Browsing by Author "Kimani, Patrick"

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    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, Jacob
    Ranks 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.
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    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, Jacob
    Ranks 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 .
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    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, Hussein
    Ranks 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.
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    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, Geoffrey
    Ranks 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.
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    Cycle Index Formulas for Dn Acting on Ordered pairs
    (UoEm, 2016-04) Muthoka, Geoffrey; Kamuti, Ireri; Kimani, Patrick; Hussein, Lao
    The 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 .
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    Modelling and Simulation of Competition for Students’ Population with Holling Type II Response
    (UoEm, 2023) Odhiambo, Brian; Ngari, Cyrus; Kimani, Patrick; Njori, Peter
    Te 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 extinct
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    On the Number of Cyclotomic Cosets and Cyclic Codes over Z13
    (UoEm, 2018-06) Hussein, Lao; Kivunge, Benard; Kimani, Patrick; Muthoka, Geoffrey
    Let 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.
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    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, Maurice
    Tourism 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.
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    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, Jane
    The 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.
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    Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups
    (UoEm, 2020) Agwanda, Siahi Maxwell; Kimani, Patrick; Kamuti, Ireri
    The 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 and