We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set Σ(f) is totally disconnected, then f is topologically conjugate to a conformal transformation.

where tn → 1 n → ∞ . We may assume that tn ≥ 1 for all n 1, 2, 3, . . . . Denote by F T the set of fixed points of T . Throughout this paper T and S : C → C are two commutative asymptotically nonexpansive mappings with asymptotical coefficients {tn} and {sn}, respectively. Suppose that F : F T ∩F S / ∅ 1, Goebel and Kirk’s theorem makes it possible . It is well known that F T and F S are convex...

If F : Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of F-coalgebras is a covariety, i.e., closed under H (homomorphic images), S (sub-coalgebras) and ∑ (sums), if and only if it can be de0ned by a set of “coequations”. Similarly, quasi-covarieties, i.e., classes closed under H and ∑ , can be characterized by implications of coequations. These res...

In a recent paper [3] it was shown that No Free Lunch results [5] hold for any subset F of the set of all possible functions from a finite set X to a finite set Y iff F is closed under permutation of X . In this article, we prove that the number of those subsets can be neglected compared to the overall number of possible subsets. Further, we present some arguments why problem classes relevant i...

Proof. [Solution due to Claudiu Raicu] Consider two polynomials f, g ∈ C[X±] such that the spherical graphs they determine intersect transversely at the points where the vectors vi intersect the sphere. This implies that T (f) and T (g) intersect transversely ([1]) at each point and T (f)∩T (g) = Σ. Using the Transverse Intersection Lemma (lemma 3.2, corollary 3.3 of [1]) we get that Σ = T (f, ...

We show that in a Banach space X every closed convex subset is strongly proximinal if and only if the dual norm is strongly sub differentiable and for each norm one functional f in the dual space X∗, JX(f) the set of norm one elements in X where f attains its norm is compact. As a consequence, it is observed that if the dual norm is strongly sub differentiable then every closed convex subset of...

Let X ⊂ R be a closed semialgebraic set and let f P(X) be the ring obtained from the characteristic function of X by the operations +, −, ∗, the half link operator and by the polynomial operations with rational coefficients which preserve finite formal sum of signs. McCrory and Parusiński proved that necessary conditions for X to be semialgebraically homeomorphic to a real algebraic set is that...

Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then f−1(y) is a Kσ set for all except countably many y ∈M , that M is also Luzin, and that the Borel classes of the sets f(F ), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the cla...

belongs to class S if f(z) is regular and schlicht in zI < 1 and a, = 1. Let R be a closed set lying in zj < 1, and let *,1,(r) be a measure function defined in the space R. Given an integer n there is a number M M. such that If()-(z) < M for all z C R and for v = O, 1, 2, ..., n when f(z) belongs to class S. Let F,(ro Ro, ..., rny ) denote a complex-valued function which is continuous together...

This is a common introduction to math.AG/0011176, math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. In these papers one studies the automorphism group G of an extension F/k of algebraically closed fields, especially in the case of countable transcendence degree and characteristic zero, its smooth linear and semi-linear representations, and their relations to algebraic geometry...