Dark adaptation of foveal cones during the cone-plateau period

Following substantial bleaching by "white" light, absolute threshold, relative spectral sensitivity and sensation of hue of monochromatic lights were measured at the central fovea during the cone-plateau period. The absolute-threshold level was found to increase and then decrease markedly, the relative spectral sensitivity remained invariant, while the sensation of hues of monochromatic lights from the long- and middle-wave regions of the spectrum changed toward hues of shorter wavelengths.     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

Dark-adaptation mechanisms of the long-wave foveal cones

The ordinary long-term rod and cone dark-adaptation curves have generally been assumed to follow a single exponential rate of recovery. However, in two previous papers on rod dark-adaptation (Stabell et al., 1986a, b), the recovery curve was found to consist of three different sections. The results of the present paper show the same type of recovery function with three different sections for the long-term dark-adaptation curve of the long-wave cone system. During the major, middle section log cone threshold, like log rod threshold, is linearly related to the logarithm of the concentration of bleached photopigment. Presupposing that the bleached cone photopigment acts as a ligand, the change in threshold level obtained during the middle section of the dark-adaptation curve is well described by the change in activity rate of an allosteric, postively cooperative enzyme built as a dimer.     

Dark adaptation of the long-wave cones at different eccentricities

Using a Wright colorimeter the ordinary long-term, long-wave cone dark-adaptation curve was measured at 0, 2, 4, 7, 17, 25, 40 and 49 degrees nasally in the visual field. In opposition to previous findings, the results show that the dark-adaptation function of the long-wave cones changes markedly when the test field is moved outward from the rod-free fovea. It is suggested that the kinetics of the long-wave cone photopigment change with eccentricity. Also, at variance with previous findings, the present curves at all eccentricities may reasonably well be interpreted as consisting of three different sections; a first section where the threshold decreases rapidly, followed by a major, approximately linear section and a terminating section that converges asymptotically towards the final level of sensitivity. This finding suggests that the dark-adaptation process of the cone system, under the given experimental conditions, is based on three somewhat different processes.     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.