In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish related Schwarz type lemmas fo
r almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce Liouville type theorems for almost holomorphic maps.
In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some
sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.
In this paper, we establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, we prove that the stochastic completeness for the heat
semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, we give some applications of these generalized maximum principles.
The friction of a nanosized sphere in commensurate contact with a flat substrate is investigated by performing molecular dynamics simulations. Particular focus is on the distribution of shear stress within the contact region. It is noticed that withi
n the slip zone, the local friction coefficient defined by the ratio of shear stress to normal pressure declines monotonically as the distance to contact center increases. With the lateral force increasing, the slip zone expands inwards from the contact edge. At the same time, the local friction coefficient at the contact edge decreases continuously, while at the dividing between the slip and stick zones keeps nearly invariant. These characteristics are distinctly different from the prediction of the conventional Cattaneo-Mindlin model assuming a constant local friction coefficient within the slip zone. An analytical model is advanced in view of such new features and generalized based on numerous atomic simulations. This model not only accurately characterizes the interfacial shear stress, but also explains the size-dependence of static friction of single nanosized asperity.
Atomistic simulations are performed to study the statistical mechanical property of gold nanoparticles. It is demonstrated that the yielding behavior of gold nanoparticles is governed by dislocation nucleation around surface steps. Since the nucleati
on of dislocations is an activated process with the aid of thermal fluctuation, the yield stress at a specific temperature should exhibit a statistical distribution rather than a definite constant value. Molecular dynamics simulations reveal that the yield stress follows a Gaussian distribution at a specific temperature. As the temperature increases, the mean value of yield stress decreases while the width of distribution becomes larger. Based on numerical analysis, the dependence of the mean yield stress on temperature can be well described by a parabolic function. Present study illuminates the statistical features of the yielding behavior of nanostructured elements.
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and compari
son theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.
Surface tension is a prominent factor for the deformation of solids at micro-/nano-scale. This paper investigates the effects of surface tension on the two-dimensional contact problems of an elastic layer bonded to the rigid substrate. Under the plan
e strain assumption, the elastic field induced by a uniformly distributed pressure within a finite width is formulated by applying the Fourier integral transform, and the limiting process leading to the solutions for a line force brings the requisite surface Greens function. For the indentation of an elastic layer by a rigid cylinder, the corresponding singular integral equation is derived, and subsequently solved by using an effective numerical method based on Gauss-Chebyshev quadrature formula. It is found from the theoretical and numerical results that the existence of surface tension strongly enhances the hardness of the elastic layer and significantly affects the distribution of contact pressure, when the size of contact region is comparable to the elastocapillary length. In addition, an approximated relationship between external load and half-width of contact is generalized in an explicit and concise form, which is useful and convenient for practical applications.
