Let $L\left( X \right)$ be the free locally convex space over a Tychonoff space $X$. Then $L\left( X \right)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L\left( D \right)$ has uncountable tightness for every uncountable discrete space $D$.

Answering a question of H. Petersson, we provide a class of examples of a pair of octonion algebras over a ring having isometric norms.

If $A$ is a $\sigma $-unital ${{C}^{*}}$-algebra and $a$ is a strictly positive element of $A$, then for every compact subset $K$ of the complete regularization Glimm$(A)$ of Prim$(A)$ there exists $\alpha \,>\,0$ such that $K\,\subset \,\{G\,\in \,\text{Glimm(}A\text{)}\,\text{ }\!\!|\!\!\text{ }\,\left\| a\,+\,G \right\|\,\ge \,\alpha \}$. This extends a result of J. Dauns to all $\sigma $-unital ${{C}^{*}}$-algebras. However, there exist a ${{C}^{*}}$-algebra $A$ and a compact subset of Glimm$(A)$ that is not contained in any set of the form $\{G\,\in \,\text{Glimm(}A\text{)}\,\text{ }\!\!|\!\!\text{ }\,\left\| a+\,G \right\|\,\ge \,\alpha \},\,a\in \,A$ and $\alpha \,>\,0$.

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if ${{\mathcal{L}}^{\infty }}\left( \mathbb{G} \right)$ contains non-zero compact operators on ${{\mathcal{L}}^{2}}\left( \mathbb{G} \right)$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where ${{\mathcal{L}}^{1}}\left( \mathbb{G} \right)$ has the Radon-Nikodym property.

We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan–Kac–Vinberg decompositions of a rational vector all lie in a single rational orbit.

We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of $L\left( E \right)$ such that $L\cap K\left( E \right)$ is weak-star dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments.

We show that the manifold resulting from $r$-surgery on the knot ${{5}_{2}}$, which is the two-bridge knot corresponding to the rational number $3/7$, has a left-orderable fundamental group if the slope $r$ satisfies $0\,\le \,r\,\le \,4$.

For relatively prime positive integers ${{u}_{0}}$ and $r$, we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$. We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$.

We give an upper bound for the number of elliptic Carmichael numbers $n\,\le \,x$ that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non $\text{CM}$). We also discuss several possible further improvements.

Let $R$ be an arbitrary ring and let ${{\left( - \right)}^{+}}\,=\,\text{Ho}{{\text{m}}_{\mathbb{Z}}}\left( -,\,{\mathbb{Q}}/{\mathbb{Z}}\; \right)$, where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers. Let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that ${{X}^{+}}\,\in \,\mathcal{D}$ for any $X\,\in \,\mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f:\,A\to \,C$ of left $R$-modules with $C\,\in \,\mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided ${{f}^{+}}:\,{{C}^{+}}\,\to \,{{A}^{+}}$ is a $\mathcal{D}$-(pre)cover of ${{A}^{+}}$. Some applications of this result are given.

A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k$-algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum global dimension is $n\,-\,1$.

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over $\mathbf{k}\,=\,\mathbb{C}$. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field $\mathbf{k}$. Secondly we give an isomorphism between the group of Cartier $b$-divisors on the birational class of $X$ and the Grothendieck group of the semigroup of subspaces of rational functions on $X$. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier $b$-divisors and their intersection theory.

In this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma \left( {R}/{I}\; \right)$ and ${{\Gamma }_{1}}\left( R \right)$, and the domination numbers of $\Gamma \left( R \right)$ and $\Gamma \left( R\left[ x,\,\alpha ,\,\delta \right] \right)$, where $R\left[ x,\,\alpha ,\,\delta \right]$ is the Ore extension of $R$, are studied.

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n\,\gg \,1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

For an $N$-step self-avoiding walk on the hypercubic lattice ${{Z}^{d}}$, we prove that the meansquare end-to-end distance is at least ${{N}^{4/(3d)}}$ times a constant. This implies that the associated critical exponent $v$ is at least $2/(3d)$, assuming that $v$ exists.

In this paper we give a characterization of a real hypersurface of Type $\left( A \right)$ in complex two-plane Grassmannians ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$, which means a tube over a totally geodesic ${{G}_{2}}\left( {{\mathbb{C}}^{m+1}} \right)$ in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$, by means of the Reeb parallel structure Jacobi operator ${{\nabla }_{\xi }}{{R}_{\xi }}\,=\,0$.

We prove the following result announced by the second and third authors: Any homogeneous, metric $ANR$-continuum is a $V_{G}^{n}$-continuum provided ${{\dim}_{G}}X\,=\,n\,\ge \,1$ and ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a ${{V}^{n}}$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional cyclic in dimension $n$ homogeneous metric continuum $X$, satisfying ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$ for some group $G$ and $n\,\ge \,1$, cannot be separated by a compactum $K$ with ${{\overset{\vee }{\mathop{H}}\,}^{n-1}}\left( K;\,G \right)\,=\,0$ and ${{\dim}_{G}}K\,\le \,n\,-\,1$. This provides a partial answer to a question of Kallipoliti–Papasoglu as to whether a two-dimensional homogeneous Peano continuum can be separated by arcs.

We study ${{L}^{p}}\to {{L}^{r}}$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

We answer a question from Raghavan and Steprāns by showing that $\mathfrak{s}\,\text{=}\,{{\mathfrak{s}}_{\omega ,\omega }}.$ Then we use this to construct a completely separable maximal almost disjoint family under $\mathfrak{s}\,\le \,\mathfrak{a}$, partially answering a question of Shelah.