Abstract: Let p be a prime number, Fp be a finite field, and let k ∈ F*p. In this paper, we consider the number of rational points onconics Cp,k: x2 − ky2 = 1 over Fp. We proved that the order of Cp,k over Fp is p-1 if k is a quadratic residue mod p and is p + 1 if k is not a quadratic residue mod p. Later we derive some resultsconcerning the sums ΣC[x]p,k(Fp) and ΣC[y]p,k(Fp), the sum of x- and y-coordinates of all points (x, y) on Cp,k, respectively.
Abstract: In view of their importance and usefulness in reliability theory and probability distributions, several generalizations of the inverse Gaussian distribution and the Krtzel function are investigated in recent years. This has motivated the authors to introduce and study a new generalization of the inverse Gaussian distribution and the Krtzel function associated with a product of a Bessel function of the third kind )(zKQ and a Z - Fox-Wright generalized hyper geometric function introduced in this paper. The introduced function turns out to be a unified gamma-type function. Its incomplete forms are also discussed. Several properties of this gamma-type function are obtained. By means of this generalized function, we introduce a generalization of inverse Gaussian distribution, which is useful in reliability analysis, diffusion processes, and radio techniques etc. The inverse Gaussian distribution thus introduced also provides a generalization of the Krtzel function. Some basic statistical functions associated with this probability density function, such as moments, the Mellin transform, the moment generating function, the hazard rate function, and the mean residue life function are also obtained.KeywordsFox-Wright function, Inverse Gaussian distribution, Krtzel function & Bessel function of the third kind.
Abstract: Decentralized eco-sanitation system is a promising and sustainable mode comparing to the century-old centralized conventional sanitation system. The decentralized concept relies on an environmentally and economically sound management of water, nutrient and energy fluxes. Source-separation systems for urban waste management collect different solid waste and wastewater streams separately to facilitate the recovery of valuable resources from wastewater (energy, nutrients). A resource recovery centre constituted for 20,000 people will act as the functional unit for the treatment of urban waste of a high-density population community, like Singapore. The decentralized system includes urine treatment, faeces and food waste co-digestion, and horticultural waste and organic fraction of municipal solid waste treatment in composting plants. A design model is developed to estimate the input and output in terms of materials and energy. The inputs of urine (yellow water, YW) and faeces (brown water, BW) are calculated by considering the daily mean production of urine and faeces by humans and the water consumption of no-mix vacuum toilet (0.2 and 1 L flushing water for urine and faeces, respectively). The food waste (FW) production is estimated to be 150 g wet weight/person/day. The YW is collected and discharged by gravity into tank. It was found that two days are required for urine hydrolysis and struvite precipitation. The maximum nitrogen (N) and phosphorus (P) recovery are 150-266 kg/day and 20-70 kg/day, respectively. In contrast, BW and FW are mixed for co-digestion in a thermophilic acidification tank and later a decentralized/centralized methanogenic reactor is used for biogas production. It is determined that 6.16-15.67 m3/h methane is produced which is equivalent to 0.07-0.19 kWh/ca/day. The digestion residues are treated with horticultural waste and organic fraction of municipal waste in co-composting plants.