Short cycles in repeated exponentiation modulo a prime

Given a prime p, we consider the dynamical system generated by repeated exponentiations modulo p, that is, by the map \({u \mapsto f_g(u)}\), where f _g (u) ≡ g ^u (mod p) and 0 ≤ f _g (u) ≤ p ? 1. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.     

Fast oxy-cope rearrangements of bis-alkynes: competition with central C-C bond fragmentation and incorporation in tunable cascades diverging from a common bis-allenic intermediate

Fast anionic oxy-Cope rearrangements of 1,5-hexadiyn-3,4-olates can be incorporated into cascade transformations which rapidly assemble densely functionalized cyclobutenes or cyclopentenones via a common bis-allenic intermediate. The competition between fragmentation, 4π-electrocyclic closure, and aldol condensation can be efficiently controlled by the nature of the acetylenic substituents. The rearrangement of bis-alkynes with two hydroxyl substituents opens a conceptually interesting entry in the chemistry of ε-dicarbonyl compounds and suggests a new approach to analogues of rocaglamide/aglafolin.     

Photoisomerization and photochemistry of matrix-isolated 3-furaldehyde

3-Furaldehyde (3FA) was isolated in an argon matrix at 12 K and studied using FTIR spectroscopy and quantum chemistry. The molecule has two conformers, with trans and cis orientation of the O=C-C=C dihedral angle. At the B3LYP/6-311++G(d,p) level of theory, the trans form was computed to be ca. 4 kJ mol(-1) more stable than the cis form. The relative stability of the two conformers was explained using the natural bond orbital (NBO) method. In fair agreement with their calculated relative energies and the high barrier of rotamerization (ca. 34 kJ mol(-1) from trans to cis), the trans and cis conformers were trapped in an argon matrix from the compound room temperature gas phase in proportion ~7:1. The experimentally observed vibrational signatures of the two forms are in a good agreement with the theoretically calculated spectra. Broad-band UV-irradiation (λ > 234 nm) of the matrix-isolated compound resulted in partial trans → cis isomerization, which ended at a photostationary state with the trans/cis ratio being ca. 1.85:1. This result was interpreted based on results of time-dependent DFT calculations. Irradiation at higher energies (λ > 200 nm) led to decarbonylation of the compound, yielding furan, cyclopropene-3-carbaldehyde, and two C(3)H(4) isomers: cyclopropene and propadiene.     

Materials for fluorescence resonance energy transfer analysis: beyond traditional donor-acceptor combinations

The use of F?rster or fluorescence resonance energy transfer (FRET) as a spectroscopic technique has been in practice for over 50 years. A search of ISI Web of Science with just the acronym "FRET" returns more than 2300 citations from various areas such as structural elucidation of biological molecules and their interactions, in vitro assays, in vivo monitoring in cellular research, nucleic acid analysis, signal transduction, light harvesting and metallic nanomaterials. The advent of new classes of fluorophores including nanocrystals, nanoparticles, polymers, and genetically encoded proteins, in conjunction with ever more sophisticated equipment, has been vital in this development. This review gives a critical overview of the major classes of fluorophore materials that may act as donor, acceptor, or both in a FRET configuration. We focus in particular on the benefits and limitations of these materials and their combinations, as well as the available methods of bioconjugation.     

Fast Sodium‐Ion Conductivity in Supertetrahedral Phosphidosilicates

Fast sodium‐ion conductors are key components of Na‐based all‐solid‐state batteries which hold promise for large‐scale storage of electrical power. We report the synthesis, crystal‐structure determination, and Na⁺‐ion conductivities of six new Na‐ion conductors, the phosphidosilicates Na₁₉Si₁₃P₂₅, Na₂₃Si₁₉P₃₃, Na₂₃Si₂₈P₄₅, Na₂₃Si₃₇P₅₇, LT‐NaSi₂P₃ and HT‐NaSi₂P₃, based entirely on earth‐abundant elements. They have SiP₄ tetrahedra assembled interpenetrating networks of T3 to T5 supertetrahedral clusters and can be hierarchically assigned to sphalerite‐ or diamond‐type structures. ²³Na solid‐state NMR spectra and geometrical pathway analysis show Na⁺‐ion mobility between the supertetrahedral cluster networks. Electrochemical impedance spectroscopy shows Na⁺‐ion conductivities up to σ (Na⁺)=4×10^?4 S cm^?1. The conductivities increase with the size of the supertetrahedral clusters through dilution of Na⁺‐ions as the charge density of the anionic networks decreases.     

Variations of wettability and protein adsorption on solid siliceous carriers grafted with poly(N-isopropylacrylamide)

Poly(N-isopropylacrylamide), a thermally responsive polymer, was end-grafted to mercaptopropyl derivatives of silica gel, plane glass sheets and glass capillary tubing by free radical polymerization of the monomer in 1,4-dioxane at 100 degrees C. The polymer monolayer attached to the glass carriers provided them with thermally controlled wettability registered by two independent methods: direct measurements of the water contact angle and capillary rise. The water contact angle changed from 54+/-3 degrees to 68+/-3 degrees in the temperature range from 20 to 50 degrees C. The polymer grafting to silica gel (pore diameter 100 A, particle size 5 microm) resulted in 15-30-fold reduction in protein adsorption on the carrier at 35 degrees C. Adsorption isotherms of myoglobin indicate completely different characters of the protein adsorption to silica gel and its polyNIPAM-grafted derivative. Cooling of the grafted carrier containing adsorbed myoglobin to 9 degrees C led to a partial release of the protein to the contacting solution, whereas heating of the system to 35 degrees C resulted in reversible binding of the protein. Adsorption of myoglobin on polyNIPAM-coated silica was ca. 2-fold higher at 35 than at 9 degrees C, most probably due to steric repulsion displayed by the swollen copolymer at the lower temperature.     

Synthesis, structure and magnetic properties of new phosphates K2Mn0.5Ti1.5(PO4)3 and K2Co0.5Ti1.5(PO4)3 with the langbeinite structure

New complex phosphates of the general formula K₂M_0.5Ti_1.5(PO₄)₃ (M=Mn, Co) have been obtained from the melting mixture of KPO₃, K₄P₂O₇, TiO₂ and CoCO₃·mCo(OH)₂ or Mn(H₂PO₄)₂ by means of a flux technique. The synthesized phosphates have been characterized by the single-crystal X-ray diffraction and the FTIR-spectroscopy. The compounds crystallize in the cubic system with the space group P2₁3 and cell parameters a=9.9030(14) Å for K₂Mn_0.5Ti_1.5(PO₄)₃ and a=9.8445(12) Å for K₂Co_0.5Ti_1.5(PO₄)₃. Both phosphates are isostructural with the langbeinite mineral and contain four formula unit K₂M_0.5Ti_1.5(PO₄)₃ per unit cell. The structure can be described using [M₂(PO₄)₃] framework composed of two [MO₆] octahedra interlinked via three [PO₄] tetrahedra. The Curie-Weiss-type behavior is observed in the magnetic susceptibility.     

Ultradoping Boron on Si(100) via Solvothermal Chemistry**

Ultradoping introduces unprecedented dopant levels into Si, which transforms its electronic behavior and enables its use as a next-generation electronic material. Commercialization of ultradoping is currently limited by gas-phase ultra-high vacuum requirements. Solvothermal chemistry is amenable to scale-up. However, an integral part of ultradoping is a direct chemical bond between dopants and Si, and solvothermal dopant-Si surface reactions are not well-developed. This work provides the first quantified demonstration of achieving ultradoping concentrations of boron (∼1e14 cm²) by using a solvothermal process. Surface characterizations indicate the catalyst cross-reacted, which led to multiple surface products and caused ambiguity in experimental confirmation of direct surface attachment. Density functional theory computations elucidate that the reaction results in direct B−Si surface bonds. This proof-of-principle work lays groundwork for emerging solvothermal ultradoping processes.