A general continued fraction expansion of 1-D complex lossless positive real function with application to 2-D

1987 ◽  
Vol 34 (9) ◽  
pp. 1121-1122
Author(s):  
H. Reddy ◽  
P. Karivaratharajan
Keyword(s):  
Real Function ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Positive Real Function

Passive Synthesis of a Class of Fractional Immittance Function Based on Multivariable Theory

2018 ◽  
Vol 27 (05) ◽  
pp. 1850074 ◽  
Author(s):  
Guishu Liang ◽  
Yongming Jing ◽  
Chang Liu ◽  
Long Ma
Keyword(s):  
Real Function ◽  
Traditional Method ◽  
Synthesis Method ◽  
Important Research ◽  
Research Topics ◽  
Positive Real ◽  
Passive Synthesis ◽  
Positive Real Function ◽  
Variable Substitution ◽  
New Variables

Passive synthesis of fractional driving-point immittance function is one of the most important research topics in fractional circuits. The synthesis method for a class of fractional driving-point immittance function is proposed in this paper. Firstly, the immittance function of passive fractional circuit is transformed to multivariable positive-real function by proper variable substitution. Secondly, to synthesize the function, except one variable, other high-order variables are transformed to the combination of products of different one-order variables by introducing new variables. Thirdly, synthesize the new function by traditional method. At last, substitute elements inversely twice and the final circuit for fractional driving-point immittance function can be achieved. The method is verified by giving synthesis demonstration.

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.

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.

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