Sur les a Algèbres Préhilbertiennes Vérifiant ∥a 2∥ ≤ ∥a∥2

Résumé.  Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ ≤ ∥a∥² pour tout . Alors A est de dimension finie dans chacun des quatre cas suivants :

Quantum superalgebra osp(2∥1)

A quantum deformation of the simplest rank-one orthosymplectic Lie superalgebra is constructed in two ways: by changing the anticommutatoin relations of the odd generators [V₊, V_–] ~ shH, and by using a constant triangular R-matrix which is an appropriate limit of the R-matrix of the supersymmetric sine-Gordon model. A relationship between these two approaches is established. The matrix elements of the odd generators and an analog of the Casimir operator are found for finite-dimensional representations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 95–106, 1988.     

On a strengthened extremal property of the Carathéodory-Fejér functions

Для заданной на едини чной окружности огра ниченной функцииω(ξ) рассматр ивается усложненная задача а ппроксимации аналит ическими функциями: $$\mathop {\inf }\limits_{\varphi \in H^\infty } \left[ {\left\| {\omega - \varphi } \right\| + \mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k \left| {\lambda _k } \right|} \right],$$ где ∥·∥ понимается вL ^∞,ε _k ≧0 — заданные чис ла, $$\mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k< + \infty ,\varphi (z) = \mathop \Sigma \limits_{k = 0}^\infty \lambda _k z^k .$$ Доказывается, что при всех достаточно малы хε _k экстремальной в этой задаче будет функция обычного наилучшего приближения (та же, что и приε _k =0,k=0, 1, ...). В частности, при $$\omega (\zeta ) = \frac{{\gamma _0 }}{{\zeta ^n }} + \frac{{\gamma _1 }}{{\zeta ^{n - 1} }} + ... + \frac{{\gamma _{n - 1} }}{\zeta }$$ экстремальной оказы вается дробь Каратео дори—Фейера. Переход к двойственн ой задаче позволяет получить т очные оценки для клас са интегралов типа Коши, выделяемого огранич ениями, наложенными на велич ины коэффициентов ря да Тейлора.     

The Intersection of a Triangular Bézier Patch and a Plane

In this paper, the problem of finding the intersection of a triangular Bézier patch and a plane is studied. For the degree that one frequently encounters in practice, i.e. $n=2,3$, an efficient and reliable algorithm is obtained, and computational steps are presented.     

An L 2-theory for a class of SPDEs driven by Lévy processes

In this paper we present an L 2-theory for a class of stochastic partial differential equations driven by Lévy processes.The coefficients of the equations are random functions depending on time and space variables,and no smoothness assumption of the coefficients is assumed.     

Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires

Summary The theoretical framework of this study is presented in Sect. 1, with a review of practical numerical methods. The linear operatorT and its approximationT _n are defined in the same Banach space, which is a very common situation. The notion of strong stability forT _n is essential and cannot be weakened without introducing a numerical instability [2]. IfT (or its inverse) is compact, most numerical methods are strongly stable. Without compactness forT(T ^–1) they may not be strongly stable [20].In Sect. 2 we establish error bounds valid in the general setting of a strongly stable approximation of a closedT. This is a generalization of Vainikko [24, 25] (compact approximation). Osborn [19] (uniform and collectivity compact approximation) and Chatelin and Lemordant [6] (strong approximation), based on the equivalence between the eigenvalues convergence with preservation of multiplicities and the collectively compact convergence of spectral projections. It can be summarized in the following way: , eigenvalue ofT of multiplicitym is approximated bym numbers, _n is their arithmetic mean.- _n and the gap between invariant subspaces are of order _n =(T-T _n)P. IfT _n ^* converges toT ^*, pointwise inX ^*, the principal term in the error on - _n is . And for projection methods, withT _n= _n T, we get the bound . It applies to the finite element method for a differential operator with a noncompact resolvent. Aposteriori error bounds are given, and thegeneralized Rayleigh quotient TP _n appears to be an approximation of of the second order, as in the selfadjoint case [12].In Sect. 3, these results are applied to the Galerkin method and its Sloan variant [22], and to approximate quadrature methods. The error bounds and the generalized Rayleigh quotient are numerically tested in Sect. 4.

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Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires

     

Branching Processes in a Lévy Random Environment

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.     

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bézier curve, usually related to an explicit function,is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized monotonicity, called direction monotonicity, is introduced for rational Bézier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine coordinate systems, and it includes the traditional monotonicity as a subcase. By means of it,proper affine coordinate systems may be chosen to make some rational Bézier curves monotonic.Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.     

The Jacobi Field of a Lévy Process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.     

On a homoclinic origin of Hénon-like maps

We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.