In this paper we present a coherent network of concepts, theorems, propositions, and corollaries that arise from a theorem. A linear combination of powers of two uniquely represents every odd number except 1. The powers of 2 are consecutive with the coefficients −1 or +1. From this representation, a set of mathematical concepts leads to a mathematical object, the octets of odd numbers. We studied the properties of octets and their relationship to the structure of odd numbers. Thus, a new framework for the study of natural numbers emerges.

For quadratic curves over finite fileds, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a quadratic curve analogue of the modular form in the Taniyama-Shimura conjecture is proposed. This modular form yields the Gaussian sum and also possesses some modular transformation structure.

The impact of Viscous dissipation on the heat and mass transfer characteristics of an Eyring Powell nanofluid flow past a horizontal circular cylinder is intensively investigated in the presence of Dufour and Soret effects. The free laminar flow is subject to a uniform transverse magnetic field. The continuity, momentum, energy, and concentrations equations are transformed into a nonlinear system of partial differential equations using appropriate non-similarity variables. The transformed system was solved numerically using the fourth order Runge Kutta method. The effect of parameters including Prandtl number, Dufour effect, Soret effect and Schmidt number were studied and presented graphically. Nusselt and Sherwood numbers have also been derived and discussed numerically.

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

We investigate two generalized forms for the recurrence relation S(n, k) = kS(n - 1, k) + S(n - 1, k - 1). From these generalized forms, we derive a new identity, for which a proof of the identity is given.

We consider the generalized dual transformation for hyperelliptic ℘ functions. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic ℘ functions have the SO(3,2) ∼= Sp(4,R)/Z2 Lie group structure. .

The nonlinear reactions-diffusion process describes a chemical reaction that involves three species, two reactions, and diffusion. The system of equations coupled with the nonlinear reaction terms with mixed Dirichlet and Neumann boundary conditions is solved analytically. The hyperbolic function method is used an approximate analytical expression of species concentrations. These analytical results are compared with numerical and previous available analytical results and are in good agreement.

In this article, we introduce the concept of bicomplex valued fuzzy b-metric spaces and a modified (ϕ, ψ) fuzzy contraction.We construct certain fixed point results in bicomplex valued fuzzy b-metric spaces. Our work is inspired by I. Demir [1] andSingh et al. [2]. Some examples are provided to validate our results. Further, we substantiate the utility of our work in identifyingthe unique solution to a system of equations emerging in dynamic programming.

In this paper, we have proposed another calculation to track down optimalanswers for the fuzzy transportation problem. Here, positioning strategy is utilized to defuzzify the Heptagonal fuzzy numbers. This article gives procedure that cuts down the optimal arrangement. The mathematical model represents the legitimacy of our proposed technique.

It is important to incorporate diverse dependence structures between stocks when managing a stock portfolio. Copulas are a useful statistical tool to capture dependence structure, dealing with both the linear and non-linear association that may occur in the tails of data. Financial time series datasets often exhibit volatility clustering that affects price forecasting accuracy. This work proposes the initial use of the principal component analysis followed by a copula and GARCH model that filters the effect of the volatility clustering in the series. For illustration, we consider ten banks from which Bank of America and PNC Financial Services Group are chosen, and then we project their future price movements through simulations. Since they are selected in terms of the principal component analysis, the procedures could help the proposed model to become a more widely used tool in forecasting financial stock performance.