Concurrency in the Linux kernel can be a contentious topic. The Linux kernel mailing list features numerous discussions related to consistency models, including those of the more than 30 CPU architectures supported by the kernel and that of the kernel itself. How are Linux programs supposed to behave? Do they behave correctly on exotic hardware? A formal model can help address such questions. Better yet, an executable model allows programmers to experiment with the model to develop their intuition. Thus we offer a model written in the cat language, making it not only formal, but also executable by the herd simulator. We tested our model against hardware and refined it in consultation with maintainers. Finally, we formalised the fundamental law of the Read-Copy-Update synchronisation mechanism, and proved that one of its implementations satisfies this law.

Concurrency in the Linux kernel can be a contentious topic. The Linux kernel mailing list features numerous discussions related to consistency models, including those of the more than 30 CPU architectures supported by the kernel and that of the kernel itself. How are Linux programs supposed to behave? Do they behave correctly on exotic hardware? A formal model can help address such questions. Better yet, an executable model allows programmers to experiment with the model to develop their intuition. Thus we offer a model written in the cat language, making it not only formal, but also executable by the herd simulator. We tested our model against hardware and refined it in consultation with maintainers. Finally, we formalised the fundamental law of the Read-Copy-Update synchronisation mechanism, and proved that one of its implementations satisfies this law.

Although the discrete Fourier transform played a seminal role in the development of modern NMR spectroscopy, it suffers from well-known limitations. Chief among them is difficulty in obtaining high-resolution spectra from short data records. A host of non-Fourier methods of spectrum analysis has been developed that avoid many of these shortcomings. A method that makes very few assumptions about the signal is maximum entropy reconstruction. The lack of assumptions makes it one of the most versatile and robust non-Fourier methods. Very efficient algorithms for maximum entropy reconstruction have been developed, and it is finding applications in NMR for stable deconvolution and nonuniform sampling. We describe the theory and application of maximum entropy reconstruction in NMR, and the relationship with other non-Fourier methods. Keywords: spectrum analysis; non-Fourier methods; maximum entropy; deconvolution; nonuniform sampling; compressed sensing

Linear prediction models each sample in a discretely sampled signal as a linear combination of the preceding samples. It implicitly corresponds to treating the signal as a sum of sinusoids and provides a straightforward means for extrapolating the signal beyond the measured interval. It was the first such method employed to ameliorate short data records in the indirect dimensions of multidimensional NMR experiments and remains one of the most widely used. In this chapter we describe the history, theory, strengths and weaknesses of linear prediction extrapolation.

Correction for 'Sparse sampling methods in multidimensional NMR' by Mehdi Mobli et al., Phys. Chem. Chem. Phys., 2012, 14, 10835-10843.

Compressed sensing (CS) has attracted a great deal of recent interest as an approach for spectrum analysis of nonuniformly sampled NMR data. Although theoretical justification for the method is abundant, it suffers from several weaknesses, among them poor convergence of some algorithms, and it remains an open question whether NMR spectra satisfy the sparsity requirements of CS theorems. The versions of CS used in NMR involve minimizing the l 1 norm of the spectrum. They bear similarity to maximum entropy (MaxEnt) reconstruction, but critical comparison of the methods can be difficult. Here we describe a formalism that places CS and MaxEnt reconstruction on equal footing, enabling critical comparison of the two methods. We also describe a new algorithm for CS that restricts the computation of the l 1 norm to the real channel for complex spectra and ensures causality. Preliminary 1D results demonstrate that this approach ameliorates some artifacts that can occur when using the l 1 norm of the complex spectrum. Copyright © 2015 John Wiley & Sons, Ltd.

•Nonuniform sampling of multidimensional NMR introduces artifacts.•Randomizing offset of columns in a uniformly undersampled grid suppresses artifacts.•Artifact suppression is exponentially related to dimensionality of sample space.•Randomizing along “quadrature phase” yields synergistic improvement.

NMR spectroscopy is one of the most powerful and versatile analytic tools available to chemists. The discrete Fourier transform (DFT) played a seminal role in the development of modern NMR, including the multidimensional methods that are essential for characterizing complex biomolecules. However, it suffers from well-known limitations: chiefly the difficulty in obtaining high-resolution spectral estimates from short data records. Because the time required to perform an experiment is proportional to the number of data samples, this problem imposes a sampling burden for multidimensional NMR experiments. At high magnetic field, where spectral dispersion is greatest, the problem becomes particularly acute. Consequently multidimensional NMR experiments that rely on the DFT must either sacrifice resolution in order to be completed in reasonable time or use inordinate amounts of time to achieve the potential resolution afforded by high-field magnets. Maximum entropy (MaxEnt) reconstruction is a non-Fourier method of spectrum analysis that can provide high-resolution spectral estimates from short data records. It can also be used with nonuniformly sampled data sets. Since resolution is substantially determined by the largest evolution time sampled, nonuniform sampling enables high resolution while avoiding the need to uniformly sample at large numbers of evolution times. The Nyquist sampling theorem does not apply to nonuniformly sampled data, and artifacts that occur with the use of nonuniform sampling can be viewed as frequency-aliased signals. Strategies for suppressing nonuniform sampling artifacts include the careful design of the sampling scheme and special methods for computing the spectrum. Researchers now routinely report that they can complete an N-dimensional NMR experiment 3(N-1) times faster (a 3D experiment in one ninth of the time). As a result, high-resolution three- and four-dimensional experiments that were prohibitively time consuming are now practical. Conversely, tailored sampling in the indirect dimensions has led to improved sensitivity. Further advances in nonuniform sampling strategies could enable further reductions in sampling requirements for high resolution NMR spectra, and the combination of these strategies with robust non-Fourier methods of spectrum analysis (such as MaxEnt) represent a profound change in the way researchers conduct multidimensional experiments. The potential benefits will enable more advanced applications of multidimensional NMR spectroscopy to study biological macromolecules, metabolomics, natural products, dynamic systems, and other areas where resolution, sensitivity, or experiment time are limiting. Just as the development of multidimensional NMR methods presaged multidimensional methods in other areas of spectroscopy, we anticipate that nonuniform sampling approaches will find applications in other forms of spectroscopy.

In Fourier transform NMR, the free induction decay is sampled at a series of uniform intervals. The Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. Obtaining high-resolution spectral estimates from short data records using the discrete Fourier transform has been a long-standing challenge. Despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions of multidimensional experiments is limited by practical constraints on measuring time. The advent of methods of spectrum analysis capable of processing nonuniformly sampled data has led to the development of a host of novel sampling strategies that enable collection of data at long evolution times, necessary for high resolution, without requiring that all intervening samples spaced at the Nyquist interval are collected. These methods enable practical high-resolution multidimensional experiments, and are especially useful at high field strengths where the Nyquist interval is shorter. Here, we describe the fundamentals of uniform and non-uniform sampling in one-and multidimensional NMR, and briefly survey non-Fourier methods of spectrum analysis suitable for nonuniformly sampled data.

Multidimensional NMR spectroscopy typically employs phase-sensitive detection, which results in hypercomplex data (and spectra) when utilized in more than one dimension. Nonuniform sampling approaches have become commonplace in multidimensional NMR, enabling dramatic reductions in experiment time, increases in sensitivity and/or increases in resolution. In order to utilize nonuniform sampling optimally, it is necessary to characterize the relationship between the spectrum of a uniformly sampled data set and the spectrum of a subsampled data set. In this work we construct an algebra of hypercomplex numbers suitable for representing multidimensional NMR data along with partial-component nonuniform sampling (i.e. the hypercomplex components of data points are subsampled). This formalism leads to a modified DFT–Convolution relationship involving a partial-component, hypercomplex point-spread function set. The framework presented here is essential for the continued development and appropriate characterization of partial-component nonuniform sampling.

Although the discrete Fourier transform played an enabling role in the development of modern NMR spectroscopy, it suffers from a well-known difficulty providing high-resolution spectra from short data records. In multidimensional NMR experiments, so-called indirect time dimensions are sampled parametrically, with each instance of evolution times along the indirect dimensions sampled via separate one-dimensional experiments. The time required to conduct multidimensional experiments is directly proportional to the number of indirect evolution times sampled. Despite remarkable advances in resolution with increasing magnetic field strength, multiple dimensions remain essential for resolving individual resonances in NMR spectra of biological macromolecues. Conventional Fourier-based methods of spectrum analysis limit the resolution that can be practically achieved in the indirect dimensions. Nonuniform or sparse data collection strategies, together with suitable non-Fourier methods of spectrum analysis, enable high-resolution multidimensional spectra to be obtained. Although some of these approaches were first employed in NMR more than two decades ago, it is only relatively recently that they have been widely adopted. Here we describe the current practice of sparse sampling methods and prospects for further development of the approach to improve resolution and sensitivity and shorten experiment time in multidimensional NMR. While sparse sampling is particularly promising for multidimensional NMR, the basic principles could apply to other forms of multidimensional spectroscopy.

Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (free induction decay) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform, the intervals are kept uniform, and the Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. With the proposal by Jeener of parametric sampling along an indirect time dimension, extension to multidimensional experiments employed the same sampling techniques used in one dimension, similarly subject to the Nyquist condition and suitable for processing via the discrete Fourier transform. The challenges of obtaining high-resolution spectral estimates from short data records were already well understood, and despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions is limited by practical constraints on measuring time. The advent of methods of spectrum analysis capable of processing nonuniformly sampled data has led to an explosion in the development of novel sampling strategies that avoid the limits on resolution and measurement time imposed by uniform sampling. In this chapter we review the fundamentals of uniform and nonuniform sampling methods in one-and multidimensional NMR.

Despite advances in resolution accompanying the development of high-field superconducting magnets, biomolecular applications of NMR require multiple dimensions in order to resolve individual resonances, and the achievable resolution is typically limited by practical constraints on measuring time. In addition to the need for measuring long evolution times to obtain high resolution, the need to distinguish the sign of the frequency constrains the ability to shorten measuring times. Sign discrimination is typically accomplished by sampling the signal with two different receiver phases or by selecting a reference frequency outside the range of frequencies spanned by the signal and then sampling at a higher rate. In the parametrically sampled (indirect) time dimensions of multidimensional NMR experiments, either method imposes an additional factor of 2 sampling burden for each dimension. We demonstrate that by using a single detector phase at each time sample point, but randomly altering the phase for different points, the sign ambiguity that attends fixed single-phase detection is resolved. Random phase detection enables a reduction in experiment time by a factor of 2 for each indirect dimension, amounting to a factor of 8 for a four-dimensional experiment, albeit at the cost of introducing sampling artifacts. Alternatively, for fixed measuring time, random phase detection can be used to double resolution in each indirect dimension. Random phase detection is complementary to nonuniform sampling methods, and their combination offers the potential for additional benefits. In addition to applications in biomolecular NMR, random phase detection could be useful in magnetic resonance imaging and other signal processing contexts.

Read-copy update (RCU) is a synchronization technique that often replaces reader-writer locking because RCU's read-side primitives are both wait-free and an order of magnitude faster than uncontended locking. Although RCU updates are relatively heavy weight, the importance of read-side performance is increasing as computing systems become more responsive to changes in their environments. RCU is heavily used in several kernel-level environments. Unfortunately, kernel-level implementations use facilities that are often unavailable to user applications. The few prior user-level RCU implementations either provided inefficient read-side primitives or restricted the application architecture. This paper fills this gap by describing efficient and flexible RCU implementations based on primitives commonly available to user-level applications. Finally, this paper compares these RCU implementations with each other and with standard locking, which enables choosing the best mechanism for a given workload. This work opens the door to widespread user-application use of RCU.

One of the stiffest challenges in structural studies of proteins using NMR is the assignment of sidechain resonances. Typically, a panel of lengthy 3D experiments are acquired in order to establish connectivities and resolve ambiguities due to overlap. We demonstrate that these experiments can be replaced by a single 4D experiment that is time-efficient, yields excellent resolution, and captures unique carbon–proton connectivity information. The approach is made practical by the use of non-uniform sampling in the three indirect time dimensions and maximum entropy reconstruction of the corresponding 3D frequency spectrum. This 4D method will facilitate automated resonance assignment procedures and it should be particularly beneficial for increasing throughput in NMR-based structural genomics initiatives.

Iterative thresholding algorithms have a long history of application to signal processing. Although they are intuitive and easy to implement, their development was heuristic and mainly ad hoc. Using a special form of the thresholding operation, called soft thresholding, we show that the fixed point of iterative thresholding is equivalent to minimum l1-norm reconstruction. We illustrate the method for spectrum analysis of a time series. This result helps to explain the success of these methods and illuminates connections with maximum entropy and minimum area methods, while also showing that there are more efficient routes to the same result. The power of the l1-norm and related functionals as regularizers of solutions to underdetermined systems will likely find numerous useful applications in NMR.