arXiv subjects
Publications and source records attributed to Harold R. Parks.
Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.
The egg-drop experiment introduced by Konhauser, Velleman, and Wagon, later generalized by Boardman, is further generalized to two additional types. The three separate types of egg-drop experiment under consideration are examined in the context of binary decision trees. It is shown that all three types of egg-drop experiment are binary decision problems that can be solved efficiently using a non-redundant algorithm -- a class of algorithms introduced here. The preceding theoretical results are applied to the three types of egg-drop experiment to compute, for each, the maximum height of a building that can be dealt with using a given number of egg-droppings.
We present an elementary proof of the generalization of the $k$-bonacci Binet formula, a closed form calculation of the $k$-bonacci numbers using the roots of the characteristic polynomial of the $k$-bonacci recursion.
We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.
Let $N\ge3$ be an integer and $B$ be a smooth, compact, oriented, $(N-2)$-dimensional boundary in ${\Bbb R}^{N}$. In 1960, H. Federer and W. Fleming proved that there is an $(N-1)$-dimensional integral current spanning surface of least area. The proof was by compactness methods and non-constructive. In 1970 H. Federer proved the definitive regularity result for such a codimension one minimizing surface. Thus it is a question of long standing whether there is a numerical algorithm that will closely approximate the area minimizing surface. The principal result of this paper is an algorithm that solves this problem. Specifically, given a neighborhood $U$ around $B$ in ${\Bbb R}^{N}$ and a tolerance $ε>0$, we prove that one can explicitly compute in finite time an $(N-1)$-dimensional integral current $T$ with the following approximation requirements: (1) spt$(\partial T)\subset U$. (2) $B$ and $\partial T$ are within distance $ε$ in the Hausdorff distance. (3) $B$ and $\partial T$ are within distance $ε$ in the flat norm distance. (4) ${\mathbb M}(T)<ε+\inf\{{\mathbb M}(S):\partial S=B\}$. (5) Every area minimizing current $R$ with $\partial R=\partial T$ is within flat norm distance $ε$ of $T$.
The authors study the classical Lagrange inversion theorem--an antecedent of the modern implicit function theorem--in the smooth case. Examples are given to show that the result is sharp.