Let f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of f$f$ with a small enough sphere S^{3}_\epsilon$S^{3}_\epsilon$ centred at the origin in \mathbb {R}^{4}$\mathbb {R}^{4}$. If f$f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

Dirichlet, Neumann and mixed boundary value problems for the equation uxx −uyy = 0 are considered for a variety of rectangular domains. Uniqueness of solutions to these non-well-posed problems is considered by separation of variables methods. The question of uniqueness is also discussed for domains other than rectangles.

A compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant with

respect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.

Let B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

We consider a class of non-linear evolution equations subject to a periodic forcing term. Using bifurcation theory we obtain results on the existence and number of periodic solutions. The theory applies to semi-linear diffusion equations defined on bounded or unbounded domains.

We present several general methods of constructing tight hypersurfaces f: Mn → En+1, n ≧ 2. We prove a smoothing lemma, which allows us to approximate tight continuous hypersurfaces by C∞ tight ones. We show that given a tight immersion or C∞-stable map f: M2 → E3 of a compact surface M2 other than S2, there is a tight C∞-stable map g: M2 # RP2:→ E3. We prove that given C∞ tight immersions f: Mn → En+1 and g: Nn → En+1 of compact n-manifolds Mn and Nn into En+1, there is a tight C∞ immersion of Mn # Nn into En+1. Two other methods involve hypersurfaces of rotation and sets in En+1 at a fixed distance from a tightly embedded n-manifold with boundary in En. One consequence of these methods is that the outer part of C∞ tight hypersurfacesf: Mn → En+1, n ≧ 3, is far more complicated than in the case of tight surfaces in En. For example, given any C∞n-manifold M with boundary tightly embedded in En, there is a tight immersion f:Nn → En+1 of a closed n-manifold having M as a topset. Kuiper's theorem describing tight immersions of surfaces into E3 does not generalise to the case of hypersurfaces f: Mn → En+1, n ≧ 3, without substantial restrictions on Mn and/or f.

The present paper is devoted to the study of the weak respectively strong convergence of solutions of variational inequalities, with non-linear partial differential operators of the generalised divergence form and of monotone type, under a perturbation of the domain of the definition. In this study there are used convergence concepts defined according to [ 22] and abstract convergence theorems given in [15 and 16].

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we have

for some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

Estimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

We consider the Null Mass nonlinear field equation (𝒫)

\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right. $$\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right.$$

{\open R}^N \setminus \Omega ${\open R}^N \setminus \Omega$

We show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

In an earlier paper (Aitken and Silverstone, 1941) the problem of estimating from sample a parameter θ of unknown value was treated by adopting two postulates for the estimating function: (i) that it should be unbiased in the linear sense; (ii) that its sampling variance should be minimal.

The Bragg two-hologram method for eliminating the unwanted image in Diffraction Microscopy is here thoroughly examined. It is shown to be of general validity for objects containing both amplitude and phase-contrast terms.

Other two-hologram methods are briefly discussed. In principle any two sufficiently distinct holograms may be used, but the Bragg 2:1 focal ratio is the simplest. The pair of holograms at ±f is also of interest, since it allows moderate degrees of phase-contrast and amplitude-contrast terms in the original object to be separated. A “three-wavelength” technique for obtaining the Bragg records is briefly outlined.

Little progress has been made in the development of a relativity theory of elasticity, although it has been realised that no disturbance can be propagated with a velocity greater than that of light. In 1917 Lorentz (1) gave a relativistic formulation of the laws of elasticity in the case of small strain and, applying the theory to the problem of a rotating, incompressible, homogeneous disc, he claimed that the radius as measured by an observer at rest on the disc undergoes a contraction. His result was accepted by Eddington (4) but was attacked by others. A great deal has been written on the subject, but it has never been pointed out that both Lorentz and Eddington were considering material in which the waves of dilatation travel with an infinite velocity. In this paper we define “incompressible” matter as that in which these waves are propagated with the velocity of light and Poisson's ratio tends to the value ½. This gives an upper limit to the modulus of compression k, which in this case is the elastic constant λ, and as a result the expansion determined by the ordinary classical theory has to be taken into account. It is found that the “relativity contraction” is exactly cancelled by the “classical expansion”. Throughout the discussion on the rotating disc the analysis is restricted to the case of small strain.

The equations of equilibrium of a continuous static distribution of matter are also investigated in the case of weak fields for which the fourth power of the density may be neglected.

In this paper, we prove that a positive perturbation T = T0 + q (q ≧ 0 and in ) of an essentially self-adjoint Schrödinger operator T0 = −Δ + q0 on is again essentially self-adjoint if T is relatively bounded with respect to T0. An application of the method of the proof to positive approximations of elements u ≧ 0 in D(T) by a positive sequence in is given.

1. Introduction.—The object of this paper is twofold: firstly, to present a table of the Chebyshev polynomials Cn(x) = 2 cos (n cos−1 ½x) for n = I(I)12 and x = o(o·o2)2, values being exact or to 10 decimals; secondly, to provide a working list of coefficients and formulæ relating to these and allied functions.

Valuable accounts of applications and properties will be found in Van der Pol and Weijers, 1933, in Lanczos, 1938, and in Szego, 1939. Further applications are indicated in the following paper, by J. C. P. Miller, which also suggests methods of reducing the inconvenience caused by the present lack of tables of the allied polynomials Sn(x). It is hoped that suitable tables will be prepared later.

We prove the absence of singular continuous spectra for Schrödinger operators − Δ + V with long-range potential V such that V and (1 + r)1+ε(∂/∂r) V is (− Δ)-compact by using a modified “Mourre type” estimate and by Kato-Lavine's H-smoothness theory.

We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.

We study the existence, nonexistence and multiplicity of non-negative solutions for the family of problems

where Ω is a bounded domain in ℝ2 and λ > 0 is a parameter. The coefficient a(x) is permitted to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the Trudinger–Moser inequality and variational methods. Note that when f(x, u) = 0 the equation is of Liouville type.