Master Constraint Operator in Loop Quantum Gravity

We introduce a master constraint operator $\hat{\mathbf{M}}$ densely defined in the diffeomorphism invariant Hilbert space in loop quantum gravity, which corresponds classically to the master constraint in the programme. It is shown that $\hat{\mathbf{M}}$ is positive and symmetric, and hence has its Friedrichs self-adjoint extension. The same conclusion is tenable for an alternative master operator $\hat{\mathbf{M'}}$, whose quadratic form coincides with the one proposed by Thiemann. So the master constraint programme for loop quantum gravity can be carried out in principle by employing either of the two operators.Comment: 11 pages, significant modification in section 2, accepted for publication in Phys. Lett.

Theoretical Study on Highly Active Bifunctional Metalloporphyrin Catalysts for the Coupling Reaction of Epoxides with Carbon Dioxide

Highly active bifunctional metalloporphyrin catalysts were developed for the coupling reaction of epoxides with CO2 to produce cyclic carbonates. The bifunctional catalysts have both quaternary ammonium halide groups and a metal center. To elucidate the roles of these catalytic groups, DFT calculations were performed. Control reactions using tetrabutylammonium halide as a catalyst were also investigated for comparison. In the present article, the results of our computational studies are overviewed. The computational results are consistent with the experimental data and are useful for elucidating the structure-activity relationship. The key features responsible for the high catalytic activity of the bifunctional catalysts are as follows: 1) the cooperative action of the halide anion (nucleophile) and the metal center (Lewis acid); 2) the near-attack conformation, leading to the efficient opening of the epoxide ring in the rate-determining step; and 3) the conformational change of the quaternary ammonium cation to stabilize various anionic species generated during catalysis, in addition to the robustness (thermostability) of the catalysts

Relativistic Harmonic Oscillator

We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in compact form, i. e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.Comment: 14 pages, extended discussion of result

Decidability for Sturmian Words

We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly ?-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words

Extracting the hierarchical organization of complex systems

Extracting understanding from the growing ``sea'' of biological and socio-economic data is one of the most pressing scientific challenges facing us. Here, we introduce and validate an unsupervised method that is able to accurately extract the hierarchical organization of complex biological, social, and technological networks. We define an ensemble of hierarchically nested random graphs, which we use to validate the method. We then apply our method to real-world networks, including the air-transportation network, an electronic circuit, an email exchange network, and metabolic networks. We find that our method enables us to obtain an accurate multi-scale descriptions of a complex system.Comment: Figures in screen resolution. Version with full resolution figures available at http://amaral.chem-eng.northwestern.edu/Publications/Papers/sales-pardo-2007.pd