A self-consistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three-dimensional Poisson equation with the particle density as the source term. This density is the product of a two-dimensional surface density and that of a one-dimensional mixed quantum state. The surface density is the solution of a drift–diffusion equation with an effective surface potential deduced from the fully three-dimensional one and which involves the diagonalization of a one-dimensional Schrödinger operator. The overall problem is viewed as a two-dimensional drift–diffusion equation coupled to a Schrödinger–Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori $L^2$ estimate provide sufficient bounds to prove existence and uniqueness of a global-in-time solution. In the case of thermodynamic equilibrium boundary data, a unique stationary solution is proven to exist. The relative entropy allows us to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low-order approximation of the relative entropy is used to prove that this convergence is exponential in time.

A theorem on the existence of moduli spaces of compact complex isotropic submanifolds in complex contact manifolds is established.

A positive answer to a question of Müller is given: any semi-perfect complete hereditary Noetherian prime ring $R$ has a weakly symmetric self-duality sending every ideal $I$ to its cycle-neighbour $X$. Consequently, the factor rings $R/I$ and $R/X$ are isomorphic without using the 1984 results of Dischinger and Müller.

Motivated by problems in the spectral theory of linear operators, we previously introduced a new concept of variation for functions defined on a non-empty compact subset of the plane. In this paper, we examine the algebras of functions of bounded variation one obtains from these new definitions for the case where the underlying compact set is either a rectangle or the unit circle, and compare these algebras with those previously used by Berkson and Gillespie in their theories of AC-operators and trigonometrically well-bounded operators.

The regular Sturm–Liouville problem

$$ \tau y:=-y''+qy=\lambda y\quad\text{on }[0,1],\ \lambda\in\CC, $$

is studied subject to boundary conditions

$$ P_j(\lambda)y'(j)=Q_j(\lambda)y(j),\quad j=0,1, $$

where $q\in L^1(0,1)$ and $P_j$ and $Q_j$ are polynomials with real coefficients. A comparison is made between this problem and the corresponding ‘reduced’ one where all common factors are removed from the boundary conditions. Topics treated include Jordan chain structure, eigenvalue asymptotics and eigenfunction oscillation.

In this paper we study the factorizable braid monoid (also known as the merge-and-part braid monoid) introduced by Easdown, East and FitzGerald in 2004. We find several presentations of this monoid, and uncover an interesting connection with the singular braid monoid. This leads to the definition of the flexible singular braid monoid, which consists of ‘flexible-vertex-isotopy’ classes of singular braids. We conclude by defining and studying the pure factorizable braid monoid, the maximal subgroups of which are (isomorphic to) quotients of the pure braid group.

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form

$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$

where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

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 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.

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

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.

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.

We study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping $F:(a,b)\to Y$ is the difference of two continuous convex operators whenever $Y$ belongs to a large class of Banach lattices which includes all $L^{p}(\mu)$ spaces ($1\leq p\leq\infty$). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.