A detailed proof of the quantum double construction is given for Z2 -graded Hopf algebras, and an explicit formula for the graded universal R−matrix is obtained in a general fashion.

By means of defined concepts all metric spaces of given weight or cardinality and their quotients are characterised. An example of a sequential space having weight w1 which is not a quotient of any metric space of weight w1 is provided. The well-known classes of sequential spaces are also obtained as images of metric spaces by particular kinds of maps.

In the first section we establish a connection between gap Tauberian conditions and isomorphic copies of Co for perfect coregular conservative BK spaces and in the second we give a characterisation of gap Tauberian conditions for strong summability with respect to a nonnnegative regular summability matrix. These results are used to show that a gap Tauberian condition for strong weighted mean summability is also a gap Tauberian condition for ordinary weighted mean summability. We also make a remark regarding the support set of a matrix and give a Tauberian theorem for a class of conull spaces.

In this note we show that a subgradient multifunction of a locally compactly Lip-schitzian mapping satisfies a closure condition used extensively in optimisation theory. In addition we derive a chain rule applicable in either separable or reflexive Banach spaces for the class of locally compactly Lipschitzian mappings using a recently derived generalised Jacobian. We apply these results to the derivation of Karush-Kuhn-Tucker and Fritz John optimality conditions for general abstract cone-constrained programming problems. A discussion of constraint qualifications is undertaken in this setting.

Rolewicz said that the norm of a Banach space (X, ║·║) has the drop property if for every closed set C disjoint from the closed unit ball B(X), there exists an x ∈ C such that co(x, B(X)) ∩ C = {x}. He then showed that if the norm of a Banach space (X, ║·║) has the drop property then it is reflexive. Later, in 1990 Giles, Sims and Yorke showed that the norm of a Banach space (X, ║·║) has the drop property if and only if the duality mapping f → D(f) from the dual sphere S(X*) into subsets of the second dual sphere S(X**) is norm upper semi-continuous and norm compact-valued on S(X*).

The existence of twist orbits and twist cycles with a given rotation number is considered for discrete dynamical systems generated by iteration of liftings of maps of the circle into itself. The class of maps for which such orbits exist for every number in the interior of the rotation set is extended to contain an important subclass of non-continuous maps.

Let B be a Banach space and X ⊂ B a normal cone such that the norm is monotone on X for the order determinated by X.

We study the sup, denoted by i(X), of the q ≥ 1 such that, for each E> 0 and each n, there are x1, …, xn in X such that:

for all a1, …, an ≥ 0, where ‖ ‖q is the norm in lq.

We prove that i(X) is the inf of the p for which we have:

The proof use a similar theorem of Kirvine, concerning Banach Riesz spaces. Here conical measures are a useful tool. We establish a link with a preceding work in which we adapt the Maurey theory factorisation of operators with values in a LP space, to the case of normal cones, contained in a Banach space.

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

This paper deals with the approximate subdifferential chain rule in a Banach space. It establishes specific results when the real-valued function is locally Lipschitzian and the mapping is strongly compactly Lipschitzian.

For a first order formula P: ∀x1, …, ∀xn, ∃y1, …, ∃ym (u(x1, …, xn, y1, …, ym) ≡ v(x1, …, xn, y1, …, ym)) where u and v are two words on the alphabet {x1, …, xn, y1, …, ym}, and a finite set E of semigroup identities with xy ≡ yx in it, we prove that it is decidable whether P follows from E, that is whether all the semigroups in the variety defined by E satisfy P.

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.

This paper investigates some of the connections between the zeros of a polynomial vector field F = (f, g): ℂ2 ← ℂ2 and the Jacobian determinant J(f, g) of f and g. As a consequence, sufficient conditions are given for F to have no zeros. In addition, in the case where F has an inverse F−1, it is proven that F−1 is also polynomial.

A new minimax inequality is first proved. As a consequence, five equivalent fixed point theorems are formulated. Next a theorem concerning the existence of maximal elements for an Lc-majorised correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium points for a non-compact one-person game and for a non-compact qualitative game with Lc-majorised correspondences are given. Using the latter result and employing an “approximation” technique used by Tulcea, we deduce equilibrium existence theorems for a non-compact generalised game with LC correspondences in topological vector spaces and in locally convex topological vector spaces. Our results generalise the corresponding results due to Border, Borglin-Keiding, Chang, Ding-Kim-Tan, Ding-Tan, Shafer-Sonnenschein, Shih-Tan, Toussaint, Tulcea and Yannelis-Prabhakar.

A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.