cite[pp. 54--128]{Geist}.

\begin{theorem} Let $E$, $kG$, $I$, $A$, and $\rho$ all be positive
constants with $E \ne kG$, and let
\begin{displaymath}
0 < \bar p_1 \le \bar p_2 \le \cdots \le \bar p_i \le \bar p_{i+1} \le
\cdots
\end{displaymath}
%\noindent 
be the positive square roots of eigenvalues for the uniform
Timoshenko
beam with free ends. Let
\begin{displaymath}
0 <  p_1 \le  p_2 \le \cdots \le  p_i \le  p_{i+1} \le \cdots
\end{displaymath} 
%\noindent 
be the positive roots of the function 
\begin{displaymath}
\sin \left ( \frac{L \rho^{1/2}}{\sqrt{E}} p \right ) 
\sin \left ( \frac{L \rho^{1/2}}{\sqrt{kG}} p \right ).
\end{displaymath}
%\noindent 
Then there exist fixed integers $\hat I_1$ and $\tilde N$ such
that if
$i > \hat I_1$, then 
\begin{equation} \label{13}
| p_i - \bar p_{i+\tilde N}| < O(1/p_i).
\end{equation}
%\noindent 
Now suppose $\{ p_{i_n} \}_{n=1}^{\infty} \subset \{ p_{i}
\}_{i=1}^{\infty}$
such that for some fixed but arbitrary $e  \in (0,1)$,
\begin{equation} \label{14}
\sin \left ( \frac{L \rho^{1/2}}{\sqrt{E}} p_{i_n} \right ) = 0
\end{equation}
%\noindent 
and
\begin{equation} \label {15}
\left | \sin \left ( \frac{L \rho^{1/2}}{\sqrt{kG}} p_{i_n} \right )
\right | > e, \ \ n = 1, 2, \ldots.
\end{equation}
%\noindent 
Then there is an  $\hat I_2$  such that $i_n > \hat I_2$
implies
\begin{equation} \label{16}
\bar p_{{i_n} + \tilde N}^2 = p_{i_n}^2 + 
\left ( 1 - \frac{1}{2} \frac{kG + E}{kG - E}  \right )
\frac{A}{I}\frac{kG}{\rho} + O (1/p_{i_n}).
\end{equation}
%\noindent 
Similarly, if $\{ p_{j_m} \}_{m=1}^{\infty} \subset \{ p_{i}
\}_{i=1}^{\infty}$ is such that 
\begin{equation} \label{17}
\sin \left ( \frac{L \rho^{1/2}}{\sqrt{kG}} p_{j_m} \right ) = 0
\end{equation}
%\noindent 
and
\begin{equation} \label{18}
\left | \sin \left ( \frac{L \rho^{1/2}}{\sqrt{E}} p_{j_m} \right )
\right | > e, \ \ m = 1, 2, \ldots,
\end{equation}
%\noindent 
then there is an  $\hat I_3$  such that $j_m > \hat I_3$
implies
\begin{equation} \label{19}
\bar p_{{j_m} + \tilde N}^2 = p_{j_m}^2 + 
\left ( 1 + \frac{1}{2} \frac{kG + E}{kG - E}  \right )
\frac{A}{I}\frac{kG}{\rho} + O (1/p_{j_m}).
\end{equation}
\end{theorem}

\section{Proof sketch}  Suppose $f$ and $g$ are defined as
\begin{align*}
f &= \left ( \frac{3r^2 - s^2}{b^2} + r^2(r^2-s^2)^2 \right ) \sin ba
\sin bB,\\
g &= \frac{2(1-\cos ba \cos bB) \sqrt{r^2s^2 - \frac{1}{b^2}}}{b^2};
\end{align*}
then $\tilde A(b)$ can be written as
\begin{displaymath}
\tilde A (b) = \frac{b^2}{\sqrt{r^2s^2 - \frac{1}{b^2}}} (f + g).
\end{displaymath}
%\noindent 
For $b > 0$, $\tilde A (b) = 0$ if and only if $f + g = 0$. It
is instructive to observe that for $b > 1/rs$, the arguments $ba$ and
$bB$ of the trigonometric functions appearing in the expressions
defining $f$ and $g$ are strictly increasing in $b$. More importantly,
as $b$ increases, $g$ converges to zero and $f$ converges to the
function $r^2(r^2\! -\! s^2)^2\! \sin ba \sin bB$. It follows then that for
large $b$, the roots of $\tilde A$ are approximated by the roots of
$\sin ba \sin bB$. In \cite{Geist} it is shown that the isolated roots
of the function $\sin ba \sin bB$, that is, those roots of $\sin ba \sin
bB$ that are separated from neighboring roots of $\sin ba \sin bB$ by
more than some arbitrarily chosen fixed distance, occur along the
$b$-axis at most an $O(1/b^2)$ distance from the nearest roots of
$\tilde A$. Furthermore, it is shown in \cite{Geist} that every root of
$\sin ba \sin bB$, regardless of whether or not it is well separated
from the neighboring roots of $\sin ba \sin bB$, occurs no further than
an $O(1/b)$ distance from the nearest zero of the function. These
results prove that near every root of $\sin ba \sin bB$, at least one
root of $\tilde A$ lies nearby.

In \cite{Geist} it is also shown (using Rouch\'e's Theorem in the
complex plane) that there exists an unbounded, strictly increasing
infinite sequence $\sigma_j$, $j = 1, 2, \ldots$, which defines a
sequence of intervals $[\sigma^* , \sigma_j ]$ in which the functions
$\tilde A$
and $\sin ba \sin bB$ have exactly the same number of zeros. Hence for
$b \in [\sigma^* , \sigma_j ]$, $j = 1, 2, \ldots$, there is a
one-to-one correspondence between the roots of $\sin ba \sin bB$ and the
roots of $\tilde A(b)$. For $b$ large enough, the roots of $\sin ba \sin
bB$ can be matched up one-to-one with the nearest roots of $\tilde A(b)$
in such a way that each root of $\sin ba \sin bB$ lies no further than
an $O(1/b)$ distance away from the corresponding nearest root of $\tilde
A$. For those roots of $\tilde A$ near isolated roots of $\sin ba
\sin bB$, the distance between such roots of $\tilde A$ and the
corresponding nearest roots of $\sin ba \sin bB$ is $O(1/b^2)$. Even
though $a$ and $B$ defined in (\ref{10a}) and (\ref{10b}) are nonlinear
functions of the frequency parameter $b$, it is possible to explicitly
determine all roots of the function $\sin ba \sin bB$. Eigenvalues
satisfying conditions (\ref{14}) and (\ref{15}) or conditions (\ref{17})
and (\ref{18}) correspond to roots of the frequency function that are
near isolated roots of $\sin ba \sin bB$. For all such eigenvalues,
formulas for the roots of $\sin ba \sin bB$ are accurate enough
approximations to the roots of $\tilde A$ that they can be used to
derive the estimates given in (\ref{16}) and (\ref{19}). 

To see why the $O(1/b^2)$ bound applies to isolated roots of $\sin ba
\sin bB$, let $b_0$ be such an isolated root; then $b_0$ is an isolated
simple root of $f$. When $b$ is near $b_0$, $df/db$ must be bounded away
from zero. Since for all $b > 1/rs$, $0 \le g(b) < 2rs/b^2$, it follows
that if $b_0$ is large enough, then $f(b_0 + \Delta b) = -g(b_0 + \Delta
b)$ for some $\Delta b = O(1/b_0^2)$, and hence that $\tilde A(b_0 +
O(1/b_0^2)) = 0$. To gain some insight into why only an $O(1/b)$ bound
(and not necessarily an $O(1/b^2)$ bound) applies to roots of $\tilde A$
near nonisolated roots of $\sin ba \sin bB$, suppose $n$ and $m$ are
positive integers such that $n - m$ is odd, and that $b_1$ satisfies
$b_1 a(b_1) = n \pi$ and that $b_2$ satisfies $b_2 B(b_2) = m \pi$,
where $b_1$ and $b_2$ are very near one another. (Nonisolated roots
$b_1$ and $b_2$ of $\sin ba \sin bB$ satisfying these requirements exist
for many choices of the beam parameters. Existence of double roots is
shown in Geist \cite{Geist}. See also Geist and McLaughlin \cite{Geist0}.)
For values of $b$ near $b_1$ and $b_2$, $g(b) \approx 2rs/b^2$, and
$df/db|_{b_i},\ i=1, 2$, is small (since $b_1$ and $b_2$ must be located
near one of the local minima of $f$). Hence, the dominant term in a
Taylor series expansion for $f$ about $b_i$, $i = 1$ or 2, will be
quadratic in $b - b_i = \Delta b$ when $b$ is small. This implies the
equation
$f(b_i + \Delta b) = -g(b_i + \Delta b) \approx -2rs/(b_i + \Delta b)^2$
will have a solution when $\Delta b \approx [O(1/b_i^2)]^{1/2} =
O(1/b_i)$,
which implies that $\tilde A(b_i + O(1/b_i)) = 0$. 

\nocite{Kruszewski}
\nocite{Trail-Nash}

\begin{thebibliography}{99}

\bibitem{Geist} Bruce Geist, \emph{The asymptotic expansion of the
eigenvalues of the Timoshenko beam}, Ph.D. Dissertation, Rensselaer
Polytechnic Institute, Troy, NY, 1994.
\bibitem{Geist0} Bruce Geist and J. R. McLaughlin, \emph{Double eigenvalues
for the uniform Timoshenko beam}, Applied Mathematics Letters,
{\bf 10} (1997), 129--134. \CMP{97:15}
\bibitem{Huang} T. C. Huang, \emph{The effect of rotatory inertia and of shear
deformation on the frequency and normal mode equations of uniform beams
with simple end conditions}, Journal of Applied Mechanics {\bf 28} (1961),
579--584.
\MR{24:B579}
\bibitem{Kruszewski} Edwin T. Kruszewski, \emph{Effect of transverse shear and
rotary inertia on the natural frequency of a uniform beam},
National Advisory Committee for Aeronautics, Technical Note no.
1909, July 1949.
\bibitem{Timoshenko1} Stephen Timoshenko, \emph{On the correction for shear of
the differential equation for transverse vibrations of prismatic bars},
Philisophical Magazine {\bf 41} (1921), 744--746.
\bibitem{Timoshenko2} \bysame, \emph{On the transverse vibrations of bars 
of uniform cross-section}, Philisophical Magazine {\bf 43} (1922), 125--131.
\bibitem{Trail-Nash} R. W. Trail-Nash and A. R. Collar, \emph{The effects 
of shear flexibility and rotatory inertia on the bending vibrations of beams},
Quart. J. Mech. Appl. Math.
{\bf 6} (1953), 186--222.
\MR{16:541f}
\end{thebibliography}

\end{document}
%End-File:
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 id OAA01322 for pub-submit@math.ams.org; Fri, 06 Mar 1998 14:58:55 -0500 (EST)
Date: Fri, 06 Mar 1998 14:58:55 -0500 (EST)
From: Alexander O Morgoulis 
Subject: *accepted to ERA-AMS, Volume 4, Number 1, 1997*
To: pub-submit@MATH.AMS.ORG
Message-id: <199803061958.OAA01322@hilbert.math.psu.edu>

%% Modified March 4, 1998 by A. Morgoulis
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