School of Pure and Applied Sciences

Exploring the Properties of Perfect Numbers and Partitions of Odd Numbers to Show the Nonexistence of Odd Perfect Numbers

Mon, 01 Jan 2024 00:00:00 GMT

Exploring the Properties of Perfect Numbers and Partitions of Odd Numbers to Show the Nonexistence of Odd Perfect Numbers Ndegwa, Duncan This study delves into the realms of number theory, specifically investigating perfect numbers and partitions of odd numbers. Perfect numbers, which are integers equal to the sum of their proper divisors, excluding themselves, have intrigued mathematicians for centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p -1) , where 𝑝 and 2p -1 are prime numbers (Mersenne primes), the existence of odd perfect numbers remains an unsolved problem. The first part of the thesis explores perfect numbers, tracing their history from ancient Greek mathematicians to modern scholars, and discussing various results and conjectures. The focus then shifts to partitions of odd numbers, which represent different ways of expressing an odd number as a sum of positive integers. The study utilizes an algorithms that has demonstrated that a positive even integer can be partitioned into all pairs of odd numbers. Using this approach, it is shown that any positive odd number 2𝑛 + 1 can be partitioned into all pairs of both odd and even numbers and from the set of these partitions, it is shown that there exist a proper subset containing all proper divisors of 2𝑛 + 1. Using the results from the partitions and the facts that there exist infinitely many odd numbers, it’s therefore verifies that odd perfect numbers do not exist.

Modelling the Impact of Screening, Treatment and Underlying Health Conditions on Dynamics of COVID-19

Mon, 01 Jan 2024 00:00:00 GMT

Modelling the Impact of Screening, Treatment and Underlying Health Conditions on Dynamics of COVID-19 Kilonzi, Jeremiah Savali Coronavirus disease is an infectious disease triggered by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) that belongs to the family of viruses that cause viral pneumonia. Despite the spreading of the COVID-19 in Kenya with a positivity of 12.9% as at 26 August 2021, there was no reliable deterministic Mathematical model that described the dynamics of COVID-19 incorporating underlying health conditions and impact of screening and treatment. In this study, we propose a SIRS (Susceptible-Infected –Recovered-Susceptible) classical mathematical model which is modified to incorporate the exposed and the treated individuals where COVID-19 is modelled. The model stratifies the population into two categories depending whether they have underlying health conditions or not, and describes disease transmission within or between the groups. Five compartments are considered in the model for each group that is; Susceptible individuals, exposed population, Infected individuals, treated population and the Recovered population. The Next generation matrix method was used to determine the basic reproduction number denoted 𝑅𝑜 of the proposed model. The results obtained indicates that the Disease Free Equilibrium is locally asymptotically stable whenever 𝑅𝑜∗ <1 and globally asymptotically stable if 𝑅0∗≤1. On the other hand, Endemic Equilibrium it is globally asymptotically stable if 𝑅𝑜∗ >1.The results obtained showed that increasing the rate of screening and treatment on the exposed population and weakening the disease transmission route between the susceptible, exposed and infected population are crucial to curb the spread of COVID-19 virus. The Government of Kenya should advocate treatment and screening of the exposed and infected individuals. Further research should consider incorporating vaccination.

Error detection and correction in the container number code

Mon, 01 Jan 2024 00:00:00 GMT

Error detection and correction in the container number code Mutugi, Praise Kiogora The global reliance on containers for transporting goods such as heavy machinery, cars, medical supplies, and spare parts underscores the importance of accurate and reliable container identification. The current system uses an 11-digit code with a modulo 11 check digit for error detection. This code is divided into four parts, a three-letter owner code, the equipment category identifier (U, J, or Z), a five-code serial number and the check digit. While this system facilitates error detection, it suffers from significant shortcomings, most notably its inability to correct errors. The 11-digit system is prone to certain error detection as well, including transposition, substitution, omission, addition, and calculation errors, which could result to cargo mis-delivery. To mitigate the short falls of the current modulo 11 container number code, we have proposed a new container identification system based on a modulo 13 code to overcome these limitations. The proposed system retains the familiar structure of the current code, consisting of a four-letter owner code, an equipment category identifier (U, J, or Z), a seven-digit serial number, with the first two digits generated by a linear congruential generator and the check digit. The modulo 13 code effectively detects substitution and transposition errors but, like its predecessor, does not offer error correction. Given these findings, the study recommends further research into error correction techniques to create a more robust and versatile container numbering system that could prevent cargo mis-delivery due to identification errors.

Light-Dependent Chloroplast Movement in Wild Strawberry (Fragaria Vesca)

Mon, 01 Jan 2024 00:00:00 GMT

Light-Dependent Chloroplast Movement in Wild Strawberry (Fragaria Vesca) Kiprono, Daisy Jepchirchir Chloroplast photorelocation is a vital organellar response that optimizes photosynthesis in plants amid fluctuating environmental conditions. Chloroplasts exhibit an accumulation response, in which they move toward weak light to enhance photoreception, and an avoidance response, in which they move away from strong light to avoid photodamage. Although chloroplast photorelocation has been extensively studied in model plants such as Arabidopsis thaliana, little is known about this process in the economically important crop strawberry. This study, investigated chloroplast photorelocation in leaf mesophyll cells of wild strawberry (Fragaria vesca), a diploid relative of commercially cultivated octoploid strawberry (F. × ananassa). The study determined the induction of avoidance & accumulation response by blue and red light, and the cold dark response induction in F. vesca. Microscopy observation revealed that the periclinal area of leaf mesophyll cells in F. vesca is considerably smaller than that of A. thaliana. Given this small cell size, chloroplast photorelocation in F. vesca was investigated by first visualizing the responses using the white and green band assays and then by measuring light transmittance in leaves and obtaining the change in transmittance with respect to change in light intensities. Weak blue light-induced the accumulation response, whereas strong blue light induced the avoidance response. When light is switch off in the light transmittance set up, the amount of light passing through drops considerably indicating a dark positioning response. Also, inter-merit light that’s unable to induce any visible change in normal temperature shifts to induce an avoidance response in cold temperature. Unexpectedly, strong red light also induced the accumulation response in F. vesca. These findings shed light on chloroplast photorelocation as an intracellular response, laying the foundation for enhancing photosynthesis and productivity in Fragaria.