In this paper, we combine group-theoretic and combinatorial techniques to study $\wedge$-transitive digraphs admitting a cartesian decomposition of their vertex set. In particular, our approach uncovers a new family of digraphs that may be of considerable interest.

Let $R$ be a commutative ring. In this paper we study the behavior of Gorenstein homological dimensions of a homologically bounded $R$-complex under special base changes to the rings $R_{x}$ and $R/xR$, where $x$ is a regular element in $R$. Our main results refine some known formulae for the classical homological dimensions. In particular, we provide the Gorenstein counterpart of a criterion for projectivity of finitely generated modules, due to Vasconcelos.

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

Let $u$ be a supertemperature on an open set $E$, and let $v$ be a related temperature on an open subset $D$ of $E$. For example, $v$ could be the greatest thermic minorant of $u$ on $D$, if it exists. Putting $w=u$ on $E\setminus D$ and $w=v$ on $D$, we investigate whether $w$, or its lower semicontinuous smoothing, is a supertemperature on $E$. We also give a representation of the greatest thermic minorant on $E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of $E$ with union $E$.  In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph ${\rm\Lambda}$ is a natural generalization of a $k$-graph. A pullback of ${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of ${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on ${\rm\Lambda}$. One of our main results in this paper shows that, for some $k$-graphs ${\rm\Lambda}$, ${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph $\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph $\text{C}^{\ast }$-algebra of its pushout and $\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph $\text{C}^{\ast }$-algebra onto it.