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Abstract. The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that all deterministic discrete Schrödinger operators which act on a random tree are almost surely selfadjoint. Furthermore, we provide several criteria of essential self-adjointness. We also address some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.

We develop a Glivenko-Cantelli theory for monotone, almost additive functions of i. i. d. sequences of random variables indexed by Z d . Under certain conditions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We apply the results to yield uniform convergence for several quantities arising naturally in statistical physics.

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Retargetable C compilers are nowadays widely used to quickly obtain compiler support for new embedded processors and to perform early processor architecture exploration. One frequent concern about retargetable compilers, though, is their lack of machine-specific code optimization techniques in order to achieve highest code quality. While this problem is partially inherent to the retargetable compilation approach, it can be circumvented by designing flexible, configurable code optimization techniques that apply to a certain range of target architectures. This paper focuses on target machines with SIMD instruction support which is widespread in embedded processors for multimedia applications. We present an efficient and quickly retargetable SIMD code optimization technique that is integrated into an industrial retargetable C compiler. Experimental results for the Philips Trimedia processor demonstrate that the proposed technique applies to real-life target machines and that it produces code quality improvements close to the theoretical limit.