Eisenstein Series and String Thresholds

We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincar\'e upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively.Comment: Latex2e, 60 pages; v2: Appendix A.4 extended, 2 refs added, thms renumbered, plus minor corrections; v3: relation (1.7) to math Eis series clarified, eq (3.3) and minor typos corrected, final version to appear in Comm. Math. Phys; v4: misprints and Eq C.13,C.24 corrected, see note adde

Biofilm growth in a biological plate tower, BPT, for VOC air pollution treatment

The growth of microorganisms in biofilters and biotrickling filters always leads to the occurrence of clogging. A good efficiency of removal makes it happen faster. To solve clogging and making that kind of process easy to operate steadily for a long time, a new concept of reactor was designed and tested with air polluted with toluene and Pseudomonas putida. The BPT is a cascade of plates to whose surface the bacteria attaches. Water flows down and air flows upwards. The reactor shows very good hydrodynamic performance and operates continuously without problem. When the thickness of the biofilm reaches the maximum value allowable, the set of plates is simply replaced by a clean one and the biomass is dealt with outside the reactor

Biodegradation of toluene in a trickling filter

A trickling filter packed with PVC 16 mm Raschig rings was used to study the degradation of toluene in a polluted air stream, by means of a bacterial biofilm of Pseudomonas putida ATCC 17484. A polluted stream was simulated by blending air with a controlled amount of toluene. The mixing was accomplished in a special mixing chamber designed for that purpose. Induction of the enzymes of the toluene degradative pathway and adaptation of the inoculum were done in batch cultures with minimum mineral media and phenol. The continuous experiments were monitored by mass spectrometry for the quantification of the various gases and of toluene removal. A 94% toluene removal was achieved with contacting times above one minute and toluene concentrations up to 400 ppm

Comment on the Adiabatic Condition

The experimental observation of effects due to Berry's phase in quantum systems is certainly one of the most impressive demonstrations of the correctness of the superposition principle in quantum mechanics. Since Berry's original paper in 1984, the spin 1/2 coupled with rotating external magnetic field has been one of the most studied models where those phases appear. We also consider a special case of this soluble model. A detailed analysis of the coupled differential equations and comparison with exact results teach us why the usual procedure (of neglecting nondiagonal terms) is mathematically sound.Comment: 9 page