共查询到20条相似文献,搜索用时 15 毫秒
1.
E. P. Golubeva 《Journal of Mathematical Sciences》1994,70(6):2059-2076
We present an algorithm that makes it possible to write out all quadratic irrationals of the form
, that have a given even period length in the continued fraction expansion. It turns out that in the expansion
相似文献
2.
Bing Li 《Journal of Mathematical Analysis and Applications》2008,339(2):1322-1331
For any real number β>1, let ε(1,β)=(ε1(1),ε2(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,
3.
R. Nair 《Monatshefte für Mathematik》1998,125(3):241-253
Supposek
n denotes either (n) or (p
n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp
k denotes thek
th rationals prime. Also let
denote the sequence of convergents to a real numberx and letc
n(x))
n=1
be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants
n(x))
n=1
by
4.
K. G. Ramanathan 《Proceedings Mathematical Sciences》1984,93(2-3):67-77
In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction
for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula. 相似文献
5.
Asymptotic expansion and continued fraction for mathieu's series 总被引:2,自引:0,他引:2
Á. Elbert 《Periodica Mathematica Hungarica》1982,13(1):1-8
6.
A piecewise linear, discontinuous one-dimensional map is analyzed combinatorically. The quasi-periodic dynamics generated by iterations are completely characterized by successive convergents of a continued fraction associated with slopes of the map. 相似文献
7.
E. P. Golubeva 《Journal of Mathematical Sciences》1999,95(3):2192-2197
We study the class number of an indefinite binary quadratic form of discriminant d based on the expansion of d into a continued fraction and single out sequences of d for which h(d) has a lower-bound extimate. Progress is made for the conjecture on the estimate of the quantity of prime discriminants d with fixed length of period of expansion of d. Bibliography: 15 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 31–45. 相似文献
8.
《Indagationes Mathematicae》2022,33(6):1189-1220
This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface , are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift. 相似文献
9.
Chu-In Charles Lee 《Annals of the Institute of Statistical Mathematics》1992,44(1):107-120
The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](x), it is simple, it converges for x>0, and it is by far the best approximation for x3. The Laplace continued fraction is rederived as an extreme case of admissible bounds of the Mills' ratio, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](x)/(x), in the family of ratios of two polynomials subject to a monotone decreasing absolute error. However, it is not optimal at any finite x. Convergence at the origin and local optimality of a subclass of admissible bounds are investigated. A modified continued fraction is proposed. It is the sharpest tail bound of the Mills' ratio, it has a satisfactory convergence rate for x1 and it is recommended for the entire range of x if a maximum absolute error of 10-4 is required.The efforts of the author were supported by the NSERC of Canada. 相似文献
10.
In this paper, we derive certain identities for the following continued of order six: 相似文献
11.
F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.Supported by The Netherlands Organization for Scientific Research (NWO). 相似文献
12.
K. R. Vasuki Abdulrawf A. A. Kahtan G. Sharath C. Sathish Kumar 《Ukrainian Mathematical Journal》2011,62(12):1866-1878
We present some new relations between a continued fraction U(q) of order 12 (established by M. S. M. Naika et al.) and U(q n ) for n = 7, 9, 11, 13: 相似文献
13.
14.
TextWe extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Göllnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v(τ) by computing the affine models of modular curves X(Γ) with Γ=Γ1(8)∩Γ0(16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(τ) is a modular unit over Z we give a new proof of the fact that the singular values of v(τ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=FWdmYvdf5Jg. 相似文献15.
Christian Faivre 《Archiv der Mathematik》1998,70(6):455-463
16.
《Journal of Number Theory》1986,23(1):55-59
We use matrices to prove two theorems in regard to the continued fraction expansion for Σk = 0∞u−2k and for Σk = 0∞u−c(k), where c(k) is any sequence of positive integers that increase quickly. 相似文献
17.
Denote by pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number x. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all x the pattern θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1, namely 0.12109?. All the four other patterns have the same asymptotic frequency 0.18945?. The constants are explicitly given. 相似文献
18.
Kh. Yo. Kuchmins’ka 《Journal of Mathematical Sciences》2012,180(1):1-14
Using estimates for the remainders of a two-dimensional continued fraction, the relation for the difference between two approximants
of such a fraction in terms of these remainders, and the majorant method, we have proposed generalizations of the Worpitzky
convergence theorem. For fractions satisfying the conditions of generalized Worpitzky theorems, we have also obtained estimates
for their convergence rate. 相似文献
19.
20.
Harald Niederreiter 《Monatshefte für Mathematik》1987,103(4):269-288
For a rational functionf/g=f(x)/g(x) over a fieldF with ged (f,g)=1 and deg (g)1 letK(f/g) be the maximum degree of the partial quotients in the continued fraction expansion off/. ForfF[x] with deg (f)=k1 andf(O)O putL(f)=K(f(x)/x
k
). It is shown by an explicit construction that for every integerb with 1bk there exists anf withL(f)=b. IfF=F
2, the binary field, then for everyk there is exactly onefF
2[x] with deg (f)=k,f(O)O, andL(f)=1. IfF
q
is the finite field withq elements andgF
q
[x] is monic of degreek1, then there exists a monic irreduciblefF
q
[x] with deg (f)=k, gcd (f,g)=1, andK(f/g)<2+2 (logk)/logq, where the caseq=k=2 andg(x)=x
2+x+1 is excluded. An analogous existence theorem is also shown for primitive polynomials over finite fields. These results have applications to pseudorandom number generation. 相似文献
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