Conformal nets I: coordinate-free nets

We describe a coordinate-free perspective on conformal nets, as functors from intervals to von Neumann algebras. We discuss an operation of fusion of intervals and observe that a conformal net takes a fused interval to the fiber product of von Neumann algebras. Though coordinate-free nets do not a priori have vacuum sectors, we show that there is a vacuum sector canonically associated to any circle equipped with a conformal structure. This is the first in a series of papers constructing a 3-category of conformal nets, defects, sectors, and intertwiners.Comment: Updated to published versio

Conformal nets II: conformal blocks

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the `bundle of conformal blocks', a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.Comment: Updated to published versio

Observations extending over three years on the amount of carbonic acid in the ground air in relation to the earth temperature at the depth of three feet with charts

The connection between certain conditions of soil and the incidence of some infectious diseases has long been recognised by Physicians but the way in which these conditions operate has been variously explained.This paper shows a correspondence between the maximum of carbonic acid and the maximum reading of the earth thermometer at three feet. Late summer, when the temperature of the soil at three feet rose to the maximum,showed also the greatest amount of carbonic acid. Throughout the entire year much rain always produced a marked temporary increase in the amount of carbonic acid. These facts probably find their explanation in the increased bacterial life called into renewed activity by the heat and moisture.Moisture, as after much rain, may also act by mechanically sealing up the pores of the soil and thus preventing the normal escape of carbonic acid into the general atmosphere. The highest reading of the earth thermometer was towards the end of August when it ranged in the neighbourhood of 56° Fahr.Edinburgh is not what is known as a "Diarrhoea Town" that is, one in which that disease plays havoc with the infantile population in late summer and early autumn, but still, it is observed that most cases of infantile diarrhoea do occur at the time corresponding with the maximum reading of the earth thermometer at three feet, as is shown by reference to the Report of the Medical Officer of Health, Edinburgh 1898.The diurnal variation in the amount of carbonic acid showed an evening excess over the morning as the year approached near to the summer solstice and beyond it to the maximum reading of the year which occurred in the last days of August. After this period a morning excess over the evening was observed as the year approached near to the winter solstice and beyond it to the minimum reading of the year which took place in the end of March. During March, April and May, and again in September, October and November, the morning and evening variations were but little marked and temporary conditions of weather probably produced them. The year may be divided as regards the morning and evening variations in carbonic acid into four periods, September to December, December to March, March to June and June to September. In the first and third of these periods the morning reading is in excess,and in the second and fourth the evening reading.Any markedly high reading was always observed to follow a high rainfall. Excessively low readings on the other hand were observed after prolonged dry weather. A temporary increase after rain was constantly observed.The remarkable rise in the amount of carbonic acid in February and March of 1895 during the period of prolonged frost, and when snow lay on the ground for about a month, is not easy of explanation. The idea suggested itself that perhaps this was due to an increase of temperature in the soil caused by the covering of snow preventing radiation of heat from the upper layers of the soil. From observations made in February 1900 I found that such a covering of snow had no effect on the ground temperature thermometer whatever. The increase in carbonic acid may be explained to some extent by the sealing up of the surface of the soil thus preventing the natural escape of carbonic acid into the atmosphere. It was not observed that severe frost, apart from a covering of snow on the ground, was followed by increase in the amount of carbonic acid. In each of the three years the highest reading was observed towards the end of August.The remarkably low reading in October 1896 and again in August 1897 followed on a period of drought. No observation was taken from February to June 1897.Much rain was invariably followed by a marked increase in the amount of carbonic acid.No difference between the morning reading and the evening reading of the earth temperature at three feet was ever observed. The observations of the ground thermometer readings were taken on three alternate days of each week. The annual range of the temperature of the earth thermometer at three feet was about 20° Fahr. The lowest reading was observed to be in the month of February when it was in the neighbourhood of 36° Fahr. and the highest reading in the month of August when it was in the neighbourhood of 56° Fahr.There is observed to be a close correspondence between the annual curve of the earth thermometer and the annual curve of the amount of carbonic acid in the soil, that is, when the earth thermometer reading falls to the lowest annual point or rises to its highest annual point the amount of carbonic acid in the soil similarly approaches its lowest and highest points.It is further to be observed by reference to the Reports of the Medical Officer of Health, Edinburgh, that there was a close correspondence between the greater number of fatal cases of Infantile Diarrhoea and the highest reading of the earth thermometer at three feet

The Effects of Interaction with Conserving Adults and Peers on the Acquisition of Conservation by Nonconservers

Nonconserving children were placed in a situation where they had to interact with either two conserving adults or two conserving children. Each triad (one nonconserver and two conservers) was asked to give judgments and explanations for conservation problems. The type of conserving explanations (invariant quantity, reversibility, and compensation) given by the conservers in the interaction were varied. Approximately one week after the interaction the nonconserving subjects were posttested and the results indicated that nonconservers increased in conservation score after interacting with conservers. Hearing different explanations did not differentially affect the scores of nonconservers. The results did indicate that invariant quantity explanations were used more often than either reversibility or compensation explanations by the subjects after the interaction. There were no differential effects when interacting with either adult or peer conservers. These results were discussed in terms of Piagetian Theory. The ramifications of these findings on our educational systems were discussed as were suggestions for future research