Boundary constructions of petals at the Wolff point in the parabolic case

In the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff point, for a holomorphic self-map f of the open unit disc Δ ⊂ ℂ of parabolic type. The result is obtained in the framework of two interesting dynamical situations which require different kinds of regularity of f at the Wolff point τ: f of non-automorphism type and or f injective of automorphism type, f∈C ^3+ɛ(τ) and . Partially supported by PRIN Proprietà geometriche delle varietà reali e complesse. Partially supported by GNSAGA of the Istituto Nazionale di Alta Matematica, Rome.     

Common fixed point results for mappings of Perov type

Fixed point results for mappings f and g are obtained where f and g are related in the sense of Das and Naik, ?iri?, Hardy and Rogers, and all are of Perov type. We consider both normal and non‐normal cones. The theory is illustrated with some examples.     

Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

Equivariance and Multiplicity for the Scalar Curvature Problem

Given (M, g ₀) a three-dimensional compact Riemannian manifold, assumed not to be conformally diffeomorphic to the standard unit 3-sphere, and G a compactsubgroup of the conformal group of (M, g ₀), we first study conditions for a smooth G-invariant function f to be the scalar curvature of a G-invariant conformalmetric to g ₀. Then, extending previous results of Hebeyand Vaugon, we study conditions for f to be the scalarcurvature of at least two conformal metrics to g ₀.     

A note on holomorphic approximation in Banach spaces

Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X. Research partially supported by NSF grant DMS0700281.     

The Kramers–;Wannier Symmetry and S-Duality in the Two-Dimensional g03A6;4 Theory

We show that the exact beta function of the two-dimensional g⁴ theory possesses two dual symmetries. These are the Kramers–Wannier symmetry d(g) and the strong–weak-coupling symmetry, or the S-duality f(g), connecting the strong- and weak-coupling domains lying above and below the fixed point g ^*. We obtain explicit representations for the functions d(g) and f(g). The S-duality transformation f(g) allows using the high-temperature expansions to approximate the contributions of the higher-order Feynman diagrams. From the mathematical standpoint, the proposed scheme is highly unstable. Nevertheless, the approximate values of the renormalized coupling constant g ^* obtained from the duality equations agree well with the available numerical results.     

Learning Functions of Few Arbitrary Linear Parameters in High Dimensions

Let us assume that f is a continuous function defined on the unit ball of ℝ ^d , of the form f(x)=g(Ax), where A is a k×d matrix and g is a function of k variables for k≪d. We are given a budget m∈ℕ of possible point evaluations f(x _i ), i=1,…,m, of f, which we are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present in this paper

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.      

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier, under some additional condition (the position of the Denjoy–Wolff point) it was shown in [13] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy–Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy–Wolff point (the case not covered in [13]) than with all the other boundary fixed points of the semigroup.     

Julia sets of permutable Holomorphic maps

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Obstruction theory and coincidences of maps between nilmanifolds

Let f, g : M N be two maps between two compact nilmanifolds with dim M dim N = n. In this paper, we show that either the Nielsen coincidence number N(f, g) = 0 or N(f, g) = R(f, g) where R(f, g) denotes the Reidemeister number of f and g. Furthermore, we show that if N(f, g) > 0 then the primary obstruction o_n(f, g) to deforming f and g to be coincidence free on the n-th skeleton of M is non-trivial.Received: 30 April 2004; revised: 20 July 2004     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f m g.     

A counter-example for the Bezout equation in the Hardy classH p

Leff ₁,f ₂ be bounded holomorphic functions in the unit disc of the complex plane ℂ. Using a recent result of S. Treil about estimates in the corona theorem, we strengthen a counterexample given by Amar, Bruna and Nicolau to the existence of functionsg ₁,g ₂ in the Hardy spaceH ^p ( ) verifying the Bezout equationf ₁ g ₁+f ₂ g ₂=1.     

Dividing Polynomials Using the Resultant Matrix

Let f, g be polynomials over a Noetherian ring A. We use the matrix coming from the resultant of f and g to get a criterion for divisibility of f by g in terms of Fitting invariants as well as a method of dividing polynomials once we know g divides f. Further we show that this is equivalent to that cokernel or the image of the multiplication-by-g map on A[X]/(f) is free. As an application we show one can test irreducibility of an integral polynomial by computing minors of a matrix.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

Berezin Transform, Mellin Transform and Toeplitz Operators

We consider functions of the form f₁[`(g)]<sub>1</sub>+h{f_1\bar g_1+h} in the range of the Berezin transform B, where f <sub>1</sub> and g <sub>1</sub> are holomorphic on the unit disk \mathbb D{\mathbb D}, and h is either harmonic or of the form f<sub>2</sub>[`(g)]₂{f_2\bar g_2} for some holomorphic functions f ₂ and g ₂ on \mathbb D{\mathbb D}. First, by using the Mellin transform, we complement Ahern’s Theorem (Ahern in J Funct Anal 215:206–216, 2004) by proving that if u ? L¹{u\in L^1} and B(u) is harmonic, then u is harmonic. Secondly, we extend Ahern’s Theorem when h is harmonic, and give very precise relations between f ₁ and f ₂, g ₁ and g ₂ when h=f₂[`(g)]₂{h=f_2\bar g_2} and g ₂(z) = z ⁿ with n ≥ 1. Finally, some applications of our results to the theory of Toeplitz operators are discussed.     

Une version feuilletée équivariante du théorème de translation de Brouwer

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f^–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any Hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.