We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$. When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

We prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.

Let $p$, $q$ be prime numbers with ${{p}^{2}}\,<\,q,\,n\,\in \,\mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $p{{q}^{n}}$ over an algebraically closed field of characteristic 0. This paper proves that $H$ must possess one of the following two structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\#kG$, where $kG$ is the group algebra of group $G$ of order $p$ and $R$ is a semisimple Yetter–Drinfeld Hopf algebra in $_{kG}^{kG}y\mathcal{D}$ of dimension ${{q}^{n}}$.

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves ${{\overline{M}}_{g,n}}$ Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the $\varphi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group, and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $\text{A}\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e{{\delta }_{0}}\left( \text{A}\,{{\star }_{\alpha }}\,G \right)e{{\delta }_{0}}$ is a field for all $e\,\in \,E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

In this paper, we characterize Jordan $*$-derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $. Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$-derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, char $R\,\ne \,2$, and let $\delta $ be a nonzero Jordan $*$-derivation of $R$. If $\delta $ is an elementary operator of $R$, then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

Let ${{H}^{2}}\left( \Omega \right)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \,\subset \,{{\mathbb{C}}^{n}}$, and let $A\,\subset \,{{L}^{\infty }}\left( \partial \Omega \right)$ denote the subalgebra of all ${{L}^{\infty }}$-functions $f$ with compact Hankel operator ${{H}_{f}}$. Given any closed subalgebra $B\,\subset \,A$ containing $C\left( \partial \Omega \right)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal{T}\left( B \right)\,\subset \,B\left( {{H}^{2}}\left( \Omega \right) \right)$. In particular, we show that every derivation on $\mathcal{T}\left( A \right)$ is inner. These results are new even for $n\,=\,1$, where it follows that every derivation on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C \right)$ is inner, while there are non-inner derivations on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C\left( \partial {{\mathbb{B}}_{n}} \right) \right)$ over the unit ball ${{\mathbb{B}}_{n}}$ in dimension $n\,>\,1$.

For $S\,\subseteq \,{{\mathbb{R}}^{n}}$ a set $C\,\subseteq \,S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\,\subseteq \,{{\mathbb{R}}^{2}}$ without an uncountable 3-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in ${{\mathbb{R}}^{2}}$ without uncountable 3-cliques in terms of clopen, ${{P}_{4}}$-free graphs on Polish spaces.

We consider the Finsler space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$ obtained by perturbing the Euclidean metric of ${{\mathbb{R}}^{3}}$ by a rotation. It is the open region of ${{\mathbb{R}}^{3}}$ bounded by a cylinder with a Randers metric. Using the Busemann–Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in ${{\overline{M}}^{3}}$. We prove that the helicoid is a minimal surface in ${{\overline{M}}^{3}}$ only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$, the only minimal surfaces in the Bonnet family with fixed axis $O{{\overline{x}}^{3}}$ are the catenoids and the helicoids.

The index of a graph $G$ is the maximum eigenvalue of its adjacency matrix $A\left( G \right)$. In this paper we characterize the extremal tree with given domination number that attains the maximum index.

Let $G$ be a finite group and $H$, $K$ two subgroups of $G$. A group $G$ is said to be a mutually $m$-permutable product of $H$ and $K$ if $G\,=\,HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group that is a mutually $m$-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Klein bottle if the inclusion $W\,\to \,M$ factors homotopically through a map to $K$. The image of ${{\pi }_{1}}\left( W \right)$ (for any base point) is a subgroup of ${{\pi }_{1}}\left( M \right)$ that is isomorphic to a subgroup of a quotient group of ${{\pi }_{1}}\left( K \right)$. Subsets of $M$ with this latter property are called ${{\mathcal{G}}_{K}}$-contractible. We obtain a list of the closed 3-manifolds that can be covered by two open ${{\mathcal{G}}_{K}}$-contractible subsets. This is applied to obtain a list of the possible closed prime 3-manifolds that can be covered by two open $K$-contractible subsets.

For Dewitt super groups $G$ modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group $BG$ compatible with the group operation on $G$, then, generically, the kernel $K$ of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra $\kappa$ has the property that for each $a\,\in \,\kappa ,\,\text{a}{{\text{d}}_{a}}$ has a zero spectrum. We also show that the exponential mapping from $\kappa$ to $K$ is surjective and that $K$ is a quotient manifold of the Banach space $\kappa$ via a lattice in $\kappa$.

Previous results by the author on the connection between three measures of noncompactness obtained for ${{\mathcal{L}}_{p}}$ are extended to regular spaces of measurable functions. An example is given of the advantages of some cases in comparison with others. Geometric characteristics of regular spaces are determined. New theorems for $\left( k,\,\beta \right)$-boundedness of partially additive operators are proved.

Dans plusieurs articles, A. R. Prince développe une représentation d’un plan projectif fini par un anneau commutatif unitaire dont les propriétés algébriques dépendent de la structure géométrique du plan. Dans un autre article, il étend cette représentation aux designs symétriques. Cependant D.-S. Yin fait remarquer que la multiplication définie dans ce cas ne peut être associative que si le design est un plan projectif. Dans cet article on mènera une étude de cette représentation dans le cas des designs symétriques. On y montrera comment on peut faire associer un anneau commutatif unitaire à tout design symétrique; on y précisera certaines de ses propriétés, en particulier, celles qui relèvent de son invariance. On caractérisera aussi les géométries projectives finies de dimension supérieure moyennant cette représentation.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares $\sum{\mathbb{R}{{\left[ \underline{X} \right]}^{2}}}$ in the polynomial ring $\mathbb{R}\left[ \underline{X} \right]\,:=\,\mathbb{R}\left[ {{X}_{1}},\,.\,.\,.\,,\,{{X}_{n}} \right]$ in the topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$, the cone consisting of all polynomials that are non-negative on the hypercube ${{\left[ -1,\,1 \right]}^{n}}$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $~{{\ell }_{1}}$-seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\,\ge \,1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}{{\left[ \underline{X} \right]}^{2d}}$ in the $\mathbb{R}\left[ \underline{X} \right]$ topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$.

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case that the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

We give some new characterizations for compactness of weighted composition operators $u{{C}_{\varphi }}$ acting on Bloch-type spaces in terms of the power of the components of $\varphi$, where $\varphi$ is a holomorphic self-map of the polydisk ${{\mathbb{D}}^{n}}$, thus generalizing the results obtained by Hyvärinen and Lindström in 2012.