On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

Isolating Segments and Anti-Periodic Solutions

 We prove a sufficient condition for the existence of 2 ⁿ (geometrically distinct) solutions of the two point boundary value problem

Stability and multiplicity of solutions for the thermistor problem

The stability of the stationary solution of the thermistor as a circuit element is studied using a Liapunov functional and the Hale–LaSalle invariance principle. The asymptotic stability of a class of periodic solutions is also considered. Received: November 22, 1999; in final form: May 23, 2001?Published online: May 29, 2002     

Eigenvalue estimates of the Dirac operator depending on the Ricci tensor

We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenb?ck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds. Received: 20 April 2001 / Published online: 5 September 2002     

Invariance of the stability of meissner’s equation under a permutation of its intervals

The aim of this paper is to study a particular aspect of the stability of Meissner’s equation, which leads to some interesting questions about the invariance of the trace of a product of matrices. Received: March 3, 2000?Published online: July 13, 2001     

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

Proprietà di 20 ordine di un riferimento isotropo

We extend to the isotropic case, by adapted non-holonomic techniques [1], 2^° order properties of a relativistic frame of reference, generalized in the polar sense [2,3]: Riemann and gravitational tensors decomposition, Lie derivative of the Ricci rotation coefficients, commutation formulae and finally Bianchi identity. All the decomposition tensors are formally invariant (as regards the standard case), by means of the longitudinal derivative extension. Received: June 14, 2000?Published online: October 2, 2001     

Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below

Let M be a complete open n-manifold with a base point p, at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface. We discuss the maximal diameter theorem for the compactification of M by attaching the ideal boundary. Under certain conditions we prove that p becomes a pole and that M is isometric to the n-model. Received: 24 September 2000; in final form: 21 November 2001 / Published online: 17 June 2002 Dedicated to Professor Su Bu-Chin on the occasion of his one hundredth birthday The work of the first author was partially supported by the Grant-in-Aid for Scientific Research, No. 12440021 and for Exploratory Research, No. 13874012     

Invariant (α, β)-metrics on homogeneous manifolds

In this paper we study invariant (α, β)-metrics on homogeneous spaces. We first give a method to construct invariant (α, β)-metrics on homogeneous spaces. Then we obtain some conditions for some special type of (α, β)-metrics to be of Berwald type and Douglas type. At last, we give a rigidity result concerning the Randers metrics and Matsumoto metrics of Berwald type on homogeneous spaces. Members of LPMC and supported by NSFC (no. 10671096) and NCET of China. Second author was corresponding author. Authors’ address: Huihui An and Shaoqiang Deng, School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China     

Invariant subsets of rank 1 manifolds

It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M. Received: 6 July 2000 / Revised version: 11 October 2001     

A Note on General Blow-Ups of Elliptic Quasi-Bundles

In this paper we give some numerical conditions for a line bundle on a general blow-up of elliptic quasi bundles to give an embedding of order k.     

On the spectral flow in Lorentzian manifolds

In this paper we use functional analytical techniques to determine the differential equation satisfied by the eigenvalues of a smooth family of Fredholm operators, obtained from the index form along a Lorentzian geodesic. The formula is then applied to the study of the evolution of the index function, and, using a perturbation argument, we prove a version of the classical Morse index theorem for stationary Lorentzian manifolds. Received: January 31, 2000; in final form: March 13, 2002?Published online: February 20, 2003 The second author is partially sponsored by CNPq (Brazil), Grant 200615/01-7.     

Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface

In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε². The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunction. Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002     

On the existence and stability of periodic orbits in non ideal problems: General results

In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E. , where , are T periodic functions of t and there is a ₀ ∈ Ω such that f ( a ₀) = 0 and f ′( a ₀) is a nilpotent matrix. When n = 3 and f (x) = (0, q (x ₃) , 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system: the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld Effect as a bifurcation of periodic orbits.     

Polyhedral results for two-connected networks with bounded rings

We study the polyhedron associated with a network design problem which consists in determining at minimum cost a two-connected network such that the shortest cycle to which each edge belongs (a “ring”) does not exceed a given length K.?We present here a new formulation of the problem and derive facet results for different classes of valid inequalities. We study the separation problems associated to these inequalities and their integration in a Branch-and-Cut algorithm, and provide extensive computational results. Received: September 1999 / Accepted: February 2002?Published online May 8, 2002     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

On the existence of periodic solutions of Rayleigh equations

In this paper, we study the existence of periodic solutions of Rayleigh equation