Properties Of Continued Fractions Research Articles

Convergence conditions for p-adic continued fractions

Continued fractions have been introduced in the field of p-adic numbers Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. In particular, an analogue of the Lagrange’s Theorem is not yet proved for any attempt of generalizing continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. Thus, it is worth to study the definition of new algorithms for p-adic continued fractions. The main condition that a new method needs to fulfill is the convergence in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} of the continued fractions. In this paper we study some convergence conditions for continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.

On Encryption with Continued Fraction

Many mathematicians have investigated the properties of continued fractions. They made continued fraction expansions of the Pi number, the golden ratio and many more special numbers. With the help of continued fractions, solutions of some Diophantine equations are obtained. In this study, encryption was made using continued fractional expansions of the square root of non-perfect-square integers. Each of the 29 letters in the alphabet is represented by the root of nonperfect square integers starting from 2. Then, continued fraction expansions of the square root of each letter’s number equivalent were calculated. Afterwards, all numbers in the continued fraction expansion were considered as an integer by removing the comma. This information was tabulated for later usage. Each word is considered as individual letters, and a space is left between the encrypted versions of each letter. After the encryption process, the process of deciphering the encrypted text was dealt with. In the deciphering process, since there is a blank between the numbers, the numbers are written as a continued fraction and the integer expansion is calculated. Later, the letter corresponding to this number was found.

On the Pillai's problem involving two linear recurrent sequences: Padovan and Fibonacci

In this paper, we find all integers $c$ having at least two representations as a difference between linear recurrent sequences. This problem is a pillai problem involving Padovan and Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies

We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non-resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the frequency vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with “holes”. If we want the set to be connected we have to require some non-resonance condition on the frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.

On solutions of the Diophantine equation $L_{n}+L_{m}=3^{a} $

Let $(L_n)_{n\geq 0}$ be the Lucas sequence given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 0$. In this paper, we are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation $L_n + L_m = 3^{a}$ in nonnegative integers $n, m,$ and $a$. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

CONTINUED FRACTIONS, INTERMEDIATE FRACTIONS AND THEIR RELATION TO THE BEST APPROXIMATIONS

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

Diophantine properties of continued fractions on the Heisenberg group

We provide several results on the Diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin’s theorem and we also show that convergents are the best rational approximations up to a constant factor of the denominator.

On the Conditional Probability of a Successful Retrial in Retrial Queues

Customer retrials are very common phenomena in industrial engineering and business management. The number of retrials required before receiving a success is an important measure for evaluating system performance. Focusing on the M/M/1 and M/M/2 retrial queues, we study the conditional probability of a successful retrial given that all previous retrials have been denied. The idea is to obtain the recursive expressions based on the properties of continued fractions and the transient solutions of systems. The result leads to computational algorithms for the probability of the number of retrials made by a blocked primary customer. Through implementation of the proposed algorithms, numerical examples and analysis are presented to show computational efficiency and properties of the conditional probability.

Open mushrooms: stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ ∊ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.

Continued fractions, the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.

On statistical properties of finite continued fractions

Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a2 + b2 ≤ R2, are studied. The main result is an asymptotic formula with two meaning terms for the quantity $$N_x (R) = \sum\limits_{_{a,b \in \mathbb{N}}^{a^2 b^2 \leqslant R^2 } } {s_x (a/b)} ,$$ where sx(a/b) = ¦{j e {1, …, s}: [0; tj, …, ts] ≤ x}¦ is the Gaussian statistic for the fraction a/b = [t0; t1, …, ts]. Bibliography: 12 titles.

Klein polyhedra for three extremal cubic forms

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms g i, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms g 1 – g 4. They also computed the “convergents” for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms g 5, – g 7 and the adjoint form g 7 * . Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincare, Brun, Parusnikov, and Bryuno, are computed. The position of the “convergents of the continued fractions” with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Euler’s and Poincare’s algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.

The Derivative of Minkowski's ?(x) Function

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it enables us to prove that its derivative, when it exists in a wide sense, can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k̄=5.31972, and ?′(x) exists, then ?′(x)=0. In the same way, if the same average is less than ḵ=2log2Φ, where Φ is the golden ratio, then ?′(x)=∞. Finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions.

On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions α=[a 0;a 1,a 2,...] is proved under growth conditions involving the denominatorsq n of the convergents and shifted partial quotientsa n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.

Model above-threshold photodetachment spectra

Model photodetachment spectra are calculated analytically for a system including an infinite number of circular states. Characteristic features of the ATD spectra are derived and interpreted based on the properties of continued fractions. Numerical results confirm a typical regular multipeak structure of the total spectra as well as of the spectra corresponding to a given net number of photons absorbed. For extremely strong fields the peaks become closer and the lines overlap.

Some metrical properties of continued fractions

Some metrical properties of continued fractions

Spectroscopic band shapes and the theory of continued fractions

Starting from some of the known properties of continued fractions and power series expansions, spectroscopic Green's functions may be recast into continued fractions. Since the algorithm so developed does not depend upon the well-known projection operator technique, it is possible to rewrite spectral functions that have been obtained by empiricism into forms that closely parallel theoretical results. Thus phenomenological parameters can be readily given microscopic interpretations.

The quadratic form x2−2 py2

Using known properties of continued fractions, we give a very simple and elementary proof of the theorem of Epstein and Rédei on the impossibility in a certain case of representing −1 by the quadratic form x 2 − 2 py 2. Two of our theorems, which concern the representation of a 2 and −2 a 2, serve to extend our method to an unknown case in which −1 is not representable.

Truncation Error Estimates for T-Fractions

Previous article Next article Truncation Error Estimates for T-FractionsThomas H. JeffersonThomas H. Jeffersonhttps://doi.org/10.1137/0706033PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Peter Henrici and , Pia Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math., 9 (1966), 120–138 10.1007/BF02166031 MR0212991 0152.15301 CrossrefISIGoogle Scholar[2] William B. Jones and , W. J. Thorn, Further properties of T-fractions, Math. Ann., 166 (1966), 106–118 10.1007/BF01361442 MR0200425 0143.29501 CrossrefISIGoogle Scholar[3] W. J. Thron, Some properties of continued fractions $1+d\sb 0z+K(z/(1+d\sb n z))$, Bull. Amer. Math. Soc., 54 (1948), 206–218 MR0024528 0039.29101 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Survey of Truncation Error Analysis for Padé and Continued Fraction ApproximantsNonlinear Numerical Methods and Rational Approximation II | 1 Jan 1994 Cross Ref A survey of truncation error analysis for Pad� and continued fraction approximantsActa Applicandae Mathematicae, Vol. 33, No. 2-3 | 1 Dec 1993 Cross Ref Truncation error bounds for modified continued fractions with applications to special functionsNumerische Mathematik, Vol. 55, No. 3 | 1 May 1989 Cross Ref Positive T-fraction expansions for a family of special functionsAnalytic Theory of Continued Fractions III | 21 October 2006 Cross Ref Continued fractions in numerical analysisApplied Numerical Mathematics, Vol. 4, No. 2-4 | 1 Jun 1988 Cross Ref Truncation error bounds for limit-periodic continued fractionsK(a n /1) with lima n =0Numerische Mathematik, Vol. 46, No. 4 | 1 Dec 1985 Cross Ref Reflections on value regions, limit regions and truncation errors for continued fractionsNumerische Mathematik, Vol. 47, No. 2 | 1 Jun 1985 Cross Ref Truncation Error Bounds for Continued Fractions $K({{a_n } / 1})$ with Parabolic Element RegionsWilliam B. Jones, W. J. Thron, and Haakon WaadelandSIAM Journal on Numerical Analysis, Vol. 20, No. 6 | 17 July 2006AbstractPDF (914 KB)MULTIPLE-POINT PADÉ TABLESPade and Rational Approximation | 1 Jan 1977 Cross Ref On a posteriori error estimatesMathematics of Computation, Vol. 31, No. 137 | 1 January 1977 Cross Ref Estimates of the speed of convergence of continued fraction expansions of functionsMathematics of Computation, Vol. 31, No. 138 | 1 January 1977 Cross Ref Truncation error analysis by means of approximant systems and inclusion regionsNumerische Mathematik, Vol. 26, No. 2 | 1 Jun 1976 Cross Ref A priori estimates for truncation error of continued fractionsK(1/b n )Numerische Mathematik, Vol. 19, No. 4 | 1 Aug 1972 Cross Ref The Padé Table and Its Relation to Certain Algorithms of Numerical AnalysisW. B. GraggSIAM Review, Vol. 14, No. 1 | 18 July 2006AbstractPDF (3619 KB)Singularities of a class of meromorphic functionsProceedings of the American Mathematical Society, Vol. 33, No. 2 | 1 January 1972 Cross Ref A Posteriori Bounds for the Truncation Error of Continued FractionsWilliam B. Jones and W. J. ThronSIAM Journal on Numerical Analysis, Vol. 8, No. 4 | 3 August 2006AbstractPDF (905 KB) Volume 6, Issue 3| 1969SIAM Journal on Numerical Analysis317-522 History Submitted:03 September 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706033Article page range:pp. 359-364ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Some properties of continued fractions

Some properties of continued fractions