Kernels of L-functions of cusp forms

We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin’s Periods Theorem. Cohen’s representation of these kernels as series is also generalized.     

Values of <Emphasis Type="Italic">L</Emphasis>-functions at integers outside the critical strip

We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form. YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund. N. Diamantis is partially supported by EPSRC grant EP/D032350/1.     

Comparative Interaction Studies of Quercetin with 2-Hydroxyl-propyl-β-cyclodextrin and 2,6-Methylated-β-cyclodextrin

Quercetin (QUE) is a well-known natural product that can exert beneficial properties on human health. However, due to its low solubility its bioavailability is limited. In the present study, we examine whether its formulation with two cyclodextrins (CDs) may enhance its pharmacological profile. Comparative interaction studies of quercetin with 2-hydroxyl-propyl-β-cyclodextrin (2HP-β-CD) and 2,6-methylated cyclodextrin (2,6Me-β-CD) were performed using NMR spectroscopy, DFT calculations, and in silico molecular dynamics (MD) simulations. Using T1 relaxation experiments and 2D DOSY it was illustrated that both cyclodextrin vehicles can host quercetin. Quantum mechanical calculations showed the formation of hydrogen bonds between QUE with 2HP-β-CD and 2,6Μe-β-CD. Six hydrogen bonds are formed ranging between 2 to 2.8 Å with 2HP-β-CD and four hydrogen bonds within 2.8 Å with 2,6Μe-β-CD. Calculations of absolute binding free energies show that quercetin binds favorably to both 2,6Me-β-CD and 2HP-β-CD. MM/GBSA results show equally favorable binding of quercetin in the two CDs. Fluorescence spectroscopy shows moderate binding of quercetin in 2HP-β-CD (520 M⁻¹) and 2,6Me-β-CD (770 M⁻¹). Thus, we propose that both formulations (2HP-β-CD:quercetin, 2,6Me-β-CD:quercetin) could be further explored and exploited as small molecule carriers in biological studies.