The Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function Γ_n, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ_n function and their applications to summation of series and infinite products.This work was supported by NFS grant CCR-0204003.2000 Mathematics Subject Classification: Primary—33E20, 33F99, 11M35, 11B73     

Intermolecular Processes in Excited Electronic States in the Gas Phase (Review)

A brief review of works on the temperature dependences of the rate constants k_q of the intermolecular processes proceeding in the excited electronic states in the gas phase is given. The dependences k_q(T) for such biomolecular processes as intermolecular vibrational energy transfer in the triplet state vibrational quasi-continuum, triplet-triplet electron excitation energy transfer, and intermolecular photoinduced electron transfer have been compared. The experimental data have shown that in the gas phase for all analyzed intermolecular processes both an increase and a decrease in k_q with increasing temperature (T) is observed, which is not associated with the specific intermolecular interactions leading to the formation of long-lived components. The change in the type of temperature dependence is due to the change in the mechanisms of the radiationless transitions with increasing density of vibrational levels in the final electronic state. The applicability of the known models based on the theory of radiationless transitions for predicting the temperature dependences k_q(T) is discussed. __________ Translated from Zhurnal Prikladnoi Spektros-kopii, Vol. 72, No. 4, pp. 429–439, July–August, 2005.     

Theoretical Study on Grating Formation in Photovoltaic Photorefractive LiNbO<Subscript>3</Subscript> Crystal with Short Light Pulses

Based on Valley’s quasi-cw approximation, the grating formation in materials exhibiting bulk photovoltaic effect on illumination with short light pulses (such as ns pulses) is investigated. We give the expression of the space-charge field, and find that it has two time constants. When considering and not considering the photovoltaic effect, the diagrams of two time constants as a function of irradiance in units of are the same. The energy required to erase a grating to 1/e of its initial value is also studied in the two cases. The two results are also the same.     

Fluorescence of Metal–Ligand Complexes of Mono- and Di-Substituted Naphthalene Derivatives

In this work, metal ion complexes for several naphthalene derivatives have been investigated. Different working pH values were chosen: 2.5 for complexes with Zr(IV), 4.0 for complexes with Fe(III), 5.0 for complexes with Al(III), and 7.5 for complexes with Cu(II). A stoichiometry of 1:1 for all complexes except two has been established by use of the Benesi–Hildebrand method and the stability constants have been calculated. All complexes between naphthalene derivatives and Cu(II) and Fe(III) show fluorescence quenching. In the case of Al(III), all complexes provided enhanced fluorescence. For Zr(IV), only the complex with 3-hydroxy-2-naphthoic acid provided enhanced fluorescence. The value of the stability constants as a function of the substituents of naphthalene derivatives has been analyzed. One can conclude that Cu(II) showed the largest binding affinity for the mono-substituted derivatives. However, Al(III) and Zr(IV) produced greater selectivity for the di-substituted derivatives. Iron(III) showed no specific binding with any of the naphthalene derivatives.     

An advanced EPR stopped-flow apparatus based on a dielectric ring resonator

A novel EPR stopped-flow accessory is described which allows time-dependent cw-EPR measurements of rate constants of reactions involving paramagnetic species after rapid mixing of two liquid reagents. The EPR stopped-flow design represents a state-of-the-art, computer controlled fluid driving system, a miniresonant EPR structure with an integrated small ball mixer, and a stopping valve. The X-band EPR detection system is an improved version of that reported by Sienkiewicz et al. [Rev. Sci. Instr. 65 (1994) 68], and utilizes a resonator with two stacked ceramic dielectric rings separated by a variable spacer. The resonator with the mode TE(H)011 is tailored particularly for conditions of fast flowing and rapidly stopped aqueous solutions, and for a high time resolution. The short distance between the ball mixer and the small EPR active volume (1.8 microl) yields a measured dead time of 330 micros. A compact assembly of all parts results in minimization of disturbing microphonics. The computer controlled driving system from BioLogic with two independent stepping motors was optimized for EPR stopped-flow with a hard-stop valve. Performance tests on the EPR spectrometer ESP 300E from BRUKER using redox reactions of nitroxide radicals revealed the EPR stopped-flow accessory as an advanced, versatile, and reliable instrument with high reproducibility.     

1H-detected double-J-modulated INEPT-INADEQUATE for simultaneous determination of one-bond and long-range carbon-carbon connectivities and the measurement of all carbon-carbon coupling constants

We describe a new NMR experiment, (1)H-detected double-J-modulated (DJM)-INEPT-INADEQUATE, for tracing out the carbon skeleton of molecules. This experiment allows simultaneous correlation of directly bonded carbon atoms and those separated by multiple bonds, while at the same time also providing the values of all J(CC) coupling constants. This is achieved by replacing both fixed carbon-carbon coupling evolution intervals of the INEPT-INADEQUATE experiment by variable time intervals, which are incremented in concert with the DQ evolution period (t(1)). We show that the analysis of the fine structure of cross-peaks in DJM-INEPT-INADEQUATE spectra leads to accurate values of coupling constants and give guidelines for the proper usage of the method. The proposed experiment is two times less sensitive that the original INEPT-INADEQUATE experiment. We show that, using a 600-MHz cryoprobe and 20 mg of a monosaccharide, spectra that are suitable for the analysis of coupling constants as small as 2 Hz can be obtained within 24 h. Instead of performing multiple experiments, a single DJM-INEPT-INADEQUATE experiment can thus provide a wealth of information for the structural analysis of small molecules.     

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λ_k}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η_j are valid. The constants η_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λ_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function. Mathematics Subject Classification (2000) 11M26.     

Flexoelectric effect modelling

A statistical theory of dipole flexoelectric (FE) polarization in liquid crystals is used to calculate temperature dependence of order parameters, elastic constants and FE coefficients. Two systems with polar wedge-shaped and banana-shaped molecules are investigated. In both cases, the FE coefficients are proportional to the dipole moment component parallel to the molecule symmetry axis. It results from the symmetries of interactions and of the Mayer function. The origin of the FE effect and microscopic pictures of the distorted phases are discussed.     

A Comparison of the Host–Guest Inclusion Complexes of 1,8-ANS and 2,6-ANS in Parent and Modified Cyclodextrins

The fluorescent molecules1-anilinonaphthalene-8-sulfonate (1,8-ANS) and2-anilinonaphthalene-6-sulfonate (2,6-ANS) areextremely sensitive to the polarity of their localenvironment, making them excellent probes for thestudy of heterogeneous systems, including cyclodextrin(CD) solutions. Both are only weakly fluorescent in ahighly polar medium, such as water, but are extremelyfluorescent in a relatively nonpolar medium, such aswithin a CD cavity. These two probes are isomers, withmajor structural differences: 1,8-ANS is much bulkierand more spherical, whereas 2,6-ANS is much morestreamlined and rod-shaped. Thus, they show majordifferences in their formation of CD inclusioncomplexes. This is reflected both in the magnitude ofthe observed fluorescence enhancement upon CDinclusion, as well as in the value of the associationconstant for complex formation. The creation of ascale for each probe for their fluorescence in CDsrelative to that in ethanol allows for directcomparisons to be made between the two probes. Theseresults are obtained and compared for the host-guestinclusion complexes of 1,8-ANS and 2,6-ANS with sixCDs: , , , and theirhydroxypropylated analogs.     

Basicity Constants of Isomeric Azafluorenes and Their Derivatives

Data are given for the first time on the measurement of basicity constants of the isomeric azafluorenes and their derivatives, which enable a study of the effect of this property on the relationship of the geometric isomers of substituted fluorenes and azomethines obtained from azafluorenes and their keto derivatives.