Articles: Department of Mathematics and Statistics
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Item Modelling and Simulation of Competition for Students’ Population with Holling Type II Response(UoEm, 2023) Odhiambo, Brian; Ngari, Cyrus; Kimani, Patrick; Njori, PeterTe increase in the country’s population attracted the establishment of more schools, both public and private schools, to cater for the increasing number of students. However, there have been dynamics of students’ population both in public and private schools through transfer from one category of school to the other, through completion of the learning period, and through dropout due to unknown reasons which have subjected both the public and private schools to compete in order to maintain a good number of students. In this work, a modifed Lotka–Volterra model of schools and nonenrolled entities population in the education system is studied. Private schools and nonenrolled entities play the role of a predator in public schools. Again, public schools and nonenrolled entities play the role of predators in private schools. Holling type II functional responses have been integrated in the analysis of the Lotka–Volterra model. Te equilibrium points are established and their stability are determined using the Routh–Hurwitz criterion and eigenvalue method. Global stability has been done for the positive equilibrium point. Bifurcation is also done around the positive equilibrium point. Finally, a graphical illustration of various parameter is derived to show their efect on schools when they are varied. It is revealed that the increase in parameters θ2 , θ3 , and η3 greatly afects the schools population as they are the ones leading to predation in school. Terefore, proper strategies should be developed to focus on reducing the mentioned parameters to avoid leading schools’ population to extinctItem Transitivity Action of An on (n=4,5,6,7) on Unordered and Ordered Quadrupples(UoEm, 2015) kimani, Gachago j; N, Kinyanjui J.; j, Rimberia; kimani, Patrick; muchemi, Jacob kiboiIn this paper, we study some transitivity action properties of the alternating group An(n=4,5,6,7 ,) acting on unordered and ordered pairs from the set X = {1,2,...,n} through determination of the number of disjoint equivalence classes called orbits.when n≤ 7 ,the alternating group acts transitively on both X (4) and X [4] . key words : Orbits ,alternating group An , An on unordered and ordered quadruples from the set X.Item Rank and Subdegrees of P GL(2, q) Acting Cosets of P GL(2, e) for q an Even Power of e(UoEm, 2019) Kimani, Patrick; Kamuti, Ireri; Rimberia, JaneThe action of projective general group on the cosets of its maximal subgroups has been studied. For instance, [9] studied the action of G on the cosets of P GL(2, e) when q is an odd prime power of e. In this paper, we determine the rank and subdegrees of the action of P GL(2, q) on the cosets of its subgroup P GL(2, e) for odd q and an even power of e. We apply the table of marks to achieve this.Item Predicting the Number of Tourists in-Flow to Kenya Using Seasonal Autoregressive Integrated Moving Average Model(UoEm, 2022-12) Gechore, Dennis; Atitwa, Edwin; Kimani, Patrick; Wanyonyi, MauriceTourism is the leading source of revenue to the Kenyan Government, contributing about 8.8% to the Kenya’s Gross Domestic Product. Based on the 2019 report released by the ministry of tourism and wildlife, tourism industry contributed approximately $7.9 billion to the Kenya’s budget. This study was therefore developed to predict the future numbers of tourists that will visit Kenya between 2023 and 2025. The Seasonal Autoregressive Integrated Moving Average time series model was applied for the prediction. The study used secondary data collected from the Ministry of Tourism and Wildlife. The data covered a period of 11 years from 2011 to 2022. The model was fitted to the real tourists’ data using the time series algorithm implemented in R statistical software. Based on the Akaike Information Criterion, the ARIMA(2,1,1)(0,1,0)12 was identified as the perfect model with minimum errors. The model passed the diagnostic test performed. Importantly, 95% confidence level prediction done for 3 years (2023-2025) using the model showed that the number of tourists expected to visit Kenya will increase significantly. Therefore, the study recommended that recreational facilities and accommodations should be maintained to cater for the high projected numbers of tourists. The study also recommended that the government of Kenya should strategize on how to beef up security to curb terrorism attacks and tribal conflicts which might discourage tourists.Item On the Number of Cyclotomic Cosets and Cyclic Codes over Z13(UoEm, 2018-06) Hussein, Lao; Kivunge, Benard; Kimani, Patrick; Muthoka, GeoffreyLet Zq be a finite field with q element and x n − 1 be a given cyclotomic polynomial. The number of cyclotomic cosets and cyclic codes has not been done in general. Although for different values of q the polynomial x n − 1 has been characterised. This paper will determine the number of irreducible monic polynomials and cyclotomic cosets of x n − 1 over Z13 .The factorization of x n − 1 over Z13 into irreducible polynomials using cyclotomic cosets of 13 modulo n will be established. The number of irreducible polynomials factors of x n − 1 over Zq is equal to the number of cyclotomic cosets of q modulo n. Each monic divisor of x n − 1 is a generator polynomial of cyclic code in Fq n . This paper will further show that the number of cyclic codes of length n over a finite field F is equal to the number of polynomials that divide x n − 1. Finally, the number of cyclic codes of length n, when n = 13k, n = 13k , n = 13k − 1, k, 13 = 1 are determine.Item Enumeration of cyclic codes over GF(17)(UoEm, 2015-05) Hussein, Lao; Kivunge, Benard; Muthoka, Geoffrey; Mwangi, PatrickIn this paper we seek the number of irreducible polynomials of xn− 1 over GF(17). We factorize Xn− 1 over GF(17)into irreducible polynomials using cyclotomic cosets of 17 modulo n . The number of irreducible polynomials factors of Xn− 1 over fq is equal to the number of q cyclotomic cosets of modulo n. Each monic divisor of Xn− 1 is a generator polynomial of cyclic code in Fqn. We show that the number of cyclic codes of length n over a finite field f is equal to Xn− 1. Lastly, the number of cyclic codes of length n , when n= 17 , = the number of polynomials that divide 17k ,n = 17k,n=17k − 1, ( , 17) = 1 are enumerated.Item Cycle Index Formulas for Dn Acting on Ordered pairs(UoEm, 2016-04) Muthoka, Geoffrey; Kamuti, Ireri; Kimani, Patrick; Hussein, LaoThe cycle index of dihedral group Dn acting on the set of the vertices of a regular n-gon was studied by Harary and Palmer in 1973 [1]. Since then a number of researchers have studied the cycle indices dihedral group acting on different sets X={1,2,...,n} and the resulting formulas Dn have found applications in enumeration of a number of items. Muthoka (2015) [2] studied the cycle index formula of the –the n vertices of a regular -gon. In this paper we study the dihedral group acting on unordered pairs from the set X={1,2,..,n} cycle index formulas of acting on ordered pairs from the set . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of and an odd value of n .Item Cycle Index Formulas for Dn Acting on Unordered Pairs(UoEm, 2015) Muthoka, Geoffrey; Kamuti, Ireri; Lao, Hussein; kimani, patrickThe cycle index of dihedral group Dn acting on the set X of the vertices of a regular -gon was studied (See [1]). In this paper we study the cycle index formulas of Dn acting on unordered pairs from the set x . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of n and an odd value of n .Item Action ofPGL (2 ,q ) on the Cosets of the Centralizer of an Eliptic Element(UoEm, 2021) Kimani, patrick; Adicka, DanielMost researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroupsCq+1. In this paper, we consider the action ofPGL (2, ) centralizer of an elliptic element which is a non maximal subgroup . In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is and the subdegrees are [1][ ] and [ + 1][ ] .Item Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets(UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, JacobRanks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 7,8,9) acting on ordered 4-element subsets and Sn (n = 8,9,10) acting on ordered 5-element subsets.Item Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples(UoEm, 2014) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, HusseinRanks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 5,6,7) and Sn (n = 6,7,8) acting on ordered pairs and triples respectively.Item Ranks and Subdegrees of External Direct Product of Cn Dr Acting on X Y(UoEm, 2020) Mutua, Felix Mutinda; Kamaku, Peter Waweru; Kimani, Patrick MwangiIn this paper, transitivity, ranks and subdegrees of the action of ex- ternal direct product of Cyclic and Dihedral group on Cartesian Product of two sets are determined. The action is proved to be transitive. Also, it's established that the rank associated with the action is n( r+1 2 ) and subdegrees are [1][n] and [2][n( r1 2 )] when r is old. Additionally, the rank of the action for the case where r is even is proved to be n( r+2 2 ) and subdegrees are [1][2n] and [2][n( r2 2 )].Item Rank and Subdegrees of PGL(2; q) Acting Cosets of PGL(2; e) for q an Even Power of e(UoEm, 2019) Kimani, Patrick Mwangi; Kamuti, Ireri; Rimberia, JaneThe action of projective general group on the cosets of its maximal subgroups has been studied. For instance, [9] studied the action of G on the cosets of PGL(2; e) when q is an odd prime power of e. In this paper, we determine the rank and subdegrees of the action of PGL(2; q) on the cosets of its subgroup PGL(2; e) for odd q and an even power of e. We apply the table of marks to achieve this.Item Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups(UoEm, 2020) Agwanda, Siahi Maxwell; Kimani, Patrick; Kamuti, IreriThe action of affine groups on Galois field has been studied. For instance, [3] studied the action of 𝐴𝑓𝑓 (𝑞 ) on Galois field 𝐺𝐹 (𝑞) for 𝑞 a power of prime 𝑝. In this paper, the rank and subdegree of the direct product of affine groups over Galois field acting on the cartesian product of Galois field is determined. The application of the definition of the product action is used to achieve this. The ranks and subdegrees are used in determination of suborbital graph, the non-trivial suborbital graphs that correspond to this action have been constructed using Sims procedure and were found to have a girth of 0, 3, 4 andItem Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets(UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, JacobRanks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group 𝑆𝑛(𝑛 = 7,8,9) acting on ordered 4-element subsets and 𝑆𝑛 (𝑛 = 8,9,10) acting on ordered 5-element subsets .Item Cycle Index Formulas for D n Acting on Unordered Pairs(Uoem, 2015) Muthoka, Geoffrey; Kamuti, Ireri; Lao, Hussein; Kimani, Patrick MwangiThe cycle index of dihedral group D n acting on the set of X the vertices of a regular n- gon was studied (See [1]). In this paper we study the cycle index formulas of D n acting on unordered pairs from the set . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of and an odd value of .1.Item Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples(Uoem, 2014) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Muthoka, GeoffreyRanks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group 𝑆𝑛(𝑛 = 5,6,7) and 𝑆𝑛 (𝑛 = 6,7,8) acting on ordered pairs and triples respectively.Item Action of PGL(2,q) on the Cosets of the Centralizer of an Eliptic Element(Uoem, 2021-08-19) Kimani, Patrick Mwangi; Adicka, DanielMost researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroups. In this paper, we consider the action of PGL(2,q) centralizer of an elliptic element which is a non maximal subgroup c(q+1). In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is q and the subdegrees are [1][2] and [q+ 1][q-2].Item MODELING DISPROPORTIONAL EFFECTS OF EDUCATING INFECTED KENYAN YOUTH ON HIV/AIDS(journal of biological systems, 2020) Ronoh, Marilyn; OGANA, WANDERA; Chirove, Faraimunashe; Wairimu, JosephineWe formulate an age and sex-structured deterministic model to assess the effect of increasing comprehensive knowledge of HIV/AIDS disease in the infected Adolescent Girls and Young Women (AGYW) and, Adolescent Boys and Young Men (ABYM) populations in Kenya. Mathematical analysis of infection through sub-network analysis was carried out to trace various infection routes and the veracity of various transmission routes as well as the associated probabilities. Using HIV data in Kenya on our model, disproportional effects were observed when dispensation of comprehensive knowledge of HIV/AIDS was preferred in one population over the other. Effective dispensation of comprehensive knowledge of HIV/AIDS in both the infected AGYW and ABYM populations significantly slows down the infection spread but may not eradicate it.Item Modeling the effects of insecticides resistance on malaria vector control in endemic regions of Kenya(Elsevier Ltd., 2018-12) Wairimu, Josephine; Ronoh, Marilyn; Malonza, David M; Chirove, Faraimunashe;We present a model to investigate the effects of vector resistance to control strategies. The model captures the development of resistance as well as loss of resistance in mosquitoes and how these affect the progress in malaria control. Important thresholds were calculated from mathematical analysis and numerical results presented. Mathematical results reveal the existence of the disease free and endemic equilibria whose existence and stability depends on the control reproduction number, Rc. The disease persist when the Rc>1 and dies out when Rc<1. Control strategies use and adherence needs to be highly efficacious to thwart the effects of insecticides resistance. Moreover, it is not enough to just eradicate resistant mosquitoes.