Through the combination of the minimum energy principle in physics and the Steiner minimal tree (SMT) theory in geometry,this paper proves a universal law for lipid nanotube networks (LNNs):at stable equilibrium state,the network of three-way lipid nanotube junctions is equivalent to a SMT.Besides,an arbitrary (usually non-equilibrium) network of lipid nanotube junctions may fission into a SMT through diffusions and dynamic self-organizations of lipid molecules.Potential applications of the law to the micromanipulations of LNNs are presented.
Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a suspended-cable-stayed beam. In the PDEs, the curvature of main cables and the deformation of cable stays are taken into account. The dynamics of the struc- ture is investigated based on the ODEs when the structure is subjected to a harmonic excitation in the presence of both high-frequency principle resonance and 1：2 internal resonance. It is found that there are typical jumps and saturation phenomena of the vibration amplitude in the struc- ture. And the structure may present quasi-periodic vibration or chaos, if the stiffness of the cable stays membrane and frequency of external excitation are disturbed.
Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a gen- eral theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived： （a） Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. （b） On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. （c） Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradi- ents form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. （d） The direction of the driving force induced by the hard curved surface is independent of the hyd
Wrinkling and buckling of nano-films on the compliant substrate are always induced due to thermal deformation mismatch.This paper proposes effective means to control the surface wrinkling of thin film on the compliant substrate,which exploits the curvatures of the curve cracks designed on the stiff film.The procedures of the method are summarized as:1)curve patterns are fabricated on the surface of PDMS(Polydimethylsiloxane)substrate and then the aluminum film with the thickness of several hundred nano-meters is deposited on the substrate;2)the curve patterns are transferred onto the aluminum film and lead to cracking of the film along the curves.The cracking redistributes the stress in the compressed film on the substrate;3)on the concave side of the curve,the wrinkling of the film surface is suppressed to be identified as shielding effect and on the convex side the wrinkling of the film surface is induced to be identified as inductive effect.The shielding and inductive effects make the dis-ordered wrinkling and buckling controllable.This phenomenon provides a potential application in the fabrication of flexible electronic devices.
This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.
This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.
This paper further extends the generalized covari ant derivative from the first covariant derivative to the sec ond one on curved surfaces. Through the linear transforma tion between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the sec ond class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric struc ture of the tensor analysis on curved surfaces are revealed.
Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.
