scholarly journals A question for iterated Galois groups in arithmetic dynamics

2020 ◽  
pp. 1-17
Author(s):  
Andrew Bridy ◽  
John R. Doyle ◽  
Dragos Ghioca ◽  
Liang-Chung Hsia ◽  
Thomas J. Tucker
Keyword(s):  
Dynamical System ◽  
Galois Groups ◽  
General Question ◽  
Arithmetic Dynamics ◽  
Polynomial Maps ◽  
Special Cases ◽  
Unicritical Polynomials

Abstract We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (1) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

scholarly journals Galois groups of iterates of some unicritical polynomials

2017 ◽  
Vol 181 (1) ◽  
pp. 57-73 ◽  
Author(s):  
Michael R. Bush ◽  
Wade Hindes ◽  
Nicole R. Looper
Keyword(s):  
Galois Groups ◽  
Unicritical Polynomials

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (2) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ‖1‖ = 1. An element u in A is said to be unitary if it is invertible and ‖u‖ = ‖u−1‖ = 1. An element h in A is said to be hermitian if ‖exp(ifh)‖ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π (x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x) = y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

scholarly journals A New Set of Stability Criteria Extending Lyapunov’s Direct Method

2020 ◽  
Author(s):  
William Li
Keyword(s):  
Dynamical System ◽  
Energy Function ◽  
Direct Method ◽  
Nonlinear Dynamical System ◽  
System Stability ◽  
Nonlinear Dynamical ◽  
Lyapunov’S Direct Method ◽  
Special Cases ◽  
Uniformly Asymptotic Stability ◽  
High Order Time

A dynamical system is a mathematical model described by a high dimensional ordinary differential equation for a wide variety of real world phenomena, which can be as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov's direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov's direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. the paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov's direct method fails.

scholarly journals A New Set of Stability Criteria Extending Lyapunov’s Direct Method

2020 ◽  
Author(s):  
William Li
Keyword(s):  
Dynamical System ◽  
Energy Function ◽  
Direct Method ◽  
Nonlinear Dynamical System ◽  
System Stability ◽  
Nonlinear Dynamical ◽  
Lyapunov’S Direct Method ◽  
Special Cases ◽  
Uniformly Asymptotic Stability ◽  
High Order Time

A dynamical system is a mathematical model described by a high dimensional ordinary differential equation for a wide variety of real world phenomena, which can be as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov's direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov's direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. the paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov's direct method fails.

SOME INITIAL THOUGHTS ON BOUNDED QUERY COMPUTATIONS OVER THE REALS

2012 ◽  
Vol 23 (07) ◽  
pp. 1499-1510
Author(s):  
KLAUS MEER
Keyword(s):  
Real Number ◽  
Implicit Function Theorem ◽  
Quantifier Elimination ◽  
Recursion Theory ◽  
General Question ◽  
Implicit Function ◽  
Basic Properties ◽  
Special Cases ◽  
New Framework

A classical theme in recursion theory is the question whether for a set A and n given elements x1,…,xn, an oracle machine having access to an oracle B can determine which of the elements xi belong to A. And if yes, how many queries are necessary? Here, B = A is possible and leads to interesting special cases of the general question In the present paper we adapt classical notions in this area of bounded query computations to real number algorithms as formalized by Blum, Shub, and Smale and analyze them in the new framework. Among the results obtained we find: A real version of a non-speedup theorem based on real quantifier elimination, some basic properties about selective real sets, and a way to construct natural terse sets in ℝ by applying the implicit function theorem. The purpose of the paper is on presenting some interesting questions and basic results as an appertizer to intensify research into this direction.

scholarly journals Dynamical System Perspective of Cosmological Models Minimally Coupled with Scalar Field

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
S. Surendra Singh ◽  
Chingtham Sonia
Keyword(s):  
Dynamical System ◽  
Scalar Field ◽  
Fixed Points ◽  
Cosmic Acceleration ◽  
Stable Model ◽  
Exponential Potential ◽  
Total Energy Density ◽  
System Perspective ◽  
Special Cases ◽  
Potential Form

The stability criteria for the dynamical system of a homogeneous and isotropic cosmological model are investigated with the interaction of a scalar field in the presence of a perfect fluid. In this paper, we depict the dynamical system perspective to study qualitatively the scalar field cosmology under two special cases, with and without potential. In the absence of potential, we get a two-dimensional dynamical system, and we study the analytical as well as geometrical behavior. For the dynamical system with potential, we analyze different potential forms: simple exponential potential form (Vϕ=Voe−λϕ), double exponential potential form Vϕ=Voexp−Aexp2αϕ, and inverse power law potential form (Vϕ=Voϕ−α). We generate an autonomous system of ordinary differential equations (ASODE) for each case by introducing new dimensionless variables and obtain respective fixed points. We also analyze the type, nature, and stability of the fixed points and how their behavior reflects towards the cosmological scenarios. Throughout the whole work, the investigation of this model has shown us the deep connection between these theories and cosmic acceleration phenomena. The phase plots of the system at different conditions and different values of γ have been analyzed in detail, and their geometrical interpretations have been studied. The perturbation plots of the dynamical system have been analyzed with emphasis on our analytical findings. We have evaluated the total energy density (Ωϕ) at the fixed points and also found out the suitable range of γ and λ for a stable model.

Unique factorisation of normal elements in polynomial rings

1988 ◽  
Vol 110 (3-4) ◽  
pp. 241-247 ◽  
Author(s):  
D.A. Jordan
Keyword(s):  
Polynomial Rings ◽  
General Question ◽  
Infinite Field ◽  
Prime Rings ◽  
Special Cases ◽  
Normal Element ◽  
Polynomial Extensions

SynopsisWe consider the class of Noetherian UFN-rings, that is, Noetherian prime rings such that every non-zero ideal contains a non-zero normal element and such that the monoid of non-zero normal elements is a unique factorisation monoid. We ask whether this class of rings, a generalisation of the commutative Noetherian unique factorisation domains (UFD), is closed under polynomial extensions. The general question is apparently difficult and remains open. However, we give positive answers in special cases, in particular, for algebras over an infinite field and for domains.

scholarly journals STRONG COMPLETENESS OF S4 FOR ANY DENSE-IN-ITSELF METRIC SPACE

2013 ◽  
Vol 6 (3) ◽  
pp. 545-570 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Metric Space ◽  
Real Line ◽  
General Question ◽  
Topological Semantics ◽  
Strong Completeness ◽  
Cantor Space ◽  
The Real ◽  
Special Cases ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.

scholarly journals Finite index theorems for iterated Galois groups of unicritical polynomials

2020 ◽  
Vol 374 (1) ◽  
pp. 733-752 ◽  
Author(s):  
Andrew Bridy ◽  
John R. Doyle ◽  
Dragos Ghioca ◽  
Liang-Chung Hsia ◽  
Thomas J. Tucker
Keyword(s):  
Finite Index ◽  
Galois Groups ◽  
Index Theorems ◽  
Unicritical Polynomials
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