We prove a conjecture of Willis saying that for each f in the Volterra algebra the sequence (‖fn‖)n is a weight sequence which is regulated at 1.

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

We give sufficient conditions for a homogeneous superposition operator to be a continuous mapping between Orlicz-Sobolev spaces. This extends a result of Marcus and Mizel concerning mappings between Sobolev spaces.

A multi-dimensional analogue of a well known inequality of Littlewood and Paley is obtained on the ball.

Let f ∈ ℤ[X] and let q be a prime power pl(l ≥ 2). Hua stated and proved that

for some unspecified constant C > 0 depending on the derivative f′ of f; M denoting the maximum multiplicity of the roots of the congruence p−tf′(x) ≡ 0 (mod p), where t is an integer chosen so that the polynomial p−tf′(x) is primitive. An explicit value for C was given by Chalk for p ≥ 3. Subsequently, Ping Ding (in two successive articles) obtained better estimates for p ≥ 2.

This article provides a better result, based upon a more precise form of Hua's main lemma, previously overlooked.

In this paper, we establish a new coincidence theorem for a Browder type set-valued mapping and an upper semi-continuous set-valued mapping with compact acyclic values in an H-space which generalises some recent results in the literature. As applications we obtain two H a type coincidence theorems and existence theorems of maximal elements for preference correspondences.

There are essentially two ways to obtain transcendence results in finite characteristic. The first, historically, is to use Ore's lemma and to prove that a series whose coefficients satisfy well-behaved divisibility properties cannot be a zero of an additive polynomial. This method is of the same kind as the method of p–automata. The second one is to try to imitate the usual methods in characteristic zero and to do transcendence theory with t–modules analogously to what we can do with algebraic groups. We want to show here that transcendence results over Fq(T) can also be obtained with the help of the variable T. If ec(z) is the Carlitz exponential function and e = ec(1), we obtain, in particular, that 1, e, …, e(p–2) (the P–2 first derivative of e with respect to T) are linearly independent over the algebraic closure of Fq(T). A corollary is that for every non-zero element α in Fq((1/T)), αpe and αec(e1/p) are transcendental over Fq(T). By changing the variable and using older results we also obtain the transcendence of ec(ω) for all ω ∈ Fq((1/T)) such that ω(T) and ω(Ti) are not zero and linearly dependent over Fq (Ti) (q > 2i + 1). Such u appear to be transcendental by the method of Mahler if i is not a power of p.

We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.

Let G be a finitely generated soluble group. Lennox and Wiegold have proved that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that (x, y) is nilpotent. The main theorem of this paper is an improvement of the previous result: we show that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that [x, ny] = 1 for some integer n = n(x, y) ≥ 0.

A ring is called s–prime if the 2-sided annihilator of a nonzero ideal must be zero. In particular, any simple ring or prime (—1, 1) ring is s–prime. Also, a nonzero s–prime right alternative ring, with characteristic ≠ 2, cannot be right nilpotent. Let R be a right alternative ring with commutators in the left nucleus. Then R is associative in the following cases: (1) R is prime, with characteristic ≠ 2, and has an idempotent e ≠ 1 such that (e, e, R) = 0. (2) R is an algebra over a commutative-associative ring with 1/6, and R is either s–prime, or R is prime and locally (—1,1).

A semigroup S is called E-solid if and only if for all idempotents e, f, g ∈ S such that e L f R g there exists an idempotent h ∈ S such that e R h L g. Each completely regular semigroup is E-solid. We characterise the idempotents of an arbitrary E-solid regular semigroup as a set with a binary operation on it satisfying a given finite set of identities.

The Schauder estimates for solutions of linear second order parabolic equations with Venttsel initial boundary conditions are proved, and existence and uniqueness of classical solutions under such an initial boundary condition are established. An application to an engineering problem is also given.

In the present paper, we consider Schrödinger operators which are formally given by . In Section 2 and 3 we prove that P has a regularly accretive extension which is a self-adjoint extension of P and it is the only self-adjoint realisation of P in L2 (RN) when satisfies = (a1, a2, …, aN) ∈ , aj, real-valued, , real-valued and the negative part V-:= max(0, -V) satisfys , with constants 0 ≤ C1 < 1, C2 ≥ 0 independent of V. In Section 4, we prove that P is essential self-adjoint on when , V sat0isfy ; V = V1 + V2, V real-valued, , i = 1, 2, V1(x) ≥ –C |x|2, for x ∈ RN with C ≥ 0 and 0 ≥ V2 ∈ KN.

For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

It is shown that, in Pettis's criterion for Bochner measurability of a vector-valued function f: S → X, scalar measurability of f can be weakened to requiring that u of be measurable for u in some subset of the dual X* separating the points of X. Even then, the separability hypotheses in Pettis's Theorem can be weakened as well.

We prove that the following ring-theoretic properties are shared by the two rings involved in a normalising extension R ⊂ S, and that these properties are inherited by any intermediate extension: semilocal, left perfect, semiprimary. This transfer fails for the nilpotency of the Jacobson radical. However, if the normalising set is a basis for the left R-module S, then the nilpotency of the Jacobson radical behaves in the same way as the three properties mentioned above.

In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.

An identity for two generalised Tribonacci sequences is obtained. Earlier identities by Pethe and Horadam follow as special cases.

We establish the existence of solutions for the elliptic systems on ℝN: such that u, v ∈ W1, 2(ℝN), where with q(x)→∞ as |x| → ∞ and (x, u, v) being superlinear or sublinear as .

The second–order behaviour of a nonsmooth convex function f is reflected by the so–called second–order subdifferential mapping ∂2f. This mathematical object has been intensively studied in recent years. Here we study ∂2f in connection with the geometric concept of “second-order normal vector” to the epigraph of f.