scholarly journals On the continued fraction expansion of the unique root in F(p) of the equation x4+x2−Tx−1/12=0 and other related hyperquadratic expansions

2012 ◽  
Vol 18 (1) ◽  
pp. 26-34
Author(s):  
A. Lasjaunias
Keyword(s):  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Unique Root

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.

scholarly journals A family of two-generator non-Hopfian groups

2017 ◽  
Vol 27 (06) ◽  
pp. 655-675
Author(s):  
Donghi Lee ◽  
Makoto Sakuma
Keyword(s):  
Continued Fraction ◽  
Small Cancellation ◽  
Continued Fraction Expansion ◽  
Link Group

We construct [Formula: see text]-generator non-Hopfian groups [Formula: see text] where each [Formula: see text] has a specific presentation [Formula: see text] which satisfies small cancellation conditions [Formula: see text] and [Formula: see text]. Here, [Formula: see text] is the single relator of the upper presentation of the [Formula: see text]-bridge link group of slope [Formula: see text], where [Formula: see text] and [Formula: see text] in continued fraction expansion for every integer [Formula: see text].

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

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