When a symmetric rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. These equations at the boundary contain a time derivative and thus are of a dynamic nature. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity and pressure fields of the fluid motion and the angular velocity of the rigid body.

In this paper we formulate the physical problem for the case of rotation about an axis of symmetry as an abstract ordinary differential equation in two Banach spaces in which the velocity field is the only unknown. To achieve this, a method for the elimination of the pressure field, which also occurs in the boundary condition, is developed. Existence and uniqueness results for the abstract equation are derived with the aid of the theory of B-evolutions and the associated theory of fractional powers of a closed pair of operators.

This paper deals with existence and non-existence of global solutions of certain fast–slow diffusion systems with nonlinear boundary conditions. Necessary and sufficient conditions for global existence of positive solutions are obtained in terms of various parameters which appear explicitly in the definition of the systems.

This paper is concerned with homogenisation processes for parametrised families of transport equations in ℝn symmetric hyperbolic systems in several space dimensions and anisotropic wave equations. The main tools for carrying out the homogenisation are the Young measures and Radon transform combined with the integral representation of holomorphic functions of Nevanlinna-Pick's type.

The bivariate series defines a partial theta function. For fixed q, θ(q, ·) is an entire function. We show that for ∣q∣ ≤ 0.108 the function θ(q, ·) has no multiple zeros.

A generalized Yosida approximation of monotone (and non-monotone) operators in Banach space is introduced. It uses a general potential that is not necessarily the square of the norm. It is therefore advisable to use it in cases where some other more convenient potentials are available, such as in Lp-spaces. As an illustration, the case of Nemyckii operators is considered.

The Hermite-Hadamard inequality not only is a consequence of convexity but also characterizes it: if a continuous function satisfies either its left-hand side or its right-hand side on each compact subinterval of the domain, then it is necessarily convex. The aim of this paper is to prove analogous statements for the higher-order extensions of the Hermite-Hadamard inequality. The main tools of the proofs are smoothing by convolution and the support properties of higher-order monotone functions.

The blood of the Invertebrata, like that of the Vertebrata, is not homogeneous. It consists of a transparent or semitransparent liquid, and a number of small, solid corpuscles which float in it.

In the higher animals the corpuscles are of two kinds—red and colourless; but in the Invertebrata there are, as a rule, only colourless corpuscles. The red blood of Annelids is different from the red blood of Vertebrates, inasmuch as the plasma is coloured, and the corpuscles are colourless in the former, while in the latter the plasma is colourless, and there are present coloured and colourless corpuscles.

A priori estimates on the solution to the complete system of equations governing a heat-conductive, viscous reactive perfect gas confined between two infinite parallel plates are derived. From these estimates, global existence of both weak and classical solutions is obtained.

We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

We introduce the concept of an elementary operator between two algebras, and thereby extend the study of operators of the form

where ai, bi are fixed elements in an algebra. Elementary operators on some classes of algebras are considered and their general form is described in some special cases.

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second-order elliptic operators, these estimates hold true on bounded C1,1 domains. For higher-order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to non-cooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations:where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

In this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.

We discuss existence and lowersemicontinuity for clusters of materials minimising an energy given by a collection of norms φij on the interfaces between regions Ri and Rj. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

We prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U, which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of U.

We consider the following semilinear elliptic equation:

where B1 is the unit ball in ℝd, d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λp,∞ such that (*) has a solution (λp,∞,Wp) satisfying lim|x|→0Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (*). It follows from the result of Dancer that (*) has an unbounded bifurcation branch of regular solutions that emanates from (λ, u) = (0, 0). Here, using the singular solution, we show that the bifurcation branch has infinitely many turning points around λp,∞ when 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in the d ≥ 11 case.

The set of effective elasticity tensors for all two-dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described. In two dimensions the effective tensors are completely characterised by bounds on their eigenvalues.

In the problem of estimating from sample the value of a parameter in a probability function new postulates are suggested of unbiased linear estimate and minimum sampling variance. A comparison is made, with illustrative examples, between this method and the principle of maximum likelihood, and ground common to the two is traversed. The new postulates are also placed in relation to the theory of sufficient statistics.

Let there be given a non-negative, quasiconvex function F satisfying the growth condition

for some p ∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if

then the variational integral

is lower semicontinuous on sequences of W1, p functions converging weakly in W1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.