Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the first model.

We study the Cauchy problem for the spatially homogeneous Landau equation for Fermi–Dirac particles, in the case of hard and Maxwellian potentials. We establish existence and uniqueness of a weak solution for a large class of initial data.

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

We prove the existence of an arbitrarily large number of periodic solutions for a class of nonlinear differential equations generalising the dynamics of a forced pendulum with small length.

We present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact Riemannian manifold with convex boundary, or on a noncompact complete manifold. Our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation, and an abstract gradient estimate for a Markov semigroup.

We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

We establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations of logistic type −Δpu = a(x)h(u) − b(x)f(u), where Δpu = div (|∇u|p−2∇u) with p > 1, h(u)/up−1 is non-increasing and f(u) is a function whose variation at infinity may be regular or rapid. In particular, our result regarding the blow-up rate reveals the main difference between regular variation function f and rapid variation function f.

In this paper we establish continuity of the interface of the weak solution to an elliptic-parabolic problem. The physical background is the theory of partially saturated fluid flows in porous media. Our method is based on the maximum principle for parabolic equations. An essential assumption is that the flow is one-dimensional.

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in ℝn. Using variational techniques and in particular a method introduced by Alikakos and Fusco we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to bulk free energies in crystals, we establish existence for N ≥ 2 in various cases of more than two minima. Finally, we obtain a sufficient condition establishing existence of connections to potentials which are not necessarily symmetric for arbitrary N and three minima.

No district of Great Britain is so rich in genera and species of Carboniferous Dipnoi and Ganoidei as that of Fife and the Lothians, and in the frequency of entire specimens of these fishes it is only approached by North Staffordshire.

Another feature of the greatest geological interest is the abundance of fish remains in estuarine strata below the horizon of the Millstone grit, whereby a means of comparing the Ganoid fish faunæ of the Upper and Lower divisions of the Carboniferous formation is afforded.

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.

In this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type. This class includes the solution map for functional differential equations and certain partial differential equations. In particular, we present necessary and sufficient conditions for one-to-oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator.

Let X be a uniformly convex Banach space, and N(X) the normal structure coefficient of X. In this paper it is proved that N(X) can be calculated by considering only sets whose points are equidistant from their Chebyshev centre. This result is applied to prove that N(LP(Ω)) = min {21−1/p, 21/p}, Ω being a σ-finite measure space. The computation of N(Lp) lets us also calculate some other coefficients related to the normal structure.

We study the Cauchy problem of certain Boussinesq-α equations in n dimensions with n = 2 or 3. We establish regularity for the solution under ▽u ∈ L1 (0, T; Ḃ0∞,∞(ℝn)). As a corollary, the smooth solution of the Leray-α–Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion ▽θ ∈ L1(0, T;L∞(ℝ2)) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.

M. G. Krein's method of directing functional is generalized in a straight-forward way to symmetric linear relations. Applications to Stieltjes differential boundary problems are indicated.

We explain the basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by the Peter-Weyl decomposition of L2() for a compact Lie group . After developing a general concept for compact groups and their homogeneous spaces, we give concrete examples for tori, which reflect the situation on ℝn, and for 2 and 3 spheres.

We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

We prove the existence of a positive solution for the Schrödinger-type equation on RN,−Δu + V(x)u = f(u) for u ∈ H1(RN), provided that The result is obtained in two cases. (a) lim|x|→ ∞V(x) = V (∞) ∈ (0,∞), f is subcritical and supt>0 2F(t)t−2 < V (∞). (b) There exist V > 0 and R* > 0 such that f(t)t−1 < V ∞ ≤ V (x) for all t > 0 and |x| ≥ R*.