References [1] Adams, W.W. and Davison, J.L., A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977) 194-198. [2] Boyarsky, A. and Góra, P., Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Birkhäuser, Boston, 1997. [3] Brezinski, C., History of Continued Fractions and Padé Approximants. Springer Series in Computational Mathematics 12, Springer-Verlag, Berlin, 1991. [4] Corless, R.M., Continued fractions and chaos, Amer. Math. Monthly 99(3) (1992

### Hendrik Jager

References [1] BOSMA,W.-JAGER, H.-WIEDIJK, F.: Some metrical observations on the approximation by continued fractions, Indag. Math. Series A 86, 1983, 281-299. [2] ERDŐS, P.: Some results on Diophantine approximation, Acta Arith. 5 (1959), 359-369. [3] ITO, SH.-NAKADA, H.: On natural extensions of transformations related to Diophantine approximations, In: Proceedings of the Conference on Number Theory and Combinatorics, Japan 1984 (Tokyo, Okayama and Kyoto, 1984), World Sci. Publ. Co., Singapore, 1985. pp. 185

### Amara Chandoul

expansion is called The simple continued fraction of x . It is customarily written x = [ a 0 ,a 1 , . . . ,a n , . ]. We call convergents of x the reduced fractions difined by: p 0 q 0 = a 0 , $$\frac{{{p}_{0}}}{{{q}_{0}}}={{a}_{0}},$$ p 1 q 1 = a 0 + 1 a 1 , … , p n q n = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + ⋯ 1 a n , ⋯ . $$\begin{align}& \frac{{{p}_{1}}}{{{q}_{1}}}={{a}_{0}}+\frac{1}{{{a}_{1}}}, \\ & \ldots \,\,\,, \\ & \frac{{{p}_{n}}}{{{q}_{n}}}={{a}_{0}}+\frac{1}{{{a}_{1}}+}\frac{1}{{{a}_{2}}+}\frac{1}{{{a}_{3}}+}\cdots \frac{1}{{{a

### Bo Li, Yan Zhang and Artur Korniłowicz

## Simple Continued Fractions and Their Convergents

The article introduces simple continued fractions. They are defined as an infinite sequence of integers. The characterization of rational numbers in terms of simple continued fractions is shown. We also give definitions of convergents of continued fractions, and several important properties of simple continued fractions and their convergents.

### Christopher J. White

, Cambridge Mathematical Library (2nd edition), Cambridge University Press, Cambridge, 1952. [5] HARMAN, G.: Metric Number Theory, In: London Mathematical Society Monographs. New Series, Vol. 18. The Clarendon Press, Oxford University Press, New York, 1998. [6] HAYNES, A.: Equivalence classes of codimension one cut-and-project sets, Ergodic Theory Dyn. Syst. 36 (2016), no. 3, 816-831. [7] KHINCHIN, A.YA.: Continued Fractions, The University of Chicago Press, Chicago, III.-London, 1964. [8] ROCKETT, A.-SZ¨USZ, P

### Yasushige Watase

. Formalized Mathematics , 1(5):887-890, 1990. [20] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics , 1(5):829-832, 1990. [21] Bo Li, Yan Zhang, and Artur Korniłowicz. Simple continued fractions and their convergents. Formalized Mathematics , 14(3):71-78, 2006. doi:10.2478/v10037-006-0009-9. [22] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics , 6(2):265-268, 1997. [23

### Krzysztof Kupiec, Monika Gwadera and Barbara Larwa

, adsorption and reaction in a catalyst: A unified method by a continued fraction for slab, cylinder and sphere geometries. Chem. Eng. J., 173, 644-650. DOI: 10.1016/j.cej.2011.08.029. Petrus R., Aksielrud G.A., Gumnicki J.M., Piątkowski W., 1998. Wymiana masy w układzie ciało stałe-ciecz. OWPR Rzeszów, 273. Płaziński W., Dziuba J., Rudziński W., 2013. Modeling of sorption kinetics: the pseudo-second order equation and the sorbate intraparticle diffusivity. Adsorption, 19, 1055-1064. DOI: 10.1007/s10450-013-9529-0. Suzuki M

### Zdzisław Ławrynowicz

## Effect of cementite precipitation on the extend of bainite transformation in fe-cr-c steel

Analytical calculations and experimental measurements of volume fraction of bainitic ferrite and volume of the untransformed austenite indicate that there is a necessity of carbides precipitation. A consequence of the precipitation of cementite during austempering is that the growth of bainitic ferrite can continue to larger extent and that the resulting microstructure is not an ausferrite but it is a mixture of bainitic ferrite, retained austenite and carbides.

### Dawoud Ahmadi Dastjerdi and Sanaz Lamei

Surface and Continued Fractions, Monatsh. Math. 133 (2001), no. 4, 295-339. [5] B. Gurevich, S. Katok, Arithmetic Coding and Entropy for the Positive Geodesic Flow on the Modular Surface, Moscow Mathematical Journal , 1 (2001), no.4, 569-582. [6] G. A. Hedlund, On the Metrical Transitivity of Geodesics on Closed Surfaces of Constant Negative Curvature, Ann. Math. 35 , (1934), 787-808. [7] S. Katok, Coding of Closed Geodesics After Gauss and Morse, Geom. Dedicata 63 (1996), 123-145. [8] S. Katok