Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove ?n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G^′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs.The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is “far” from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.     

Self-Assembled Monolayers of Nitron: Self-Activated and Chemically Addressable N-Heterocyclic Carbene Monolayers with Triazolone Structural Motif

N-heterocyclic carbenes (NHCs) have emerged as a unique molecular platform for the formation of self-assembled monolayers (SAMs) on various surfaces. However, active carbene formation requires deprotonation of imidazolium salt precursors, which is mostly facilitated by exposure of the salt to exogenous base. Base residues were found to be adsorbed on the metal surface and hindered the formation of well-ordered carbene-based monolayers. Herein, we show that nitron, a triazolone-based compound that freely tautomerizes to a carbene, can spontaneously self-assemble into monolayers on Pt and Au surfaces, which obviates the necessity for base-induced deprotonation for active carbene formation. SAMs of nitron were found to be thermally stable and could not be displaced by thiols, and thus their high chemical stability was demonstrated. The amino group in surface-anchored nitron was shown to be chemically available for S_N2 reactions, and makes surface-anchored nitron a chemically addressable cross-linking reagent for surface modifications.     

The Best Constant in the Embedding of WN,1(ℝN) into L ∞ ( ℝ N ) $L^{\infty }(\mathbb {R} ^{N})$

Potential Analysis - We compute the best constant in the embedding of $W^{N,1}(\mathbb {R} ^{N})$ into $L^{\infty }(\mathbb {R} ^{N})$ , extending a result of Humbert and Nazaret in dimensions one...     

Tree-indexed random walks on groups and first passage percolation

Summary Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.Partially supported by a grant from the Landau Center for Mathematical AnalysisPartially supported by NSF grant DMS-921 3595     

ω‐saturated quasi‐minimal models of Th(ℚω, +, σ, 0)

We show that (ℚ^ω, +, σ, 0) is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface‐plasmon wavefront and spectral shaping by near‐field holography

Surface‐plasmon‐polariton waves are two‐dimensional electromagnetic surface waves that propagate at the interface between a metal and a dielectric. These waves exhibit unusual and attractive properties, such as high spatial confinement and enhancement of the optical field, and are widely used in a variety of applications, such as sensing and subwavelength optics. The ability to precisely control the spatial and spectral properties of the surface‐plasmon wave is required in order to support the growing interest in both research and applications of plasmonic waves, and to bring it to the next level. Here, we review the challenges and methods for shaping the wavefront and spectrum of plasmonic waves. In particular, we present the recent advances in plasmonic spatial and spectral shaping, which are based on the realization of plasmonic holograms for the optical nearfield.

     

Cone-and-plate apparatus: a compact system for studying well-characterized turbulent flow fields

We have investigated the turbulent flow field inside a cone-and-plate Couette flow apparatus (CPA). The CPA presents a compact, easily varied and easily measured flow system with well-characterized turbulent and laminar flow fields and a good full turbulent spectrum. The CPA can also produce the same time-average stress for both laminar and turbulent regimes. The apparatus consisted of a shallow 2° cone of 0.75 m radius rotating above a flat plate in air. A high-resolution X-wire probe was used to measure the turbulent and time-mean flow properties. A single parameter , which is a measure of the centrifugal forces acting on the moving fluid, governs the degree of secondary flow and turbulence in the device. In the fully turbulent region, R > 4, the mean velocity profile has steep gradients near each surface and is nearly uniform in the central core; and the normalized azimuthal velocity closely follows the ubiquitous law of the wall near the plate. The turbulence intensity exhibits two peaks at the edges of the core flow, and the frequency spectrum of velocity fluctuations near the plate resembles that found in isotropic turbulence. This study also suggests that the rotational symmetry with the stationary bottom plate is a much more convenient experimental geometry than a circular Couette flow.     

The antitumour drug 7‐ethyl‐10‐hydroxycamptothecin monohydrate and its solid‐state hydrolysis mechanism on heating

7‐Ethyl‐10‐hydroxycamptothecin [systematic name: (4S)‐4,11‐diethyl‐4,9‐dihydroxy‐1H‐pyrano[3′,4′:6,7]indolizino[1,2‐b]quinoline‐3,14(4H,12H)‐dione, SN‐38] is an antitumour drug which exerts activity through the inhibition of topoisomerase I. The crystal structure of SN‐38 as the monohydrate, C₂₂H₂₀N₂O₅·H₂O, reveals that it is a monoclinic crystal, with one SN‐38 molecule and one water molecule in the asymmetric unit. When the crystal is heated to 473 K, approximately 30% of SN‐38 is hydrolyzed at its lactone ring, resulting in the formation of the inactive carboxylate form. The molecular arrangement around the water molecule and the lactone ring of SN‐38 in the crystal structure suggests that SN‐38 is hydrolyzed by the water molecule at (x, y, z) nucleophilically attacking the carbonyl C atom of the lactone ring at (x − 1, y, z − 1). Hydrogen bonding around the water molecules and the lactone ring appears to promote this hydrolysis reaction: two carbonyl O atoms, which are hydrogen bonded as hydrogen‐bond acceptors to the water molecule at (x, y, z), might enhance the nucleophilicity of this water molecule, while the water molecule at (−x, y + , −z), which is hydrogen bonded as a hydrogen‐bond donor to the carbonyl O atom at (x − 1, y, z − 1), might enhance the electrophilicity of the carbonyl C atom.     

Spectral estimation and analysis of LDA data in pulsatile flow through heart valves

The Laser Doppler Anemometry (LDA) measurement technique presents an inherent difficulty when spectral analysis is applied to it. The random nature of the LDA signal prohibits sampling at regular, equi-spaced, time instants. Irregular sampling presents additional variability of the spectral estimator. In order to reduce this variability, spectral analysis of LDA data is performed according to the method of direct Fourier transform of short blocks of data, as suggested by Gaster and Roberts. The LDA data is measured in a flow field distal to prosthetic heart valves with varying degrees of stenosis. The spectral estimates of velocity data sampled during the rapid closure stage of the valve are achieved with excellent frequency resolution. Important and useful information about dominant frequency peaks and preferred modes which exist in the flow, otherwise smeared or concealed in the spectral contents, are then derived from the spectral information. These modes are quantitatively analysed in light of vortex formation and related flow mechanisms. Comparative studies of normal and stenosed valves show that the preferred modes are governed by the valve geometry and dynamic behavior and are correlated to the severity of the stenosis.