### The Spherical Landau Problem

, 2001

"... The magnetization for electrons on a two-dimensional sphere, under a spherically symmetrical normal magnetic field has been studied in the large field limit. This allows us to use an Euclidean approximation for low energies electron states getting an analytical solution for the problem and avoiding ..."

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the difficulties of quantization on a curved manifold. At low temperatures our results are exact and allow direct comparisson with the planar Landau case. In this temperature limit we

*compute*the magnetization and show it exhibit an oscillatory de Hass-Van Alphen type of behaviour.###
Nuclear Physics B *Proceedings* Supplement – preprint (2014) 1–42 Nuclear Physics B *Proceedings* Supplement Universal Aspects of QCD-like TheoriesI

"... In these lectures I review some basic examples of how the concepts of universality and scaling can be used to study aspects of the chiral and the deconfinement transition, if not in QCD directly but in QCD-like theories. As an example for flavor dynamics I discuss a quark-hadron model to describe th ..."

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the phase diagram of two-color QCD with the functional renormalization group. Universal aspects of deconfinement are illustrated mainly in the

*2*+ 1 dimensional SU(N) gauge theories with second order transition where many exact results from spin models can be exploited.### A Precise Error Bound for Quantum Phase Estimation

, 2011

"... Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can be f ..."

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Quantum phase estimation is one of the key algorithms in the field of quantum

*computing*, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can### Authors:

, 1997

"... 1 Dedicated to Ludwig Elsner on the occasion of his sixtieth birthday We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these s ..."

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1 Dedicated to Ludwig Elsner on the occasion of his sixtieth birthday We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transforma-tions. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in in-tersections of these classes and their Schur-like forms. Such multistructered matrices arise in applications from quantum physics and quantum chemistry. 1

### RICE UNIVERSITY Regime Change: Sampling Rate vs. Bit-Depth in Compressive Sensing

, 2011

"... The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by exploiting inherent structure in natural and man-made signals. It has been demon-strated that structured signals can be acquired with just a small number of linear measurements, on the order of t ..."

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The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by exploiting inherent structure in natural and man-made signals. It has been demon-strated that structured signals can be acquired with just a small number of linear measurements, on the order of the signal complexity. In practice, this enables lower sampling rates that can be more easily achieved by current hardware designs. The primary bottleneck that limits ADC sam-pling rates is quantization, i.e., higher bit-depths impose lower sampling rates. Thus, the decreased sampling rates of CS ADCs accommodate the otherwise limiting quantizer of conventional ADCs. In this thesis, we consider a different approach to CS ADC by shifting towards lower quantizer bit-depths rather than lower sampling rates. We explore the extreme case where each measurement is quantized to just one bit, representing its sign. We develop a new theoretical framework to analyze this extreme case and develop new algorithms for signal reconstruction from such coarsely quantized measurements. The 1-bit CS framework leads us to scenarios where it may be more appropriate to reduce bit-depth instead of sampling rate. We find that there exist two distinct regimes of operation that correspond to high/low signal-to-noise ratio (SNR). In the measurement

### by an NSERC Discovery Grant.

, 2014

"... We study Ginzburg–Landau equations for a complex vector order parameter Ψ = (ψ+, ψ−) ∈ C2. We consider the Dirichlet problem in the disk in R2 with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that t ..."

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We study Ginzburg–Landau equations for a complex vector order parameter Ψ = (ψ+, ψ−) ∈

*C**2*. We consider the Dirichlet problem in the disk in R*2*with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find