A new proof of the Gasca - Maeztu conjecture for n = 5

A new proof of the Gasca-Maeztu conjecture for n=4

This preprint is in final form. It appeared in: J. Approx. Theory 159 (2009), pp. 224–242. In the Chung-Yao construction of poised nodes for bivariate polynomial interpolation [4], the interpolation nodes are intersection points of some lines. The BerzolariRadon construction [1,8] seems to be more general, since in this case the nodes of interpolation lie (almost) arbitrarily on some lines. In ...

A New Proof of the Collatz Conjecture

This paper is a revised version of a previous paper by the author that, after publication, was found to contain an error. This paper supersedes and replaces the previous paper. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N0 to the Collatz Conjecture is hypothesized ; the existence of N0 implies that N...

A new proof of Friedman's conjecture

We give a new proof of Friedman’s conjecture that every uncountable ∆1 set of reals has a member of each hyperdegree greater than or equal to the hyperjump.

A New Short Proof of Kneser's Conjecture

A new proof of Vázsonyi's conjecture

We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n − 2. The proof avoids the ball polytopes used in the original proofs by Grünbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane. Let S be a set of n points of diameter D in Rd. Define the diamet...