A recurrence relation for the square root

The behavior of the sequence x_{n + 1} = x_n(3N − x_n²)/2N is studied for N > 0 and varying real x₀. When 0 < x₀ < (3N)^1/2 the sequence converges quadratically to N^1/2. When x₀ > (5N)^1/2 the sequence oscillates infinitely. There is an increasing sequence β_r, with β₋₁ = (3N)^1/2 which converges to (5N)^1/2 and is such that when β_r < x₀ < β_{r + 1} the sequence {x_n} converges to (−1)^rN^1/2. For x₀ = 0, β₋₁, β₀,… the sequence converges to 0. For x₀ = (5N)^1/2 the sequence oscillates: x_n = (−1)ⁿ(5N)^1/2. The behavior for negative x₀ is obtained by symmetry.     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.     

Some theorems on graphs and posets

In this journal, Leclerc proved that the dimension of the partially ordered set consisting of all subtrees of a tree T, ordered by inclusion, is the number of end points of T. Leclerc posed the problem of determining the dimension of the partially ordered set P consisting of all induced connected subgraphs of a connected graph G for which P is a lattice.In this paper, we prove that the poset P consisting of all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimension n where n is the number of noncut vertices in G whether or not P is a lattice. We also determine the dimension of the distributive lattice of all subgraphs of a graph.     

Poisson structures due to Lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.     

Synthesizing 0.9PZN–0.1BT by mechanically activating mixed oxides

0.9Pb(Zn_1/3Nb_2/3)–0.1BaTiO₃ (0.9PZN–0.1BT) of perovskite structure has been successfully prepared by mechanically activating mixed oxides of PbO, ZnO, Nb₂O₅, BaO and TiO₂. The novel mechanochemical technique skips the phase-forming calcination step at an intermediate temperature that is always required in both the conventional solid state reaction and chemistry-based precursor routes. Ultrafine 0.9PZN–0.1BT particles of perovskite structure were formed when the constituent oxides were mechanically activated for more than 10 h. The powder was sintered to a density of 96% theoretical density at 1100°C for 1 h. The sintered 0.9PZN–0.1BT exhibits perovskite structure and a peak dielectric constant of 8800 at the Curie temperature of 60°C when measured at a frequency of 100 Hz.     

On the joint distribution of two discrete random variables

Summary LetX, Y be two discrete random variables with finite support andX≧Y. Suppose that the conditional distribution ofY givenX can be factorized in a certain way. This paper provides a method of deriving the unique form of the marginal distribution ofX (and hence the joint distribution of (X, Y)) when partial independence only is assumed forY andX−Y.     

Eigenfunctions on symmetric spaces with distribution-valued boundary forms

A characterization is given for those eigenfunctions of invariant differential operators on symmetric spaces of noncompact type which are representable as generalized Poisson integrals of distributions on the boundary, the criterion being that the function grow no faster than some power of the exponential of the distance from the origin. For symmetric spaces of arbitrary rank, the result is proved in one direction only, namely, that the Poisson integral of a distribution satisfies the growth condition; however, for rank one symmetric spaces, the converse is also shown to be true.     

On the totality of sequences in topological vector spaces

Price and Zink [Ann. of Math.82 (1965), 139–145] gave necessary and sufficient conditions for the existence of a multiplier m so that {mφ_n}₁^∞ is total; that is, the linear span is dense in L²[0, 1], thus answering a question raised by Boas and Pollard [Bull. Amer. Math. Soc.54 (1948), 512–522]. Using techniques similar to those of Price and Zink, it is shown that this result can be extended to more general spaces. Indeed, if X is either a separable Fréchet space or a complete separable p-normed space (0 < p ? 1), then the existence of a continuous linear operator A so that {Aφ_n}₁^∞ spans a dense subspace is implied by the existence of a nested, equicontinuous family of commuting projections which in addition has the properties that the union of their ranges is dense and that, for each projection, the projection of the original sequence is total in the projected space. Conversely, in a Banach space, it is shown that if such an operator exists and is 1-1 and scalar, then such a family of projections also exists. Further, it is shown that the above considerations extend the first half of the Price-Zink result to L^p[0, 1] (0 < p < ∞) and the other half to L^p[0, 1] (1 ? p < ∞).