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Nouvelles formulations intégrales pour les problèmes de diffraction d'ondes

David P. Levadoux, Bastiaan L. Michielsen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all...

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

We prove that on ℝ N , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if ℝ N has an n-dimensional subspace whose orbit under T ∈ ( ℝ N ) is dense in ℝ N , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ∈ ( ℝ N ) is strongly n-supercyclic if ℝ N has an n-dimensional subspace whose orbit under T is dense in ℙ ₙ ( ℝ N ) , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite...

n-supercyclic operators

Nathan S. Feldman (2002)

Studia Mathematica

We show that there are linear operators on Hilbert space that have n-dimensional subspaces with dense orbit, but no (n-1)-dimensional subspaces with dense orbit. This leads to a new class of operators, called the n-supercyclic operators. We show that many cohyponormal operators are n-supercyclic. Furthermore, we prove that for an n-supercyclic operator, there are n circles centered at the origin such that every component of the spectrum must intersect one of these circles.

Null spaces and ranges of polynomials of operators.

Manuel González (1988)

Publicacions Matemàtiques

We give an elementary proof of the fact that given two polynomials P, Q without common zeros and a linear operator A, the operators P(A) and Q(A) verify some properties equivalent to the pair (P(A),Q(A)) being non-singular in the sense of J.L. Taylor. From these properties we derive expressions for the range and null space of P(A) and spectral mapping theorems for polynomials fo continuous (or closed) operators in Banach spaces.

Numerical considerations of a hybrid proximal projection algorithm for solving variational inequalities

Christina Jager (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, some ideas for the numerical realization of the hybrid proximal projection algorithm from Solodov and Svaiter [22] are presented. An example is given which shows that this hybrid algorithm does not generate a Fejér-monotone sequence. Further, a strategy is suggested for the computation of inexact solutions of the auxiliary problems with a certain tolerance. For that purpose, ε-subdifferentials of the auxiliary functions and the bundle trust region method from Schramm and Zowe [20]...

Numerical index of vector-valued function spaces

Miguel Martín, Rafael Payá (2000)

Studia Mathematica

We show that the numerical index of a c 0 -, l 1 -, or l ∞ -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and L 1 ( μ , X ) (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.

Numerical index with respect to an operator

Mohammad Ali Ardalani (2014)

Studia Mathematica

We introduce new concepts of numerical range and numerical radius of one operator with respect to another one, which generalize in a natural way the known concepts of numerical range and numerical radius. We study basic properties of these new concepts and present some examples.

Numerical precision for differential inclusions with uniqueness

Jérôme Bastien, Michelle Schatzman (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is 1 / 2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl’s rheological model, our estimates in maximum norm do not depend on spatial dimension.

Numerical precision for differential inclusions with uniqueness

Jérôme Bastien, Michelle Schatzman (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is 1/2 in general and 1 when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl's rheological model, our estimates in maximum norm do not depend on spatial dimension. ...

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