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Kovič, and implicitly Ufnarovski andÅhlander, defined a notion of arithmetic partial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equations , we give a complete solution, and for others, we extend previously known results. For example, we(More)
The incorporation of nonnegativity constraints in image reconstruction problems is known to have a stabilizing effect on solution methods. In this paper, we both demonstrate and provide an explanation of this phenomena when the image reconstruction problem of interest has least squares form. The benefits of using this natural constraint suggest the(More)
We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.