These experiments were begun in January last at the request of Professor Tait, and have been carried on since then in his laboratory. The subject suggested itself to Professor Tait during his experiments on the effects of great pressures on thermometers carried out for the “Challenger” Expedition Commission, and it was indeed by part of the apparatus purchased by him for this purpose that the experiments were made. The method adopted was of an indirect nature, and consisted in determining first the change of temperature produced by a sudden change of pressure to which the water was subjected.

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation $\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in $\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate

$$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$

$q \gt 1+{N}/{2}$

${\open R}^N$

$\partial _p\Omega _T$

$\Omega _T$

$\omega \in L^1(\Omega _T)$

$\omega \ges 0$

$q=1+{N}/{2}$

$\Vert f \Vert_{q,\Omega _T}$

$$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$

We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

In this paper, we study divergence properties of the Fourier series on Cantor-type fractal measure, also called the mock Fourier series. We give a sufficient condition under which the mock Fourier series for doubling spectral measure is divergent on a set of strictly positive measure. In particular, there exists an example of the quarter Cantor measure whose mock Fourier sums are not almost everywhere convergent.

Existence and uniqueness theorems are proved for the solution of a Dirichlet-Neumann-Third mixed boundary value problem for the Helmholtz equation in ℝ3. The proofs make use of an equivalent system of two integral equations of the second kind.

A sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.

If X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

Since the delivery of my presidential address (1) in July I have assembled an amount of supplementary information regarding “the Chemical Society instituted in the beginning of the Year 1785”. This, together with a brief description of some other chemical societies of the revolutionary period, forms the basis of the present paper.

First of all, it will be expedient to furnish a complete list of the dissertations read before the Society during 1785–86 and included in the first volume of its Proceedings, appending short comments with respect to the communicators or their topics when anything of special interest arises.

The aim of this paper is to characterise sets of anisotropic weighted capacity zero. In this we generalise previous known results for the isotropic equivalent. A particular case of this zero capacity set is used to generalise removable singularity results for weak solutions of degenerate quasilinear parabolic equations and for their elliptic equivalent when its structure is still essentially isotropic, with the anisotropy confined to the mixed norms of the generalised Lebesgue spaces involved.

Let P be a set of primes. A nilpotent space X is called “P-universal”, if its P-localization canbe obtained as a direct limit over a sequence of selfmaps of X. A nilpotent fibration is called “fibrewiseP-universal”, if its fibrewise P-localization can be obtained in a similar way as a direct limit of fibrewise maps. In this paper, the following results are proved. Let π:E → B be a nilpotent fibration of connected finite or cofinite spaces. If π is fibrewise P-universal for some proper subset P of the set of primes, then its minimal model (in the sense of D. Sullivan's rational homotopy theory) admits a certaintype of weight decomposition. But the existence of such a weight decomposition implies only that there exists a finite set of primes, such that IT is fibrewise P-universal for all P with . In the absolute case however, i.e.B is a point, the set P can be chosen as the empty set. Thus a result announced by R. Body and D. Sullivan is recovered.

Since the year 1889, when the last of a series of six papers with a title similar to the above appeared, further research has led to the discovery of a number of writings belonging to the period then dealt with, viz., 1693–1844. Of those an account is now given before proceeding to the papers of later date than 1844.

The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.

In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.

This paper gives sufficient conditions ensuring that a non-linear control system of the form

is controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.