Coloured Letters and Numbers (CLaN): A reliable factor-analysis based synaesthesia questionnaire

Synaesthesia is a heterogeneous phenomenon, even when considering one particular sub-type. The purpose of this study was to design a reliable and valid questionnaire for grapheme-colour synaesthesia that captures this heterogeneity. By the means of a large sample of 628 synaesthetes and a factor analysis, we created the Coloured Letters and Numbers (CLaN) questionnaire with 16 items loading on 4 different factors (i.e., localisation, automaticity/attention, deliberate use, and longitudinal changes). These factors were externally validated with tests which are widely used in the field of synaesthesia research. The questionnaire showed good test–retest reliability and construct validity (i.e., internally and externally). Our findings are discussed in the light of current theories and new ideas in synaesthesia research. More generally, the questionnaire is a useful tool which can be widely used in synaesthesia research to reveal the influence of individual differences on various performance measures and will be useful in generating new hypotheses.

Legendre Polynomials and Angular Momentum

An introduction to Legendre polynomials as precursor to studying angular momentum in quantum chemistry,

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

The American cotton spinner and managers' and carders' guide : a practical treatise on cotton spinning : giving the dimensions and speed of machinery, draught and twist calculations, etc., with notices of recent improvements : together with rules and examples for making changes in the size and numbers of roving and yarn /

Cover title: American cotton-spinner.

Of men and numbers; the story of the great mathematicians

Mode of access: Internet.

Leviticus and Numbers /

Mode of access: Internet.

Tacit collusion, firm asymmetries and numbers:evidence from EC merger cases

This paper estimates the implicit model, especially the roles of size asymmetries and firm numbers, used by the European Commission to identify mergers with coordinated effects. This subset of cases offers an opportunity to shed empirical light on the conditions where a Competition Authority believes tacit collusion is most likely to arise. We find that, for the Commission, tacit collusion is a rare phenomenon, largely confined to markets of two, more or less symmetric, players. This is consistent with recent experimental literature, but contrasts with the facts on ‘hard-core’ collusion in which firm numbers and asymmetries are often much larger.

Tacit vs. overt collusion firm asymmetries and numbers:what's the evidence?

It is conventional wisdom that collusion is more likely the fewer firms there are in a market and the more symmetric they are. This is often theoretically justified in terms of a repeated non-cooperative game. Although that model fits more easily with tacit than overt collusion, the impression sometimes given is that ‘one model fits all’. Moreover, the empirical literature offers few stylized facts on the most simple of questions—how few are few and how symmetric is symmetric? This paper attempts to fill this gap while also exploring the interface of tacit and overt collusion, albeit in an indirect way. First, it identifies the empirical model of tacit collusion that the European Commission appears to have employed in coordinated effects merger cases—apparently only fairly symmetric duopolies fit the bill. Second, it shows that, intriguingly, the same story emerges from the quite different experimental literature on tacit collusion. This offers a stark contrast with the findings for a sample of prosecuted cartels; on average, these involve six members (often more) and size asymmetries among members are often considerable. The indirect nature of this ‘evidence’ cautions against definitive conclusions; nevertheless, the contrast offers little comfort for those who believe that the same model does, more or less, fit all.

Tacit collusion, firm asymmetries and numbers:evidence from EC merger cases

The purpose of this paper is to identify empirically the implicit structural model, especially the roles of size asymmetries and concentration, used by the European Commission to identify mergers with coordinated effects (i.e. collective dominance). Apart from its obvious policy-relevance, the paper is designed to shed empirical light on the conditions under which tacit collusion is most likely. We construct a database relating to 62 candidate mergers and find that, in the eyes of the Commission, tacit collusion in this context virtually never involves more than two firms and requires close symmetry in the market shares of the two firms.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45