1. CJM 2015 (vol 67 pp. 1247)
 Barros, Carlos Braga; Rocha, Victor; Souza, Josiney

Lyapunov Stability and Attraction Under Equivariant Maps
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed
with
admissible families of open coverings. Assume that $\mathcal{S}$ is a
semigroup acting on both $M$ and $N$. In this paper we study the behavior of
limit sets, prolongations, prolongational limit sets, attracting sets,
attractors and Lyapunov stable sets (all concepts defined for the action of
the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations
from $M$ to $N$.
Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spaces Categories:37B25, 37C75, 34C27, 34D05 

2. CJM 2015 (vol 67 pp. 1270)
 Carcamo, Cristian; Vidal, Claudio

Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.
Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov sense Categories:34D20, 34E10 

3. CJM 2014 (vol 67 pp. 1065)
4. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

5. CJM 2012 (vol 65 pp. 808)
 Grandjean, Vincent

On Hessian Limit Directions along Gradient Trajectories
Given a nonoscillating gradient trajectory $\gamma$ of a real analytic function $f$,
we show that the limit $\nu$ of the secants at the limit point
$\mathbf{0}$
of $\gamma$ along the trajectory
$\gamma$ is an eigenvector of the limit of the direction of the
Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$
along $\gamma$. The same holds true at infinity if the function is globally subanalytic. We also deduce
some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is
of metric nature and still holds in a general Riemannian analytic setting.
Keywords:gradient trajectories, nonoscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx 

6. CJM 2011 (vol 64 pp. 961)
 Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim

Densities of Short Uniform Random Walks
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic
to expect similar results for more than five steps. New results are also
presented concerning the moments of uniform random walks and, in particular,
their derivatives. Relations with Mahler measures are discussed.
Keywords:random walks, hypergeometric functions, Mahler measure Categories:60G50, 33C20, 34M25, 44A10 

7. CJM 2010 (vol 62 pp. 261)
8. CJM 2009 (vol 62 pp. 74)
 Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre

Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces
We present the explicit formulas for the projectors on the generalized
eigenspaces associated with some eigenvalues for linear neutral functional
differential equations (NFDE) in $L^{p}$ spaces by using integrated
semigroup theory. The analysis is based on the main result
established elsewhere by the authors and results by Magal and Ruan
on nondensely defined Cauchy problem.
We formulate the NFDE as a nondensely defined Cauchy problem and obtain
some spectral properties from which we then derive explicit formulas for
the projectors on the generalized eigenspaces associated with some
eigenvalues. Such explicit formulas are important in studying bifurcations
in some semilinear problems.
Keywords:neutral functional differential equations, semilinear problem, integrated semigroup, spectrum, projectors Categories:34K05, 35K57, 47A56, 47H20 

9. CJM 2007 (vol 59 pp. 393)
 Servat, E.

Le splitting pour l'opÃ©rateur de KleinGordon: une approche heuristique et numÃ©rique
Dans cet article on \'etudie la diff\'erence entre les deux
premi\`eres valeurs propres, le splitting, d'un op\'erateur de
KleinGordon semiclassique unidimensionnel, dans le cas d'un
potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une
petite barri\`ere de potentiel, B. Helffer et B. Parisse ont obtenu
des r\'esultats analogues \`a ceux existant pour l'op\'erateur de
Schr\"odinger. Dans le cas d'une grande barri\`ere de potentiel, on
obtient ici des estimations des tranform\'ees de Fourier des fonctions
propres qui conduisent \`a une conjecture du splitting. Des calculs
num\'eriques viennent appuyer cette conjecture.
Categories:35P05, 34L16, 34E05, 47A10, 47A70 

10. CJM 2007 (vol 59 pp. 127)
 Lamzouri, Youness

Smooth Values of the Iterates of the Euler PhiFunction
Let $\phi(n)$ be the Euler phifunction, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$smooth,
conditionally on a weak form of the ElliottHalberstam conjecture.
Categories:11N37, 11B37, 34K05, 45J05 

11. CJM 2006 (vol 58 pp. 726)
 Chiang, YikMan; Ismail, Mourad E. H.

On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
We show that the value distribution (complex oscillation) of
solutions of certain periodic second order ordinary differential
equations studied by Bank, Laine and Langley is closely
related to confluent hypergeometric functions, Bessel functions
and Bessel polynomials. As a result, we give a complete
characterization of the zerodistribution in the sense of
Nevanlinna theory of the solutions for two classes of the ODEs.
Our approach uses special functions and their asymptotics. New
results concerning finiteness of the number of zeros
(finitezeros) problem of Bessel and Coulomb wave functions with
respect to the parameters are also obtained as a consequence. We
demonstrate that the problem for the remaining class of ODEs not
covered by the above ``special function approach" can be
described by a classical Heine problem for differential
equations that admit polynomial solutions.
Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble Categories:34M10, 33C15, 33C47 

12. CJM 2006 (vol 58 pp. 449)
 Agarwal, Ravi P.; Cao, Daomin; Lü, Haishen; O'Regan, Donal

Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$Laplacian Equations
Positive solutions are obtained for the boundary value problem
\[\begin{cases}
(  u' ^{p2}u')'
=\lambda f( t,u),\;t\in ( 0,1) ,p>1\\
u( 0) =u(1) =0.
\end{cases}
\]
Here $f(t,u) \geq M,$ ($M$ is a positive constant)
for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$.
We will show the existence of two positive
solutions by using degree theory together with the upperlower
solution method.
Keywords:one dimensional $p$Laplacian, positive solution, degree theory, upper and lower solution Category:34B15 

13. CJM 2004 (vol 56 pp. 310)
 Llibre, Jaume; Schlomiuk, Dana

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebrogeometric concepts of divisor and
zerocycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integervalued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.
Categories:34C40, 51F14, 14D05, 14D25 

14. CJM 2003 (vol 55 pp. 724)
 Cao, Xifang; Kong, Qingkai; Wu, Hongyou; Zettl, Anton

SturmLiouville Problems Whose Leading Coefficient Function Changes Sign
For a given SturmLiouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled selfadjoint boundary condition and those for two
corresponding separated selfadjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated selfadjoint boundary condition can
be numbered in terms of the Pr\"ufer angle; and our inequalities can
then be used to index the eigenvalues for any coupled selfadjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
SturmLiouville problems, and its range as a function on the space of
selfadjoint boundary conditions. We also relate this indexing scheme
to the number of zeros of eigenfunctions. In addition, we
characterize the discontinuities of each eigenvalue under a different
indexing scheme.
Categories:34B24, 34C10, 34L05, 34L15, 34L20 

15. CJM 2003 (vol 55 pp. 3)
 Baake, Michael; Baake, Ellen

An Exactly Solved Model for Mutation, Recombination and Selection
It is well known that rather general mutationrecombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a
multiple tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time,
with independent mutation events at the sites, and crossover events
between them. It admits a closed solution of the corresponding
differential equation on the basis of the original state space, and
also closed expressions for the linkage disequilibria, derived by means
of M\"obius inversion. As an extra benefit, the approach can be extended
to a model with selection of additive type across sites. We also derive
a necessary and sufficient criterion for the mean fitness to be a Lyapunov
function and determine the asymptotic behaviour of the solutions.
Keywords:population genetics, recombination, nonlinear $\ODE$s, measurevalued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 

16. CJM 2002 (vol 54 pp. 1187)
 Cobo, Milton; Gutierrez, Carlos; Llibre, Jaume

On the Injectivity of $C^1$ Maps of the Real Plane
Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Categories:34D05, 54H20, 58F10, 58F21 

17. CJM 2002 (vol 54 pp. 1142)
18. CJM 2002 (vol 54 pp. 1038)
 Gavrilov, Lubomir; Iliev, Iliya D.

Bifurcations of Limit Cycles From Infinity in Quadratic Systems
We investigate the bifurcation of limit cycles in oneparameter
unfoldings of quadractic differential systems in the plane having a
degenerate critical point at infinity. It is shown that there are
three types of quadratic systems possessing an elliptic critical point
which bifurcates from infinity together with eventual limit cycles
around it. We establish that these limit cycles can be studied by
performing a degenerate transformation which brings the system to a
small perturbation of certain wellknown reversible systems having a
center. The corresponding displacement function is then expanded in a
Puiseux series with respect to the small parameter and its
coefficients are expressed in terms of Abelian integrals. Finally, we
investigate in more detail four of the cases, among them the elliptic
case (BogdanovTakens system) and the isochronous center
$\mathcal{S}_3$. We show that in each of these cases the
corresponding vector space of bifurcation functions has the Chebishev
property: the number of the zeros of each function is less than the
dimension of the vector space. To prove this we construct the
bifurcation diagram of zeros of certain Abelian integrals in a complex
domain.
Categories:34C07, 34C05, 34C10 

19. CJM 2002 (vol 54 pp. 897)
 Fortuny Ayuso, Pedro

The Valuative Theory of Foliations
This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
Categories:12J20, 13F30, 16W60, 37F75, 34M25 

20. CJM 2002 (vol 54 pp. 648)
 Yuan, Wenjun; Li, Yezhou

Rational Solutions of PainlevÃ© Equations
Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$,
$\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the
necessary and sufficient conditions for the existence of rational
solutions of equation~(P$_6$) in term of special relations among the
parameters. The number of distinct rational solutions in each case is
exactly one or two or infinite. And each of them may be generated by
means of transformation group found by Okamoto [7] and B\"acklund
transformations found by Fokas and Yortsos [4]. A list of rational
solutions is included in the appendix. For the sake of completeness,
we collected all the corresponding results of other five Painlev\'e
equations (P$_1$)(P$_5$) below, which have been investigated by many
authors [1][7].
Keywords:PainlevÃ© differential equation, rational function, BÃ¤cklund transformation Categories:30D35, 34A20 

21. CJM 2000 (vol 52 pp. 248)
 Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.

Spectral Problems for NonLinear SturmLiouville Equations with Eigenparameter Dependent Boundary Conditions
The nonlinear SturmLiouville equation
$$
(py')' + qy = \lambda(1  f)ry \text{ on } [0,1]
$$
is considered subject to the boundary conditions
$$
(a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j =
0,1.
$$
Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending
on the independent variable $x$ alone, while $f$ depends on $x$,
$y$ and $y'$. Results are given on existence and location of sets
of $(\lambda,y)$ bifurcating from the linearized eigenvalues, and
for which $y$ has prescribed oscillation count, and on completeness
of the $y$ in an appropriate sense.
Categories:34B24, 34C23, 34L30 

22. CJM 1998 (vol 50 pp. 497)
23. CJM 1998 (vol 50 pp. 412)
 McIntosh, Richard J.

Asymptotic transformations of $q$series
For the $q$series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$
we construct a companion $q$series such that the asymptotic
expansions of their logarithms as $q\to 1^{\scriptscriptstyle }$
differ only in the dominant few terms. The asymptotic expansion
of their quotient then has a simple closed form; this gives rise
to a new $q$hypergeometric identity. We give an asymptotic
expansion of a general class of $q$series containing some of
Ramanujan's mock theta functions and Selberg's identities.
Categories:11B65, 33D10, 34E05, 41A60 

24. 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 

25. 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 
