This paper introduces the INTERVAL unilateral QUADRATIC MATRIX EQUATION, AX2 + BX + C = 0 and attempts to  nd various analytical results on its AE-solution sets in which A; B and C are known real INTERVAL matrices, while X is an unknown MATRIX. These results are derived from a generalization of some results of Shary. We also give su cient conditions for non-emptiness of some quasi-solution sets, provided that A is regular. As the most common case, the united solution set has been studied and two direct methods for computing an outer estimation and an inner estimation of the united solution set of an INTERVAL unilateral QUADRATIC MATRIX EQUATION are proposed. The suggested techniques are based on nonlinear programming as well as sensitivity analysis.

In this research paper, we present an exact MATRIX form analytical solution of the multi-dimensional generalized Langevin EQUATION with QUADRATIC potentials. Our investigation provides detailed expressions for the two-dimensional probability distribution and extends the understanding of the dynamics governed by harmonic potentials. By utilizing the inverse Laplace transformation, we offer a precise method to solve these EQUATIONs, corroborated by specific examples. This study contributes to the fundamental understanding of stochastic processes in multi-dimensional systems with harmonic potentials and clarifies the limitations of our approach. While the findings are specific to QUADRATIC potentials, they provide a robust framework for exploring related phenomena within this context.

In this paper, we have obtained and presented new relativistic stellar configurations considering an anisotropic fluid distribution with a charge distribution and a gravitational potential ()Zx that depends on an adjustable parameter. The QUADRATIC EQUATION of state based on Feroze and Siddiqui viewpoint is used for the matter distribution. The new solutions can be written in terms of elementary and polynomial functions. We have investigated that the radial pressure, metric coefficients, energy density, anisotropy factor, charge density, mass function have been well defined and are regular in the interior of these new models, which satisfy all physical properties in a realistic star.

In this paper, we introduce an efficient method for solving the QUADRATIC Riccati differential EQUATION. In this technique, combination of Laplace trans-form and new homotopy perturbation methods (LTNHPM) are considered as an algorithm to the exact solution of the nonlinear Riccati EQUATION. Un-like the previous approach for this problem, so-called NHPM, the present method, does not need the initial approximation to be defined as a power series. Four examples in different cases are given to demonstrate simplicity and efficiency of the proposed method.

In this article, we characterize the involutiveness of the linear combination of the form a1 A1+a2 A2 when a1; a2 are nonzero complex numbers, A1 is a QUADRATIC or tripotent MATRIX, and A2 is arbitrary, under certain properties imposed on A1 and A2.