A confinement problem for a linearly elastic Koiter's shell

In this Note, we propose a natural two-dimensional model of “Koiter's type” for a general linearly elastic shell confined in a half space. This model is governed by a set of variational inequalities posed over a non-empty closed and convex subset of the function space used for modeling the corresponding “unconstrained” Koiter's model. To study the limit behavior of the proposed model as the thickness of the shell, regarded as a small parameter, approaches zero, we perform a rigorous asymptotic analysis, distinguishing the cases where the shell is either an elliptic membrane shell, a generalized membrane shell of the first kind, or a flexural shell. Moreover, in the case where the shell is an elliptic membrane shell, we show that the limit model obtained via the asymptotic analysis of our proposed two-dimensional Koiter's model coincides with the limit model obtained via a rigorous asymptotic analysis of the corresponding three-dimensional “constrained” model.     

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.     

The modified model of Koiter’s type for nonlinearly elastic shell

In this paper, we proposed a modified model of Koiter’s type for nonlinearly elastic shell. The change of metric tensor and the change of curvature tensor play an important role in constructing the linearly and nonlinearly elastic shell model of Koiter’s type. The approximate expressions of them once were proposed by Ciarlet. In this paper, the exact full expressions of the change of metric tensor and the change of curvature tensor are provided by tensor analysis. The former coincides with Ciarlet’s expression. And the latter is more exact than Ciarlet’s expression. Thus the modified model is better than Ciarlet’s model. At the same time, a numerical experiment of special hemispherical shell is provided to validate the modified model of Koiter’s type.     

A variant of Nitsche's method

We present a method that aims to reconcile Nitsche's method with the traditional finite element method ('weak' versus 'strong implementation' of essential boundary conditions). We retain the original idea of a variational formulation based on an extended energy, but replace the original boundary terms by domain terms involving weak derivatives. The solution of the proposed method coincides, for the Poisson problem, with the one of the traditional method, which in particular shows monotonicity under the standard angle condition for the Courant element. For more general second-order problems, it allows for the weighting of boundary terms inherent to Nitsche's method. This is of particular interest for singularly perturbed problems.     

A class of Orlicz-Sobolev spaces with applications to variational problems involving nonlinear Hill's equations

Necessary and sufficient conditions are proved for a $\text{b}$⁽²⁾-Young function G (with independent variable t) to be convex (resp. concave) in t² in terms of inequalities between the second derivative of G and the first derivative of its Legendre transform G? (with independent variable s). It is then proven that a Young function G is convex (resp. concave) in t² if and only if G? is concave (resp. convex) in s². These results, along with another set of inequalities for functions G convex (resp. concave) in t², allow the proof of the uniform convexity and thereby of the reflexivity with respect to Luxemburg's norm $\text{∥f∥}{}_{\mathrm{G}}\text{=}\text{inf}\text{{k > 0: ∝}{}_{\mathrm{\Omega }}\text{dξ G(f}\text{(ξ)}\text{k}\text{) ? 1}}$ of the Orlicz space $\text{L}{}_{\mathrm{G}}\text{(Ω)}$ over an open domain Ω ?$\text{R}$_N with Lebesgue measure dξ. When applied to $\text{G(t) =}\text{¦t¦}{}^{\mathrm{p}}\text{p}$ and $\text{G}\text{?}\text{(s) =}\text{¦s¦}{}^{\mathrm{p\prime }}\text{p′}$ with p^?1 + (p′)^?1 = 1, the preceding results lead to the shortest proof to date of two Clarkson's inequalities and of the reflexivity of L^p-spaces for 1 < p < +∞. Finally, some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear Hill's equations; these variational problems are formulated on suitable Orlicz-Sobolev spaces $\text{W}{}^{\mathrm{m}}\text{L}{}_{\mathrm{G}}\text{(Ω)}$ and thereby allow for nonlinear terms which may grow faster than any power of the variable.     

Justification of two-dimensional nonlinear dynamic shell equations of Koiter's type

In this paper we give the justification of a two-dimensional nonlinear dynamic shells model of Koiter's type.     

The membrane shell model in nonlinear elasticity: A variational asymptotic derivation

Summary We consider a shell-like three-dimensional nonlinearly hyperelastic body and we let its thickness go to zero. We show, under appropriate hypotheses on the applied loads, that the deformations that minimize the total energy weakly converge in a Sobolev space toward deformations that minimize a nonlinear shell membrane energy. The nonlinear shell membrane energy is obtained by computing the Γ-limit of the sequence of three-dimensional energies. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

分布视角下的异质性群体收入分配格局研究

本文运用反映不同收入群体异质性的混合分布测度方法估计收入分布,对收入分布拟合效果的评判方法进行了探讨,在利用假设检验思想从收入变动角度给出一种收入群体界定标准的基础上,从规模上刻画我国收入分配格局,并运用夏普里分解规则对影响格局形成的宏观因素进行了解构。对CHNS(1989-2011)住户调查数据实证分析的结果表明,近年来城乡居民收入分布逐渐呈多峰特征且城镇更为明显;城镇和农村居民不同收入群体间的相互作用相当,低收入群体内部收入的增长可有效促进自身规模的缩小,中收入群体内部收入的增长虽同样有助于低收入群体规模的缩小但对自身群体规模变动影响不大,高收入群体内部收入的增长则将有助于自身规模的增大,而各群体内收入分配差距的扩大多会阻碍"橄榄型"格局的形成。     

A remark on Hilbert's matrix

We exhibit a Jacobi matrix T which has simple spectrum and integer entries, and 0 commutes with Hilbert's matrix. As an application we replace the computation of the eigenvectors of Hilbert's matrix (a very ill-conditioned problem) by the computation of the eigenvectors of T (a nicely stable numerical problem).     

A short proof of Meyniel's theorem

Meyniel's theorem states that a strict diconnected digraph has a directed Hamilton cycle if d(u) + d(v) ? 2n ? 1 for every pair u, v of nonadjacent vertices. We give short proof of this theorem.     

A short proof of Minc's conjecture

A short proof is given of the following conjecture of Minc, proved in 1973 by Brègman. Let A be a n × n ? (0, 1)-matrix with r_i ones in row i. Then per .     

A short proof of Kneser's conjecture

In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale.     

A simplified proof of parikh's theorem

A strengthened form of the pumping lemma for context-free languages is used to give a simple proof of Parikh's Theorem.