Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into , so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v∈S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of three-polytopes into . We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds. The first author was partially supported by the National Science Foundation, and he acknowledges the Mathematisches Forschungsinstitut Oberwolfach for a stimulating research environment during the week-long program on Topological and Geometric Combinatorics (April, 2003). Most of this work was completed while the first author was at the Massachusetts Institute of Technology (Cambridge, MA) and the Mathematical Sciences Research Institute (Berkeley, CA). The second author was partially supported by the National Security Agency and the National Science Foundation. He thanks the organizers of the “Second Geometry Meeting dedicated to A.D. Aleksandrov” held at the Euler International Mathematical Institute in St. Petersburg (June, 2002), where these results were originally presented.     

Charge-transfer-to-solvent-driven dissolution dynamics of I- (H2O)2-5 upon excitation: excited-state ab initio molecular dynamics simulations

In contrast to the extensive theoretical investigation of the solvation phenomena, the dissolution phenomena have hardly been investigated theoretically. Upon the excitation of hydrated halides, which are important substances in atmospheric chemistry, an excess electron transfers from the anionic precursor (halide anion) to the solvent and is stabilized by the water cluster. This results in the dissociation of hydrated halides into halide radicals and electron-water clusters. Here we demonstrate the charge-transfer-to-solvent (CTTS)-driven femtosecond-scale dissolution dynamics for I-(H2O)n=2-5 clusters using excited state (ES) ab initio molecular dynamics (AIMD) simulations employing the complete-active-space self-consistent-field (CASSCF) method. This study shows that after the iodine radical is released from I-(H2O)n=2-5, a simple population decay is observed for small clusters (2 相似文献   

Distribution of luminescence excitation energy between Eu<Superscript>3+</Superscript> and Tb<Superscript>3+</Superscript> ions fixed in a perfluorosulfonic membrane

Ion-exchange modification of MF-4SK perfluorosulfonic membranes with Eu³⁺ and Tb³⁺ cations was realized. The state of cations in the membrane was studied by X-ray photoelectron spectroscopy. Special features of the luminescence of the resulting systems point to preferential excitation energy transfer from europium to terbium.     

Modeling positrons in molecular electronic structure calculations with the nuclear-electronic orbital method

The nuclear-electronic orbital (NEO) method was modified and extended to positron systems for studying mixed positronic-electronic wavefunctions, replacing the mass of the proton with the mass of the positron. Within the modified NEO framework, the NEO-HF (Hartree-Fock) method provides the energy corresponding to the single-configuration mixed positronic-electronic wavefunction, minimized with respect to the molecular orbitals expressed as linear combinations of Gaussian basis functions. The electron-electron and electron-positron correlation can be treated in the NEO framework with second-order perturbation theory (NEO-MP2) or multiconfigurational methods such as the full configuration interaction (NEO-FCI) and complete active space self-consistent-field (NEO-CASSCF) methods. In addition to implementing these methods for positronic systems, strategies for calculating electron-positron annihilation rates using NEO-HF, NEO-MP2, and NEO-FCI wavefunctions were also developed. To apply the NEO method to the positronium hydride (PsH) system, positronic and electronic basis sets were optimized at the NEO-FCI level and used to compute NEO-MP2 and NEO-FCI energies and annihilation rates. The effects of basis set size on NEO-MP2 and NEO-FCI correlation energies and annihilation rates were compared. Even-tempered electronic and positronic basis sets were also optimized for the e+LiH molecule at the NEO-MP2 level and used to compute the equilibrium bond length and vibrational energy.     

Environmental effects on the geometric phase

The behavior of the geometric phase gained by a single spin-1/2 nucleus immersed into a thermal or a squeezed environment is investigated. Both the time dependence of the phase and its value at infinity are examined against several physical parameters. It is observed that for some intermediate ranges of the temperature and the coupling strength, the presence of squeezing enhances the geometric phase.     

Radiometric detector options to aid in DOE high activity waste tank in situ characterization efforts

Radioactive waste cleanup and subsequent closure of waste storage tanks is currently underway at the Savannah river site, prompting the need to characterize the residual contents (heels) of the tanks. Occasionally, results from laboratory analyses indicate alternative sub-sampling strategies are needed, resulting in repetitive efforts to sample and analyze tank bottoms. The development of a system for in situ tank analyses using a radiometric probe, which could be lowered into a waste tank, could aid in identifying waste structures on tank bottoms requiring further sampling and characterization. Ideally, the probe would provide information for determining which structures were higher in concentrations of actinides and fission products characteristic of DOE high level waste (HLW) heels. Although only a limited set of isotopes can be measured directly without extensive radiochemical separations, the low-energy photon spectra of HLW do offer some intriguing possibilities for characterization using a radiometric probe. One possibility for obtaining a low-energy photon spectrum in the presence of high levels of interfering radiation would be to design a probe primarily based upon recently developed technology from Amptek Inc. Such a detector would be relatively insensitive to the high photon background, which would paralyze conventional gamma probes (i.e. sodium iodide) subjected to the same radiological conditions. The prototype detector is capable of successfully obtaining high resolution measurements at very high count rates (in excess of 500,000 counts per second). An overview of measurements obtained from various HLW samples using the prototype Amptek detector, as well as some additional detector technologies, which could enhance this prototype, will be discussed.     

Photochromic properties of perfluorosulfonic membranes modified by sodium molybdate

Transfer of a perfluorosulfonic membrane from the hydrogen form into the sodium form provides accessibility of pore space to sodium molybdate aqueous solution. Charge transfer spectra make it possible to judge the course of polymerization of capsulated molybdate ions and its intensification when the membranes are held in HCl vapor. The UV irradiation of the modified membranes causes a progressing absorption in the visible spectral region. The rate and depth of photoreduction increase as the concentration and degree of polymerization of molybdate ions increase. The scheme of the process was offered, which is based on taking into account a possibility of delocalization of 4d ¹ electrons of molybdenum(V) over the system of metal-metal bonds.     

A model of the multiple adsorption of hydrogen atoms on the surface of carbon nanotubes

A model of the multiple adsorption of atomic hydrogen on the surface of single-walled carbon nanotubes of the zigzag and arm-chair types was constructed. The adsorption model is based on the Anderson periodic model. An analytic equation for the band structure of carbon nanotubes with adsorbed hydrogen atoms was obtained, and the special features of this structure were studied. The dependence of the band structure of carbon nanotubes on the concentration of adsorbed hydrogen atoms was analyzed. The model constructed can be used to study adsorption of other univalent atoms on the surface of carbon particles.     

Results of prescribing Q-curvature on S n

Using a flow approach, generalizing previous results from Chen and Xu [12] and Ho [20], an existence result of prescribing Q-curvature on the even-dimensional standard sphere S ⁿ as a given positive function f was obtained under suitable assumptions on the Morse index of f. In this note, we prove a more general existence result by assuming only that f is positive somewhere.