Searcharxiv⌕ Search

arXiv subjects

Colin Defant

Publications and source records attributed to Colin Defant.

At least 109 records · Page 6Linked to original sources
arXiv2017

Binary Codes and Period-2 Orbits of Sequential Dynamical Systems

Let $[K_n,f,π]$ be the (global) SDS map of a sequential dynamical system (SDS) defined over the complete graph $K_n$ using the update order $π\in S_n$ in which all vertex functions are equal to the same function $f\colon\mathbb F_2^n\to\mathbb F_2^n$. Let $η_n$ denote the maximum number of periodic orbits of period $2$ that an SDS map of the form $[K_n,f,π]$ can have. We show that $η_n$ is equal to the maximum number of codewords in a binary code of length $n-1$ with minimum distance at least $3$. This result is significant because it represents the first interpretation of this fascinating coding-theoretic sequence other than its original definition.

math.CO↗
arXiv2017

On the Density of Ranges of Generalized Divisor Functions with Restricted Domains

We begin by defining functions $σ_{t,k}$, which are generalized divisor functions with restricted domains. For each positive integer $k$, we show that, for $r>1$, the range of $σ_{-r,k}$ is a subset of the interval $\displaystyle{\left[1,\frac{ζ(r)}{ζ((k+1)r)}\right)}$. After some work, we define constants $η_k$ which satisfy the following: If $k\in\mathbb{N}$ and $r>1$, then the range of the function $σ_{-r,k}$ is dense in $\displaystyle{\left[1,\frac{ζ(r)}{ζ((k+1)r)}\right)}$ if and only if $r\leqη_k$. We end with an open problem.

math.NT↗
arXiv2017

Unitary Cayley Graphs of Dedekind Domain Quotients

If $X$ is a commutative ring with unity, then the unitary Cayley graph of $X$, denoted $G_X$, is defined to be the graph whose vertex set is $X$ and whose edge set is $\{\{a,b\}\colon a-b\in X^\times\}$. When $R$ is a Dedekind domain and $I$ is an ideal of $R$ such that $R/I$ is finite and nontrivial, we refer to $G_{R/I}$ as a \emph{generalized totient graph}. We study generalized totient graphs as generalizations of the graphs $G_{\mathbb{Z}/(n)}$, which have appeared recently in the literature, sometimes under the name \emph{Euler totient Cayley graphs}. We begin by generalizing to Dedekind domains the arithmetic functions known as Schemmel totient functions, and we use one of these generalizations to provide a simple formula, for any positive integer $m$, for the number of cliques of order $m$ in a generalized totient graph. In particular, we prove that the number of cliques of order $m$ in $G_{\mathbb Z/(n)}$ is \[\prod_{k=1}^m\frac{S_{k-1}(n)}{k},\] where $S_r$ is the $r^{\text{th}}$ Schemmel totient function. We then proceed to determine many properties of generalized totient graphs such as their clique numbers, chromatic numbers, chromatic indices, clique domination numbers, and (in many, but not all cases) girths. We also determine the diameter of each component of a generalized totient graph. We correct one erroneous claim about the clique domination numbers of Euler totient Cayley graphs that has appeared in the literature and provide a counterexample to a second claim about the strong domination numbers of these graphs.

math.CO↗
arXiv2017

Poset Pattern-Avoidance Problems Posed by Yakoubov

Extending the work of Yakoubov, we enumerate the linear extensions of comb posets that avoid certain length-$3$ patterns. We resolve many of Yakoubov's open problems and prove both of the conjectures from her paper.

math.CO↗
arXiv2016

Anti-Power Prefixes of the Thue-Morse Word

Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1w_2\cdots w_k$, where $w_1,w_2,\ldots,w_k$ are distinct words of the same length. They defined $AP(x,k)$ to be the set of all positive integers $m$ such that the prefix of length $km$ of the word $x$ is a $k$-anti-power. Let ${\bf t}$ denote the Thue-Morse word, and let $\mathcal F(k)=AP({\bf t},k)\cap(2\mathbb Z^+-1)$. For $k\geq 3$, $γ(k)=\min(\mathcal F(k))$ and $Γ(k)=\max((2\mathbb Z^+-1)\setminus\mathcal F(k))$ are well-defined odd positive integers. Fici et al. speculated that $γ(k)$ grows linearly in $k$. We prove that this is indeed the case by showing that $1/2\leq\displaystyle{\liminf_{k\to\infty}}(γ(k)/k)\leq 9/10$ and $1\leq\displaystyle{\limsup_{k\to\infty}}(γ(k)/k)\leq 3/2$. In addition, we prove that $\displaystyle{\liminf_{k\to\infty}}(Γ(k)/k)=3/2$ and $\displaystyle{\limsup_{k\to\infty}}(Γ(k)/k)=3$.

math.CO↗
arXiv2015

On Ranges of Variants of the Divisor Functions that are Dense

For a real number $t$, let $s_t$ be the multiplicative arithmetic function defined by $\displaystyle{s_t(p^α)=\sum_{j=0}^α(-p^t)^j}$ for all primes $p$ and positive integers $α$. We show that the range of a function $s_{-r}$ is dense in the interval $(0,1]$ whenever $r\in(0,1]$. We then find a constant $η_A\approx1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\displaystyle{\left(\frac{1}{ζ(r)},1\right]}$ if and only if $r\leq η_A$. We end with an open problem.

math.NT↗
arXiv2015

An Anti-Ramsey Problem Concerning Complete Bipartite Graphs

We consider quadruples of positive integers $(a,b,m,n)$ with $a\leq b$ and $m\leq n$ such that any proper edge-coloring of the complete bipartite graph $K_{m,n}$ contains a rainbow $K_{a,b}$ subgraph. We show that any such quadruple with $a\leq m$ and $n>(a^2-a+1)(b-1)$ satisfies this property. We also show that the quadruple $(2,3,3,6)$ satisfies this property. We end with a conjecture.

math.CO↗
arXiv2015

Upper Bounds for Stern's Diatomic Sequence and Related Sequences

Let $(s_2(n))_{n=0}^\infty$ denote Stern's diatomic sequence. For $n\geq 2$, we may view $s_2(n)$ as the number of partitions of $n-1$ into powers of $2$ with each part occurring at most twice. More generally, for integers $b,n\geq 2$, let $s_b(n)$ denote the number of partitions of $n-1$ into powers of $b$ with each part occurring at most $b$ times. Using this combinatorial interpretation of the sequences $s_b(n)$, we use the transfer-matrix method to develop a means of calculating $s_b(n)$ for certain values of $n$. This then allows us to derive upper bounds for $s_b(n)$ for certain values of $n$. In the special case $b=2$, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_bϕ}}=\frac{(b^2-1)^{\log_bϕ}}{\sqrt 5}}$.

math.CO↗
arXiv2015

Multiperfect Numbers in Certain Quadratic Rings

Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$-powerfully $t$-perfect numbers in these rings. This definition serves to extend the concept of multiperfect numbers that have been defined and studied in the integers. At the end of the paper, as well as at various points throughout the paper, we point to some potential areas for further research.

math.NT↗
arXiv2015

An Extension of the Abundancy Index to Certain Quadratic Rings

We begin by introducing an extension of the traditional abundancy index to imaginary quadratic rings with unique factorization. After showing that many of the properties of the traditional abundancy index continue to hold in our extended form, we investigate what we call $n$-powerfully solitary numbers in these rings. This definition serves to extend the concept of solitary numbers, which have been defined and studied in the integers. We end with some open questions and a conjecture.

math.NT↗
arXiv2015

On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_m$, we define two new functions, denoted $R_m$ and $H_m$, that arise from iterating $L_m$. Roughly speaking, $R_m$ counts the number of iterations of $L_m$ needed to reach either $0$ or $1$, and $H_m$ takes the value (either $0$ or $1$) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer $m$, the function $H_m$ is completely multiplicative. We then introduce an iterate summatory function, denoted $D_m$, and define the terms $D_m$-deficient, $D_m$-perfect, and $D_m$-abundant. We proceed to prove several results related to these definitions, culminating in a proof that, for all positive even integers $m$, there are infinitely many $D_m$-abundant numbers. Many open problems arise from the introduction of these functions and terms, and we mention a few of them, as well as some numerical results.

math.NT↗
arXiv2015

On the Density of Ranges of Generalized Divisor Functions

The range of the divisor function $σ_{-1}$ is dense in the interval $[1,\infty)$. However, the range of the function $σ_{-2}$ is not dense in the interval $\displaystyle{\left[1,\frac{π^2}{6}\right)}$. We begin by generalizing the divisor functions to a class of functions $σ_{t}$ for all real $t$. We then define a constant $η\approx 1.8877909$ and show that if $r\in(1,\infty)$, then the range of the function $σ_{-r}$ is dense in the interval $[1,ζ(r))$ if and only if $r\leqη$. We end with an open problem.

math.NT↗
arXiv2015

A Note about Iterated Arithmetic Functions

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $α$, $f(p^α)<p^α$ and $f(p)\vert f(p^α)$. Suppose also that any prime that divides $f(p^α)$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.

math.NT↗
arXiv2015

An Arithmetic Function Arising from the Dedekind $ψ$ Function

We define $\overlineψ$ to be the multiplicative arithemtic function that satisfies \[\overlineψ(p^α)=\begin{cases} p^{α-1}(p+1), & \mbox{if } p\neq 2; \\ p^{α-1}, & \mbox{if } p=2 \end{cases}\] for all primes $p$ and positive integers $α$. Let $λ(n)$ be the number of iterations of the function $\overlineψ$ needed for $n$ to reach $2$. It follows from a theorem due to White that $λ$ is additive. Following Shapiro's work on the iterated $φ$ function, we determine bounds for $λ$. We also use the function $λ$ to partition the set of positive integers into three sets $S_1,S_2,S_3$ and determine some properties of these sets.

math.NT↗
arXiv2014

On 2-powerfully Perfect Numbers in Three Quadratic Rings

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, the three imaginary quadratic rings with unique factorization in which $2$ is not a prime.

math.NT↗
arXiv2014

On Sparsely Schemmel Totient Numbers

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers $n$ satisfying $ϕ(n)<ϕ(m)$ for all integers $m>n$. We define a sparsely Schemmel totient number of order $r$ to be a positive integer $n$ such that $S_r(n)>0$ and $S_r(n) n$ with $S_r(m)>0$. We then generalize some of the results of Masser and Shiu.

math.NT↗
arXiv2014

On Schemmel Nontotient Numbers

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^α)=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{α-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $α$. The function $S_1$ is simply Euler's totient function $ϕ$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient numbers generalize to results concerning Schemmel nontotient numbers. We also invoke Zsigmondy's Theorem in order to generalize a result due to Mendelsohn.

math.NT↗