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We study an abstract second-order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying a numerical scheme based on time discretization. We show that the sequence of approximate solutions converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.
In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.
This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a Noetherian ring. Their definition is in terms of maps to free modules, and we give an intrinsic definition using divided powers.
For a von Neumann subalgebra and any two elements a, b ∈ A with a normal, such that the corresponding derivations da and db satisfy the condition for all x ∈ A, there exists a completely bounded (a)ʹ-bimodule map such that . (In particular, db(A) ⊆ da(A).) Moreover, if A is a factor, then can be taken to be normal and these equalities hold on instead of just on A. This result is not true for general (even primitive) C*-algebras A.