Exact NMR simulation of protein-size spin systems using tensor train formalism

We introduce a new method, based on alternating optimization, for compact representation of spin Hamiltonians and solution of linear systems of algebraic equations in the tensor train format. We demonstrate the method's utility by simulating, without approximations, a 15N NMR spectrum of ubiquitin—a protein containing several hundred interacting nuclear spins. Existing simulation algorithms for the spin system and the NMR experiment in question either require significant approximations or scale exponentially with the spin system size. We compare the proposed method to the Spinach package that uses heuristic restricted state space techniques to achieve polynomial complexity scaling. When the spin system topology is close to a linear chain (e.g., for the backbone of a protein), the tensor train representation is more compact and can be computed faster than the sparse representation using restricted state spaces

f(R) theories

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.Comment: 156 pages, 14 figures, Invited review article in Living Reviews in Relativity, Published version, Comments are welcom

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Numerical solution of a class of third order tensor linear equations

We propose a new dense method for determining the numerical solution to a class of third order tensor linear equations. The approach does not require the use of the coefficient matrix in Kronecker form, thus it allows the treatment of structured very large problems. A particular version of the method for symmetric matrices is also discussed. Numerical experiments illustrate the properties of the proposed algorithm

Large-scale patterns in community structure of benthos and fish in the Barents Sea

Biogeographical patterns have an ecological basis, but few empirical studies possess the necessary scale and resolution relevant for investigation. The Barents Sea shelf provides an ideal study area, as it is a transition area between Atlantic and Arctic regions, and is sampled by a comprehensive survey of all major functional groups. We studied spatial variation in species composition of demersal fish and benthos to elucidate how fish and benthos communities co-varied in relation to environmental variables. We applied co-correspondence analysis on presence–absence data of 64 fishes and 302 benthos taxa from 329 bottom trawl hauls taken at the Barents Sea ecosystem survey in August–September 2011. We found highly significant similarities in the spatial pattern of distribution of benthos and fishes, despite their differences in motility and other ecological traits. The first common ordination axis separated boreal species in the south-west (Atlantic temperate water) from Arctic species in the north-east (Arctic cold water, ice-covered in winter). The second common axis separated shallow bank species from species found in deep basins and trenches. Our results show that fish and benthos communities had a similar relationship to the environmental gradients at the scale of hundreds to thousands of kilometres. We further discussed how fish–benthos interactions vary between sub-regions in the Barents Sea based on species traits and a food web topology for the Barents Sea. This study forms a basis for further investigations on links between fish and benthos communities in the Barents Sea