Extremal Graphs for Homomorphisms II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize . We prove that if H is loop‐threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges that maximizes . Similarly, we show that loop‐quasi‐threshold image graphs have quasi‐threshold extremal graphs. In the case , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for . Also in this article, using similar techniques, we determine a set of extremal graphs for “the fox,” a graph formed by deleting the loop on one of the end‐vertices of . The fox is the unique connected loop‐threshold image graph on at most three vertices for which the extremal problem was not previously solved.     

The use of antenna theory to calculate the electric fields in a thermal field emission metal whisker diode

In recent years, research groups have used metal-metal point contact diodes for frequency mixing and detection of infrared laser radiation. It has been postulated that the mechanism for the nonlinear current-voltage characteristic of the diode is the tunneling of electrons through an intermediate oxide film from the whisker tip to the metal base, i.e., the configuration is considered to be a metal-oxide-metal (MOM) tunneling junction. Several features of the diodes' operation create considerable doubt concerning the applicability of the MOM tunneling mechanism. Analysis of the available data led us to postulate an alternate solid state mechanism, namely a thermally enhanced field emission process. Such emission would be a consequence of the immersion of the whisker in the laser radiation resulting in (1) conduction heating which induces thermionic emission and (2) generation of an electric field at the tip necessary for electron tunneling. In an earlier paper, we calculated the power absorbed by the cylindrical shank of a point contact diode in an infrared radiation field. Using the absorbed power as a source, detailed calculations were made of the laser induced temperature distributions on the diode; more approximate treatments were used to obtain the electric fields developed on the tip. Values of the computed temperature and field parameters for tungsten were found to be consistent with a thermal field emission process. In this paper we present a more rigorous calculation of the voltages and fields induced on different metal whisker tips by the incident laser radiation. Linear antenna theory is used to describe the receiving properties of the diode. The actual pointed geometry of the diode tip has been taken into account using Schelkunoff's theory of the conical antenna. The electric fields at the tip are found to be comparable with those necessary for field emission. The highest fields are established on gold tips, consistent with the experiments of Green et al. who found the best responsivity occurs with gold-gold contacts. Finally we discuss the significance of the experimental results of Young et al. on metal-vacuum-metal tunneling characteristics to the MOM tunneling hypothesis.     

Static structure factor and electronic transport properties of liquid rubidium from a first principles pseudopotential calculation

A first principles nonlocal pseudopotential calculation is used to obtain the ion-ion potential for liquid rubidium of 45°C. This is then used in a Monte Carlo simulation to determine the liquid structure S(q) and transport properties. It is found that the accuracy of the S(q), in the low q-region, is crucial to the calculation of electronic transport properties.     

Reactivity modes of bridged bimetallic [(η-C5H5)Fe(CO)]2-μ-dppe with electrophiles. Preparation and reactions of bimetallic hydride complexes

The complex [(η⁵-C₅H₅)Fe(CO)]₂-μ-dppe (dppe = ethane-1,2-bisdiphenylphosphide) (I) reacts with electrophiles through a η-CO and forms Lewis acid O-Adducts with alkylating reagents (giving cationic μ₂-alkoxycarbyne compounds) or with alkulaluminum compounds. Treatment of I with acid affords a stable μ₂-hydride salt (IV), [CpFe(CO)]₂(μ₂-dppe)⁺, which serves as an intermediate in the stepwise hydrogenation (reversibly) of I to a bridged bimetallic dihydride, [CpFe(CO)H]₂-μ₂-dppe. This dihydride serves as a hydride donor, regenerating IV, towards Ph₃c⁺ or CpFe(CO)₂(η²-CH₂---CH₂)⁺ hydride acceptors. The necessity of the μ₂-dppe as a “mechanical linkage” in facilitating some bimetalic reactions is also established.     

Thermal field emission as a mechanism for infrared laser light detection in metal whisker diode

The non-linear current-voltage characteristic of thermally enhanced field emission is proposed to explain the operation of a metal-metal point contact diode used for laser harmonic frequency generation and frequency mixing in the infrared region. This mechanism can explain several experimental observations which appear inconsistent with the previous analysis based on a planar metal-oxide-metal tunneling geometry.     

Nanocompatible chemistry toward fabrication of target-specific gold nanoparticles

Nanocompatible chemistry which utilizes a novel nontoxic phosphino amino acid as a reducing agent has resulted in the development of therapeutically useful gold nanoparticles under biologically benign media. Stabilization of gold nanoparticles by the edible gum arabic matrix has provided an effective pathway toward in vivo stable target-specific gold nanoparticles.     

Vibrational spectroscopy study of hydrothermally produced scorodite (FeAsO4·2H2O), ferric arsenate sub‐hydrate (FAsH; FeAsO4·0.75H2O) and basic ferric arsenate sulfate (BFAS; Fe[(AsO4)1−x(SO4)x(OH)x]·wH2O)

Three crystalline ferric arsenate phases: (1) scorodite; FeAsO₄·2H₂O, (2) ferric arsenate sub‐hydrate (FAsH; FeAsO₄·0.75H₂O) and (3) basic ferric arsenate sulfate (BFAS; Fe[(AsO₄)_1−x(SO₄)_x(OH)_x]·wH₂O) synthesized by hydrothermal precipitation (175–225 °C) from Fe(III)‐AsO₄³⁻–SO₄²⁻ solutions have been investigated via Raman and infrared spectroscopies. The spectroscopic nature of these high‐temperature Fe(III)‐ AsO₄³⁻–SO₄²⁻ phases has not been extensively studied despite their importance to the hydrometallurgical industrial processing of precious metal (Au and Cu) arsenic sulfidic ores. It was found that scorodite, FAsH and BFAS all gave rise to very distinct arsenate, sulfate and hydroxyl vibrations. In scorodite and FAsH, the distribution of the internal arsenate modes was found to be distinct, with the factor effect being more predominant in the crystal system. For the crystallographically unknown BFAS phase, vibrational spectroscopy was used to monitor the arsenate ↔ sulfate solid solution behavior that occurs in this phase where the molecular symmetry of arsenate and sulfate in the crystal structure is reduced from an ideal T_d to a distorted T_d or C₂/C_2v symmetry. With the new collected vibrational data of the pure phases, the use of attenuated total reflectance infrared (ATR‐IR) spectroscopy was finally extended to investigate the nature of the arsenate in an industrial residue generated by pressure oxidation of a gold ore, where it was found that the arsenate was present in the form of BFAS. Copyright © 2009 John Wiley & Sons, Ltd.