We obtain the generalized codimension-$p$Cauchy–Kovalevsky extension of the exponential function $e^{i\langle\underline{y},\underline{t}\rangle}$ in R^{$m$}=R^{$p$}⊕R^{$q$}, where$p$>1, $\underline{y},\underline{t}\in\mathbf{R}^{q}$ , and prove the corresponding codimension-$p$Paley–Wiener theorems.

We study the tropical lines contained in smooth tropical surfaces in ℝ^{3}. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit RankineHugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( \sqrt t ) for large time, and the front fluctuation scaled by \sqrt t converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root <i>t</i> normalized front standard

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