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.

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.

Reasons, impossibility and efficient steps: reply to Heuer

Ulrike Heuer argues that there can be a reason for a person to perform an action that this person cannot perform, as long as this person can take efficient steps towards performing this action. In this reply, I first argue that Heuer’s examples fail to undermine my claim that there cannot be a reason for a person to perform an action if it is impossible that this person will perform this action. I then argue that, on a plausible interpretation of what ‘efficient steps’ are, Heuer’s claim is consistent with my claim. I end by showing that Heuer fails to undermine the arguments I gave for my claim.