Using Fractionation and Diffusion Ordered Spectroscopy to Study Lignin Molecular Weight.

Recent reports demonstrate that applications of the biopolymer lignin can be helped by the use of a fraction of the lignin which has an optimal molecular weight range. Unfortunately, the current methods used to determine lignin's molecular weight are inconsistent or not widely accessible. Here, an approach that relies on 2D DOSY NMR analysis is described that provides a measure of lignin's molecular weight. Consistent results were obtained using this well-established NMR technique across a range of lignins.     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

A class of explicitly resolvable evolution equations

We consider a class of evolution equations with “scalar nonlinearities” and the associated steady equations. An explicit representation of solutions is obtained in terms of the solution of a scalar nonlinear functional differential equation. Convergence to an equilibrium solution is discussed.     

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem

Conclusions We have investigated solutions of equation (3) when ² is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and ² = _i ² we have exactly two nontrivial solutions which bifurcate to the right of _i ² ; these solutions are shown to exist in an interval ( _i ² , _i ² + ₀). The method of Section 3 may then be used to extend these two solutions to the right of _i ² + ₀ providing that ²= _i ² + ₀ is not an eigenvalue of the linear operator (13) evaluated at = ± ₁. Either a solution can be uniquely extended, or there exists a value of ²where the bifurcation method must be applied again³.While the method used here gives the exact number of solutions bifurcating from _i ² , other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels [4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval _i ² < _i ^+1/2 remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner [7] and Coffman [8].     

Lower bounds to eigenvalues of quantum mechanical systems

Procedures are presented which are expected to produce good numerical lower bounds to eigenvalues for Hamiltonians having continuous spectra. Further, it is observed how these lower bounds can be utilized in other estimation methods.     

Explicity resolvable equations with functional non-linearities

We consider the non-linear evolution equation μdu/dt+Bu+n(u)L(u)=h, where n(u) is a functional, and introduce assumptions which allow its explicit solution. Several concrete applications of our procedure are given.     

Bifurcation of special periodic solutions to delay-differential equations

Summary We consider a class of forced delay differential equations in which the delay is given by /2 and investigate the problem of finding its special periodic solutions. We first approximate these by a Rayleigh-Ritz-Galerkin sequence. Our second method introduces an averaged model thought to give a qualitative approximation to the solution behavior. The effects of cubic and quintic nonlinearities are compared.

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Zusammenfassung Wir untersuchen eine Klasse von forcierten Differentialgleichungen mit Zeitverzögerung /2 und suchen nach speziellen periodischen Lösungen. Zunächst approximieren wir Lösungen durch eine Folge von Rayleigh-Ritz-Galerkin-Näherungen. Als zweite Methode benutzen wir ein Mittelungsverfahren zur qualitativen Approximation des Lösungsverhaltens. Die Effekte von Nichtlinearitäten dritter und fünfter Ordnung werden verglichen.

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Supported in part by the DAAD/CONICYT Professor Exchange Program.

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Supported in part by CONICYT (Grant 89-576) and University of Concepción (Grant 20-12-17).

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Supported in part by DICYT at the University of Santiago (Grant 16-14).