Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

Limit cycles and singular point quantities for a 3D Lotka-Volterra system

Four limit cycles are constructed for a three dimensional Lotka-Volterra system. This gives a good example to the cyclicity of 3D Lotka-Volterra systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. What is worth mentioning is that the expressions of focal values are simpler, and the formula is readily done with using computer symbol operation system such as Mathematica due to its linearity.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

Continuity,integrability and integral equivalence

Summary A class of continuous functions is characterized in terms of integrability and integral equivalence. Entrata in Redazione il 28 giugno 1971.     

On a singular point with one zero characteristic root

A sufficiently smooth real autonomous system of differential equations with an isolated singular point on the plane is considered. A new method for investigating the behavior of trajectories of the system in the neighborhood of this point is suggested.     

Quantum integrability of Beltrami-Laplace operator as geodesic equivalence

Given two Riemannian metrics on a closed connected manifold , we construct self-adjoint differential operators such that if the metrics have the same geodesics then the operators commute with the Beltrami-Laplace operator of the first metric and pairwise commute. If the operators commute and if they are linearly independent, then the metrics have the same geodesics. Received: 11 February 2000; in final form: 20 August 2000/ Published online: 17 May 2001     

On the center-focus problem for systems with a degenerate singular point

For an analytic system with nonzero cubic part and with a degenerate monodromic singular point, we indicate an algorithm for constructing the asymptotic expansion of the Poincaré map. We present closed-form expressions for the first four focus quantities and find two necessary conditions for a center.     

On a class of singular biharmonic problems involving critical exponents

This paper deals with the following class of singular biharmonic problems

     

Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.     

Analytic integrability and characterization of centers for nilpotent singular points

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form where P_n and Q_n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y²+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see [6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.     

On a singular and nonhomogeneous N-Laplacian equation involving critical growth

In this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form