Singular structure of Toda lattices and cohomology of certain compact Lie groups

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $\tau$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $\tau$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics

USE OF DERIVATIVE SPECTROPHOTOMETRY IN THE ANALYSIS OF SEMI-SOLID PHARMACEUTICAL FORMS WITH MONARDA ESSENTIAL OIL

Universitatea de Stat de Medicină şi Farmacie „Nicolae Testemiţanu”, Chişinău, Republica MoldovaIntroducere. Aplicarea metodelor cromatografice pentru analiza produselor semisolide este limitată de prezența substanțelor lipofile și/sau polimere, care pot modifica ireversibil coloanele analitice, iar metodele selective de preparare a probelor majorează timpul și costul analizei. Scopul lucrării. Elaborarea unei metode simple de analiză pentru formele farmaceutice semisolide cu ulei volatil de monardă, bazate pe proprietățile spectrale ale substanțelor active. Material și Metode. Spectrofotometru UV-VIS „Lambda-25”” (Perkin Elmer), ulei volatil de monardă, substanțe de referință timol și carvacrol, substanțe auxiliare, etanol. S-au înregistrat spectrele UV ale substanțelor individuale, uleiului volatil, formelor farmaceutice (gel, unguent și pastă) și amestecurilor placebo. Rezultate. Spectrele UV ale substanțelor individuale se suprapun aditiv, iar utilizarea derivatei a 2 majorează considerabil selectivitatea determinării compușilor activi. Spectrul derivat de ordinul 2 al tuturor obiectelor testate prezintă 3 maxime la lungimile de undă 270±2, 278±2 și 287±2 nm și 2 minime la 273±2 și 281±2 nm, care și au fost incluse în testul de identificare. Pentru dozare s-a propus varianta simplificată a spectrofotometriei derivate după 3 puncte, utilizând în calitate de răspuns analitic expresiile: 2·A (281)-A (291)-A (245) pentru gel și 2·A (282)-A (292)-A (255) pentru unguent și pastă. Concluzii. Aplicând spectrofotometria derivată de ordinul 2 s-au elaborat tehnici de lucru pentru identificarea și dozarea principiilor active uleiului volatil de monardă în formele farmaceutice semisolide, ce pot fi incluse în documentația analitico-normativă.Background. The application of chromatographic methods for the analysis of semi-solid products is limited by the presence of lipophilic and/or polymeric substances, which can irreversibly alter the analytical columns, while selective sample preparation methods increase the time and cost of the analysis. Objective of the study. Development of a simple analytical method for semi-solid pharmaceutical forms with Monarda essential oil, based on the spectral properties of active substances. Material and Methods. “Lambda-25” UV-VIS spectrophotometer (Perkin Elmer), Monarda essential oil, thymol and carvacrol reference substances, excipients, ethanol. UV spectra of individual substances, essential oil, pharmaceutical forms (gel, ointment, and paste), and placebo mixtures were recorded. Results. The UV spectra of the individual substances overlap additively, and the use of the 2nd derivative considerably increases the selectivity of the determination of the active compounds. The 2nd order derivative spectrum of all tested objects has 3 maxima at wavelengths 270±2, 278±2, and 287±2 nm and 2 minima at 273±2 and 281±2 nm, which were included in the identification test. For assay, a simplified 3-point variant of derivative spectrophotometry was proposed, using as an analytical response the expressions: 2·A (281)-A (291)-A (245) for gel and 2·A (282)-A (292)-A (255) for ointment and paste. Conclusions. Applying 2nd order derivative spectrophotometry, working techniques were developed to identify and quantify the active principles of Monarda essential oil in semi-solid pharmaceutical forms, which can be included in the analytical-normative documentation