BASIC programs for the quantitative analysis of Gaussian curves.

A satisfiability-unsatisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.

Mathematical Computing↗

PubMed2006

Surface-enhanced Raman spectroscopy and density functional theory study on 4,4'-bipyridine molecule.

The molecular geometry and vibrational frequencies of 4,4'-bipyridine (BPE) in the ground state were calculated using density functional theory (DFT) methods (B3LYP) with 6-31++G(d,p) basis set. The optimized geometric bond lengths and bond angles are obtained by DFT employing the hybrid of Beckes non-local three parameter exchange and correlation functional and Lee-Yang-Parr correlation functional. Fourier transform infrared (FT-IR), Fourier transform Raman (FT-Raman) and near-infrared surface-enhanced Raman scattering (NIR-SERS) spectra of BPE on the silver foil substrate have been recorded. All FT-IR, FT-Raman and NIR-SERS band were assigned on the basis of the B3LYP/6-31++G(d,p) method. The vibrational frequencies obtained by DFT(3LYP) are in good agreement with observed results. The NIR-SERS of BPE excited by 1064nm laser line is little difference with that excited by visible laser line. This phenomenon is result to the increase of the contribution of CHEM enhancement effect. Surface selection rules derived from the electromagnetic enhancement model were employed to infer the orientations of BPE on the silver foil substrate surface. Some vibrational frequency which are sensitive to the planar or non-planar structure of BPE, and to the dihedral angle were concluded.

Mathematical Computing↗