This article will begin with the claim that Hamilton spent a great deal of time trying to figure out the three-dimensional complex numbers. He was never able to accomplish that.

Complex numbers are in the form a+bi where ‘a’ is a real part and ‘bi’ an imaginary with 𝑖= √−1 The motive behind the claim is that both 𝑖2 and 𝑗2= −1; The failure may also have been caused by the lack of a proper definition for the field of complex numbers. To address this issue, the author of this article offers his own definition of the field of complex numbers with key vectots i,j,k taking values (-1,0,1) respectively Of course the field of complex numbers remains unchanged with 𝑋 ⃗⃗⃗ =𝑥+𝑦𝑖+𝑧𝑗 under transformation becoming 𝑋 ⃗⃗⃗ =𝑥𝑖+𝑦𝑗+𝑧𝑘 vector but it has it’s correspondence in the field of real numbers and it’s a vector (-1,0,1) The entire process of transition between fields, in author’s opinion, is possible thanks to the matrix of transformation It’s form has already been explored earlier on by the same author in his ‘Ternary Mathematics and 3D Placement of Logical Elements Justification’.

Professor Juan Weisz (Doctor of Philosophy, Northeastern University, Argentina) generously proposed the concept of field transition, which allows for conversion between imaginary and real numbers without altering the field’s structure or the relationships between its constituent parts. In essense it should work for all entries just the same way it works for the entries of real numbers The fact that serves as the proof is in 𝐴′𝐴=1 and it works well for both Real and Imaginary number fields which is what we are aiming to prove.

This study evaluates the performance of various fuzzy membership functions (MFs) in predicting volume and bedload rate using sediment data from a bathymetric survey at Ikpoba Dam. Twelve cases with different membership functions: Gaussian, triangular, trapezoidal, and bell-shape were tested across different epochs. The models were assessed based on Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²) values for both training and testing datasets. The Gaussian membership function (Gaussmf), with 7 membership functions and 200 training epochs, outperformed the others, achieving the lowest RMSE of 0.568 (training) and 0.579 (testing), MAE of 0.437 (training) and 0.445 (testing), and highest R² values of 0.914 (training) and 0.932 (testing) for volume prediction. For bedload rate, it also achieved the lowest RMSE of 0.509 (training) and 0.517 (testing), MAE of 0.391 (training) and 0.397 (testing), and highest R² values of 0.9354 (training) and 0.9496 (testing). In contrast, the Trapezoidal membership function (Trapmf) showed the worst performance with RMSE values of 0.874 (training) and 0.905 (testing), MAE values of 0.652 (training) and 0.677 (testing), and R² values of 0.812 (training) and 0.804 (testing). These results emphasize the significance of membership function selection and training epochs in optimizing fuzzy models for environmental and geospatial applications.

Aims/ Objectives: The aim of the article is to provide the proof for use of trits in coding of the signal for Ternary processors The author substantiates the aforementioned use by formula and properties of Ternary Algebra In order to do so an analysis of the incoming signal is done and the wave is presented in the form of 𝑓(𝑥)=𝑎 sin⁡(bx+C)+D By converting this formula we obtain numeric value of the energy conveyed by the standing wave The same numeric value is the mathematical representation of a signal in the Ternary circuits (Pierce, 1973)

Volterra integral equations (VIEs) are extensively utilized across scientific, technological and engineeringdisciplines for modeling systems influenced by past behaviors or interactions. This study focuses on solvingnonlinear VIEs using a combination of power series and trigonometrically fitted functions. The analysis of thebasic properties of the Volterra scheme were numerically analyzed. The Volterra scheme is efficient, accurateand particularly effective for nonlinear problems. Numerical applications demonstrate the scheme’s accuracyin solving second-kind nonlinear Volterra integral equations, validating its robustness and computationaladvantages.

In the ordinal theory, maximum utility is achieved by determining the tangent point of the budget constraint line to one of the difference curves of equal satisfaction of a consumer. The model is a simplified model to determine the choices of buying two products such that maximum consumer satisfaction is reached. Market conditions frequently change due to inflation, deflation or price fluctuations. It may also happen that the individual income may change in time. In both cases, the budget constraint line shifts to the right or left. In this work, budget constraint line changes are modeled by assuming changes in the intercepts of the line. In the first scenario, the ratio of intercept changes are proportional in time while in the second scenario, the intercept changes are modeled by arbitrary functions of time. For both cases, by combining the optimal points, maximum utility curves are formed. Numerical examples are treated to outline the calculations.

In recent times, the conditional marginal modeling approach has emerged as a new area in longitudinal studies; a new multivariate Poisson regression model has been proposed for count data, the model’s validity was assessed using simulation techniques, followed by fitting the model to real data from the Health and Retirement Study, and the correlation coefficients of the response variable’s impact among the first, second, and third follow-up were estimated.

This paper attempts to describe the Fibonacci numbers, their qualities, completeness, and mathematical completeness; the Fibonacci sequence may be seen in a number of stunning natural events; as a result, this paper begins by introducing the Fibonacci sequence and then goes on to discuss some of its properties; the Fibonacci sequence is mathematically complete, as well as having natural and artistic manifestations; the series itself is infinite, and any positive integer may be expressed as the sum of individual Fibonacci numbers; the Fibonacci sequence and golden ratio act as testimonies; the Fibonacci sequence, which has ancient mathematical foundations, continues to fascinate our minds and show subtle patterns in a variety of fields; as we interpret the complexities of these mathematical masterworks, we develop a better understanding of how mathematics and the world around us.

The approximate equivalence of functions is defined with respect to Taylor series expansions first. Based on the approximate equivalence, approximation of a given function with other functional series is discussed. The theory is first applied to root finding iteration algorithms. Then the solution of ordinary differential equations is treated as the next application. Several numerical sample problems are considered. It is shown that the formalism can be effectively used in determining the roots of algebraic equations or finding the solutions of differential equations. To increase the precision of the solutions, higher order approximations should be taken.

This paper considers the conventional logistic model and how to obtain the solutions of fractional differential equations. It examines using a hybrid of Sumudu transform method, which is an approximate analytical method for solving fractional differential equations that are associated with time delay. Furthermore, the paper introduces the fractional Caputo-Fabrizio derivative and proportional time delay into the conventional logistic model to propose a general and more logistic model for the population growth. The paper considers different cases of the newly introduced general logistic model and using a hybrid of Sumudu transform method, their solutions were obtained. Using MATLAB, it displays and compares the behaviour of different cases of the general logistic model with fractional Caputo-Fabrizio derivative and proportional time delay.

The use of the linear programming technique has increased in a number of applications, including the traffic light control, the allocation of aircraft over various routes, the management of water quality in warehouses, caterers, personnel, and advertising media selection problems. It has recently had a significant impact on research in the fields of agriculture, animal husbandry, and livestock and it may be used to determine the components of fish feed for fish farmers in order to increase fish productivity. Fish farmers in Bangladesh’s Jashore district feed their fish using a traditional manner, although contemporary fish feed is made to stringent nutrient criteria that are important for fish growth and increased animal productivity. This study discusses the formulation of fish feed using a flexible tool known as the “linear programming technique”.