On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

Detection of corrosion beneath a paint layer by use of spectral polarization optical imaging

The spectral and polarization properties of scattered light were used to image corrosion beneath the surface of a painted aluminum plate. The quality of imaging of the corrosion on the metal surface was significantly enhanced by the spectral polarization optical imaging technique. Depolarization scattered light was used to detect and image corrosion beneath the paint layer.     

Form factors,branching ratio and forward–backward asymmetry in B→K<Subscript>1</Subscript>ℓ<Superscript>+</Superscript>ℓ<Superscript>-</Superscript> decays

We study long-distance effects in the rare exclusive semileptonic decays B→K₁ℓ⁺ℓ^-, where K₁ is the axial vector meson. The form factors describing the meson transition amplitudes of the effective Hamiltonian are calculated using the Ward identities, which are then used to calculate the branching ratio and the forward–backward asymmetry in these decay modes. The zero of the forward–backward asymmetry is of special interest and provides us with a precision test of the standard model.     

A sagittally focusing double‐multilayer monochromator for ultrafast X‐ray imaging applications

The development of a sagittally focusing double‐multilayer monochromator is reported, which produces a spatially extended wide‐bandpass X‐ray beam from an intense synchrotron bending‐magnet source at the Advanced Photon Source, for ultrafast X‐ray radiography and tomography applications. This monochromator consists of two W/B₄C multilayers with a 25 Å period coated on Si single‐crystal substrates. The second multilayer is mounted on a sagittally focusing bender, which can dynamically change the bending radius of the multilayer in order to condense and focus the beam to various points along the beamline. With this new apparatus, it becomes possible to adjust the X‐ray beam size to best match the area detector size and the object size to facilitate more efficient data collection using ultrafast X‐ray radiography and tomography.     

Locking bandwidth and birefringence effects for polarized optical injection in vertical-cavity surface-emitting lasers

The aim of this paper is to investigate the effects of external optical injection taking account of polarization and electron spin properties in vertical-cavity surface-emitting lasers (VCSELs). Using external polarized injection we seek the locked phases and amplitudes of specific polarized fields in terms of injection level and frequency detuning, taking account of two kinds of distinguishable carrier density (spin-up and spin-down). For the conventional form of optical injection without taking account of spin-polarized fields there are three fundamental equations describing the carrier density, field amplitude and phase. However, by using the spin flip model (SFM), the combined effect of polarized fields along two perpendicular crystal axes and electron spin properties results in six equations. We analyse the conditions for stable locking and also the influence of birefringence effects on the stability map of detuning versus optical injection for both cases of injection polarized parallel and perpendicular to the lasing mode of the solitary VCSEL. For given values of pumping and spin relaxation rate there is a minimum birefringence rate for orthogonal injection. Above this value three regions of elliptical polarization are found in the stability map, namely “quasi-stability” (QS), “coupled limit cycle” (CLC) and “coupled chaos” (CC). The three regions of linear polarization, namely chaos, limit cycle and stability, are reduced in area compared to the case of parallel injection. For orthogonal injection it is found that increased birefringence or reduced spin relaxation rate causes the stable locking region to begin at higher injected power and frequency detuning.     

Effect of the spectral profile of a deriving field on the bistability of a dissipative medium

 Through the differential equation describing the behavior of the nonlinear polarization of a medium with respect to an incident field, the Maxwell field equations and the boundary conditions of the field inside a resonator, the relationship between the output and the incident fields is obtained, describing a bistability phenomena. The effect of the spectral profile of the incident field and its spectral halfwidth on the bistability phenomena is studied.     

Facile one‐pot method of initiator fixation for surface‐initiated atom transfer radical polymerization on carbon hard spheres

An efficient and novel one‐pot process is developed to immobilize the atom transfer radical polymerization (ATRP) initiators onto the surface of fully pyrolyzed carbon hard spheres (CHSs) via a radical trapping process from the in situ thermal decomposition of bis(bromomethylbenzoyl)peroxide. The CHSs do not require any additional preparative treatment prior to the initiator immobilization. Styrene and methyl methacrylate are polymerized onto initiator‐immobilized CHSs by surface‐initiated atomic transfer radical polymerization (SI‐ATRP). Samples are characterized using Fourier transform infrared, thermogravimetric analysis, scanning electron microscopy, and transmission electron microscopy. These methods of characterization confirmed that all the CHSs are coated with a uniform layer of grafted polymer. This efficient, one‐pot immobilization of ATRP‐initiators represents an exceptionally simple route for the rapid preparation of various polymer‐coated carbon‐based nanomaterials using SI‐ATRP. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 3314–3322     

Host–guest system of etodolac in native and modified β-cyclodextrins: preparation and physicochemical characterization

Etodolac, being a practically insoluble candidate, exhibits certain toxic effects and a limited bioavailability. Upon chronic use, it causes gastro-intestinal injury and increases the risk of ulcer complications. The approach of this study was to improve the physicochemical properties of the drug utilizing complexation phenomenon with β-, methyl-β- and hydroxypropyl-β-cyclodextrins, which may enhance the aqueous solubility and dissolution rate of etodolac, in an effort to increase oral bioavailability. In certain instances, this approach can be used to increase drug solubility, improve organoleptic properties and maximize the gastrointestinal tolerance by reducing drug irritation after oral administration. Differential UV measurements as well as continuous variation plots revealed the formation of equimolar complex with hydroxypropyl-β-cyclodextrin and 1:2 complexes with β-cyclodextrin and its methyl derivative. Differential scanning calorimetry (DSC), X-ray and FT-IR measurements were applied to prove inclusion complex formation and characterize the complexes. These results lend support to the idea that solubilization of etodolac is mainly related to inclusion complex formation and to a lesser extent to cyclodextrin aggregates. Understanding the factors that influence the performance of etodolac, will allow us to state that molecular encapsulation of the drug and other modifications with appropriate hydroxylation or methylation of parent β-cyclodextrin is able to overcome its problems and facilitate safe and efficient delivery of the drug.     

Phosphonobetaine/sulfur dioxide copolymer by Butler’s cyclopolymerization process

The zwitterionic monomer, ethyl 3-(N,N-diallylammonio)propanephosphonate and sulfur dioxide were cyclocopolymerized in DMSO using azobisisobutyronitrile or ammonium persulfate as initiators to afford a pH-responsive polyphosphonobetaine/SO₂ (PPB/SO₂) copolymer. The polymers, on treatment with HCl and NaOH, gave the aqueous solutions of the corresponding cationic polyphosphononic acid (CPP) and anionic polyphosphonate (APP). The solution properties of the PPB having two pH-responsive functionalities were investigated in detail by potentiometric and viscometric techniques. Basicity constants of the amine and phosphonate groups in APP were found to be “apparent” and as such follow the modified Henderson–Hasselbalch equation. The incorporation of SO₂ moiety has resulted in the decrease of basicity constant of the nitrogens in the copolymer by staggering ?2 units of log K in compare to that of the corresponding homopolymer. The basicity difference is expected to have an effect on the chelating properties of the polymers. In contrast to many polycarbo- and -sulfobetaines, the PPB was all found to be soluble in salt-free water as well as in salt (including Ca²⁺ and Li⁺)-added solutions. The PPB demonstrated ‘antipolyelectrolyte’ viscosity behavior and found to have higher viscosity values in LiCl than in NaCl or NaI.