The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

On invariant subspaces and eigenfunctions of regular Hecke operators on spaces of multiple theta constants

Invariant subspaces and eigenfunctions of regular Hecke operators acting on spaces spanned by products of even number of Igusa theta constants with rational characteristics are constructed. For some of the eigenfunctions of genuses g=1 and g=2, corresponding zeta functions of Hecke and Andrianov are explicitly calculated.     

Measurement of uncertainty in peptide molecular weight determination using size-exclusion chromatography with multi-angle laser light-scattering detection and matrix-assisted laser desorption/ionization time-of-flight mass spectrometry

The molecular weight of biopolymers such as peptides or proteins is vital information for understanding their physical/chemical properties. The matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF-MS) and size-exclusion chromotography-multi-angle laser light scattering (SEC-MALLS) techniques each with its advantages and limitations were used for molecular weight determination of the SPf66 peptide. The precision of each method was studied using a two-factor fully-nested design with all the analyses performed by the same operator on a single instrument. The overall precision corresponded to the time-different intermediate precision (i.e. time and repeatability conditions). In the case of SEC-MALLS, all potential uncertainty components were carefully evaluated in an Ishikawa diagram, then included and mathematically combined with the uncertainty arising from the accuracy assessment to provide the overall uncertainty. In this case, the refractive-index increment with solute concentration value (dn/dc) provided the most significant contribution to the combined uncertainty. A method for its quantitation is proposed. The accuracy of method B (SEC-MALLS) against reference method A (MALDI-TOF-MS) was assessed using the interval testing hypothesis to limit the risk of unacceptable bias. The results indicate that the bias of B is higher than the limit established at 5%, and is therefore not traceable to A under the studied conditions.     

Breather trapping and breather transmission in a DNA model with an interface

We study the dynamics of moving discrete breathers in an interfaced piecewise DNA molecule. This is a DNA chain in which all the base pairs are identical and there exists an interface such that the base pairs dipole moments at each side are oriented in opposite directions. The Hamiltonian of the Peyrard-Bishop model is augmented with a term that includes the dipole-dipole coupling between base pairs. Numerical simulations show the existence of two dynamical regimes. If the translational kinetic energy of a moving breather launched towards the interface is below a critical value, it is trapped in a region around the interface collecting vibrational energy. For an energy larger than the critical value, the breather is transmitted and continues travelling along the double strand with lower velocity. Reflection phenomena never occur. The same study has been carried out when a single dipole is oriented in opposite direction to the other ones. When moving breathers collide with the single inverted dipole, the same effects appear. These results emphasize the importance of this simple type of local inhomogeneity as it creates a mechanism for the trapping of energy. Finally, the simulations show that, under favorable conditions, several launched moving breathers can be trapped successively at the interface region producing an accumulation of vibrational energy. Moreover, an additional colliding moving breather can produce a saturation of energy and a moving breather with all the accumulated energy is transmitted to the chain.     

The equivalent potential of water molecules for electronic structure of cysteine

In order to get more reliable electronic structure of protein in aqueous solution, it is necessary to construct a simple, easy-use equivalent potential of water molecules for protein's electronic structure calculation. The first-principles, all-electron, ab initio calculations have been performed to construct the equivalent potential of water molecules for the electronic structure of Cys. The process consists of three steps. First, the electronic structure of the cluster containing Cys and water molecules is calculated. Then, based on the structure, the electronic structure of Cys with the potential of water molecules is calculated using the self-consistent cluster-embedding method. Finally, the electronic structure of Cys with the potential of dipoles is calculated. The dipoles are adjusted so the electronic structure of Cys with the potential of dipoles is close to that of water molecules. The calculations show that the major effect of water molecules on Cys' electronic structure is lowering the occupied electronic states by about 0.032 Ry, and broadening energy gap by 16%. The effect of water molecules on the electronic structure of Cys can be simulated by dipoles potential.     

Elastic charge density representation of the interaction via the nematic director field

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.     

Renormalized solutions of elliptic equations with Robin boundary conditions

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type{-div(B(x, u)▽u) = f in ?,u = 0 on Γ_0,B(x, u)▽u·n→+γ(x)h(u) =g on Γ_1,where f and g are the element of L~1(?) and L~1(Γ_1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.     

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.     

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.