Plane partitions IV: A conjecture of Mills—Robbins—Rumsey

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ _n (x, m) The special caseZ _n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ _n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ _n (1,m) given previously Many results for the₃ F ₂ hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ _n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ _n (2,m) In the final analysis, one would like a combinatorial explanation ofZ _n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture     

Comparing ordered-entry queues with heterogeneous servers

We study a queueing system withm exponential servers with distinct service rates. Jobs arrive at the system following an arbitrary point process. Arrived jobs receive service at the first unoccupied server (if any) according to an entry order , which is a permutation of the integers 1, 2,...,m. The system has a finite buffer capacity. When the buffer limit is reached, arrivals will be blocked. Blocked jobs will either be lost or come back as New arrivals after a random travel time. We are concerned with the dynamic stochastic behavior of the system under different entry orders. A partial ordering is established among entry orders, and is shown to result in some quite strong orderings among the associated stochastic processes that reflect the congestion and the service characteristics of the system. The results developed here complement existing comparison results for queues with homogeneous servers, and can be applied to aid the design of conveyor and communication systems.     

The synthesis and transition temperatures of some trans-4-alkylcyclohexylethyl-substituted 2,3-difluorobiphenyls

Terphenyls with two lateral ortho-fluoro-substituents have proved to be excellent host materials for ferroelectric (S_C*) mixtures. The compounds reported here are biphenyls with the same arrangement of lateral substituents but with a trans-4-alkylcyclohexylethyl moiety as one of the terminal substituents. Such three ring systems retain the ability to generate the S_C mesophase and have low melting points. Low temperature lithiation procedures were used to prepare phenylboronic acids, which were then used in palladium catalysed cross-coupling procedures to prepare the desired compounds. The effect of molecular structure on the mesophase types and thermal stabilities is discussed and comparisons are made with analogous terphenyls and biphenyls with open chain terminal substituents.     

Raman H-bond pair volume for water

The dispersion of the H-bond pair volume Delta V over the decoupled OD and coupled OH-stretching contours from HDO in H(2)O was determined from Raman intensities at pressures to 9700 bar at 301 K. The dispersion of Delta V was determined from -RT[partial differential ln(I(i)/I(REF))/ partial differential P](T) versus omega (in cm(-1)), where i refers to omega's over the stretching contours and I(REF) refers to the reference intensity at the isosbestic frequency. The maximum H-bond pair volume (defined for breakage) is 1.4+/-0.1 cm(3)/mol H-bond, which corresponds to the volume difference between a large dispersion maximum at 2,675 cm(-1) (near the OD stretch omega of HDO in dense supercritical water) and a large, broad minimum centered near 2,375 cm(-1) (just below the OD stretch omega of HDO in lda ice). The average DeltaV is 0.71+/-0.10 cm(3)/mol H-bond. Other minima near 2,625 cm(-1) (OD) and 3550 cm(-1) (OH) refers to bent H-bonds whose angles are approximately 150 deg. Isothermal pressurization of water lowers the molal volume by decreasing the concentration of long, weak H-bonds, and increasing the concentrations of bent H-bonds and short, strong, linear H-bonds. Such bending, shortening, and strengthening produces freezing to ice VI near 10 kbar at 301 K. The isobaric temperature derivative of the maximum H-bond volume is (partial differential Delta V/partial differential T)(P)< or =(2-5) x 10(-3) cm(3)/deg mol H-bond. The OH enthalpy dispersion curve for saturated NaBF(4) in water, yields a large maximum at 3,530-3,540 cm(-1) indicating that BF(4) (-) interacts preferentially with the dangling or "free" OH groups of water thus producing weak, strongly bent H-bonds having angles similar to those of the 3,550 cm(-1) high-pressure H-bond bending feature.     

Solution-phase, triangular ag nanotriangles fabricated by nanosphere lithography

A novel method to produce solution-phase triangular silver nanoparticles is presented. Ag nanoparticles are prepared by nanosphere lithography and are subsequently released into solution. The resulting nanoparticles are asymmetrically functionalized to produce either single isolated nanoparticles or dimer pairs. The structural and optical properties of Ag nanoparticles have been characterized. Mie theory and the Discrete Dipole Approximation method (DDA) have been used to model and interpret the optical properties of the released Ag nanoparticles.     

Large-scale synthesis of globotriose derivatives through recombinant E. coli

The carbohydrate chains decorating cell membranes and secreted proteins participate in a range of important biological processes. However, their ultimate significance and possible therapeutic potential have not been fully explored due to the lack of economical methods for their production. This study is an example of the use of a genetically engineered bacterial strain in the preparation of diverse oligosaccharides. Based on an ex vivo biosynthetic pathway, an artificial gene cluster was constructed by linking the genes of five associated enzymes on a plasmid vector. This plasmid was inserted into the E. coli NM522 strain to form globotriose-producing cells ('superbug' pLDR20-CKTUF). The specific strain was conveniently applied to the synthesis of globotriose trisaccharide and its derivatives, as potential neutralizers for Shiga toxin. This work demonstrates a novel and economical method for generating ligand diversity for carbohydrate drug development.     

Acrylamide analysis: assessment of results from six rounds of Food Analysis Performance Assessment Scheme (FAPAS) proficiency testing

Six proficiency tests have now been completed in an ongoing program of the UK Food Analysis Performance Assessment Scheme (FAPAS) for the analysis of acrylamide in a range of food matrixes. Homogeneous test material samples were requested by laboratories throughout the world, with 29 to 45 submitting results for each test. Results were analyzed by appropriate statistical procedures, and z-scores were awarded for reported values. In the absence of both legislation and collaborative trial data, the target standard deviation was derived from the Horwitz equation, although it is acknowledged that there is a need to establish a "fit for purpose" target standard deviation specifically for acrylamide analysis. Participants were encouraged to use the analytical method routinely used in their own laboratory and to provide details of their procedure. Close examination of the data submitted indicates that performance is generally acceptable in terms of accuracy. There is no significant difference between results submitted by gas chromatography and liquid chromatography (GC and LC) methods, and no method dependency on the use of internal standards or sample size. However, choice of extraction solvent may be important, with indications that plain water is an acceptable extraction method. There is evidence from the most recent test that direct (underivatized) GC methodology may present problems, but more data are required and this aspect will be monitored in the continuing proficiency testing program.     

Theory of absorption and emission of radiation in molecular collisions: Semiclassical model

A semiclassical transition amplitude is derived for electronically inelastic molecular collisions in the presence of an intense electromagnetic field. The calculation of this amplitude involves the integration of classical trajectories for nuclei moving on effective potential energy surfaces. These surfaces, which exhibit an avoided crossing due to the coupling through the dipole interaction around the resonance nuclear configuration, are analytically continued to their complex intersection points, where trajectories switch surfaces. Other parts of these surfaces are similar to the field-free adiabatic surfaces, where one is shifted by ?ω. Metastable states, formed at some collision energies, are suggested in the field which are absent in the field-free system.     

Thermal addition of dibenzoylacetylene to cycloheptatriene and the photo-rearrangement of the thermal adduct

The reaction of cycloheptatriene (CHT) with dibenzoylacetylene (DBA), in refluxing toluene gave a mixture of products consisting of 6,7-dibenzoyltricyclo[3.2.2.0^2.4]nona-6,8-diene 6 and 3,4-dibenzoylbicyclo [3.2.2]nona-2,6,8-triene 7. Neat heating of a mixture of CHT and DBA around 95–100° gave besides, 6 and 7, small amounts of two dimers of DBA. Photolysis of 6 in benzene solution gave a lactonic, rearrangement product. 25. The structures of the various products have been established and reasonable mechanisms for their formation have been suggested.     

Cyclopentadienyl-ruthenium and -osmium chemistry: XXVIII. Reactions and isomerisation of 1,2-bis(methoxycarbonyl)ethenyl complexes: X-ray structures of Ru{Z)-C(CO2Me)CH(CO2Me)} -(CO)(PPh3)(η-C5H5) · 0.5EtOH,Ru{(E)-C(CO2Me)CH(CO2Me)}(dppe)(η-C5H5) and Ru{C(CO2Me)C(CO2Me)C(CO2Me)CH(CO2Me)} - (PPh3)(η-C5H5)

A reinvestigation of the reaction between C₂(CO₂Me)₂ and RuH(PPh₃)₂(η-C₅H₅) and some related complexes is reported. Initial cis addition is followed by conversion into the trans isomer. In the case of the bis-(PPh₃) complex, isomerisation is followed by chelation of the ester CO group with concomitant displacement of one PPh₃ligand. The resulting chelate complex reacts with CO or CNBu^t to give the (Z)-RuC(CO₂Me)CH(CO₂Me) complexes; the (E)-isomer of the carbonyl complex is obtained by addition of C₂(CO₂Me)₂to RuH(CO)(PPh₃)(η-C₅H₅). The ^1Hand ¹³C NMR spectra are not a reliable guide to assignment of the stereochemistry of the vinyl group. Other products isolated from the initial reaction are the bis-insertion product $\text{Ru{C(CO}{}_2\text{Me)C(CO}{}_2\text{Me)C(CO}{}_2\text{Me)}$CH(CO₂Me)} -(PPh₃)(η-C₅H₅) and the 1/2 PPh₃/C₂(CO₂Me)₂ adduct. The molecular structures of Ru{(Z)-C(CO₂Me)CH(CO₂Me)}(CO)(PPh₃(η-C₅H₅) · 0.5EtOH, Ru{(E)-C(C₂Me)CH(CO₂Me)}(dppe)(η-C₅H₅) and $\text{Ru{C(CO}{}_2\text{Me)C(CO}{}_2\text{Me)C(CO}{}_2\text{-Me)}$CH(CO₂Me)}(PPh₃)(η-C₅H₅) have been determined. The cis isomer is monoclinic, space group P2₁,with a 9.328(8), b 17.385(10), c 10.356(7) Å, β 101.78(3)° and Z = 2; 2107 data with I ≥ 2.5σ(I) were refined to R = 0.076 R_w = 0.085. The trans isomer is triclinic, space group P$\text{1}$, with a 10.404(7) b 11.221(6), c 13.230(9) Å, α 92.67(5), β 110.56(5), γ 106.21(5)° and Z = 2; 2520 data with I ≥ 2.5σ(I) were refined to R = 0.055 R_w = 0.068. The butadienyl complex is monoclinic, space group P2₁/a, with a 19.655(8), b 8.674(4), c 21.060(5) Å, β 116.22(3)° and Z = 4; 2724 data with I ≥ 2.5σ(I) were refined to R = 0.042, R_w = 0.047.