We consider a molecule constrained to a hypersurface S in the conguration space R^m. In order to derive an expression for the mean force acting along the constrained coordinate we decompose the molecular vector field, and single out the direction of the respective coordinate utilising the structure of affine connections. By these means we reconsider the well-known results derived by Sprik et al. [1] and Darve et al. [2]; we gain concise geometrical insight into the different contributions to the force in terms of molecular potential, mean curvature, and the connection 1-form of the normal bundle over the submanifold S. Our approach gives rise to a Hybrid Monte-Carlo based algorithm that can be used to compute the averaged force acting on selected coordinates in the context of thermodynamic free energy statistics.

We study Onsager's free energy functional for nematic liquid crystals with an orientation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of orientations near the critical and zero temperatures.

An efficient second-order stable numerical method is presented to solve the model partial differential equations of thermal glass fiber processing. The physical process and structure of the model equations are described first. The numerical issues are then clarified. The heart of our method is a MacCormack scheme with flux limiting. The numerical method is validated on a linearized isothermal model and by comparison with known exact stationary solutions. The numerical method is then generalized to solve the equations of motion of thermal glass fiber drawing, exhibiting order of convergence. Further, the nonlinear PDE scheme is benchmarked against an independent linearized stability analysis of boundary value solutions near the onset of instability, which demonstrates the efficiency of the method.

A discrete auditory transform (DAT) from sound signal to spectrum is presented and shown to be invertible in closed form. The transform preserves energy, and its spectrum is smoother than that of the discrete Fourier transform (DFT) consistent with human audition. DAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. It is found that DAT can gain 3 to 5 decibel (dB) in signal to noise ratio (SNR) over DFT except when the noise level is relatively low.

We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically. We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics. We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum. We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization. Ground, symmetric and central vortex states, as well as their energy con.gurations, are reported in 2D and 3D for a rotating BEC. Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime. Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.

In a weakly nonlinear model equation for capillary-gravity water waves on a two-dimensional free surface, we show, numerically, that there exist localized solitary traveling waves for a range of parameters spanning from the long wave limit (with Bond number B>1/3, in the regime of the Kadomtsev-Petviashvilli-I equation) to the wavepacket limit (B>1/3, in the Davey-Stewartson regime). In fact, we show that these two regimes are connected with a single continuous solution branch of nonlinear localized solitary solutions crossing B=1/3.