26. CJM 1998 (vol 50 pp. 40)
 Engliš, Miroslav; Peetre, Jaak

Green's functions for powers of the invariant Laplacian
The aim of the present paper is the computation of Green's functions
for the powers $\DDelta^m$ of the invariant Laplace operator on rankone
Hermitian symmetric spaces. Starting with the noncompact case, the
unit ball in $\CC^d$, we obtain a complete result for $m=1,2$ in
all dimensions. For $m\ge3$ the formulas grow quite complicated so
we restrict ourselves to the case of the unit disc ($d=1$) where
we develop a method, possibly applicable also in other situations,
for reducing the number of integrations by half, and use it to give
a description of the boundary behaviour of these Green functions
and to obtain their (multivalued) analytic continuation to the
entire complex plane. Next we discuss the type of special functions
that turn up (hyperlogarithms of Kummer). Finally we treat also
the compact case of the complex projective space $\Bbb P^d$ (for
$d=1$, the Riemann sphere) and, as an application of our results,
use eigenfunction expansions to obtain some new identities involving
sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the
polylogarithm function. The case of Green's functions of powers of
weighted (no longer invariant, but only covariant) Laplacians is
also briefly discussed.
Keywords:Invariant Laplacian, Green's functions, dilogarithm, trilogarithm, Legendre and Jacobi polynomials, hyperlogarithms Categories:35C05, 33E30, 33C45, 34B27, 35J40 

27. CJM 1997 (vol 49 pp. 1066)
 Shibata, Tetsutaro

Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity
We consider the multiparameter nonlinear SturmLiouville problem
$$\displaylines{
u''(x)  \sum_{k=1}^m\mu_k u(x)^{p_k} + \sum_{k=m+1}^n
\mu_ku(x)^{p_k} = \lambda u(x)^q, \quad x \in I := (1,1), \cr
u(x) > 0, \quad x \in I, \cr
u(1) = u(1) = 0,\cr}$$
where $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n)
\in \bar{R}_+^m \times R_+^{nm} \bigl(R_+ := (0, \infty)\bigr)$
and $\lambda \in R$ are parameters. We assume that
$$1 \le q \le p_1 < p_2 < \cdots < p_n < 2q + 3.$$
We shall establish an asymptotic formula of
variational eigenvalue $\lambda = \lambda(\mu,\alpha)$ obtained
by using LjusternikSchnirelman theory on general level set
$N_{\mu,\alpha} (\alpha > 0:$ parameter of level set).
Furthermore, we shall give the optimal condition of
$\{(\mu, \alpha)\}$, under which $\mu_i (m + 1 \le i \le n:
\hbox{\rm fixed})$ dominates the asymptotic behavior of
$\lambda(\mu,\alpha)$.
Categories:34B15, 34B25 

28. CJM 1997 (vol 49 pp. 583)
 Pal, Janos; Schlomiuk, Dana

Summing up the dynamics of quadratic Hamiltonian systems with a center
In this work we study the global geometry of planar quadratic
Hamiltonian systems with a center and we sum up the dynamics of
these systems in geometrical terms. For this we use the
algebrogeometric concept of multiplicity of intersection
$I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$,
$Q(x,y,z) = 0$ at a point $p$ of the plane. This is a
convenient concept when studying polynomial systems and it
could be applied for the analysis of other classes of nonlinear
systems.
Categories:34C, 58F 

29. CJM 1997 (vol 49 pp. 338)
 Rousseau, C.; Toni, B.

Local bifurcations of critical periods in the reduced Kukles system
In this paper, we study the local bifurcations of critical periods
in the neighborhood of a nondegenerate centre of the reduced Kukles
system. We find at the same time the isochronous systems. We show
that at most three local critical periods bifurcate from the
ChristopherLloyd centres of finite order, at most
two from the linear isochrone and at most one critical period from the
nonlinear isochrone. Moreover, in all cases, there exist
perturbations which lead to the maximum number of critical
periods. We determine the isochrones, using the method of Darboux:
the linearizing transformation of an isochrone is derived from the
expression of the first integral.
Our approach is a combination of computational algebraic techniques
(Gr\"obner bases, theory of the resultant, Sturm's algorithm), the
theory of ideals of noetherian rings and the transversality theory
of algebraic curves.
Categories:34C25, 58F14 

30. CJM 1997 (vol 49 pp. 212)
 Coll, B.; Gasull, A.; Prohens, R.

Differential equations defined by the sum of two quasihomogeneous vector fields
In this paper we prove, that under certain hypotheses,
the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$,
$\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are
quasihomogeneous vector fields, has at most two limit cycles.
The main tools used in the proof are the generalized polar
coordinates, introduced by Lyapunov to study the stability of degenerate
critical points, and the analysis of the derivatives of the Poincar\'e
return map. Our results generalize those obtained for polynomial
systems with homogeneous nonlinearities.
Categories:34C05, 58F21 
