An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Computer Theorem Proving and HoTT

Theorem-proving is a one-player game. The history of computer programs being the players goes back to 1956 and the ‘LT’ LOGIC THEORY MACHINE of Newell, Shaw and Simon. In game-playing terms, the ‘initial position’ is the core set of axioms chosen for the particular logic and the ‘moves’ are the rules of inference. Now, the Univalent Foundations Program at IAS Princeton and the resulting ‘HoTT’ book on Homotopy Type Theory have demonstrated the success of a new kind of experimental mathematics using computer theorem proving.

Parallel editing, multi-positionality and maximalism: cosmopolitan effects as explored in some art works by Melanie Jackson and Vivienne Dick

Garfield produces a critique of neo-minimalist art practice by demonstrating how the artist Melanie Jackson’s Some things you are not allowed to send around the world (2003 and 2006) and the experimental film-maker Vivienne Dick’s Liberty’s booty (1980) – neither of which can be said to be about feeling ‘at home’ in the world, be it as a resident or as a nomad – examine global humanity through multi-positionality, excess and contingency, and thereby begin to articulate a new cosmopolitan relationship with the local – or, rather, with many different localities – in one and the same maximalist sweep of the work. ‘Maximalism’ in Garfield’s coinage signifies an excessive overloading (through editing, collage, and the sheer density of the range of the material) that enables the viewer to insert themselves into the narrative of the work. In the art of both Jackson and Dick Garfield detects a refusal to know or to judge the world; instead, there is an attempt to incorporate the complexities of its full range into the singular vision of the work, challenging the viewer to identify what is at stake.

A proof of van Aubel's theorem using orthogonal vectors

We show how two linearly independent vectors can be used to construct two orthogonal vectors of equal magnitude in a simple way. The proof that the constructed vectors are orthogonal and of equal magnitude is a good exercise for students studying properties of scalar and vector triple products. We then show how this result can be used to prove van Aubel's theorem that relates the two line segments joining the centres of squares on opposite sides of a plane quadrilateral.