Plethystic Vertex Operators and Boson-Fermion Correspondences

We study the algebraic properties of plethystic vertex operators, introduced in J. Phys. A: Math. Theor. 43 405202 (2010), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape \pi. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each \pi, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.Comment: 21 pages, LaTeX. Minor typos corrected. Added brief survey of related work and new reference

The Hopf Algebra Structure of the Character Rings of Classical Groups

The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym$ of symmetric functions. Here we study the character rings \CO and \CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \CO and \CSp also admit natural Hopf algebra structures that are isomorphic to that of \CGL, and hence to \Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \CO and \CSp analogous to those used for \Sym. A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the \CGL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^* and \CSp^* are identified. Finally, the Hopf algebra of the universal rational character ring \CGLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.Comment: 38 pages, uses pstricks; new version is a major update, new title, new material on rational character

Acidified and ultrafiltered recovered coagulants from water treatment works sludge for removal of phosphorus from wastewater

This study used a range of treated water treatment works sludge options for the removal of phosphorus (P) from primary wastewater. These options included the application of ultrafiltration for recovery of the coagulant from the sludge. The treatment performance and whole life cost (WLC) of the various recovered coagulant (RC) configurations have been considered in relation to fresh ferric sulphate (FFS). Pre-treatment of the sludge with acid followed by removal of organic and particulate contaminants using a 2kD ultrafiltration membrane resulted in a reusable coagulant that closely matched the performance FFS. Unacidified RC showed 53% of the phosphorus removal efficiency of FFS, at a dose of 20 mg/L as Fe and a contact time of 90 min. A longer contact time of 8 h improved performance to 85% of FFS. P removal at the shorter contact time improved to 88% relative to FFS by pre-acidifying the sludge to pH 2, using an acid molar ratio of 5.2:1 mol H+:Fe. Analysis of the removal of P showed that rapid phosphate precipitation accounted for >65% of removal with FFS. However, for the acidified RC a slower adsorption mechanism dominated; this was accelerated at a lower pH. A cost-benefit analysis showed that relative to dosing FFS and disposing waterworks sludge to land, the 20 year WLC was halved by transporting acidified or unacidified sludge up to 80 km for reuse in wastewater treatment. A maximum inter-site distance was determined to be 240 km above the current disposal route at current prices. Further savings could be made if longer contact times were available to allow greater P removal with unacidified RC

Coagulant recovery and reuse for drinking water treatment

Coagulant recovery and reuse from waterworks sludge has the potential to significantly reduce waste disposal and chemicals usage for water treatment. Drinking water regulations demand purification of recovered coagulant before they can be safely reused, due to the risk of disinfection by-product precursors being recovered from waterworks sludge alongside coagulant metals. While several full-scale separation technologies have proven effective for coagulant purification, none have matched virgin coagulant treatment performance. This study examines the individual and successive separation performance of several novel and existing ferric coagulant recovery purification technologies to attain virgin coagulant purity levels. The new suggested approach of alkali extraction of dissolved organic compounds (DOC) from waterworks sludge prior to acidic solubilisation of ferric coagulants provided the same 14:1 selectivity ratio (874 mg/L Fe vs. 61 mg/L DOC) to the more established size separation using ultrafiltration (1285 mg/L Fe vs. 91 mg/L DOC). Cation exchange Donnan membranes were also examined: while highly selective (2555 mg/L Fe vs. 29 mg/L DOC, 88:1 selectivity), the low pH of the recovered ferric solution impaired subsequent treatment performance. The application of powdered activated carbon (PAC) to ultrafiltration or alkali pre-treated sludge, dosed at 80 mg/mg DOC, reduced recovered ferric DOC contamination to <1 mg/L but in practice, this option would incur significant costs. The treatment performance of the purified recovered coagulants was compared to that of virgin reagent with reference to key water quality parameters. Several PAC-polished recovered coagulants provided the same or improved DOC and turbidity removal as virgin coagulant, as well as demonstrating the potential to reduce disinfection byproducts and regulated metals to levels comparable to that attained from virgin material

Reuse of recovered coagulants in water treatment: An investigation on the effect coagulant purity has on treatment performance

Coagulant recovery offers many potential benefits to water treatment, by reducing chemical demand and waste production. The key obstacle to successful implementation is achieving the same levels of treatment quality and process economics as commercial coagulants. This study has evaluated the selectivity of pressure-filtration in the role of a low-cost coagulant recovery technology from waterworks sludge. The treatment performance of the purified recovered coagulant was directly compared to fresh and raw recovered coagulants. DOC and turbidity removal by recovered coagulants was close to that of commercial coagulants, indicating that coagulant can be successfully recovered and regenerated by acidifying waterworks sludge. However, performance was less consistent, with a much narrower optimum charge neutralisation window and 10–30% worse removal performance under optimum conditions. This inferior performance was particularly evident for recovered ferric coagulants. The impact of this was confirmed by measuring THM formation potential and residual metals concentrations, showing 30–300% higher THMFPs when recovered coagulants were used. This study confirms that pressure-filtration can be operated on an economically viable basis, in terms of mass flux and fouling. However, the selectivity currently falls short of the purity required for potable treatment, due to incomplete rejection of sludge contaminants

Lie Markov models with purine/pyrimidine symmetry

Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines -- that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra. The whole list of Lie Markov models with purine/pyrimidine symmetry is available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.Comment: 32 page

Chord diagrams and BPHZ subtractions

The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three point vertex in $\phi^3$ theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory.Comment: 12 pages, 5 Postscript figures, LaTex 2