Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

Experimental determination of rheological properties of polydimethylsiloxane

A series of linear polydimethylsiloxane of different molecular weight and with reduced polydispersity were prepared by partial fractionation of commercial products. The rheological functions, i.e. zero shear viscosity (η₀), first and second normal stress coefficients (Ψ₁₀ and Ψ₂₀), of the materials were experimentally measured by conventional rotational rheometers and by a rotation rod apparatus. The relationships between molecular structure (molecular weight and polydispersity index) and rheological functions are presented and discussed on the basis of equations proposed in the literature. Zero shear viscosity data conform to the well-known dependence on a power 3.5 of molecular weight. However, a consistently stronger influence of molecular weight over Ψ₁₀ and Ψ₂₀ is found. The influence of polydispersity over Ψ₁₀ is also analyzed.     

Nilpotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of a finite semigroup S is the size of a minimal generating set of S, and the nilpotent rank of S is the size of a minimal generating set of S consisting of nilpotents in S. A partition of a q-element subset of the set X_n = {1,2,..., n} is said to be of type τ if the sizes of its classes form the partition τ of the positive integer q ≤ n. A non-trivial partition τ of q consists of k < q elements. For a non-trivial partition τ of q < n, the semigroup S(τ), generated by all the transformations with kernels of type τ, is nilpotent-generated. We prove that if τ is a non-trivial partition of q < n, then the rank and the nilpotent rank of S(τ) are both equal to the number of partitions X_n of type τ.     

Constructing near spanning trees with few local inspections

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded‐degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph consisting of edges (for a prespecified constant ), where the decision for different edges should be consistent with the same subgraph . Can this task be performed by inspecting only a constant number of edges in G ? Our main results are:

• We show that if every t‐vertex subgraph of G has expansion then one can (deterministically) construct a sparse spanning subgraph of G using few inspections. To this end we analyze a “local” version of a famous minimum‐weight spanning tree algorithm.
• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3‐regular graphs of high girth, in which every t‐vertex subgraph has expansion . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

© 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 183–200, 2017     

A Direct Construction of Nontransitive Dice Sets

In this paper, we give a direct construction for a set of dice realizing any given tournament T. The construction for a tournament with n vertices requires dice with n sides if n is odd, sides if n is divisible by 4, and sides if mod 4. This appears to be the most efficient general construction to date. Our construction relies only on a standard construction from graph theory.     

Effect of oxidized silicon (SiOx) surfaces functionalization on real‐time PCR by Lab‐on‐a‐chip microdevices

There is a great need to improve the biocompatibility of silicon‐based lab‐on‐chip substrate materials for reliable quantitative analysis of biological solutions. These advanced microdevice surfaces need not only be biocompatible but also have surfaces of defined wettability characteristics. The inhibition of biomolecular activity due to microdevice surface interaction is common and can result in inaccurate results or decreased reaction yields. In this work we investigate different techniques for the chemical functionalization of oxidized silicon (SiO_x) surfaces in order to: (i) obtain defined hydrophobic/hydrophilic surfaces; and (ii) increase the efficiency of performing Real‐Time Polymerase Chain Reaction (PCR) on a silicon‐based lab‐on‐chip. Silicon oxide surfaces are functionalized by grafting alkylic chain silanes and poly(ethylene glycol) (PEG) chains to the surfaces, rendering them hydrophobic or hydrophilic. Functionalized surfaces are characterized through contact angle and atomic force microscopy (AFM) measurements, showing stable hydrophobic surfaces with contact angles of 69–78° and layer thicknesses of 11–15 Å and hydrophilic surfaces displaying contact angles of 5–6° and thicknesses of 22–52 Å. PCR experiments carried out directly on bare silicon oxide lab‐on‐chip surfaces show low yields of DNA amplification. Hydrophobic surfaces decrease the inhibition of PCR. Hydrophilic surfaces are a major improvement on the bare silicon oxide exhibiting the same maximum reaction yield as obtained with a standard thermocycler. We have found that the best results are associated with PEG modified surfaces, which prove very suitable for the fabrication of reliable PCR silicon lab‐on‐chips. Copyright © 2011 John Wiley & Sons, Ltd.     

Enantioselective total syntheses of (-)-taiwaniaquinone H and (-)-taiwaniaquinol B by iridium-catalyzed borylation and palladium-catalyzed asymmetric α-arylation

We report a concise, enantioselective total synthesis of (-)-taiwaniaquinone H and the first enantioselective total synthesis of (-)-taiwaniaquinol B by a route that includes asymmetric palladium-catalyzed α-arylation of a ketone with an aryl bromide that was generated by sterically controlled halogenation via iridium-catalyzed C-H borylation. This asymmetric α-arylation creates the benzylic quaternary stereogenic center present in the taiwaniaquinoids. The synthesis was completed efficiently by developing a Lewis acid-promoted cascade to construct the [6,5,6] tricyclic core of an intermediate common to the synthesis of a number of taiwaniaquinoids. Through the preparation of these compounds, we demonstrate the utility of constructing benzylic quaternary stereogenic centers, even those lacking a carbonyl group in the α-position, by asymmetric α-arylation.     

Accelerating chemical reactions: exploring reactive free-energy surfaces using accelerated ab initio molecular dynamics

A biased potential molecular dynamics simulation approach, accelerated molecular dynamics (AMD), has been implemented in the framework of ab initio molecular dynamics for the study of chemical reactions. Using two examples, the double proton transfer reaction in formic acid dimer and the hypothetical adiabatic ring opening and subsequent rearrangement reactions in methylenecyclopropane, it is demonstrated that ab initio AMD can be readily employed to efficiently explore the reactive potential energy surface, allowing the prediction of chemical reactions and the identification of metastable states. An adaptive variant of the AMD method is developed, which additionally affords an accurate representation of both the free-energy surface and the mechanism associated with the chemical reaction of interest and can also provide an estimate of the reaction rate.