Oscillations in the height of the Yule tree and application to the binary search tree

For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave in the following sense: there exist continuous functions and such that: where and is the height of the Yule tree. We also shows that similar oscillations occur for , when f is in a large class of functions. This process is classically related to the binary search tree, thus yielding analogous results for the height and for the saturation level of the binary search tree. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 90–120, 2017     

Action-FRET of a Gaseous Protein

Mass spectrometry is an extremely powerful technique for analysis of biological molecules, in particular proteins. One aspect that has been contentious is how much native solution-phase structure is preserved upon transposition to the gas phase by soft ionization methods such as electrospray ionization. To address this question—and thus further develop mass spectrometry as a tool for structural biology—structure-sensitive techniques must be developed to probe the gas-phase conformations of proteins. Here, we report Förster resonance energy transfer (FRET) measurements on a ubiquitin mutant using specific photofragmentation as a reporter of the FRET efficiency. The FRET data is interpreted in the context of circular dichroism, molecular dynamics simulation, and ion mobility data. Both the dependence of the FRET efficiency on the charge state—where a systematic decrease is observed—and on methanol concentration are considered. In the latter case, a decrease in FRET efficiency with methanol concentration is taken as evidence that the conformational ensemble of gaseous protein cations retains a memory of the solution phase conformational ensemble upon electrospray ionization.

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Bounded Correctors in Almost Periodic Homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.     

h-Principles for the Incompressible Euler Equations

In Dissipative Euler Flows and Onsager’s Conjecture. arxiv.1205.3626, preprint, 2012, De Lellis and Székelyhidi construct Hölder continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus ${{\mathbb T}^3}$ . The construction consists of adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the H ^?1-norm by exact solutions. Furthermore, we prove that the flows thus constructed on ${{\mathbb T}^3}$ are genuinely three-dimensional and are not trivially obtained from solutions on ${{\mathbb T}^2}$ .     

Synthesis and multinuclear NMR characterization of iminophosphoranyl phosphines and silanes

Tris(iminotriphenylphosphoranyl)phosphonium chloride 2 , tris(iminotriphenylphosphoranyl)phosphine 3 , bis(iminotriphenylphosphoranyl)dichlorosilane 4 , tris(iminotriphenylphosphoranyl)-chlorosilane 5 and -hydrogenosilane 6 have been prepared by reacting the iminotriphenylphosphorane 1 with trichlorophosphine or the corresponding chlorosilanes. A multinuclear NMR study of these compounds is presented. © 1996 John Wiley & Sons, Inc.     

Strong intramolecular O—H⋯O hydrogen bonds in quinaldic acid N‐oxide and picolinic acid N‐oxide

The title compounds contain very short intramolecular hydrogen bonds of the type C—O—H?O—N. The O?O distances are 2.425 (2) Å in picolinic acid N‐oxide (2‐carboxy­pyridine N‐oxide), C₆H₅NO₃, (I), and 2.435 (2) Å in quinaldic acid N‐oxide (2‐carboxy­quinoline N‐oxide), C₁₀H₇NO₃, (II). In (II), this is associated with slight molecular distortion from planarity, while in (I), such an effect cannot be observed because the mol­ecule crystallizes on a mirror plane.     

Cobalt‐Mediated Radical Polymerization of Vinyl Acetate in Miniemulsion: Very Fast Formation of Stable Poly(vinyl acetate) Latexes at Low Temperature

Summary: Poly(vinyl acetate) macroinitiators end‐capped by a Co(acac)₂ complex (PVAc–Co(acac)₂), prepared in bulk by cobalt‐mediated radical polymerization (CMRP), are used for the controlled radical polymerization of vinyl acetate in miniemulsion to give high‐molecular‐weight polymers and high monomer conversion. Stable poly(vinyl acetate) latexes with solid contents ranging from 25 to 30 wt.‐% are prepared within unusually short reaction times (∼1 h) at low temperatures (0–30 °C).

SEC chromatograms for the PVAc–Co(acac)₂ macroinitiator and PVAc latex obtained under ultrasonication for 6 min at 0 °C (79% monomer conversion).     

Stable P‐Heterocyclic Carbenes: Scope and Limitations

The conjugate acids (PHCH⁺s) of P‐heterocyclic carbenes (PHCs) are prepared by formal [3+2] cycloaddition of a 1,3‐diphosphaallyl or 1,3‐phosphinophosphenium cation with various nitriles. The effect of the phosphorus substituent on the fate of the cyclization and on that of the counteranion and base in the subsequent deprotonation reaction are reported. Two PHCs that are indefinitely stable in the solid state are described. In solution, one of them, made from acetonitrile, undergoes a facile [3+2] cycloreversion, whereas the other, based on dimethyl cyanamide, is stable, presumably owing to its zwitterionic structure, which involves a tricoordinate pentavalent phosphorus atom. The reactivity of PHCs is strongly driven by the high electrophilicity of the phosphorus centers, as demonstrated by their reactivity with water and benzaldehyde. Although both PHCs reported in this paper are direct analogues of the least‐basic NHCs, their basicity is comparable to those of the more strongly basic NHCs (as determined by comparison of the carbonyl stretching frequencies of their corresponding cis‐[RhCl‐(CO)₂(L)] complexes).