Expand all Collapse all | Results 1 - 25 of 25 |
1. CJM Online first
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, non-compact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 |
2. CJM 2012 (vol 65 pp. 808)
On Hessian Limit Directions along Gradient Trajectories Given a non-oscillating 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 eigen-vector 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 sub-analytic. 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, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx |
3. CJM 2011 (vol 64 pp. 961)
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 |
4. CJM 2010 (vol 62 pp. 261)
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials |
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials No abstract.
Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Problem Categories:34M10, 33C15, 33C47 |
5. CJM 2009 (vol 62 pp. 74)
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces |
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 non-densely defined Cauchy problem.
We formulate the NFDE as a non-densely 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 semi-linear problems.
Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectors Categories:34K05, 35K57, 47A56, 47H20 |
6. CJM 2007 (vol 59 pp. 393)
Le splitting pour l'opÃ©rateur de Klein--Gordon: 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
Klein--Gordon semi-classique 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 |
7. CJM 2007 (vol 59 pp. 127)
Smooth Values of the Iterates of the Euler Phi-Function Let $\phi(n)$ be the Euler phi-function, 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 Elliott--Halberstam conjecture.
Categories:11N37, 11B37, 34K05, 45J05 |
8. CJM 2006 (vol 58 pp. 726)
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials |
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 zero-distribution 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
(finite-zeros) 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 |
9. CJM 2006 (vol 58 pp. 449)
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations Positive solutions are obtained for the boundary value problem
\[\begin{cases}
-( | u'| ^{p-2}u')'
=\lambda f( t,u),\;t\in ( 0,1) ,p>1\\
u( 0) =u(1) =0.
\end{cases}
\]
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 upper-lower
solution method.
Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution Category:34B15 |
10. CJM 2004 (vol 56 pp. 310)
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 16-th
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 algebro-geometric concepts of divisor and
zero-cycle 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 integer-valued 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 |
11. CJM 2003 (vol 55 pp. 724)
Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign For a given Sturm-Liouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled self-adjoint boundary condition and those for two
corresponding separated self-adjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated self-adjoint 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 self-adjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
Sturm-Liouville problems, and its range as a function on the space of
self-adjoint 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 |
12. CJM 2003 (vol 55 pp. 3)
An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination 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, measure-valued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |
13. CJM 2002 (vol 54 pp. 1187)
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 |
14. CJM 2002 (vol 54 pp. 1142)
Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions |
Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions Form domains are characterized for regular $2n$-th order differential
equations subject to general self-adjoint boundary conditions
depending affinely on the eigenparameter. Corresponding modes of
convergence for eigenfunction expansions are studied, including
uniform convergence of the first $n-1$ derivatives.
Categories:47E05, 34B09, 47B50, 47B25, 34L10 |
15. CJM 2002 (vol 54 pp. 897)
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 |
16. CJM 2002 (vol 54 pp. 1038)
Bifurcations of Limit Cycles From Infinity in Quadratic Systems We investigate the bifurcation of limit cycles in one-parameter
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 well-known 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 (Bogdanov-Takens 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 |
17. CJM 2002 (vol 54 pp. 648)
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 |
18. CJM 2000 (vol 52 pp. 248)
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions |
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions The nonlinear Sturm-Liouville 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 |
19. CJM 1998 (vol 50 pp. 497)
Morse index of approximating periodic solutions for the billiard problem. Application to existence results |
Morse index of approximating periodic solutions for the billiard problem. Application to existence results This paper deals with periodic solutions for the billiard problem in a
bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular
solutions of Lagrangian systems with a potential well. We give a
precise link between the Morse index of approximate solutions
(regarded as critical points of Lagrangian functionals) and the
properties of the bounce trajectory to which they converge.
Categories:34C25, 58E50 |
20. CJM 1998 (vol 50 pp. 412)
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 |
21. CJM 1998 (vol 50 pp. 40)
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 rank-one
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 (multi-valued) 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 |
22. CJM 1997 (vol 49 pp. 1066)
Multiparameter Variational Eigenvalue Problems with Indefinite Nonlinearity We consider the multiparameter nonlinear Sturm-Liouville 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 $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n)
\in \bar{R}_+^m \times R_+^{n-m} \bigl(R_+ := (0, \infty)\bigr)$
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 Ljusternik-Schnirelman 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 |
23. CJM 1997 (vol 49 pp. 583)
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
algebro-geometric 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 |
24. CJM 1997 (vol 49 pp. 338)
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
Christopher-Lloyd 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 |
25. CJM 1997 (vol 49 pp. 212)
Differential equations defined by the sum of two quasi-homogeneous 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
quasi-homogeneous 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 non-linearities.
Categories:34C05, 58F21 |