Randomized Rounding for the Largest Simplex Problem

The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem is known to be $\mathrm{NP}$-hard to approximate within a factor of $c^{j}$ for some constant $c > 1$. Our algorithm also gives a factor $e^{j + o(j)}$ approximation for the problem of finding the principal $j\times j$ submatrix of a rank $d$ positive semidefinite matrix with the largest determinant. We achieve our approximation by rounding solutions to a generalization of the $D$-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid problem. Our arguments give a short and simple proof of a restricted invertibility principle for determinants

On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page

Computational toolbox for ultrastructural quantitative analysis of filament networks in cryo-ET data

A precise quantitative description of the ultrastructural characteristics underlying biological mechanisms is often key to their understanding. This is particularly true for dynamic extra- and intracellular filamentous assemblies, playing a role in cell motility, cell integrity, cytokinesis, tissue formation and maintenance. For example, genetic manipulation or modulation of actin regulatory proteins frequently manifests in changes of the morphology, dynamics, and ultrastructural architecture of actin filament-rich cell peripheral structures, such as lamellipodia or filopodia. However, the observed ultrastructural effects often remain subtle and require sufficiently large datasets for appropriate quantitative analysis. The acquisition of such large datasets has been enabled by recent advances in high-throughput cryo-electron tomography (cryo-ET) methods. However, this also necessitates the development of complementary approaches to maximize the extraction of relevant biological information. We have developed a computational toolbox for the semi-automatic quantification of filamentous networks from cryo-ET datasets to facilitate the analysis and cross-comparison of multiple experimental conditions. GUI-based components simplify the manipulation of data and allow users to obtain a large number of ultrastructural parameters describing filamentous assemblies. We demonstrate the feasibility of this workflow by analyzing cryo-ET data of untreated and chemically perturbed branched actin filament networks and that of parallel actin filament arrays. In principle, the computational toolbox presented here is applicable for data analysis comprising any type of filaments in regular (i.e. parallel) or random arrangement. We show that it can ease the identification of key differences between experimental groups and facilitate the in-depth analysis of ultrastructural data in a time-efficient manner

Constraints on Natural MNS Parameters from |U_e3|

The MNS matrix structure emerging as a result of recent neutrino measurements strongly suggests two large mixing angles (solar and atmospheric) and one small angle (|U_e3| << 1). Especially when combined with the neutrino mass hierarchy, these values turn out to impose rather stringent constraints on possible flavor models connecting the three active fermion generations. Specifically, we show that an extremely small value of |U_e3| would require fine tuning of Majorana mass matrix parameters, particularly in the context of seesaw models.Comment: 21 pages, ReVTeX, 2 .eps figure files, updated references and acknowledgment

Characterizing Operations Preserving Separability Measures via Linear Preserver Problems

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see in particular that for k at least 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.Comment: 16 pages, typos corrected, references added, proof of Theorem 4.3 simplified and clarifie

Natural preconditioning and iterative methods for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends

Factors Affecting European Farmers’Participation in Biodiversity Policies

This article reports the major findings from an interdisciplinary research project that synthesises key insights into farmers’ willingness and ability to co-operate with biodiversity policies. The results of the study are based on an assessment of about 160 publications and research reports from six EU member states and from international comparative research.We developed a conceptual framework to systematically review the existent literature relevant for our purposes. This framework provides a common structure for analysing farmers’ perspectives regarding the introduction into farming practices of measures relevant to biodiversity. The analysis is coupled and contrasted with a survey of experts. The results presented above suggest that it is important to view support for practices oriented towards biodiversity protection not in a static sense – as a situation determined by one or several influencing factors – but rather as a process marked by interaction. Financial compensation and incentives function as a necessary, though clearly not sufficient condition in this process