scholarly journals Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1268
Author(s):  
Azmeer Nordin ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Zeta Function ◽  
Complex Analysis ◽  
Discrete Dynamical System ◽  
Zeros Of Polynomials ◽  
Asymptotic Results ◽  
Square Roots ◽  
Shift Spaces ◽  
Closed Orbits ◽  
Counting Functions ◽  
Asymptotic Behaviours

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

scholarly journals Orbit Growth of Periodic-Finite-Type Shifts via Artin–Mazur Zeta Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 685
Author(s):  
Azmeer Nordin ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Zeta Function ◽  
Finite Type ◽  
Discrete Dynamical System ◽  
Closed Orbits ◽  
Counting Functions ◽  
Type Shift

The prime orbit and Mertens’ orbit counting functions describe the growth of closed orbits in a discrete dynamical system in a certain way. In this paper, we prove the asymptotic behavior of these functions for a periodic-finite-type shift. The proof relies on the meromorphic extension of its Artin–Mazur zeta function.

Null trajectories for the symmetrized Hurwitz zeta function

2006 ◽  
Vol 463 (2077) ◽  
pp. 303-319 ◽  
Author(s):  
Ross C McPhedran ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Hurwitz Zeta Function ◽  
Trigonometric Functions ◽  
Asymptotic Results ◽  
End Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

Computing complex zeros of polynomials and power series

2011 ◽  
Author(s):  
Jean-Marc Couveignes
Keyword(s):  
Power Series ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Zeros Of Polynomials ◽  
Complex Data ◽  
Square Roots ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Zeros

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.

Poroelastic acoustic wave trains excited by harmonic line tractions

2007 ◽  
Vol 464 (2090) ◽  
pp. 491-511 ◽  
Author(s):  
Vladimir Gerasik ◽  
Marek Stastna
Keyword(s):  
Complex Analysis ◽  
Near Field ◽  
Plane Boundary ◽  
Open Boundary ◽  
Far Field ◽  
S Wave ◽  
Asymptotic Results ◽  
Dimensional Boundary ◽  
Wave Trains ◽  
Biot’S Equations

A two-dimensional boundary-value problem for a porous half-space with an open boundary, described by the widely recognized Biot's equations of poroelasticity, is considered. Using complex analysis techniques, a general solution is represented as a superposition of contributions from the four different types of motion corresponding to P1, P2, S and Rayleigh waves. Far-field asymptotic solutions for the bulk modes, as well as near-field numerical results, are investigated. Most notably, this analysis reveals the following: (i) a line traction generates three wave trains corresponding to the bulk modes, so that P1, P2 and S modes emerge from corresponding wave trains at a certain distance from the source, (ii) bulk modes propagating along the plane boundary are subjected to geometric attenuation, which is found quantitatively to be x −3/2 , similar to the classical results in perfect elasticity theory, (iii) the Rayleigh wave is found to be predominant at the surface in both the near (due to the negation of the P1 and S wave trains) and the far field (due to geometric attenuation of the bulk modes), and (iv) the recovery of the transition to the classical perfect elasticity asymptotic results validates the asymptotics established herein.

scholarly journals A novel approach to the Lindelöf hypothesis

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
A S Fokas
Keyword(s):  
Integral Equation ◽  
Zeta Function ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Linear Integral Equation ◽  
Asymptotic Results ◽  
Novel Approach ◽  
Double Exponential ◽  
Lindelöf Hypothesis ◽  
Riemann Zeta

Abstract Lindelöf’s hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann’s zeta function $\zeta (1/2+it)$ is of order $O(t^{\varepsilon })$ for any $\varepsilon>0$. It is well known that for large $t$, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum. However, since such estimates cannot yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf’s hypothesis. Here a completely different approach is introduced. In particular, a novel linear integral equation is derived for $|\zeta (\sigma +it)|^2, \ 0<\sigma <1$ whose asymptotic analysis yields asymptotic results for a certain Riemann zeta-type double exponential sum. This sum has the same structure as the sum describing the leading asymptotics of $|\zeta (\sigma +it)|^2$, namely it involves $m_1^{-\sigma -it}m_2^{-\sigma -it}$, but its summation limits are different than those of the sum corresponding to $|\zeta (\sigma +it)|^2$. The analysis of the above integral equation requires the asymptotic estimation of four different integrals denoted by $I_1,I_2,\tilde{I}_3,\tilde{I}_4$, as well as the derivation of an exact relation between certain double exponential sums. Here the latter relation is derived, and also the rigourous analysis of the first two integrals $I_1$ and $I_2$ is presented. For the remaining two integrals, formal results are only derived that suggest a possible roadmap for the derivation of rigourous asymptotic results of the above double exponential sum, as well as for other sums associated with $|\zeta (\sigma +it)|^2$. Additional developments suggested by the above novel approach are also discussed.

scholarly journals Generalizations of a cotangent sum associated to the Estermann zeta function

2016 ◽  
Vol 18 (01) ◽  
pp. 1550078 ◽  
Author(s):  
Helmut Maier ◽  
Michael Th. Rassias
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Positive Measure ◽  
Asymptotic Formulas ◽  
Asymptotic Results ◽  
Rate Of Growth

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure [Formula: see text] on [Formula: see text], with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order [Formula: see text], as a function of [Formula: see text].

AN “i” FOR ANi: SINGULAR TERMS, UNIQUENESS, AND REFERENCE

2012 ◽  
Vol 5 (3) ◽  
pp. 380-415 ◽  
Author(s):  
STEWART SHAPIRO
Keyword(s):  
Philosophy Of Language ◽  
Complex Analysis ◽  
Singular Term ◽  
Definite Descriptions ◽  
Singular Terms ◽  
Square Roots ◽  
Donkey Anaphora ◽  
Unique Object ◽  
Logical Semantic

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of −1 are indiscernible: anything true of one of them is true of the other. So how does the singular term ‘i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and the philosophy of language, I suggest thatifunctions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages.

scholarly journals On the Riemann zeta-function I

1976 ◽  
Vol 15 (2) ◽  
pp. 161-211
Author(s):  
Masako Izumi ◽  
Shin-ichi Izumi
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Complex Analysis ◽  
Approximation Formula ◽  
Classical Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

We prove an approximation formula for the Riemann zeta function. We show that a classical theorem:uniformly in the domain ½ ≤ σ < 1, is an immediate consequence of our approximation formula. Our method is real and free from complex analysis.

scholarly journals The Schottky–Klein prime function and counting functions for Fenchel double crosses

2021 ◽  
Author(s):  
M. Pollicott
Keyword(s):  
Nineteenth Century ◽  
Ergodic Theory ◽  
Hyperbolic Geometry ◽  
Complex Analysis ◽  
Thermodynamic Formalism ◽  
Counting Problem ◽  
Counting Functions ◽  
Prime Function

AbstractWe relate the classical nineteenth century Schottky–Klein function in complex analysis to a counting problem for pairs of geodesics in hyperbolic geometry studied by Fenchel. We then solve the counting problem using ideas from ergodic theory and thermodynamic formalism.

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