First integrals of the axisymmetric shape equation of lipid membranes

The shape equation of lipid membranes is a fourth-order partial differential equation.Under the axisymmetric condi-tion,this equation was transformed into a second-order ordinary differential equation(ODE)by Zheng and Liu(Phys.Rev.E 482856(1993)).Here we try to further reduce this second-order ODE to a first-order ODE.First,we invert the usual process of variational calculus,that is,we construct a Lagrangian for which the ODE is the corresponding Euler-Lagrange equation.Then,we seek symmetries of this Lagrangian according to the Noether theorem.Under a certain restriction on Lie groups of the shape equation,we find that the first integral only exists when the shape equation is identical to the Will-more equation,in which case the symmetry leading to the first integral is scale invariance.We also obtain the mechanical interpretation of the first integral by using the membrane stress tensor.

Exact Solutions of the Equilibrium Shape Equations in a General WLC Model for DNA Forms

In this letter,we are going to use a geometrical approach to describe the free energy of DNA structures.The exact solutions of the equilibrium shape equations in a general WLC model for DNA forms by using the Feoli's formalism [A.Feoli,et al.,Nucl.Phys.B 705(2005) 577] are studied.Then,the free energy of transition between Band Z-DNA is calculated in this formalism.

SOLUTION OF DIFFERENT HOLES SHAPE BORDERS OF FIBRE REINFORCED COMPOSITE PLATES BY INTEGRAL EQUATIONS

Accurate boundary conditions of composite material plates with different holes are founded to settle boundary condition problems of complex holes by conformal mapping method upon the nonhomogeneous anisotropic elastic and complex function theory. And then the two stress functions required were founded on Cauchy integral by boundary conditions. The final stress distributions of opening structure and the analytical solution on composite material plate with rectangle hole and wing manholes were achieved. The influences on hole-edge stress concentration factors are discussed under different loads and fiber direction cases, and then contrast calculates are carried through FEM.

曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

Study on Headland-Bay Sandy Coast Stability in South China Coasts

Headland-bay beach equilibrium planform has been a crucial problem abroad to long-term sandy beach evolution and stabilization, extensively applied to forecast long-term coastal erosion evolvement and the influences of coastal engineering as well as long-term coastal management and protection. However, little concern focuses on this in China. The parabolic relationship is the most widely used empirical relationship for determining the static equilibrium shape of headland-bay beaches. This paper utilizes the relation to predict and classify 31 headland-bay beaches and concludes that these bays cannot achieve the ultimate static equilibrium planform in South China. The empirical bay equation can morphologically estimate beach stabilization state, but it is just a referential predictable means and is difficult to evaluate headland-bay shoreline movements in years and decades. By using Digital Shoreline Analysis System suggested by USGS, the rates of shoreline recession and accretion of these different headland-bay beaches are quantitatively calculated from 1990 to 2000. The conclusions of this paper include that （a） most of these 31 bays maintain relatively stable and the rates of erosion and accretion are relatively large with the impact of man-made constructions on estuarine within these bays from 1990 to 2000; （b） two bays, Haimen Bay and Hailingshan Bay, originally in the quasi-static equilibrium planform determined by the parabolic bay shape equation, have been unstable by the influence of coastal engineering; and （c） these 31 bays have different recession and accretion characters occuning in some bays and some segments. On the one hand, some bays totally exhibit accretion, but some bays show erosion on the whole. Shanwei Bay, Houmen Bay, Pinghai Bay and Yazhou Bay have the similar planforms, characterized by less accretion on the sheltering segment and bigger accretion on the transitional and tangential segments. On the other hand, different segments of some bays have two dissimilar evolvement characters. Dacheng Bay, Shenquan Bay, Hudong Bay, Wukan Bay, Fengjia Bay, Wuchang Bay, Lingshui Bay and Tufu Bay produce accretion on the tangential segment, erosion on the transitional segment and accretion on the sheltering segment. However, Guang＇ao Bay, Haimen Bay, Jinghai Bay, Sanya Bay（a）, Dajiao Bay, Hailingshan Bay, Hebei Bay, Fuhu Bay, Shuidong Bay, Wangcun Bay and Bomao Bay generate erosion on the tangential part, accretion on the transitional part and accretion on the sheltering part. It seems to imply some relations between headland-bay beach evolvement and controls on headland-bay beaches, which may possibly to classify headland-bay beach types and should be further studied.

Crystallization Kinetics of Pr8Fe86-xZrxB6 Amorphous Alloys

Two experimental methods were adopted to verify the correctness and practicability of the shape meter method: one is to roll aluminum plate, calculate the shape stiffness of mill and rolled piece, and then measure aluminum plate crown to verify shape stiffness equation; the other is to calculate the measured off line data of hot continuous roll and verify the shape mathematical model for measuring and controlling by self adaptation method.

Exact Solutions of Schr¨odinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form,V（r,θ）=V（r）＋（h^2/2M r^2）[（βsin^2θ＋γcos^2θ＋2λ）/sinθcosθ]^2 and its exact solutions are presented via the Nikiforov–Uvarov method.The angle dependent part V（θ）=（h^2/2M r^2）[（βsin^2θ＋γcos^2θ＋λ）/sinθcosθ]^2,which is reported for the first time embodied the novel angle dependent（NAD）potential and harmonic novel angle dependent potential（HNAD）as special cases.We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

Plate Shape Control Theory and Experiment for 20-high Mill

Roll flattening theory is an important part of plate shape control theories for 20-high mill. In order to improve the accuracy of roll flattening calculation for 20-high mill, a new and more accurate roll flattening model was proposed. In this model, the roll barrel was considered as a finite length semi-infinite body. Based on the boundary integral equation method, the numerical solution of the finite length semi-infinite body under the distributed force was obtained and an accurate roll flattening model was established. Coupled with roll bending model and strip plastic deformation, a new and more accurate plate control model for 20-high mill was established. Moreover, the effects of the first intermediate roll taper angle and taper length were analyzed. The tension distribution calculated by analytical model was consistent with the experimental results.