Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function

We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar ${\cal N} = 4$ supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory's two-loop MHV amplitudes---considered as functions, symbols, and at the level of their Lie cobracket---and recount how the `nonclassical' part of these amplitudes can be decomposed into specific functions evaluated on the $A_2$ or $A_3$ subalgebras of Gr$(4,n)$. We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the $D_5$ and $A_5$ subalgebras of Gr$(4,7)$, and that these decompositions are themselves decomposable in terms of the same $A_4$ function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.Comment: 68 pages, 7 figures, 6 table

Toward gas-phase controlled mass transfer in micro-porous membrane contactors for recovery and concentration of dissolved methane in the gas phase

A micro-porous hollow fibre membrane contactor (HFMC) operated in sweep-gas mode has been studied to enable the recovery of dissolved methane from water in concentrated form. At high sweep-gas flow rates, up to 97% dissolved methane removal efficiency is achievable which is sufficient to achieve carbon neutrality (around 88%). An increase in methane composition of the recovered sweep-gas was achievable through two primary mechanisms: (i) an increase in liquid velocity which improved dissolved methane mass transfer into the gas phase; and (ii) a reduction in gas flow which lowered dilution from the receiving gas phase. It was posited that further refinement of the methane content was provided through counter-diffusion of the nitrogen sweep-gas into the liquid phase. Within the boundary conditions studied, the methane composition of the recovered gas phase exceeded the threshold for use in micro-turbines for electricity production. However, reducing the gas-to-liquid ratio to enhance gas phase methane purity introduced gas-phase controlled mass transfer which constrained removal efficiency. Whilst this reduction in removal efficiency can be compensated for by extending path length (i.e. more than one module in series), it is suggested that the gas-phase controlled conditions encountered were also a product of poor shell-side dispersion rather than an approach toward the limiting theoretical gas-to-liquid ratio. This implies that further optimisation can be ascertained through improved membrane contactor design. Importantly, this study demonstrates that micro-porous hollow fibre membrane contactors provide a compact process for recovery of dissolved methane in sufficient concentration for re-use

Biogas upgrading by chemical absorption using ammonia rich absorbents derived from wastewater

The use of ammonia (NH3) rich wastewaters as an ecological chemical absorption solvent for the selective extraction of carbon dioxide (CO2) during biogas upgrading to ‘biomethane’ has been studied. Aqueous ammonia absorbents of up to 10,000 gNH3 m−3 demonstrated CO2 absorption rates higher than recorded in the literature for packed columns using 20,000–80,000 g NH3 m−3 which can be ascribed to the process intensification provided by the hollow fibre membrane contactor used in this study to support absorption. Centrifuge return liquors (2325 g m−3 ionised ammonium, NH4+) and a regenerant (477 gNH4+ m−3) produced from a cationic ion exchanger used to harvest NH4+ from crude wastewater were also tested. Carbon dioxide fluxes measured for both wastewaters compared reasonably with analogue ammonia absorption solvents of equivalent NH3 concentration. Importantly, this demonstrates that ammonia rich wastewaters can facilitate chemically enhanced CO2 separation which eliminates the need for costly exogenic chemicals or complex chemical handling which are critical barriers to implementation of chemical absorption. When testing NH3 analogues, the potential to recover the reaction product ammonium bicarbonate (NH4HCO3) in crystalline form was also illustrated. This is significant as it suggests a new pathway for ammonia separation which avoids biological nitrification and produces ammonia stabilised into a commercially viable fertiliser (NH4HCO3). However, in real ammonia rich wastewaters, sodium bicarbonate and calcium carbonate were preferentially formed over NH4HCO3 although it is proposed that NH4HCO3 can be preferentially formed by manipulating both ion exchange and absorbent chemistry

Quantifying the loss of methane through secondary gas mass transport (or 'slip') from a micro-porous membrane contactor applied to biogas upgrading

Secondary gas transport during the separation of a binary gas with a micro-porous hollow fibre membrane contactor (HMFC) has been studied for biogas upgrading. In this application, the loss or ‘slip' of the secondary gas (methane) during separation is a known concern, specifically since methane possesses the intrinsic calorific value. Deionised (DI) water was initially used as the physical solvent. Under these conditions, carbon dioxide (CO2) and methane (CH4) absorption were dependent upon liquid velocity (VL). Whilst the highest CO2 flux was recorded at high VL, selectivity towards CO2 declined due to low residence times and a diminished gas-side partial pressure, and resulted in slip of approximately 5.2% of the inlet methane. Sodium hydroxide was subsequently used as a comparative chemical absorption solvent. Under these conditions, CO2 mass transfer increased by increasing gas velocity (VG) which is attributed to the excess of reactive hydroxide ions present in the solvent, and the fast conversion of dissolved CO2 to carbonate species reinitiating the concentration gradient at the gas-liquid interface. At high gas velocities, CH4 slip was reduced to 0.1% under chemical conditions. Methane slip is therefore dependent upon whether the process is gas phase or liquid phase controlled, since methane mass transport can be adequately described by Henry's law within both physical and chemical solvents. The addition of an electrolyte was found to further retard CH4 absorption via the salting out effect. However, their applicability to physical solvents is limited since electrolytic concentration similarly impinges upon the solvents' capacity for CO2. This study illustrates the significance of secondary gas mass transport, and furthermore demonstrates that gas-phase controlled systems are recommended where greater selectivity is required

Effect of cholesterol and cholesterol analogues on platelet function

This thesis reports studies carried out to investigate further the effect of alteration of the platelet cholesterol content on platelet function. Recent research has shown that cholesterol - enriched human platelets were capable of synthesising greater quantities of pro- aggregating compounds compared to control and cholesterol- depleted platelets. Phospholipase A2 activity in the platelet membrane may be an important rate limiting step in the metabolism of arachidonic acid to pro- aggregating compounds for example thromboxane A2. Thus in the present project, cholesterol - enriched platelets have been investigated with particular reference to phospholipase A2 activity.Rat platelets were enriched with and depleted of cholesterol by in vitro incubation with cholesterol -rich and cholesterol -poor phospholipid liposome suspensions respectively. Aggregation studies shoved that cholesterol- enriched rat platelets (in platelet -liposome mixtures) were more sensitive to collagen and ADP induced aggregation. Phospholipase A2 assays were carried out by analysing the conversion of phosphatidylcholine containing a radiolabelled sn -2 fatty acid. Rat platelets were resuspended once, and no difference in phospholipase A2 activity was detected between cholesterol- enriched and cholesterol normal platelets.Rabbits were fed a cholesterol -rich diet for 4 weeks. Platelets isolated from cholesterol fed rabbits had a significantly higher cholesterol :phospholipid molar ratio than platelets from rabbits fed a normal diet. The once resuspended cholesterol- enriched rabbit platelets showed significantly higher phospholipase A2 activity compared to control platelets treated in the same way. Also, the cholesterol- enriched rabbit platelets were observed to have a more active arachidonic acid metabolic pathway, as assayed by measuring arachidonic acid induced thromboxane A2 and MDA production.Platelets were also examined from hypercholesterolaemic human subjects and compared with platelets from normal human subjects. In these experiments, no differences were detected in cholesterol content or phospholipase A2 activities of crude platelet membrane fractions prepared from these platelets. This may have been due to this assay not being suitable for the very labile human platelet phospholipase A2 activity.Crude platelet membrane fractions were also prepared from rat platelets with altered cholesterol content. Phospholipase A2 assays consistently showed that the activity was significantly higher in cholesterol -enriched rat platelet membrane fractions than in control platelet membrane fractions. These results and those from the rabbit platelet experiments suggested that platelet hyper sensitivity induced by cholesterol -enrichment may be mediated partly through hyperactivity of the membrane bound phospholipase A2.Rat platelets were also enriched with cholesterol analogues which were different from cholesterol only in that they had side chains of reduced length. The analogues studied were pregn -5 -en- 30-ol, chol- 5- en- 3ß -ol, and 27 norcholest- 5- en- 3/ -ol. These analogues were readily taken up by the platelets on incubation with analogue loaded liposome suspensions. Aggregation tests and scanning electron microscopy indicated that major changes occurred in the physiology of platelets enriched with analogues which had a side chain 3 or more carbon atoms shorter than cholesterol. Enrichment of rat platelets with the analogue which had a side chain shorter by one carbon atom had the same effects as cholesterol- enrichment. These platelets became hypersensitive to collagen induced aggregation.The results presented in this thesis suggest that the activity of phospholipase A2 may be influenced by the membrane cholesterol content, and clearly indicate a requirement by the platelet membrane of a sterol of precise dimensions to maintain normal platelet function

Controlling shell-side crystal nucleation in a gas-liquid membrane contactor for simultaneous ammonium bicarbonate recovery and biogas upgrading

A gas–liquid hollow fibre membrane contactor (HFMC) process has been introduced for carbon dioxide (CO2) separation from biogas where aqueous ammonia (NH3) is used to chemically enhance CO2 absorption and initiate heterogeneous nucleation of the reaction product ammonium bicarbonate at the membrane–solvent interface. Aqueous ammonia absorbents (2–7 M) were initially used in single pass for CO2 separation from a synthetic biogas where nucleation of ammonium bicarbonate crystals was observed at the perimeter of the micropores. Recirculation of the aqueous ammonia absorbent encouraged the growth of ammonium bicarbonate crystals on the shell-side of the membrane that measured several microns in diameter. However, at high aqueous NH3 concentrations (3–7 M), lumen side crystallisation occurred and obstructed gas flow through the lumen of the HFMC. The suggested mechanism for lumen-side crystallisation was absorbent breakthrough into the lumen due to pore wetting which was promoted by low absorbent surface tension at high NH3 concentration. Preferential shell-side nucleation can therefore be promoted by (1) raising surface tension of the absorbent and (2) selection of a membrane with a more regulated pore shape than the PTFE membrane used (d/L 0.065) as both actions can diminish solvent ingress into the pore. This was evidenced using 2 M NH3 absorbent where shell-side crystallisation was evidenced without the onset of lumen side crystallisation. Raising surface tension through the inclusion of salt into the chemical absorbent also promoted greater CO2 flux stability. Importantly, this study demonstrates that chemically enhanced HFMC are an attractive prospect for gas–liquid separation applications where reaction product recovery offers further economic value

A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $\phi^4$ theory that saturate our predicted bound in rigidity at all loop orders.Comment: 5+2 pages, 11 figures, infinite zoo of Calabi-Yau manifolds. v2 reflects minor changes made for publication. This version is authoritativ

Bootstrapping a Five-Loop Amplitude Using Steinmann Relations

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude.Comment: 5 pages, 2 figures, 1 impressive table, and 2 ancillary files. v2: a few clarifications and references added; version to appear in PR

The Double Pentaladder Integral to All Orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.Comment: 70 pages, 3 figures, 4 tables; v2, minor typo corrections and clarification

Heptagons from the Steinmann Cluster Bootstrap

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal ($\bar{Q}$) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.Comment: 43 pages, 2 figures. v2: typos corrected; version to appear in JHE