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% AMS # 28B05, 26A39, 46G10
%%%%From: VALENTIN A. SKVORTSOV
%%%%To: Brian Thomson
%%%Subject: Paper by Skvortsov and Solodov (final version)
%%%%Date: Thursday, April 30, 1998 4:15 AM
\documentclass{rae}
\usepackage{amsmath,amsthm,amssymb,enumerate}
\markboth{V.~A.~Skvortsov and A.~P.~Solodov}{A Variational integral for Banach--valued functions}
%\coverauthor{Valentin A.~Skvortsov and Alexei P.~Solodov}
%\covertitle{A Variational Integral for Banach--Valued Functions}
\received{April 16,1998}
\MathReviews{28B05, 26A39, 46G10}
\keywords{Henstock integral, McShane integral, variational integral,
Bochner integral, Banach-valued functions}
\firstpagenumber{1}
\markboth{V.~A.~Skvortsov and A.~P.~Solodov}{A Variational Integral for Banach--Valued Functions}
\author{Valentin A.~Skvortsov and Alexei P.~Solodov\thanks{Supported by RFFI (Grants 96-01-00332 and
96-15-96073)},
Mechanics and Mathematics Department, Moscow State University, 119899,
Russia, e-mail: {\tt vaskvor@nw.math.msu.su}}
\title{A VARIATIONAL INTEGRAL FOR BANACH--VALUED FUNCTIONS}
%%%%%%Put Author's Definitions Below Here %%%%%%%%%%%
\def\VarM{\mathop{{\mbox{\large V}_M}}}
\def\VarH{\mathop{{\mbox{\large V}_H}}}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{remark}{Remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\begin{abstract}
It is shown that
for Banach--space--valued
functions the variational Henstock integral
is
equivalent to the Henstock integral if and only if the range space
is of a finite
dimension.
The same is true for the equivalence of the variational McShane
integral
and the McShane integral.
\end{abstract}
\bigskip
There are various ways to define the Henstock integral. The
original
Hens\-tock--Kurzweil definition is based on generalized Riemann sums
(the $H$-in\-teg\-ral). For real--valued functions this integral is
known
to be equivalent to the variational Henstock integral (the
$V$-integral, see \cite{ost}) and to the Den\-joy--Per\-ron integral
(the
$D_*$-integral, see \cite{g1}). The first of this equivalence is a
corollary of the so--called Saks--Henstock Lemma (see Lemma \ref{sh}
below).
Here we are considering Henstock type integral for
Banach--space--valued
functions. It was noticed by S.~S.~Cao in \cite{cr} that for such
functions
Saks--Hens\-tock Lemma might fail to be true. Because of that for some
spaces
the $V$-integral is not equivalent to the $H$-integral. It is natural
to
ask what is a characterization of those Banach spaces for which
such
equivalence holds.
We are showing here that
for Banach--space--valued
functions the $V$-integral
is
equivalent to the $H$-integral if and only if the range space
is of a finite
dimension.
At the same time for any Banach space the $V$-integral is
equivalent to the Denjoy--Bochner integral.
Similar problems are considered for the variational McShane integral.
First we recall some notations and definitions.
We denote by $X$ a Banach space with the norm $\|\cdot\|$,
by ${{\mathbb R}}$ the real line,
by $[a,b]$ a closed interval on the line, and by $|E|$
Lebesgue
measure of a set $E$.
Let $\mathcal I$ be a collection
of all closed intervals that are contained in $[a,b]$.
A~collection $T$ of pairs $(\Delta_k,\xi_k)\in
\mathcal I\times[a,b],\; i=1,\ldots, n$,
is called
a \emph{partition} of the interval $[a,b]$ if the
intervals
$\Delta_i$ and $\Delta_j$ are non-overlapping for $i\neq j$, and
$\bigcup_{k=1}^n \Delta_k=[a,b]$.
Let $\delta:[a,b]\longrightarrow (0,\infty) $ be a positive
function defined on $[a,b]$.
A partition
$T$ of
$[a,b]$ is called \emph{Henstock $\delta$-fine} if every pair
$(\Delta,\xi)\in T$ satisfies
$$\xi\in\Delta\subset\bigl(\xi-
\delta(\xi),\xi+\delta(\xi)\bigl).$$
\begin{definition}\label{hen} (\cite{fre}).
A function $f:[a,b]\longrightarrow X$ is called
\emph{Henstock integrable ($H$-integrable)} on a closed interval
$[a,b]$ with integral value $I\in X$ if for every $\varepsilon>0$
there
exists a positive function $\delta:[a,b]\longrightarrow
(0,\infty) $
such that for every Henstock
$\delta$-fine partition $T$ of $[a,b]$
$$\Bigl\|\sum_T f(\xi_k)|\Delta_k|-I\Bigl\|<\varepsilon.$$
We denote $I=(H)\int_a^b f\, dt$.
\end{definition}
A partition
$T$ of
$[a,b]$ is called \emph{McShane $\delta$-fine} ($\delta$ being a
positive function on $[a,b]$) if every pair
$(\Delta,\xi)\in T$ satisfies
$$\Delta\subset\bigl(\xi-\delta(\xi),\xi+\delta(\xi)\bigl).$$
\begin{definition} \label{mc} (\cite{g2}).
A function $f:[a,b]\longrightarrow X$ is called
\emph{McShane integrable ($M$-integrable)} on a closed interval
$[a,b]$
with integral value $I\in X$ if for every
$\varepsilon>0$ there exists a positive function $\delta: [a,b]
\longrightarrow (0,\infty) $
such
that for every McShane $\delta$-fine partition $T$ of
$[a,b]$
$$\Bigl\|\sum_T f(\xi_k)|\Delta_k|-I\Bigl\|<\varepsilon.$$
We denote $I=(M)\int_a^b f\, dt$.
\end{definition}
It is clear that $M$-integrability implies $H$-integrability.
Since $H$-integ\-ra\-bi\-lity and $M$-integrability on $[a,b]$
imply integrability on any subinterval
$\Delta\subset[a,b]$, we can define \emph{the indefinite
$H$-integral}
and \emph{the indefinite $M$-integral} by putting
$F(\Delta)=(H)\int_\Delta
f\, dt$ $\bigl(F(\Delta)=(M)\int_\Delta f\, dt\bigl)$.
Let $\Phi:\mathcal I\times[a,b]\longrightarrow X$ be an interval-
-point
function. \emph{The Henstock} and \emph{the
McShane variations} of
$\Phi$ are defined as
\begin{align*}
\VarH(\Phi)=\inf_{\delta}\sup_T\sum_T\bigl\|\Phi(\xi_k,\Delta_k
\bigl\|\\ \intertext{(sup is taken over all Henstock
$\delta$--fine partitions T and inf is taken over all positive
functions $\delta$ on $[a,b]$), and}
\VarM(\Phi)=\inf_{\delta}\sup_T\sum_T\bigl\|\Phi(\xi_k,\Delta_k
\bigl\|
\end{align*}
(sup is taken over all McShane $\delta$--fine partitions $T$ and
inf is taken over all positive functions $\delta$ on $[a,b]$).
Note that any interval function $\Phi:\mathcal I\longrightarrow
X$
can be considered as an interval--point function dependent only on
the
first argument.
The following two definitions of variational integrals are
natural
extensions of the definitions for the real-valued case (see
\cite{ost}).
Functions $\Phi_1,\Phi_2:\mathcal I\times[a,b]\longrightarrow
X$ are
said to be \emph{Henstock variationally equivalent} if
$\VarH(\Phi_1-\Phi_2)=0.$
\begin{definition} \label{henv}
A function $f:[a,b]\longrightarrow X$ is called \emph{Henstock
variationally integrable ($V$-integrable)} on $[a,b]$ if there
exists
an additive interval function $F:\mathcal I\longrightarrow X$ such
that the
interval--point function $f(t)|\Delta|$ and $F(\Delta)$ are
Henstock
variationally equivalent,
$F(\Delta)$ being \emph{the indefinite
$V$-integral} of $f$.
\end{definition}
Functions $\Phi_1,\Phi_2:\mathcal I\times[a,b]\longrightarrow X$
are
said to be \emph{McShane variationally equivalent} if
$\VarM(\Phi_1-\Phi_2)=0.$
\begin{definition} \label{mcv}
A function $f:[a,b]\longrightarrow X$ is called \emph{McShane
variationally integrable ($MV$-integrable)} on $[a,b]$ if there
exists
an additive interval function $F:\mathcal I\longrightarrow X$
such that
the interval--point function $f(t)|\Delta|$ and $F(\Delta)$ are
McShane
variationally equivalent,
$F(\Delta)$ being \emph{the indefinite
$MV$-integral} of~$f$.
\end{definition}
\begin{definition} \label{acc} (\cite{g3}).
A function $F:\mathcal I \longrightarrow X$
is said to be \emph{$AC$-function} on a set $E\subset[a,b]$ if
for every $\varepsilon>0$ there exists
$\delta>0$ such that for every collection
of non-overlapping closed intervals $\left\lbrace
\Delta_i\right\rbrace_{i=1}^n$ with the end points belonging to
$E$
and with $\sum_{i=1}^n |\Delta_i|<\delta$,
we have $\sum_{i=1}^n F(\Delta_i)<\varepsilon$.
\end{definition}
\begin{definition}\label{b} (\cite{g3}).
A function $f:[a,b]\longrightarrow X$ is said to be \emph{Bochner
integrable ($B$-integrable)} on $[a,b]$, if there exists a
function $F:\mathcal I\longrightarrow X$ that is $AC$ on
$[a,b]$ and such
that it is differentiable a.~e.~and $F'(t)=f(t)$ a.~e. on
$[a,b]$,
$F(\Delta)$ being \emph{the indefinite $B$-integral} of $f$.
\end{definition}
\begin{definition}\label{ac*} (\cite{nav}).
A function $F:\mathcal I \longrightarrow X$
is said to be \emph{$AC^*$-function} on a set $E\subset[a,b]$ if
for every $\varepsilon>0$ there exists
$\delta>0$ such that for every collection
of non-overlapping closed intervals $\left\lbrace
\Delta_i\right\rbrace_{i=1}^n$ with one of the end points
belonging to
$E$ and with $\sum_{i=1}^n |\Delta_i|<\delta$, we have
$\sum_{i=1}^n F(\Delta_i)~<~\varepsilon$.
\end{definition}
It is clear that for $E=[a,b]$ the class of $AC^*$-functions
coincides
with the class of $AC$-functions.
\begin{definition} \label{acg} (\cite{nav}).
A function
$F:\mathcal I\longrightarrow X$ is said to be
\emph{$ACG^*$-function} on a
set $E\subset[a,b]$ if $E$ can be represented as a union of a
sequence
of sets such that $F$ is $AC^*$-function on each of them.
\end{definition}
\begin{definition} \label{db} (\cite{nav}).
A function $f:[a,b]\longrightarrow X$ is said to be
\emph{Denjoy--Bochner integrable ($D_*B$-integrable)} on
$[a,b]$, if
there exists an $ACG^*$-function $F:\mathcal I\longrightarrow X$
such
that it is differentiable a.~e.~and $F'(t)=f(t)$ a.~e. on
$[a,b]$,
$F(\Delta)$ being \emph{the indefinite $D_*B$-integral} of $f$.
\end{definition}
The Definitions \ref{acc} and \ref{ac*} --- \ref{db} are
extensions of
the respective definitions for the real--valued case (see \cite{s}).
The following proposition is a direct corollary of the
definitions.
\begin{proposition}\label{h-v}
If $f:[a,b]\longrightarrow X$ is $V$-integrable ($MV$-integrable)
on $[a,b]$ then it is also $H$-integrable ($M$-integrable)on
$[a,b]$
and the indefinite integrals coincide. \end{proposition}
The following assertion is known as Saks--Henstock Lemma for
real--valued functions and is easily extended to the case of
vector--valued functions with range spaces being spaces of finite
dimensions.
\begin{lemma}\label{sh}\emph{(\cite{cb}).}
Let $X$ be a Banach space of a finite dimension.
If a function
$f:[a,b]\longrightarrow X$ is $H$-integrable ($M$-integrable)
with the indefinite integral $F:\mathcal I\longrightarrow X$ then
for every $\varepsilon>0$ there exists a function
$\delta:[a,b]\longrightarrow (0,\infty)$ such that for every
Henstock
(McShane) $\delta$-fine partition $T$ of $[a,b]$
$$\sum_T\bigl\|f(\xi_k)|\Delta_k|-
F(\Delta_k)\bigl\|<\varepsilon.$$
\end{lemma}
S.~S.~Cao (see \cite{cb}) introduced a definition of
$HL$-integrability
of a function $f:[a,b]\longrightarrow X$ which is a restriction of
$H$-integrability by the requirement that the assertion of Lemma
\ref{sh}
is valid for $f$. It is proved in \cite{solf} that the
$HL$-in\-teg\-ral is equivalent to the $D_*B$-integral. (The
same
equivalence was stated in~\cite{nav}, but there was some gap in
the
proof which was overcome in \cite{solf}.) Since $V$-integrability
of a
function $f:[a,b]\longrightarrow X$ is obviously equivalent to
the
assertion of Saks--Henstock Lemma we get
\begin{theorem}\label{v-db}
Let $X$ be a Banach space.
For functions taking values in $X$ the
$V$-in\-teg\-ral is equivalent to the $D_*B$-integral.
\end{theorem}
Analogous fact for the $MV$-integral is the following one.
\begin{theorem}\label{mv-b}
Let $X$ be a Banach space. For functions taking values in $X$
the $MV$-in\-teg\-ral is equivalent to the $B$-integral.
\end{theorem}
The
proof is the same as above. It is enough to use \cite{chin}
instead of
\cite{solf}.
Proposition \ref{h-v} and Theorems \ref{v-db} and \ref{mv-b} imply
\begin{proposition}\label{db-h}
If a function $f:[a,b]\longrightarrow X$ is $D_*B$-integrable
($B$-in\-teg\-rable) it is also $H$-integrable ($M$-integrable)
and the
indefinite integrals coincide.
\end{proposition}
Now we consider the relation between the $H$-integral (the
$M$-integral) and the $V$-integral (the $MV$-integral). Our aim
in the
rest of the paper is to prove the following theorem.
\begin{theorem}\label{main}
Consider functions on $[a,b]$ taking values in a fixed Banach
space
$X$. Then the $V$-integral (the $MV$-integral) is equivalent to
the
$H$-integral (the $M$-integral) on this class of functions if and
only
if $X$ is of a finite dimension.
\end{theorem}
\begin{proof}
The sufficiency follows easily from Lemma \ref{sh}. The proof of
the necessity is based on a geometric idea (see \cite{solm})
which in
turn follows from the construction by A.~Dvoretzky and
C.~A.~Rogers
used in \cite{dr} to show that in every
infinite--dimensional
Banach space there exists a series that is unconditionally but
not
absolutely convergent.
\begin{lemma}\label{l1}\emph{(\cite{dr})}.
Let $B$ be a body in ${{\mathbb R}}^n$ which
is convex and has the origin as a center and let $r$ be an integer
with
$1\leq r\leq n$. Then there exist $r$ vectors
$A_1,A_2,\ldots,A_r\in{\mathbb R}^n$ on the boundary of $B$ such that
if
$\lambda_1,\lambda_2,\ldots,\lambda_r$ are any $r$ real numbers then
$$\sum_{i=1}^r \lambda_i A_i\in \lambda B,
\text { where }
\lambda^2=\left(2+\frac{r(r-1)}n\right)\sum_{i=1}^r \lambda_i^2.$$
($\lambda B$ is the set $\{\lambda x,\; x\in B\}$.)
\end{lemma}
\begin{lemma}\label{l2} Let $X$ be a Banach space of the
infinite dimension. Then for every natural number $r$
there exist unit vectors $x_1,x_2,\ldots,x_r\in X$ such that
for every numbers
$\theta_1,\theta_2,\ldots,\theta_r$
with
$|\theta_i|\leq 1$, ${1\leq i\leq r},$
\begin{equation*} \label{*0}
\Bigl\|\sum_{i=1}^r \theta_i x_i\Bigl\|^2\leq 3r.
\end{equation*}
\end{lemma}
\begin{proof} Since $X$ is of the infinite dimension, for any
$n$ there exist linear independent vectors $z_1,z_2,\ldots,z_n\in
X$.
Take~$n=r(r-1)$.
Consider the set of vectors
$z=\sum_{i=1}^n\mu_i z_i$,
where
$\mu_i$ are numbers such that $\|z\|\leq 1$.
In Euclidean space with the norm generated by vectors
$(\mu_1,\mu_2,\dots,\mu_n)$ they form a convex body $B$ having the
origin as a center.
According to Lemma \ref{l1} there exist vectors
$x_1,x_2,\ldots,x_r$ on the boundary of a set $B$ with the
following
property: for every numbers $\theta_1,\theta_2,\ldots,\theta_r$
with
$|\theta_i|\leq 1$, ${1\leq i\leq r},$ \begin{equation} \label{*}
\sum_{i=1}^r \theta_i x_i\in \theta B,\text{ where }
\theta^2=3\sum_{i=1}^r \theta_i^2\leq 3r. \end{equation} Since
for all
$i=1,2,\ldots,r$ vectors $x_i$ belong to the boundary of $B$
they have
unit norm in $X$. Since $\|z\|\leq 1$ for all $z\in B$ it
follows from \eqref{*} that $$\Bigl\|\sum_{i=1}^r \theta_i
x_i\Bigl\|^2\leq 3r.$$ \end{proof}
Now we can complete the proof of the necessity in Theorem
\ref{main}.
For simplicity we suppose $[a,b]=[0,1]$. Let $C$ be the Cantor
ternary
set, $(a_i^r,b_i^r)$, $r\geq 0$, $1\leq i\leq 2^r$, being the
intervals
of rank $r$ contiguous to $C$ (we have $b_i^r-a_i^r=3^{-r-1}$)
and
$d_i^r$ being the middle points of the intervals $(a_i^r,b_i^r)$.
Assume that $X$ is of the infinite dimension.
According to Lemma \ref{l2}, for every $r$ we may construct
vectors
$x_1^r,x_2^r,\ldots,$ $x_{2^r}^r\in X$ such that
\begin{equation*}
\label{1*}
\|x_i^r\|=\frac 1{2^r},\; 1\leq i\leq 2^r,
\end{equation*}
and for every numbers
$\theta_1^r,\theta_2^r,\ldots,\theta_{2^r}^r$,
with $|\theta_i^r|\leq 1,
1\leq i\leq 2^r$,
\begin{equation*}
\label{1**}
\Bigl\|\sum_{i=1}^{2^r} \theta_i^r x_i^r\Bigl\|^2\leq \frac
3{2^r}.
\end{equation*}
Define the function $f:[0,1]\longrightarrow X$ in the following
way $$f(t)=\left\{\begin{aligned} 0,& \text{ if } t\in C \text{
or
$t=d_i^r$, }& r\geq 0, 1\leq i\leq 2^r,\\ 2\cdot 3^r x_i^r,&
\text{ if }
t\in (a_i^r,d_i^r),& r\geq 0, 1\leq i\leq 2^r,\\ -2\cdot 3^r
x_i^r,&
\text{ if } t\in (d_i^r,b_i^r),& r\geq 0, 1\leq i\leq 2^r.\\
\end{aligned}\right.$$
It is proved in \cite{solm} that this function is $M$-integrable
(and
consequently $H$-in\-teg\-rable) and its indefinite integral
$F(\Delta)$ is not an $ACG^*$-function. Hence in view of the
Proposition
\ref{db-h} function $f$ is not $D_*B$-integrable (and is not
$B$-integrable) and therefore Theorems \ref{v-db} and \ref{mv-b} imply
that
$f$ is not $V$-integrable and is not $MV$-in\-teg\-rable. This
completes
the
proof. \end{proof}
\begin{thebibliography}{99}
\bibitem{nav}
Jr.~S.~R.~Canoy and M.~P.~Navarro, \emph{A Denjoy--type integral
for
Banach--valued functions}, Rend.~Circ.~Mat.~Palermo \textbf{44}
(1995),
no.~2, 330-336.
\bibitem{cr}
S.~S.~Cao, \emph{Banach--valued {Henstock} integration}, Real
Analysis
Exchange \textbf{19} (1993-94), no.~1, 34.
\bibitem{cb}
S.~S.~Cao, \emph{The {Henstock} integral for Banach--valued
functions},
SEA Bull.\ Math. \textbf{16} (1992), no.~1, 35-40.
\bibitem{chin}
W.~Congxin and Y.~Xiaobo, \emph{A Riemann--type definition of the
{Bochner} integral}, J.\ Math.\ Study \textbf{27} (1994), no.~1,
32-36.
\bibitem{dr} A.~Dvoretzky and C.~A.~Rogers, \emph{Absolute and
unconditional convergence in normed linear spaces}, Proc.\ Nat.\
Acad.\
Sci.\ USA \textbf{36} (1950), no.~3, 192-197.
\bibitem{fre} D.~H.~Fremlin, \emph{The Henstock and {McShane}
integrals of vector--valued functions}, Illinois J. Math. \textbf{38}
(1994), no.~3, 471-479.
\bibitem{g1}
R.~Gordon, \emph{Equivalence of the generalized {Riemann} and
restricted {Denjoy} integral}, Real Analysis Exchange \textbf{12}
(1986-87), no.~2, 551-574.
\bibitem{g3} R.~Gordon, \emph{The {Denjoy} extension of the
Bochner, Pettis and Dunford integrals}, Studia Math.
\textbf{92}
(1989), 73-91.
\bibitem{g2} R.~Gordon, \emph{The {McShane}
integration of Banach--valued functions}, Illinois J. Math.
\textbf{34}
(1990), 557-567.
\bibitem{ost}
K.~M.~Ostaszewski, \emph{Henstock integral in the plane}, Mem.~of
Amer.~Math. Soc. \textbf{63}(353) (1986), 1-106.
\bibitem{s}
S.~Saks, \emph{Theory of the integral}, Dover, New York, 1964.
\bibitem{solm}
A.~P.~Solodov, \emph{Henstock and McShane integrals for
Banach--valued
functions}, Mat.~Zametki (to appear).
\bibitem{solf}
A.~P.~Solodov,
\emph{Riemann--type definition for the restricted Denjoy--Bochner
integral}, Fundamental and Applied Mathematics (to appear).
\end{thebibliography}
\end{document}
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{\huge R}{\Large EAL} {\huge A}{\Large NALYSIS} {\huge E}{\Large
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Professor
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{Alexei P.~Solodov\\
Mechanics and Mathematics Department\newline Moscow State University\newline 119899\newline
Russia}
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{A Variational Integral for Banach--Valued Functions}
{Valentin A.~Skvortsov and Alexei P.~Solodov}
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{\huge R}{\Large EAL} {\huge A}{\Large NALYSIS} {\huge E}{\Large
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Professors
{Valentin A.~Skvortsov and Alexei P.~Solodov\\
Mechanics and Mathematics Department\newline Moscow State University\newline 119899\newline
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Please make corrections in red on the enclosed copy and return it to
me within about two weeks, {\bf but no later than December 21.}.
%We are encouraging authors to include a short abstract of their work
%to be included at the beginning of the paper. For short articles
%this may not be appropriate, however. If you would like to include
%an abstract, please include the text with your return note.
%Please include your email addresses if you have them.
%Too, we would like to include the AMS subject classification
%number(s), and a list of keywords and phrases with your paper.
%Please fill these in below.
We prefer an e-mail response ({\bf in addition to the return of the
galley sheets}) if possible.
The {\it Real Analysis Exchange} charges \$15 per page to those whose research
is supported by a grant to which publication costs can be charged. Please
request an invoice when you return your gallies.
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\vbox{
Sincerely,
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%
Paul D. Humke, Editor \\
%Jimmy Peterson, Asst. Editor \\
% Ben J. Grommes, Asst. Editor \\
Phone: 507-646-3113 Fax: 507-646-3116 \\
e-mail: analysis@stolaf.edu}
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