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We define Fermat type partial differentiaJ equations by einalogy with Fermat type functional equations. We utilize some existing results for the functional equations to describe entire and meromorphic solutions for the partiaJ differential equations. Mathematics Subject Classification: 35F99, 35G20The purpose of this article is to study the complex-analytic solutions of certain nonlinear first-order partial differential equations of the form (1) (£","« = l i=i using an analogy with the Fermat type functional equation (2) dn.n.f ••= ^{fir = 1, i=l where u : C" C, G C, /i : C C, and m, n > 2. If we take C, one can think of equation (1) aa a special case of equation (2), hence making it possible to obtain results for equation (1) from existing results for equation (2). There are a number of results concerning the entire and meromorphic solutions in C of equation (2) and its relatives (see references). Such functional equations were originally introduced by analogy with Fermat's last theorem. In this article, we show how to apply some of the results for such equations.

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The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.