Box-counting dimension is one of the most widely used fractal dimensions, because it is relatively easy for empirical estimation. Unfortunately, the task is simple only if one operates on oneor twodimensional data. Otherwise, the amount of computation grows exponentially with the number of space coordinates.A newmethod of estimation of the box-counting dimension of multivariate objects using sp...

We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E., Ren, and Vanden-Eijnden (J. Chem. Phys., vol. 126 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional “ridges” of saddle-continua. We then ap...

In this article, we present a summary of the current status of the study of the transport of electrons confined to one dimension in very low disorder GaAs-AlGaAs heterostructures. By means of suitably located gates and application of a voltage to 'electrostatically squeeze' the electronic wave functions, it is possible to produce a controllable size quantization and a transition from two-dimens...

We report benchmark results for one-dimensional (1D) atomic and molecular systems interacting via the Coulomb operator |x|(-1). Using various wavefunction-type approaches, such as Hartree-Fock theory, second- and third-order Møller-Plesset perturbation theory and explicitly correlated calculations, we study the ground state of atoms with up to ten electrons as well as small diatomic and triatom...

The three-body problem in one-dimension with a repulsive inverse square potential between every pair was solved by Calogero. Here, the known results of supersymmetric quantum mechanics are used to propose a number of new three-body potentials which can be solved algebraically. Analytic expressions for the eigenspectrum and the eigenfunctions are given with and without confinement.

In recent years, the quasi-Monte Carlo approach for pricing high-dimensional derivative securities has been used widely relative to other competitive approaches such as the Monte Carlo methods. Such success can be, in part, attributed to the notion of effective dimension of the finance problems. In this paper, we provide additional insight on the connection between the effective dimension and t...