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An analysis of a simply supported rectangular elastic plate forced into bending vibrations by the application of time harmonic voltages to piezoelectric actuators attached to its bottom and top surfaces is performed by using the equations of linear elasticity. The actuators have been modeled as thin surface films and mixed edge conditions are employed to simulate simple supports. This paper gives a method for specifying the optimal ‘twist vectors’ at grid points for an interpolating surface to a rectangular network of curves. These twists are uniquely determined by minimizing an approximate energy form of the surface and can be obtained by solving a well-defined linear system. We show that there is a class of risk lovers who optimally choose a positive level of self-protection investment. In most cases, a risk lover increases his self-protection investment Nike Jordan Heels Canada as he becomes less Nike Air Jordan 3 downside risk averse. The paper deals with the influence of the vertical angle of a tool on the resultant tool mark. Experiments were carried out using five screwdrivers on aluminium surfaces, the vertical angle of each screwdriver being varied from 90° to 25° in steps of 5°. It was found that differences in vertical angle greater than 15° may cause non-identity between two marks from the same tool.