In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d > 0 and order h with a generic projective differential hyperplane is shown to be an irreducible projective differential variety of dimension d − 1 and order h. Based on the generic intersection theo...

In 1893 J. J. Sylvester [8] posed the following celebrated problem: Given a finite collection of points in the affine plane, not all lying on a line, show that there exists a line which passes through precisely two of the points. Sylvester’s problem was reposed by Erdős in 1944 [4] and later that year a proof was given by Gallai [6]. Since then, many proofs of the Sylvester-Gallai Theorem have ...

In 1929, the KKMmap was introduced by Knaster et al. [13] and it provides the foundation for many well-known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan-Browder’s fixed point theorem, Nash’s equilibrium theorem, HartmanStampacchia’s variational inequality theorem and many others (see [1, 2, 5–12, 14–17]). The central idea of applying KKM theory to prove that a family ...

In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient intersection about s(1 ≤ s ≤ n) algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems ( such as Bez...

In this paper we introduce projective geometry and one of its important theorems. We begin by defining projective space in terms of homogenous coordinates. Next, we define homgenous curves, and describe a few important properties they have. We then introduce Bezout’s Theorem, which asserts that the number of intersection points of two homogenous curves is less than or equal to the product of th...

If each four spheres in a set of five unit spheres in R have nonempty intersection, then all five spheres have nonempty intersection. This result is proved using Grace’s theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results; including Schläfli’s double six theorem and modified version of Lie’s line-sphere tra...

It is known from a previous paper [3] that Katona’s Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing–pulling method of that paper. Here we add a third proof via a new (k,k+1)-shifting technique, whose impact will be exploared elsewhere. The fourth and last of our proofs ...