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

Decomposition of fractional quantum Hall states: New symmetries and approximations

We provide a detailed description of a new symmetry structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall states first obtained in Ref. 1, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The symmetry rules in Ref. 1 as well as the ones we obtain for the spin singlet states allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size). We show that these overlaps reach unity in the thermodynamic limit even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state which can be written as an expectation value of parafermionic operators.Comment: 22 pages, 8 figure

Recurrence for discrete time unitary evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte

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

Twin Analyses of Fatigue

Abstract Prolonged fatigue equal to or greater than 1 month duration and chronic fatigue equal to or greater than 6 months duration are both commonly seen in clinical practice, yet little is known about the etiology or epidemiology of either symptom. Chronic fatigue syndrome (CFS), while rarer, presents similar challenges in determining cause and epidemiology. Twin studies can be useful in elucidating genetic and environmental influences on fatigue and CFS. The goal of this article was to use biometrical structural equation twin modeling to examine genetic and environmental influences on fatigue, and to investigate whether these influences varied by gender. A total of 1042 monozygotic (MZ) twin pairs and 828 dizygotic (DZ) twin pairs who had completed the University of Washington Twin Registry survey were assessed for three fatigue-related variables: prolonged fatigue, chronic fatigue, and CFS. Structural equation twin modeling was used to determine the relative contributions of additive genetic effects, shared environmental effects, and individual-specific environmental effects to the 3 fatigue conditions. In women, tetrachoric correlations were similar for MZ and DZ pairs for prolonged and chronic fatigue, but not for CFS. In men, however, the correlations for prolonged and chronic fatigue were higher in MZ pairs than in DZ pairs. About half the variance for both prolonged and chronic fatigue in males was due to genetic effects, and half due to individual-specific environmental effects. For females, most variance was due to individual environmental effects

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