Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

Controlled Radical Polymerization

Various methods which lead to the control of molecular weight and polydispersities, and which allow for the preparation of block copolymers by radical polymerization are discussed. Thermal polymerizationof styrenes in the presence of stable radicals, polymerization of vinyl acetate and methyl methacrylate in the presence of chromium complexed by macrocyclic ligands polymerization of vinyl acetate initiated by organoaluminum compounds complexed by dipyridyl and activated by stable radicals, as well as in the presence of phosphites, are described in detail.     

Estimates for Wallis’ ratio and related functions

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.     

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).     

The Space of Inextensional Displacements for a Partially Clamped Linearly Elastic Shell with an Elliptic Middle Surface

We consider a linearly elastic shell with an “elliptic” middle surface, clamped along a portion of its lateral face and subjected to body forces. Under weak regularity assumptions on the middle surface, we prove that the space of linearized inextensional displacements is reduced to zero, by using unique continuation results. Consequently, when the thickness of the shell goes to zero, the limit of the average with respect to the thickness of the three-dimensional displacement vector solves the “generalized membrane” shell model, according to the terminology introduced by P.G. Ciarlet and the first author. This revised version was published online in August 2006 with corrections to the Cover Date.     

An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors

Let $\text{Ω}$ be a bounded open connected subset of $\text{R}{}^{\mathrm{n}}$ with a Lipschitz-continuous boundary and let $\text{Θ}\text{∈}\text{C}{}^1\text{(}\text{Ω}\text{;}\text{R}{}^{\mathrm{n}}\text{)}$ be a deformation of the set $\text{Ω}$ satisfying $\text{det}\text{?Θ}\text{>0}$ in $\text{Ω}$. It is established that there exists a constant $\text{C(}\text{Θ}\text{)}$ with the following property: for each deformation $\text{Φ}\text{∈H}{}^1\text{(Ω;}\text{R}{}^{\mathrm{n}}\text{)}$ satisfying $\text{det}\text{?Φ}\text{>0}$ a.e. in $\text{Ω}$, there exist an n×n rotation matrix $\text{R}\text{=}\text{R}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ and a vector $\text{b}\text{=}\text{b}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ in $\text{R}{}^{\mathrm{n}}$ such that

The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Remarks on a lemma by Jacques-Louis Lions

Let Ω   be a bounded and connected open subset of R^N${\mathbb{R}}^N$ with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω  . It is shown in this Note that a fundamental characterization of the space L²(Ω)$L^2(\Omega )$ due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.     

Recovery of Immersions from Their Metric Tensors and Nonlinear Korn Inequalities: A Brief Survey

The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω  R~n→ E~n with prescribed metric tensor field(g ij) : Ω→ S~n, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.     

A Method for Determining Model‐Structure Errors and for Locating Damage in Vibrating Systems

A method is proposed for the determination of natural frequencies and mode shapes of a system which is constrained so that unknown stiffnesses are replaced by rigid connections. The constraint is not imposed physically but only in mathematics so that the behaviour of the constrained system is inferred from the unconstrained measurements. Since stiffnesses which are made rigid cannot experience any elastic strain they can have no effect on the inferred measurements. A procedure for comparing the inferred measurements with similarly constrained finite element predictions can be used to determine modelstructure errors. Damage, such as a crack in a beam, can be located by comparing the inferred measurements from the structure in its undamaged and current states. It is demonstrated how unmeasured rotations may be constrained by using rigidbody modes and a reduction/expansion transformation from a finite element model.Sommario. Viene proposto un metodo per la determinazione delle frequenze proprie e dei modi di vibrazione di un sistema vincolato in modo tale che alcuni elementi elastici siano sostituiti da collegamenti rigidi. Il vincolo non viene imposto fisicamente, ma solo matematicamente, e pertanto il comportamento del sistema vincolato viene dedotto dalle misure sul sistema non vincolato. Poiché gli elementi che sono resi rigidi non possono subire alcuna deformazione elastica, essi non hanno certamente alcun effetto sulle misure dedotte per il sistema vincolato. Una procedura che mette a confronto le misure dedotte per il sistema vincolato con le previsioni fornite da un modello ad elementi finiti con analoghi vincoli, può essere utilizzata per determinare errori nella struttura del modello. Danni del tipo di una cricca su una trave possono essere localizzati confrontando le misure dedotte – per sistemi analogamente vincolati – da quelle effettuate sulla struttura non danneggiata e sulla struttura danneggiata. Si dimostra come si possono imporre vincoli sulle rotazioni (non misurate) utilizzando i modi di corpo rigido dell'elemento e una tecnica di riduzione/espansione dei gradi di libertà di un modello ad elementi finiti.     

Ramanujan’s estimate for the gamma function via monotonicity arguments

The aim of this paper is to extend and refine an approximation formula of the gamma function by Ramanujan.