Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

On the direct calculation of the free energy of quantization for molecular systems in the condensed phase

Using the path integral formalism or the Feynman-Hibbs approach, various expressions for the free energy of quantization for a molecular system in the condensed phase can be derived. These lead to alternative methods to directly compute quantization free energies from molecular dynamics computer simulations, which were investigated with an eye to their practical use. For a test system of liquid neon, two methods are shown to be most efficient for a direct evaluation of the excess free energy of quantization. One of them makes use of path integral simulations in combination with a single-step free energy perturbation approach and was previously reported in the literature. The other method employs a Feynman-Hibbs effective Hamiltonian together with the thermodynamic integration formalism. However, both methods are found to give less accurate results for the excess free energy of quantization than the estimate obtained from explicit path integral calculations on the excess free energy of the neon liquid in the classical and quantum mechanical limit. Suggestions are made to make both methods more accurate.     

Plumbagin: A New Route to the Electroanalytical Determination of Cystine

The interaction between 5‐hydroxy‐1,4‐naphthoquinone and cysteine was found to give a clearly resolved voltammetric signature that could be used as an electrochemical assay for the thiol. A structure‐function study assessed the nature of the resulting voltammetric profile and the highly selective response to cysteine. The translation of the system as the basis of an assay for the determination of cystine was investigated using mercaptopropanol as the in situ reducing agent. The lack of response to the latter and high recovery performance of the label opens up a new direction for the one pot, mercury free determination of cystine.     

A new xanthone from the bark extract of Rheedia acuminata and antiplasmodial activity of its major compounds

Bioassay-guided fractionation of the ethyl acetate bark extract of Rheedia acuminata led to the isolation of the new compound 1,5,6-trihydroxy-3-methoxy-7-geranyl-xanthone, together with four known compounds. These compounds were tested in vitro for their antiplasmodial activity on a chloroquine-resistant strain of Plasmodium falciparum (FcB1) and for their cytotoxicity against the human diploid embryonic lung cell line MRC-5.     

Lost in Translation: Examining Translation Errors Associated With Mathematical Representations

Translation errors and conceptual misunderstandings made by students translating among graphical, tabular, and symbolic representations of linear functions were examined. The study situated student errors in the context of the “Translation‐Verification Model” developed specifically for the purpose of explaining student behavior during the process of translating relationships from one mathematical representation to another. Three distinct error types were identified to explain student performance. An examination of the error types revealed that specific translation errors tend to occur at different stages of the translation process. Translation errors are also related to “attribute density,” the amount of information inherently encoded in a given representation. The findings of the study have implications for teaching linear relationships—student weaknesses and strengths are identified.