http://ir.mu.ac.ke:8080/jspui/handle/123456789/61 2026-08-05T08:57:09Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10414 Construction of symmetric third order difference operators on a Hilbert space Sorowen, Ben; Nyamwala, Fredrick; Ambogo, D.O. We have constructed a symmetric third order difference equation and studied the absolutely continuous spectrum of the self adjoint subspace extension generated using the subspace and decomposition theories. In particular, we have shown that under some decay and growth conditions, the self-adjoint subspace extension exists with discrete spectrum on l2(N) while the absolutely continuous spectrum contained on half line of multiplicity one exists on l2(Z) if we use decomposition technique. We have given a number of examples to reinforce our results. 2025-12-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10413 Generalization of a formula for marked plane trees Nyariaro, Albert Oloo; Okoth, Isaac Owino; Nyamwala, Fredrick Oluoch Deutsch, Munarin and Rinaldi derived a formula for counting marked plane trees while investigating the enumeration of skew Dyck paths. The formula involves the Catalan numbers, which count plane trees among other classical combinatorial structures. In this paper, we generalize their result by enumerating families of noncrossing trees and recently introduced -dimensional plane trees in which certain edges are marked. The resulting formulas are shown to also count: noncrossing trees that allow multi-edges, plane trees in which each internal vertex has outdegree at most 3 and some edges may be marked, and ternary trees in which certain edge type are coloured using two colours. These generalizations provide new combinatorial interpretations and extend the scope of the original enumeration. 2026-03-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10386 Prevalence, determinants, and wealth-related inequalities in malaria among under five children in Uganda: evidence from the malaria indicator survey Mwebesa, Edson; Agbi, Delight Mawufemor; Isiko, Isaac; Nwankwo, Gideon Ikemdinachi; Obiora, Rejoice Uche; Olubiyi, Bisola Folusho; Natuhamya, Charles Background Malaria remains a leading cause of morbidity and mortality among children under five in Uganda. Despite national control efforts, significant disparities and inequalities in prevalence persist across regions, residences, and mean socio-economic status. This study examines the socio-demographic factors and wealth-related inequalities associated with malaria among children under five in Uganda. Methodology A secondary analysis of data from the Uganda Malaria Indicator Survey (UMIS) 2018–2019 was con- ducted. A sample of 4,600 children with malaria test results was included in the study. The distribution of malaria prevalence across socio-demographic factors was analysed using cross-tabulations and a chi-squared test. Malaria- related inequalities were measured using equity plots and the concentration index (CIX). A multilevel logistic regres- sion model was used to examine the relationship between malaria prevalence and associated factors. The results are presented as adjusted odds ratios (aORs) with 95% confidence intervals (CI). Results Almost twenty-three in every 100 children under five had malaria infection (95% CI: 19.2–27.0). Regional vari- ations in malaria prevalence and wealth-related inequalities were observed. The multilevel model identified several significant independent factors: older child age (aOR = 2.12, 95% CI: 1.51–2.96, P < 0.001), child’s anemia (aOR = 3.16, 95% CI: 2.33–4.29, P < 0.001), and larger household size (aOR = 1.98, 95% CI: 1.13–3.45, P < 0.05) were positively associ- ated with malaria in children under five years in Uganda. A negative concentration index (CIX = − 0.334, P < 0.001) was also observed, indicating that higher malaria prevalence is concentrated among children in the poorest wealth quintile. Conclusion Malaria prevalence in Uganda is associated with a complex interplay of socioeconomic and geographic factors. The substantial disparities observed highlight the need for tailored public health strategies designed for high- burden regions and vulnerable communities to reduce disease burden effectively 2026-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10358 Modification of automobile shock absorbers thickness and weight parameters for adaptation on poorly maintained roads via mathematical modelling Owino, Benard Ouma; Kweyu, Cleophas M.; Rotich, Titus Chebion Uneven oscillatory forces applied to an object cause vibration, which can lead to noise, discomfort, and mechanical wear. The optimization of shock absorber thickness and weight is critical for enhancing vehicle performance on poorly maintained roads. In automobiles, vibrations due to irregular road surfaces have been mitigated using shock absorbers to enhance performance, comfort, and control. The main aim of this study was to determine the effect of automotive weights and shock absorber thickness on automobile vibrations. The objectives of this study were; to formulate a mathematical model describing the effect of automobile weight on unsteady vibrations; to generate numerical solutions of the mathematical model equations for the shock absorber modified to reduce vibrations at the mounting points; and to analyze from the numerical solution the effects of increase in velocity and weight on the automobile unsteady spring vibration. A mathematical model was developed, utilizing the central difference scheme for discretization and solved using the Jacobian iterative method, with stability conditions implemented. Through computational simulations, various thickness and weight configurations were tested under diverse road conditions. The results indicate a significant improvement in shock absorber performance, with optimized configurations that increasing shock absorber thickness reduced vibration amplitude by approximately 35%, while increased vehicle weight amplified vibrations by 20%, necessitating thicker shock absorbers for stability. Higher speeds above 80 km/h intensified unsteady vibrations on poorly maintained roads. The study concludes that modifying shock absorber thickness and optimizing weight distribution can effectively minimize automobile vibrations. 2026-02-01T00:00:00Z