We are concerned with variational properties of a fold energy for a unit, dilation-invariant gradient field (called a cluster) in the unit disc. We show that boundedness of a fold energy implies $L^{1}$-compactness of clusters. We also show that a fold energy is $L^{1}$-lower semicontinuous. We characterize absolute minimizers. We also give a sequence of stationary states and discuss its stability. Surprisingly, the stability depends upon $q$, the power of modulus of the jump discontinuities, in the definition of the fold energy.

In this paper the problem of finding the impermeable parachute of optimum shape in a subsonic gas flow has been solved. The effects of compressibility have been taken into account out by making use of ideas of Chaplygin's approximate method. A formula for the drag coefficient has been deduced and by comparison with exact solutions it has been demonstrated that for bluff bodies its related error is no more than 0.5% for any Mach number at infinity less than unity. On the basis of the formula a functional for the drag force has been constructed and its global maximum has been found analytically. It has been shown that the optimum shape is slightly affected by compressibility.

The Kuramoto–Sivashinsky-perturbed Korteweg–de Vries (KS–KdV) equation

$$\partial_tu=-\partial_x^3u-\tfrac{1}{2}\partial_x(u^2)-\varepsilon(\partial_x^2+\partial_x^4)u,$$

with $0<\varepsilon\ll1$ a small parameter, arises as an amplitude equation for small amplitude long waves on the surface of a viscous liquid running down an inclined plane in certain regimes when the trivial solution, the so-called Nusselt solution, is sideband unstable. Although individual pulses are unstable due to the long-wave instability of the flat surface, the dynamics of KS–KdV is dominated by travelling pulse trains of $O(1)$ amplitude. As a step toward explaining the persistence of pulses and understanding their interactions, we prove that for $n=1$ and $2$ the KdV manifolds of $n$-solitons are stable in KS–KdV on an $O(1/\varepsilon)$ time-scale with respect to $O(1)$ perturbations in $H^n(\mathbb{R})$.

We consider a singularly perturbed convection–diffusion equation, $-\varepsilon\Delta u+\bm{v}\cdot\bm{\nabla}u=0$, defined on a rectangular domain $\varOmega\equiv\{(x,y)\mid 0\leq x\leq\pi a,0\leq y\leq\pi\}$, $a>0$, with Dirichlet-type boundary conditions discontinuous at the points $(0,0)$ and $(\pi a,0)$: $u(x,0)=1$, $u(x,\pi)=u(0,y)=u(\pi a,y)=0$. An asymptotic expansion of the solution is obtained from a series representation in two limits, namely when the singular parameter $\varepsilon\to0^+$ (with fixed distance to the points $(0,0)$ and $(\pi a,0)$), and when $(x,y)\to(0,0)$ or $(x,y)\to(\pi a,0)$ (with fixed $\varepsilon$). It is shown that the first term of the expansion at $\varepsilon=0$ contains a linear combination of error functions. This term characterizes the effect of the discontinuities on the $\varepsilon$-behaviour of the solution $u(x,y)$ in the boundary or the internal layers. On the other hand, near the points of discontinuity $(0,0)$ and $(\pi a,0)$, the solution $u(x,y)$ is approximated by a linear function of the polar angle.

This paper is concerned with solutions of the complex Korteweg–de Vries (KdV) equation. We achieve two goals. First, we prove local well-posedness results for the complex KdV equation on a line, in a periodic domain and in a finite domain. These results are in line with the local well-posedness theory for the real KdV equation. Second, we establish a rigorous connection between the local (in space) regularity of the real part and that of the imaginary part of any solution to the complex KdV equation. This result partly validates the numerical observation that the real and imaginary parts of a singular solution of the complex KdV equation blow up at the same point and at the same time.

This paper is concerned with the positive solutions of the boundary-value problem \[ \left\{{\begin{array}{@{}l@{}}{\varepsilon u^{''} - \sigma (u) = -\gamma,}\\{u(0) = u(1) = 0,}\\ \end{array}}\right.\] where $\varepsilon$ is a small positive parameter and $\gamma$ is a positive constant. The nonlinear term $\sigma(u)$ behaves like a cubic; it vanishes only at $u=0$, where $\sigma'(0) > 0$ and $\sigma''(0) < 0$. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given $\gamma$, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of $\gamma$. Our approach is again based on the shooting method used previously by Ou & Wong (Stud. Appl. Math.112 (2004), 161-200).

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.

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.

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.

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.