This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$ , where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

We give an explicit way of writing down a minimal set of generators for the canonical ideal of a nondegenerate curve, or of a more general smooth projective curve in a toric surface, in terms of its defining Laurent polynomial.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$ , there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$ , $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$ . Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$ , where $\mathsf{D}(G)$ denotes the Davenport constant of $G$ . The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$ . Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

Let ${\rm\Delta}:G\rightarrow \text{GL}(n,K)$ be an absolutely irreducible representation of an arbitrary group  $G$ over an arbitrary field  $K$ ; let ${\it\chi}:G\rightarrow K:g\mapsto \text{tr}({\rm\Delta}(g))$ be its character. In this paper, we assume knowledge of ${\it\chi}$ only, and study which properties of ${\rm\Delta}$ can be inferred. We prove criteria to decide whether ${\rm\Delta}$ preserves a form, is realizable over a subfield, or acts imprimitively on  $K^{n\times 1}$ . If $K$ is finite, we can decide whether the image of ${\rm\Delta}$ belongs to certain Aschbacher classes.

We introduce an approximation property ( ${\mathcal{K}}_{\mathit{up}}$ -AP, $1\leq p<\infty$ ), which is weaker than the classical approximation property, and discover the duality relationship between the ${\mathcal{K}}_{\mathit{up}}$ -AP and the ${\mathcal{K}}_{p}$ -AP. More precisely, we prove that for every $1<p<\infty$ , if the dual space $X^{\ast }$ of a Banach space $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP, then $X$ has the ${\mathcal{K}}_{p}$ -AP, and if $X^{\ast }$ has the ${\mathcal{K}}_{p}$ -AP, then $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP. As a consequence, it follows that every Banach space has the ${\mathcal{K}}_{u2}$ -AP and that for every $1<p<\infty$ , $p\neq 2$ , there exists a separable reflexive Banach space failing to have the ${\mathcal{K}}_{\mathit{up}}$ -AP.

Let ${\it\gamma}$ be a hyperbolic closed orbit of a $C^{1}$ vector field $X$ on a compact $C^{\infty }$ manifold $M$ and let $H_{X}({\it\gamma})$ be the homoclinic class of $X$ containing ${\it\gamma}$ . In this paper, we prove that if a $C^{1}$ -persistently expansive homoclinic class $H_{X}({\it\gamma})$ has the shadowing property, then $H_{X}({\it\gamma})$ is hyperbolic.

Thestudy concerns semistability and stability of probability measures on a convex cone, showing that the set $S(\boldsymbol{{\it\mu}})$ of all positive numbers $t>0$ such that a given probability measure $\boldsymbol{{\it\mu}}$ is $t$ -semistable establishes a closed subgroup of the multiplicative group $R^{+}$ ; semistability and stability exponents of probability measures are positive numbers if and only if the neutral element of the convex cone coincides with the origin; a probability measure is (semi)stable if and only if its domain of (semi-)attraction is not empty; and the domain of attraction of a given stable probability measure coincides with its domain of semi-attraction.

For a positive integer $n\geq 2$ , let $M_{n}$ be the set of $n\times n$ complex matrices and $H_{n}$ the set of Hermitian matrices in $M_{n}$ . We characterize injective linear maps ${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$ satisfying

$$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$

$A_{k}\in H_{m_{k}}$

$k=1,\dots ,l$

$l,m_{1},\dots ,m_{l}\geq 2$