The Mahler measure of a multivariable polynomial is given by an integral over the unit torus, while the areal Mahler measure, defined by Pritsker, is given by an integral over the product of unit disks. It is well-known that the classical Mahler measure is invariant under the change of variables $x\mapsto x^r$, where r is an integer, but this is not the case for the areal Mahler measure. In this note we investigate how the areal Mahler measure changes with this transformation and provide some specific examples.

The rank of a d-dimensional polytope P is defined by $F-(d+1)$, where F denotes the number of facets of P. In this paper, we focus on the toric rings of $(0,1)$-polytopes with small rank. We study their normality, the torsion-freeness of their divisor class groups, and the classification of their isomorphism classes.

In this paper, we aim to reveal the inherent structure of two important families of operators: the complex skew-symmetric compact operators and the self-commutators of a Toeplitz operator. More specifically, we first completely characterize when a general compact operator is complex skew-symmetric, and use a constructive way to obtain a skew-symmetric canonical form of such an operator. Then we obtain a neat rank-one decomposition of the self-commutators of the Toeplitz operator whose symbol is a bilateral rational function on the Hardy space of the unit disk. As an application of our main results, the complex skew-symmetry of the self-commutators of such a Toeplitz operator is studied. In particular, we completely characterize the corresponding problem for the Toeplitz operator whose symbol is a Laurent polynomial.

Let $\mathit{\sigma }=\{{\mathit{\sigma }}_j:j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroups

$$X=X_0\le X_1\le \cdots \le X_n=G$$

such that, for each $1\le i\le n-1$, $X_{i-1}⊴X_i$ or $X_i∕{(X_{i-1})}_{X_i}$ is a ${\mathit{\sigma }}_{j_i}$-group for some $j_i\in J$. Skiba studied the main properties of σ-subnormal subgroups in finite groups and showed that the set of all σ-subnormal subgroups plays a very relevant role in the structure of a finite soluble group. In a previous paper, we laid the foundation of a general theory of σ-subnormal subgroups (and σ-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behavior of the join of σ-subnormal subgroups: in finite groups, σ-subnormal subgroups form a sublattice of the lattice of all subgroups, but this is no longer true for arbitrary locally finite groups. This situation is very similar to that concerned with subnormal subgroups; therefore, as in the case of subnormal subgroups, it makes sense to study the class ${\mathfrak{S}}_{\mathit{\sigma }}^{\mathrm{\infty }}$ (resp. ${\mathfrak{S}}_{\mathit{\sigma }}$) of locally finite groups in which the join of (resp. of finitely many) σ-subnormal subgroups is σ-subnormal. In particular, the aim of this paper is to study how much one can extend a group in one of these classes before going outside the same class (see, for example, Theorems 3.6, 3.8, 5.5, and 5.7). Furthermore, some σ-subnormality criteria for the join of two σ-subnormal subgroups are obtained: for example, similar to a celebrated theorem of Williams, we give necessary and sufficient conditions for a join of two σ-subnormal subgroups to always be σ-subnormal; as a consequence, we show that the join of two orthogonal σ-subnormal subgroups is σ-subnormal (this is the analog of a result of Roseblade).