The aim of this paper is to introduce a technique for describing trajectories of systems of ordinary differential equations (ODEs) passing near saddle-fixed points. In contrast to classical linearization techniques, the methods of this paper allow for perturbations of the underlying vector fields. This robustness is vital when modelling systems containing small uncertainties, and in the develop...

The configuration of regular 3-forms in dimension 6 is generalized to n-forms in dimension 2n. The algebras of complex, paracomplex, and dual numbers are systematically used. The automorphism groups of all forms are determined. In the last decade there has arisen interest in exterior forms of higher degree, first of all in forms of degree 3 (see [2] and [3]). It is known (see [4] and [1]) that ...

In this work we consider the class of the compact connected three-dimensional manifolds with positive constant curvature, also known as the three-dimensional spherical space forms. These spaces, or subclasses like generalized quaternions or lens spaces, appear in many different contexts in topology and geometry, and have been completely classified; it is thus natural to ask if we can also count...

We consider a class of linear ordinary differential equations determined by a modular form of weight one, and construct vector-valued modular forms of weight two by using solutions of such differential equations.

A method for constructing the syzygies of vector-valued bilinear forms is given. Explicit formulas for the generators of the relations among the invariant rational functions are listed. These formulas have applications in the geometry of Riemannian submanifolds and in CR geometry. 1 Preliminaries 1.

Let V be a representation of the modular group Γ of dimension p. We show that the Z-graded space H(V ) of holomorphic vector-valued modular forms associated to V is a free module of rank p over the algebra M of classical holomorphic modular forms. We study the nature of H considered as a functor from Γ-modules to graded M-lattices and give some applications, including the calculation of the Hil...

Let R = k[x1, . . . , xr] denote the polynomial ring in r variables over a field k, with maximal ideal M = (x1, . . . , xr), and let V ⊂ Rj denote a vector subspace of the space Rj of degree-j homogeneous elements of R. We study three related algebras determined by V . The first is the ancestor algebra Anc(V ) = R/V whose defining ancestor ideal V is the largest graded ideal of R such that V ∩ ...

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

The present paper is devoted to studying a class of smoothly (C∞) finitely determined vector fields on R3. Given any such generic local system of the form ẋ = Ax + · · · , where A is a 3 × 3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C∞ classification. We also give a list of the simplest normal...