arXiv subjects

Russell Jay Hendel

Publications and source records attributed to Russell Jay Hendel.

9 recordsLinked to original sources

arXiv2026

Proof of Convergence of a Laplace Expansion Algorithm For Calculating Recursions Satisfied by a Family of Determinants

In Evans and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e., terminated) in the particular case studied, and the paper presented an open question on when such a procedure converges in general. This paper proves the convergence of a Laplace expansion procedure for an arbitrary family of determinants of banded, square, Toeplitz matrices. A comparison of the procedure presented in this paper, the paper by Evans and Hendel, and a paper by Jia, Yang, and Li is presented.

math.CO↗

arXiv2025

Runs, Squares, Palindromes, and Unbordered Factors of a Family of Binary Pattern Sequences with the All-One Pattern

This paper presents results on maximal runs, order of squares, palindromes, and unbordered factors of members of the family of binary pattern sequences with the all-one pattern. Restricting ourselves to binary pattern sequences with the all-one pattern with at least three ones, five categories of maximal run lengths and 3 categories of orders of squares are presented, palindromes with locally maximal length as well as palindromes with the second to fifth-largest palindrome lengths are described, and unbordered factors of lengths powers of two are presented. Interestingly, the characteristic functions of specified prefixes of sequences of the 2-kernel of these sequences can be formulated using the Vile and Jacobsthal sequences. Both Mathematica and Walnut are employed for exploratory pattern analysis. Proofs are based on a correspondence between binary strings under concatenation and integers under addition and multiplication. It is observed that this correspondence seems most efficacious for proofs of theorems whose statements are classified at low levels in the arithmetic hierarchy.

cs.FL↗

arXiv2025

Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree

Barret, Evans, and Francis conjectured that if $G$ is the straight linear 3-tree with $n$ vertices and $H$ is the straight linear 3-tree with $n+1$ vertices then \[\lim_{n\rightarrow \infty} r_{H} (1, n+1) - r_G(1,n) = \frac{1}{14},\] where $r_G(u,v)$ and $r_H(u,v)$ are the resistance distance between vertices $u$ and $v$ in graphs $G$ and $H$ respectively. In this paper, we prove the conjecture by looking at the determinants of deleted Laplacian matrices. The proof uses a Laplace expansion method on a family of determinants to determine the underlying recursion this family satisfies and then uses routine linear algebra methods to obtain an exact Binet formula for the $n$-th term.

math.CO↗

arXiv2025

An Introductory Survey of Recursions in the Computation of Resistance Distance

This paper presents an introduction and expository account of a beautiful, current, and active application of recursions to the computation of resistance distance. Resistance distance, also referred to as effective resistance, is a well-known graph metric that arises naturally by considering a graph as an electrical circuit; heuristically resistance distance measures both the number of paths between two vertices in a graph and the cost of each path. This topic finds applications in a rich array of fields including social, biological, ecological, and transportation networks, chemistry, graph theory, numerical linear algebra, and engineering. A variety of methods are used in the field to determine resistance distance including recursive, mathematical, and graphical techniques. Sequences familiar to the readers of the Fibonacci Quarterly such as the Fibonacci and Lucas sequences appear quite often in results in the literature. Twenty five to forty years ago there were a handful of papers on resistance that appeared in the Fibonacci Quarterly and the Proceedings and recently papers on the subject have appeared again. It is hoped that this introductory expository account will interest readers of the Quarterly to renew interest in this current and active field.

math.HO↗

arXiv2024

Limiting Behavior of Resistances in Triangular Graphs

Barrett et al studied resistance labels of electrical circuits whose underlying graphs when embedded in the Cartesian plane has the form of an $n$-grid, $n$ rows of upright triangles. Proofs in Barrett introduced a row-reduction algorithm which uses series, $Δ$--Y, and Y--$Δ$ electric transformations to transform an $n$-grid into an $n-1$ grid with equivalent resistances between specified nodes. This paper explores this row-reduction algorithm computationally. The introductory part of the paper presents several conjectures supported by numerical evidence, showing that repeated application of the row-reduction algorithm to an initial $n$-grid uniformly labeled 1 asymptotically produces triangular grids whose sides are labeled with rational multiples of $\frac{1}{e};$ moreover, the ratio of specified consecutive edges in the row-reduced grids are asymptotically described by four rational functions. The main part of this paper studies a family of graphs whose edge labels are determined using these limiting edge-ratios functions arising in the conjectures. The main result proven is that these $n$-grids and their repeated reductions under the row-reduction algorithm possess vertical and rotational symmetries and satisfy the relationships captured by the four edge-ratio functions. Thus, the limiting edge-ratio relationships are local algebraic relationships mirroring the global vertical and rotational symmetries possessed by the underlying graph. Additionally, because row-reduction is local (in contrast to the combinatoric Laplacian which is global) the paper is able to introduce a mechanical verification method of proof for assertions about effective resistance identities.

math.CO↗

arXiv2022

A System of Four simultaneous Recursions: Generalization of the Ledin-Shannon-Ollerton Identity

This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all the metallic sequences. The results give closed formulas for the sum of products of powers of the first $n$ integers with the first $n$ members of the metallic sequence. Three key innovations of this paper are i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; ii) use of the shift operation to prove equality of polynomials; and iii) new OEIS sequences arising from the coefficients of the four polynomial families satisfying the 4 simultaneous recursions.

math.CO↗

arXiv2022

Sums of Squares: Methods for Proving Identity Families

This paper presents both a method and a result. The result presents a closed formula for the sum of the first $m+1,m \ge 0,$ squares of the sequence $F^{(k)}$ where each member is the sum of the previous $k$ members and with initial conditions of $k-1$ zeroes followed by a 1. The generalized result includes the known result of sums of squares of the Fibonacci numbers and a recent result of Schumaker on sums of squares of Tribonacci numbers. To prove the identities uniformly for all $k,$ the Algebraic Verification method is presented which reduces proof of an identity to verification of the equality of finitely many pairs of finite-degree polynomials, possibly in several variables. Several other papers proving families of identities are examined, and it is suggested that the collection of the uniform proof methods used in these papers could produce a new trend in stating and proving identities.

math.NT↗

arXiv2021

A Method for Uniformly Proving a Family of Identities

This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the family of recursive sequences $F^{(k)}_n = \sum_{i=1}^k F^{(k)}_{n-i}, n \ge 0, k \ge 2,$ with $n$ a parameter varying over integers, and $k$ a parameter indexing members of the family. The main theorem states $ F^{(k)}_n = \sum_{j=1}^k P_{k,j} F^{(k)}_{n-jk},$ with $P$ a recursive triangle satisfying the triangle recursion $P_{i,j}=2P_{i-1,j}- P_{i-1,j-1},$ with appropriate initial conditions. The proof of the theorem exploits the fact that characteristic polynomials of identities are divisible by the characteristic polynomial of the recursion generating the underlying sequence.

math.CO↗

arXiv2021

Recursive Triangles Appearing Embedded in Recursive Families

We continue the work begun in OEIS sequence A332636 which presents recursive sequences that have triangles that appear embedded in them. This paper i) generalizes the main result presented in A332636, ii) provides a complete set of definitions and underlying concepts, and iii) provides a complete proof.

math.NT↗