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The mathematical physics underlying the adsorption and subsequent desorption of dissolved oxygen (DO) in a water body subject to effluent loading had been rarely investigated. The current state of play in this field although reflects use of different analysis, the combine use of hat matrix and bootstrapping techniques to study the phenomenon of chemical adsorption and desorption of DO at molecular level in a polluted waterbody has not been thoroughly investigated. This study seeks to use a matrix projector, H-hat (Ĥ ), to cast virtual spectral rays on pollutant loadings in a water body and in the process unravel the dynamics of chemical and biological gravitation of dissolved oxygen towards constituents of effluent pollutants in water body. This approach is anchored on the ordinary least squares methodology of multivariate linear regression. The method hypothecated is studded by a mathematical physics analysis of the phenomenon. Bootstrapping was used to establish means and variances of regression parameters, and subsequently, the confidence intervals of point estimates of parameters. Tricking technique adopted facilitated the development of extreme values of the dissolved oxygen and hence the supremum and infimum of assimilative capacity of the river which fluctuates with intensity of effluent loadings and season of the year (rainy, dry, and harmattan seasons). The result of bootstrapping revealed that assimilative capacity fluctuated widely from the values detected by point estimates of regression parameters thus suggesting that tricking of regression parameters, in turn, tunes up the regression model, and hence, fine tunes the value of assimilative capacity through necessary adjustments of model parameters. The results of this study obviates the need to deploy eudiometer for laborious direct measurement of dissolved oxygen in a body of polluted water. Thus an elegant technique for crossing the stream where it is shallowest has been developed in this study. The method is considered as a great improvement on previous approaches that seem to dawdle. http://dx.doi.org/10.4314/njt.v35i1.28

The assimilative capacity of a water body requires knowledge of the river characteristics, pattern of effluent loading, river hydraulic resilience and far-field mixing to effectively determine it, hence it is considered appropriate that diverse models and approaches are utilized in its determination’’. This study evaluated the assimilative capacity of Ikpoba River using different approaches namely: homogeneous differential equation, ANOVA/Duncan Multiple rage test, first and second order differential equations, correlation analysis, Eigen values and eigenvectors, multiple linear regression, bootstrapping and far-field mixing Analytics. In the rainy, dry, and harmattan seasons the values of assimilative capacity for Guinness, Ewa road and UBTH point sources of effluent discharge are as follows: 1.07 x 10 10 kg/day, 1.07 x 10 10 kg/day and 1.29x10 10 kg/day; 8.1x10 9 kg/day, 9.07x10 9 kg/day and 1.05x 10 10 kg/day;1.28x10 10 kg/day, 1.03x10 9 kg/day and 1.95x10 10 kg/day in that order. It was also discovered that by model tricking, assimilative capacity had a range of values that varied from 231.09mg/l upper limit to -222.27mg/l lower limit at 95%significance level and beyond this range, the model developed will breakdown, and its robustness weakened by poor predictive capacity. The upper control limit for each of the various pollutants considered indicates the saturated value of the dissolved oxygen (DO), and the negative lower control limits depict dissolved oxygen debt whereby the DO falls below critical (anoxia) to harsh critical level. At this level any reaeration or reoxygenation effort is not practically palpable until recovery is made up to critical level. The eclectic approaches adopted have crossed the stream where it is shallowest. http://dx.doi.org/10.4314/njt.v35i1.27

The sheer display of reprehensible propensity on the part of public hospitals, abattoirs, breweries and city dwellers at large to discharge untreated waste, debris, scum and, in particular, municipal and industrial effluents into Ikpoba River has morphed into a situation whereby the assimilative capacity of the river has reached a record level. The seeming quietism and clinical posture of the relevant Environmental Agency to this pathological condition is sickening and in any case seen as remiss on their part. This paper seeks to determine the assimilative capacity of the river with a view to articulating policy-proposal stream in the instance that its value is unsafe for ichthyofaunal beings and humans. A combination of basic Streeter-Phelp equation and monograph of Fair et al. were used for the determination of the assimilative capacity and the maximum allowable biological oxygen demand (BOD) load discharged into the river from three identified point sources of wastewater and for three different seasons in Nigeria (i.e. rainy, dry and harmattan seasons). Results indicate that in all the segments studied, the Fair ratio (self-purification factor) is less than unity, indicating the predominance of deoxygenation rate over reaeration rate. Reaeration rate ranges over all real values from 0.641day- 1 to 0.693day- 1 , while deoxygenation rate ranges between 0.718 day- 1 and 0.839 day -1 ; a condition which indicates a poor assimilative capacity potential. Moreover, a higher assimilative capacity was obtained for rainy season for all the segments studied, with sampling point 3 (Guinness-harmattan and dry season) showing an overall worse assimilative capacity and poor maximum allowable BOD of 8.1x10 9 , 1.28x10 10 and 5.60x10 9 kg/day respectively. In a way, the determination of the assimilative capacity of Ikpoba River is a landmark in the history of the river pollution. It beggars belief that we live down the harrowing pathological condition of the river without recognizing its wider implications. http://dx.doi.org/10.4314/njt.v34i1.17