On the digraph of a unitary matrix

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.Comment: 6 page

The approach to criticality in sandpiles

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

Molecular simulations for dynamic nuclear polarization in liquids: a case study of TEMPOL in acetone and DMSO

A computational strategy for calibrating, validating and analyzing molecular dynamics (MD) simulations to predict dynamic nuclear polarization (DNP) coupling factors and relaxivities of proton spins is presented. Simulations of the polarizing agent TEMPOL in liquid acetone and DMSO are conducted at low (infinite dilution) and high (1 M) concentrations of the free radical. Because DNP coupling factors and relaxivities are sensitive to the time scales of the molecular motions, the MD simulations are calibrated to reproduce the bulk translational diffusion coefficients of the pure solvents. The simulations are then validated by comparing with experimental dielectric relaxation spectra, which report on the rotational dynamics of the molecular electric dipole moments. The analysis consists of calculating spectral density functions (SDFs) of the magnetic dipole–dipole interaction between the electron spin of TEMPOL and nuclear spins of the solvent protons. Here, MD simulations are used in combination with an analytically tractable model of molecular motion. While the former provide detailed information at relatively short spin–spin distances, the latter includes contributions at large separations, all the way to infinity. The relaxivities calculated from the SDFs of acetone and DMSO are in excellent agreement with experiments at 9.2 T. For DMSO we calculate a coupling factor in agreement with experiment while for acetone we predict a value that is larger by almost 50%, suggesting a possibility for experimental improvement

Mod/Resc Parsimony Inference

We address in this paper a new computational biology problem that aims at understanding a mechanism that could potentially be used to genetically manipulate natural insect populations infected by inherited, intra-cellular parasitic bacteria. In this problem, that we denote by \textsc{Mod/Resc Parsimony Inference}, we are given a boolean matrix and the goal is to find two other boolean matrices with a minimum number of columns such that an appropriately defined operation on these matrices gives back the input. We show that this is formally equivalent to the \textsc{Bipartite Biclique Edge Cover} problem and derive some complexity results for our problem using this equivalence. We provide a new, fixed-parameter tractability approach for solving both that slightly improves upon a previously published algorithm for the \textsc{Bipartite Biclique Edge Cover}. Finally, we present experimental results where we applied some of our techniques to a real-life data set.Comment: 11 pages, 3 figure

Encoded loop-lanthanide-binding tags for long-range distance measurements in proteins by NMR and EPR spectroscopy

We recently engineered encodable lanthanide binding tags (LBTs) into proteins and demonstrated their applicability in Nuclear Magnetic Resonance (NMR) spectroscopy, X-ray crystallography and luminescence studies. Here, we engineered two-loop-LBTs into the model protein interleukin-1β (IL1β) and measured [superscript 1]H, [superscript 15]N-pseudocontact shifts (PCSs) by NMR spectroscopy. We determined the Δχ-tensors associated with each Tm[superscript 3+]-loaded loop-LBT and show that the experimental PCSs yield structural information at the interface between the two metal ion centers at atomic resolution. Such information is very valuable for the determination of the sites of interfaces in protein–protein-complexes. Combining the experimental PCSs of the two-loop-LBT construct IL1β-S2R2 and the respective single-loop-LBT constructs IL1β-S2, IL1β-R2 we additionally determined the distance between the metal ion centers. Further, we explore the use of two-loop LBTs loaded with Gd[superscript 3+] as a novel tool for distance determination by Electron Paramagnetic Resonance spectroscopy and show the NMR-derived distances to be remarkably consistent with distances derived from Pulsed Electron–Electron Dipolar Resonance.German Science Foundation (collaborative research centers 807 and 902)National Science Foundation (U.S.) (Grant MCB 0744415

Dynamic nuclear polarization at high magnetic fields in liquids

High field dynamic nuclear polarization spectrometer for liquid samples have been constructed. ► The field dependence of the Overhauser DNP efficiency has been measured for the first time up to 9.2 T. ► High DNP enhancements for liquid samples have been observed at high magnetic fields. ► The enhancements have been compared with results from NMRD, MD and theoretical models. ► Coherent and relaxation effects within fast magnetic field changes have been analyzed