Random Digit Representation of Integers

Modular exponentiation is core to today\u27s main stream public key cryptographic systems. In this article, we generalize the classical fractional $w$NAF method for modular exponentiation -- the classical method uses a digit set of the form $\{1,3,\dots,m\}$ which is extended here to any set of odd integers of the form $\{1,d_2,\dots, d_n\}$. We give a formula for the average density of non-zero terms in this new representation and discuss its asymptotic behavior when those digits are randomly chosen from a given set. We also propose a specific method for the precomputation phase of the exponentiation algorithm

The Supervision of Speech Production

Introduction Since the emergence of phonology as a component within the generative paradigm of linguistics there has been discussion concerning how it relates to phonetics. At least two views of the relationship are possible: € Phonology describes the same data as phonetics, but whereas phonetics models physical detail phonology models abstract relationships holding between units within the data. These units are defined within the general framework of linguistic theory. € Phonology stands logically prior to phonetics, and the output of its processes comprises the input to phonetics. Phonetics itself is a processing component producing an output. The model can be regarded as static, involving no temporal relationship (the usual view in linguistics), or it can be regarded as dynamic. The dynamic view is often taken when the theory is used to support work in an allied area, for example in speech technology, where temporal as well as logical relationships between processes are important. It is usually understood that phonetic processes are of little concern to linguistics, and that by the output of phonology all linguistic processing is complete. Phonetic realisation of phonological 'requirements' is thought of as a passive process involving no cognitive processing, and introducing nothing new of linguistic consequence or interest. For example, phonologists working on language acquisition are interested in the phonetic constraints on what can be acquired by a child, but are not concerned with the detail of phonetic processes. The most extreme form of this position would be that speech processing of a cognitive nature falls within the province of phonology, but that all physical processing falls within the province of phonetics. Since, by definition, language is a cognitive system it can have nothing formal to do with phonetics, except in its trivial realisational rôle. There has recently been a trend toward a more phonetic approach to phonology, 1,2 and a greater awareness among phoneticians of the abstractions ofphonology3 and why they are necessary. This paper will consider how ARTICULATORY PHONOLOGY might handle some phonetic data, and attempt to deal with some problems that the data points up in the theory

Measurement as an ontological scalpel

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