Enumeration of Factorizations in the Symmetric Group: From Centrality to Non-centrality

The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}

Character-theoretic Techniques for Near-central Enumerative Problems

The centre of the symmetric group algebra $\mathbb{C}[\mathfrak{S}_n]$ has been used successfully for studying important problems in enumerative combinatorics. These include maps in orientable surfaces and ramified covers of the sphere by curves of genus $g$, for example. However, the combinatorics of some equally important $\mathfrak{S}_n$-factorization problems forces $k$ elements in $\{1,...,n\}$ to be distinguished. Examples of such problems include the star factorization problem, for which $k=1,$ and the enumeration of 2-cell embeddings of dipoles with two distinguished edges \cite{VisentinWieler:2007} associated with Berenstein-Maldacena-Nastase operators in Yang-Mills theory \cite{ConstableFreedmanHeadrick:2002}, for which $k=2.$ Although distinguishing these elements obstructs the use of central methods, these problems may be encoded algebraically in the centralizer of $\mathbb{C}[\mathfrak{S}_n]$ with respect to the subgroup $\mathfrak{S}_{n-k}.$ We develop methods for studying these problems for $k=1,$ and demonstrate their efficacy on the star factorization problem. In a subsequent paper \cite{JacksonSloss:2011}, we consider a special case of the the above dipole problem by means of these techniques

The chronostratigraphy of a Holocene Barrier Estuary: Lake Illawarra NSW

Valley-fill sequences, preserved in topographic lows associated with incised valley systems, potentially preserve a record of Holocene sea level fluctuations. A detailed litho- and biostratigraphy of the Holocene barrier estuary, Lake Illawarra, New South Wales has been constructed. Forty kilometres of seismic surveys, forty-one vibracores, supplemented by auger drill holes and trenches, and faunal analysis provides the data for this investigation. A detailed chronology of the infilling of the barrier estuary has been established using 115 aspartic acid derived ages and six radiocarbon ages. The results provide a detailed chronology for the deposition of marine transgressive deposits, barrier growth, and the subsequent development of the estuarine back-barrier environment. The results from Lake Illawarra indicate that the generalised evolution of the barrier estuary occurred in five geomorphologically distinct phases associated with rising sea levels following the last glacial maximum (LGM)

Extended residence times for foraminifera in a marine-influenced terrestrial archaeological deposit and implications for palaeoenvironmental reconstruction

Accelerator Mass Spectrometry (AMS) radiocarbon dating and taphonomic grading was undertaken on foraminifera preserved in the archaeological shell matrix site of Thundiy, Bentinck Island, southern Gulf of Carpentaria, Australia. Foraminifera were assigned to one of six taphonomic grades ranging from pristine to severely abraded. AMS dating demonstrates a weak relationship between preservation status and age. Foraminifera ages are inconsistent with multiple ages on marine shell from the same deposit implying significant sediment transport system residence ages (the time between death of the organism and final deposition) for foraminifera in the deposit. Results demonstrate that foraminifera cannot be assumed to be contemporary with other components of the sedimentary context in which they occur, indicating that caution is required in interpreting chronologies and palaeoenvironmental records based on foraminifera recovered from highly dynamic depositional settings. Findings point to the potential of foraminifera AMS dating of coastal archaeological deposits to contribute to evaluations of site integrity and chrono-stratigraphic analyses

Mid-Holocene Aboriginal occupation of offshore islands in northern Australia? A reassessment of Wurdukanhan, Mornington Island, southern Gulf of Carpentaria, Australia

Claims for mid-Holocene Aboriginal occupation at the shell matrix site of Wurdukanhan, Mornington Island, Gulf of Carpentaria, Australia, are reassessed through an analysis of the excavated assemblage coupled with new surveys and an extensive dating program. Memmott et al. (2006, pp. 38, 39) reported basal ages of c.5000–5500 years from Wurdukanhan as ‘the oldest date yet obtained for any archaeological site on the coast of the southern Gulf of Carpentaria’ and used these dates to argue for ‘a relatively lengthy occupation since at least the mid-Holocene’. If substantiated, with the exception of western Torres Strait, these claims make Mornington Island the only offshore island used across northern Australia in the mid-Holocene where it is conventionally thought that Aboriginal people only (re)colonised islands after sea-level maximum was achieved after the mid-Holocene. Our analysis of Wurdukanhan demonstrates high shellfish taxa diversity, high rates of natural shell predation and high densities of foraminifera throughout the deposit demonstrating a natural origin for the assemblage. Results are considered in the context of other dated shell matrix sites in the area and a geomorphological model for landscape development of the Sandalwood River catchment

Environmental context for late Holocene human occupation of the South Wellesley Archipelago, Gulf of Carpentaria, northern Australia

A 2400 year record of environmental change is reported from a wetland on Bentinck Island in the southern Gulf of Carpentaria, northern Australia. Three phases of wetland development are identified, with a protected coastal setting from ca. 2400 to 500 years ago, transitioning into an estuarine mangrove forest from ca. 500 years ago to the 1940s, and finally to a freshwater swamp over the past+60 years. This sequence reflects the influence of falling sea-levels, development of a coastal dune barrier system, prograding shorelines, and an extreme storm (cyclone) event. In addition, there is clear evidence of the impacts that human abandonment and resettlement have on the island's fire regimes and vegetation. A dramatic increase in burning and vegetation thickening was observed after the cessation of traditional Indigenous Kaiadilt fire management practices in the 1940s, and was then reversed when people returned to the island in the 1980s. In terms of the longer context for human occupation of the South Wellesley Archipelago, it is apparent that the mangrove phase provided a stable and productive environment that was conducive for human settlement of this region over the past 1000 years

Dataset of characteristic remanent magnetization and magnetic properties of early Pliocene sediments from IODP Site U1467 (Maldives platform)

This data article describes data of magnetic stratigraphy and anisotropy of isothermal remanent magnetization (AIRM) from "Magnetic properties of early Pliocene sediments from IODP Site U1467 (Maldives platform) reveal changes in the monsoon system" [1]. Acquisition of isothermal magnetization on pilot samples and anisotropy of isothermal remanent magnetization are reported as raw data; magnetostratigraphic data are reported as characteristic magnetization (ChRM).info:eu-repo/semantics/publishedVersio

Onset of Mediterranean outflow into the North Atlantic

Sediments cored along the southwestern Iberian margin during Integrated Ocean Drilling Program Expedition 339 provide constraints on Mediterranean Outflow Water (MOW) circulation patterns from the Pliocene epoch to the present day. After the Strait of Gibraltar opened (5.33 million years ago), a limited volume of MOW entered the Atlantic. Depositional hiatuses indicate erosion by bottom currents related to higher volumes of MOW circulating into the North Atlantic, beginning in the late Pliocene. The hiatuses coincide with regional tectonic events and changes in global thermohaline circulation (THC). This suggests that MOW influenced Atlantic Meridional Overturning Circulation (AMOC), THC, and climatic shifts by contributing a component of warm, saline water to northern latitudes while in turn being influenced by plate tectonics

Magnetic properties of early Pliocene sediments from IODP Site U1467 (Maldives platform) reveal changes in the monsoon system

We report a study of the magnetic stratigraphy and the anisotropy of isothermal remanent magnetization of Pliocene sediments from International Ocean Discovery Program (IODP) Site U1467 drilled in the Maldives platform (Indian Ocean) during Exp. 359. Magnetic stratigraphy gives a precise record of geomagnetic reversals of the early Pliocene from approximately 5.3 Ma to 3.1 Ma providing a detailed age model in an interval where the biostratigraphic record is scarce. We use the anisotropy of isothermal remanent magnetization (AIRM) to investigate the statistical orientation of fine magnetic particles and provide data on the strength and direction of bottom currents during the early Pliocene. The strength of bottom currents recorded by the AIRM, shows a prominent increase at the top of Chron C3n.1n (about 4.2 Ma), and the current direction (NE - SW) is consistent with that of modern instrumental measurements. Since bottom currents in the Maldives are driven by the monsoon, we speculate that the 4.2 Ma increase of bottom currents could mark the onset of the present-day setting, probably related to the coeval uplift phase of the Himalayan plateau