In this paper a new fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values is established in the setting of an abstract convex structure – called a locally G-convex space, which generalises usual convexity such as locally convex H-spaces, locally convex spaces (locally H-convex spaces), hyperconvex metric spaces and locally convex topological spaces. Our fixed point theorem includes corresponding Fan-Glicksberg type fixed point theorems in locally convex H-spaces, locally convex spaces, hyperconvex metric space and locally convex spaces in the existing literature as special cases.

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:

(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;

(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;

(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.

In the proof, we use some properties of commutative rings associated to multilinear maps.

We use a nonabelian exterior product to strengthen two old and basic results on the Schur multiplier of a (central) quotient of a direct product of groups.

We show that every f: ℂ2 → ℂ which is a rational polynomial map with irreducible fibres is a coordinate.

In the present paper we consider the approximation of |x|α on [−1, 1] by interpolating polynomials and we establish upper bounds for the approximation error. It turns out that these bounds, apart from the constants, are of the best possible order. We compare our results with estimates for the best approximation, established by Bernstein.

For an n-times differentiable function f(x) whose derivatives f(j)(xj) at x = xj, j = 0, 1, …, n are specified, we introduce a sequence of fundamental polynomials to interpolate f(x) with a remainder as

The remainder R(x|x0, x1,… xn) is given in an integral form and Lagrange's form.

In addition, by introducing orthogonality of Sobolev type we verify the best optimality of the approximations and interpret the fundamental polynomials as a kind of Sobolev orthogonal polynomial.