Abstract: Nonlinear problems arise in most of the scientific fields. In general, such behavior is represented by a nonlinear equation, whose solution is sought. Analytical and numerical methods have been applied to the solution of this class of equations, notwithstanding, in cases where highly nonlinear phenomena are analyzed, the number of iterations and computational effort necessary to achieve the minimum required accuracy is very high. Lagrange´s Inversion Theorem (LIT) has been applied to solve this kind of problems analytically, giving the solution as an infinite power series. This way, the accuracy can be as high as necessary by taking more terms from the series solution, which is easily computationally implemented. Also, in some cases it is possible to relate the series obtained to the expansion of special and elementary functions, which enables one to exactly solve the desired equation. In the present review paper, a total of eleven applications have been discussed in order to show the role of LIT in various areas of nonlinear sciences.

Abstract: The Fibonacci sequence is famous for possessing wonderful and amazing properties. In this paper, we present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers and related identities consisting even and odd terms. Binet’s formula will employ to obtain the identities.

Abstract: In the manuscript is proved that parallel postulate is only in Plane (three points only) and is based on the four Postulates for Constructions, where all properties of Euclidean geometry compactly exist as Extrema on points, lines, planes, circles and spheres. Projective, Hyperbolic and Elliptic geometry is proved to be an Extrema (deviations) in Euclidean geometry where on them Einstein's theory of general relativity is implicated approximately to the properties of physical space.

Abstract: A glass system with chemical formula: xLi₂O– (50_x) B₂O₃–40PbO–10P₂O₅ mole % is prepared to be used as radiation shield. The mass attenuation coefficient and half value layer of the glass system to gamma rays have been measured experimentally and compared with those determined from theoretical calculations using the mixture rule of WinXCom program. A database of effective mass removal cross-sections for fast neutrons is also introduced in this work.

Abstract: The influence of the geometric parameters of two-link manipulator on the energy expenditure necessary for the implementation of a given movement is investigated. It is proposed approximation-compensatory approach to replace the active actuators in the joints of the manipulator on semi-passive and entirely passive actuators. Advantages and disadvantages of such replacement are analyzed and the fields of use of such manipulators are proposed.