We study the spectrum $\mathit{\sigma }(T)$ of a weighted composition operator T induced by a weight $m\in \mathrm{Hol}(\mathbb{D})$ and a holomorphic self-map φ on the unit disc, which is not an elliptic automorphism. If φ has a unique fixed point in $\mathbb{D}$, we show that $\mathit{\sigma }(T)$ is a bounded discrete set such that $\mathit{\sigma }(T)\setminus \{0\}$ is a set of eigenvalues with multiplicity one. If φ has a Denjoy–Wolff point α on the unit circle, we first prove that the point spectrum is $\mathbb{C}\setminus \{0\}$ whenever $m\ne 0$ is constant. Moreover, the multiplicity of each eigenvalue is infinite. Then we describe classes of m for which the point spectrum of T is either empty or equal to $\mathbb{C}\setminus \{0\}$.

For a given closed oriented surface Σ, we introduce a chord index homomorphism from a subgroup of $H_1(\mathrm{\Sigma },\mathbb{Z})$ to the group of chord indices of a knot K in $\mathrm{\Sigma }\times I$. Some knot invariants derived from this homomorphism are discussed.

In this article, we define restricted log-exp-analytic functions as compositions of log-analytic functions and exponentials of locally bounded functions. We prove that the derivative of a restricted log-exp-analytic function is again restricted log-exp-analytic and that such a function exhibits strong quasianalytic properties. We establish the parametric version of Tamm’s theorem for this class of functions.

In this paper, we discuss the primality of weakly connected collections of cells, showing that the ideal generated by inner 2-minors attached to a weakly connected and simple collection of cells is the toric ideal of the edge ring of a weakly chordal bipartite graph. As an application of this result, we characterize the primality of the polyomino ideals of weakly closed paths—a new class of nonsimple polyominoes.

We study the minimal primary decomposition of completely t-spread lexsegment ideals via simplicial complexes. We determine some algebraic invariants of such a class of t-spread ideals. Hence, we classify all t-spread lexsegment ideals which are Cohen–Macaulay.

We define recurrence for a compactly generated para-topological group G acting continuously on a locally compact Hausdorff space X with $dimX=0$, and then we show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions are pairwise equivalent: (i) this dynamic is pointwise recurrent, (ii) X is a union of G-minimal sets, (iii) the G-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous. Consequently, if this dynamic is pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and nonconnected G-flow has a nontrivial equicontinuous pointwise regularly almost periodic factor.

In the present paper, we’ll introduce quantities measuring how far a (Schauder) basis is from being boundedly complete or shrinking. These quantities will be proved to really measure nonbounded completeness or nonshrinkingness of bases by investigating many bases. As applications, they will be used to prove quantitative versions of the well-known relationships between shrinking bases and boundedly complete bases due to R. C. James.