A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.

In this paper we define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.

Let a group G be covered by finitely many disjoint cosets of subgroups Gi. We study conditions which imply that the subgroups Gi are conjugate in G.

We compare the definitions of analyticity of vector-valued functions and their connections with the topological tensor products of non-locally convex spaces.

Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

The cyclotomic identity

where and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Let E be a real Banach space with a uniformly convex dual, and let K be a nonempty closed convex and bounded subset of E. Suppose T: K → K is a continuous monotone map. Define S: K → K by Sx = f – Tx for each x in K and define the sequence iteratively by x0 ∈ K, xn+1 = (1 – Cn)xn + CnSxn, n ≥ 0, where is a real sequence satisfying appropriate conditions. Then, for any given f in K, the sequence converges strongly to a solution of x + Tx = f in K. Explicit error estimates are also computed. A related result deals with iterative solution of nonlinear equations of the dissipative type.

Let f(z) be an n–Valued algebroid function of finite lower order. In the present paper, we give a spread relation of f(z) and some applications of the spread relation.

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

In this paper we investigate the relationships among the notions of minimality, characteristic 0, equicontinuity, Lipschitz stability and isometry in dynamical systems. Examples are provided to show that the results obtained are sharp.

Let B be the Bloch space of all those functions f holomorphic in the open unit disc D of the complex plane satisfying . We establish sufficient conditions for the boundedness of functions f belonging to B satisfying a certain uniform radial boundedness condition, and, by introducing a wide class of subsets E of ∂D, which we call negligible sets for boundedness, we show that if f ∈ B and there is a constant K > 0 such that , then f is bounded in D. Hence a significant extension of a theorem of Goolsby is obtained.

We give sufficient conditions for C∞ vector field systems on Rn with genus g = 1 to be diffeomorphic to a contact structure. The diffeomorphism is explicitly constructed and used to give the most general integral submanifolds for the systems. Finally the implications of these results for integrable hyperbolic partial differential equations in the plane is discussed.

Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X ⊆ F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).

A family 𝒟 of sets is said to be uniformly deep of depth d if and only if for every x ∈ ∪ 𝒟 there are exactly d distinct 𝒜 ∈ 𝒟 for which x ∈ 𝒜.

We establish the following result, and extend our ideas to general Euclidean spaces.

Variational inequality theory provides techniques for solving a variety of applied problems in science and engineering. Recently Noor considered some interesting general nonlinear and linear variational inequalities in a series of papers and proved the existence and uniqueness of solutions by a fixed point technique developed by Glowinski, Lions and Tremolieres and also by a fixed point technique of Lions and Stampacchia. But there are several inaccuracies in his proofs and here they have been removed and correct formulation of the theorems are stated and proved and relationships are clearly shown. The existence of solution necessitates an additional condition in one case, and less condition in the other, but uniqueness can be proved without the condition that the operator be antimonotone.

In this paper we study the isometric deformations of surfaces in E3, which preserve the lines of curvature. We call a surface M and LC-surface if it admits a non-trivial deformation of this type. We distinguish three types of LC-surfaces, and obtain some new results about these three types of surfaces.

A distance is defined on the quotient of the set of submanifolds of a Euclidean space, with respect to similarity. It is then related to a previously defined function which captures the metric behaviour of paths.

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

The weighted Bergman space Ap, α, 0 < p < 1, a > −1 of analytic functions on the unit disc Δ in C is an F-space. We determine the dual of Ap, α explicitly.

Let G be a finite group. It is proved that the localised Buinside ring Ωp(G) is of finite representation type if and only if for each p–perfect subgroup H of G, , where means the conjugacy class of K.