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 condition, this equation was transformed into a second-order ordinary differential equation(ODE) by Zheng and Liu(Phys. Rev.E 48 2856(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 Willmore 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.     

Radiative heat transfer in axisymmetric domains of complex shape with Fresnel boundaries

A detailed study of the peculiarities of the radiative heat transfer (RHT) in axisymmetric domains bounded with Fresnel surfaces is undertaken. The analytical (exact) solutions of the RHT problem in conical and cylindrical domains with refractive index more than unity were obtained for a variety of absorption coefficient and geometrical parameters of the domains. It is shown that due to Fresnel reflections the net radiative flux strongly varies over the base of cone and cylinder. The difference in RHT processes for the cases of constant reflectivity of the boundaries and that calculated by Fresnel formula is demonstrated. The influence of specular reflection at the crystal side surface on the shape of the solid/liquid interface in growing bismuth germanate crystals is shown.     

Analytical expressions for electrostatics of graphene structures

This study focuses on electrostatics of various graphene structures as graphene monolayer, graphene nanoribbons, as well as multi-layer graphene or graphene flakes. An atomistic moment method based on classical electrostatics is utilized in order to evaluate the charge distribution in each nanostructure. Assuming a freestanding graphene structure in an infinite or in a semi-infinite space limited by a grounded infinite plane, the effect of the length, width, number of layers and position of the nanostructure on its electrostatic charge distributions and total charge and capacitance is examined through a parametric analysis. The results of the present show good agreement with corresponding available data in the literature, obtained from different theoretical approaches. Performing nonlinear regression analysis on the numerical results, where it is possible, simple analytical expressions are proposed for the total charge and charge distribution prediction based on structure geometry.     

Analytical expressions for polar magnetoptics in magnetic multilayers

This paper is a comprehensive study of different formalisms used to simulate polar magneto-optical (MO) effects in multilayers. A development of the 4 × 4 matrix calculation to first order in off-diagonal elements of the permittivity tensor gives original analytical expressions. They allow to express the MO effects in multilayers as a sum of contributions from individual layers. Each contribution consists of two components assigned to the interface and propagation effects. The explicit expressions for multilayers containing up to seven layers, together with those obtained in the frame of an ultrathin layer approximation, are shown to be useful both in a qualitative comprehension and in quantitative simulations of the experimental MO Kerr spectra of a wide variety of ultrathin magnetic structures. They can be used to separate the different contributions arising from the interfaces due to intermixing or spin polarization (including interface roughness) as well as to simulate magnetically coupled multilayers. As an example, the analytical expression is provided for a system consisting of two ultrathin films separated by a non-magnetic spacer and deposited on a non-magnetic substrate. The capabilities of the analytical expressions are demonstrated on MO spectroscopy in a broad spectral range of some well characterized ultrathin magnetic structures. This work has been partially supported by HCM programme “Magnetic properties of novel ultrathin films” (project number ERBCIPDCT940622), NATO Grant “High Technology” (LG931415) and Grant Agency of Czech Republic (# 202/97/1180 and # 202/00/0761).     

Analytical expressions for the evaluation of vacuum-polarization potentials in muonic atoms

Exact analytical expressions in terms of known transcendental functions are given for the evaluation of the Uehling vacuum-polarization potential induced by a static spherically-symmetrric charge distribution.     

平顶光束K参数的解析传输方程

从Li提出的平顶光束模型和Collins公式出发,推导出平顶光束通过近轴ABCD光学系统K参数的解析传输公式。平顶光束的K参数与光束阶数M、瑞利长度zR和传输矩阵元A/B有关。另一方面,基于强度矩的传输方程,通过繁冗的积分,也推导出K参数的解析传输方程。大量数值计算结果证明二者所得结果相同。对平顶光束K参数的传输特性用数值计算例作了说明。     

平顶光束K参数的解析传输方程

 从Li提出的平顶光束模型和Collins公式出发,推导出平顶光束通过近轴ABCD光学系统K参数的解析传输公式。平顶光束的K参数与光束阶数M、瑞利长度z_R和传输矩阵元A/B有关。另一方面,基于强度矩的传输方程,通过繁冗的积分,也推导出K参数的解析传输方程。大量数值计算结果证明二者所得结果相同。对平顶光束K参数的传输特性用数值计算例作了说明。     

Analytical solutions for axisymmetric vibration of loudspeaker cones in transitional range

The loudspeaker cone vibration in the transitional frequency range in which the governing equations have transition or turning points is studied by applying the uniformly valid solutions obtained in a previous paper. The solutions are presented analytically as well as the resonant and anti-resonant frequency equations, the associated patterns and the axial admittance. A new frequency equation for the non-bending effect is obtained. The results agree well with numerical results.     

Free vibration of composite membranes with arbitrary shape

This paper is concerned with a method for solving problems of a composite membrane with an arbitrarily shaped outer boundary and an arbitrarily shaped inner boundary. The boundary conditions and the conditions of continuity are satisfied directly by using a Fourier expansion collocation method which has been given by Nagaya. The general equation for finding the natural frequencies of the composite membranes has been presented. As examples, numerical calculations have been carried out for composite polygonal membranes, composite elliptical membranes and composite circular membranes with eccentric circular boundaries.     

Shapes of nearly cylindrical, axisymmetric bilayer membranes

Shapes of nearly cylindrical sections of axisymmetric phospholipid membranes are studied theoretically. Describing the shape of such sections by their deviation from a reference cylinder, the well-established shape equation for axisymmetric bilayer membranes is expanded in terms of this deviation, and it is then solved analytically. The phase diagram shows the resulting stationary shapes as functions of system parameters and external conditions, i.e., the pressure difference across the membrane, the membrane tension, the difference between the tensions of the two monolayers, and the axial force acting on the vesicle. The accuracy of the approximate analytical solution is demonstrated by comparison with numerical results. The obtained analytical solution allows to extend the analysis to include shapes where numerical methods have failed. Received 27 September 2000 and Received in final form 26 March 2001     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Model for the dynamics of two interacting axisymmetric spherical bubbles undergoing small shape oscillations

Interaction between acoustically driven or laser-generated bubbles causes the bubble surfaces to deform. Dynamical equations describing the motion of two translating, nominally spherical bubbles undergoing small shape oscillations in a viscous liquid are derived using Lagrangian mechanics. Deformation of the bubble surfaces is taken into account by including quadrupole and octupole perturbations in the spherical-harmonic expansion of the boundary conditions on the bubbles. Quadratic terms in the quadrupole and octupole amplitudes are retained, and surface tension and shear viscosity are included in a consistent manner. A set of eight coupled second-order ordinary differential equations is obtained. Simulation results, obtained by numerical integration of the model equations, exhibit qualitative agreement with experimental observations by predicting the formation of liquid jets. Simulations also suggest that bubble-bubble interactions act to enhance surface mode instability.     

Analytical expressions of non-relativistic static multipole polarizabilities for hydrogen-like ions

Analytical formulas for the static multipole polarizabilities of hydrogen-like ions are derived by using the analytical wave functions and the reduced Green function and by applying a numerical fitting procedure.Our results are then applied to the studies of blackbody radiation shifts to atomic energy levels at different temperatures.Our analytical results can be served as a benchmark for other theoretical methods.     

The shape of an axisymmetric bubble in uniform motion

We consider in a frame fixed to a bubble translating with steady speedU, the inviscid, axisymmetric, irrotational motion of the liquid past it. If all speeds are normalized byU and lengths by {ie437-1}, whereT is the surface tension of the liquid-bubble interface, it can be shown that the unknown bubble shape and field depend on a single parameter {ie437-2} alone, where the pressures are the ones in the bubble and far away respectively. WhenΓ is very large the bubble is almost spherical in shape while for Γ<- Γ^* ≈ -0.315, bubbles whose exteriors are simply connected do not exist. We solve the non-linear, free boundary problem for the whole range Γ^* < Γ < ∞ by the use of an analytical representation for the bubble shape, a surface singularity method to compute potential flows and a generalized Newton’s method to continue inΓ. Apart from providing explicit representations for bubble shapes and detailed numerical values for the bubble parameters, we show that the classical linearized solution for largeΓ is a very good approximation, surprisingly, to as low values of Γ as 2. We also show that Miksiset al [1] is inaccurate over the whole range and in serious error for large and smallΓ. These have been corrected.     

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein--Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex (EHC) function method is used to study further the stationary axisymmetric Einstein--Maxwell theory with $p$ Abelian gauge fields (EM-$p$ theory, for short). Two EHC structural Riemann--Hilbert (RH) transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-$p$ theory. This symmetry group is verified to have the structure of semidirect product of Kac--Moody group $\widehat{SU(p+1,1)}$ and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author's paper by a different scheme. This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.     

Analytical expressions for and geometrical meaning of the comparative indices of recording media

The indices for characterizing the performance of holographic recording media introduced recently by Tai and Yu can be related to the parameters of the optimal linear transfer function (OLTF). These parameters have been obtained in an analytical form by the author for the recording material with an exponential transfer curve and for the holographic setup using two plane waves and for the case of the fourier lensless hologram. The expressions for the indices in terms of the OLTF parameters are given and their geometrical meaning in relation to the OLTF is pointed out for the two holographic setups mentioned.     

Analytical expressions for multiplicity distributions and deviations from KNO scaling

A simple method is presented for deducing multiplicity distributions from a few multiplicity moments. The essence of the method is the straightforward trick of converting a discrete distribution to a continuous distributions by interpolation. Interpolation changes the moments by calculable amounts, which depend only on the interpolation procedure and not on the multiplicity distribution. This device makes it possible to give a simple analytical expression for the multiplicity distribution, even at rather low energies, in terms of the measured moments. A quantitative discussion of KNO scaling, and deviations from scaling, is given, based on the measured moments.