The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.
Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.
We consider circulant graphs having vertices, with prime. To any such graph we associate a certain number , that we call type of the graph. We prove that for the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.
We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis in ,...
*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important...
We tackle R. V. Kadison’s similarity problem (i.e. any bounded representation of any unital C*-algebra is similar to a *-representation), paying attention to the class of C*-unitarisable groups (those groups G for which the full group C*-algebra C*(G) satisfies Kadison’s problem) and thereby we establish some stability results for Kadison’s problem. Namely, we prove that inherits the similarity problem from those of the C*-algebras A and B, provided B is also nuclear. Then we prove that G/Γ is...