For a century, one of the most famous problems in mathematics was to prove the four-color theorem.In 1912, George Birkhoff proposed a way to tackling the four-color conjecture by introduce a function P(M, t), defined for all positive integer t, to be the number of proper t-colorings of a map M. This function P(M, t)in fact a polynomial in t is called chromatic polynomial of M. If one could prove that P(M, 4)>0 for all maps M, then this would give a positive answer to the four-color problem. In this book, we have proved the following results: (1)Recursive form of the chromatic polynomials of...

Numerical simulations of sheet metal forming process based on finite element method (FEM) is widely applied for its powerful capability in forming process prediction. Since there are parameters which could affect the result of forming process, it becomes important to approach a set of parameters to improve the formability. In the present work, first, a comprehensive literature review was made for different optimization methodologies in sheet metal forming process. Then we proposed an optimization methodology using Response Surface methodology (RSM) and Genetic Algorithm (GA) for the...

Modal parameters of spacecraft structures are important indicators of their dynamic characteristics which can be used for vibration reduction and control. In the present research modal parameters of a rocket are identified by using continuous wavelet transformation. The advantages of the wavelet transformation for the modal parameters identification over other methods is study. Through finite element method transient dynamic response of a system is determined then continuous wavelet transformation is applied to find the modal parameters like modal frequency, mode shapes and damping ratio.

Composite stiffened panels have been used in aerospace industries for past decades. Due to light weight nature composite panels are prone to buckling. In this study buckling load for Ortho-grid panel and Iso-grid panel is determined. This was accomplished by developing a composite laminate for both skin and stiffener using MSC PATRAN/NASTRAN was used for linear buckling analysis of composite panels. The buckling loads of two panels were determined based on finite element analysis results, including geometric dimension, thickness of the skin, number of laminae, ply stacking sequence, thickness...

This book is concern with the rate of infestation of intestinal helminth parasites in sheeps of Peshawar. Which shows the effect of these parasites on the health of sheep. Sheep is a very important and valuable animal in our society. Most of the sheeps lose its weight and value due high burden of worms. You will find the way of transmission of worms from one animal to another. How can we deworm these parasites and valuate our sheeps? Different and common helminth parasites and there effects on different lifestyle of sheeps, can be easily known. Most of the sheeps owner may be intense with...