We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

Let $f$ be a transcendental meromorphic function with at least one direct tract. In this note, we investigate the structure of the escaping set which is in the same direct tract. We also give a theorem about the slow escaping set.

We describe a general framework for realistic analysis of sorting algorithms, and we apply it to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort, BubbleSort). Usually the analysis deals with the mean number of key comparisons, but here we view keys as words produced by the same source, which are compared via their symbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n2/4 and BubbleSort to n2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ (n log2n), Θ(n2), Θ (n2 log n). In these three cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely entropy and coincidence) with respect to the algorithm.

The paper examines a higher-order ordinary differential equation of the form $\mathcal{P}[u]\,{:=} \sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u \,{=}\, \lambda{u}, x\in[0,b)$, where $D\,{=}\,i({d}/{dx})$, and where the coefficients $a_{jk}$, $j,k\in[0,m]$, with $a_{mm}\,{=}\,1$, satisfy certain regularity conditions and are chosen so that the matrix $(a_{jk})$ is hermitean. It is also assumed that $m\,{>}\,1$. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients $a_{jk}$, $j,k\in[0,m]$, $j+k\neq{2m}$, as well as $b$ and the boundary conditions at 0 and at $b$ (if any).

A partition of the class of all Hermite–Biehler entire functions of finite order into subclasses ${\cal P}_\kappa$ is introduced. A given function $E(z)$ belongs to ${\cal P}_\kappa$ if and only if $-z^{-1}\log E(z)\in{\mc N}_\kappa$, where the class ${\mc N}_\kappa$ is a well studied family of meromorphic functions on the upper half plane, which originates from operator theoretic problems. It is also proved that the subclasses ${\cal P}_\kappa$ are stable under bounded type perturbation.

Let where {λn}n ∈ Ζ is a sequence of real numbers such that |λn — n| ≤ Δ for some Δ > 0 and all n ∈ ℤ . Extending an obvious property of sin πz to which the function G reduces when Δ = 0 we show that is bounded by a constant independent of n. The result is then applied to a problem concerning derivative sampling in one and several variables.

We adopt the terminology and notations of [5]. If f ∈ Bτ is an entire function of exponential type τ bounded on the real axis then we have the complementary interpolation formulas [1, p. 142-143]

and

where t, γ are reals and