In this paper, we establish a Farkas-Minkowski type alternative theorem under the assumption of nearly semiconvexlike set-valued maps. Based on the alternative theorem and some other lemmas, we establish necessary optimality conditions for set-valued vector optimisation problems with extended inequality constraints in a sense of weak E-minimisers.

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

If the indeterminate X in a Fréchet algebra A of power series is a power series generator for A, then either A is the algebra of all formal power series or is the Beurling-Fréchet algebra on non-negative integers defined by a sequence of weights. Let the topology of A be defined by a sequence of norms. Then A is an inverse limit of a sequence of Banach algebras of power series if and only if each norm in the defining sequence satisfies certain closability condition and an equicontinuity condition due to Loy. A non-Banach uniform Fréchet algebra with a power series generator is a nuclear space. A number of examples are discussed; and a functional analytic description of the holomorphic function algebra on a simply connected planar domain is obtained.

Every subgroup of finite index of the modular group PSL (2, ℤ) has a signature consisting of conjugacy-invariant integer parameters satisfying certain conditions. In the case of genus zero, these parameters also constitute a prescription for the degree and the orders of the poles of a rational function F with the property:

Functions correspond to subgroups, and we use this to establish necessary and sufficient conditions for existence of subgroups with a certain subclass of allowable signatures.

In this paper, we consider the growth of meromorphic solutions of nonlinear differential equations of the form L (f) + P (z, f) = h (z), where L (f) denotes a linear differential polynomial in f, P (z, f) is a polynomial in f, both with small meromorphic coefficients, and h (z) is a meromorphic function. Specialising to L (f) − p (z) fn = h (z), where p (z) is a small meromorphic function, we consider the uniqueness of meromorphic solutions with few poles only. Our results complement earlier ones due to C.-C. Yang.

In this paper we describe when two Toeplitz operators Tf and Tg on the Bergman space of the bidisc commute, where f = f1 + 2, g = g1 + ḡ2, fi, gi ∈ H∞(D2)(i = 1, 2).

The object of this note is to present a brief derivation of the main result of Andrews, Paule and Riese (2001).

Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.

Hardy's uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces.