Let $L^{2}=L^{2}(D,rdrd\theta/\pi)$ be the Lebesgue space on the open unit disc $D$ and let $L_{a}^2=L^{2}\cap\mathrm{Hol}(D)$ be a Bergman space on $D$. In this paper, we are interested in a closed subspace $\mathcal{M}$ of $L^{2}$ which is invariant under the multiplication by the coordinate function $z$, and a Hankel-type operator from $L_{a}^2$ to $\mathcal{M}^\bot$. In particular, we study an invariant subspace $\mathcal{M}$ such that there does not exist a finite-rank Hankel-type operator except a zero operator.

In this paper we study the multiplicity of solutions of the quasilinear elliptic system

\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\}\end{equation} \tag{S$_\lambda$}

where $\varOmega$ is a strip-like domain and $\lambda>0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.

The homogeneous Riemannian structures on the three-dimensional Berger spheres, their corresponding reductive decompositions and the associated groups of isometries are obtained. The Berger 3-spheres are also considered as homogeneous almost contact metric manifolds.

Second-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at $\infty$. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at $\infty$.

We consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

We investigate the role of projections in norming a $C^*$-algebra by a type II1 subfactor. Applications are given for factors with property $\varGamma$ and for free-product factors.

We present an abstract approach to the construction of holomorphic functional calculi for unbounded operators and apply it to the special case of sectorial operators. In effect, we obtain a calculus for a much larger class of functions than was known before, including certain meromorphic functions. We discuss the role of topology. Then we prove in detail a composition rule $(f\circ g)(A)=f(g(A))$ which is the main result of the paper. This is done in such a way that the proof can easily be transferred to functional calculi for other classes of operators.

A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

We investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

given a compact $n$-dimensional immersed riemannian manifold $m^n$ in some euclidean space we prove that if the hausdorff dimension of the singular set of the gauss map is small, then $m^n$ is homeomorphic to the sphere $s^n$.

also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with a small set of points of zero gauss–kronecker curvature are topologically the sphere minus a finite number of points. a characterization of the $2n$-catenoid is obtained.

An optimization problem for the fundamental eigenvalue $\lam_0$ of the Laplacian in a planar simply-connected domain that contains $N$ small identically-shaped holes, each of radius $\eps\ll 1$, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue $\lam_0$ is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains $N$ small traps. For small hole radii $\eps$, a two-term asymptotic expansion for $\lam_0$ is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations $x_{i}$, for $i=1,\ldots,N$, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize $\lam_0$. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes $\lam_0$. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize $\lam_0$. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer–Meinhardt reaction-diffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg–Landau theory of superconductivity.

A free boundary problem arising in a model for inviscid, incompressible shallow water entry at small deadrise angles is derived and analysed. The relationship between this novel free boundary problem and the well-known viscous squeeze film problem is described. An inverse method is used to construct explicit solutions for certain body profiles and to find criteria under which the splash sheet can ‘split’. A variational inequality formulation, conservation of certain generalized moments and the Schwarz function formulation are introduced.

Mathematical models for the prediction of the hydrodynamic pressure distribution and the force on a body entering liquid are investigated. Particular attention is paid to analytical models which are based on the velocity potential given by the classical Wagner theory. Formal use of the Wagner theory provides the loads on an entering body, which are higher than the measured ones. To improve the predictions, the higher order terms in the Bernoulli equation are taken into account within the generalized Wagner model and the Logvinovich model. It is shown that the Logvinovich model corresponds better to the experimental data than the generalized Wagner model. A rational derivation of the Logvinovich model is given in the paper for the two-dimensional case. The analytical models are tested against both numerical and experimental results.

We consider short-time existence, uniqueness, and regularity for a moving boundary problem describing Stokes flow of a free liquid drop driven by surface tension. The surface tension coefficient is assumed to be a nonincreasing function of the surfactant concentration, and the surfactant is insoluble and moves by convection along the boundary. The problem is reformulated as a fully nonlinear, nonlocal Cauchy problem for a vector-valued function on a fixed reference manifold. This problem is, in general, degenerate parabolic. Existence and uniqueness results are obtained via energy estimates in Sobolev spaces of sufficiently high order. In the two-dimensional case, the problem is strictly parabolic, and we prove instantaneous smoothing of the free boundary, using maximal regularity results in little Hölder spaces.

We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven extensional flow for the Upper Convected Maxwell and Oldroyd-B constitutive models. Assuming the axial stress in the filament has a spatial dependence provides the simplest coupling of viscoelastic effects to the motion of the filament, and yields a closed system of ODEs with an exact solution for the stretch rate and filament thickness satisfied by both constitutive models. This viscoelastic solution, which is a generalization of the exact solution for Newtonian filaments, converges to the Newtonian power-law scaling as $t \rightarrow \infty$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. We compare the viscoelastic solution to measurements of the thinning filament that forms behind a falling drop for several semi-dilute (strongly-viscoelastic) polymer solutions. We find the exact solution correctly predicts the time-dependence of the filament diameter in all of the experiments. As $t \rightarrow \infty$, observations of the filament thickness follow the Newtonian scaling $1/\sqrt{t}$. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.

We consider two-dimensional bubbles in a corner flow in a Hele–Shaw cell of a viscous incompressible fluid that occupies the complement to a bubble. We discuss the governing equations, some basic properties of the free interface of the bubbles, their geometry, and construct explicit solutions that present asymmetric long bubbles analogous to the famous Saffman–Taylor fingers in a wedge of arbitrary angle $\alpha\in (0,2\pi)$.

We examine a number of initial boundary value problems for a paradigm sixth-order degenerate parabolic equation of the form $h_t=(h^n h_{xxxxx})_x$ which arises when considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. Analytical and numerical methods are exploited to characterise the solutions, which turn out to be rather sensitive to the value of $n$.

We use lubrication theory on the flow equations for nematic liquid crystals to derive a simple model describing the evolution of the film height under gravity, in the case of finite surface “anchoring energy” at the free surface and at the rigid substrate. This means that the molecules of the nematic have a preferred alignment at interfaces, modelled by a single-well potential surface energy (first introduced by Rapini & Papoular [9]). This paper generalises the earlier work of Ben Amar & Cummings, in which the orientation of the nematic liquid crystal molecules is effectively specified at both surfaces (strong anchoring; isotropic surface tension). Additional terms, analogous in some sense to Marangoni terms, are introduced into the PDE governing the film height evolution. The stability of the derived model is considered, and stability criteria are presented and discussed. The existence of static, drop-like solutions to the model is also briefly considered.