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).

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.

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.

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.

In this research, we introduce Banach space–valued $H^p$ spaces with $A_p$ weight and prove the following results: Let $\mathbb{A}$ and $\mathbb{B}$ be Banach spaces, and let T be a convolution operator mapping $\mathbb{A}$-valued functions into $\mathbb{B}$-valued functions—that is,

$$Tf(x)={\int }_{{\mathbb{R}}^n}K(x-y)\cdot f(y)dy,$$

where K is a strongly measurable function defined on ${\mathbb{R}}^n$ such that $\Vert K(x){\Vert }_{\mathbb{B}}$ is locally integrable away from the origin. Suppose that w is a positive weight function defined on ${\mathbb{R}}^n$ and that

Then there exists a positive constant $C_3$ such that

$$\Vert Tf{\Vert }_{L_{\mathbb{B}}^1(w)}\le C_3\Vert f{\Vert }_{H_{\mathbb{A}}^1(w)}$$

for all $f\in H_{\mathbb{A}}^1(w)$.

Let $w\in A_1$. Assume that $K\in L_{\mathrm{loc}}({\mathbb{R}}^n\setminus \{0\})$ satisfies

$$\Vert K\ast f{\Vert }_{L_{\mathbb{B}}^2(w)}\le C_1\Vert f{\Vert }_{L_{\mathbb{A}}^2(w)}$$

and

$${\int }_{|x|\ge C_2|y|}{\Vert K(x-y)-K(x)\Vert }_{\mathbb{B}}w(x+h)dx\le C_{3}w(y+h)(\forall y\ne 0,\forall h\in {\mathbb{R}}^n)$$

for certain absolute constants $C_1$, $C_2$, and $C_3$. Then there exists a positive constant C independent of f such that

$$\Vert K\ast f{\Vert }_{L_{\mathbb{B}}^1(w)}\le C\Vert f{\Vert }_{H_{\mathbb{A}}^1(w)}$$

for all $f\in H_{\mathbb{A}}^1(w)$.

In a recent work it was proved that any immersed hypersurface $M^n$ of a space form evolves through the mean curvature flow ($\mathcal{MCF}$) by parallel hypersurfaces if and only if $M^n$ is an isoparametric hypersurface. The goal of this article is to extend the quoted work to immersed hypersurface $M^n$ of ${\mathbb{Q}}_c^n\times \mathbb{R}$ and ${\mathbb{Q}}_c^n\times {\mathbb{S}}^1$ under a geometric condition of the tangential component of $\frac{\partial }{\partial t}$. More exactly, supposing that this tangential component is a principal direction of the second fundamental form, we will show that such a hypersurface evolves through the $\mathcal{MCF}$ by parallel hypersurfaces if and only if $M^n$ is also an isoparametric hypersurface. Moreover, we will prove that any embedded convex hypersurface $M^n$ of a sphere ${\mathbb{S}}^{n+1}$ evolves through the inverse mean curvature flow ($\mathcal{IMCF}$) by parallel hypersurfaces if and only if $M^n$ is an umbilic hypersurface.

This paper is motivated by Phong and Stein’s paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular integrals on the Hardy spaces.

This paper establishes the Stein–Weiss inequalities on local Morrey spaces with variable exponents. It is presented as the boundedness of the fractional integral operators on power-weighted local Morrey spaces with variable exponents with different power weights. As a consequence of the main result, we obtain some Poincaré inequalities for local Morrey spaces with variable exponents.