Partially ordered knapsack and applications to scheduling

In the partially ordered knapsack problem (POK) we are given a set N of items and a partial order ?_P on N. Each item has a size and an associated weight. The objective is to pack a set N^′⊆N of maximum weight in a knapsack of bounded size. N^′ should be precedence-closed, i.e., be a valid prefix of ?_P. POK is a natural generalization, for which very little is known, of the classical Knapsack problem. In this paper we present both positive and negative results. We give an FPTAS for the important case of a two-dimensional partial order, a class of partial orders which is a substantial generalization of the series-parallel class, and we identify the first non-trivial special case for which a polynomial-time algorithm exists. Our results have implications for approximation algorithms for scheduling precedence-constrained jobs on a single machine to minimize the sum of weighted completion times, a problem closely related to POK.     

Data parallel computing for network-structured optimization problems

Data level parallelism is a type of parallelism whereby operations are performed on many data elements concurrently, by many processors. These operations are (more or less) identical, and are executed in a synchronous, orderly fashion. This type of parallelism is used by massively parallel SIMD (i.e., Single Instruction, Multiple Data) architectures, like the Connection Machine CM-2, the AMT DAP and Masspar, and MIMD (i.e., Multiple Instruction, Multiple Data) architectures, like the Connection Machine CM-5. Data parallelism can also be described by a theoretical model of computation: the Vector-Random Access Machine (V-RAM).In this paper we discuss practical approaches to the data-parallel solution of large scale optimization problems with network—or embedded-network—structures. The following issues are addressed: (1) The concept of dataparallelism, (2) algorithmic principles that lead to data-parallel decomposition of optimization problems with network—or embedded-network—structures, (3) specific algorithms for several network problems, (4) data-structures needed for efficient implementations of the algorithms, and (5) empirical results that highlight the performance of the algorithms on a data-parallel computer, the Connection Machine CM-2.     

Robust Optimization for Power Systems Capacity Expansion under Uncertainty

We develop a robust optimization model for planning power system capacity expansion in the face of uncertain power demand. The model generates capacity expansion plans that are both solution and model robust. That is, the optimal solution from the model is ‘almost’ optimal for any realization of the demand scenarios (i.e. solution robustness). Furthermore, the optimal solution has reduced excess capacity for any realization of the scenarios (i.e. model robustness). Experience with a characteristic test problem illustrates not only the unavoidable trade-offs between solution and model robustness, but also the effectiveness of the model in controlling the sensitivity of its solution to the uncertain input data. The experiments also illustrate the differences of robust optimization from the classical stochastic programming formulation.     

Assigning vibrational spectra of ferryl-oxo intermediates of cytochrome C oxidase by periodic orbits and molecular dynamics

Complexity is inherent in biological molecules not only because of the large number of atoms but also because of their nonlinear interactions responsible for chaotic behaviours, localized motions, and bifurcation phenomena. Thus, versatile spectroscopic techniques have been invented to achieve temporal and spatial resolution to minimize the uncertainties in assigning the spectra of complex molecules. Can we associate spectral lines to specific chemical bonds or species in a large molecule? Can energy stay localized in a bond for a substantial period of time to leave its spectroscopic signature? These longstanding problems are investigated by studying the resonance Raman spectra of ferryl-oxo intermediates of cytochrome c oxidase. The difference spectra of isotopically substituted ferryl oxygen ((16)O minus (18)O) in the cytochrome c oxidase recorded in several laboratories show one or two prominent positive peaks which have not been completely elucidated yet. By applying the hierarchical methods of nonlinear mechanics, and particularly the study of periodic orbits in the active site of the enzyme, in conjunction with molecular dynamics calculations of larger systems which include the embraced active site by the protein and selected protonated/deprotonated conformations of amino acids, we translate the spectral lines to molecular motions. It is demonstrated that for the active site stable periodic orbits exist for a substantial energy range. Families of periodic orbits which are associated with the vibrations of Fe(IV)=O bond mark the regions of phase space where nearby trajectories remain localized, as well as assign the spectral bands of the active site in the protein matrix. We demonstrate that proton movement adjacent to active site, which occurs during the P --> F transition, can lead to significant perturbations of the Fe(IV)=O isotopic difference vibrational spectra in cytochrome c oxidase, without a change in oxidation state of the metal sites. This finding links spectroscopic characteristics to protonation events occurring during enzymatic turnover.     

Molecular modeling of liquid crystalline self-organization of fullerodendrimers: columnar to lamellar phase transitions driven by temperature and/or concentration changes

The molecular cubic-block model [ J. Chem. Phys. 2005, 123, 164904 ] is used to study a class of poly(benzyl ether) fullerodendrimers that have recently been reported to form columnar liquid crystal phases. In agreement with experiment, the model-molecules are found to self-assemble into columns which form hexagonal or rectangular lattices. The columnar cross sections are elongated in the rectangular phase. Transitions to the isotropic phase, either directly or through the intermediate formation of smectic phases, have been found. The effects of dissolving small amounts of nonbonded fullerene molecules have been explored. The results predict that the fullerene solutes restrict the range of stability of the columnar phase and may induce transitions from the columnar to the smectic or the isotropic phase.     

Molecular dynamic simulations of carbon nanotubes in CO2 atmosphere

This work investigates by means of molecular dynamics the filling of carbon nanotubes by carbon dioxide molecules. Nanotubes of various sizes are simulated and the resulting CO₂ density calculated. The effects of various CO₂ models are also investigated. The results show that the carbon dioxide molecules have a natural tendency to fill the nanotubes and the final CO₂ concentration inside the nanotube can be approximately 100 times (depending on diameter and CO₂ model) higher than that of the external atmosphere.     

Probing the environment of cu(b) in heme-copper oxidases

Time-resolved step-scan FTIR (TRS2-FTIR) and density functional theory have been applied to probe the structural dynamics of CuB in heme-copper oxidases at room temperature. The TRS2-FTIR data of cbb3 from Pseudomonas stutzeri indicate a small variation in the frequency of the transient CO bound to CuB in the pH/pD 7-9 range. This observation in conjunction with density functional theory calculations, in which significant frequency shifts of the nu(CO) are observed upon deprotonation and/or detachment of the CuB ligands, demonstrates that the properties of the CuB ligands including the cross-linked tyrosine, in contrast to previous reports, remain unchanged in the pH 7-9 range. We attribute the small variations in the nu(CO) of CuB to protein conformational changes in the vicinity of CuB. Consequently, the split of the heme Fe-CO vibrations (alpha-, beta-, and gamma-forms) is not due to changes in the ligation and/or protonation states of the CuB ligands or to the presence of one or more ionizable groups, as previously suggested, but the result of global protein conformational changes in the vicinity of CuB which, in turn, affect the position of CuB with respect to the heme Fe.     

On the Melvin–Morton–Rozansky conjecture

We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Oblatum 28-VII-1994 & 5-XI-1995 & 22-XI-1995     

Non-linear dynamics of the photodissociation of nitrous oxide: equilibrium points, periodic orbits, and transition states

The diffuse vibrational bands, observed in the ultraviolet photodissociation spectrum of nitrous oxide by exciting the molecule in the first (1)A' state, have recently been attributed to resonances localized mainly in the NN stretch and bend degrees of freedom. To further investigate the origin of this localization, fundamental families of periodic orbits emanating from several stationary points of the (1)A' potential energy surface and bifurcations of them are computed. We demonstrate that center-saddle bifurcations of periodic orbits are the main mechanism for creating stable regions in phase space that can support the partial trapping of the wave packet, and thus they explain the observed spectra. A non-linear mechanical methodology, which involves the calculation of equilibria, periodic orbits, and transition states in normal form coordinates, is applied for an in detail exploration of phase space. The fingerprints of the phase space structures in the quantum world are identified by solving the time dependent Schro?dinger equation and calculating autocorrelation functions. This demonstrates that different reaction channels could be controlled if exact knowledge of the phase space structure is available to guide the initial excitation of the molecule.     

Enhancing the stiffness of electrospun nanofiber scaffolds with a controlled surface coating and mineralization

A new method was developed to coat hydroxyapatite (HAp) onto electrospun poly(lactic-co-glycolic acid) (PLGA) nanofibers for tendon-to-bone insertion site repair applications. Prior to mineralization, chitosan and heparin were covalently immobilized onto the surface of the fibers to accelerate the nucleation of bone-like HAp crystals. Uniform coatings of HAp were obtained by immersing the nanofiber scaffolds into a modified, 10-fold-concentrated simulated body fluid (m10SBF) for different periods of time. The new method resulted in thicker and denser coatings of mineral on the fibers compared to those produced by previously reported methods. Scanning electron microscopy measurements confirmed the formation of nanoscale HAp particles on the fibers. A mechanical property assessment demonstrated a higher stiffness with respect to previous coating methods. A combination of the nanoscale fibrous structure and bonelike mineral coating could mimic the structure, composition, and function of mineralized tissues.