Given a left Noetherian ring $R$, we give a necessary and sufficient condition in order that a complex of $R$-modules be DG-injective. Using this result we prove that if $(K_i)_{i\in I}$ is a family of DG-injective complexes of left $R$-modules and $K$ is the $\aleph_1$-product of $(K_i)_{i\in I}$ (i.e. $K\subset\prod_{i\in I}K_i$ is such that, for each $n$, $K^n\subset\prod_{i\in I}K_i^n$ consists of all $(x_i)_{i\in I}$ such that $\{i\mid x_i\neq0\}$ is at most countable), then $K$ is DG-injective.

In this paper we establish a multiplicity result for a second-order non-autonomous system. Using a variational principle of Ricceri we prove that if the set of global minima of a certain function has at least $k$ connected components, then our problem has at least $k$ periodic solutions. Moreover, the existence of one more solution is investigated through a mountain-pass-like argument.

Proper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

We provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group $\uqg$, with $L$-operators as generators and relations ruled by an $R$-matrix. We look at $\uqg$ as being generated by the quantum Borel subalgebras $U_q(\mathfrak{b}_+)$ and $U_q(\mathfrak{b}_-)$, and use the standard presentation of the latter as quantum function algebras. When $\mathfrak{g}=\mathfrak{gl}_n$, these Borel quantum function algebras are generated by the entries of a triangular $q$-matrix. Thus, eventually, $U_q(\mathfrak{gl}_n)$ is generated by the entries of an upper triangular and a lower triangular $q$-matrix, which share the same diagonal. The same elements generate over $\Bbbk[q,q^{-1}]$ the unrestricted $\Bbbk [q,q^{-1}]$-integral form of $U_q(\mathfrak{gl}_n)$ of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for $\mathfrak{g}=\mathfrak{sl}_n$ too.

We solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

We investigate the asymptotic behaviour of the entropy numbers of the compact embedding $B^{s_1}_{p_1,q_1}(\mathbb{R}^d,w_1)\hookrightarrow B^{s_2}_{p_2,q_2}(\mathbb{R}^d,w_2)$. Here $B^s_{p,q}(\mathbb{R}^d,w)$ denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.

We study algebraicity and transcendency of certain basic special values of the double sine functions due to Hölder and Shintani by employing the zeta regularized product expressions.

We study topological properties of the correspondence of prime spectra associated with a non-commutative ring homomorphism $R\rightarrow S$. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of $\spec R$ and $\spec S$; these functors arise naturally from restriction and extension of scalars. When $R$ and $S$ are left Noetherian, adjointness occurs only for centralizing and ‘nearly centralizing’ homomorphisms.

We find positive solutions to a nonlinear equation of Klein–Gordon type. Our analysis is carried out by truncating the related functional and estimating mountain pass solutions by Moser’s iterative scheme.

Let $X\subset\mathbb{R}^2$ be the graph of a Pfaffian function $f$ in the sense of Khovanskii. Suppose that $X$ is non-algebraic. This note gives an estimate for the number of rational points on $X$ of height less than or equal to $H$; the estimate is uniform in the order and degree of $f$.

We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface $S^*$ obtained by deleting a closed set from one original surface $S$. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

An elegant result of Ryan gives a characterization of weakly compact operators from a Banach space $A$ into $c_{0}(X)$, the space of null sequences in a Banach space $X$. It would be a useful tool if the analogue of Ryan’s result were valid when $c_{0}(X)$ is replaced by $c(X)$, the space of convergent sequences in $X$. This seems plausible and has been assumed to be true by some authors. Unfortunately, it is false in general; Ylinen has produced a counterexample. But when $A$ is a $C^*$-algebra, or, more generally, when the dual of $A$ is weakly sequentially complete, we show that the desired extension of Ryan’s result does hold. The latter result turns out to be ‘best possible’.

We construct moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a ‘jump’ phenomenon in a family of automorphism groups of supersingular K3 surfaces with a constant Néron–Severi lattice.

This paper deals with the existence of positive radial solutions for the quasilinear system $\text{div}(|\nabla u_i|^{p-2}\nabla u_i)+\lambda f^i(u_1,\dots,u_n)=0$, $|x|\lt1$, $u_i(x)=0$, on $|x|=1$, $i=1,\dots,n$, $p\gt1$, $\lambda>0$, $x\in\mathbb{R}^N$. The $f^i$, $i=1,\dots,n$, are continuous and non-negative functions. Let $\bm{u}=(u_1,\dots,u_n)$, $\|\bm{u}\|=\sum_{i=1}^n|u_i|$,

$$ f_0^i=\lim_{\|\bm{u}\|\to0}\frac{f^i(\bm{u})}{\|\bm{u}\|^{p-1}}, $$

$i=1,\dots,n$, $\bm{f}=(f^1,\dots,f^n)$, $\bm{f}_0=\sum_{i=1}^nf_0^i$. We prove that the problem has a positive solution for sufficiently small $\lambda>0$ if $\bm{f}_0=\infty$. Our methods employ a fixed-point theorem in a cone.