Impact of anisotropy on the structure and dynamics of ionic liquids: a computational study of 1-butyl-3-methyl-imidazolium trifluoroacetate

The complex ionic network of 1-butyl-3-methyl-imidazolium trifluoroacetate was simulated by means of the molecular dynamics methods over a time period of 100 ns. The influence of the anisotropy of the shape and charge distribution of both the cations and the anions on the local (molecular) and global (collective) structure and dynamics is analyzed. The distance-dependent g coefficients of the orientational probability function g(r,Omega) were found to be an excellent way to interpret local structure. Thereby, the combination and interrelation of individual g coefficients elucidate the mutual orientation. Dynamics at the molecular level is characterized by the time correlation function of the center-of-mass corrected molecular dipole moment mucm. Upon uniting the set of molecular dipoles to a single collective rotational dipole moment, MD, dynamics on a global level is studied. Decomposing into subsets of cations and anions respective self terms as well as the prominent cross term can be extracted. This decomposition also enables a detailed peak assignment in dielectric spectra.     

Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.     

A Low-Complexity Divide-and-Conquer Method for Computing Eigenvalues and Eigenvectors of Symmetric Band Matrices

A framework for an efficient low-complexity divide-and-conquer algorithm for computing eigenvalues and eigenvectors of an n × n symmetric band matrix with semibandwidth b n is proposed and its arithmetic complexity analyzed. The distinctive feature of the algorithm—after subdivision of the original problem into p subproblems and their solution—is a separation of the eigenvalue and eigenvector computations in the central synthesis problem. The eigenvalues are computed recursively by representing the corresponding symmetric rank b(p–1) modification of a diagonal matrix as a series of rank-one modifications. Each rank-one modifications problem can be solved using techniques developed for the tridiagonal divide-and-conquer algorithm. Once the eigenvalues are known, the corresponding eigenvectors can be computed efficiently using modified QR factorizations with restricted column pivoting. It is shown that the complexity of the resulting divide-and-conquer algorithm is O (n ² b ²) (in exact arithmetic).This revised version was published online in October 2005 with corrections to the Cover Date.     

Exact solutions for the collaborative pickup and delivery problem

In this study we investigate the decision problem of a central authority in pickup and delivery carrier collaborations. Customer requests are to be redistributed among participants, such that the total cost is minimized. We formulate the problem as multi-depot traveling salesman problem with pickups and deliveries. We apply three well-established exact solution approaches and compare their performance in terms of computational time. To avoid unrealistic solutions with unevenly distributed workload, we extend the problem by introducing minimum workload constraints. Our computational results show that, while for the original problem Benders decomposition is the method of choice, for the newly formulated problem this method is clearly dominated by the proposed column generation approach. The obtained results can be used as benchmarks for decentralized mechanisms in collaborative pickup and delivery problems.