Sir Henry Thomas De la Beche and the founding of the British Geological Survey

The founding of the Geological Survey by Henry De la Beche in 1835 is a key event in the history of British geology. Yet the Survey’s initiation actually began three years earlier when De la Beche secured financial assistance from the Board of Ordnance to map the geology of Devon at a scale of one inch to the mile. The British Geological Survey has thus been in existence for at least 175 years and can justly claim to be the world’s oldest continuously functioning geological survey organisation. There were early government-funded geological surveys also in France, the United States, Ireland and Scotland. De la Beche’s notable success both in launching and sustaining the Geological Survey demanded a good deal of diplomacy, determination and deviousness! Even so, the Survey was nearly brought to an untimely end in 1837 when De la Beche was publicly criticised for his interpretation, based on lithology and field relations, of the difficult Culm strata of north Devon. The resolution of the ‘Devonian Controversy’ led to a fundamental change in geological practice, in which the value of fossils as stratigraphic markers, founded on an acceptance of organic change over time, was established beyond question. Fortunately the Survey survived its early trauma and De la Beche went on to extend his influence with the expansion of the Museum of Economic Geology (also formed in 1835), and the establishment of the Mining Record Office and the School of Mines

Structure of measures in Lipschitz differentiability spaces

We prove the equivalence of two seemingly very di erent ways of generalising Rademacher's theorem to metric measure spaces. The rst was introduced by Cheeger and is based upon di erentiation with respect to another, xed, chart func- tion. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very rich structure of Lipschitz curves, analogous to the di erentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz di erentiability spaces

Cheeger's differentiation theorem via the multilinear Kakeya inequality

Suppose that $(X,d,\mu)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincar\'e inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbourhoods of Lipschitz graphs to show that any non-trivial measure with $n$ independent Alberti representations has Hausdorff dimension at least $n$.Comment: 14 page

Modelling circumstellar discs with 3D radiation hydrodynamics

We present results from combining a grid-based radiative transfer code with a Smoothed Particle Hydrodynamics code to produce a flexible system for modelling radiation hydrodynamics. We use a benchmark model of a circumstellar disc to determine a robust method for constructing a gridded density distribution from SPH particles. The benchmark disc is then used to determine the accuracy of the radiative transfer results. We find that the SED and the temperature distribution within the disc are sensitive to the representation of the disc inner edge, which depends critically on both the grid and SPH resolution. The code is then used to model a circumstellar disc around a T-Tauri star. As the disc adjusts towards equilibrium vertical motions in the disc are induced resulting in scale height enhancements which intercept radiation from the central star. Vertical transport of radiation enables these perturbations to influence the mid-plane temperature of the disc. The vertical motions decay over time and the disc ultimately reaches a state of simultaneous hydrostatic and radiative equilibrium.Comment: MNRAS accepted; 15 pages; 17 figures, 4 in colou

Structure of measures in Lipschitz differentiability spaces

We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces

Andrew Templeman and Strata Smith

About geologist and bibliophile Andrew Templeman and his unfortunate death from carbon dioxide poisoning

The Besicovitch Federer projection theorem is false in every infinite dimensional Banach space

We construct a purely unrectifiable set of finite H 1-measure in every infinite-dimensional separable Banach space X whose image under every 0 ≠ x* ∈ X* has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch–Federer projection theorem in infinitedimensional Banach spaces