Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called isometric flow has the distinctive feature of preserving the underlying metric induced by that $\mathrm{G}_2$-structure, so it can be used to evolve a $\mathrm{G}_2$-structure to one with the smallest possible torsion in a given metric class. This flow is built upon the divergence of the full torsion tensor of the flowing $\mathrm{G}_2$-structures in such a way that its critical points are precisely $\mathrm{G}_2$-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant $\mathrm{G}_2$-structures defined on simply connected solvable Lie groups previously studied and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.

Let $B$ be a right coideal subalgebra of a Hopf algebra $H$. We construct a left $H^*$-module structure on the Hochschild cohomology of $B$ with coefficients in $B\otimes M$ where $M$ is a $B$-bimodule as well as a right $H$-comodule satisfying a compatible condition. We prove that the structure is a canonical invariant which coincides with one given by Krähmer (2012) for quantum homogeneous spaces. In particular, Krähmer's group-like element turns out to be determined by $B$ uniquely, which plays an important role in describing the dualizing complex of $B$.

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^\mu>\mu^+$ then the almost strong polarized partition relation $\binom{\mu^+}{\mu}\rightarrow \binom{\tau}{\mu}_{\theta}$ for every ordinal $\tau$ smaller that $\mu^+$ and for every $\theta$ smaller than cf$(\mu) $. We obtain an optimal positive relation under $2^\mu=\mu^+$, as after collapsing $2^\mu$ to $\mu^+$ this positive relation is preserved.

We introduce a class of homeomorphisms between metric spaces, which are locally biHölder continuous maps. Then an embedding result between Besov spaces induced by locally biHölder continuous maps between Ahlfors regular spaces is established, which extends the corresponding result of Björn, Björn, Gill, and Shanmugalingam. Furthermore, an example is constructed to show that our embedding result is more general. We also introduce a geometric condition, named as uniform boundedness, to characterize when a quasisymmetric map between uniformly perfect spaces is locally biHölder continuous.

Luc Menichi showed that the BV algebras on $H_\bullet(L S^2;{\mathbb Z}_2)[-2]$ coming from string topology and the one on $HH^\bullet(H^\bullet( S^2;{\mathbb Z}_2), H^\bullet( S^2;{\mathbb Z}_2))$ using Poincaré duality on $H^\bullet( S^2;{\mathbb Z}_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincaré duality structure with higher homotopies. This Poincaré duality (with higher homotopies) on cohomology is induced by a local Poincaré duality (with higher homotopies) on the cochain level.

Starting from involutive BE algebras, we redefine the orthomodular algebras, by introducing the notion of implicative-orthomodular algebras. We investigate properties of implicative-orthomodular algebras, and give characterizations of these algebras. Then we define and study the notions of filters and deductive systems, and characterize certain classes of filters. Furthermore, we introduce and characterize the commutative deductive systems in implicative-orthomodular algebras. We also show that any deductive system determines a congruence, and conversely, for any congruence we can define a deductive system in an implicative-orthomodular algebra. We define the quotient implicative-orthomodular algebra with respect to the congruence induced by a deductive system, and prove that a deductive system is commutative if and only if all deductive systems of the corresponding quotient algebra are commutative.

Using monotonicity rules, we establish the monotonicity of three kinds of functions involving the Gaussian hypergeometric function. This refines some existing results. We also present some bounds for the zero-balance Gaussian hypergeometric function.