Algebraic results for the values ϑ3(mτ)\vartheta_3(m\tau) and ϑ3(nτ)\vartheta_3(n\tau) of the Jacobi theta-constant

Let $\vartheta_3(\tau) = 1+2\sum_{\nu=1}^{\infty} e^{\pi i\nu^2\tau}$ denote the classical Jacobi theta-constant. We prove that the two values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ are algebraically independent over $\mathbb{Q}$ for any $\tau$ in the upper half-plane such that $q=e^{\pi i\tau}$ is an algebraic number, where $m,n\geq2$ are distinct integers

Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps

The problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d. We consider the case where the traps are initially randomly distributed in space, with uniform density rho, and derive upper and lower bounds for the probability Q(t) (averaged over all particle and trap trajectories) that the particle survives up to time t. We show that, for 1<=d<2, the bounds converge asymptotically to give $Q(t) \sim exp(-\lambda_d t^{d/2})$ where $\lambda_d = (2/\pi d) sin(\pi d/2) (4\pi D)^{d/2} \rho$ and D is the diffusion constant of the traps, and that $Q(t) \sim exp(- 4\pi\rho D t/ln t)$ for d=2. For d>2 bounds can still be derived, but they no longer converge for large t. For 1<=d<=2, these asymptotic form are independent of the diffusion constant of the particle. The results are compared with simulation results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev. E v65 050101 (2002)] which is described in detail. Deviations from the predicted asymptotic forms are found to be large even for very small values of Q(t), indicating slowly decaying corrections whose form is consistent with the bounds. We also present results in d=1 for the case where the trap densities on either side of the particle are different. For this case we can still obtain exact bounds but they no longer converge.Comment: 13 pages, RevTeX4, 6 figures. Figures and references updated; equations corrected; discussion clarifie

Reorganization Energy for Internal Electron Transfer in Multicopper Oxidases.

We have calculated the reorganization energy for the intramolecular electron transfer between the reduced type 1 copper site and the peroxy intermediate of the trinuclear cluster in the multicopper oxidase CueO. The calculations are performed at the combined quantum mechanics and molecular mechanics (QM/MM) level, based on molecular dynamics simulations with tailored potentials for the two copper sites. We obtain a reorganization energy of 91-133 kJ/mol, depending on the theoretical treatment. The two Cu sites contribute by 12 and 22 kJ/mol to this energy, whereas the solvent contribution is 34 kJ/mol. The rest comes from the protein, involving small contributions from many residues. We have also estimated the energy difference between the two electron-transfer states and show that the reduction of the peroxy intermediate is exergonic by 43-87 kJ/mol, depending on the theoretical method. Both the solvent and the protein contribute to this energy difference, especially charged residues close to the two Cu sites. We compare these estimates with energies obtained from QM/MM optimizations and QM calculations in a vacuum and discuss differences between the results obtained at various levels of theory

Photoinduced Excited State Electron Transfer at Liquid/Liquid Interfaces

Several aspects of the photoinduced electron transfer (ET) reaction betweencoumarin 314 (C314) and N,N-dimethylaniline (DMA) at the water/DMA interface areinvestigated by molecular dynamics simulations. New DMA and water/DMA potentialenergy surfaces are developed and used to characterize the neat water/DMA interface.The adsorption free energy, the rotational dynamics and the solvation dynamics of C314at the liquid/liquid interface are investigated and are generally in reasonable agreementwith available experimental data. The solvent free energy curves for the ET reactionbetween excited C314 and DMA molecules are calculated and compared with thosecalculated for a simple point charge model of the solute. It is found that thereorganization free energy is very small when the full molecular description of the soluteis taken into account. An estimate of the ET rate constant is in reasonable agreement withexperiment. Our calculations suggest that the polarity of the surface “reported” by thesolute, as reflected by solvation dynamics and the reorganization free energy, is strongly solute-dependent

Transcendence over meromorphic functions

DOI: 10.1017/S000497271700019