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.     

Tailor-made derivatizing agents for carbonyl compounds using liquid chromatography

A new approach for the synthesis and application of tailor-made derivatizing agents for liquid chromatography is presented. The s-triazine ring serves as the backbone for these substances. Synthesis is performed in three steps based on substitution reactions of cyanuric chloride. A chromophor or fluorophor, a reactive group and a third moiety intended to modify the polarity of the derivatizing agent are coupled to cyanuric chloride. The three groups may be selected freely from molecules which carry functional groups which might be linked to cyanuric chloride. As an example, the synthesis and application of a respective reagent for the determination of aldehydes and ketones using HPLC with fluorescence detection is carried out. Baseline separation of the derivatives and low limits of detection (1.5×10⁻⁹ mol/l for formaldehyde, 2.0×10⁻⁹ mol/l for acetaldehyde and 3.0×10⁻⁹ mol/l for p-tolualdehyde) are obtained with the new versatile reagent.     

Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.

     

Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.     

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. Received: 8 January 1998 / Revised: 9 March 1998 / Accepted: 15 March 1998     

Basis properties of eigenfunctions of the -Laplacian

For , the eigenfunctions of the non-linear eigenvalue problem for the -Laplacian on the interval are shown to form a Riesz basis of and a Schauder basis of whenever .

     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada     

MMNTP-a new tailor-made modular derivatization agent for the selective determination of isocyanates and diisocyanates

The synthesis of a new tailor-made derivatization agent for the selective determination of (di)isocyanates is presented. Starting from cyanuric chloride, the reagent 4-methoxy-6-(4-methoxy-1-naphthyl)-1,3,5-triazine-2-(1-piperazine)(MMNTP) is synthesized by subsequent substitution of the three chlorine atoms. This new derivatization agent and the five urea derivatives of phenylisocyanate (PI), hexamethylene-diisocyanate (HDI), toluene-2,4-diisocyanate (2,4-TDI), toluene-2,6-diisocyanate (2,6-TDI) and methylenebisphenyl-4,4-diisocyanate (MDI) show good spectroscopic properties with small compound-to-compound variabilities (RSD([epsilon])= 5.3 %, RSD(relative fluorescence)= 9.4 %). Therefore, using UV detection, a single calibration is needed for the quantification of all diisocyanates and isocyanates respectively. For separation and analysis a HPLC method with a RP column and a binary gradient is presented. All derivatives are separated and show low limits of detection. In addition to the good spectroscopic properties and low limits of detection, good reactivity for the derivatizations at room temperature is observed. The aromatic diisocyanates can be measured immediately whereas aliphatic diisocyanates need 2 h incubation. These advantages make MMNTP a powerful and versatile derivatization agent for (di)isocyanates which is demonstrated by a real sample with solid phase sampling, where the reagent is coated on a sorbent.     

Syn/anti isomerization of 2,4-dinitrophenylhydrazones in the determination of airborne unsymmetrical aldehydes and ketones using 2,4-dinitrophenylhydrazine derivation

Aldehydes and ketones readily react with 2,4-dinitrophenylhydrazine (2,4-DNPH) to form the corresponding hydrazones. This reaction has been frequently used for the quantification of airborne carbonyl compounds. Since unsymmetrical aldehydes and ketones are known to form isomeric 2,4-dinitrophenylhydrazones (syn/ anti-isomers), the influence of isomerization on the practicability and accuracy of the 2,4-DNPH-method using 2,4-dinitrophenylhydrazine-coated solid sorbent samplers has been studied with three ketones (methyl ethyl ketone (MEK), methyl isopropyl ketone (MIPK), and methyl isobutyl ketone (MIBK)). With all three ketones the reaction with 2,4-DNPH resulted in mixtures of the isomeric hydrazones which were separated by HPLC and GC and identified by mass spectroscopy and ¹H nuclear magnetic resonance spectroscopy. The isomers show similar chromatographic behaviour in HPLC as well as in GC, thus leading to problems in quantification and interpretation of chromatographic results.     

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer.We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids.In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.