On infinite-dimensional differential equations

Differential equations ·x(t) = f(x(t), t) are exhibited in a general infinite-dimensional Banach space, failing each of the following in turn. (i) The set S_t of solution values x(t) from a given point x(0) is compact. (ii) S_t is connected. (iii) Any point on the boundary ?S_t of S_t can be reached by a solution x with x(s) ??S_s, 0 ? s ? t.     

Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem $\text{T}{}_{\mathrm{m}}\text{x}{}_{\mathrm{m}}\text{=}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}\text{x}{}_{\mathrm{m}}$, 0 ≠ x_m?H_m, m = 1 … k. (1) Here T_m and V_mn are self-adjoint operators on Hilbert spaces H_m. The index is based on the number of negative eigenvalues of T_m ? ∑_{n = 1}^kλ_nV_mn and on the sign of the determinant δ₀ with (m, n)th entry (x_m, V_mnx_m). We assume that certain cofactors of δ₀ are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ₀ is positive.     

Analytical reliability of carbonyl compound determination using 1,5-dansylhydrazine-derivatization

Solid sorbents coated with the fluorescent reagent 5-dimethylaminonaphthalene-1-sulfohydrazide (dansylhydrazine, DNSH) have been used for derivatization and quantitative determination of airborne carbonyl compounds, for example in investigations on atmospheric pollution. The evaluation of this derivatization reaction presented here revealed that, for several reasons, it may not be recommended when sampling is performed with impingers containing liquid reagent solutions. Derivative yields came out to be strongly influenced by water and phosphoric acid which are essential for sufficient derivatization rates, but also responsible for the degradation of derivatives. Even at water and acid concentrations considered as an optimal compromise between accelerating and degrading effects, the analytical reliability of the method can only be guaranteed under controlled laboratory conditions. The reduced or lacking reactivity of DNSH towards aromatic aldehydes or aliphatic and aromatic ketones is an additional finding discarding the DNSH method for routine air monitoring at least when impingers are used for sampling.     

A canonical form for self-adjoint pencils in Hilbert space

We characterize those pencils P=A–B of operators on a separable Hilbert space H for which a linear homeomorphism D of H exists satisfying the following: (i) H decomposes into a direct sum F+G, orthogonal in a (perhaps indefinite) inner product induced by A, where F is finite dimensional (ii) D^*PD=P_F(I_G–C) where P_F is a (congruence) canonical form for the general self-adjoint pencil on F, and C is a bounded self-adjoint operator on G. For a given P, explicit constructions are given for C, D, F, G and P_F.     

On Carathéodory's conditions for the initial value problem

We prove a local existence theorem of Carathéodory-Goodman type for where instead of being continuous in we require only that it have no ``downward discontinuities.'

     

On the interpretation of data from Converging Flow Rheometers

A reappraisal of data obtained from a Converging Flow Rheometer (CFR) is presented, together with new results for a specific polymer solution.Particular emphasis is placed on the interpretation of the experimental pressure data in terms of a planar extensional viscosity. It is suggested that previous interpretations, while yielding viscosity levels that appear reasonable, nevertheless fail to give the qualitative behaviour that might be expected on the basis of predictions from well accepted constitutive models. This, in the authors' opinions, arises because certain fluids, i.e. those that are highly tension-thickening, cease to flow in accordance with the assumed kinematics at high flow rates.By adapting a recently proposed approximate analysis for flow through a contraction it is shown that better qualitative behaviour, for the planar extensional viscosity, can be obtained from the Converging Flow Rheometer.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.