We show that the space of smooth maps from a compact smooth manifold into another smooth manifold can be endowed with the structure of a smooth manifold if we use the Γ-differentiation of Yamamuro. We then generalise the Smale Density Theorem to mappings between these manifolds.

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

The motion of a slender body parallel and very close to a flat interface which separates two immiscible liquids of differing density and viscosity is considered for very small Reynolds numbers. Approximate analytical expressions are obtained for the distribution of forces acting on the slender body. The limiting case of a rigid plane wall yields results obtained previously.

The following theorem proved in this paper is a generalization of a result of Jakob Nielsen. Suppose G is a group of linear fractional transformations acting on the unit disc D in the complex plane; suppose also that each element of G, except the identity, is either a hyperbolic or a parabolic transformation. Then any homeomorphism h of the open disc onto itself which satisfies the functional equation hg = g′h, for some automorphism g → g′ of G, has a unique extension to a homeomorphism of D onto itself.

We generalise the standard notion of transversality to maps between manifolds modelled on locally convex spaces and prove that all standard transversal theorems remain valid for this generalised notion.

As first applications, we use our generalised Transversal Density Theorem to prove several simple “generic” results for local smooth maps which parallel the usual ones.

An operator T on a Hilbert space is in the class of k-quasihyponormal operators Q(k), if T*k(T*T−TT*)Tk ≥ 0. It is shown that if T is in Q(k) and S is normal such that TX = XS, where X is one to one with dense range, then T is normal; and is unitarily equivalent to S. It is proved that S can be replaced by a cohyponormal operator, if T in Q(1) is one to one. It is also shown that two quasisimilar operators in Q(k) have equal spectra, and every reductive operator quasisimilar to a normal operator is normal.

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

Expanding maps of the interval with unique turning points have periodic points of period 2n · 3 for some n and therefore are chaotic.

The asymptotic behavior of nonoscillatory solutions of nth order nonlinear functional differential equations

is investigated. Sufficient conditions are provided which ensure that all nonoscillatory solutions approach zero as t → ∞.

We apply Keller's method to the study of differential calculus in Frechet spaces and establish an inverse mapping theorem. A special case of this theorem is similar to a theorem of Yamamuro.

The mechanical procedure of paper folding generates an uncountable family of infinite sequences of fold patterns. We obtain the associated Fourier series and show that the sequences are almost periodic and hence deterministic. Further, we show that paper folding numbers defined by the sequences are all transcendental.

We prove that a regular double Stone algebra is protective in the category of Stone algebras if and only if its centre is a projective Boolean algebra and its dense set is countably generated as a filter. It follows that every countable regular double Stone algebra is projective as a Stone algebra.

A class of rings in which each member is the extension of a nilpotent torsion ring by a semisimple semiartinian ring is presented.