All grown up? The fate after 15 years of a quarter of a million UK firms born in 1998

The theory of firm growth is in a rather unsatisfactory state. However, the analysis of large firm-level datasets which have become available in recent years allows us to begin building an evidence base which can, in turn, be used to underpin the development of more satisfactory theory. Here we study the 239 thousand UK private sector firms born in 1998 over their first 15 years of life. A first, and quite striking, finding is the extraordinary force of mortality. By age 15, 90% of the UK firms born in 1998 are dead, and, for those surviving to age 15, the hazard of death is still about 10% a year. The chance of death is related to the size and growth of firms in an interesting way. Whilst the hazard rate after 15 years is largely independent of size at birth, it is strongly affected by the current (age 14) size. In particular, firms with more than five employees are half as likely to die in the next year as firms with less than five employees. A second important finding is that most firms, even those which survive to age 15, do not grow very much. By age 15 more than half the 26,000 survivors still have less than five jobs. In other words, the growth paths – what we call the ‘growth trajectories’ – of most of the 26,000 survivors are pretty flat. However, of the firms that do grow, firms born smaller grow faster than those born larger. Another striking finding is that growth is heavily concentrated in the first five years. Whilst growth does continue, even up to age 15, each year after age five it involves only a relatively small proportion of firms. Finally, there are two groups of survivors which contribute importantly to job creation. Some are those born relatively large (with more than 20 jobs) although their growth rate is quite modest. More striking though, is a very small group of firms born very small with less than five jobs (about 5% of all survivors) which contribute a substantial proportion (more than one third) of the jobs added to the cohort total by age 15

Justifying definitions in mathematics: going beyond Lakatos

This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world justification, condition justification, and redundancy justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show how various kinds of justification can be found and can be reasonable, and they fail to acknowledge the interplay among the different kinds of justification

Evidence for the deterministic or the indeterministic description? A critique of the literature about classical dynamical systems

It can be shown that certain kinds of classical deterministic and indeterministic descriptions are observationally equivalent. Then the question arises: which description is preferable relative to evidence? This paper looks at the main argument in the literature for the deterministic description by Winnie (The cosmos of science-essays of exploration. Pittsburgh University Press, Pittsburgh, pp 299-324, 1998). It is shown that this argument yields the desired conclusion relative to in principle possible observations where there are no limits, in principle, on observational accuracy. Yet relative to the currently possible observations (of relevance in practice), relative to the actual observations, or relative to in principle observations where there are limits, in principle, on observational accuracy the argument fails. Then Winnie's analogy between his argument for the deterministic description and his argument against the prevalence of Bernoulli randomness in deterministic descriptions is considered. It is argued that while the arguments are indeed analogous, it is also important to see they are disanalogous in another sense

Ehrenfest and Ehrenfest-Afanassjewa on why Boltzmannian and Gibbsian calculations agree

The relation between the Boltzmannian and the Gibbsian formulations of statistical mechanics is one of the major conceptual issues in the foundations of the discipline. In their celebrated review of statistical mechanics, Paul Ehrenfest and Tatiana Ehrenfest-Afanassjewa discuss this issue and offer an argument for the conclusion that Boltzmannian equilibrium values agree with Gibbsian phase averages. In this paper, we analyse their argument, which is still important today, and point out that its scope is limited to dilute gases

Cardiodynamic Complexity: Electrocardiographic Characterization of Arrhythmic Foci

International audienceThe electrical activity of the heart consists of nonlinear interactions emerging as a complex system. As such, proper characterization of arrhythmias and arrhythmogenic areas requires nonlinear analysis methods. Electrocardiographic imaging provides a full spatiotemporal picture of the electric potential on the human epicardium. Rhythm reflects the connection topology of the pacemaker cells driving it. Hence, characterizing the attractors as nonlinear, effective dynamics can capture the key parameters without imposing any particular microscopic model on the empirical signals. A dynamic phase-space reconstruction from an appropriate embedding can be made robust and numerically stable with the presented method. We show that both the phase-space descriptors and those of the a priori unrelated singularity analysis are able to highlight the arrhythmogenic areas on cases of atrial fibrillation