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Stationary wave functions of an electron with discrete spectrum in a hydrogen or a hydrogen-like atom are studied. In a Euclidean 3-space with coordinates x1, x2, x3, a spherical coordinate system is introduced. If the magnetic quantum number m vanishes, then, at the extremal points of the function, the sectional curvature of the hypersurface satis...
For the Monge-Ampère Z xx Z yy -Z xy 2 =b 20 x 2 +b 11 xy+b 02 y 2 +b 00 we study the question of existence of a solution Z(x,y) in the class of polynomials, where Z(x,y) is the graph of a convex surface. If Z is a polynomial of odd order, a solution does not exist. If Z is a polynomial of fourth order and 4b 20 b 02 -b 11 2 >0, a solution also doe...
For the Monge-Ampere equation ZxxZyy-Z2xy = b20x2+b11xy + b02y2 + 600 we consider the question on the existence of a solution Z(x, y) in the class of polynomials such that Z = Z(x, y) is a graph of a convex surface. If Z is a polynomial of odd degree, then the solution does not exist. If Z is a polynomial of 4-th degree and 4b20b02 - b211 > 0, then...