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    Determining Equations for a Wave Equation Arising Due to Collapse of Shafts in Power Transmission System
    (Asian Journal of Mathematics and Computer Research, 2025-05-10) Nyangwe G. O.; Omolo O.; Aminer T. J. O.
    The study of wave propagation in mechanical systems plays a crucial role in understanding the dynamic behavior of components under stress or failure. One such scenario is the collapse of shafts in power transmission systems, where the sudden failure of structural components can trigger wave-like disturbances that propagate through the system, potentially causing further damage and system-wide disruptions. The ability to predict and analyze these wave phenomena is essential for ensuring the stability, safety, and efficiency of power transmission networks. In this paper, we study a fourth-order nonlinear wave equation which arises due to collapse of shafts in power transmission systems. Lie symmetry analysis is used to derive the corresponding determining equations that describe the symmetries of the system, which can then be used to reduce the problem to lower-order equations, find exact solutions, or gain insights into the underlying dynamics. By applying Lie group analysis, we systematically prolonged the corresponding infinitesimal generator and applied the symmetry condition to the given wave equation to yield the required determining equations. These equations enable a pathway to exact solutions and system simplification. The work reinforces the value of symmetry-based methods in understanding the dynamic behavior of mechanical systems under collapse-induced wave propagation.
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    Item type: Item , Access status: Open Access ,
    Estimation of Radii of Regions of Starlikeness and Spirallikeness for Analytic Mappings
    (CSRS Publication, 2026-01-25) Nyawalo, Mofat; Aminer, Titus; Okelo, Benard
    Many researchers in complex analysis have invested time particularly in investigating geometric properties like starlikeness and spirallikeness of analytic mappings on the unit disk. Finding the exact lengths of the radii in the starlike and or spirallike regions for these functions is very difficult since these shapes are not regular. This therefore requires that an estimation of the radii of these regions be carried out. This research seeks to estimate the radii within which analytic mappings in the unit disk remain starlike or spirallike. This note derives sharp radii estimates and constructs extremal functions achieving these estimates. The methodology involved using techniques of differential subordination, coefficient estimates, distortion theorems, and subordination principles. Moreover, algorithm development techniques were used as well as numerical methods to come up with pictorial representation of starlikeness and spirallikeness regions. Advancement of theoretical development of geometric function theory and also in providing sharp radii bounds useful in modeling involving conformal maps which enhances applications in engineering, fluid dynamics, and signal theory requires the results generated in this work.
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    Computational Modeling of Vibronic Transitions and Stokes Shift in Poly(3-hexylthiophene) Aggregates: Insights into Molecular Ordering and Optical Behavior
    (Research Square, 2026-01-26) Agumba, John Onyango; Barasa, Godfrey Okumu
    Conjugated polymers such as poly(3-hexylthiophene) (P3HT) exhibit optical and electronic properties that depend strongly on molecular ordering and aggregation. This study employed Gaussian-based simulations to model the absorption and emission spectra of P3HT under varying temperature (25-125°C), crystallization time (1-8 h), concentration (0.5-2.5 a.u.), and solvent conditions. In solution, P3HT showed a broad π-π* band at 455 nm with a weak 495 nm shoulder, while aggregates exhibited sharp vibronic peaks at 520 nm, 560 nm, and 610 nm. Increasing temperature and concentration produced red-shifts, spectral narrowing, and higher 0-2/0-1 ratios, signifying enhanced π-π stacking and molecular order. Poor-solvent systems yielded stronger aggregation with 0-2/0-1 ratios of 1.2-1.3 compared to 0.8-0.9 in good solvents. Calculated Stokes shifts of 0.35 eV (solution) and 0.477 eV (aggregate) confirm exciton relaxation and interchain coupling, providing quantitative insight into P3HT’s structure-property relationships for organic optoelectronic design.
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    On geometrical Aspects of Various Operators and their Orthogonality in Complex Normed Spaces
    (CSRS Publication, 2025-11) Otae, Lamech; Okelo, Benard
    Studies involving orthogonality of operators is an area with various applications with regard to the ever dynamic and emerging technological research outputs. In normed spaces (NS) there are different types of orthogonality. Useful results have come up where operators possessing given conditions are chosen for Range-Kernel orthogonality to be established. However, most of the results have been focussing on one type of orthogonality called the Birkhoff-James which we have given more results on. In this paper we give results on various notions of orthogonality by considering certain geometrical aspects in NS.
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    Nonlinear Geometry of Norm-Attaining Functionals: Variational Principles, Subdifferential Calculus, and Polynomial Optimization in Locally Convex Spaces
    (2025-04-16) Mogoi, N. Evans; Moraa, Priscah
    We develop a unified theory of norm-attainment for nonlinear functionals in locally convex spaces, extending classical results to sublinear, quasiconvex, and polynomial settings. Our main contributions include: (1) nonlinear Bishop-Phelps theorems establishing density of norm-attaining functionals, (2) a subdifferential characterization of attainment via interiority conditions, (3) a KreinMilman principle for convex functionals on compact sets, and (4) a complete solution to the polynomial norm-attainment problem through tensor product geometry. The work combines innovative applications of Choquet theory, variational analysis, and complex-geometric methods to reveal new connections between functional analysis and optimization. Key applications address stochastic variational principles and reproducing kernel Hilbert space optimization, with tools applicable to PDE constraints and high-dimensional data science. These results collectively bridge fundamental gaps between linear and nonlinear functional analysis while providing fresh geometric insight into infinite-dimensional phenomena.