Some new applications of Weinstein-Aronszajn's determinants-Part 1

Summary The title's determinants (or matrices) emerged for the first time when its eponyms looked for the eigenvalues of finite dimensionally perturbated linear operators describing for instance oscillations of a continuum. The literature emphasized only slightly till this time the physical sense of such perturbations, considering them mainly as a device for a systematic improvement of eigenvalues of continuous systems. Realizing that a coupled continuous and discrete system (according to this paper's terminology: combined or hybrid) can be, with a suitable application of Dirac's distribution and its distributional derivative, a physical picture of a finite dimensionally perturbated linear operator, several main chapters of the oscillation theory of such systems can be developed. Such ones are e.g. forming of the frequency equation regarding nonpersistent and persistent eigenvalues, application of operator truncation, treatment of a vibrating beam with several supports as a perturbated problem, and some theorems of Rayleigh. Part 2 will show some further applications.

---

Einige neue Anwendungen der Weinstein-Aronszajnschen Determinanten — Teil 1

Übersicht Diese Determinanten (oder Matrizen) tauchten auf, als ihre Namensgeber Eingenwerte endlichdimensional gestörter linearer Operatoren suchten, die z. B. Schwingungen von Kontinua beschreiben. Die Literatur betonte bisher wenig den physikalischen Inhalt solcher Störungen und betrachtete diese in erster Linie als ein Mittel zur methodischen Eingrenzung der Eigenwerte kontinuierlicher Systeme. Mit der Erkenntnis, daß ein mit einem diskreten System gekoppeltes Kontinuum (mit der Terminologie dieser Arbeit gemischt oder hydrid genannt) ein physikalisches Bild der linearen, endlich-dimensional gestörten Operatoren sein kann (bei geeigneter Anwendung der Diracschen Distribution und ihrer distributionalen Ableitung), können mehrere Kapitel der Schwingungstheorie der hybriden Systeme ausgebaut werden. Solche sind z. B. die Aufstellung der Frequenzgleichung mit Rücksicht auf unveränderte und veränderte Eigenwerte, und die Anwendung der verkürzten Operatoren endlicher Näherung, die Behandlung des mehrfeldrigen schwingenden Balkens als ein Störungsproblem. Einige weitere Anwendungsmöglichkeiten werden im zweiten Teil gezeigt.

---

---

Dedicated to Professor W. H. Wittrick, pioneer of frequency counting formulas     

Microextraction of Reseda luteola-Dyed Wool and Qualitative Analysis of Its Flavones by UHPLC-UV,NMR and MS

Detailed knowledge on natural dyes is important for agronomy and quality control as well as the fastness, stability, and analysis of dyed textiles. Weld (Reseda luteola L.), which is a source of flavone-based yellow dye, is the focus of this study. One aim was to reduce the required amount of dyed textile to ≤50 μg for a successful chromatographic analysis. The second aim was to unambiguously confirm the identity of all weld flavones. By carrying out the extraction of 50 μg dyed wool with 25 μL of solvent and analysis by reversed-phase UHPLC at 345 nm, reproducible chromatographic fingerprints could be obtained with good signal to noise ratios. Ten baseline separated peaks with relative areas ≥1% were separated in 6 min. Through repeated polyamide column chromatography and prepHPLC, the compounds corresponding with the fingerprint peaks were purified from dried weld. Each was unequivocally identified, including the position and configuration of attached sugars, by means of 1D and 2D NMR and high-resolution MS. Apigenin-4′-O-glucoside and luteolin-4′-O-glucoside were additionally identified as two trace flavones co-eluting with other flavone glucosides, the former for the first time in weld. The microextraction might be extended to other used dye plants, thus reducing the required amount of precious historical textiles.     

New Members of the RuSr 2 RECu 2 O 8 Family of Phases (RE=Rare Earth) Obtained at High Pressure and Temperature

At room pressure, Sm, Eu and Gd seem to be the only RE elements that accept to enter into the structure of RuSr 2 GdCu 2 O 8 . However high pressure and high temperatures allow one to replace gadolinium by other lanthanide cations. We have been successful in replacing Gd by Y, La, Pr, Nd, Tb, Dy, Ho and Er. X-ray diffraction patterns fitted by the Rietveld method show that the average crystal structure of RuSr 2 RECu 2 O 8 is tetragonal with P 4/ mmm space group symmetry. Micro-Raman experiments have been performed on RuSr 2 GdCu 2 O 8 samples synthesized at both high and room pressure; the resulting spectra are consistent with the same structure, since there are no major changes in the low-frequency range.     

Cauchy structures in closure and proximity spaces

Research supported by Hungarian National Foundation for Scientific Research, grant no. 2114.