Level curves of open polynomial functions on the real plane

Let f : f ² → R be an open polynomial function. Thenf changes sign across V(f) (alternatively around a singular point of V(f)) and the function c : R → N expressing the number f(λ) of connected components of the λ-level curve of f is lower semicontinuous; it has a removable singilarity at every value λ which is critical and is not a real critical value at infinity for f.     

Periodic points and rotation numbers for area preserving diffeomorphisms of the plane

Letf be an orientation preserving diffeomorphism ofR ² which preserves area. We prove the existence of infinitely many periodic points with distinct rotation numbers around a fixed point off, provided only thatf has two fixed points whose infinitesimal rotation numbers are not both 0. We also show that if a fixed pointz off is enclosed in a “simple heteroclinic cycle” and has a non-zero infinitesimal rotation numberr, then for every non-zero rational numberp/q in an interval with endpoints 0 andr, there is a periodic orbit inside the heteroclinic cycle with rotation numberp/q aroundz.     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.     

Taylor polynomials and non-homogeneous blow-ups

The graph of a function f is subjected to non-homogeneous dilatations around (x₀;f(x₀)), related to the Taylor expansion of f at x₀. Some natural questions about convergence are considered and answered. Finally, it is provided a counterexample to a statement which was presumed to be true in former literature.Mathematics Subject Classification (1991): Primary 41A10, 49Q15, 53A05, 54G20; Secondary 28A75, 28A78, 28A33, 54C20     

Approaches to Effective Semi-Continuity of Real Functions

For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo(f) is recursively enumerably open in dom(f) × ?; we call f lower semi-computable of type two, if its closed epigraph Epi(f) is recursively enumerably closed in dom(f) × ?; we call f lower semi-computable of type three, if Epi(f) is recursively closed in dom(f) × ?. We show that type one and type two semi-computability are independent and that type three semi-computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.     

On furstenberg’s characterization of harmonic functions on symmetric spaces

We provide a necessary and sufficient condition on a radial probability measureμ on a symmetric space for whichf =f *μ, f bounded, implies thatf is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that iff is a bounded function on a symmetric space which satisfiesf =f *μ for some radialabsolutely continuous probability measureμ, thenf is harmonic.     

Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

We investigateV _f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq. It has been shown that iff is not bijective, thenV _f ≤q−(q−1)/n. Polynomials do exist which essentially achieve that bound. We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV _f ≤(5/6)q. We consider related problems for curves and higher dimensional varieties. This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well. Both authors partially supported by the NSF.     

On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f ₀ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f ₀. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f ₀ is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than \frac12{\frac{1}{2}}. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.     

A Reconstruction Theorem for Riemannian Symmetric Spaces of Noncompact Type

Let f be a rapidly decreasing radial function on a Riemannian symmetric space of noncompact type whose spherical Fourier transform has compact support. We prove a reconstruction theorem which recovers f from the values of an integral operator applied to f on a discrete subset. When G/K is of the complex type we prove a sampling formula recovering f from its own values on a discrete subset. We give explicit results for three dimensional hyperbolic space.     

Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere

We characterize Hopf hypersurfaces inS ⁶ as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS ⁶.     

Derivatives of Computable Functions

As is well known the derivative of a computable and C¹ function may not be computable. For a computable and C∞ function f, the sequence {f⁽ⁿ⁾} of its derivatives may fail to be computable as a sequence, even though its derivative of any order is computable. In this paper we present a necessary and sufficient condition for the sequence {f⁽ⁿ⁾} of derivatives of a computable and C^∞ function f to be computable. We also give a sharp regularity condition on an initial computable function f which insures the computability of its derivative f′.     

Sum List Coloring Block Graphs

A graph is f-choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. We characterize f-choosable functions for block graphs (graphs in which each block is a clique, including trees and line graphs of trees). The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f. The sum choice number of any graph is at most the number of vertices plus the number of edges. We show that this bound is tight for block graphs.Acknowledgments. Partially supported by a grant from the Reidler Foundation. The author would like to thank an anonymous referee for useful comments.     

On convergence of solutions to difference equations with additive perturbations

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behaviour of the map function f at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map f at zero exceeds one or, independently of the behaviour of f at zero, when the amplitudes are not square summable.     

A characterization of the disc via a Hessian equality

LetM be a bounded open plane domain. Let f be a continuous function on the closure of M, 3-times continuously differentiable in M, which vanishes on the boundary. Polterovich and Sodin proved that the values of f cannot exceed the norm of the Hessian of f, averaged over the entire domain M. In this work we study the equality case for this inequality. We show that equality holds if and only if M is an open disc and f belongs to a special class of radial functions. We also give an upper bound for f.     

Coupling the Proximal Point Algorithm with Approximation Methods

We study the convergence of a diagonal process for minimizing a closed proper convex function f, in which a proximal point iteration is applied to a sequence of functions approximating f. We prove that, when the approximation is sufficiently fast, and also when it is sufficiently slow, the sequence generated by the method converges toward a minimizer of f. Comparison to previous work is provided through examples in penalty methods for linear programming and Tikhonov regularization.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.      

Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization

We study a generalization of the nonderivative discrete gradient method of Bagirov et al. for minimizing a locally Lipschitz function f on ℝ ⁿ . We strengthen the existing convergence result for this method by showing that it either drives the f-values to −∞ or each of its cluster points is Clarke stationary for f, without requiring the compactness of the level sets of f. Our generalization is an approximate bundle method, which also subsumes the secant method of Bagirov et al.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory.