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In this paper, we consider the problem that the maximal order considers the groups that consist of transformations we called NG-Transformation on a nonempty set
A has no bijection as its element. We found that the order of these groups is not greater than (n-1)!. In addition, we will prove our result by showing that any kind of
NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A.

We give necessary and sufficient conditions for a linear connection without torsion to come from a Riemannian metric. The nullity space and the image space of the curvature are involved in the formulation of the results.

Mathematics Subject Classification (2010) 53XX, 53C08, 53B05.

Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.

Mathematical modeling:

• is indispensable in many applications

• gives direction for the problem solutions

• prepares the way for better designing or control system

Learning about mathematical modeling is an important step for a theoretical mathematical training to an application oriented mathematical expertise, and makes the student fit for mustering the challenges of our modern technological culture. One of the most important mathematical model is Spruce Budworm model for researchers as well as for students. Understanding the dynamic of spruce budworm is very important for the protection of spruce and fir trees of Canada and Northern Minnesota (also in recent time Indian Himalayan range forest). This model was designed to identify the (a) critical factors that affect the Budworm population dynamics (b) to evaluate the effect of budworm population on the growth and yield of wood industry and also the present loss claimed by irruptions done by Budworm (c) to formulate mathematical model and to find out the steady state and the existence of the steady state and steady state analysis. The bifurcation analysis and the hysteresis effect of the model has been discussed. Analysis of the equilibrium stability and examination of amplitudes and periodic oscillations are conducted, and the effect of Budworm control, immature population control and predation by the birds are assessed.