Graded Jacobi operators on the algebra of differential forms

One-to-one correspondences are established between the set ofall nondegenerate graded Jacobi operators of degree -1 defined onthe graded algebra of differential forms on a smooth, oriented,Riemannian manifold M, the space of bundle isomorphisms , and the space of nondegenerate derivations of degree 1 havingnull square. Derivations with this property, andJacobi structures of odd -degree are also studied throughthe action of the automorphism group of .     

Linear-quadratic control and quadratic differential forms for multidimensional behaviors

This paper deals with systems described by constant coefficient linear partial differential equations (nD-systems) from a behavioral point of view. In this context we treat the linear quadratic control problem where the performance functional is the integral of a quadratic differential form. We look for characterizations of the set of stationary trajectories and of the set of local minimal trajectories with respect to compact support variations, turning out that they are equal if the system is dissipative. Finally we provide conditions for regular implementability of this set of trajectories and give an explicit representation of an optimal controller.     

Limit algebras of differential forms in non-commutative geometry

Given a C*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω_∞ A and Ω_ε A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω_∞ A induced by a K-cycle on A are considered. The GNS-representation of Ω_∞ A defined by a d-dimensional non-commutative volume integral on a d ⁺-summable K-cycle on A is realized as the representation induced by the left action of A on Ω*A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).     

Polynomial normal forms of constrained differential equations with three parameters

We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow–fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables.     

Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction

For finite-dimensional bifurcation problems, it is well known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called “radial” part and an “angular” part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings. In this paper, we are interested in the case where such bifurcations occur in retarded functional differential equations, and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay-differential equations. Our analysis allows us to recover and considerably generalize recent results by Faria and Magalhães [T. Faria, L.T. Magalhães, Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. Dynam. Differential Equations 8 (1996) 35-70] and by Buono and Bélair [P.-L. Buono, J. Bélair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations 189 (2003) 234-266].     

Central limit theorem for quadratic forms for sparse tables

Sufficient conditions for asymptotic normality for quadratic forms in {n_t − np_t} are given, where {n_t} are the observed counts with expected cell means {np_t}. The main result is used to derive asymptotic distributions of many statistics including the Pearson's chi-square.     

Inequalities for two type potential operators on differential forms

The purpose of this paper is to derive some strong-type inequalities for convolution type potential operator applied on differential forms. Caccioppoli-type inequalities for integral type potential operator acting on A-harmonic tensor are also obtained.     

An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem

In this paper, an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions. These conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

一类微分系统的极限环分布情况

用定性分析和数值判定方法,对一类微分系统x=y,y=x(l-bx2) (α-cx2)y(其中l＞0,b＞0,c≠0)的极限环分布情况进行了研究,得出该系统有3个极限环,并且给出了该系统所有极限环的分布情况.     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

A modularity criterion for Klein forms, with an application to modular forms of level 13

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ₁(N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ₁(13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).     

Multivalued solutions of second-order differential equations

Multivalued (not set-valued as in the theory of differential inclusions!) solutions of ordinary differential equations (ODE) appear naturally in geometrical and physical problems in which the independent and dependent variablesx, y are geometric coordinates of a current point on the sought-for curve. This note contains some simple results concerning smooth multivalued solutions of real second-order ODE resolved with respect toy″; the special role of equations of the third degree with respect toy′ is underlined. The method of investigation is based on combining ODEs fory(x) andx(y). Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 871–878, December, 1999.     

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.     

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.