This paper defines and explains, in simple and clear terms, key concepts of descriptive statistics and connects statistical theory with applied statistical practice through real cases. It begins by clarifying the meaning of statistics and the basic notions used daily in many fields, while distinguishing statistics as a science and an art from statistics understood as data and measures. The study adopts the following definition: “Statistics is simply an art and a science that allows for the collection, analysis, presentation, and interpretation of data for the purpose of making decisions about any phenomenon under study.” To illustrate these concepts, practical applications were carried out using real data from two institutions: AL BANK and a non-governmental organization. For the calculations and applications, EXCEL, EVIEWS, and R STUDIO were used. In the AL BANK case, relative frequencies were calculated step by step with the three software programs and then verified manually using the mathematical frequency formula. In Excel, the process involved entering the data, computing the total number of observations, and calculating the relative frequencies. In EVIEWS, the observations were entered, descriptive statistical measures including the total were obtained, and relative frequencies were then calculated. In R STUDIO, the data were entered as a vector and relative frequencies were computed. The three methods produced identical results, showing that 66% of AL BANK employees are single, 31% are married, 2% are widowed, and 1% is divorced. In the non-governmental organization case, the variable “business opening” was used to calculate cumulative frequencies. The procedure consisted of constructing the absolute frequency table, the relative frequency table, and then the cumulative absolute and cumulative relative frequency tables. The analysis showed that, in terms of cumulative absolute frequency, ten individuals reported that they sometimes or often open their businesses. Regarding relative frequencies, six out of sixteen individuals reported often opening their businesses (37.5%), while 25% sometimes open their businesses, 25% rarely open them, and 12.5% never open them. These findings highlight the importance of frequency as the one of the fundamental concepts of applied statistics.
Mathematical model is formulated using Partial differential equations, and analyzed to describe the dynamics of wave propagation with application to landslide. Its occurrence leads to loss of lives of people and their properties. This study aims at understanding the dynamics of landslide wave propagation, in order to make adequate projection in an attempt to mitigate the risks involved in the occurrence. The objective of this study is; to develop a mathematical model to describe the dynamics of a landslide wave propagation. Numerical solutions were carried out using Runge-Kutta algorithm’s inbuilt in MATLAB, to simulate the current and future dynamics of the model. The mathematical model was used to simulate and analyze landslide wave propagation phenomena in affected areas. The results of this study showed that the maximum amplitude of 0.49m is achieved at an inclination of 50° and gradually drops down to 0.45m at a slope of 19.2°. This defines the region where inhabitants should be relocated to avoid loss of lives and destruction of property during the occurrence of landslide. The high amplitude occurs in the wave propagation region, necessitating the implementation of control strategies along the same region.
Time and reality are central concepts in both philosophy and physics. While classical mechanics describes the universe as deterministic, where the present emerges from past conditions, quantum theory introduces probabilistic interpretations suggesting inherent uncertainty at microscopic scales. These two perspectives have led to an ongoing debate regarding whether reality is fundamentally deterministic, probabilistic, or a combination of both. The Zia Theory of Temporal Reality (ZTTR) integrates these frameworks by proposing that the present state of reality is determined by accumulated past causes, while the future evolves probabilistically under uncertainty. To formalize this concept, the Zia Unified Continuity Equation (ZUCE) is expressed as a stochastic model where the state of reality represents a combination of initial conditions, integrated causal influences, and stochastic fluctuations represented by a Wiener process. To enhance generality and address limitations of fixed functional assumptions, the uncertainty term is further extended as a generalized function, allowing different forms such as diffusion-like, logarithmic, or nonlinear growth depending on system dynamics. In this framework, deterministic causal accumulation represents the structured continuity of past influences, while the stochastic component reflects uncertainty arising from complexity, interaction, and incomplete knowledge of system variables. This interpretation aligns with developments in statistical mechanics, information theory, and complex systems research. The applicability of the model is illustrated through conceptual and numerical examples drawn from physical processes, biological systems, and learning dynamics. These examples demonstrate that deterministic historical structures and stochastic variations can coexist within a single temporal framework. The Zia Theory of Temporal Reality thus provides a generalized and mathematically consistent perspective on temporal evolution and contributes to interdisciplinary research in physics, complexity science, and artificial intelligence.
The commencement of conventions which are the stationary and oscillatory in the study of the influence of temperature-dependent internal heating on magnetoconvection in a rotating Darcy porous layer was systematically examined. The linear stability analysis was scrutinized with the boundary condition of the problem being free-free. A modified Boussinesq approximation is included in the momentum equation and the Coriolis force was taken into consideration. The given non-dimensional equations were linearized and the normal mode technique was used to obtain marginal stationary and the marginal oscillatory convection. The criteria for the onset of convection in the system are derived analytically. The effects of the parameters: heat source, , magnetic field, , rotation, and ratio of viscosities, on the onset of convection are presented and analyzed graphically in details. The investigation showed that the result of raising magnetic field, rotation and ratio of viscosity slowed down the commencement of the convections, that is both the stationary and oscillatory convections, hence making the system steady. In the other hand, increasing heat source parameter, catalyzes the commencement of convection and destabilizes the mechanism. Prandtl number was found to slow the onset of oscillatory convection. Hence we can infer that magnetic field parameter, Ha, rotation parameter, T D , and the Prandtl number, Pr, are balancing factors, while the heat source parameter, ℽ, speeds up the commencement of convection.
In this article we pursue the aim to define (and we prove) a model of quantized co-observability and co-objectivity of the language-linguistic knowledge space proper to framing (typical), antiframing (autistic) duality along with the proper space of co-semioticity (explanability/observability). Thus, a first distinction between typified propositional duality of text/field (relying on ACE sequences) and the underlying tensegrity of both text/its individuation (text/individual structures) provides an inviariant model of balance grounding in the reducibility of mathematical spaces (propositional rational cuts as duality, Dedekind reals on predicative recursion...) and the widening of their epistemic-mathematical assumptions. From understandable/non-understandable duality of sentences (syntax/semantics), a double verifiability of algebraic stress on tensegrity frameworks and its graph choosability verifies an odd/even graph of in-out polynomially/non-polynomially complete setting of co-observability/co-objectivity in phase with brain-consciousness mathematical spaces for language (higher and lower). Secondly, thanks to the hypothesis of epistemic numbers and modularity, we advance the concept of brain-consciousness semiome throughout distributional tests leading to establish maps of scientific (object-based) and semiotic (explanability/interpretability) directions. Henceforth, co-observation/co-objectification of language-linguistic autism proposes a robust and nuanced epistemic-mathematical frame aware of the overlapping issues dealt with from other linguistic and non-linguistic methodologies on autism and its language conjecture.
In this paper, we introduce two new operations on Min-max Intuitionistic Fuzzy Graphs, namely Parallel and Series connections along with some of its basic properties. Parallel and Series connections between two Min-max Intuitionistic Fuzzy Graphs are constructed and illustrated with relevant examples. Parallel connections and Series connections are obtained by deleting any two edges and introducing a new edge. Definition of Max-max Intuitionistic Fuzzy Graph is also introduced. Construction of Parallel and Series connections between two Min-max Intuitionistic Fuzzy Graphs gives rise to Max-max IFG which is a new type of Intuitionistic Fuzzy Graph where both the membership and non-membership functions of some of the edges are less than or equal to maximum of membership and non-membership functions of their respective incident vertices. It is shown that the number of edges in a Parallel connection is twice the product of edges in the two Min-max Intuitionistic Fuzzy Graphs whereas in a Series connection it is four times the product of edges in the two Min-max Intuitionistic Fuzzy Graphs. The study reveals that any two electrical circuits with its enclosed components (resistors, capacitors etc.) can be represented as vertices of two Min-max Intuitionistic Fuzzy Graphs and their Parallel and Series connections can be generated.
This paper investigates the geometric properties of normalised Bessel functions of the first kind, focusing on the radii of convexity and uniform convexity. Using analytic techniques such as logarithmic differentiation and properties of zeros of the Bessel function. Additionally, we provide new proofs related to the order convexity and compare our results with existing results to reveal interesting relationships, and we provide a graphical interpretation that supports the analytical findings.
This study introduces a fractional-order HIV-1 infection model formulated with the Caputo derivative, combining the effects of antiretroviral therapy and a saturating cytotoxic T lymphocyte (CTL) immune response. The fractional formulation allows the inclusion of memory-dependent viral dynamics and the saturation function provides a biologically consistent representation of immune regulation. The basic reproduction number R0 is derived and used to determine the threshold behavior of the system. A theoretical analysis is carried out to prove existence and uniqueness of solutions and to investigate both local and global stability of the disease-free equilibrium. The endemic equilibrium is also derived to offer a deeper understanding of long-term infection dynamics. To obtain approximate solutions for the nonlinear system the semi analytical methods, the Differential Transform Method (DTM), Adomian Decomposition Method (ADM), and Homotopy Perturbation Method (HPM), are applied. A fractional predictor–corrector scheme is applied and compared with the semi analytical solutions. Numerical experiments show the influence of the fractional order, therapeutic effectiveness and immune parameters on disease evolution. The results indicate that decreasing the fractional order significantly alters the transient dynamics of viral load and CD4+ T-cell populations, demonstrating that fractional-order modeling provides a more flexible and realistic framework for describing HIV-1 dynamics and evaluating treatment effects.
This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.
In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.
This paper introduces and investigates the concepts of strong hub sets and the strong hub number in hypergraphs, extending the notion of hub sets defined for graphs. For a hypergraph H = (V, E), a subset S ⊆ V (H) is said to be a strong hub set if, for every pair of distinct vertices u, v ∈ V (H) − S, either u and v are adjacent or there exists a strong S-hyperpath joining them. The minimum cardinality of such a set is called the strong hub number of H, denoted by h∗(H). Fundamental properties of strong hub sets are established, and relationships between the strong hub number and various hypergraph parameters such as the domination number and connectivity are explored. The effects of vertex deletion and weak deletion on h∗(H) are studied, leading to several sharp bounds and recursive characterizations. In particular, it is shown that under weak deletion of a non cut vertex, the strong hub number remains invariant, while deletion of a cut vertex reduces it according to the structure of the resulting components. Special attention is devoted to hypertrees, where structural simplicity allows a precise characterization. This characterization provides an exact formula for h∗(H) in acyclic hypergraphs and establishes the foundational link between connectivity and hub-based path structure in hypertrees.
Group decision-making is now an essential approach in our daily lives. It plays a crucial role in the decision-making process. This compels certain human structures or decision-makers to seek external assistance in order to reach a consensus that is accepted by all stakeholders. This is why many multi-criteria decision-making methods have been developed and are widely used to clarify complex decision-making situations where intuition alone is insufficient. Among these existing methods, a new one has recently been developed, the scientific validity of which has been proven: the MACBEV method. It is obtained by hybridizing the EVAMIX method and the VMAVA+ voting method. The collective aggregation method based on the EVAMIX method and the VMAVA+ voting method (MACBEV) is one of these very recent methods that generates good properties but is unfortunately used to solve problems with small datasets where calculations are performed manually. Given the importance of the MACBEV method, it is essential to develop a computer program to broaden its scope. This will facilitate its application to concrete cases. In this work, we propose an algorithm and a computer program for this method that efficiently solves group decision problems, particularly large-scale problems whose manual processing is impractical. We then conduct a theoretical and graphical complexity study to demonstrate the efficiency of our program. Our computer model has been applied to large-scale data problems, and this has produced satisfactory results.
The aim of present paper is to find the frequencies using the mathematical model: Effect of parabolically varying non-homogeneity on thermally induced vibration of orthotropic trapezoidal plate with thickness varies parabolically in both directions. In the above model both thickness and density varies parabolically. Rayleigh-Ritz method is used to solve the governing differential equation for maximum strain energy and maximum kinetic energy for orthotropic trapezoidal plate. Two term deflection function corresponding to clamped-simply supported clamped-simply supported (C-S-C-S) boundary condition is defined by the product of the equation of the prescribed continuous piecewise boundary shape. The effect of frequencies for first and second mode investigated with the variations in structural parameters such as taper constant, non-homogeneity constant, aspect ratio and thermal gradient respectively. Differential equations for maximum strain energy and maximum kinetic energy for orthotropic trapezoidal plate are solved using the mathematica software. All the results are calculated with great accuracy and are displayed graphically. To validate the model all the results are also compared with the preexisting literature and fit well. So by developing such type of model we increase sustainability and also reduce the environmental hazards.
This study investigates the three-dimensional flow and heat transfer characteristics of a Jeffrey nanofluid flowing through a stretching channel under the influence of a Lorentz force generated by an applied magnetic field. The Jeffrey fluid model is a significant non-Newtonian fluid model that accounts for both relaxation and retardation effects, which are important in describing the viscoelastic behavior of complex fluids. The incorporation of magneto-hydrodynamic (MHD) effects enables the analysis of electrically conducting fluids subjected to magnetic forces, which are widely encountered in industrial and engineering applications such as cooling systems, polymer processing, and biomedical devices. The analysis further considers nonlinear thermal radiation to accurately represent heat transfer at high temperature conditions. In addition, the Soret and Dufour effects are included to examine cross-diffusion phenomena between heat and mass transfer processes. The Soret effect describes mass diffusion caused by temperature gradients, whereas the Dufour effect represents energy flux generated due to concentration gradients. These coupled transport mechanisms significantly influence the thermal and concentration boundary layers. The governing nonlinear PDEs are transformed into ODEs using suitable similarity transformations and solved numerically. The effects of various controlling physical parameters on velocity, temperature, and concentration distributions are examined in detail. The numerical results reveal that an increase in the Dufour number enhances thermal energy transport, leading to higher temperature and velocity profiles while reducing concentration distribution. Conversely, increasing the Soret number strengthens mass diffusion induced by temperature gradients, thereby improving concentration and velocity distributions within the boundary layer region.
The large amount of information nowadays requires building Data Centers and implementation of optimization models for storing and transferring data. The requirement of limited time of processing the network requests, are needed proper ways of redirection of data and application of algorithms programmatically in different levels. Before this, a data has to be gathered under different circumstances and to be checked if the information has been transferred successfully or not and then based on the results the counts of successful and not successful outcomes to presented and compared with predicate theory. There are many probability models, with which can be analyzed and predicted future events. In this article with the Theory of Index matrices, Graph theory and Theory of probabilities will be analyzed a stochastic process for modeling the times at which flows of a network enter a system. Because the network traffic depends on time, different scenarios of communication durations such as intrinsic time interval and endogenous jump time, will be considered and evaluated if they perform a certain condition. The most proper results of the experiments, which will be calculated with linear and exponential functions and represented with different Index matrices, can be used in machine learning of Data Center Networks.
Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.
This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.
The fractal dimension is the basic notion for describing structures that have a scaling symmetry. In finance, multi-fractality is one of the well known facts which characterized non-trivial properties of financial time series. The stock price (or index) fluctuations can be described in terms of long-range temporal correlations by a spectrum of the Holder exponents and a set of fractal dimensions. To forecast the market risk, assessing the stock price indices is the foundation. Multi-fractal has lots of advantages when explaining the volatility of the stock prices. The asset price returns are multi-period market depending on market scenarios which are the measure points. In this work, we use some tools of multi-fractal analysis to derive the worth growth rate of an investor’s portfolio for particular and general cases. For the particular case, we considered the situation when the mean interest rate of some stocks does not depend on other stocks in the market. That is, an investor has invested his money in a stock with a linear mean return. Under the general case, we considered a market comprising some units of assets in long position and a unit of the option in short position. Using Ito’s formula on the present value of the market, we derived the growth rate of investor’s portfolio. Our model equations, which are based on multiplicative processes, capture all the features of the returns. They are tested using data from Zenith Bank of Nigeria stock prices. From our graphs, the worth of investment grows as stock price increases and also decreases with stock price.
A theoretical analysis is made on the unsteady stagnation point flow of a conducting fluid over a flat stretching surface in the presence of magnetic field with chemically reactive species concentration and mass diffusion under Soret and Dufour effects. The governing partial differential equations of continuity, momentum, energy and concentration have been converted to self-similar unsteady equations by using similarity transformations and solved numerically by the Runge-Kutta algorithm with Newton iteration in double precisions along with the shooting method across the boundary layer for the whole transient domain from the initial state to the final steady state flow. The effects of existing flow parameters viz Soret number, Dufour number, chemical reaction parameter, Darcy number and magnetic parameter are shown graphically for the dimensionless velocity, temperature and concentration of the conducting fluid.The velocity of the conducting fluid is seen to decrease across the boundary layer with increasing the magnetic parameter, Prandtl number, Schmidt number and chemical reaction parameter; and the velocity profiles are seen to increase with increasing the thermal Grashof number, mass Grashof number, Soret number, stretching parameter and Dufour number. The temperature is seen to decrease with increasing the Prandtl number, Soret number, stretching parameter, and the same temperature are found to increase with increasing Dufor number and Chemical reaction parameter across the boundary layer.In the same way, the concentration is seen to reduce with increasing Schmidt number, Dufour number, stretching parameter,chemical reaction parameter and but concentration increases with increasing Soret number. Further more, numerical results for the skin friction, Nusselt number and Sherwood number are tabulated for various flow parameters.It is clearly observed from the result that a smooth transition of flow of conducting fluid is seen from unsteady stage to the final steady stage. Skin friction decreases with the increasing magnetic parameter, Schmidt number, Prandtl number and chemical reaction parameter and same skin friction increases with the increasing Soret numbe, Dufour number, thermal and concentration buoyancy parameters, Darcy number, stretching parameter and dimensionless time. Nusselt number is seen to increase with increasing the value of Soret number, Prandtl number, stretching parameter, dimensionless time and the same Nusselt number decreases with increasing the value of Dufour number, chemical reaction parameter and Schmidt number. Sherwood number is seen to decrease with increasing the value of Soret number, Dufour number, Prandtl number and dimensionless time and is seen to increase with increasing Chemical reaction parameter, stretching parameter and Schmidt number. The results obtained in this investigation are seen good agreement with earlier published results in some particular cases.
The dynamic mechanism comprising an enzymatic reaction and the diffusion of reactants and products inside a glucose-sensitive composite membrane is described using a mathematical model created by Abdekhodaie and Wu. A set of non-linear steady-state reaction-diffusion equations is presented in this theoretical model. These equations have been meticulously and accurately solved analytically, considering the concentrations of glucose, oxygen, and gluconic acid, using a novel approach of Akbari Ganji and differential transform methods. The high level of agreement between these analytical results and the numerical results for steady-state conditions is a testament to the model's precision. A numerical simulation was produced via the precise and widely used MATLAB software. A comprehensive graphic representation of the model's various kinetic parameters' effects has also been provided. Additionally, a theoretical analysis of the kinetic parameters, such as the maximal reaction velocity (Vmax) and the Michaelis-Menten constants (Kg and Kox) for oxygen and glucose, pH profiles with membranes is presented. This expressed model is incredibly helpful when creating glucose-responsive composite membranes for closed-loop insulin delivery.
