On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances

Cytochrome cM decreases photosynthesis under photomixotrophy in Synechocystis sp. PCC 6803

Photomixotrophy is a metabolic state that enables photosynthetic microorganisms to simultaneously perform photosynthesis and metabolism of imported organic carbon substrates. This process is complicated in cyanobacteria, since many, including Synechocystis sp. PCC 6803, conduct photosynthesis and respiration in an interlinked thylakoid membrane electron transport chain. Under photomixotrophy, the cell must therefore tightly regulate electron fluxes from photosynthetic and respiratory complexes. In this study, we demonstrate, via characterization of photosynthetic apparatus and the proteome, that photomixotrophic growth results in a gradual inhibition of QA- reoxidation in wild-type Synechocystis, which largely decreases photosynthesis over 3 d of growth. This process is circumvented by deleting the gene encoding cytochrome cM (CytM), a cryptic c-type heme protein widespread in cyanobacteria. The ΔCytM strain maintained active photosynthesis over the 3-d period, demonstrated by high photosynthetic O2 and CO2 fluxes and effective yields of PSI and PSII. Overall, this resulted in a higher growth rate compared to that of the wild type, which was maintained by accumulation of proteins involved in phosphate and metal uptake, and cofactor biosynthetic enzymes. While the exact role of CytM has not been determined, a mutant deficient in the thylakoid-localized respiratory terminal oxidases and CytM (ΔCox/Cyd/CytM) displayed a phenotype similar to that of ΔCytM under photomixotrophy. This, in combination with other physiological data, and in contrast to a previous hypothesis, suggests that CytM does not transfer electrons to these complexes. In summary, our data suggest that CytM may have a regulatory role in photomixotrophy by modulating the photosynthetic capacity of cells

Distances sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from \cite{IKP01} and \cite{IKT01} that if $K$ is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then $L^2(K)$ does not possess an orthogonal basis of exponentials

Random matrix analysis of localization properties of Gene co-expression network

We analyze gene co-expression network under the random matrix theory framework. The nearest neighbor spacing distribution of the adjacency matrix of this network follows Gaussian orthogonal statistics of random matrix theory (RMT). Spectral rigidity test follows random matrix prediction for a certain range, and deviates after wards. Eigenvector analysis of the network using inverse participation ratio (IPR) suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix, whereas few eigenvalues are localized. Based on these IPR calculations, we can divide eigenvalues in three sets; (A) The non-degenerate part that follows RMT. (B) The non-degenerate part, at both ends and at intermediate eigenvalues, which deviate from RMT and expected to contain information about {\it important nodes} in the network. (C) The degenerate part with $zero$ eigenvalue, which fluctuates around RMT predicted value. We identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties

Sum-product formulae

This is a survey on sum-product formulae and methods. We state old and new results. Our main objective is to introduce the basic techniques used to bound the size of the product and sumsets of finite subsets of a field