Assessment of trace metals in sewage water and sludge from River Kubanni drainage basin

The concentrations of trace metals in sewage water and sludge samples from River Kubanni drainage basin in Zaria City, Nigeria were investigated in this study. The drainage basin is utilized as a source for irrigation water, during dry seasons. The sewage water quality characteristics in three month sampling periods, that is, February - April, 2008 (peak of dry season and period of intensive usage of the sewage water), the speciation of metals in the sewage sludge from the drainage basin, and the risk to sewage water column contamination were evaluated. The sewage water quality characteristics were mostly beyond the recommended irrigation water standards by the food and agriculture organization (FAO) and United State environmental protection agency (USEPA) except for zinc and nickel. In addition, the average values of Cd, Cu, Pb, Cl- and NO3- in sewage water samples analyzed were higher than the respective reference values for irrigation water. To study the speciation of metals in sewage sludge, five metals (Zn, Ni, Cu, Pb and Cd) in the sludge were subjected to sequential extractions. The metals analyzed were distributed in both the non-residual and residual phases. Total extractable trace metals in sewage sludge were: Zn (403.3 mg/kg dry weight), Ni (184.2 mg/kg dry weight), Cu (303.4 mg/kg dry weight), Pb (129.0 mg/kg dry weight) and Cd (19.7 mg/kg dry weight). However, there was low risk to sewage water contamination based on the calculated individual contamination factors (ICF) obtained for sewage sludge from the trace metal sequential extractions. From the calculated individual contamination factors, Ni and Zn followed by Cd and Pb posed the highest risk to sewage water contamination. Based on this study, the human health is at risk, since sewage water from the drainage basin has been the source for irrigation water during dry seasons, which might lead to trace metal ingestion by soil and subsequently by vegetables. Thus, this might become important pathways of human exposure to metal contamination.Key words: Trace metals, speciation, contamination factor, sewage water and sludge

Solutions of Chi-square Quantile Differential Equation

The quantile function of probability distributions is often sought after because of their usefulness. The quantile function of some distributions cannot be easily obtained by inversion method and approximation is the only alternative way. Several ways of quantile approximation are available, of which quantile mechanics is one of such approach. This paper is focused on the use of quantile mechanics approach to obtain the quantile ordinary differential equation of the Chi-square distribution since the quantile function of the distribution does not have close form representations except at degrees of freedom equals to two. Power series, Adomian decomposition method (ADM) and differential transform method (DTM) was used to find the solution of the nonlinear Chi-square quantile differential equation at degrees of freedom equals to two. The approximate solutions converge to the closed (exact) solution. Furthermore, power series method was used to obtain the solutions for other degrees of freedom and series expansion was obtained for large degrees of freedom

Ordinary Differential Equations of the Probability Functions of the Weibull Distribution and their Application in Ecology

Weibull distribution has been applied to many areas in ecological studies and engineering. Application of the Weibull and other probability distributions in ecology are mainly in fitting ecological data which is very vital in revealing latent characteristics of the object of study. The use of the ordinary differential equations (ODE) in fitting has not been studied in ecological studies. Ordinary differential calculus was used to obtain the homogenous ODE of the probability density function (PDF), quantile function (QF), survival function (SF), inverse survival function (ISF), hazard function (HF) and reversed hazard function (RHF) whose solutions are their respective functions of the Weibull distribution. Different classes of ODEs were obtained. The novelty of this proposed method is applied to radiation data

Quantile Approximation of the Chi–square Distribution using the Quantile Mechanics

In the field of probability and statistics, the quantile function and the quantile density function which is the derivative of the quantile function are one of the important ways of characterizing probability distributions and as well, can serve as a viable alternative to the probability mass function or probability density function. The quantile function (QF) and the cumulative distribution function (CDF) of the chi-square distribution do not have closed form representations except at degrees of freedom equals to two and as such researchers devise some methods for their approximations. One of the available methods is the quantile mechanics approach. The paper is focused on using the quantile mechanics approach to obtain the quantile density function and their corresponding quartiles or percentage points. The outcome of the method is second order nonlinear ordinary differential equation (ODE) which was solved using the traditional power series method. The quantile density function was transformed to obtain the respective percentage points (quartiles) which were represented on a table. The results compared favorably with known results at high quartiles. A very clear application of this method will help in modeling and simulation of physical processes

BOUNDARY PROPERTIES OF BOUNDED INTERVAL SUPPORT PROBABILITY DISTRIBUTIONS

This paper explores the properties of probability distributions as the random variables that defined those distributions approaching their bounded interval support. The models under study are: Kumaraswamy, Kumaraswamy Kumaraswamy, Kumaraswamy with beta and Kumaraswamy with beta distributions. The behavior of the probability density function of the random variables differs greatly at both the lower and the upper boundary points of the support. The results displayed in this research are the same for all the aforementioned pdfs and their cumulative distribution functions, survival functions and hazard functions. The results agreed with some well-known results in the literature. The probability density function, cumulative distribution function, survival function and hazard function approximate to the different values at the boundary points as the support approaches the boundary point

Classes of Ordinary Differential Equations Obtained for the Probability Functions of Linear Failure Rate and Generalized Linear Failure Rate Distributions

The linear failure rate (hazard) and generalized linear failure rate (hazard) distributions are uniquely identified by their linear hazard functions. In this paper, homogenous ordinary differential equations (ODES) of different orders were obtained for the probability functions of linear failure rate and generalized linear failure rate distributions. This is possible since the aforementioned probability functions of the distributions are differentiable and the former distribution is a particular case of the later. Differentiation and modified product rule were used to derive the required ODEs, whose solutions are the respective probability functions. The different conditions necessary for the existence of the ODEs were obtained and it is in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODEs obtained. This method provides new ways of classifying and approximating other probability distributions apart from one considered in this research. Algorithms for implementation can be helpful in improving the results

Classes of Ordinary Differential Equations Obtained for the Probability Functions of Kumaraswamy Distribution

In this paper, differential calculus was used to obtain the ordinary differential equations (ODE) of the probability density function (PDF), Quantile function (QF), survival function (SF), inverse survival function (ISF), hazard function (HF) and reversed hazard function (RHF) of Kumaraswamy distribution. The parameters and support that define the distribution inevitably determine the nature, existence, uniqueness and solution of the ODEs. The method can be extended to other probability distributions, functions and can serve an alternative to estimation and approximation. Computer codes and programs can be developed and used for the implementation

Properties of Sequences Generated by Summing the Digits of Cubed Positive Integers

Having established some properties of sequences generated by summing the digits of squared positive integers (Okagbue et al, 2015), we go a step further to explore the properties and characteristics of sequences generated by summing the digits of cubed positive integers. The results are different from summing the digits of squared positive integers. Two distinct sequences were obtained: one generated by summing the digits of cubed positive integers and the other sequence as the complement of the first but the domain remains the positive integers. The properties of these two sequences are discussed. The properties include their decompositions, subsequences, algebraic, additive, multiplicative, divisibility, uniqueness and ratios

Classes of Ordinary Differential Equations Obtained for the Probability Functions of Exponentiated Frĕchet Distribution

In this work, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the exponentiated Frĕchet distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are a new ways of understanding the nature of the probability functions that characterize the distribution. The method can be extended to other probability distributions and can serve an alternative to approximation