On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n−O(1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.     

Use of sol-gel entrapment techniques for the resolution of itaconic acid biosynthetic pathway in Aspergillus terreus

A new technique of investigation of labile metabolic pathways using immobilization of the pathway in sol-gel derived silicate matrices was demonstrated. The biosynthetic pathway of itaconic acid in Aspergillus terreus is believed to proceed through decarboxylation of cis-aconitate catalyzed by an unstable enzyme aconitate decarboxylase, E.C. 4.1.1.6. Stabilization of this pathway in sol-gel derived silicate matrix enabled the elucidation of the correct sequence of biosynthetic steps. The decarboxylation of cis-aconitate does not lead directly to itaconate, but rather to citraconate. The latter is then isomerized to itaconate by a previously unknown enzyme.     

Infrared laser enhancement of the reaction of sulfur hexafluoride with sodium vapor

The vibrational energy dependence of the rate of the gas phase reaction of Na with SF₆ has been determined in a diffusion cloud experiment using CO₂ laser excitation. The “conversion efficiency” of ca. 40% for vibrational energy suggests a preference for vibrational over the translational energy when compared with “prior” statistical expectation.     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

Effect of the Solvent on the Conformation of Isolated MEH‐PPV Chains Intercalated Into SnS2

Photophysical processes in conjugated polymers are influenced by two competing effects: the extent of excited state delocalization along a chain, and the electronic interaction between chains. Experimentally, it is often difficult to separate the two because both are controlled by chain conformation. Here we demonstrate that it is possible to modify intra‐chain delocalization without inducing inter‐chain interactions by intercalating polymer monolayers between the sheets of an inorganic layered matrix. The red‐emitting conjugated polymer, MEH‐PPV, is confined to the interlayer space of layered SnS₂. The formation of isolated polymer monolayers between the SnS₂ sheets is confirmed by X‐ray diffraction measurements. Photoluminescence excitation (PLE) and photoluminescence (PL) spectra of the incorporated MEH‐PPV chains reveal that the morphology of the incorporated chains can be varied through the choice of solvent used for chain intercalation. Incorporation from chloroform results in more extended conformations compared to intercalation from xylene. Even highly twisted conformations can be achieved when the incorporation occurs from a methanol:chloroform mixture. The PL spectra of the MEH‐PPV incorporated SnS₂ nanocomposites using the different solvents are in good agreement with the PL spectra of the same solutions, indicating that the conformation of the polymer chains in the solutions is retained upon intercalation into the inorganic host. Therefore, intercalation of conjugated polymer chains into layered hosts enables the study of intra‐chain photophysical processes as a function of chain conformation.     

Synthesis and evaluation of a pseudocyclic tristhiourea-based anion host

A novel methodology for the evaluation of receptor arrangement in structurally flexible anion chemosensors was developed and applied to map the binding site of a new pseudocyclic tristhiourea chemosensor (6). The syntheses of 6 and related macrocyclic chemosensor 10 (a model of the folded monomeric structure of 6) are reported. Both chemosensors were evaluated by titration with a variety of structurally different anions in CH3Cl and DMSO, showing a common preference for F-, CH3CO2-, and H2PO4-. However, within this group of anions, the binding patterns of the chemosensors differed, indicating dissimilarity in the arrangement of the binding sites of 6 and 10.     

Cycle Factors and Renewal Theory

For which values of k does a uniformly chosen 3‐regular graph G on n vertices typically contain n/k vertex‐disjoint k ‐cycles (a k ‐cycle factor)? To date, this has been answered for k = n and for k ? log n ; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability (w.h.p.) most vertices do not lie on k ‐cycles. A major role in our study of this problem is played by renewal processes without replacement, where one wishes to estimate the probability that in a uniform permutation of a given set of positive integers, the partial sums hit a designated target integer. Using sharp tail estimates for these renewal processes, which may be of independent interest, we settle the cycle factor problem completely: the “threshold” for a k ‐cycle factor in G as above is κ ₀ log₂ n with . To be precise, G contains a k ‐cycle factor w.h.p. if and w.h.p. does not contain one if . Thus, for most values of n the threshold concentrates on the single integer K ₀(n ). As a byproduct, we confirm the “comb conjecture,” an old problem concerning the embedding of certain spanning trees in the random graph (n,p ).© 2015 Wiley Periodicals, Inc.     

Privileged users in zero-error transmission over a noisy channel

The k-th power of a graph G is the graph whose vertex set is V(G) ^k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim _k→∞ α(G ^k )^1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = Σ _i C _i , describes a setting when there are t ≥ 2 senders, each with his own channel C _i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = Σ _i G _i . It is well known that c(G) ≥ Σ _i c(G _i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C _i to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F ∈ F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k − 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp vertices contains all t colors. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research partially supported by a Charles Clore Foundation Fellowship.     

Detection and quantification through a lipid membrane using the molecularly controlled semiconductor resistor

The detection of covalent and noncovalent binding events between molecules and biomembranes is a fundamental goal of contemporary biochemistry and analytical chemistry. Currently, such studies are performed routinely using fluorescence methods, surface-plasmon resonance spectroscopy, and electrochemical methods. However, there is still a need for novel sensitive miniaturizable detection methods where the sample does not have to be transferred to the sensor, but the sensor can be brought into contact with the sample studied. We present a novel approach for detection and quantification of processes occurring on the surface of a lipid bilayer membrane, by monitoring the current change through the n-type GaAs-based molecularly controlled semiconductor resistor (MOCSER), on which the membrane is adsorbed. Since GaAs is susceptible to etching in an aqueous environment, a protective thin film of methoxysilane was deposited on the device. The system was found to be sensitive enough to allow monitoring changes in pH and in the concentration of amino acids in aqueous solution on top of the membrane. When biotinylated lipids were incorporated into the membrane, it was possible to monitor the binding of streptavidin or avidin. The device modified with biotin-streptavidin complex was capable of detecting the binding of streptavidin antibodies to immobilized streptavidin with high sensitivity and selectivity. The response depends on the charge on the analyte. These results open the way to facile electrical detection of protein-membrane interactions.