A Prüfer angle approach to semidefinite Sturm–Liouville problems with coupling boundary conditions

It is shown how to reduce the periodic/antiperiodic Sturm–Liouville problems to analysis of the Prüfer angle. This provides an alternative to the more usual approaches via operator theory or the Hill discriminant in the definite case, and leads to new results in the semidefinite case. An extension to more general coupling boundary conditions is also given.     

Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter

Three inverse problems for a Sturm-Liouville boundary value problem −y″+qy=λy, y(0)cosα=y′(0)sinα and y′(1)=f(λ)y(1) are considered for rational f. It is shown that the Weyl m-function uniquely determines α, f, and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.     

A hierarchy of Sturm–Liouville problems

Sturm–Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classesof these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to thoseof the ‘usual’ Sturm–Liouville problem which has constant boundary conditions.In fact it will be shown how these classes can be converted into each other, andinto the ‘usual’ Sturm–Liouville problem, by means of transformations preserving all but finitely many eigenvalues. Copyright © 2003 John Wiley & Sons, Ltd.     

Erratum

The problem of extending a space-time through a boundary point motivates the study of space-times where the metric is not necessarily twice differentiable. In order to correct the differentiability classes used in the author's previous work, it is shown, first, that unique timelike geodesies exist provided only that the Riemann tensor and the first derivatives of the metric are bounded; and, secondly, that a space-time can be extended subject to Hölder continuity of the Riemann tensor. Also discussed are the possible reductions that might be made in the level of differentiability.     

The range of the p-Laplacian

We prove that for λ ≥ 0, p ≥ 3, there exists an open ball B L²(0,1) such that the problem

− (|u′|^p−2 u′)′ − λ|u|^p−2u = f, in (0,1)

, subject to certain separated boundary conditions on (0,1), has a solution for f B.     

Elastico-viscous squeeze films: Part 2. superimposed rotation

The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t_{$\text{1}\text{2}$} sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied.The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading.The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.     

Elastico-viscous squeeze films: Part 3. the torsional-balance rheometer

The observation made in Part 2 that squeezing flow with a superimposed rotation results in an equilibrium situation with the applied load just balancing the normal stresses generated in the test fluid is used to develop a new technique (the Torsional-Balance Rheometer) for measuring the viscometric functions of elastic liquids.The Rheometer utilizes conventional torsional flow and its novel feature is that the applied load is fixed and the associated shear rate at the rim determined, in contrast to the usual situation where the shear rate is fixed and the total normal force measured.It is argued that the Torsional Balance has significant advantages over other rheometers in the very high shear-rate range, since the normal stresses being measured themselves supply a mechanism for keeping the top plate (which is free to float on the test fluid) at a constant separation from the rotating bottom plate, hence allowing very small gaps to be considered. Consistent data are shown to be possible for shear rates in excess of 10⁵ s^?1.     

Bounded variation evolution equations

A “fundamental theory” is presented for the equation x(t) = ∫₀^tq(x(s), s) ds where the integral is Stieltjes and x is of bounded variation with values in Rⁿ. This includes the ordinary differential equation (o.d.e.) case with impulses. The principal conclusion is that the corresponding conditions for Carathéodory's o.d.e. problem carry over almost unchanged to the more general case. Areas treated include existence (local and global), uniqueness, dependence, integral funnels, stability, and Picard iterates.     

Synthesis of 8-Aza-2′-deoxyadenosine and related 7-Amino-3H-1,2,3-triazolo[4,5-d]pyrimidine 2′-Deoxyribofuranosides: Stereoselective glycosylation via the nucleobase anion

The synthesis of 8-aza-2′-deoxyadenosine ( = 7-amino-3H-1,2,3 triazolo[4,5-d]pyrimidine N³-(2′-deoxy-β-D-ribofuranoside); 1 ) as well as the N²- and N¹-(2′-deoxy-β-D-ribofuranosides) 2 and 3 is described. Glycosylation of the anion of 7-amino-3H-1,2,3-triazolo[4,5-d]pyrimidine ( 6 ) in DMF yielded three regioisomeric protected 2′-deoxy-β-D-ribofuranosides, i.e. the N³-, N²-, and N⁴-glycosylated isomers 7 (14%), 9 (11%), and 11 (3%), respectively, together with nearly equal amounts of their α-D-anomers 8 (13%), 10 (12%), and 12 (4%; Scheme 1). The reaction became Stereoselective for the β-D-nucleosides if the anion of 7-methoxy-3H-1,2,3-triazolo[4,5-d]pyrimidine ( 13 ) was glycosylated in MeCN: only the N³-, N², and N¹-(2′-deoxy-β-D-nucleosides) 14 (29%), 15 (32%), and 16 (23%), respectively, were formed (Scheme 2). NH₃ Treatment of the methoxynucleosides 14–16 afforded the aminonucleosides 1–3 . The anomeric configuration as well as the position of glycosylation were determined by combination of ¹³ C-NMR , ¹ H-NMR , and 1D-NOE difference spectroscopy. Compound 1 proved to be a substrate for adenosine deaminase, whereas the regioisomers 2 and 3 were not deaminated.     

Evaluation of a passive sampler for airborne formaldehyde

Summary A diffusive sampler for the large scale routine determination of airborne formaldehyde was developed. Formaldehyde is sampled in a badge-type passive sampler containing a 2,4-dinitrophenylhydrazine-coated filter paper as sampling layer. Formaldehyde is immediately converted to the corresponding hydrazone, which, after desorption with acetonitrile, is separated and quantified by gradient HPLC using UV detection at 345 nm. Calibration was done via an active sampling method and showed an excellent, time- and concentration-independent linear performance of the diffusive samplers. A detection limit of about 0.05 ml/m³·h (ppm·h) and a relative standard deviation of about 10% ensured a good analytical reliability. By testing the influence of air movements at the sampler surface, a minimum air velocity of 10 cm/s was found necessary to ensure representative sampling.