Then, by using the conclusion in Equation (29), the discrete scalar wave equation with a boundary term can be written as:
Here, [bar.K] is an [N.sub.1] x [N.sub.1] matrix with only [N.sub.1,[partial derivative]] nonzero rows, similar with [[partial derivative].sub.n][bar.g] introduced for scalar wave. The (i,j) element [[[bar.K]].sub.i,j] associated with j-th primal edge and i-th dual face at surface, which only has one edge [L.sup.[partial derivative].sub.i] on the surface, is defined as:
Mann, "Scalar wave falloff in asymptotically anti-de Sitter backgrounds," Physical Review D, vol.
Mann, "Scalar wave falloff in topological black hole backgrounds," Physical Review D, vol.
This description above of in-and out-waves is almost identical to the quantum waves of the electron that can be obtained rigorously using a scalar wave equation in Section H.
Wolff, (6,7) Mead, (8) and Haselhurst (13) explored the Scalar Wave Equation and found that its solutions form a quantum-wave structure, possessing all the electron's experimental properties, eliminating the paradoxes of quantum mechanics and cosmology.
Quantum waves exist in space and are solutions of a scalar wave equation