We study transcendental entire maps f of finite order, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B of an attracting fixed point of f. Then the Julia set of f consists of disjoint curves tending to infinity (hairs), attached to the unique point accessible from B (endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class (i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

We consider classes of second order boundary value problems with a nonlinearity f(t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞ and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.

From Sturmian and Christoffel words we derive a strictly increasing function Δ:[0,∞) → . This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of Δ distinguishes some irrationality measures of real numbers.

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-DimsE = DimsE for all integers s > 0 and all analytic sets E ⊆ RN.

The aim of this paper is to prove that P-DimsE = DimsE for all real numbers s > 0 and analytic sets E ⊆ RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

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We consider certain sets of non-normal continued fractions for which the asymptotic frequencies of digit strings oscillate in one or other ways. The Hausdorff dimensions of these sets are shown to be the same value 1/2 as long as they are non-empty. An interesting example among them is the set of “extremely non-normal continued fractions” which was previously conjectured to be of Hausdorff dimension 0.

We present an example of an entire function with a Baker domain such that the complement of the set of singular values of the function contains annuli of arbitrarily large width. This is the first known example of a function with this property.

An asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.