The volume inside a black hole

The horizon (the surface) of a black hole is a null surface, defined by those hypothetical “outgoing” light rays that just hover under the influence of the strong gravity at the surface. Because the light rays are orthogonal to the spatial two-dimensional surface at one instant of time, the surface area of the black hole is the same for all observers (i.e. the same for all coordinate definitions of “instant of time”). This value is 4π(2Gm/c ²)² for nonspinning black holes, with G = Newton’s constant, c = speed of light, and m = mass of the black hole. The three-dimensional spatial volume inside a black hole, in contrast, depends explicitly on the definition of time, and can even be time dependent, or zero. We give examples of the volume found inside a standard, nonspinning spherical black hole, for several different standard time-coordinate definitions. Elucidating these results for the volume provides a new pedagogical resource of facts already known in principle to the relativity community, but rarely worked out.     

Zeta functions of finite graphs and coverings, III

A graph theoretical analog of Brauer-Siegel theory for zeta functions of number fields is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.     

Voltage effects on the nanoelectrospray characteristics in fully voltage-controlled atomisation of gold nanocolloids

The voltage effects on fully voltage-controlled nanoelectrospraying of aqueous gold nanocolloids were investigated. In the nanoelectrospray using 30 μm nozzle, with an increasing of voltage two stable spray regimes of pulsating and cone-jet mode were clearly observed by a combination of current measurement and fast imaging technology. In this nanoelectrospray uniquely determined by the electrical field voltage the current-voltage characteristics were analysed and evaluated based on an equivalent circuit method. At high field in cone-jet regime all the equivalent resistances derived by fitting appear close to a value of 0.53 ± 0.03 GΩ, showing independence to the conductivity of the nanocolloids. In low field pulsating regime a high pulsation frequency up to 100 kHz with relatively stable current pulse was exhibited in all the gold nanocolloids. A linear relationship between the DC components in the pulsating current and the voltage was found and the DC equivalent resistance obtained from the fitting varies between 0.90 and 1.47 GΩ. A strong correlation between the pulsating properties and the conductivity of the colloids was identified.     

Synthesis of 6- and 7-membered cyclic enaminones: scope and mechanism

Six- and seven-membered cyclic enaminones can be prepared using common, environmentally benign reagents. Amino acids are used as synthetic precursors allowing diversification and the incorporation of chirality. The key reaction in this multistep process involves deprotection of Boc-amino ynones and subsequent treatment with methanolic K(2)CO(3) to induce cyclization. A β-amino elimination side reaction was identified in a few labile substrates that led to either loss of stereochemical purity or degradation. This process can be mitigated in specific cases using mild deprotection conditions. NMR and deuterium-labeling experiments provided valuable insight into the workings and limitations of this reaction. Although disguised as a 6-endo-dig cyclization, the reagents employed in the transformation play a direct role in bond-making and bond-breaking, thus changing the mode of addition to a 6-endo-trig cyclization. This method can be used to construct an array of monocyclic and bicyclic scaffolds, many of which are found in well-known natural products (e.g., indolizidine, quinolizidine, and Stemona alkaloids).     

Synthesis of MSn(OH)<Subscript>6</Subscript> (where M = Mg,Ca, Zn,Mn, or Cu) materials at room temperature

Applying a metathesis approach (MCl₂·xH₂O+Na₂SnO₃·xH₂O→MSn(OH)₆+NaCl+xH₂O), Schoenfliesite-type materials with general formula MSn(OH)₆ (where M=Ca, Cu, Mg, Mn or Zn) were synthesized at room-temperature. The high lattice-energy of the by-product alkali halide NaCl drives the reaction in the forward direction leading to the formation of the desired materials. The materials synthesized were characterized by powder X-ray diffraction (XRD) and scanning electron microscopy (SEM) to elucidate structural and micro structural features.     

Understanding the synthesis of DEGVE pulsed plasmas for application to ultra thin biocompatible interfaces

There is interest in the development of novel surface treatments for biocompatibility and non-fouling behaviors on various surfaces of in vivo devices. Polyethylene glycol thin films have shown promise as non-fouling passivation layers for such devices. Studies of the surface chemistry and non-fouling effectiveness of plasma deposited di(ethylene glycol) vinyl ether (DEGVE) films have observed that non-fouling performance is maximized when plasma deposition occurs at low values of average power, (<5 W). [Y.J. Wu, R.B. Timmons, J.S. Jen, Frank E. Molock, Non-fouling surfaces produced by gas phase pulsed plasma polymerization of an ultra low molecular weight ethylene oxide containing monomer, Colloids and Surfaces B: Biointerfaces 18 (2000) 235–248.] Chemical properties of plasma deposited films were directly attributed to the complex interactions occurring within the gas phase. In order to better understand the deposition process, as well as the significance of the conclusions drawn by Wu et al. [Y.J. Wu, R.B. Timmons, J.S. Jen, Frank E. Molock, Non-fouling surfaces produced by gas phase pulsed plasma polymerization of an ultra low molecular weight ethylene oxide containing monomer, Colloids and Surfaces B: Biointerfaces 18 (2000) 235–248.] an investigation of the gas phase behavior in DEGVE pulsed plasma discharges was performed. Infrared spectra were used to characterize the chemical composition and dissociative behavior of DEGVE plasmas across a range of average powers. This allowed for the construction of a dissociative model of the DEGVE monomer in the plasma discharge. Analysis of the observed dissociative pattern demonstrates the presence of key daughter species which would account for the observations made on deposited DEGVE films by Wu et al. [Y.J. Wu, R.B. Timmons, J.S. Jen, Frank E. Molock, Non-fouling surfaces produced by gas phase pulsed plasma polymerization of an ultra low molecular weight ethylene oxide containing monomer, Colloids and Surfaces B: Biointerfaces 18 (2000) 235–248.].     

Synthesis of (±)-oleocanthal via a tandem intramolecular Michael cyclization-HWE olefination

A synthesis of racemic oleocanthal has been accomplished in 11 steps from 1,3 propanediol by a key tandem intramolecular Michael cyclization-Horner-Wadsworth-Emmons olefination.