Warburton, N. (2012). Vprasanje umetnosti [The art question]. Afterword study by Marjan Simenc. Ljubljana: Sophia. [Book review]

Book review of: Warburton, N. (2012). Vprasanje umetnosti [The art question]. Afterword study by Marjan Simenc. Ljubljana: Sophia. 158 p., ISBN 978-961-6768-48-1

Continuity of the Peierls barrier and robustness of laminations

We study the Peierls barrier for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start with the case of a fixed local potential and derive an estimate for the difference of the periodic Peierls barrier and the Peierls barrier of a general rotation number in a given point. A similar estimate was obtained by Mather in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that the Peierls barrier is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers for a given rotation number, is open in the C1-topology.Comment: 20 pages, submitted to Ergodic Theory and Dynamical System

Jordan triple product homomorphisms on Hermitian matrices to and from dimension one

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$\Phi(ABA) = \Phi(A)\Phi(B)\Phi(A)$$ from the set of all Hermitian $n \times n$ complex matrices to the field of complex numbers. Further we characterise all Jordan triple product homomorphisms from the field of complex or real numbers or the set of all nonnegative real numbers to the set of all Hermitian $n \times n$ complex matrices.Comment: 10 page

Jordan triple product homomorphisms on Hermitian matrices of dimension two

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$\Phi(ABA) = \Phi(A)\Phi(B)\Phi(A)$$ on the set of all Hermitian $2 \times 2$ complex matrices.Comment: 34 page