A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.Comment: 8 page

On certain other sets of integers

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).Comment: 29 pp. Corrected typos. Added definitions for some non-standard notation and remarks on lower bound

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Diquat Derivatives: Highly Active, Two-Dimensional Nonlinear Optical Chromophores with Potential Redox Switchability

In this article, we present a detailed study of structure−activity relationships in diquaternized 2,2′-bipyridyl (diquat) derivatives. Sixteen new chromophores have been synthesized, with variations in the amino electron donor substituents, π-conjugated bridge, and alkyl diquaternizing unit. Our aim is to combine very large, two-dimensional (2D) quadratic nonlinear optical (NLO) responses with reversible redox chemistry. The chromophores have been characterized as their PF_6^− salts by using various techniques including electronic absorption spectroscopy and cyclic voltammetry. Their visible absorption spectra are dominated by intense π → π^* intramolecular charge-transfer (ICT) bands, and all show two reversible diquat-based reductions. First hyperpolarizabilities β have been measured by using hyper-Rayleigh scattering with an 800 nm laser, and Stark spectroscopy of the ICT bands affords estimated static first hyperpolarizabilities β_0. The directly and indirectly derived β values are large and increase with the extent of π-conjugation and electron donor strength. Extending the quaternizing alkyl linkage always increases the ICT energy and decreases the E_(1/2) values for diquat reduction, but a compensating increase in the ICT intensity prevents significant decreases in Stark-based β_0 responses. Nine single-crystal X-ray structures have also been obtained. Time-dependent density functional theory clarifies the molecular electronic/optical properties, and finite field calculations agree with polarized HRS data in that the NLO responses of the disubstituted species are dominated by ‘off-diagonal’ β_(zyy) components. The most significant findings of these studies are: (i) β_0 values as much as 6 times that of the chromophore in the technologically important material (E)-4′-(dimethylamino)-N-methyl-4-stilbazolium tosylate; (ii) reversible electrochemistry that offers potential for redox-switching of optical properties over multiple states; (iii) strongly 2D NLO responses that may be exploited for novel practical applications; (iv) a new polar material, suitable for bulk NLO behavior

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes

Cholesterol Depletion in Adipocytes Causes Caveolae Collapse Concomitant with Proteosomal Degradation of Cavin-2 in a Switch-Like Fashion

Caveolae, little caves of cell surfaces, are enriched in cholesterol, a certain level of which is required for their structural integrity. Here we show in adipocytes that cavin-2, a peripheral membrane protein and one of 3 cavin isoforms present in caveolae from non-muscle tissue, is degraded upon cholesterol depletion in a rapid fashion resulting in collapse of caveolae. We exposed 3T3-L1 adipocytes to the cholesterol depleting agent methyl-β-cyclodextrin, which results in a sudden and extensive degradation of cavin-2 by the proteasome and a concomitant movement of cavin-1 from the plasma membrane to the cytosol along with loss of caveolae. The recovery of cavin-2 at the plasma membrane is cholesterol-dependent and is required for the return of cavin-1 from the cytosol to the cell surface and caveolae restoration. Expression of shRNA directed against cavin-2 also results in a cytosolic distribution of cavin-1 and loss of caveolae. Taken together, these data demonstrate that cavin-2 functions as a cholesterol responsive component of caveolae that is required for cavin-1 localization to the plasma membrane, and caveolae structural integrity

Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratorily versatile bacterium Shewanella oneidensis

© The Author(s), 2012. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Nucleic Acids Research 40 (2012): 7132-7149, doi:10.1093/nar/gks467.The capacity of microorganisms to respond to variable external conditions requires a coordination of environment-sensing mechanisms and decision-making regulatory circuits. Here, we seek to understand the interplay between these two processes by combining high-throughput measurement of time-dependent mRNA profiles with a novel computational approach that searches for key genetic triggers of transcriptional changes. Our approach helped us understand the regulatory strategies of a respiratorily versatile bacterium with promising bioenergy and bioremediation applications, Shewanella oneidensis, in minimal and rich media. By comparing expression profiles across these two conditions, we unveiled components of the transcriptional program that depend mainly on the growth phase. Conversely, by integrating our time-dependent data with a previously available large compendium of static perturbation responses, we identified transcriptional changes that cannot be explained solely by internal network dynamics, but are rather triggered by specific genes acting as key mediators of an environment-dependent response. These transcriptional triggers include known and novel regulators that respond to carbon, nitrogen and oxygen limitation. Our analysis suggests a sequence of physiological responses, including a coupling between nitrogen depletion and glycogen storage, partially recapitulated through dynamic flux balance analysis, and experimentally confirmed by metabolite measurements. Our approach is broadly applicable to other systems.Office of Science (BER), U.S. Department of Energy [DE-FG02-07ER64388 to D.S. and DE-FG02- 08ER64511 to M.H.S.]; National Aeronautics and Space Administration, NASA Astrobiology Institute [NNA08CN84A to D.S.]

On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic

Conserved synteny at the protein family level reveals genes underlying Shewanella species’ cold tolerance and predicts their novel phenotypes

© The Authors 2009. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License. The definitive version was published in Functional & Integrative Genomics 10 (2010): 97-110, doi:10.1007/s10142-009-0142-y.Bacteria of the genus Shewanella can thrive in different environments and demonstrate significant variability in their metabolic and ecophysiological capabilities including cold and salt tolerance. Genomic characteristics underlying this variability across species are largely unknown. In this study, we address the problem by a comparison of the physiological, metabolic, and genomic characteristics of 19 sequenced Shewanella species. We have employed two novel approaches based on association of a phenotypic trait with the number of the trait-specific protein families (Pfam domains) and on the conservation of synteny (order in the genome) of the trait-related genes. Our first approach is top-down and involves experimental evaluation and quantification of the species’ cold tolerance followed by identification of the correlated Pfam domains and genes with a conserved synteny. The second, a bottom-up approach, predicts novel phenotypes of the species by calculating profiles of each Pfam domain among their genomes and following pair-wise correlation of the profiles and their network clustering. Using the first approach, we find a link between cold and salt tolerance of the species and the presence in the genome of a Na+/H+ antiporter gene cluster. Other cold-tolerance-related genes include peptidases, chemotaxis sensory transducer proteins, a cysteine exporter, and helicases. Using the bottom-up approach, we found several novel phenotypes in the newly sequenced Shewanella species, including degradation of aromatic compounds by an aerobic hybrid pathway in Shewanella woodyi, degradation of ethanolamine by Shewanella benthica, and propanediol degradation by Shewanella putrefaciens CN32 and Shewanella sp. W3-18-1.This research was supported by the U.S. Department of Energy (DOE) Office of Biological and Environmental Research under the Genomics: GTL Program via the Shewanella Federation consortium