In this work, we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential. The essence of the scheme is to introduce several additional auxiliary variables and design some corresponding auxiliary ODEs to reformulate the system into an equivalent form so that the explicit discretization for the nonlinear terms can also achieve unconditional energy stability. Moreover, the scheme has a full decoupling structure and can avoid calculating variable-coefficient systems. The advantage of this scheme is its high efficiency and ease of implementation, that is, only by solving two independent linear biharmonic equations with constant coefficients for each phase-field variable, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. We strictly prove that the fully discrete energy stability that the scheme holds and give a detailed step-by-step implementation process. Further, numerical experiments are carried out in 2D and 3D to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.
Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. We prove that the fact of imposing specific properties on the second maximal A-invariant subgroups of G determines that G is either soluble or isomorphic to a few non-soluble groups such as PSL(2, 5) or SL(2, 5).
In CNC machining, the tool path planning of the cutter plays an important role. In this paper, we generate a space-filling and continuous tool path for free-form surface represented by the triangular mesh with a confined scallop height. The tool path is constructed from connected Fermat spirals (CFS) but with fewer inflection points. Comparing with the newly developed CFS method, only about half of the number of inflection points are involved. Moreover, the kinematic constraints are simultaneously taken into account to increase the feedrates in machining. Finally, we use a micro-line trajectory technique to smooth the tool path. Experimental results and physical cutting tests are provided to illustrate and clarify our method.
A Frölicher-type inequality for Bott-Chern cohomology and its relation with -lemma were introduced in [
We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of branched along six stable hyperplanes.