Peter Nickolas

On the topology of free paratopological groups

Abstract Let FP ( X ) be the free paratopological group on a topological space X. For n ∈ N , denote by FP n ( X ) the subset of FP ( X ) consisting of all words of reduced length at most n, and by i n the natural mapping from ( X ⊕ X − 1 ⊕ { e } ) n to FP n ( X ) . In this paper a neighbourhood base at the identity e in FP 2 ( X ) is found. A number of characterisations are then given of the circumstances under which the natural mapping i 2 : ( X ⊕ X d − 1 ⊕ { e } ) 2 → FP 2 ( X ) is a quotient mapping, where X is a T 1 space and X d − 1 denotes the set X − 1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T 1 space. Several specific spaces and classes of spaces are also examined. For example, i 2 is a quotient mapping for every countable subspace of R , i 2 is not a quotient mapping for any uncountable compact subspace of R , and it is undecidable in ZFC whether an uncountable subspace of R exists for which i 2 is a quotient mapping.

Uniformities and uniformly continuous functions on locally connected groups

We show that the left and the right uniformities on a locally connected topological group G coincide if and only if every left uniformly continuous real-valued function on G is right uniformly continuous.

The Qu-Prolog unification algorithm: formalisation and correctness

Abstract Qu-Prolog is an extension of Prolog which performs meta-level computations over object languages, such as predicate calculi and λ-calculi, which have object-level variables, and quantifier or binding symbols creating local scopes for those variables. As in Prolog, the instantiable (meta-level) variables of Qu-Prolog range over object-level terms, and in addition other Qu-Prolog syntax denotes the various components of the object-level syntax, including object-level variables. Further, the meta-level operation of substitution into object-level terms is directly represented by appropriate Qu-Prolog syntax. Again as in Prolog, the driving mechanism in Qu-Prolog computation is a form of unification, but this is substantially more complex than for Prolog because of Qu-Prologs greater generality, and especially because substitution operations are evaluated during unification. In this paper, the Qu-Prolog unification algorithm is specified, formalised and proved correct. Further, the analysis of the algorithm is carried out in a framework which straightforwardly allows the ‘completeness’ of the algorithm to be proved: though fully explicit answers to unification problems are not always provided, no information is lost in the unification process.

Breaking and repairing an approximate message authentication scheme

Traditional hash functions are designed to protect against even the slightest modification of a message. Thus, one bit changed in a message would result in a totally different message digest when a hash function is applied. This feature is not suitable for applications whose message spaces admit a certain fuzziness, such as multimedia communications or biometric authentication applications. In these applications, approximate hash functions must be designed so that the distance between messages are proportionally reflected in the distance between message digests. Most of the previous designs of approximate hash functions employ traditional hash functions. In an ingenious approximate message authentication scheme for an N-ary alphabet recently proposed by Ge, Arce and Crescenzo, the approximate hash functions are based on the majority selection function. This scheme is suitable for N-ary messages with arbitrary alphabet size N. In this paper, we show a hidden property of the majority selection function, whi...

LOCAL COMPACTNESS IN FREE TOPOLOGICAL GROUPS

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω iff A2(X) is locally compact iff F2(X) is locally compact iff X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F (X) is locally compact for each n ∈ ω iff F4(X) is locally compact iff Fn(X) has pointwise countable type for each n ∈ ω iff F4(X) has pointwise countable type iff X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω iff A2(X) has pointwise countable type iff F2(X) has pointwise countable type iff there exists a compact set C of ∗The second author wishes to thank the first author, and his department, for hospitality extended during the course of this work, and acknowledges the financial support of a Quality Fund grant from the Institute for Mathematical Modelling and Computational Systems (IMMaCS) at the University of Wollongong. †AMS classification numbers: 22A05, 54H11, 54A25, 54D30, 54D45

Distance geometry in quasihypermetric spaces. II

Let

The completeness of functional logic

(X, d)

On a conjecture of Higgins

be a compact metric space and let

A performance study of FileServer System on the AppleTalk Personal Network

\mathcal{M}(X)

Some Remarks on the Logic of Gong, Needham and Yahalom

denote the space of all finite signed Borel measures on