Studying the application of extreme value theory, statistical methods, and machine learning techniques in describing various climatological extremes. Using of Lidstone’s polynomials and Green’s functions obtaining Jensen-Mercer inequality for n-convex functions and studying its applications. Hermite-Hadamard type inequalities for higher order convex functions and the general weighted two-point integral formula involving w-harmonic sequences of functions are investigated. In special cases, Hermite-Hadamard type estimates are derived for various classical quadrature formula, such as the Gauss-Legendre two-point quadrature formula, and the Gauss-Chebyshev two-point quadrature formula of the first and of the second kind. Also, weighted three-point integral quadrature formula is estimated by the new bounds using the integral identity involving w-harmonic sequences of functions. Obtained results are applied for establishing new estimates for the Legendre-Gauss three point quadrature formula using specific form of the weight function w. We have introduced the Levinson functional on time scales using integral inequality of Levinson’s type in the terms of delta- integral for convex (concave) functions on time scale sets and investigated its properties as superadditivity and monotonicity. Obtained properties are used to derive the bounds of the given Levinson’s functional and those results provides a refinement and the converse of known Levinson’s inequality on time scales. Further, we defined new types of functionals using weighted generalized and power means on time scales and proved their properties which can be employed in future works to obtain refinements and converses of known integral inequalities on time scales. Studying the application of Hurewicz theorem in the coarse shape theory. Studying the exactness of the coarse shape group and the coarse shape homology group sequences. Studying of asymptotic expansion and approximations of special functions. Studying Fibonacci and Lucas sequences and related (infinite) matrices and determinant Studying systems of linear congruences and modulo m, modulo multiple m₁ , …, m_k , and modulo a prime power, Z-score application to child pedestrian traffic safety analysis.