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Let sq(n) denote the sum of the digits in the q-ary ex-pansion of an integer n. In 1978, Stolarsky showed that lim inf n→∞ s2(n
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Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n²). We extend this study to the more general case of generic q and polynomials p(n), and... more
Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n²). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi’s result. We also give a more detailed analysis of the special case p(n) = n², looking at the subsets of n where sq(n) = sq(n²) = k for fixed k.
We study growth rates of random Fibonacci sequences of a particular structure. A random Fibonacci sequence is an integer sequence starting with $1,1$ where the next term is determined to be either the sum or the difference of the two... more
We study growth rates of random Fibonacci sequences of a particular structure. A random Fibonacci sequence is an integer sequence starting with $1,1$ where the next term is determined to be either the sum or the difference of the two preceding terms where the choice of taking either the sum or the difference is chosen randomly at each step. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and $G_n$ is the $n$th term of the resulting random Fibonacci sequence, then \begin{equation*} \lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation*} exists. We extend her results to recurrences of the form $G_{m+2} = \alpha_m G_{m+1} \pm G_{m}$ if the choices of pluses and minuses, and of the $\alpha_m$ follow a balancing word type pattern.
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In 2003 Cohen and Iannucci introduced a multiplicative arithmetic function D by assigning D(p a ) = ap ai1 when p is a prime and a is a positive integer. They deflned D 0 (n) = n and D k (n) = D(D ki1 (n)) and they called fD k (n)g 1=0... more
In 2003 Cohen and Iannucci introduced a multiplicative arithmetic function D by assigning D(p a ) = ap ai1 when p is a prime and a is a positive integer. They deflned D 0 (n) = n and D k (n) = D(D ki1 (n)) and they called fD k (n)g 1=0 the derived sequence of n. This paper answers some open questions about the function D and its iterates. We show how to construct derived sequences of arbitrary cycle size, and we give examples for cycles of lengths up to 10. Given n, we give a method for computing m such that D(m) = n, up to a square free unitary factor.
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We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on... more
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.
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As is well-known, the ratio of adjacent Fibonacci numbers tends to � = (1+ √ 5)/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three �
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We show that the number $\alpha=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ satisfy the additive relation... more
We show that the number $\alpha=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ satisfy the additive relation $\alpha_1+\alpha_2=\alpha_3+\alpha_4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $\alpha_1 = \alpha_2 + \alpha_3+\alpha_4$ or $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 =0$ cannot be solved in conjugates of a Pisot number $\alpha$. We also show that the roots of the Siegel's polynomial $x^3-x-1$ are the only solutions to the three term equation $\alpha_1+\alpha_2+\alpha_3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $\alpha_1=\alpha_2+\alpha_3$.
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Given a monic degree N polynomial f(x) ∈ Z[x] and a non-negative integer l, we may form a new monic degree N polynomial fl(x) ∈ Z[x] by raising each root of f to the lth power. We generalize a lemma of Dobrowolski to show that if m < n... more
Given a monic degree N polynomial f(x) ∈ Z[x] and a non-negative integer l, we may form a new monic degree N polynomial fl(x) ∈ Z[x] by raising each root of f to the lth power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p divides the resultant of fpm and fpn . We then consider the function (j, k) 7→ Res(fj , fk) mod p . We show that for fixed p and m that this function is periodic in both j and k, and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.
In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$. In particular, we conjecture that for $q \in (1,2)$ we have $|q'| \geq \frac{\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$.... more
In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$. In particular, we conjecture that for $q \in (1,2)$ we have $|q'| \geq \frac{\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$. Further, for $m \geq 3$, we conjecture that for all Pisot numbers $q \in (m, m+1)$ we have $|q'| \geq \frac{m+1-\sqrt{m^2+2m-3}}{2}$. A similar conjecture if made for $m =2$. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by $\beta$, whose conjugate is the reciprocal of a Pisot number.
We build upon previous work on the densities of uniform random walks in higher dimensions, exploring some properties of the even moments of these densities and extending a result about their modularity.
We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| 2$ on the unit circle $\partial... more
We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| 2$ on the unit circle $\partial \mathcal{D}$. This polynomial is of degree $38$ and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional $(k, n)$ with $k \in \{1, 2, 3, n-3, n-2, n-1\}$, for which no such $\{0, 1\}$--polynomial of degree $n$ exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for $\{-1, 1\}$ polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of $N(f)$ in the set of Newman and Littlewood polynomials.
We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is similar to Jn, the Jordan block of 1 of size n. If M = J2, we classify digit sets of size 2 allowing... more
We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is similar to Jn, the Jordan block of 1 of size n. If M = J2, we classify digit sets of size 2 allowing representation of the whole Z. For Jn with n ≥ 3, it is shown that three digits suffice to represent all of Z. For bases similar to Jn, at most n digits are required, with the exception of n = 1. Moreover, the language of strings representing the zero vector with M = J2 and the digits (0,±1) is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.
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We introduce a parameter space containing all algebraic integers β ∈ (1, 2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia... more
We introduce a parameter space containing all algebraic integers β ∈ (1, 2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution ν β . This allows us to show that dimH(ν β ) = 1 for all β with representations in certain open regions of the parameter space.
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Let d and q be positive integers, and consider representing a positive integer n in base d and digits 0, 1, … , q − 1. If q < d, then not all positive integers can be represented. If q = d, then every positive integer can be... more
Let d and q be positive integers, and consider representing a positive integer n in base d and digits 0, 1, … , q − 1. If q < d, then not all positive integers can be represented. If q = d, then every positive integer can be represented in exactly one way. If q > d, then there may be multiple ways of representing an integer n. Let fd, q(n) be the number of representations of n in base d and digits 0, 1, … , q − 1. In this paper, we will look at the asymptotics of fd, q(n) as n → ∞. They depend in a rather strange way on the generalized Thue–Morse sequence.
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Consider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such... more
Consider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval. Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff...
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Research Interests: Applied Mathematics, Computational Physics, Finite Element Methods, Finite element method, Convergence, and 15 moreFinite Element, Algorithm, Classical Conditioning, Fourier Analysis, Convergence Rate, asymptotic Analysis, Difference Equations, Filtering, Differential equation, Difference equation, Boundary Value Problem, Fourier Series, Burgers equation, Condition number, and Elliptic equation
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This paper demonstrates how the Groebner Basis Algorithm can be used for finding the bifurcation points in the generalized Mandelbrot set. It also shows how resultants can be used to find components of the generalized Mandelbrot set.
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Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_\beta0.81$ for all Pisot $\beta$, and improve this lower bound for... more
Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_\beta0.81$ for all Pisot $\beta$, and improve this lower bound for certain ranges of $\beta$. Our method is computational in nature.
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Abstract. This paper is based on a talk given at WWCA (Waterloo Work- shop on Computer Algebra) held at Wilfird Laurier University, April 2006. This paper gives a history of beta-expansions, and surveys some of the com- putational aspects... more
Abstract. This paper is based on a talk given at WWCA (Waterloo Work- shop on Computer Algebra) held at Wilfird Laurier University, April 2006. This paper gives a history of beta-expansions, and surveys some of the com- putational aspects of beta-expansions. Special attention is given to how these beta-expansions relate to Pisot and Salem numbers. This paper also gives an
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Let n be a positive integer. We consider the Sylvester Resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultant's height.... more
Let n be a positive integer. We consider the Sylvester Resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultant's height. If f is a cubic polynomial, we find tight asymptotics for the resultant's height.
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ABSTRACT
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The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with ±1 coefficients and degree at most... more
The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with ±1 coefficients and degree at most 72 are also determined. 1.
We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set... more
We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with $m$-fold sums of Cantor sets with ratio of dissection $1/R$ for integer $R\leq m$. We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all bu...
This paper examines properties of the zeros of poly- nomials with restricted coecients. In particular we study the case when the coecients are restricted to the roots of unity and possibly zero. The methods used in this paper are... more
This paper examines properties of the zeros of poly- nomials with restricted coecients. In particular we study the case when the coecients are restricted to the roots of unity and possibly zero. The methods used in this paper are adaptations of methods used by Odlyzko and Poonen in "Zeros of Polynomials with 0, 1 Coecients ". The main result of
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Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 2005, Melfi examined the structure of $n$ such that $s_2(n) = s_2(n^2)$. We extend this study to the more general case of generic $q$ and polynomials... more
Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 2005, Melfi examined the structure of $n$ such that $s_2(n) = s_2(n^2)$. We extend this study to the more general case of generic $q$ and polynomials $p(n)$, and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case $p(n) = n^2$, looking at the subsets of $n$ where $s_q(n) = s_q(n^2) = k$ for fixed $k$. Comment: 16 pages