In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set So f size q + 1 in the classical project...
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In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set So f size q + 1 in the classical projective plane PG(2, q), where the intersection distribution of Sindicates the intersection pattern between Sand the lines in PG(2, q). The second one relates to a polynomial f over a finite field F-q, where the intersection distribution of f records an overall distribution property of a collection of polynomials {f(x) + cx vertical bar c is an element of F-q}. These two perspectives are closely related, in the sense that each polynomial produces a (q + 1)-set in a canonical way and conversely, each (q+ 1)-set with certain property has a polynomial representation. Indeed, the intersection distribution provides a new angle to distinguish polynomials over finite fields, based on the geometric property of the corresponding (q+ 1)-sets. Among the intersection distribution, we identify a particularly interesting quantity named non-hitting index. For a point set S, its non-hitting index counts the number of lines in PG(2, q) which do not hit S. For a polynomial fover a finite field F-q, its nonhitting index gives the summation of the sizes of qvalue sets {f(x) + cx vertical bar x is an element of F-q}, where c is an element of F-q. We derive lower and upper bounds on the non-hitting index and show that the non-hitting index contains much information about the corresponding set and the polynomial. More precisely, using a geometric approach, we show that the non-hitting index is sufficient to characterize the corresponding point set and the polynomial, when it is very close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the non-hitting index and the intersection distribution of several families of point sets and polynomials. As an application, we consider the determination of the sizes of Kakeya sets in affine planes. The polynomi
In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes minimal surfaces with the induced metric. Using this characterization equation we study various...
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In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.
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