Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function \(\psi \in W^{m,p}({\mathbb{R}}^d)\) such that if \(\widehat{\psi}(0) = 1\) and ψ satisfies the Strang–Fix conditions to order m ? 1, then a scale averaged approximation formula holds for all \(f \in W^{m,p}({\mathbb{R}}^d)\) :

$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$

The dilations { a _j } are lacunary, for example a _j =  2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in \({W^{m - 1,p}({\mathbb{R}}^d)}\) the scale averaging is unnecessary and one has the simpler formula \(f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)\) . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system \(\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}\) in \(W^{m,p}({\mathbb{R}}^d)\) . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

     

Use of a simple surface-active initiator in controlled/living free-radical miniemulsion polymerization under AGET and ARGET ATRP conditions

This communication describes the first example of the efficient use of a simple amphiphilic molecule as both a surfactant and an initiator in the miniemulsion polymerization of methyl methacrylate under AGET and ARGET ATRP conditions.     

Gold nanoparticle aggregation-based highly sensitive DNA detection using atomic force microscopy

The potential ability of atomic force microscopy (AFM) as a quantitative bioanalysis tool is demonstrated by using gold nanoparticles as a size enhancer in a DNA hybridization reaction. Two sets of probe DNA were functionalized on gold nanoparticles and sandwich hybridization occurred between two probe DNAs and target DNA, resulting in aggregation of the nanoparticles. At high concentrations of target DNA in the range from 100 nM to 10 μM, the aggregation of gold nanoparticles was determined by monitoring the color change with UV-vis spectroscopy. The absorption spectra broadened after the exposure of DNA–gold nanoparticles to target DNA and a new absorption band at wavelengths >600 nm was observed. However, no differences were observed in the absorption spectra of the gold nanoparticles at low concentrations of target DNA (10 pM to 10 nM) due to insufficient aggregation. AFM was used as a biosensing tool over this range of target DNA concentrations in order to monitor the aggregation of gold nanoparticles and to quantify the concentration of target DNA. Based on the AFM images, we successfully evaluated particle number and size at low concentrations of target DNA. The calibration curve obtained when mean particle aggregate diameter was plotted against concentration of target DNA showed good linearity over the range 10 pM to 10 nM, the working range for quantitative target DNA analysis. This AFM-based DNA detection technique was three orders of magnitude more sensitive than a DNA detection method based on UV-vis spectroscopy.     

A new dirhodium tetraacetate carbenoid: Synthesis, crystal structure and catalytic application

A new dirhodium tetraacetate II involving 1,3-bis(2,6-diisopropylphenyl)-imidazol-2-ylidene I was synthesized and characterized by general spectroscopic tools in the solution state as well as single X-ray crystallographic analysis in the solid state. The catalytic activity of dirhodium tetraacetate carbenoid II was tested for the allylic oxidation, and the improved reactivity to the allylic oxidation was observed compared to that of Rh₂(OAc)₄. The different electrochemical properties of dirhodium tetraacetate carbenoid II and Rh₂(OAc)₄ were compared via cyclic voltammetry.     

Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on L₁(\mathbbR)L_{1}({\mathbb{R}}), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel.     

Multichannel vector sum phase shifter

A novel multichannel vector sum phase shifter that is suitable for phased array antenna applications is demonstrated. Each channel is implemented using a distinct optical wavelength. Selective control of each channel is performed using an acousto-optic polarization coupler. The concept is successfully demonstrated for two individually controlled channels. For each channel, a continuously variable frequency linear phase shift is demonstrated between DC and 7 GHz, with the phasing range exceeding 100 degrees.     

Impurity-induced anisotropic semiconductor-semimetal transition in monolayer biased black phosphorus

Taking into account the electron-impurity interaction within the continuum approximation of tight-binding model, the Born approximation, and the Green's function method, the main features of anisotropic electronic phase transition are investigated in monolayer biased black phosphorus (BP). To this end, we concentrated on the disordered electronic density of states (DOS), which gives useful information for electro-optical devices. Increasing the impurity concentration in both unbiased and biased impurity-infected single-layer BP, in addition to the decrease of the band gap, independent of the direction, leads to the midgap states and an extra Van Hove singularity inside and outside of the band gap, respectively. Furthermore, strong impurity scattering potentials lead to a semiconductor-semimetal transition and one more Van Hove singularity in x-direction of unbiased BP and surprisingly, this transition does not occur in biased BP. We found that there is no phase transition in y-direction. Since real applications require structures with modulated band gaps, we have studied the influence of different bias voltages on the disordered DOS in both directions, resulting in the increase of the band gap.     

Combined effect of the perpendicular magnetic field and dilute charged impurity on the electronic phase of bilayer AA-stacked hydrogenated graphene

We address the electronic phase engineering in the impurity-infected functionalized bilayer graphene with hydrogen atoms (H-BLG) subjected to a uniform Zeeman magnetic field, employing the tight-binding model, the Green's function technique, and the Born approximation. In particular, the key point of the present work is focused on the electronic density of states (DOS) in the vicinity of the Fermi energy. By exploiting the perturbative picture, we figure out that how the interaction and/or competition between host electrons, guest electrons, and the magnetic field potential can lead to the phase transition in H-BLG. Furthermore, different configurations of hydrogenation, namely reduced table-like and reduced chair-like, are also considered when impurities are the same and/or different. A comprehensive information on the various configurations provides the semimetallic and gapless semiconducting behaviors for unfunctionalized bilayer graphene and H-BLGs, respectively. Further numerical calculations propose a semimetal-to-metal and gapless semiconductor-to-semimetal phase transition, respectively, when only turning on the magnetic field. Interestingly, the results indicate that the impurity doping alone affects the systems as well, leading to semimetal-to-metal and no phase transition in the pristine system and hydrogenated ones, respectively. However, the combined effect of charged impurity and magnetic field shows that the pristine bilayer graphene is not influenced much as the functionalized ones and phase back transitions appear. Tuning of the electronic phase of H-BLG by using both types of electronic and magnetic perturbations play a decisive role in optical responses.