Pisot Numbers from {0, 1}-Polynomials

2010 ◽  
Vol 53 (1) ◽  
pp. 140-152
Author(s):  
Keshav Mukunda
Keyword(s):  
Unit Disk ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Pisot Number ◽  
Open Unit ◽  
Closed Unit Disk ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Salem Number ◽  
Newman Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
Author(s):  
J. P. Bell ◽  
K. G. Hare
Keyword(s):  
Algebraic Integer ◽  
Full Rank ◽  
Pisot Number ◽  
List Type ◽  
Algebraic Integers ◽  
Pisot Numbers ◽  
Cyclotomic Numbers ◽  
Special Cases ◽  
Fixed Order ◽  
Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

scholarly journals On the lines passing through two conjugates of a Salem number

2008 ◽  
Vol 144 (1) ◽  
pp. 29-37 ◽  
Author(s):  
ARTŪRAS DUBICKAS ◽  
CHRIS SMYTH
Keyword(s):  
Algebraic Number ◽  
General Position ◽  
Imaginary Axis ◽  
Pisot Number ◽  
Pisot Numbers ◽  
Parallel Lines ◽  
Salem Number ◽  
Pass Through

AbstractWe show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d2/2]-d+2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ $\overline{\Q}$ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number $\al_1\,{=}\,(1+\sqrt{3+2\sqrt{5}})/2$ shows that this equation has such a solution.

scholarly journals On the co-factors of degree 6 Salem number beta expansions

2020 ◽  
pp. 1-26
Author(s):  
Jacob J. Stockton
Keyword(s):  
Computational Methods ◽  
Pisot Number ◽  
Salem Numbers ◽  
Positive Proportion ◽  
Salem Number ◽  
Almost All

For [Formula: see text], a sequence [Formula: see text] with [Formula: see text] is the beta expansion of [Formula: see text] with respect to [Formula: see text] if [Formula: see text]. Defining [Formula: see text] to be the greedy beta expansion of [Formula: see text] with respect to [Formula: see text], it is known that [Formula: see text] is eventually periodic as long as [Formula: see text] is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with [Formula: see text]. We give examples of numbers with [Formula: see text] whose expansions have period and preperiod lengths exceeding [Formula: see text], yet are still eventually periodic.

scholarly journals A note on a result of A. J. Lohwater and George Piranian

1975 ◽  
Vol 56 ◽  
pp. 171-174
Author(s):  
G. L. Csordas
Keyword(s):  
Unit Disk ◽  
Boundary Behavior ◽  
Single Point ◽  
Open Unit ◽  
Outer Function ◽  
Open Unit Disk ◽  
Inner Functions ◽  
Bounded Analytic Functions ◽  
Closed Unit Disk ◽  
Cluster Set

Let I denote the set of all inner functions in H∞, where H∞ is the Banach algebra of all bounded analytic functions on the open unit disk D. Let I* denote the set of all functions f(z) in H∞ for which the cluster set C(f,α) at any point α on the circumference C = {α| |α| = 1} is either the closed unit disk |w| ≤ 1 or else a single point of modulus one. Clearly, I is a subset of I*. In [3] the author has proved that I is properly contained in I*. Recently, Lohwater and Piranian [7] have shown that there is an outer function in I*. The purpose of this note is to point out some applications of this result. In particular we shall show in Theorem 2.3 that there exist outer functions whose boundary behavior is similar to that of inner functions.

scholarly journals Some Properties for Strong Differential Subordination of Analytic Functions Involving Ruscheweyh Derivative Operator

2020 ◽  
pp. 63-70
Author(s):  
Asraa Abdul Jaleel Husien
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Derivative Operator ◽  
Closed Unit Disk ◽  
Ruscheweyh Derivative Operator ◽  
Ruscheweyh Derivative ◽  
Strong Differential Subordination ◽  
Differential Subordinations

In this paper, we introduce and study some properties for strong differential subordinations of analytic functions associated with Ruscheweyh derivative operator defined in the open unit disk and closed unit disk of the complex plane.

scholarly journals Some Properties for Strong Differential Subordination of Analytic Functions Associated with Wanas Operator

2020 ◽  
pp. 29-38
Author(s):  
Abbas Kareem Wanas ◽  
Pall-Szabo Agnes Orsolya
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination ◽  
Differential Subordinations

In this paper, by making use of Wanas operator, we derive some properties related to the strong differential subordinations of analytic functions defined in the open unit disk and closed unit disk of the complex plane.

scholarly journals The Pisot conjecture for -substitutions

2016 ◽  
Vol 38 (2) ◽  
pp. 444-472 ◽  
Author(s):  
MARCY BARGE
Keyword(s):  
Dynamical System ◽  
Discrete Spectrum ◽  
Minimal Polynomial ◽  
Pisot Number ◽  
Companion Matrix ◽  
Pisot Numbers ◽  
Almost Everywhere ◽  
One To One

We prove the Pisot conjecture for$\unicode[STIX]{x1D6FD}$-substitutions: if$\unicode[STIX]{x1D6FD}$is a Pisot number, then the tiling dynamical system$(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$associated with the$\unicode[STIX]{x1D6FD}$-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

scholarly journals Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 27
Author(s):  
Hari Mohan Srivastava ◽  
Ahmad Motamednezhad ◽  
Safa Salehian
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Expansion Method ◽  
Upper Bounds ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Faber Polynomial ◽  
The Subject ◽  
Polynomial Expansion Method

In this paper, we introduce a new comprehensive subclass ΣB(λ,μ,β) of meromorphic bi-univalent functions in the open unit disk U. We also find the upper bounds for the initial Taylor-Maclaurin coefficients |b0|, |b1| and |b2| for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients |bn|(n≧1) for functions in the subclass ΣB(λ,μ,β) by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

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