This study identifies critical parameter thresholds associated with climate regime shifts and extreme events in the Kenyan Arid and Semi-Arid Lands (ASALs) using a calibrated stochastic coupled Lorenz system. The analysis combines bifurcation theory, stability analysis, numerical simulation, extreme-value statistics, and critical-slowing-down indicators to investigate transitions in the model dynamics. For the classical Lorenz parameters (\(\sigma\), \(\beta\)) = (10, 8/3), the non-zero equilibria lose stability through a subcritical Hopf bifurcation at the analytically derived critical value pH = 470/19 ≈ 24.74. The homoclinic bifurcation occurs at approximately phom ≈ 13.926, marking the emergence of a non-attracting chaotic invariant set and transient chaotic dynamics, while an attracting chaotic set emerges near pA ≈ 24.06. The calibrated ASAL parameter values are then examined relative to these dynamical thresholds. The stochastic extension is further used to investigate noise-induced transitions, synchronisation extreme-value behaviour, and early-warning indicators. The results demonstrate how critical thresholds in the reduced-order model can provide a mathematical framework for analysing nonlinear climate variability and extreme-event behaviour in Kenyan ASALs.
Meromorphic functions and L-functions are central objects in value-distribution theory, where uniqueness questions are commonly studied through shared values and differential or difference expressions. This study investigates uniqueness relations between a nonconstant meromorphic function having finitely many poles and an L-function when associated difference-differential polynomials share a small function with a prescribed weight. The formulation incorporates the higher-order difference operator and considers expressions generated from powers of the functions, the factor involving the function minus one, and the corresponding difference term. The analysis is developed within Nevanlinna theory using weighted sharing, truncated counting functions, deficiency-type quantities, and growth properties of L-functions. Sufficient lower-bound conditions are obtained for the combined integer parameters under the three cases in which the sharing weight satisfies l ≥ 2, l = 1, or l = 0. Under the stated hypotheses, the relevant kth derivatives of the paired difference-differential polynomials either have product equal to the square of the shared small function, coincide, or yield a relation of the form f = tL subject to the corresponding algebraic restriction on the constant t. Corollaries for the cases n = 0 and m = 0 provide analogous conclusions for reduced forms. The results extend the manuscript’s earlier uniqueness formulations while remaining restricted to the stated assumptions on the functions, sharing weight, and integer parameters.
Linear hypergraph set-indexers (LHSIs) associate a graph with a vertex hypergraph and an induced edge hypergraph through injective set-valuations and symmetric-difference edge labels. This study examines structural properties of graphs under LHSIs, with particular emphasis on conditions under which the associated vertex and induced edge hypergraphs are isomorphic. It is shown that a connected graph admitting such an LHSI must be unicyclic and that the cardinality of each edge in the associated hypergraphs is at most two. A construction is established for connected unicyclic graphs in which every vertex has odd degree. The study further identifies graph classes for which the lower and upper LHSI numbers coincide. In particular, for complete bipartite graphs Km,n with m, n ≥ 4, both parameters equal m + n. Exact upper LHSI numbers are also determined for cycles: IUL (C3) = 3, IUL (C4) = 6, and IUL (Cn) = 2n for n ≥ 5. The realizability of finite linear hypergraphs as vertex hypergraphs of LHSIs is also investigated. A necessary and sufficient condition is formulated in terms of a spanning subgraph of the complete graph on the hyperedge set whose symmetric-difference edge labels are distinct and themselves form a linear hypergraph. These results clarify several structural restrictions inherent in LHSI constructions.
In this paper, we study a viscosity implicit midpoint rule for approximating fixed points of Reich–Suzuki-type nonexpansive mappings in a real Hilbert space. Motivated by the viscosity midpoint methods introduced by Vaishnav et al. and Hu for quasi-nonexpansive and nonexpansive mappings, we extend the framework to a broader class of nonlinear mappings. Under suitable assumptions on the control parameters and involved operators, we establish strong convergence results for the proposed iterative scheme. The obtained limit is characterized as the unique solution of an associated variational inequality problem.
This study establishes fixed point results for expansive self-maps in complete extended b-metric spaces, where the multiplicative factor in the triangle inequality is governed by a variable control function. Because repeated use of the extended triangle inequality can amplify orbitwise distances, the analysis incorporates an orbit-bound condition, or an analogous summability requirement, to control the distortion generated by backward preimage iterations. Under surjectivity and suitable expansive inequalities, existence and uniqueness results are developed for a single mapping, a two-term condition, a multi-term coefficient condition, and a pair of mappings with a common fixed point. The proofs construct preimage sequences and show geometric decay of successive gaps; completeness then yields convergence, while the expansive conditions establish the fixed-point property and uniqueness. Several examples of extended b-metric spaces are provided to illustrate the framework. A numerical example based on an affine map demonstrates the convergence of the backward sequence towards its analytic fixed point. The manuscript further considers an affine scaling-plus-bias operator in a finite-dimensional setting and relates its equilibrium to the proposed expansive fixed point framework under the stated boundedness, invertibility, and orbit conditions. Overall, the results extend the manuscript’s expansive-mapping analysis from metric and b-metric settings to the extended b-metric setting while retaining explicit control of the variable distortion.
Typhoid fever remains a significant public health challenge, particularly in developing countries with inadequate sanitation infrastructure. The emergence and spread of drug-resistant typhoid fever strains have complicated treatment, leading to prolonged illness, higher healthcare costs, and sustained transmission within communities. This growing resistance underscores the need for effective treatment approaches and disease control strategies. This study develops a mathematical model of typhoid fever transmission that incorporates antibiotic resistance. The model categorises infected individuals into drug-sensitive and drug-resistant typhoid fever strains. The impact of treatment modification through different therapeutic options is examined to assess its effect on the prevalence of both sensitive and resistant strains. The model is analysed qualitatively, and the basic reproduction number, R0 , is derived as the sum of two reproduction numbers, R0s and Rr0, representing the transmission contributions of the sensitive and resistant strains, respectively. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. Sensitivity analysis is conducted to identify the key parameters that influence typhoid fever transmission and persistence. Numerical simulations were performed to validate the analytical results, which demonstrated that typhoid vaccination, the use of appropriate treatment adjustment using first-line and second-line antibiotics, and improved hygiene and sanitation practices significantly reduce the prevalence of both drug-sensitive and drug-resistant strains, as well as the overall infection burden. These findings highlight the effectiveness of integrated prevention and treatment strategies in mitigating antibiotic resistance and enhancing typhoid fever control in the community
Generative AI inference presents a scheduling problem that differs from conventional compute workloads because requests vary substantially in service time and their memory requirements increase dynamically during autoregressive decoding. This study formulates generative AI serving as a memory-constrained, variable-size batch queue and develops a modified Pollaczek–Khinchine waiting-time approximation that includes a memory-contention penalty. It also proposes Memory-Aware Adaptive Batching (MAAB), an online admission-control policy that admits requests only when a probabilistic memory-headroom constraint is satisfied. MAAB combines projected peak memory demand with a size-aware priority score and an ageing mechanism to limit starvation. The framework was evaluated using discrete-event simulation with Poisson arrivals, shifted log-normal output lengths, and accelerator memory provisioned for heterogeneous request sizes. Four policies were compared: first-come-first-served, shortest-job-first, static batching, and MAAB. Across 50,000 completed requests and 20 replications, MAAB achieved a mean waiting time of 5.8 s, a 95th-percentile waiting time of 14.2 s, memory utilisation of 88.7%, three out-of-memory retries, and throughput of 58.6 jobs per minute. Relative to first-come-first-served scheduling, MAAB reduced mean waiting time by up to 68%, reduced out-of-memory-triggered retries by more than 95%, and increased throughput by 58%. These results support memory-aware, size-adaptive queueing as a practical approach to improving latency, utilisation, and memory safety in generative AI serving.
Background: Forest canopies strongly influence atmospheric airflow, turbulence generation, heat exchange, water transport, and carbon dioxide distribution, thereby regulating the local climate. However, accurately representing airflow dynamics within forests remains challenging because of vegetation drag and turbulent mixing. Aims: This study developed a mathematical model based on the Navier–Stokes equations to investigate airflow behaviour within forest canopies and assess its influence on local climate dynamics. Study Design: This was a computational fluid dynamics (CFD)-based modelling study employing the Reynolds-averaged Navier–Stokes (RANS) equations coupled with the standard k–ε turbulence model. Place and Duration of Study: Department of Mathematics and Computer Science, University of Eldoret, Kenya, between July 2025 and April 2026. Methodology: The incompressible Navier–Stokes equations were used to model airflow within and above forest canopies. Vegetation effects were represented using a canopy drag-force term based on leaf area density. Turbulence was simulated using the standard k–ε model, while additional transport equations described temperature, water vapour, and carbon dioxide dynamics. The governing equations were discretised using the Finite Volume Method (FVM) and solved numerically in MATLAB. Simulations were performed for dense, medium, and sparse canopy configurations over a 30 m computational domain. Results: Airflow velocity increased with height in all canopy configurations, with dense canopies showing the greatest attenuation. At canopy height, velocities were approximately 1.45 m/s, 1.95 m/s, and 2.65 m/s for dense, medium, and sparse canopies, respectively. Turbulent kinetic energy (TKE) peaked near the canopy top, reaching approximately 66 m²/s², 44 m²/s², and 22 m²/s², respectively. Temperature increased with height, while moisture and carbon dioxide concentrations decreased because of enhanced turbulent mixing. Dense canopies retained higher moisture and carbon dioxide levels than medium and sparse canopies. Conclusion: Forest canopy density significantly influenced airflow structure, turbulence production, and scalar transport. Dense canopies provided stronger microclimatic regulation through enhanced momentum attenuation, moisture retention, and carbon storage. The developed modelling framework provides a useful tool for studying canopy–atmosphere interactions and local climate dynamics.
Reliable Origin–Destination (OD) demand information is fundamental to transport planning, fleet scheduling, infrastructure investment, and policy formulation. However, in many developing cities, particularly those dominated by informal paratransit systems, comprehensive passenger-movement data are often unavailable, making conventional demand estimation difficult. This study developed and validated an integrated Gravity–Poisson framework for the synthetic estimation of passenger demand along the Nairobi CBD–Ongata Rongai–Kiserian public transport corridor, a major commuter route characterised by severe data limitations and highly directional travel patterns. The framework combines a doubly constrained gravity model for OD trip distribution with a Poisson stochastic process for modelling passenger arrivals. Spatial interactions were estimated using demographic and geospatial proxy variables derived from population distributions and network distances, while model calibration was achieved through parameter optimisation and the Iterative Proportional Fitting Procedure (IPFP). Theoretical properties of the framework, including the existence and uniqueness of balanced OD solutions, positivity of passenger flows, trip conservation, convergence of the balancing algorithm, distance elasticity, and parameter identifiability, were formally established. Model performance was evaluated using mean absolute error (MAE), root mean square error (RMSE), sensitivity analysis, and benchmarking against XGBoost and Long Short-Term Memory (LSTM) machine-learning models. The results revealed a highly concentrated morning commuter flow towards the Nairobi CBD, with approximately 97% of corridor demand converging at the Railways terminal. Calibration yielded a distance-decay parameter of β = 0.1000, indicating relatively weak sensitivity of commuter demand to travel distance. The balanced OD matrix satisfied all theoretical conservation and positivity conditions, while the Poisson arrival model indicated that rainfall, holidays, demonstrations, and network disruptions reduced passenger arrivals. Model calibration substantially improved predictive accuracy, and the sensitivity analysis supported the robustness of the framework under varying operational conditions. The study demonstrates that reliable and operationally meaningful transport-demand information can be generated even in environments where conventional OD survey data are unavailable. Beyond providing a practical decision-support tool for public transport planning and fleet allocation, the framework contributes to the theoretical foundations of synthetic demand estimation by integrating spatial interaction theory and stochastic arrival modelling within a mathematically rigorous and computationally validated framework. The approach offers a scalable methodology for transport-demand analysis and mobility planning in rapidly urbanising, data-constrained regions.
Background: Anaerobic digestion (AD) is a proven technology for renewable bio-methane production, but digester efficiency is often limited by poor hydrodynamic mixing rather than by microbial kinetics alone; most existing models, however, assume idealised, fully homogeneous reactors. Objective: This study investigates the influence of hydrodynamics on mixing efficiency and bio-methane production potential in anaerobic digesters using mathematical modelling. Methods: A two-dimensional incompressible Navier–Stokes model was coupled with a tracer advection–diffusion equation to simulate slurry flow and mixing behaviour. The governing equations were non-dimensionalised using the Reynolds and Péclet numbers, discretised using the finite difference method, and solved numerically in MATLAB. An optimisation framework that treated inlet velocity as the control variable, together with an adjoint sensitivity analysis, was used to evaluate and improve mixing efficiency. Results: At a Reynolds number of 2100, the flow exhibited transitional characteristics, with a dead zone fraction of approximately 35.1%. Velocity contours revealed limited circulation, whereas the tracer distribution showed a non-uniform concentration pattern across the domain. The dead zone fraction declined exponentially as Re increased, with values above 4000 projected to reduce it below 15%. At Pe = 10,000, transport was strongly advection-dominated, and the adjoint sensitivity analysis identified the inlet/impeller region as offering the greatest leverage over mixing performance. Conclusion: Hydrodynamic conditions play a critical role in determining mixing efficiency and, consequently, bio-methane production potential. The developed model provides a computationally efficient framework for analysing and optimising anaerobic digester performance and offers a foundation for future integration with biochemical reaction models.
This study proposes a closed-system linear ordinary differential equation (ODE) model to analyse and predict the dynamics of liquid assets and outstanding loans in credit union liquidity risk management. Using 40 quarterly observations (2015–2024) obtained from the Central Finance Facility (CFF) of the Ghana Co-operative Credit Unions Association (CUA), the study formulates a two-state deterministic model governed by the asset-to-loan transition rate α, the loan-to-asset recovery rate β, and the loan default rate δ. Closed-form solutions are derived using the Laplace transform technique, and discrete quarterly prediction equations are obtained using a first-order Taylor expansion. Stability analysis confirms that the system is asymptotically stable when α, β, and δ are positive. Model performance is evaluated using root mean square error (RMSE), mean absolute percentage error (MAPE), and the Scatter Index (SI). The optimal asset prediction model corresponds to α = 0.0869 (MAPE = 14.27%), and the optimal loan model to ω = 0.91 (RMSE = 225,548.63). Additionally, three machine-learning models—Support Vector Regression (SVR), XGBoost, and Random Forest—are employed to generate eight-quarter-ahead forecasts (2025–2026). SVR achieves the best out-of-sample performance for forecasting both assets (MAPE = 2.92%) and loans (MAPE = 3.70%). The study concludes that the ODE-based closed liquidity model provides an analytically rigorous and practically interpretable framework for credit union liquidity planning, and that SVR-based forecasting offers a useful complement for short-term projections.
We introduce a new graph structure, called the arithmetic divisor graph GAD(\(\mathbb{Z}_n\)), defined on \(\mathbb{Z}_n\) by declaring two vertices adjacent whenever the difference of their standard representatives divides their sum modulo n. We establish several fundamental structural properties of GAD(\(\mathbb{Z}_n\)): it is connected with radius 1, and it is complete if and only if n ≤ 4, while for n > 4 it has diameter 2. We further show that GAD(\(\mathbb{Z}_n\)) is Hamiltonian for all n ≥ 3, and we investigate its degree bounds, clique structure, and adjacency behavior. In particular, we obtain necessary conditions for adjacency among units, expressed in terms of congruence restrictions modulo prime divisors of n, leading to a complete characterization for odd moduli. Our resultsreveal a strong interaction between additive and multiplicative structures in \(\mathbb{Z}_n\), providing a new perspective on arithmetic graphs.
This article proposes a unified mathematical framework combining the theory of fuzzy vector spaces and that of fuzzy topologies to model complex systems characterized by uncertainty and structural imprecision. We formally introduce the notion of a fuzzy topological vector space and study its fundamental algebraic and topological properties, including the continuity of vector operations, fuzzy separation axioms, and fuzzy compactness and connectivity. A constructive approach based on α-cuts is developed to establish a rigorous link between fuzzy structures and classical topological vector spaces. Furthermore, we extend this framework to fuzzy functional analysis, fuzzy Sobolev spaces, fuzzy partial differential equations, and fuzzy dynamical systems. Potential applications in optimal control, image processing, artificial intelligence, and mathematical physics are also discussed. This work constitutes a theoretical contribution towards the coherent integration of algebra, topology and uncertainty, paving the way for the development of new mathematical tools for the analysis of complex systems. Future work could focus on integrating fuzzy topological vector spaces with fuzzy differential geometry, fuzzy manifolds, fuzzy neural networks, and advanced numerical methods, in order to develop a comprehensive analytical and computational theory.
This paper presents a high-resolution h-adaptive Finite Element Method (FEM) framework for simulating transient heat conduction in laser-based additive manufacturing (AM). The approach solves the 2D heat equation with a moving Gaussian heat source, using residual-based a posteriori error estimation and Dörfler marking to drive local mesh refinement. Linear triangular elements and implicit Euler time integration are combined with nearest-neighbour solution prolongation on adaptively regenerated meshes. Validation against manufactured solutions demonstrates exponential convergence in both L² and energy norms. Applied to a moving heat source problem, the method accurately captures melt pool dynamics while minimizing computational overhead. Numerical results highlight up to 90% reduction in computational cost compared to uniform refinement. This makes the method well-suited for predictive thermal modelling and real-time simulation in industrial AM applications.
This study formulates and analyses a reaction-diffusion predator-prey model incorporating logistic prey growth, Holling type II predation, stage structure in the predator population, prey-taxis of adult predators, and a fixed maturation delay. The model is posed on a bounded spatial domain with homogeneous Neumann boundary conditions. Basic qualitative properties of the system are investigated, including non-negativity of solutions and an exponential bound for the total population under appropriate parameter restrictions. Three spatially homogeneous equilibria are identified: the extinction equilibrium, the prey-only equilibrium, and a coexistence equilibrium. A threshold quantity for predator invasion is derived using a next-generation matrix approach, and its dependence on key biological parameters is discussed through normalized sensitivity indices. Local stability is examined in both non-spatial and spatial settings. The analysis indicates that prey-taxis can influence the onset of spatial instability at the coexistence equilibrium, while the maturation delay may contribute to temporal oscillatory behaviour through a Hopf-type mechanism. Conditions for Turing instability and delay-induced oscillation are presented in terms of model parameters. A Lyapunov-Krasovskii functional approach is also used to discuss sufficient conditions for global stability of the equilibria. The results suggest that the combined effects of directed predator movement and predator maturation delay can produce qualitatively distinct dynamical regimes, including stable coexistence, stationary spatial patterns, temporal oscillations, and combined spatio-temporal dynamics. The findings provide a theoretical framework for understanding how stage structure, taxis-driven movement, and delay may interact in spatial predator-prey systems.
We know formula for roots of general polynomial equations of degree two, three and four, also it has been proven that formula for general quintic could not written in the form of radicals, as on today there does not exist formula for the root of general quintic, In this article we shall find formula for roots of general equation of degree 5 which is not based on radicals.
Let V4 = {0, a, b, c} be the Klein-4-group with the elements a, b, c have order 2 and 0 be the identity element. Let G = (V (G),E(G)) be a simple, connected, finite and undirected graph. Let f : E(G) → V4∖{0} be an edge labeling and f+ : V (G) → V4 denotes the induced vertex labeling of f defined by f+(u) = \( \begin{array}{c} \sum\\uv\epsilon E(G) \end{array}\) f(uv) for all u ∈ V (G). Then f+ again induces an edge labeling f++ : E(G) → V4 defined by f++(uv) = f+(u)+f+(v), for all uv ∈ E(G). A graph G = (V (G),E(G)) is said to be an edge induced V4-magic graph (Libeeshkumar and Kumar, 2020a), if there exists an edge labeling f for which the function f++ is a constant function. The function f, so obtained is called an Edge Induced V4-Magic Labeling (EIML) of G. The present paper discusses some results related to the EIML of line graphs and provides a characterization of the EIML of line graphs for certain well-known named graphs.
Multi server queueing models with consultation by one well trained and experienced main server to the fellow servers play an important role in many modern service systems. In this paper, we analyze a two-server queueing model, where the main server provides consultation support in Markovian environment to the regular server. The main server serves customers directly but also extends consultation to the regular server with preemptive priority over its own customer service. The service of the customer at the main server will be interrupted due to higher-priority consultations. Both the number of interruptions at the main server and the number of consultations to the regular server are limited to finite upper bounds. The consultations are arised from M Markovian environmental factors which are related through a transition probability matrix F. The arrival process and consultation requirement are governed by two mutually independent Poisson processes. The service times at the main server and the regular server are modelled as mutually independent phase type distributions. The duration of the threshold clock is exponentially distributed whereas that of the super clock is a phase-type distributed parameter. The stability condition for the system is established and some performance measures are investigated through numerical analysis.
The George Pólya method strengthens mathematical problem-solving by guiding students through structured thinking, encouraging independence, and improving overall learning outcomes. This study evaluated the extent to which the Polya method improved students’ mathematical problem-solving skills. It specifically assessed the respondents’ competency, performance, and challenges encountered in the four phases of Polya’s approach: understanding the problem, devising a plan, carrying out the plan, and looking back. It determines whether the Polya method is an effective teaching and learning approach that enhances students' problem-solving skills. The research employed a one-group pre-posttest mixed-methods design. The research uses a one-block comprising 34 students. This study administered a pre-test and post-test questionnaire for the quantitative part and later identified the challenges respondents encountered using Polya's stages through qualitative interviews via audio call. This study was conducted at Western Philippines University over three months (September 2025 to November 2025) during the first semester of the 2025-2026 academic year. The research instrument used in this study comprised two components: the first assessed respondents’ problem-solving ability across the Polya stages; it consisted of 10 arithmetic problem-solving questions, 10 for each pre-test and post-test, and underwent face and content validity testing. Using SPSS software, the acquired data were examined utilizing the Wilcoxon Signed Rank test, mean, standard deviation, and percentage. For the qualitative part, the research interviewed ten (10) students who scored below the third quartile regarding the challenges they encountered at each stage of the Polya method, and the data were interpreted using thematic coding. The majority of respondents were categorized as Level 1 (Needs Improvement) in the pre-test findings, indicating low skill across all stages of Polya's problem-solving process. The respondents' performance improved significantly across the following stages of the Polya method: understanding the problem, devising a plan, and carrying out the plan after the intervention (i.e., the Polya method). Even though there was progress, many respondents still struggled with looking backstage. The mean score percentage increased from 33.62% in the pre-test to 75.41% in the post-test, indicating a significant improvement in the respondents’ mathematical problem-solving skills. The results indicated a significant difference between pre-test and post-test scores across all four stages of Polya’s technique, yielding a p-value = 0.00, which is less than the 0.05 level of significance (p < 0.05). Polya’s approach to problem-solving is a useful way to teach and raise students’ proficiency in solving real-world mathematics problems. Nonetheless, ongoing practice and instructional support are advised, especially to enhance students' reflective thinking and assessment skills during the looking-back stage of Polya’s problem-solving process. Strengthening students' creative problem-solving skills develops more competent and independent problem-solvers and, on a larger scale, potentially improves students’ performance in mathematics on regional, national, and international mathematics proficiency assessments.
In this this study, we introduced a novel deterministic Coefficient of Intensity derived from nano-topological boundary expansions to isolate the true structural damage caused by symptom removal. Because purely deterministic point-estimates are highly vulnerable to natural sample variance in small data set, we then applied the Bayesian theorem by mapping this coefficient to a Beta-Binomial conjugate model. This Bayesian approach allows us to incorporate statistical uncertainty and generate a posterior distributions that bridge topological approximation with predictive reliability. Statistical hazard intensity is evaluated through Joint Posterior Dominance (computed via Monte Carlo integration) and Interval Displacement (using the Inverse Regularized Beta Function). The proposed Bayesian-Topological model successfully resolves the structural blindness of classical rough set measures. By mathematically filtering out natural sample variance, it proves definitively that Low Platelets is the highly significant driving hazard in Dengue diagnostics. This framework provides a transparent, computationally efficient, and mathematically verified mechanism to rank hazard intensity for clinical decision support and Machine Learning expert systems.
