3,585 research outputs found
Constructive Theory of Banach algebras
We present a way to organize a constructive development of the theory of
Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We
illustrate this by giving elementary proofs of Wiener's result on the inverse
of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we
show how this can be used in a localic, or point-free, description of the
spectrum of a Banach algebra
Reverse-Safe Data Structures for Text Indexing
We introduce the notion of reverse-safe data structures. These are data structures that prevent the reconstruction of the data they encode (i.e., they cannot be easily reversed). A data structure D is called z-reverse-safe when there exist at least z datasets with the same set of answers as the ones stored by D. The main challenge is to ensure that D stores as many answers to useful queries as possible, is constructed efficiently, and has size close to the size of the original dataset it encodes. Given a text of length n and an integer z, we propose an algorithm which constructs a z-reverse-safe data structure that has size O(n) and answers pattern matching queries of length at most d optimally, where d is maximal for any such z-reverse-safe data structure. The construction algorithm takes O(n ω log d) time, where ω is the matrix multiplication exponent. We show that, despite the n ω factor, our engineered implementation takes only a few minutes to finish for million-letter texts. We further show that plugging our method in data analysis applications gives insignificant or no data utility loss. Finally, we show how our technique can be extended to support applications under a realistic adversary model
Two-dimensional arrays with maximal complexity
We present natural bounds for the complexity function of two-dimensional arrays, and we study the shape of the maximal complexity function. Some problems concerning the existence of maximal arrays are discussed
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