Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the noncommutative...

Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. ...

We prove a diagonalisation theorem for the tautological, or generalised Miller-Morita-Mumford classes of compact, smooth, simply-connected definite $4$-manifolds. Our result can be thought as families version Donaldson's theorem. our using Bauer-Furuta cohomotopy refinement Seiberg-Witten theory. use main to deduce various results concerning tautological such In particular, we completely determ...

[26] argues that the diagonal argument of the number theorist Cantor can be used to elucidate issues that arose in the socialist calculation debate of the 1930s. In particular he contends that the diagonal argument buttresses the claims of the Austrian economists regarding the impossibility of rational planning. We challenge Murphy’s argument, both at the number theoretic level and from the sta...

A new method of constructing an infinite set of exact eigenfunctions of 1–d interacting spinless Fermionic systems is derived. Creation and annihilation operators for the interacting system are found and thereby the many–body Hamiltonian is diagonalised. The formalism is applied to several examples. One instance is the theory of Jack polynomials. For the Calogero-Moser-Sutherland Hamiltonian a ...

“Tight binding” has existed for many years as a convenient and transparent model for the description of electronic structure in molecules and solids. It often provides the basis for construction of many body theories such as the Hubbard model and the Anderson impurity model. Slater and Koster call it the tight binding or “Bloch” method and their historic paper provides the systematic procedure ...

We present detailed calculations of low-energy spin dynamics in the " ferric wheel " systems Na:Fe6 and Cs:Fe8 in a magnetic field. We compute by exact diagonalisation the low-energy spectra and matrix elements for total-spin and Néel-vector components, and thus the time-dependent correlation functions of these operators. Comparison of our results with the semiclassical theory of coherent quant...

The optical conductivity of charge carriers coupled to quantum phonons is studied in the framework of the one-dimensional spinless Holstein model. For one electron, variational diagonalisation yields exact results in the thermodynamic limit, whereas at finite carrier density analytical approximations based on previous work on single-particle spectral functions are obtained. Particular emphasis ...

We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at arbitrary filling. Although the system does not possess obvious symmetries except translation invariance, the spectrum presents many multiplets with degeneracies of high order. This behaviour is explained by a hidden symmetry property of the Bethe Ansatz. Combi...