"Exact WKB integration'' of polynomial 1D Schr\"odinger (or Sturm-Liouville) problem

We review an "exact semiclassical" resolution method for the general stationary 1D Schr\"odinger equation with a polynomial potential. This method avoids having to compute any Stokes phenomena directly; instead, it basically relies on an elementary Wronskian identity, and on a fully exact form of Bohr--Sommerfeld quantization conditions which can also be viewed as a Bethe-Ansatz system of equations that will "solve" the general polynomial 1D Schr\"odinger problem.Comment: latex txt12.tex, 4 files, 3 figures, 18 pages Differential equations and Stokes phenomenon Groningen, The Netherlands May 28-30 2001 [SPhT-T01/146

An Overview of Multi-Processor Approximate Message Passing

Approximate message passing (AMP) is an algorithmic framework for solving linear inverse problems from noisy measurements, with exciting applications such as reconstructing images, audio, hyper spectral images, and various other signals, including those acquired in compressive signal acquisiton systems. The growing prevalence of big data systems has increased interest in large-scale problems, which may involve huge measurement matrices that are unsuitable for conventional computing systems. To address the challenge of large-scale processing, multiprocessor (MP) versions of AMP have been developed. We provide an overview of two such MP-AMP variants. In row-MP-AMP, each computing node stores a subset of the rows of the matrix and processes corresponding measurements. In column- MP-AMP, each node stores a subset of columns, and is solely responsible for reconstructing a portion of the signal. We will discuss pros and cons of both approaches, summarize recent research results for each, and explain when each one may be a viable approach. Aspects that are highlighted include some recent results on state evolution for both MP-AMP algorithms, and the use of data compression to reduce communication in the MP network

Spectral Efficiency of One-Bit Sigma-Delta Massive MIMO

We examine the uplink spectral efficiency of a massive MIMO base station employing a one-bit Sigma-Delta ( \Sigma \Delta ) sampling scheme implemented in the spatial rather than the temporal domain. Using spatial rather than temporal oversampling, and feedback of the quantization error between adjacent antennas, the method shapes the spatial spectrum of the quantization noise away from an angular sector where the signals of interest are assumed to lie. It is shown that, while a direct Bussgang analysis of the \Sigma \Delta approach is not suitable, an alternative equivalent linear model can be formulated to facilitate an analysis of the system performance. The theoretical properties of the spatial quantization noise power spectrum are derived for the \Sigma \Delta array, as well as an expression for the spectral efficiency of maximum ratio combining (MRC). Simulations verify the theoretical results and illustrate the significant performance gains offered by the \Sigma \Delta approach for both MRC and zero-forcing receivers

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

A Mathematical Analysis of the Axial Anomaly

As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We use the Batalin-Vilkovisky (BV) formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. Using these methods, we formalize two conventional interpretations of the axial anomaly, the first as a violation of current conservation at the quantum level and the second as the obstruction to the existence of a well-defined fermionic partition function. Moreover, in the formalism of Costello and Gwilliam, anomalies are measured by cohomology classes in a certain obstruction-deformation complex. Our main result shows that---in the case of the axial symmetry---the relevant complex is quasi-isomorphic to the complex of de Rham forms of the spacetime manifold and that the anomaly corresponds to a top-degree cohomology class which is trivial if and only if the index of the corresponding Dirac operator is zero.Comment: Version 3 differs from version 2 only in the metadata. The title and abstract in the metadata have been corrected to match those appearing in the document itsel

Discrete quantum spectrum of black holes

The quantum genesis of Hawking radiation is a long-standing puzzle in black hole physics. Semi-classically one can argue that the spectrum of radiation emitted by a black hole look very much sparse unlike what is expected from a thermal object. It was demonstrated through a simple quantum model that a quantum black hole will retain a discrete profile, at least in the weak energy regime. However, it was suggested that this discreteness might be an artifact of the simplicity of eigen-spectrum of the model considered. Different quantum theories can, in principle, give rise to different complicated spectra and make the radiation from black hole dense enough in transition lines, to make them look continuous in profile. We show that such a hope from a geometry-quantized black hole is not realized as long as large enough black holes are dubbed with a classical mass area relation in any gravity theory ranging from GR, Lanczos-Lovelock to f(R) gravity. We show that the smallest frequency of emission from black hole in any quantum description, is bounded from below, to be of the order of its inverse mass. That leaves the emission with only two possibilities. It can either be non-thermal, or it can be thermal only with the temperature being much larger than 1/M.Comment: Matches the published versio