On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3. Both authors are supported by the NSF grants DMS-0308985 and DMS-0456185. Yaroslav Vorobets is supported by a Clay Research Scholarship.     

Rigidity of Branch Groups Acting on Rooted Trees

Automorphisms of groups acting faithfully on rooted trees are studied. We find conditions under which every automorphism of such a group is induced by a conjugation from the full automorphism group of the rooted tree. These results are applied to known examples such as Grigorchuk groups, Gupta–Sidki group, etc.     

Biosensors based on enzyme field-effect transistors for determination of some substrates and inhibitors

This paper is a review of the authors' publications concerning the development of biosensors based on enzyme field-effect transistors (ENFETs) for direct substrates or inhibitors analysis. Such biosensors were designed by using immobilised enzymes and ion-selective field-effect transistors (ISFETs). Highly specific, sensitive, simple, fast and cheap determination of different substances renders them as promising tools in medicine, biotechnology, environmental control, agriculture and the food industry.The biosensors based on ENFETs and direct enzyme analysis for determination of concentrations of different substrates (glucose, urea, penicillin, formaldehyde, creatinine, etc.) have been developed and their laboratory prototypes were fabricated. Improvement of the analytical characteristics of such biosensors may be achieved by using a differential mode of measurement, working solutions with different buffer concentrations and specific agents, negatively or positively charged additional membranes, or genetically modified enzymes. These approaches allow one to decrease the effect of the buffer capacity influence on the sensor response in an aim to increase the sensitivity of the biosensors and to extend their dynamic ranges.Biosensors for the determination of concentrations of different toxic substances (organophosphorous pesticides, heavy metal ions, hypochlorite, glycoalkaloids, etc.) were designed on the basis of reversible and/or irreversible enzyme inhibition effect(s). The conception of an enzymatic multibiosensor for the determination of different toxic substances based on the enzyme inhibition effect is also described.We will discuss the respective advantages and disadvantages of biosensors based on the ENFETs developed and also demonstrate their practical application.     

Potentiometric determination of diclofenac in pharmaceutical formulation by membrane electrode based on ion associate with base dye

The characteristics,performance and application of membrane electrode based on ion associate of diclofenac with base dye Safranine T are described.The electrode response to diclofenac has the sensitivity of 47±1.0 mV decade~(-1)over the range of 5×10~(-5)to 5×10~(-2)mol/L at pH 6-12,and the detection limit of 3.2×10~(-5)mol/L.The electrode is easy assembled at a relatively low cost has fast response time(2-4 s)and can be used for a period up to 3.5 months without any considerable divergence in potential.The proposed sensor displayed good selectivity for diclofenac in the presence of different substances.It was used to determine diclofenac in pharmaceuticals by means of the standard addition method.     

Phosphorus-based p-tert-butylcalix[5]arene ligands

New calix[5]arene trivalent phosphorus derivatives have been synthesized which should be excellent ligands with which to study and control the interaction of a ligand atom with a metal. The larger cavity of the calix[5]arene (compared to calix[4]arene) provides a good balance between constraint and flexibility. Treatment of p-tert-butylcalix[5]arene with 2 equiv of either tris(dimethylamino)phosphine or dichlorophenylphosphine inserts two RP moieties into the calix[5]arene framework to give calix[5](PR)2(OH) (1, R = Me2N; 2, R = Ph). Further treatment of 1 with 4 equiv of HCl gives calix[5](PCl)2(OH) (3). Heating a solution of the monophosphorus compound calix[5](PNMe2)(OH)3 (4) releases dimethylamine to yield both monomeric calix[5](P)(OH)2 (6) and dimeric [calix[5](P)(OH)2](2) (7), the latter having a tubelike geometry. X-ray crystallographic studies confirm the structures and show that 1 and 2 have approximate cone conformations while 3 has an approximate 1,2-alternate conformation. The orientations of the phosphorus lone pairs and oxygen atoms in all derivatives provide a framework for both soft and hard ligand interactions within the calix[5]arene.     

Boronate Ligands in Materials: Determining Their Local Environment by Using a Combination of IR/Solid‐State NMR Spectroscopies and DFT Calculations

Boronic acids (R‐B(OH)₂) are a family of molecules that have found a large number of applications in materials science. In contrast, boronate anions (R‐B(OH)₃^?) have hardly been used so far for the preparation of novel materials. Here, a new crystalline phase involving a boronate ligand is described, Ca[C₄H₉‐B(OH)₃]₂, which is then used as a basis for the establishment of the spectroscopic signatures of boronates in the solid state. The phase was characterized by IR and multinuclear solid‐state NMR spectroscopy (¹H, ¹³C, ¹¹B and ⁴³Ca), and then modeled by periodic DFT calculations. Anharmonic OH vibration frequencies were calculated as well as NMR parameters (by using the Gauge Including Projector Augmented Wave—GIPAW—method). These data allow relationships between the geometry around the OH groups in boronates and the IR and ¹H NMR spectroscopic data to be established, which will be key to the future interpretation of the spectra of more complex organic–inorganic materials containing boronate building blocks.     

Electromagnetic wormholes and virtual magnetic monopoles from metamaterials

We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity epsilon and magnetic permeability micro, which allow one to construct devices that function as invisible tunnels. These allow EM wave propagation between the regions at the two ends of a tunnel, but the tunnels themselves and the regions they enclose are not detectable to lateral EM observations. Such devices act as wormholes with respect to Maxwell's equations and effectively change the topology of space vis-à-vis EM wave propagation. We suggest several applications, including devices behaving as virtual magnetic monopoles, invisible cables, and scopes for MRI-assisted surgery.     

Tunable soliton self-bending in optical lattices with nonlocal nonlinearity

We address the phenomenon of soliton self-bending in Kerr-type nonlocal nonlinear media with an imprinted transverse periodic modulation of the linear refractive index. We show that the imprinted optical lattice makes possible to control the mobility of soliton by varying the depth and the frequency of the linear refractive index modulation.     

Resonant mode oscillations in modulated waveguiding structures

We put forward the concept of resonant, Rabi-like oscillatons and adiabatic transitions between confined light modes in properly modulated multimode waveguides. The phenomenon is shown to take place in both the linear and the nonlinear regimes. In addition, we find that the mode transitions occur not only in simple geometries, but also in complex confining multimode structures. The phenomenon is analogous to the familiar stimulated state transitions that occur in multilevel quantum systems.     

Observation of higher-order solitons in defocusing waveguide arrays

We observe experimentally higher-order solitons in waveguide arrays with defocusing saturable nonlinearity. Such solitons can comprise several in-phase bright spots and are stable above a critical power threshold. We elucidate the impact of the nonlinearity saturation on the domains of existence and stability of the observed complex soliton states.