The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.

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 consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.

We show that for a given base $b$ and a proper subset $E\subset\{0,\dots,b-1\}$, $\#E\ltb-1$, the set of numbers $x\in[0,1]$ that have no digits from $E$ in their expansion to base $b$ consists almost exclusively of $S^*$-numbers of type at most $\min\{2,\log b/\log(b-\#E)\}$. We also give upper bounds on the Hausdorff dimension of some exceptional sets.

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.

We discuss the existence of breather solutions for a discrete nonlinear Schrödinger equation in an infinite $N$-dimensional lattice, involving site-dependent anharmonic parameters. We give a simple proof of the existence of (non-trivial) breather solutions based on a variational approach, assuming that the sequence of anharmonic parameters is in an appropriate sequence space (decays with an appropriate rate). We also give a proof of the non-existence of (non-trivial) breather solutions, and discuss a possible physical interpretation of the restrictions, in both the existence and non-existence cases.

We study quantum deformations of Catalan’s constant, Mahler’s measure and the double sine function. We establish quantum deformations of basic relations between these three objects.

We construct a spectral sequence converging to the $E_{2}$-term of the Bousfield–Kan spectral sequence (BKSS) for a wide variety of homology theories. Using this, the $E_{2}$-term of the BKSS based on $K(1)$-theory for the odd spheres is computed and the unstable $K(1)$-completion is computed.

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 $f$.

A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

We investigate the multidimensional non-isentropic Euler–Poisson (or full hydrodynamic) model for semiconductors, which contain an energy-conserved equation with non-zero thermal conductivity coefficient. We first discuss existence and uniqueness of the non-constant stationary solutions to the corresponding drift–diffusion equations. Then we establish the global existence of smooth solutions to the Cauchy problem with initial data, which are close to the stationary solutions. We find that these smooth solutions tend to the stationary solutions exponentially fast as $t\rightarrow+\infty$.

The aim of this note is to characterize 2-local automorphisms and derivations on matrix rings over finite-dimensional division rings.

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’.

The paper deals with problems of the type $-\Delta u+a(x)u=|u|^{p-2}u$, $u\gt0$, with zero Dirichlet boundary condition on unbounded domains in $\mathbb{R}^N$, $N\geq2$, with $a(x)\geq c\gt0$, $p\gt2$ and $p\lt2N/(N-2)$ if $N\geq3$. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential $a(x)$ has $k$ suitable ‘bumps’ and the domain has $h$ suitable ‘holes’, it is proved that the problem has at least $2(h+k)$ positive solutions ($h$ or $k$ can be zero). The multiplicity results are obtained under a geometric assumption on $\varOmega$ at infinity which ensures the validity of a local Palais–Smale condition.

We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let $(A_{\alpha},T_{\alpha,\beta})_{(\varLambda,\le)}$ be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let $A$ be the projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,\alpha$}}} A_\alpha$. We prove that, for the continuous cyclic cohomology $HC^*$ and continuous periodic cohomology $HP^*$ of $A$ and $A_\alpha$, $\alpha\in\varLambda$, for all $n\ge0$, $HC^n(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HC^n(A_\alpha)$, the inductive limit of $HC^n(A_\alpha)$, and, for $k=0,1$, $HP^k(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HP^k(A_\alpha)$. For a projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,m$}}}A_m$ of a countable reduced projective system $(A_m,T_{m,\ell})_{\mathbb{N}}$ of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of $A$ and $A_m$, $m\in\mathbb{N}$. For example, we show the exactness of the following short sequence for all $n\ge0$:

$$ 0\rightarrow {\lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}}}^{1}HC_{n+1}(A_m)\rightarrow HC_n(A)\rightarrow \lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}} HC_{n}(A_m)\rightarrow0. $$

We present a class of Fréchet algebras $A$ for which the continuous periodic cohomology $HP^k(A)$, $k=0,1$, is isomorphic to the continuous cyclic cohomology $HC^{2\ell+k}(A)$ starting from some integer $\ell$. We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally $m$-convex algebras.

We establish results concerning the detailed asymptotic structure of the eigenvalues of a class of regular and singular Sturm–Liouville problems as the domain size shrinks to zero. This is motivated by the recent paper of Bailey et al. and provides an extensive generalization of the results therein.

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.

It is known that if $a\in\mathbb{C}\setminus(-\infty,-\tfrac14]$ and $a_n\to a$ as $n\to\infty$, then the infinite continued fraction with coefficients $a_1,a_2,\dots$ converges. A conjecture has been recorded by Jacobsen et al., taken from the unorganized portions of Ramanujan’s notebooks, that if $a\in(-\infty,-\tfrac14)$ and $a_n\to a$ as $n\to\infty$, then the continued fraction diverges. Counterexamples to this conjecture for each value of $a$ in $(-\infty,-\tfrac14)$ are provided. Such counterexamples have already been constructed by Glutsyuk, but the examples given here are significantly shorter and simpler.

In this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, $u_t+u_{x}=\lambda f(u)/(\int_{0}^1f(u)\,\mathrm{d} x)^{2}$, together with initial and boundary conditions. It is known that, for $f(s)$, $-f'(s)$ positive and $\int_0^\infty f(s)\,\mathrm{d} s\lt\infty$, there exists a critical value of the parameter $\lambda>0$, say $\lambda^\ast$, such that for $\lambda>\lambda^\ast$ there is no stationary solution and the solution $u(x,t)$ blows up globally in finite time $t^\ast$, while for $\lambda\leq\lambda^\ast$ there exist stationary solutions. Moreover, the solution $u(x,t)$ also blows up for large enough initial data and $\lambda\leq\lambda^\ast$. Thus, estimates for $t^\ast$ were found either for $\lambda$ greater than the critical value $\lambda^\ast$ and fixed initial data $u_0(x)\geq0$, or for $u_0(x)$ greater than the greatest steady-state solution (denoted by $w_2\geq w^*$) and fixed $\lambda\leq\lambda^\ast$. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given $\lambda$, $\lambda^*$ and $0\lt\lambda-\lambda^*\ll1$, estimates of the following form were found: upper bound $\epsilon+c_1\ln[c_2(\lambda-\lambda^*)^{-1}]$; lower bound $c_3(\lambda-\lambda^*)^{-1/2}$; asymptotic estimate $t^\ast\sim c_4(\lambda-\lambda^\ast)^{-1/2}$ for $f(s)=\mathrm{e}^{-s}$. Moreover, for $0\lt\lambda\leq\lambda^*$ and given initial data $u_0(x)$ greater than the greatest steady-state solution $w_2(x)$, we have upper estimates: either $c_5\ln(c_6A^{-1}_0+1)$ or $\epsilon+c_7\ln(c_8\zeta^{-1})$, where $A_0$, $\zeta$ measure, in some sense, the difference $u_0-w_2$ (if $u_0\to w_2+$, then $A_0,\zeta\to0+$). $c_i\gt0$ are some constants and $0\lt\epsilon\ll1$, $0\ltA_0,\zeta$. Some numerical results are also given.