On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Application of Legendre Polynomials Correction Before Component Quantification of Biological Mid-infrared Spectra

Abstract

Quantitative determination from infrared spectra still faces problems that are not completely resolved to date. One of these problems is the deformation of baselines.

We have tested the Legendre baseline correction method by subtracting from a spectrum the low-degreed terms from a Legendre polynomial function which is obtained by the decomposition of the spectrum.

As opposed to the filtered spectra with n=0 which remained very similar to the original spectra, the filtered spectra, as from n=2, exhibit a great amount of distorsion. This result suggests that it is as from n=2, that the terms that are meaningful to the signal (not just to baseline), are to be substracted.

Mathematical methods such as Principal Component Analysis and Principal Component Regression are used to analyse quantitatively components from biological samples. The thus predicted sucrose content values for values of n that range between 0 and 8 (n={0, 2, 4, 6, 8}) show that the best results are obtained for n=4. The mean and standard deviation values of the difference between reference and predicted values varies between 8.5 × 10^?2 and 6.6 × 10^?2 and between 0. 224 and 0. 195 respectively; these values are minimum when the n=4 term is subtracted from the spectra.

This processing hence, sensibly improves the quantification of components from Mid-FTIR spectra of biological solutions. This procedure would be interesting in cases where the precision of quantitative measures is essential.     

Sectorial extensions for ultraholomorphic classes defined by weight functions

We prove an extension theorem for ultraholomorphic classes defined by so-called Braun–Meise–Taylor weight functions ω and transfer the proofs from the single weight sequence case from V. Thilliez to the weight function setting. We are following a different approach than the results obtained in a recent paper by the authors, more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.     

一类四阶微积分方程的Legendre-Galerkin谱逼近

针对研究吊桥模型而建立的四阶微积分方程, 提出Legendre谱逼近法进行求解.构造迭代算法来求解得到的线性系统, 证明了迭代格式的收敛性, 对问题进行了误差分析.数值算例验证了迭代的收敛性和方法的高精度.     

极坐标系下的Legendre谱元方法求解Poisson-型方程

r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR(Legendre Gauss Radau)积分点,其他单元径向使用LGL(Legendre Gauss Lobatto)积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。     

Hellmann–Feynman connection for the relative Fisher information

The (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI–Euler theorem are self-consistently derived from the Hellmann–Feynman theorem. These new reciprocity relations generalize the RFI–Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI–LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf’s, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.     

Meson contributions in the Nambu-Jona-Lasinio model

We discuss the problem of calculating corrections to the mean-field approximation in the Nambu-Jona-Lasinio model. To calculate such corrections, we propose using the method of the Legendre transformation with respect to a bilocal source, which allows taking symmetry constraints related to the chiral Ward identity into account effectively. Using the proposed method, we determine the corrections to the quark propagator and the two-particle quark function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 81–95, April, 2009.     

On an identity associated with Weil's estimate and its applications

In this paper we study the fourth mean value of generalized quadratic Gauss sums, and get an interesting identity associated with Weil's estimate,