Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a tr...

We show that the roots of any smooth curve of polynomials with real roots only can be parametrized twice differentiable (but not better). In [1] we claimed that there exists a smooth curve of polynomials of degree 3 for which no C-parametrization of the roots exists. Unfortunately there was an error in the calculation of b3 and we have been informed by Jacques Chaumat and Anne-Marie Chollet in ...

A general H∞ controller design technique is proposed for scalar linear systems, based on properties of positive polynomial matrices. The order of the controller is fixed from the outset, independently of the order of the plant and weighting functions. A sufficient LMI condition is used to overcome non-convexity of the original design problem. The key design step, as well as the whole degrees of...

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. Si denotes a generalized 3-star, all three maximal paths starting in Z have exactly i + 1 vertices (including Z). Let M be a surface of Euler characteristic χ(M) ≤ 0, and m(M) := b 5+ √ 49−24χ(M) 2 c. We prove: (1) Let k ≥ 1, d ≥ m(M) be integers. Each polyhedral map G on M with a k-path (on k ...

We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa’s formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

(Algebra) It’s a complex projective manifold C of dimension one. As such, it has a field of rational functions K(C) of transcendence degree one over the field C of scalars. The curve can be recovered from the field as the set of discrete valuation rings of K/C, which has the “natural” structure of a projective manifold. There is a canonical line bundle ωC , whose degree captures the genus via d...

Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function which agrees with the characteristic function of A on its domain D, and furthermore D has density 1, i.e. limn→∞ |{k < n : k ∈ D}|/n = 1. A set A is called co...

1. We consider a polynomial f of degree d _> 2, and we denote its n-th iteration by fn . The results of the theory of iterations that are used in the present article may be found in [1, 2]. A root of the equation f n z = z is called a periodic point (with period n). The quantity X(z) = ~ log I(f~)'(z)l is the characteristic exponent of this point. When the Julia set of the polynomial f is conne...