We define the class of \textit{easily-representable groups} as the class of those finitely presented groups $\Gamma $ admitting an \textit{inverse representation} (which, roughly, is a map from some $2$-complex to a certain singular 3-manifold $M^3 (\Gamma )$ associated to $\Gamma$, satisfying several topological properties) for which the set of double points is closed. Our main result is that easily-representable groups are {\sc qsf} (i.e. quasi-simply filtered).

We discuss different mixing properties for triples of finite von Neumann algebras $B\subset N\subset M$, and we introduce families of triples of groups $H<K<G$ whose associated von Neumann algebras $L(H)\subset L(K)\subset L(G)$ satisfy $\mathcal{N}_{L(G)}(L(H))''=L(K)$. It turns out that the latter equality is implied by two conditions: the equality $\mathcal{N}_G(H)=K$ and the above mentioned mixing properties. Our families of examples also allow us to exhibit examples of pairs $H<G$ such that $L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))''$.

We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on $\mathbb S_3$. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.

We give an upper bound for the number of nested ovals of a real tetragonal curve $R$ embedded in an Hirzubruch surface in terms of the genus of the curve.

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\mathbb D$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$\left |f'(z)\left (\frac{z}{f(z)} \right )^{2}-1\right | <\lambda ~\mbox{ for $z\in \mathbb D$}, $$ and some $\lambda \in (0,1]$. In this paper, among other things, we study a ``harmonic mean'' of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=\frac{1}{2}\left( \frac{z}{f(z)}+\frac{z}{g(z)} \right), $$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.

$m$-infrabarrelledness, in the context of locally convex algebras, is considered to prove results previously obtained for barrelled algebras. Thus, any unital commutative $m$-infrabarrelled advertibly complete and pseudo-complete locally $m$-convex algebra with bounded elements has the $Q$-property; hence, it is functionally continuous (: all characters are continuous). In the framework of commutative $GB^{\ast }$-algebras with jointly continuous multiplication and bounded elements, the notions {\em $m$-infrabarrelled algebra} and {\em $C^{\ast }$-algebra} coincide. In unital uniform locally $m$-convex algebras, $m$-infrabarrelledness is equivalent to the Banach algebra structure, modulo pseudo-completeness. Moreover, $m$-infrabarrelledness for locally $A$-convex algebras (in particular, $A$-normed ones) is also examined.

We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.

Suppose that $X$ is a subspace of a Tychonoff space $Y$. Then the embedding mapping $e_{X, Y}: X\rightarrow Y$ can be extended to a continuous monomorphism $\hat{e}_{X, Y}: AP(X)\rightarrow AP(Y)$, where $AP(X)$ and $AP(Y)$ are the free Abelian paratopological groups over $X$ and $Y$, respectively. In this paper, we mainly discuss when $\hat{e}_{X, Y}$ is a topological monomorphism, that is, when $\hat{e}_{X, Y}$ is a topological embedding of $AP(X)$ to $AP(Y)$.

Standard irreducible representations of the group $\mathrm{SL}(2,\mathbb R)$ on coefficients of homogeneous polynomials in two variables are studied in a new context. It is proved that any standard representation of $\mathrm{SL}(2,\mathbb R)$ on $\mathbb R^{n+1}$ induces an involutive rational mapping of an open dense subset of $\mathbb R^{n+1}$ onto itself. Examples in low dimensions are presented. We also construct formal involutive rational mappings with ``arbitrary complexity''.

Let $\mathcal{F}$ be a family of holomorphic functions defined in $D \subset C$, and let $ k, m, n, p $ be four positive integers with $ \frac{k+p+1}{m}+\frac{p+1}{n} < 1 $. Let $\psi (\not \equiv 0, \infty )$ be a meromorphic function in $ D $ and which has zeros only of multiplicities at most $ p $. Suppose that, for every function $ f \in \mathcal{F} $, (i) $ f $ has zeros only of multiplicities at least $ m $; (ii) all zeros of $ f^{(k)}-\psi(z) $ have multiplicities at least $ n $; (iii) all poles of $ \psi $ have multiplicities at most $ k $, and (iv) $ \psi(z) $ and $ f(z) $ have no common zeros, then $\mathcal{F}$ is normal in $ D $.

Recent papers in physics literature have introduced spinor Ginzburg-Landau model for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors. In this paper, we study the spatial behavior of interacting components of Spinor Ginzburg-Landau model. We prove the interspecies interaction leads to extinction, that is, configurations where one or more densities are null.

We give a construction that associates to a pointed topological space $(X,x_0)$ a homotopy invariant $\,_2\kappa^4$ which we call the secondary invariant. This construction can be seen a ``3-type" generalization of the classical $k$-invariant.