A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find some finiteness theorems which ensure 2-universality of Hermitian lattices over several imaginary quadratic number fields.

We show that a Dicke-type non-Hermitian Hamiltonian admits entirely real spectra by mapping it to the "dressed Dicke model" through a similarity transformation. We find a positive-definite metric in the Hilbert space of the non-Hermitian Hamiltonian so that the time evolution is unitary and allows a consistent quantum description. We then show that this non-Hermitian Hamiltonian describing nond...

In this dissertation we firstly present a brief review of the Standard Model of particle physics, focussing on aspects of the model that prove most relevant in the context of explaining the observed oscillation of neutrinos. The phenomenology of neutrino oscillations is then introduced, and different possible extensions to the Standard Model that would allow for non-zero neutrino masses, thereb...

We introduce the signals found in radar systems, and how the information they contain about targets can be used for fuzing. We then describe a new wavelet-based model for the random radar reflections from the ocean surface, which are a major source of noise in naval operations. Next we show ways of improving the performance of wavelet denoising when the corrupting noise is fractal. We then desc...

A recent development in studies of random non-Hermitian quantum systems is reviewed. Delocalization was found to occur under a sufficiently large constant imaginary vector potential even in one and two dimensions. The phenomenon has a physical realization as flux-line depinning in type-II superconductors. Relations between the delocalization transition and the complex energy spectrum of the non...

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient...

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient...

We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (...