operator theory; finite difference method; variational iteration method; a posteriori error estimation; boundary value problems; Banach spaces; adaptive mesh refinement.Edem, Inemesit DavidThis paper proposes a unified operator-theoretic framework for analyzing the Finite Difference Method (FDM) and the Variational Iteration Method (VIM) for solving nonlinear second-order boundary value problems (BVPs). The VIM iteration is interpreted as a Picard-type fixed-point iteration in a Banach space, and a computable a priori error bound is established under a Lipschitz condition on the nonlinearity. Concurrently, FDM is treated as a finite-rank projection method, and its convergence is demonstrated via consistency and stability. The main contribution of this work is a novel hybrid a posteriori error estimator that uses the continuous VIM iterate to bound the global discretization error of the FDM solution without requiring additional PDE solves or mesh-doubling, requiring only the VIM iteration already computed as part of the error estimation process. Unlike traditional residual-based estimators, the proposed estimator relies solely on the discrepancy between the smooth semi-analytical approximation and the discrete numerical solution. Theoretical convergence rates and the efficiency of the estimator are validated through numerical experiments on a nonlinear BVP with a known exact solution and a non-solvable linear BVP. The results indicate that the hybrid estimator provides a reliable and computationally efficient indicator for adaptive mesh refinement.
African Journal of Mathematics, Statistics and Computer Science · 2026Research publicationInemesit D. Edem, Donald Goodhead, Utomobong M. OkonImage denoising is a classical inverse problem in which the recovery of a clean image from its noisy observation is complicated by ill-posedness. This paper develops a functional-analytic framework in which a broad class of denoising methods is expressed as the iteration of firmly nonexpansive or quasi-nonexpansive operators on a Hilbert space. Total variation regularization, wavelet shrinkage, projection onto convex sets (POCS), projection onto epigraph sets (PESC), and regularization by denoising via fixed-point projection (RED-PRO) are each shown to fit this structure, and their convergence is obtained from a single Krasnosel'skii–Mann argument. The proximal nature of wavelet shrinkage is distinguished from metric projection onto the \(\ell^1\)-ball, and the demicontractive hypothesis underlying RED-PRO is stated explicitly. Numerical experiments on standard test images show that RED-PRO with a BM3D denoiser attains the highest PSNR at every noise level tested, that Chambolle's TV projection follows within \(1.2\)–\(1.7\) dB, and that POCS and wavelet shrinkage trade accuracy for speed. The framework clarifies the operator-theoretic content of each method and provides a common language for their analysis and comparison.
Ktrend - International Journal of Computational Mathematics and Scientific Computing · 2026Research publicationOkeke Ikenna StephenA feasible manufacturing adjustment can preserve mandatory commitments while losing most of the contribution attainable through reoptimization. This paper studies nominal-surplus scaling under simultaneous capacity losses and increasing production floors. For arbitrary finite product and resource counts with nonnegative resource use, it derives the exact feasible stress interval and a finite breakpoint audit of the largest scaled-plan regret. A two-product family with strictly positive resource coefficients and positive commitments proves that the optimum-to-scaled-value ratio can grow without bound. A dual-certificate tracing procedure supplies an interval-wide regret bracket without enumerating every dual vertex. The generic tracing principle is established parametric optimization; its specialization here couples commitment-adjusted scaling with a complete regret audit. An explicit two-resource model also gives exact product-mix transitions, shadow prices and coefficient-uncertainty limits. Experiments cover 90 manufacturing instances and compare oracle tracing, direct optimizer grids and independently enumerated dual envelopes. The trace requires a mean of 3.58 LP queries on these instances; 62 have an interior regret maximum, and a 101-point grid understates the largest regret by up to 0.4425 accounting units. These synthetic results support complete-path auditing rather than extrapolation from a nominal shadow price.
Ktrend - International Journal of Computational Mathematics and Scientific Computing · 2026Research publicationOkeke Ikenna StephenHarmonic-mean transportation rules can generate feasible schedules without providing a cost guarantee. This paper studies the rule that selects the active row or column of largest harmonic mean and allocates to its cheapest cell. An explicit positive-cost family has unbounded approximation ratio, and a perturbed family shows that the failure persists throughout a full-dimensional neighbourhood with strict deciding choices. For every route-cost perturbation of magnitude at most 0.01, the ratio is bounded below by \((M+11.97)/13.03\) for \(M\geq32\). A weighted potential-correction problem then computes the strongest transportation dual bound obtainable by downward movement of supplied candidate potentials. Its bipartite packing dual supplies an independent repair certificate, and the resulting bound dominates a scalar correction. Computation includes 300 assignment instances, 600 perturbation tests and 120 rectangular transportation instances. The weighted correction improves the scalar bound on all rectangular cases, with a mean correction-loss ratio of approximately 0.372. The worst-case theorem concerns the specified initial rule; it does not apply to subsequent exact improvement. The study combines a perturbation-stable negative result with an auditable certificate framework. All numerical data are synthetic, and the underlying LP duality is established.
Ktrend - International Journal of Mathematics and Statistics (IJMS) · 2026Research publicationTiminibife N. Charles, T. I. AyakemeAccurate exchange-rate forecasting is important for financial planning, international trade, investment decisions and macroeconomic management. This study develops and compares three time-series forecasting approaches for the Naira–Pound Sterling exchange rate: autoregressive integrated moving average (ARIMA), autoregressive integrated moving average with exogenous variables (ARIMAX), and ARIMA combined with generalized autoregressive conditional heteroskedasticity (ARIMA-GARCH). A synthetic monthly dataset comprising 180 observations from January 2008 to December 2022 was generated specifically for methodological and forecasting-model evaluation. The synthetic observations are not presented as official historical observations. The analysis uses ARIMA(1,1,1) as the benchmark model, while ARIMAX incorporates the interest-rate differential, inflation differential, crude-oil price and Nigerian foreign-exchange reserves. ARIMA-GARCH combines an ARIMA conditional-mean specification with a GARCH(1,1) conditional-variance specification. The first 144 observations were used for model estimation and the final 36 observations were reserved for out-of-sample forecasting. Forecasting accuracy was evaluated using mean absolute error (MAE), root mean square error (RMSE) and mean absolute percentage error (MAPE). The simulated results show that ARIMAX achieved the lowest RMSE of 7.3931, while ARIMA recorded the lowest MAE and MAPE of 6.6043 and 3.4729%, respectively. ARIMA-GARCH produced results very close to the ARIMA benchmark.
Ktrend - International Journal of Mathematics and Statistics (IJMS) · 2026stochastics; SEIARV; intervention; COVID-19; modellingAniayam B. Okrinya, Timinibife N. CharlesThis study presents a stochastic SEIARV model for analysing COVID-19 transmission and vaccination intervention. The model incorporates symptomatic and asymptomatic infectious individuals, demographic recruitment and mortality, disease-induced mortality, incomplete vaccine protection, and stochastic fluctuations in epidemic dynamics. The basic reproduction number, $R_0$, is derived using the next-generation matrix approach, and the disease-free and endemic equilibria are analysed. The stochastic model is solved numerically using the Euler--Maruyama method, and multiple sample paths are generated to examine uncertainty in epidemic trajectories. Numerical results show that vaccination reduces both the reproduction number and the peak infectious population. In particular, increasing the vaccination rate from the baseline level to five times its value reduces $R_0$ below unity and substantially decreases the epidemic peak. The results demonstrate the importance of vaccination in reducing COVID-19 transmission and highlight the value of stochastic modelling in capturing variability and uncertainty in epidemic outcomes.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026Caputo derivative; HIV dynamics; CD4+ T cells; local stability; hybrid activation; bounded neural modifier.Okeke Ikenna StephenA commensurate Caputo model of HIV infection of CD4+ T cells is analysed by separating biologically admissible equilibria from algebraic roots. Building on the fractional model of Okeke et al. [1] and the hybrid activation construction of Essang et al. [2], this study introduces bounded neural modifiers of infection and viral production rates. Explicit equilibrium expressions and an infection threshold are derived. The disease-free equilibrium is locally asymptotically stable below the threshold, while a unique positive endemic equilibrium exists and is locally asymptotically stable above it for every common order in $(0,1]$. A recalculation using the parameter values reported in [1] produces a stable endemic equilibrium, contrary to the published numerical classification. A conservative bound on neural modifiers provides a sufficient condition for local disease-free stability across fixed covariate settings. A completed numerical study reports dimensionless fractional trajectories, threshold surfaces, Jacobian spectra and time-step refinement. These results concern mathematical equilibria rather than clinical treatment efficacy.
African Journal of Mathematics, Statistics and Computer Science · 2026structural identifiability; fractional HIV model; hybrid activation; inverse problem; slope bounds; physics-informed neural networksOkeke Ikenna StephenThe recovery of biological parameters from fractional HIV trajectories is examined alongside the mathematical properties of hybrid neural activations. Using the model of Okeke et al. [1] and the five-component activation mixture of Essang et al. [2], an exact parameter symmetry is derived: proliferation rate, healthy-cell death rate and carrying capacity cannot all be recovered separately from the model trajectories without additional information. A reduced parametrisation eliminates this redundancy. Conditional identifiability of the reduced coefficients is established through full-rank trajectory regressors for a known fractional order and fully observed states. The hybrid activation derivative is corrected, and global scalar slope bounds are proved. These bounds do not guarantee preservation of gradients through arbitrary network depth. A fractional residual learning framework is specified, and completed synthetic experiments examine parameter symmetry, regressor singular values, observation-window conditioning and slope products. The study establishes mathematical and computational limitations rather than empirical superiority of a trained learning algorithm.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026Accreditation; Quality assurance; Machine learning; Higher education; Early warning; NigeriaOsemengbe Oyaimare Uddin, Susan Konyeha, Glory Nosa EdegbeProgramme accreditation by the National Universities Commission (NUC) is the main external quality assurance mechanism in Nigerian universities, but it is periodic and retrospective: weaknesses are usually discovered during the visit rather than before it. This study aimed to develop and evaluate a machine-learning (ML) framework that predicts accreditation outcomes from routinely collected pre-visit indicators so that institutions can remediate deficiencies early. Because programme-level accreditation records are not publicly available, we built a reproducible synthetic panel of 4,321 accreditation visits to 1,515 programmes in 149 simulated universities (2014–2024), with outcomes (full, interim, denied) generated by NUC-style scoring rules. Fifteen indicators covering staffing, curriculum, facilities, library, funding, research and employer rating, together with ownership, discipline and prior status, were used as predictors. Logistic regression (LR), support vector machine, multilayer perceptron, random forest and extreme gradient boosting (XGBoost) were trained on 2014–2021 visits with university-grouped cross-validation and tested on 2022–2024 visits. LR performed best on the temporal test set (accuracy 0.773; macro-F1 0.708, 95% CI 0.669–0.744; area under the curve [AUC] 0.908), exceeding a prior-status rule (macro-F1 0.537). For the binary task of identifying programmes at risk of not receiving full accreditation, LR achieved an AUC of 0.900 and good calibration; the 20% of programmes with the highest predicted risk included 52 of the 53 denied programmes. Proportion of PhD-holding staff and laboratory provision were the most influential predictors. ML-based risk scoring could support continuous, pre-emptive quality monitoring, but validation on real NUC data is required before operational use.
African Journal of Mathematics, Statistics and Computer Science · 2026Research publicationTaiwo Wale Ayanniyi, Efosa Michael OgbeideMissing values in rainfall datasets reduce the reliability of statistical inference and undermine decision-making in agriculture, climate studies, and environmental management. This study evaluated the performance of five imputation techniques; Expectation-Maximization (EM), Multiple Imputation (MI), Regression Imputation (RI), Bootstrap Expectation-Maximization (BEM), and Random Forest (RF) using the 2019 Nigerian rainfall dataset obtained from the National Bureau of Statistics. The methods were compared using Raw Bias, Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Variance to assess estimation accuracy, predictive performance, and stability. The results revealed that MI produced the lowest bias (-0.003), making it the most suitable method for minimizing systematic estimation error. In contrast, RF achieved the highest predictive accuracy, recording the lowest MSE (0.9530) and RMSE (0.9762). Although BEM exhibited the lowest variance (0.9844), indicating greater stability, it was associated with relatively high bias, limiting its overall effectiveness. The findings demonstrate that no single method is universally optimal; rather, the choice of imputation technique should be guided by the primary analytical objective. RF is recommended for applications requiring high predictive accuracy, whereas MI is preferable when unbiased parameter estimation is essential. The study provides empirical evidence to support the adoption of robust imputation techniques by agencies such as the Nigerian Meteorological Agency (NiMet), thereby improving the quality of national climate databases and strengthening evidence-based agricultural and environmental decision-making.
Ktrend - International Journal of Mathematics and Statistics (IJMS) · 2026Rhotrix semigroup; extended Green relations; matrix monoid; finite field; rank pair.O. G. UdoakaLet $q$ be a prime power and let $T_m(q)$ be the monoid of order-$(2m-1)$ Rhotrix arrays whose row–column multiplication corresponds to componentwise multiplication in $M_m(\mathbb F_q)\times M_{m-1}(\mathbb F_q)$. The extended Green relations $\mathcal L^*$ and $\mathcal R^*$ are characterized by the row and column spaces of both matrix components. Regularity implies $\mathcal L^*=\mathcal L$ and $\mathcal R^*=\mathcal R$, while $\mathcal D^*=\mathcal D=\mathcal J$ is indexed by the pair of component ranks. For each rank pair, closed formulas are derived for the sizes and numbers of $\mathcal L^*$-, $\mathcal R^*$-, and $\mathcal H^*$-classes, the number of idempotents, and the order of every maximal subgroup. The principal ideal order, all two-sided ideals, and a rank-generating polynomial are determined. An exhaustive computation for $T_2(2)$ confirms the formulas and the translation-kernel definitions. These results provide an explicit finite-field structure theory for the specified row–column Rhotrix monoid.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026Research publicationOsifo Nosayaba Blessing, John Temitope OgbitiRansomware poses an escalating threat to healthcare institutions, where attacks disrupt clinical workflows, compromise patient data, and directly endanger patient safety. Existing detection systems are reactive, binary in classification, and lack contextual awareness of hospital-specific risk. This paper presents the design and implementation of Leadeks REWS, an AI-powered, hospital-aware ransomware early warning system that classifies system behaviour into three stages: normal, pre-encryption, and encryption, using a Random Forest ensemble classifier trained on dynamic behavioural features. The system identifies the affected hospital subsystem (Electronic Health Records, Picture Archiving and Communication Systems, or Administrative Systems), assigns a clinical risk level, and surfaces confidence-scored alerts through a real-time Flask and MongoDB dashboard with stage-specific mitigation recommendations. A rule-based Hospital Security Monitor additionally detects brute-force, privilege-escalation, and cross-system attacks within three interactive hospital subsystems. The Random Forest model, trained on an 80/20 stratified split of a 2,000-sample dynamic ransomware dataset, achieved 91% overall classification accuracy and 87% recall for the pre-encryption stage, outperforming Logistic Regression, single Decision Tree, and Support Vector Machine baselines. Twenty-five structured test cases and a formal evaluation against eight non-functional requirements confirm that the system delivers low-latency, reliable, and clinically prioritised ransomware detection suited to resource-constrained hospital environments.
Ktrend - International Journal of Computer Science and Artificial Intelligence (IJCSAI) · 2026stochastic processes; bank stress testing; geometric Brownian motion; exponential decay; stock pricesInnocent Uchenna Amadi, Nkemdinim Kaydee Odu-Ndom, Celestine NelsonThis paper applies an exponential-decay geometric Brownian motion (GBM) model to illustrate how shocks and sustained downward pressure may affect Nigerian bank stock prices. The standard GBM drift is adjusted from $\mu$ to $\mu-k$, where $k>0$ is the decay rate. Brownian-motion paths illustrate the source of uncertainty, while simulated stock-price paths demonstrate how assets with identical initial values can diverge under volatility. Sensitivity analysis shows that the relationship between $k$ and $\mu$ determines the behavior of the expected price: the mean grows when $\mu>k$, remains constant when $\mu=k$, and declines exponentially when $\mu\mu$, the mean-price half-life is $\ln(2)/(k-\mu)$. The model therefore offers a transparent scenario-generation tool for regulators and risk managers. It is an illustrative stress-testing model rather than an empirically calibrated model of any named Nigerian bank.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026stochastic fluctuations; bank valuation; geometric Brownian motion; inflation; long-term growth; stock prices.Innocent Uchenna Amadi, Tombari Stephen Ban, Nkemdinim Kaydee Odu-NdomThis study investigates the dynamics of Nigerian bank stock prices using an exponential-growth geometric Brownian motion framework. The model extends the classical geometric Brownian motion (GBM) by incorporating a growth-adjustment parameter $k$ in the drift term, allowing it to capture both deterministic long-term growth and stochastic fluctuations. Three Brownian-motion paths are first presented to illustrate the underlying randomness, followed by sample price paths demonstrating how the same growth rate can produce both upward and downward trajectories depending on the shock realizations. A sensitivity analysis is then conducted to assess the effect of $k$ on the simulated paths. The results show that higher values of $k$ amplify exponential growth or decay and increase dispersion in absolute valuation. This framework provides insight into how long-term growth expectations interact with short-term volatility in the Nigerian banking sector.
African Journal of Mathematics, Statistics and Computer Science · 2026leadership dynamics; financial performance; differential transform method; mathematical modellingS.O. EdekiLeadership is widely known to be the key to organizational effectiveness and firm financial performance, but the situation is complicated by the fact that leadership style, leadership behavior, employee experience, organizational state, and financial performance are not fully mathematically correlated. In this paper, two basic contributions are made. First, a mathematical model based on differential equations is developed to describe the dynamics of leadership and organization, and a rigorous mathematical proof is provided for the existence of a repeatable solution. Secondly, the model is analyzed and interpreted analytically and numerically using numerical solutions obtained from the differential transform method. In this technique, the set of differential equations is converted into a more convenient form. Since the coupled system of differential equations is not analytically tractable, the Differential Transform Method and numerical simulations are used to obtain approximate analytical solutions of the main equations. Lastly, the results are explored in terms of the evolution of leadership and its impact on employee, contextual, and financial outcomes. The results show that adaptive leadership strategies lead to better employee outcomes and make organizations more resilient and improve long-term financial performance compared to a focus on short-term wins. Therefore, the model proposed in this study is designed to be a quantitative tool for tracking the leadership process and a mathematical basis for managing organizations in the era of modernism.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026Research publicationEtim Uduak JamesOrdinary differential equations provide a common language for representing rates of change in chemical, biological, epidemiological, and financial systems. This study develops and computationally examines four model families drawn from physical and life-science applications: the dimensionless Lengyel–Epstein model for the chlorine dioxide–iodine–malonic acid reaction; a six-compartment demographic, exposure, infection, and AIDS-progression model motivated by delayed first sexual intercourse; the classical susceptible–infectious–removed epidemic model; and exponential growth and radioactive-decay models. Equilibria and local stability conditions are derived analytically, while numerical solutions are obtained with adaptive Runge–Kutta integration. For the chemical model with illustrative parameters $a=12$ and $b=0.30$, the positive equilibrium is unstable and the numerical trajectory approaches sustained oscillation. The delayed-intercourse model is locally asymptotically stable when the feedback between the sexually active and under-age compartments is weaker than total demographic removal, specifically when $(d_1+m_1)(d_2+m_2+b_2)>b_1m_1$. For the epidemic illustration, the effective transmission rate is $0.8$ per day, the recovery rate is $0.125$ per day, and $R_0=6.4$; the infectious population peaks at approximately $554$ persons near day $13$ in a population of $1,000$. Continuous $5%$ financial growth increases $20,000$ monetary units to $23,236.68$ after three years, whereas an $800,\mathrm{mg}$ bismuth-210 sample with a five-day half-life declines to $12.5,\mathrm{mg}$ after $30$ days. The results demonstrate how a shared differential-equation framework supports model formulation, stability analysis, simulation, and transparent comparison across distinct applications. All numerical outcomes are illustrative and are not fitted to clinical or laboratory observations.
Ktrend - International Journal of Computational Mathematics and Scientific Computing · 2026Research publicationAbosede Oyebukola Adenipebi, Mohammed Mubashiru Olayiwola Babatunde, Abari Ayodeji OlasunkanmiThe specification and implementation of teachers’ professional responsibilities are increasingly recognised as factors associated with student success. This study examined the relationship between the dimensions of teachers’ job descriptions, specifically planning, organising, lesson delivery, and classroom management, and students’ academic performance in public junior secondary schools in Lagos State, Nigeria. The study adopted a descriptive survey and correlational research design. The target population comprised 8,865 teachers in 327 public junior secondary schools across the state’s six Education Districts. A multi-stage sampling procedure was used. Of 563 questionnaires administered, 551 were valid for analysis, representing a 97.9% response rate. Data were collected using the Teachers’ Job Description Questionnaire (TJDQ), a 20-item instrument on a 4-point Likert scale, while students’ academic performance was measured using Basic Education Certificate Examination (BECE) results obtained through a Record Observation Format. The TJDQ achieved a Cronbach alpha reliability coefficient of 0.85. Data analysis involved descriptive statistics and Pearson Product-Moment Correlation at the .05 level of significance. The findings revealed a strong, positive, and statistically significant relationship between teachers’ job description and students’ academic performance $r=.711$, $N=551$, $p<.001$. Among the four dimensions, planning and lesson delivery had the highest mean scores $M=3.34$, followed closely by classroom management $M=3.33$ and organising $M=3.30$. The study concluded that clearer and more effective execution of prescribed instructional duties was associated with higher student achievement. Recommendations include the formal codification of teachers’ job descriptions by educational authorities and the provision of regular professional development focused on the core instructional dimensions.
Ktrend - International Journal of Education and Educational Research (IJEER) · 2026conformal mapping; Joukowski transformation; circular domains; airfoil profile; complex analysis; potential flow; aerodynamics.Etim Uduak JamesConformal mapping is one of the central geometric techniques of complex analysis because it permits a complicated planar domain to be represented by a simpler one while preserving local angles. This paper develops the analytic conditions for conformality from the principal linear part of a differentiable complex mapping, relates the Cauchy--Riemann equations to local rotation and dilation, and presents explicit mappings between standard domains. Particular attention is given to the Joukowski transformation and its constructive action on circular boundaries. Actual mappings are worked out for the upper half-plane and unit disk, for a strip and half-plane, and for circles mapped by the Joukowski transformation into a line segment, an ellipse, and an airfoil-like profile. The derivations are accompanied by graphical realizations and a discussion of potential-flow aerodynamics. The results make explicit the connection between the analytic formula, its critical points, transformed geometry, and the distinction between local conformality and global one-to-one behavior.
African Journal of Mathematics, Statistics and Computer Science · 2026soft set theory; soft algebra; soft group; soft ring; soft semiring; soft lattice; uncertainty; decision making; parameter reduction.Udoaka, O. G.Soft set theory is a parameterized framework for the representation of uncertainty in situations where the available information is incomplete, qualitative, or difficult to encode by a probability distribution or membership function. In this paper, I develop a unified treatment of the basic algebra of soft sets and examine how classical algebraic structures can be represented through parameter-dependent approximations. I formulate the principal soft-set operations on compatible parameter domains, establish elementary structural properties, and study soft groups, soft rings, soft semirings, and soft lattices. I further show that, for a fixed parameter set, suitable families of soft subsets inherit commutative idempotent monoid and distributive lattice-type structures under parameterwise union and intersection. To demonstrate the applied value of the framework, I construct a multi-parameter decision model for selecting an alternative from a finite universe. The model combines binary soft information, parameter reduction, and weighted choice values. An illustrative house-selection example shows how the same soft information can be converted into a transparent ranking procedure while preserving the semantic role of each parameter. The analysis emphasizes that the validity of algebraic identities depends on the adopted definitions and parameter domains, and therefore distinguishes fixed-domain operations from extended operations. The paper provides a concise bridge between the foundational theory, algebraic interpretation, and decision-making use of soft sets.
African Journal of Mathematics, Statistics and Computer Science · 2026Research publicationD. GOODHEAD, I. D. EDEMThis paper presents a unified coefficient-recurrence framework for obtaining series solutions of second-order linear ordinary differential equations with variable coefficients. The standard power-series method, applicable at ordinary points, and the Frobenius method, applicable at regular singular points, are treated as special cases of a common series ansatz $y(x)=(x-x_0)^r\sum_{n=0}^{\infty}a_n(x-x_0)^n$, where $r=0$ recovers the ordinary case and $r$ is determined by an indicial equation in the singular case. General recurrence relations for analytic coefficient functions are derived, conditions under which series terminate to yield polynomial solutions are established, and resonance in the integer-difference root case is characterized. Carefully selected examples illustrate the framework, including polynomial solutions, fractional exponents, resonance with logarithmic terms, and the Bessel equation. The analysis provides a systematic structural comparison of the two methods in terms of recurrence order, termination, resonance, and computational complexity.
Ktrend - International Journal of Mathematics and Statistics (IJMS) · 2026Research publicationOrdu Isaac Chioma, Florence Nne AgwuLoanwords are an integral part of language development, particularly in multilingual societies where speakers are in constant contact with other languages. As languages evolve, lexical items are borrowed to meet new communicative, cultural, and social needs. This study identifies emerging loanwords in Nigerian English and examines how they enter Nigerian English, with particular attention to indigenous Nigerian languages within the Port Harcourt speech community. The study is anchored in Terence Odlin’s theory of language transfer, especially the concept of substratum transfer, which explains the influence of a source or native language on a recipient language [16]. A qualitative survey design was adopted, with data collected through random sampling from language use on WhatsApp and Facebook. The findings show that many emerging loanwords in Nigerian English originate from Igbo, Hausa, Yoruba, and Nigerian Pidgin. Some retain their meanings in both donor and recipient languages, whereas others undergo semantic change after entering Nigerian English. Of the 108 items examined, 30 exhibit semantic change and 78 retain their donor-language meanings. The study recommends greater scholarly attention to emerging loanwords because they contribute to the continuing lexical expansion and localization of Nigerian English.
Ktrend - International Journal of English, Communication and Applied Linguistics · 2026wealth; stochastic systems; financial mathematics; periodic events; stocks.Nwagor PetersFinancial-market dynamics are inherently uncertain, and stochastic differential equations provide a natural framework for assessing the evolution of investment wealth under such conditions. This study formulates two stochastic systems for measuring the wealth of corporate investors by incorporating expected stock returns, intrinsic growth rates, interest-rate parameters, volatility, periodic effects, and random market fluctuations. The systems are solved analytically through Itô's lemma after logarithmic transformation of the wealth processes, yielding explicit expressions for the fourth and fifth corporate investors. For the fourth corporate investor, the stochastic wealth process is expressed as $$ V_4(t)=V_{40}\exp\left[\left(\mu\alpha_4-\beta_4-\frac{1}{2}\sigma^2\right)t+\sigma W_t^4\right], $$ while the corresponding wealth process for the fifth corporate investor is $$ V_5(t)=V_{50}\exp\left[\left(K\tanh(\alpha_5)-\beta_5-\frac{1}{2}\sigma^2\right)t+\sigma W_t^5\right]. $$ Numerical evaluations are used to examine the effects of intrinsic growth, interest rates, and stock volatility on portfolio values. The results show that increases in intrinsic growth rates generally increase investor wealth, whereas higher interest-rate parameters reduce wealth. Increased volatility lowers wealth under the non-periodic specification and makes wealth more sensitive to market fluctuations when periodic effects are incorporated. Surface-view representations further illustrate the response of investor wealth to changes in the principal model parameters. The findings provide a quantitative basis for corporate investment decisions under time-varying and uncertain market conditions and suggest that stochastic delay and periodic extensions may offer useful directions for subsequent research.
Ktrend – Nigerian Journal of Mathematical and Computational Sciences · 2026wealth; stochastic systems; investments; interest rates; stock pricesNwagor PetersUncertainty in stock-market conditions makes the quantitative assessment of corporate investment portfolios an important problem in financial modelling. This study develops a system of stochastic differential equations for estimating the wealth dynamics of three corporate investors under changing market conditions. The model incorporates expected stock returns, intrinsic growth rates, interest-rate effects, stock-price volatility, and random market fluctuations. The stochastic wealth processes are solved analytically using Itô's lemma, leading to explicit solutions of the general form $$ V_i(t)=V_{i0}\exp\left[\left(\mu\alpha_i-\beta_i-\frac{1}{2}\sigma^2\right)t+\sigma W_i(t)\right],\qquad i=1,2,3. $$ Numerical evaluations are performed to determine the effects of the principal model parameters on corporate-investor portfolio values. The results show that increases in the intrinsic growth-rate parameters $\alpha_i$ are associated with higher terminal wealth, whereas increases in the interest-rate parameters $\beta_i$ reduce portfolio wealth. The volatility parameter $\sigma$ affects both the deterministic Itô correction $-\frac{1}{2}\sigma^2$ and the stochastic component $\sigma W_i(t)$, thereby influencing the distribution and sensitivity of terminal wealth. Under the parameter values considered, the second corporate investor records the largest wealth values among the three investors. The stochastic framework provides a useful mathematical approach for assessing corporate-investor portfolio values and evaluating the effects of key financial parameters under uncertain stock-market conditions.
African Journal of Mathematics, Statistics and Computer Science · 2026Research publicationObataze J. Akpoyovwere, Aruoriwo Grace OkparaveroContraceptive access and informed use are essential components of reproductive health, enabling women to prevent unintended pregnancies, achieve desired birth spacing, and exercise informed reproductive choices. Despite increasing awareness of family-planning services in Nigeria, contraceptive acceptance and consistent utilization remain uneven across communities. This study assessed knowledge, attitudes, acceptance, utilization, communication patterns, and perceived barriers to contraceptive practices among women of childbearing age in Sapele Community, Delta State, Nigeria. A descriptive cross-sectional survey was conducted among 109 women aged 15–49 years selected through a multistage sampling procedure. Data were collected using a structured, pre-tested questionnaire with good internal consistency (Cronbach’s α = 0.85) and analysed using frequencies and percentages. Although 69 respondents (63.3%) had heard about contraceptive practices, 69 (63.3%) were classified as having poor method-specific knowledge, indicating a considerable gap between general awareness and practical contraceptive literacy. Fifty-one respondents (46.8%) reported ever or current contraceptive use, whereas 58 (53.2%) reported no use, and only 13 (11.9%) reported always using contraception. Condoms were the most frequently reported method (18.3%), followed by intrauterine devices (9.2%), injectables (approximately 8–9%), implants (5.5%), and oral contraceptive pills (4.6%). Attitudes toward contraception were predominantly unfavourable: 55.0% did not consider contraception important for women’s health, 50.5% did not regard its use as morally acceptable, and 64.2% did not believe that contraception was effective in preventing pregnancy. Fear of side effects (31.2%) was the leading reported barrier, followed by inadequate contraceptive knowledge (27.5%), partner opposition (22.9%), and religious or cultural beliefs (18.3%). Furthermore, 62.4% had never discussed contraception with a partner or healthcare provider, while 40.4% reported difficulty accessing contraceptive services. The findings demonstrate that general awareness alone does not necessarily translate into adequate knowledge, favourable attitudes, acceptance, or consistent contraceptive use. Strengthening method-specific education, nurse-led counselling, side-effect management, confidential reproductive-health communication, culturally responsive community engagement, reliable access to a broad range of contraceptive methods, and appropriate partner involvement may improve informed contraceptive decision-making and sustained utilization among women of reproductive age in Sapele.
Ktrend - International Journal of Medical and Health Sciences (IJMHS) · 2026 