(2+)-replication and the Baby Monster

The definitions of replicable and completely replicable functions are intimately related to the Hecke operators for the modular group. We define the notions of "$(2+)$-replicable" and "completely $(2+)$-replicable" functions by considering the Hecke operators for $\Gamma_0(2)^+$. We prove that the McKay-Thompson series for $2\cdot\mathbb{B}$, as computed by H\"ohn, are completely $(2+)$-replicable

The interpretation and use of numerically-quantified expressions

This thesis presents a novel pragmatic account of the meaning and use of numerically-quantified expressions. It can readily be seen that quantities can typically be described by many semantically truthful expressions – for instance, if "more than 12" is true of a quantity, so is "more than 11", "more than 10", and so on. It is also intuitively clear that some of these expressions are more suitable than others in a given situation, a preference which is not captured by the semantics but appears to rely upon on wider-ranging considerations of communicative effectiveness. Motivated by these observations, I lay out a set of criteria that are demonstrably relevant to the speaker's choice of utterance in such cases. Observing further that it is typically impossible to satisfy all these criteria with a single utterance, I suggest that the speaker's choice of utterance can be construed as a problem of multiple constraint satisfaction. Using the formalism of Optimality Theory (Prince and Smolensky 1993), I proceed to specify a model of speaker behaviour for this domain of usage. The model I propose can be used to draw predictions both about the speaker's choice of utterance and the hearer's interpretation of utterances. I discuss the relation between these two aspects of the model, showing how constraints on the speaker's choice of utterance are predicted to make pragmatic enrichments available to the hearer. I then consider applications of this idea to specific issues that have been discussed in the literature. Firstly, with respect to superlative quantifiers, I show how this model provides an alternative account to that of Geurts and Nouwen (2007), building upon that offered by Cummins and Katsos (2010), and I present empirical evidence in its favour. Secondly, I show how this model yields the novel prediction that comparative quantifiers give rise to implicatures that are conditioned both by granularity and by prior mention of the numeral, and demonstrate these implicatures empirically. Finally I discuss the predictions that the model makes about the frequency of quantifiers in corpora, and investigate their validity. I conclude that the model presented here proves its worth as a source of hypotheses about speaker and hearer behaviour in the numerical domain. In particular, it serves as a way to integrate insights from distinct domains of enquiry including psycholinguistics, theoretical semantics and numerical cognition. I discuss the claim of this model to psychological plausibility, its relation to existing approaches, and its potential utility when applied to broader domains of language use.This work was supported by a University of Cambridge (Trinity College) Domestic Research Studentship