Bounded type Siegel disks of finite type maps with few singular values

Let U be an open subset of the Riemann sphere \(\hat {\mathbb{C}}\). We give sufficient conditions for which a finite type map f: U → \(\hat {\mathbb{C}}\) with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-?wi?tek. We also give sufficient conditions for which, instead, Δ has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-?wi?tek.     

Exploring the origin of tip‐enhanced Raman scattering; preparation of efficient TERS probes with high yield

Tip‐enhanced Raman scattering (TERS) spectroscopy is a promising technique for nanoscale chemical analysis. However, there are several challenges preventing widespread application of this technology, including reproducible fabrication of efficient TERS probes. These problems reflect a lack of clear understanding of the origins of, and the parameters influencing TERS. It is believed that the coating characteristics at the apex of the tip have a major effect on the near‐field optical enhancement and thus the TERS activity of a metalized probe. Here we show that the aspect ratio of the tip can play a significant role in the efficiency of TERS probes. We argue that the electrostatic field arising from the lightning‐rod effect has a substantial role in the observed TERS effect. This argument is supported by ‘edge‐enhanced Raman scattering’ which is shown for a noble metal film. Furthermore, it is reported that an associated tip‐surface‐enhanced Raman scattering effect can be achieved by using a TERS‐inactive metalized probe on a surface‐enhanced Raman spectroscopy‐inactive roughened surface. This observation can be explained by an interparticle enhancement of the electromagnetic field. Copyright © 2011 John Wiley & Sons, Ltd.     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

Compact formulations of the Steiner Traveling Salesman Problem and related problems

The Steiner Traveling Salesman Problem (STSP) is a variant of the TSP that is particularly suitable when routing on real-life road networks. The standard integer programming formulations of both the TSP and STSP have an exponential number of constraints. On the other hand, several compact formulations of the TSP, i.e., formulations of polynomial size, are known. In this paper, we adapt some of them to the STSP, and compare them both theoretically and computationally. It turns out that, just by putting the best of the formulations into the CPLEX branch-and-bound solver, one can solve instances with over 200 nodes. We also briefly discuss the adaptation of our formulations to some related problems.     

Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

Strange products of projections

Let H be an infinite-dimensional Hilbert space. We show that there exist three orthogonal projections X ₁,X ₂,X ₃ onto closed subspaces of H such that for every 0 ≠ z ₀ ∈ H there exist k ₁, k ₂, · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z _n = X _kn z _n ?1 does not converge in norm.     

Stereocontrolled Photooxygenations–A Valuable Synthetic Tool§

The stereochemical course of the singlet-oxygen ene reaction with acyclic olefins may be controlled if in the substrate conformational fixation (1,3-allylic strain) an allyl-ic substituent for interaction with the attacking oxygen enophile aligns. Various substrates were chosen to elucidate the features of the olefin that are necessary to control the sense (threo versus erythro) and the extent of the ii-facial preference of the singlet-oxygen attack. Depending on the electronic properties of the double bond and the nature of the allylic substituent, threo or erythro selectivity may be imposed through hydrogen bonding, electrostatic and steric effects and stereoelectronic alignment. Such directing properties, especially that of the hydroxy group, were also confirmed in the other reaction modes of singlet oxygen, namely the [4+2] cycloaddition to chiral naphthylenic alcohols and the [2+2] cycloaddition to an adamantylidene-substituted allylic alcohol. The syntheses of the natural products Merucathin and Iso-dihydromahubanolide B are two examples in which such stereocontrolled photooxygenations have been used as key steps to build up the required chirality diastereose-lectively.     

Preparation of optically active allylic hydroperoxy alcohols and 1, 3-diols by enzyme-catalyzed kinetic resolution and photooxygenation of chiral homoallylic alcohols

All four possible enantiomers of the 3-hydroperoxy-4-penten-1-ols 2a, b and their corresponding 4-pentene-1,3-diols 4a,b have been prepared for the first time in high enantiomeric purity (up to 98% ee) and in preparative amounts according to two distinct ways: First the photooxygenation of the racemic homoallylic alcohols 1 gave the racemic hydroperoxy alcohols 2, which have subsequently been kinetically resolved by horseradish peroxidase (HRP); alternatively, first the lipase-catalyzed resolution afforded the optically active homoallylic alcohols 1 and subsequent photooxygenation led to the enantiomerically enriched hydroperoxy alcohols 2.