We present evidence revealing that an object with specific properties can exhibit multiple stable falling postures at low Reynolds numbers. By scrutinizing the force equilibrium relationship of a fixed object at various attack angles and Reynolds numbers, we introduce a methodology that can obtain the stable falling postures of the object. This method saves computational resources and more intuitively presents the results in the full parameter domain. Our findings are substantiated by free-fall tests conducted through both physical experiments and numerical simulations, which validate the existence of multiple stable solutions in accordance with the interpolation results obtained with fixed objects. Additionally, we quantify the abundance and distribution patterns of stable falling postures for a diverse range of representative shapes. This discovery highlights the existence of multiple stable solutions that are universally present across objects of different shapes. The implications of this research extend to the design, stability control and trajectory prediction of all free and controlled flights in both air and water.

The main aim of this paper is to establish the Lipschitz continuity of the $(K,K^{\prime })$-quasiconformal solutions of the Poisson equation ${\rm\Delta}w=g$ in the unit disk $\mathbb{D}$.

Numerical error caused by “ghost forces” in a quasicontinuum method is studied in the context of dynamic problems. The error in the discrete W1,∞ norm is analyzed for the time scale (ε) and the time scale (1) with ε being the lattice spacing.

We present a multiscale method for the modeling of dynamics of crystalline solids. The method employs the continuum elastodynamics model to introduce loading conditions and capture elastic waves, and near isolated defects, molecular dynamics (MD) model is used to resolve the local structure at the atomic scale. The coupling of the two models is achieved based on the framework of the heterogeneous multiscale method (HMM) and a consistent coupling condition with special treatment of the MD boundary condition. Application to the dynamics of a brittle crack under various loading conditions is presented. Elastic waves are observed to pass through the interface from atomistic region to the continuum region and reversely. Thresholds of strength and duration of shock waves to launch the crack opening are quantitatively studied and related to the inertia effect of crack tips.

We will present a general formalism for deriving boundary conditions for molecular dynamics simulations of crystalline solids in the context of atomistic/continuum coupling. These boundary conditions are modeled by generalized Langevin equations, derived from Mori-Zwanzig's formalism. Such boundary conditions are useful in suppressing phonon reflections, and maintaining the system temperature.

In this paper, we get a necessary and sufficient condition for the normalizers of higher dimensional Kleinian groups to be discrete. Also we obtain a necessary and sufficient condition for the isomorphisms between two higher dimensional Kleinian groups induced by quasiconformal mappings to be the same.