### Browsing by Department "Department of Mathematics and Applied Mathematics"

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- ItemOpen Access1+1+2 covariant approach to gravitational lensing in f(R) gravity(2009) Nzioki, Anne Marie; Dunsby, PeterIn this thesis, we develop the 1 + 1 + 2 formalism, a technique originally devised for General Relativity, to treat spherically symmetric spacetimes in for fourth order theories of gravity. Using this formalism, we derive equations for a static and spherically symmetric spacetime for general f(R) gravity. We apply these master eqautions to derive some exact solutions, which are used to gain insight on Birkhoff's theorem in this framework. Additionally, we derive a covariant form of the lensing angle for a specific spherically symmetric solution in f(R) = Rn gravity.
- ItemOpen AccessThe 2017 Horse Mackerel Updated Assessment(2017-09) Johnston, S J; Butterworth, Doug SFISHERIES/2016/SEP/SWG-DM/51 provided a full description of the current stock assessment model for the South African Horse Mackerel. FISHERIES/2016/SEP/SWG-DM/52 reported updated 2016 assessments for a number of model variants where the assessments were extended to allow for better fits to the low CPUE values (2014-2015). This document here provides updated 2017 assessment results for a number of model variants which follow on from the variants that were explored in 2016 and take the data from 2016 now available into account.
- ItemOpen Access2018 GLMM-standardised Tristan lobster CPUE compared with the current OMP target levels (Itar)(2018-05) Johnston, Susan; Butterworth, Doug SThis document reports the 2018 GLMM-standardised Tristan lobster CPUE compared with the current OMP target levels that are in place for each island.
- ItemOpen AccessThe 2018 Operation Management Procedure for the South African Merluccius paradoxus and M. capensis Resources(2019) Ross-Gillespie, Andrea; Butterworth, Doug S; Glazer, Jean; Fairweather, TraceySpecifications and projection results for the 2018 Operational Management Procedure used for setting South African hake Total Allowable Catches are provided along with various background information, including details of the metarule processes.
- ItemOpen AccessThe 2018 Operational Management Procedure for the South African Merluccius paradoxus and M. capensis resources.(2018-10) Ross-Gillespie, Andrea; Butterworth, Doug S; Glazer, Jean; Fairweather, TraceySpecifications and projection results for the 2018 OMP used for setting SA hake TACs are provided along with various background information, including details of the metarule process.
- ItemOpen AccessThe 2018 Operational Management Procedure for the South African sardine and anchovy resources(2018) de Moor, CarrynThe Operational Management Procedure (OMP) to be used to recommend total allowable catches and bycatches for sardine and anchovy in South African waters has been updated and is detailed herein
- ItemOpen AccessThe 2018 south coast rock lobster reference case operating model(2019) Johnston, Susan; Butterworth, Doug SThis document details the current 2018 Reference Case Operating Model which is to be proposed to be used for simulation testing of a new OMP for South Coast rock lobster.
- ItemOpen Access2018 Updated horse mackerel assessments(2018-10) Johnston, Susan; Butterworth, Doug SThe assessments for the two hypotheses (reduced catchability and a one-off additional mortality event) for the recent reduced Desert Diamond catch rate are updated to take one further yearâ€™s data into account. The 2017 CPUE somewhat increased above its 2016 value, and now almost returned to the values typical before the 2014 low. If the same basis as in the recent past is used to make management recommendations with an unchanged effort limitation, the total allowable midwater catch would be set at 23 851 MT. However the assessment results, given this data update, are suggestive of the catchability hypothesis now being more plausible than the additional mortality hypothesis. This in turn suggests that consideration should be given to some increase in the cap on effort, which would lead to a concomitant increase in the midwater TAC.
- ItemOpen Access2018 updated south coast rock lobster assessment results(2019) Johnston, Susan; Butterworth, Doug SThe 2017 assessment of the resource for South Coast rock lobster is updated given the two further years of catch-at-length data and one further year of CPUE data now available. Recruitment is estimated to have decreased over the last two seasons for which this is now estimable (2007 and 2008). The spawning biomass trajectory decreases slightly over recent years. Current spawning biomass is estimated to be 29% of K. The Reference Case (RC) MSY estimate is 358 MT at a Bsp/K of 0.29. Fmsy is estimated to be 0.29, though MSY-related estimates are heavily dependent on assumptions made about the stock-recruitment relationship. The RC model estimates that the resource is currently at Bmsy.
- ItemOpen Access2018 updated south coast rock lobster assessment results(2018) Johnston, Susan; Butterworth, Doug SThe 2017 assessment of the resource is updated given two further years of catch-at-length data now available. Recruitment is estimated to have increased over the last two seasons compared to the previous three seasons when recruitment was estimated to be poor. The spawning biomass trajectory decreases slightly over recent years. Current spawning biomass is estimated to be 27% of K. The RC MSY estimate is 259 MT at a Bsp/K of 0.27. Fmsy is estimated to be 0.112. The RC model thus estimates that the resource is currently slightly above Bmsy.
- ItemOpen Access2018 updated west coast rock lobster assessments(2018-11) Johnston, Susan; Butterworth, Doug SA new set of 2018 assessments for all five super-areas for the west coast rock lobster resource has recently been completed. These provide the bases for projecting the resource forwards (in each super-area) under alternative future catches to in turn provide a basis for management advice, as is reported in MARAM/IWS/2018/WCRL/P4. Limitations of the analysis are also discussed.
- ItemOpen Access2020 sole dynamic Schaefer production model results(2020) Glazer, Jean; Butterworth, DougThe sole resource is modelled by a dynamic Schaefer production model which allows for a drop in the value of the intrinsic growth rate parameter from 2000 onwards. The model is fit to the available CPUE and survey abundance indices. These data are not sufficiently informative to be able to distinguish amongst fairly wide ranges of pre- and post-2000 intrinsic growth rate parameters. Nevertheless, all suggest that the sole resource has never been substantially depleted (being well above its MSY level), and furthermore that the current replacement yield and MSY are reasonably robustly estimated in the ranges of 275-350 and 490-720 mt respectively. Given that this update 2020 assessment indicates slightly lower values for resource productivity than in 2019, consideration should be given to some reduction in the current TAC of 502 mt.
- ItemOpen AccessA (giant) void is not mandatory to explain away dark energy with a LemaÃ®tre-Tolman model(2010) CÃ©lÃ©rier, M-N; Bolejko, K; KrasiÅ„ski, AContext. LemaÃ®tre-Tolman (L-T) toy models with a central observer have been used to study the effect of large scale inhomogeneities on the SN Ia dimming. Claims that a giant void is mandatory to explain away dark energy in this framework are currently dominating.
- ItemOpen AccessA combinatorial interpretation for Schreyer's tetragonal invariants(2015) Castryck, Wouter; Cools, FilipSchreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers b(1) and b(2), associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width.
- ItemOpen AccessA contribution to the foundations of the theory of Quasifibration(1995) Witbooi, Peter Joseph; Hardie, K. A.The concept of quasifibration was invented by Dold and Thom (DT]. May (M2] approached quasifibrations from a new angle, making use of n-equivalences. This dissertation presents a study of the notion of n-equivalences and related types of maps. The first of our two main goals is to prove a result, Theorem 5.1, which generalizes the fundamental theorem (DT; Satz 2.2] by Dold and Thom on globalization of quasifibrations. Secondly we show that by means of adjunction or clutching constructions, this theorem enables us to retrieve the famous results of James (J2; Theorem 1.2 and Theorem 1.3] in his work on suspension of spheres. The results of James appear in the thesis as Theorem 13.8. For some of the applications we need a generalized version of n-equivalence. This generalization entails replacing, in the definition of n-equivalence, the isomorphisms by isomorphisms modulo a suitable Serre class [Se] of abelian groups. For the sake of having the thesis self-contained, we include a formal discussion of localization of 1-connected spaces and Serre classes of abelian groups. This summarizes the scope of the thesis. More detail on the content of the thesis will be given after we have sketched a historical perspective on quasifibrations.
- ItemOpen AccessA disformally coupled quintessence mimicking the Î›CDM background(2022) Dusoye, Avishek; Dunsby, Peter Klaus; de la Cruz-Dombriz, Alvaro; Dunsby, P K S; Nunes, N JAlthough the currently-accepted Concordance model of the Universe has been very successful observationally, it cannot resolve two main issues. Firstly, it cannot untangle the unknown nature of the cosmological constant in the Einstein Field Equations, which is responsible for the accelerated cosmological expansion. Secondly, it cannot explain the Ïƒ8 tension, which occurs because the constraints upon galactic clustering by the Cosmic Microwave Background Planck experiments diverge from the large-scale measurement by the Dark Energy Survey. As an alternative to the cosmological constant, this thesis will be using a scalar field, namely the quintessence. Our studied cosmological model assumes that the quintessence is coupled with a generic fluid. It also assumes a theory of gravity with two geometries. The gravitational geometry describes the curvature of space-time while the physical geometry describes the propagation of matter fields. The conformal transformation, which relates the gravitational metric and the physical metric, is extended here to a disformal transformation. In this thesis, the disformally coupled quintessence model mimics the expansion history of the Concordance model, in order to reproduce its observational success and yet have additional degrees of freedom to attempt to address those two issues. Using this approach, the quintessential potential is not specified. The dynamical system for our model is analysed using phase portraits for various studied scenarios. We investigate the expansion history of the DCQ model, where the quintessence couples disformally with dark matter (Scenario I). Our investigation confirms that the quintessential mass influences the disformal characteristics of the dynamical system. Furthermore, the evolution of the density perturbations for the disformally coupled dark matter is reviewed. A disformal effect due to the quintessential mass is seen in the growth rate of the cosmological structures on large scales. The disformal parameter renders no appreciable effect on the evolution of total matter perturbation. A Bayesian analysis of the relevant parameters for the perturbative model (i.e., conformal parameter and quintessential mass) is then carried out using the Redshift Space Distortion data to constrain the best-fit parameters, which might elucidate the Ïƒ8 tension. The best fit set of parameters indicates that the data prefers the model to behave conformally.
- ItemOpen Access
- ItemOpen AccessA History of recent bases for management and the development of a species-combined Operational Management Procedure for the South African hake resource(2008) Rademeyer, R; Butterworth, Doug S; PlagÃ¡nyi, Ã‰va EThe bases for historical catch limits placed on the hake fishery are reviewed in brief for earlier years and then in some depth over the period from 1991 when the Operational Management Procedure (OMP) approach was introduced for this fishery. The new OMP implemented from 2007 was the first to be based on the use of rigorous species-disaggregated assessments of the resource as Operating Models. The paper describes the Reference Set and range of robustness trials, together with the associated Operating Models, which were used for the simulation testing of the new OMP. Performance statistics for a number of candidate OMPs are compared, and the two key trade-off decisions in the selection process discussed (substantial Merluccius paradoxus and catch per unit effort [CPUE] recovery, and total allowable catch [TAC] stability constraints). Details of the OMP adopted and how its formulae depend on recent trends in CPUE and survey estimates of abundance are provided. OMP-2007, which is tuned to a median 20-year recovery target of 20% of pristine spawning biomass for M. paradoxus and a 50% increase in CPUE over the next 10 years, has been adopted for recommending hake TACs over the 2007â€“2010 period until the next scheduled major review. A set of general guidelines adopted for the process of possible overruling of recommendations from OMPs or bringing forward their reviews within an otherwise intended four-year cycle is detailed.
- ItemOpen AccessA Lagrangian formulation of a theory of a scalar field superfluid dark matter(2022) Tema, Seturumane; Larena, Julien; Osano, BobIn this thesis we discuss the dynamics of the relativistic Lagrangian of the theory of dark matter superfluidity. The second and third chapters of the thesis are a review. In the fourth chapter we show that a complex scalar field whose dynamics are dictated by such a Lagrangian, models dust in the background universe on cosmological scale. Prior to our calculations, the theory was shown to model dust on cosmological scale and a superfluid on galactic scale in the non-relativistic case [1]. This project, extends the non-relativistic theory, to include the relativistic background. We continued, using the relativistic Lagrangian, to investigate how perturbations of such a theory grow in a perturbed universe, and found that the density contrast (of the theory) is constant when the complex scalar field is not coupled to baryons in the weak-field limit.
- ItemOpen AccessA mathematical model of a grazing system including beetles(1982) Xulu, Sibusiso Southwell[PAGES SCANNED INCORRECTLY] A number of mathematical studies of herbivore (predator)- grass (prey) models (e.g. Noy-Meir 1975,_ 1976,1978; Wielgolaski 1973; Christian et al. 1978) have been developed. With the exception of a few (Jackson 1980), in all these models the accumulation of dung in pastures is regarded as being negligible. In reality herb~vore dung may cover a non-negligible area of pasture and thus sensibly reduce the carrying capacity of the grazing land. Waterhouse (1974) found that an average of 12 dung pads are dropped by a single bovine every day. According to MacLusky (1960) cattle dung blankets an area of approximately 6600sq.cm per cow p.er day. If the pads are not disposed of, and taking into account the ensuing unpalatable reduced by (Waterhouse, rank growth, the effective area of pasture is thereby each animal by about 20% of an acre per year 1974). Hence one can give a conservative estimation that one million cattle may be putting out of service as much as 100000 acres of grazing land per annum, an area which can feed about 30000 cattle.