This is a critical research work on numbers, their distribution; even and uneven, and how it affects their multiplication. This work contains underlying principles of numbers and how to increase their rate of multiplicity, which can be applied in mathematics, sciences, economics and basic life style. CONTENTS: THE LAW OF EXPONENTIAL MULTIPLICITY. FIRST LAW SECOND LAW THIRD LAW INFERENCE MATHEMATICAL PROOF BIOLOGICAL PROOF CONCLUSION REFERENCES.
The vibrations of axially functionally graded (AFG) beams are examined incorporating shear deformation and an imperfection. A coupled nonlinear model is developed in the Hamiltonian framework for an AFG imperfect beam. Different sources of displacement nonlinearities are involved in this problem, namely, the geometrical nonlinearities due to the large deformations, and the nonlinearities due to the presence of a geometrical imperfection along the length of the beam. Expressions for the virtua...
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