We construct several examples of Jacobson radical rings which are not squares of other rings.

We show that L∞(μ,X) has the Dunford-Pettis property for some classical Banach spaces including L1(μ), C (K), the disc algebra A and H∞.

In [3] for general integral metric q an integral extension of Lebesgue power was discussed. In this paper we introduce the abstract Daniell-Loomis spaces Rp, p real, 0 < p < ∞, of q-measurable functions with finite “p-norm”, and study their basic properties.

The existence of globally attractive order intervals for some strongly monotone discrete dynamical systems in ordered Banach spaces is first proved under some appropriate conditions. With the strict sublinearity assumption, threshold results on global asymptotic stability are then obtained. As applications, the global asymptotic behaviours of nonnegative solutions for time-periodic parabolic equations and cooperative systems of ordinary differential equations are discussed and some biological interpretations and concrete application examples are also given.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

We give necessary and sufficient conditions on the semigroup S for the Jacobson radical to be S-invariant.

Consider the singular boundary value problem (r(x′))′ + f(t, x) = 0, 0 < t < 1. We give necessary and sufficient conditions for this problem to have solutions. In addition, a uniqueness result is obtained.

The main aim of this paper is to introduce new ways of constructing flat projective planes from spherical circle planes and cylinder circle planes. We shall also touch upon topics that have natural connections with our constructions, like the construction of spreads of pseudolines from flat projective planes, the extendability of partial spherical circle planes to spherical circle planes and giving examples of sets of 2-arcs that determine flat projective planes and ℝ2-planes.

We give an algorithm by which one can compute, using only rational numbers, the continued fraction and more generally the Jacobi-Perron algorithm expansion of real algebraic numbers.

Let K1 and K2 be extension fields over a field K with char K = p > 0. Assume L = K1(x1) = K2(x2) ⊃ K where xi is transcendental over Ki, for i = 1, 2. In this paper we prove that if K1 is a perfect field, then K1 = K2.

In this paper, we establish the generalized discriminant theorem governing uniqueness of symmetric polynomials in n variables of least total degree dn, k which vanish when any k + 1 of the variables are equal.

Strong solvability and uniqueness in the Sobolev space W2, q(Ω), q > n, are proved for the oblique derivative problem

assuming the coefficients of the quasilinear elliptic operator to be Carathéodory functions, aij ∈ VMO∩L∞ with respect to x, and b to grow at most quadratically with respect to the gradient.

A convex function f is said to be pointed if its epigraph has a recession cone which is pointed. Partial pointedness of f refers to the case in which such a recession cone is only partially pointed. In this note we show that the degree of pointedness of f is related to the “thickness” of the effective domain of the conjugate function f*.

The study of nonclassical solutions for elliptic and parabolic PDE's often involves the use of regularisation processes such as the sup- and inf-convolutions. In this note we study the behaviour of these regularised functions near the boundary of the domain, and derive constraints on the appropriate second-order sub- and superdifferentials on and near the boundary. Potential applications to regularity results are also noted.