\magnification=\magstep1
\def\giorno{3 June 96}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\D{{\cal D}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\I{{\cal I}}
\def\J{{\cal J}}
\def\K{{\cal K}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\Q{{\cal Q}}
\def\R{{\bf R}} %% reals
\def\T{{\rm T}} %% tangent
\def\S{{\cal S}}
\def\V{{\cal V}}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\ga{\gamma}
\def\de{\delta} %% NON ridefinire come \d !!!!
\def\eps{\varepsilon}
\def\phi{\varphi}
\def\la{\lambda}
\def\ka{\kappa}
\def\s{\sigma}
\def\z{\zeta}
\def\om{\omega}
\def\th{\theta}
\def\vth{\vartheta}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\La{\Lambda}
\def\Om{\Omega}
\def\Th{\Theta}
\def\pa{\partial}
\def\pd{\partial}
\def\d{{\rm d}} %% derivative
\def\xd{{\dot x}}
\def\yd{{\dot y}}
\def\grad{\nabla} %% gradient
\def\lapl{\triangle} %% laplacian
\def\ss{\subset}
\def\sse{\subseteq}
\def\Ker{{\rm Ker}}
\def\Ran{{\rm Ran}}
\def\ker{{\rm Ker}}
\def\ran{{\rm Ran}}
\def\iff{{\rm iff\ }}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\=#1{\bar #1}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\def\"#1{\ddot #1}
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\ref#1{[#1]}
\font \petit = cmr9
{\nopagenumbers
~\vskip 3 truecm
{\bf POINCARE' RENORMALIZED FORMS}
\footnote{}{{\tt \giorno }}
\vskip 3 truecm
Giuseppe Gaeta
Department of Mathematics
Loughborough University
Loughborough LE11 3TU (England)
{\tt G.Gaeta@lboro.ac.uk}
\vfill\parindent=0pt
{\bf Summary.} {In Poincar\'e Normal Form theory, one considers a series
of transformations generated by homogeneous polynomials obtained as
solution of the homological equation; such solutions are unique up to
terms in the kernel of the homological operator. Careful consideration
of the higher order terms generated by polynomials differing for a term
in this kernel leads to the possibility of further reducing the Normal
Form expansion of a formal power series, in a completely algorithmic way.
The algorithm is also applied to planar vector fields whose linear part
has eigenvalues $\la = \pm i$.}
\vfill\eject}
\pageno=1
\parindent=0pt
\parskip=10pt
{\bf Introduction.}
The theory of Normal Forms, first introduced by Poincar\'e, is in
many aspects central to the study of nonlinear dynamical systems; in
particular, its original version -- in which we are interested here --
deals with expansion of a system of ${\cal C}^\infty $ ODEs in $R^n$
(equivalently, of a ${\cal C}^\infty$ vector field on a $n$-dimensional
smooth manifold) in the neighbourhood of an equilibrium point, say the
origin $x=0$, and one aims at a classification of these up to
formal\footnote{$^1$}{By ``formal'', it is meant that these are defined
by series, and we do not consider the problem of convergence of such
series.} near-identity changes of coordinates.
It is well known \ref{1-3} that a system of ODEs (having the origin as
equilibrium point) can be taken into Normal Form by means of the
Poincar\'e algorithm, i.e. by performing a series of near-identity
changes of coordinates (Poincar\'e transformations) of the form $x' = x
+ h_k (x)$ where the $h_k (x)$ are homogeneous vector polynomials; these
are chosen as solution of a certain equation, the {\it homological
equation}, and such solutions are unique up to terms in the kernel of the
relevant {\it homological operator}.
In this way one manages to eliminate, order by order, all the {\it
nonresonant} terms. However, each of these changes of coordinates
produces new terms of higher order; those which are nonresonant will
then be disposed of by successive Poincar\'e transformations, while if
resonant terms are generated in this way, we are not able to eliminate
them by the Poincar\'e algorithm.
It should also be stressed that, while two vector polynomials $h_k (x) $
and $h'_k (x)$ which differ only by an element in the kernel of the
homological operator produce the same transformation on terms of order
$k$, they will in general give raise to different higher order terms;
in particular, we can in this way generate different higher order
resonant terms.
It is then clear that, unless we give a prescription
concerning the projection of $h_k (x)$ on the kernel of the homological
operator $\L_0$ (e.g. that $h_k (x) \in \[ \ker (\L_0 ) \]^\perp$), the
Normal Form is not uniquely defined.
Here we argue that this should not be seen as a drawback, but as an
advantage: indeed, a careful exploitation of the higher order terms
generated in each Poincar\'e transformation can lead to a remarkable
simplification of the Normal Form unfolding; this simplification is
specially important for systems satisfying resonance
relations\footnote{$^2$}{Here the $\la_k$ are the eigenvalues of the
matrix describing the linearization of the system around the equilibrium
point, and $m_k$ are nonnegative integers.} $\sum_k m_k \la_k = \la_r$
in a large or even infinite (as e.g. in the classical case where $\la_k =
\pm i$) number.
In order to present the argument, and the completely defined algorithm
which implements it, we need to consider the usual Poincar\'e scheme,
or actually the Lie-Poincar\'e one \ref{4,5}, paying special attention
to the higher order terms generated in each change of coordinates. Thus,
we will shortly go over the Poincar\'e and Lie-Poincar\'e methods
providing the relevant detailed formulas, in sections 1-5.
We do have a special advantage in considering Lie-Poincar\'e
transformations, as for these we have closed formulas expressing
transformed vector fields in terms of iterated commutators, essentially
the Baker-Campbell-Hausdorff formula; although this is a classical
result, we reproduce it in the appendix for completeness.
We briefly discuss the problem of non-unicity of Poincar\'e Normal Forms
(in section 6), and in sections 7-9 we implement this discussion by
providing an algorithm to reduce a system to a form which amounts
essentially to applying iteratively the Poincar\'e normalization
procedure first on the system, then on its normal form, then on this
``normalized normal form'', and so on (through $m$ normalizations if we
want to reduce the terms of order up to $m$), each time taking into
account terms of higher order; due to this, we call the final form thus
obtained the {\bf Poincar\'e renormalized form} for the system. It
should be stressed that in actual computation one would prefer to
proceed order by order, and this can actually be done; thus, we present
our algorithm directly in this form, i.e. we take sequentially the terms
of order 1,2,... to their renormalized form.
Finally, in section 10 we apply the algorithm to the case, mentioned
above, of planar vector fields having the generator of rigid rotations
as linear part: in this case the relevant matrix $A$ has eigenvalues
$\la_k = \pm i$, and thus an infinite number of satisfied resonance
relations and of terms appearing in the Poincar\'e normal form
unfolding, parametrized by two infinite sequences of real numbers.
However, the renormalized form unfolding constains only three nonlinear
terms -- i.e. three free real parameters -- at most.
%\vfill\eject
\bigskip
{\bf 1. General setting}
The Poincar\'e theory of Normal Forms \ref{1-3} for dynamical systems,
i.e. for first order autonomous smooth ODEs of the form
$$ {\dot x} = f(x) \qquad \qquad , ~ x \in R^n ~,~ f : R^n \to R^n
\eqno(1.1) $$
or equivalently for vector fields
$$ X = \sum_{i=1}^n \ f^i (x) {\pa \over \pa x^i} \ , \eqno(1.2) $$
is based on systematically employing near-identity changes of
coordinates with homogeneous vector polynomial functions as generator.
One is interested in $f$ being a formal power series, i.e.
$$ f(x) = \sum_{k=0}^\infty f_k (x) \eqno(1.3) $$
with $f_k (x)$ homogeneous of order $(k+1)$ in the $x$.
We denote by $V$ the set of vector formal power series $f:R^n \to R^n$
which have the origin as a fixed point, and by $V_k \ss V $ the set of
polynomial vector functions homogeneous of order $(k+1)$; obviously,
$$ V \ = \ \sum_{k=0}^\infty {}^\oplus \ V_k \ . \eqno(1.4) $$
It will be useful to define the bracket $\{ . , . \} : V \times V \to V$
given by
$$ \{ f , g \} = (f \cdot \grad ) g - (g \cdot \grad
) f \equiv f^i {\pa g \over \pa x^i} - g^i {\pa f \over \pa x^i} \ ;
\eqno(1.5) $$
this expresses the Lie commutator of vector fields when we look at the
component of vector fields in the $x$ coordinates; that is, for
$X=f^i \pa_i$ and $Y=g^i \pa_i$, we have $[X,Y] = h^i \pa_i$ with $h =
\{ f , g \}$. Notice that $$ \{ . , . \} \ : \ V_k \times V_m \ \to \
V_{k+m} \ . \eqno(1.6) $$
The (standard) homological operator $\L_0$ can be defined in terms of
this bracket, as $\L_0 (.) = \{ f_0 , . \}$; by (1.6), $\L_0 : V_k \to
V_k$.
In the following, we will need (linear) operators acting between the
spaces $V_k$, and in particular we will have to consider the
complementary sets of the ranges of such operators; it is thus
convenient to introduce a scalar product in $V$ (actually, in each of
the $V_k$), so that we can consider the adjoint operators.
It turns out that the convenient scalar product is defined as follows
\ref{6,7}. First of all, we notice that each of the $V_k$ is a
finite dimensional vector space. In each of these, we can choose a basis
$e_{\mu , j} (x)$ of functions which have components
$$ e_{\mu,j}^i (x) = x^\mu \de_{i,j} = (x^1)^\mu_1 ... (x^n)^\mu_n
\de_{i,j} \ ; \eqno(1.7) $$
we define then a scalar product in $V_k$ as
$$ \( e_{\nu , j } , e_{\mu , i} \) \ = \ < \mu , \nu > \ \de_{i,j} \ ,
\eqno(1.8) $$
where $<.,.>$ is the Bargman \ref{7,8} scalar product
$$ < \mu , \nu > = \[ \pa_\mu x^\nu \]_{x=0} = \prod_{i=1}^n (\mu_i ! )
\de_{\mu_i , \nu_i } \ . \eqno(1.9) $$
The scalar product in $V$ is then naturally defined in terms of these as
$\( f, g \) = \sum_k \( f_k , g_k \)$.
\vfill\eject
\bigskip
{\bf 2. Poincar\'e transformations.}
One considers then near-identity changes of coordinates of the form
$$ x = y + h_k (y) \quad \quad , \quad h_k \in V_k \ , \eqno(2.1) $$
also called Poincar\'e transformations. We denote by $\Ga$ the jacobian
of the change of coordinates, i.e. $\Ga^i_{~j} = {\pa h_k^i / \pa
y^j}$. Under the change of coordinates (2.1), our system
(1.1) is transformed into $$ {\dot y} = \[ I + \Ga \]^{-1} \ f \( y + h_k
(y) \) \ . \eqno(2.2) $$
For $y$ -- and therefore $x$ -- small enough, $\La = (I + \Ga )^{-1}$
does surely exist, and we can write it in a power series as $ \La \equiv
\( I + \Ga \)^{-1} = \sum_{m=0}^\infty \[ (-1)^m \( \Ga \)^m \]$.
Similarly, we can expand $f_m (y + h_k (y) ) $ as a power series; we
write $J = (j_1 , ... , j_n )$, $|J| = \sum_i j_i$. With this multiindex
notation, $\pa_J := \pa_1^{j_1} ... \pa_n^{j_n}$, and similarly $h^J_k
:= (h^1_k)^{j_1} ... (h^n_k)^{j_n}$. We define the operators
$ \Phi_h^r = (1 / r!) \sum_{|J|=r} \( h^J \cdot \pa_J \)$
(representing all the partial derivatives of order $|J|$), and in terms
of these the system (2.2) can then be written as
$$ {\dot y} \ = \
\sum_{m=0}^\infty \ \sum_{r=0}^\infty \ \sum_{s=0}^\infty \ \[ (-1)^s \
\Ga^s \Phi_{h_k}^{r-s} \] \ f_m (y) \ . \eqno(2.3) $$
Thus we see that, in a Poincar\'e transformation with generator $h_k
\in V_k$ each term $f_m$ is transformed into a new $\~f_m$
given\footnote{$^3$}{Here the square brackets in $[m/k]$ denotes integer
part.} by
$$ \~f_m \ = \ f_m \ + \ \sum_{p=1}^{[m/k]} \[ \sum_{s=0}^p \
(-1)^s \Ga^s \Phi_{h_k}^{p-s} \] f_{m-kp} \ . \eqno(2.4) $$
One could, in principle, also obtain explicit formulas for terms of all
degrees, but these become quickly too involved to be of practical use.
We notice, however, that the terms of degree smaller than $k$ are
not changed at all,
$$ \~f_m = f_m \quad \forall m < k \ , \eqno(2.5) $$
and the terms of degree $k \le m < 2k$ are changed according to
$$ \~f_{k+\nu} = f_{k+\nu} + \[ \Phi_h - \Ga \] f_\nu \quad (0 < \nu < k)
\ . \eqno(2.6) $$
\vfill\eject
\bigskip
{\bf 3. Transformation to Poincar\'e normal form.}
The transformation to Poincar\'e normal form is given by a well known
algorithm, which is just the same if we consider the Poincar\'e or the
Lie (see next section) form for the changes of coordinates. Indeed, in
both cases we have that for the transformation with generator $h_k \in
V_k$ it results $\~f_m = f_m $ for $mFrom (4.4) it is easy to derive formulas for the decomposition of $\~f$
into homogeneous factors, i.e. for $\~f = \sum_m \~f_m$. We introduce
the notation $\H (.) = \{ h , . \}$, and with this we have
$$ \~f_m \ = \ \sum_{s=0}^{[m/k]} {1 \over s !} \H^s \( f_{m - sk} \) \
. \eqno(4.5) $$
Notice that we have written $[m/k]$ for the integer part of $(m/k)$,
and defined $\H^0 (f) = f$.
\bigskip
{\bf 5. The homological operators}
We will define a series of homological operators $\L_k$ associated to
$f$ in a way slightly different from the customary one; the usual
homological operator, which we will denote by $\L_0$, will however be
just the standard one. This definition will suit our way of proceeding,
based on Poincar\'e-Lie transformations and thus on (4.4).
For $f \in V$, $f = \sum f_k$, we define the Lie operator $\F : V \to V$
associated to $f$ as $\F = \{ f , . \}$; clearly we can write
$$ \F \ = \ \sum_{k=0}^\infty \ \{ f_k , . \} \ \equiv \
\sum_{k=0}^\infty \ \L_k \ . \eqno(5.1) $$
The operators $\L_k = \{ f_k , . \}$ defined in (5.1) are called the
series of homological operators associated to $f$; the operator $\L_0$
coincides with the usual homological operator considered in Poincar\'e
Normal Form theory. Notice that, by (1.6), $\L_k : V_m \to V_{m+k}$. We
also denote by $\L_{k,m}$ the restriction of $\L_k$ to $V_m$.
It should be stressed that the homological operators do not permit to
describe (4.4), or (4.5), in full generality: they are only related to
the first nontrivial term in (4.5). However, it will turn out that, in
the procedure we employ in the following, a suitable choice of the $h$
permits to analyze iterated Poincar\'e-Lie transformations in terms of
the $\L_k$ alone.
%\vfill \eject
\bigskip
{\bf 6. Non-unicity of Poincar\'e normal forms.}
In the Poincar\'e procedure\footnote{$^7$}{Here, by this we mean
indifferently the usual Poincar\'e scheme, or the Poincar\'e-Lie one.},
shortly described above, one has no need to keep track of the effect of
the transformation generated by $h_k \in V_k$ on terms of higher order:
indeed, this will generate additional terms in $V_s$, in principle at
all the higher orders $s > k$, but these can then be disposed of by the
successive Poincar\'e transformations with generator $h_s$.
This point should be considered with some extra care: indeed, while the
terms generated by $\L_0 (h_s )$ are in $\ran (\L_0 ) \cap V_s = \ran
\( \L_{0,s} \)$, those appearing as ``higher order terms'' due to the
transformation generated by $h_k$ (with $k~~From (10.8), it appears that, choosing $\a = - b_2 / (4 b_1 )$ in
$h_2^{(2)}$, we can still reduce $f_4$ to $r^4 a_2 x$
(no reduction at all would be possible if $a_1 = b_1 = 0$).
When it comes to considering $f_6$, (10.10) shows that choosing $\a = -
b_3 / (4 b_1)$ in $h_4^{(3)}$ we can arrive to $f_6^{(3)} = r^6 a_3 x$.
We can then proceed further in the renormalization; $\L_3 \equiv 0$,
and thus the next -- and last possible -- step will be
$ f_6^{(5)} = f_6^{(3)} - \L_4 \( h_2^{(4)} \) $,
where $h_2^{(4)} \equiv h = r^2 ( \a I + \b A ) x$, due to the
condition $h \in \ker (\L_0 )$. Recall however that we also have to ask
$h \in \ker (\L_2 )$: this condition is readily see to be equivalent to
$a_1 \b = b_1 \a$; with our present assumptions, this means that $\a =
0$. Thus, we cannot eliminate $f_6^{(3)}$.
We could then check explicitely that the higher order terms, i.e. the
$f_{2k}$ with $k \ge 4$, can be completely eliminated.
Rather than going on with discussion of more and more degenerate cases,
we will give a general criterion and an inductive proof of it.
{\bf Lemma 2.} {\it Let the vector formal power series $f : R^2 \to R^2$
be given by $f(x) = Ax + \sum_{k=1}^\infty r^{2k} \[ a_k I + b_k A
\] x$; let $\mu$ be the lowest number such that $a_\mu \not= 0$, and
$\nu$ the lowest number such that $b_\nu \not=0$, so that $f(x)$ can be
written as
$$ f(x) \ = \ Ax \ + \ \sum_{k=\mu}^\infty \ r^{2k} a_k x \ + \
\sum_{k=\nu}^\infty \ r^{2k} b_k Ax \ . \eqno(10.13) $$
Then, the Poincar\'e renormalized form of $f$ up to any given order $n$
is given by
$$ f^* (x) \ = \ Ax + r^{2\mu} a_\mu x + r^{2\nu} \b A x + r^{4\mu} \a x
\ , \eqno(10.14) $$
where $a_\mu \not= 0$ is the same as in (10.13), and the $\a$, $\b$
could (possibly, but not necessarily) vanish. In particular, if $\nu >
\mu$, then $\b = 0$.}
{\tt Proof.} To see that this is true, it is convenient to use the
vector fields notation, with $X = f^i \pa_i = \sum_k X_k $, and $X_k =
f^i_k \pa_i$.
It is useful to consider the vector
fields $D$ and $R$ corresponding respectively to dilations and
rotations in $R^2$, i.e.
$$ D = x_1 \pa_1 + x_2 \pa_2 \quad , \quad R = -x_2 \pa_1 + x_1 \pa_2 \
; \eqno(10.15) $$
moreover, we consider the vector fields $ Z_k = r^k D$ and
$Y_k = r^k R$ (for $k$ even); these satisfy
$$ \[ a Z_k + b Y_k , \a Z_m + \b Y_m \] = (m-k) a \a Z_{(m+k)} + (m a
\b - k b \a ) Y_{(m+k)} \ . \eqno(10.16) $$
With the notation introduced above, the effect of $\L_k
(h_m )$ with $h_m \in \ker (\L_0 )$ can be computed via
$$ \[ a Z_k + b Y_k , \a Z_m + \b Y_m \] \ = \ (m-k) a \a Z_{k+m} \ + \
(m a \b - k b \a ) Y_{k+m} \ . \eqno(10.17) $$
First of all, we notice that we can eliminate all the terms $a_p Z_p$
in $f$, except the one for $p=2 \mu$: indeed, it suffices to choose
each time a $h_{p-\mu} = r^{2(p-\mu )} \( \a I + \b A \) x$ with $\a =
a_p / \( (p-2\mu ) a_\mu \)$. Notice that by a suitable choice of $\b$
(in particular, $\b = 0$ if $b_\mu = 0$) we can always manage to do
this without modifying the term $b_p Y_p$. Let us then assume we
eliminate first all the terms $a_p Z_p$ (except $p=\mu$ and possibly
$p=2\mu$) up to $p=n$.
Let us now look at the terms $b_p Y_p$ with $p$ greater than the
smaller of $\mu$ and $\nu$: it is clear, again by (10.17), that these
can be eliminated via the term $\L_\mu (h_{p-\mu} ) $ by choosing $\b =
b_p / \( (p-\mu ) a_\mu \)$ (if $\mu < \nu$), or via the term $\L_\nu
(h_{p-\nu} ) $ by choosing $\a = - b_p / \( (p-\nu ) b_\nu \)$ (if $\nu
< \mu$). Notice that if $\nu \le \mu$, the term $b_\nu Y_\nu$ cannot be
eliminated. $\odot$
\vfill\eject
{\bf Appendix.}
In this appendix, we shortly go over the Poincar\'e-Lie transformation,
and the derivation of (4.4); we will follow the discussion given in
\ref{5}.
We recall that in this case the change of coordinates is given by
(4.2), and that this transforms $X$ into $\~X$ given by (4.3) \ref{5}.
As mentioned in section 4, $\~X$ can now be explicitely computed by the
Baker-Campbell-Haussdorf formula \ref{5,16}, as
$$ \~X = \sum_{n=0}^\infty {(-1)^n \la^n \over n!} X^{(n)} \eqno(A.1) $$
where the $X^{(n)}$ are determined recursively by $X^{(n+1)} = \[
X^{(n)} , H_k \]$, with $X^{(0)} = X$.
We can thus consider a one-parameter family of vector fields $X_\la$,
where $X_0 = X$ and $X_1 = \~X$; this satisfies ${d X_\la / d \la } =
\[ H_k , X_\la \]$.
Correspondingly, we write $x_\la$ for $e^{-\la H_k} x$ (i.e. the
transformed coordinates, see (4.2), corresponding to $\la$), and $ X_\la
= f^i_\la (x_\la ) (\pa / \pa x_\la^i)$; the $f^i_\la$'s satisfy then $$
{d f^i_\la \over d \la } \ = \ \{ h_k , f_\la \}_\la^i \ , \eqno(A.2) $$
where $\{ .,. \}_\la$ is the bracket $\{.,.\}$ in the $x_\la$
coordinates, i.e. $\{ f,g \}_\la = f^j (\pa g / \pa x^j_\la ) - g^j
(\pa f / \pa x^j_\la )$.
If we consider the power series expansion of $f$, and writing for ease
of notation $f(x,\la ) = f_\la (x_\la )$ and $\{.,.\}$ for
$\{.,.\}_\la$, we have
$$ {\pa f^i_m (x , \la ) \over \pa \la } \ = \ \{ h_k , f_{m-k} \}^i \
. \eqno(A.3) $$
The $\~X = X_1$ is then written in the $\~x$ coordinates as $\~X = f^i
(x , 1 ) (\pa / \pa {\~x}^i ) \equiv \~f^i (\~x ) (\pa / \pa {\~x}^i
)$; the $\~f$ correspond to the solution of (A.2) for $\la = 1$.
These can be expressed by means of the BCH formula: indeed, from (A.1)
and the recursion relation for $X^{(n)}$, we have immediately that
$$ f (x, \la ) \ = \ \sum_{n=0}^\infty \ \[ {(-1)^n \la^n \over n!}
\ \varphi^{(n)} (x) \] \eqno(A.4) $$
with $\varphi^{(0)} (x) = f (x,0)$ and $\varphi^{(n+1)} = \{
\varphi^{(n)} , h_k \}$.
>From this, we have indeed, with $\H (.) = \{ h , . \}$,
$$ f_\la \ = \ \sum_{n=0}^\infty {\la^n \over n!} \ \H^n (f) \ \ ,
\eqno(A.5) $$
and for $\la=1$, i.e. for $\~f (x) = f (x,1)$, this is just (4.4).
\vfill\eject
~\bigskip\bigskip
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\bigskip\bigskip\bigskip\parskip=6pt
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Pergamon, London, 1959
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