A Saito-Tomita-Lusin theorem for JB*-triples and applications

A theorem of Lusin is proved in the non-ordered context of JB*-triples. This is applied to obtain versions of a general transitivity theorem and to deduce refinements of facial structure in closed unit ballls of JB*-triples and duals.

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Euler's theorem under the microscope

In this article Geoff Tennant summarises the first half of Imre Lakatos's seminal 1976 book, "Proofs and refutations: the logic of mathematical discovery". Implications are drawn for the classroom treatment of proof.

Untitled (Abstraction of Labour Time/ External Recurrence/Monad)

Commissioned print. Artist of the Month Club: February, 2010. January Curator: Mark Beasley. Invisible Exports Gallery, New York. Archival Inkjet Print on metallic silver polyester, 841 x 643mm. Edition of 50 + 10ap. Subsequently exhibited in the following exhibition: 'A Unicorn Basking in the Light of Three Glowing Suns' The Devos Art Museum School of Art & Design at Northern Michigan University October 8 – November 14, 2010 Curated by Anthony Elms and Philip von Zweck

Untitled (Abstraction of labour time eternal recurrence monad)

An AHRC funded project titled: Picturing ideas? Visualising and Synthesising Ideas as art (2009-10). Outputs including: 4 exhibitions; 4 publications; 3 papers; 2 largescale backlit digital prints; 1 commissioned print. (See Additional Information) ----ABSTRACT: Utilising the virtuality of digital imagery this practice-led project explored the possibility of the cross-articulation between text and image and the bridging or synthesising potential of the visual affect of ideas. A series of digital images were produced 'picturing' or 'visualising' philosophical ideas derived from the writings of the philosopher Giles Deleuze, as remodellings of pre-existing philosophical ideas; developed through dialogues and consultation with specialists in the fields from which the ideas were drawn (philosophy, psychology, film) as well as artists and theorists concerned with ideas of 'mental imagery' and visualisation. Final images were produced as a synthesis (or combination) of these visualisations and presented in the format of large scale, backlit digital prints at a series of prestigious international exhibitions (see details above). Evaluation took the form of a four page illustrated text in Frieze magazine (August 2009) and three papers delivered at University of Ulster, Goldsmiths College of Art and Loughborough University. The project also included the publication of a catalogue essay (EAST 09) and an illustrated poem (in the Dark Monarch publication). A print version of the image was commissioned by Invisible Exports Gallery, New York and subsequently exhibited in The Devos Art Museum, School of Art & Design at Northern Michigan University and in a publication edited by Cedar Lewisohn for Tate Publishing. The project was funded by an AHRC practice-led grant (17K) and Arts Council of England award (1.5K). The outputs, including high profile, publicly accessible exhibitions, prestigious publications and conference papers ensured the dissemination of the research to a wide range of audiences, including scholars/researchers across the arts and humanities engaged in practice-based and interdisciplinary theoretical work (in particular in the fields of contemporary art and art theory and those working on the integration of art and theory/philosophy/psychology) but also the wider audience for contemporary art.

Thermodynamic derivation of the fluctuation theorem and Jarzynski equality

A thermodynamic expression for the analog of the canonical ensemble for nonequilibrium systems is described based on a purely information theoretical interpretation of entropy. It is shown that this nonequilibrium canonical distribution implies some important results from nonequilibrium thermodynamics, specifically, the fluctuation theorem and the Jarzynski equality. Those results are therefore expected to be more widely applicable, for example, to macroscopic systems.

Crooks' fluctuation theorem for a process on a two dimensional fluid field

We investigate the behavior of a two-dimensional inviscid and incompressible flow when pushed out of dynamical equilibrium. We use the two-dimensional vorticity equation with spectral truncation on a rectangular domain. For a sufficiently large number of degrees of freedom, the equilibrium statistics of the flow can be described through a canonical ensemble with two conserved quantities, energy and enstrophy. To perturb the system out of equilibrium, we change the shape of the domain according to a protocol, which changes the kinetic energy but leaves the enstrophy constant. We interpret this as doing work to the system. Evolving along a forward and its corresponding backward process, we find numerical evidence that the distributions of the work performed satisfy the Crooks relation. We confirm our results by proving the Crooks relation for this system rigorously.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.