In this paper we begin a systematic study of smooth tight maps with singularities of surfaces into E3, particularly C∞-stable maps. The class of C∞-stable tight maps of surfaces into E3 is much larger and richer than the class of C∞ (or even topological) tight immersions. We describe the general structure of C∞-stable tight maps of surfaces into E3. We show that, given any integer n ≧ 2 and a compact surface X other than the sphere or the projective plane, there is a C∞ -stable tight map X → E3 with exactly n topcycles. This is very different from the situation for tight topological immersions, where Cecil and Ryan have shown that the number α(f) of topcycles of a map f: X → E3, X a compact surface other than S2, satisfies the bound 2 ≦ α(f) ≦ 2 − Euler number of X. We prove also an analogue of the Cecil–Ryan result for C∞-stable maps.

This paper is concerned with some properties of an ordinary symmetric matrix differential expression M, denned on a certain class of vector-functions, each of which is defined on the real line. For such a vector-function F we have M[F] = −F“ + QF on R, where Q is an n × n matrix whose elements are reasonably behaved on R. M is classified in an equivalent of the limit-point condition at the singular points ± ∞, and conditions on the matrix coefficient Q are given which place M, when n> 1, in the equivalent of the strong limit-point for the case n = 1. It is also shown that the same condition on Q establishes the integral inequality for a certain class of vector-functions F.

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

Integral operators are used to solve the direct and inverse problems of the scattering of acoustic waves by a spherically stratified inhomogeneous medium of compact support. The results are valid for all values of the wave number and an arbitrarily large index of refraction. In the limiting case of small wave number or small inhomogeneities the results are in agreement with those of Rorres and Born.

This work is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form

$$ \mathcal{L}u+|u|^{q-1}u \leq\mathcal{L}v+|v|^{q-1}v, $$

where $q>0$ is a given real number and $\mathcal{L}$ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form given by the relation

$$ \mathcal{L}=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\bigg[a_{ij}(x)\frac{\partial}{\partial x_j}\bigg]. $$

We assume that $n\geq2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots,n$, are measurable bounded functions on $\mathbb{R}^n$ such that $a_{ij}(x)=a_{ji}(x)$ and that the corresponding quadratic form is non-negative. The results obtained in this work were announced by the author in 2005.

We use the extended multiple Riccati equations expansion method to construct a series of double-soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear Jaulent-Miodek PDEs. With the help of symbolic computation using Maple and Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combinations of trigonometric periodic functions and hyperbolic function solutions, various combinations of trigonometric periodic functions and rational function solutions, and various combinations of hyperbolic functions and rational function solutions.

A standard method of constructing Steiner triple systems of order 19 from the Steiner triple system of order 9 gives rise to 212 different such systems. It is shown that there are just three isomorphism classes amongst these systems. Representatives of each isomorphism class are described and the orders of their automorphism groups are determined.

The oscillatory behaviour of quasilinear hyperbolic equations of the form

is studied using a Sturmian-type comparison theorem. We assume that for some function ψ′(x)≦0 and ψ′(x≦0.) The existence of the first nodal line of u is then inferred from the existence of that of the solution ν of

with .Some results of Pagan are improved using this approach and a problem posed by him is also studied.

The presence of a non-uniform distribution of temperature in an elastic solid gives rise to an additional term in the generalized Hooke's Law connecting the stress and strain tensors and to a term involving the time rate of change of the dilatation in the equation governing the conduction of heat in the solid. The present paper is concerned with the effects produced by these additional terms in two simple situations. In the first, the elastic solid is regarded as being of infinite extent and the distribution of temperature in the solid is produced by heat sources whose strength may vary with time. In the second, the solid is supposed to be semi-infinite and to be deformed by prescribed variations in the temperature of the bounding plane and by heat sources within itself.

A scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

In this paper, we study global weak solutions for elastic equations with damping using the compensated compactness method. When the two end states at ± ∞ are not equal, the selfsimilar solutions for the corresponding parabolic equation are used to get the entropic estimates for both the L∞ and L2 cases.

In this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.

The existence of periodic solutions is proved for first order vector ordinary and functional differential equations when the right-hand side satisfies a one-sided growth restriction of Wintner type together with some conditions of asymptotic nature. Special cases in the line of Landesman-Lazer and of Winston are explicited.

We investigate equilibrium configurations for a polymer-stabilized liquid-crystal material subject to an applied magnetic field. The configurations are determined by energy minimization, where the energies of the system include those of bulk, surface and external field. The Euler–Lagrange equation is a nonlinear partial differential equation with nonlinear boundary conditions defined on a perforated domain modelling the cross-section of the liquid-crystal–polymer-fibre composite. We analyse the critical values for the external magnetic field representing Fredericks transitions and describe the equilibrium configurations under any magnitude of the external field. We also discuss the limit of the critical values and configurations as the number of polymer fibres approaches infinity. In the case where, away from the boundary of the composite, the fibres are part of a periodic array, we prove that non-constant configurations develop order-one oscillations on the scale of the array's period. Furthermore, we determine the small-scale structure of the configurations as the period tends to zero.

A Laplace transform is developed for functions which are holomorphic in the complex wedge Ω = {z| |arg (z)| < σ}, where 0 ≦ σ < π. The resulting transform will be holomorphic in a complementary wedge of the form Ωa = {a+ z||arg (z)|< (π/2) +σ for some a. This Laplace transform is shown to be an isomorphism between two appropriate spaces. The spanning properties of sets of the form {eλkS}k∊I in the domain space are studied. These results are then applied to a control problem.

Following an earlier paper (Curle 1978) we consider a compressible laminar boundary layer with uniform pressure when the distance x along the wall satisfies x < x0 and a prescribed large adverse pressure gradient when x > x0. The viscosity and absolute temperature are again taken to be proportional, but the Prandtl number is no longer assumed to be unity. After applying the Illingworth-Stewartson transformation, the transformed external velocity u1(x) is chosen so that

is large and constant, where Ts is the stagnation temperature, Tw is the (constant) Wall temperature and u0 is the upstream value of u1(x).

The flow reacts to this sharp pressure rise mainly in a thin inner sublayer, so inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature.

The skin friction, heat transfer, displacement thickness and momentum thickness are determined as functions of , andinvolve two parameters B1, B2, which depend upon the Mach number and the walltemperature. Detailed numerical calculations are presented here for σ = 0.72. In particular, it is seen that the heat transfer rate varies roughly like σ⅓ except near to separation, where it varies like σ¼.

This is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that if

then almost all these power series take every complex value infinitely often in the unit disk.

We consider steady potential hydrodynamic-Poisson systems with a dissipation term (viscosity) proportional to a small parameter v in a two- or three-dimensional bounded domain. We show here that for any smooth solution of a boundary value problem which satisfies that the speed, denoted by |∇φv|, has an upper coarse bound , uniform in the parameter v, then a sharper, correct uniform bound is obtained: the viscous speed |∇φv| is bounded pointwise, at points x0 in the interior of the flow domain, by cavitation speed (given by Bernoulli's Law at vacuum states) plus a term of that depends on . The exponent is β = 1 for the standard isentropic gas flow model and β = 1/2 for the potential hydrodynamic Poisson system. Both cases are considered to have a γ-pressure law with 1<γ<2 in two space dimensions and 1 < γ< 3/2 in three space dimensions. These systems have cavitation speeds which take not necessarily constant values. In fact, for the potential hydrodynamic-Poisson systems, cavitation speed is a function that depends on the potential flow function and also on the electric potential.

In addition, we consider a two-dimensional boundary value problem which has been proved to have a smooth solution whose speed is uniformly bounded. In this case, we show that the pointwise sharper bound can be extended to the section of the boundary ∂Ω\∂3Ω, where ∂3Ω is called the outflow boundary. The exponent β varies between 1 and 1/8 depending on the location of x0 at the boundary and on the curvature of the boundary at x0. In particular, our estimates apply to classical viscous approximation to transonic flow models.

We characterize the boundedness of the Hardy operator between weighted amalgams, a problem studied, but not completely solved, by C. Carton-Lebrun, H. P. Heinig and S. C. Hofmann. We also characterize the weighted weak-type inequalities for modified Hardy operators on amalgams.

§ 1. Let L2(0, ∞) denote the Hilbert space of Lebesgue measurable, integrable-square functions on the half-line [0, ∞).

Integral operators of the form

acting on the space L2 (0, ∞) occur in the theory of ordinary differential equations; see for example the book by E. C. Titchmarsh [4; § 2.6]. It is important to establish when operators of this kind are bounded; see the book by A. E. Taylor [3; § 4.1 and §§4.11, 4.12 and § 4.13].