scholarly journals On a θ-Weyl sum

1973 ◽  
Vol 52 ◽  
pp. 163-172 ◽  
Author(s):  
Yoshinobu Nakai
Keyword(s):  
Error Term ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Weyl Sum

We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood [4], and after them, by Behnke [1] and [2], Mordell [9], Wilton [11] and others. The reader will find its history in [7] and in the comments of the Collected Papers [4]. Here we show that the sum can be expressed explicitly, together with an error term O(N1/2), using the regular continued fraction expansion of α. As the statements have complications we will divide them into two theorems. In the followings all letters except ϑ, i, σ, ζ, χ and those in 3° are real, N is a positive real, and always k, n, a, A, B, C, D and E denote integers. The author expresses his thanks to Professor Tikao Tatuzawa and Professor Tomio Kubota for their encouragements.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

scholarly journals Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

2018 ◽  
Vol 27 (11) ◽  
pp. 1850170 ◽  
Author(s):  
Georgia Tsirimokou ◽  
Aslihan Kartci ◽  
Jaroslav Koton ◽  
Norbert Herencsar ◽  
Costas Psychalinos
Keyword(s):  
Comparative Study ◽  
Fractional Order ◽  
High Performance ◽  
Continued Fraction ◽  
Transfer Functions ◽  
Operational Amplifiers ◽  
Current Conveyors ◽  
Continued Fraction Expansion ◽  
Current Feedback ◽  
Discrete Component

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order differentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order differentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup’s approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of differentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy offered by the Continued Fraction Expansion and the Oustaloup’s approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational amplifiers, operational transconductance amplifiers, current conveyors, and current feedback operational amplifiers realized in commercially available discrete-component IC form, are compared in terms of the most important performance characteristics. The most suitable of them are further compared using the OrCAD PSpice software.

Sign in / Sign up

Export Citation Format

Share Document