According to Felix Klein's Erlanger program (1872), a (classical) geometry is the study of properties of a space X invariant under a group G of transformations of X. In practice G will be a Lie group which acts transitively on X, so that X is represented as a homogeneous space G=H, where H G is a closed subgroup. For example Euclidean geometry is the geometry of n-dimensional Euclidean space R ...

No single natural representation exists for visualizing all of the information in a vector eld. As a result, numerous methods have been developed, each emphasizing diierent aspects of the ow. This paper presents a method of ow visualization that highlights directional information in the eld by overloading feature glyphs with directional information information through a technique similar to bum...

In this paper we give an extension of Barbashin-Krasovski-LaSalle Theorem to a class of timevarying dynamical systems, namely the class of systems for which the restricted vector eld to the zero-set of the time derivative of the Liapunov function is time invariant and this set includes some trajectories. Our goal is to improve the su cient conditions for the case of uniform asymptotic stability...

We prove two statements. The rst one is a conjecture of Ian Hughes which states that if f1; : : : ; fn are primary invariants of a nite linear group G, then the least common multiple of the degrees of the fi is a multiple of the exponent of G. The second statement is about vector invariants: If G is a permutation group and K a eld of positive characteristic p such that p divides jGj, then the i...

Exactly 125 years ago G. Peano introduced the modern concept of vectors in his 1888 book "Geometric Calculus According to the Ausdehnungslehre (Theory of Extension) of H. Grassmann". Unknown to Peano, the young British mathematician W. K. Cli ord (1846-1879) in his 1878 work "Applications of Grassmann's Extensive Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern ...

Lattices over number elds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, short (pseudo)-basis is important in many applications. In this article, we provide the rst algorithm that computes such a short (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohs...

In this paper we are going to study the chain of some distributions required by nonlinear control problems associated to homogeneous polynomial nonlinear systems. The idea is due to Ruiz and Nijmeijer and the basic facts are strongly inspiered by their paper. The distribution studied here is the controlled invariant distribution used in the problems of local disturbance decoupling and input-out...

We present a symbolic algorithm to solve for the zeros of a polynomial vector eld equivariant with respect to a nite subgroup of O(n). We prove that the module of equivariant polynomial maps for a nite matrix group is Cohen-Macaulay and give an algorithm to compute a fundamental basis. Equivariant normal forms are easily computed from this basis. We use this basis to transform the problem of nd...

Behaviour of a weekly self-interacting scalar eld with a small mass in the de Sitter background is investigated using the stochastic approach (including the case of a double-well interaction potential). Existence of the de Sitter invariant equilibrium quantum state of the scalar eld in the presence of interaction is shown for any sign of the mass term. The stochastic approach is further develop...

We study the Ginzburg-Landau energy of superconductors submitted to a possibly non-uniform magnetic eld, in the limit of a large Ginzburg-Landau parameter. We prove that the induced magnetic elds associated to minimizers of the energy-functional converge as ! +1 to the solution of a free-boundary problem. This free boundary-problem has a nontrivial solution only when the applied magnetic eld is...