Continued Fraction Research Articles

ON THE SHORTEST DISTANCE FUNCTION IN CONTINUED FRACTIONS

AbstractLet $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$ . Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function $$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$ which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function.

Short-term wind speed prediction based on fractal dimension-variational mode decomposition and general continued fraction

Accurate wind speed prediction is significant to maintain the stable power system operation and enhance wind power utilization efficiency. However, most of wind speed prediction models cannot well fit the variation pattern of wind series due to its random and chaotic characteristics. This study proposes a novel wind speed prediction model incorporating fractal dimension (FD) and variational mode decomposition (VMD) and general continued fraction (GCF). More specifically, the fractal feature of wind speed series is analyzed, and the parameter determination strategy for VMD based on fractal feature of wind series is developed to overcome the problem of ambiguous mode parameter K for VMD, so that the relatively regular modes can be extracted from the raw wind series. To well capture the variation characteristics of wind series, a novel GCF model is derived on the basis of the inverse difference quotient theory, and the structure parameters of GCF model are determined by the bats algorithm (BA). To verify the accuracy and stability of the proposed model, several experiments involving comparisons with some mainstream models are executed on five wind speed datasets, and the proposed model outperforms comparative models with mean absolute percentage error reduction of 0.48 %–74.44 %. The results of the experiments indicate the proposed model has great capacity of capturing the variation characteristics of wind speed series.

Continued fractions with bounded even-order partial quotients

Continued fractions with bounded even-order partial quotients

Increasing rate of weighted product of partial quotients in continued fractions

Let [a1(x),a2(x),…,an(x),…] be the continued fraction expansion of x∈[0,1). In this paper, we study the increasing rate of the weighted product ant0(x)an+1t1(x)⋯an+mtm(x) ,where ti∈R+(0≤i≤m) are weights. More precisely, let φ:N→R+ be a function with φ(n)/n→∞ as n→∞. For any (t0,…,tm)∈R+m+1 with ti≥0 and at least one ti≠0(0≤i≤m), the Hausdorff dimension of the set E̲({ti}i=0m,φ)=x∈[0,1):lim infn→∞logant0(x)an+1t1(x)⋯an+mtm(x)φ(n)=1 is obtained. Under the condition that (t0,…,tm)∈R+m+1 with 0<t0≤t1≤⋯≤tm, we also obtain the Hausdorff dimension of the set E¯({ti}i=0m,φ)=x∈[0,1):lim supn→∞logant0(x)an+1t1(x)⋯an+mtm(x)φ(n)=1.

Hausdorff dimension of sets with restricted, slowly growing partial quotients

I. J. Good [Proc. Cambridge Philos. Soc. 37 (1941), pp. 199–228] showed that the set of irrational numbers in ( 0 , 1 ) (0,1) whose partial quotients a n a_n tend to infinity is of Hausdorff dimension 1 / 2 1/2 . A number of related results impose restrictions of the type a n ∈ B a_n\in B or a n ≥ f ( n ) a_n\geq f(n) , where B B is an infinite subset of N \mathbb N and f f is a rapidly growing function with n n . We show that, for an arbitrary B B and an arbitrary f f with values in [ min B , ∞ ) [\min B,\infty ) and tending to infinity, the set of irrational numbers in ( 0 , 1 ) (0,1) such that \[ a n ∈ B , a n ≤ f ( n ) for all n ∈ N , and a n → ∞ as n → ∞ a_n\in B,\ a_n\leq f(n)\text { for all $n\in \mathbb N$, and }a_n\to \infty \text { as }n\to \infty \] is of Hausdorff dimension τ ( B ) / 2 , \tau (B)/2, where τ ( B ) \tau (B) is the exponent of convergence of B B .

On the periodicity of an algorithm for p-adic continued fractions

In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), for generating continued fractions in the field of p-adic numbers mathbb Q_p. First of all, we obtain an analogue of the Galois’ Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in mathbb Q_p that have a periodic expansion with period of length 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).

On sums of partial quotients in Hurwitz continued fraction expansions

On sums of partial quotients in Hurwitz continued fraction expansions

On the existence of numbers with matching continued fraction and base b expansions

On the existence of numbers with matching continued fraction and base b expansions

Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers

Let θ=[0;a 1 ,a 2 ,⋯] be the continued fraction expansion of an irrational real number θ∈(0,1). It is well-known that the characteristic Sturmian word of slope θ is the limit of a sequence of finite words (M k ) k≥0 , with M k of length q k (the denominator of the k-th convergent to θ) being a suitable concatenation of a k copies of M k-1 and one copy of M k-2 . Our first result extends this to any Sturmian word s. Let b≥2 be an integer. Our second result gives the continued fraction expansion of any real number ξ whose b-ary expansion is a Sturmian word s over the alphabet {0,b-1}. This extends a classical result of Böhmer who considered only the case where s is characteristic. As a consequence, we obtain a formula for the irrationality exponent of ξ in terms of the slope and the intercept of s.

Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

AbstractJoint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells, batteries, and supercapacitors. To increase the reliability of time–frequency analysis, a theoretical correlation between frequency‐domain stationary analysis and time‐domain transient analysis is urgently required. The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional‐order models to integer‐order models and from high‐ to low‐order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process. The following work has been carried out: (i) the model‐reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique, respectively; (ii) the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis; (iii) the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured; and (iv) the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium‐ion batteries, supercapacitors, and solid oxide fuel cells. In turn, the numerical validation has demonstrated the powerful function of the joint time–frequency analysis. The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time‐domain transient analysis and frequency‐domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

Principal bundle groupoids, their gauge group and their nerve

We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle gerbe over a Lie groupoid. In that case, the resulting bundle gerbe is isomorphic as a groupoid to a partial quotient of the PB groupoid. We describe the nerves of PB groupoids and their partial quotients, which are simplicial objects in the category of principal bundles. Applying this construction enables us to define the inner transformation group of the nerve of a partial quotient groupoid and to describe the transformations of the corresponding bundle gerbe.

Analogue of Ramanujan’s function k(τ) for the cubic continued fraction

We study the modularity of the function [Formula: see text], where [Formula: see text] is Ramanujan’s cubic continued fraction. It is an analogue of Ramanujan’s function [Formula: see text], where [Formula: see text] is the Rogers–Ramanujan continued fraction. We first prove the modularity of [Formula: see text] and express [Formula: see text] and [Formula: see text] in terms of [Formula: see text]. Subsequently, we find modular equations of [Formula: see text] of level [Formula: see text] for every positive integer [Formula: see text] by using affine models of modular curves. Finally, we demonstrate that the value of [Formula: see text] generates the ray class field over an imaginary quadratic field modulo 2 for some [Formula: see text] in an imaginary quadratic field.

A note on the length of some finite continued fractions

In this paper, based on a 2008 result of Lasjaunias, we compute the lengths of simple continued fractions for some rational numbers whose numerators and denominators are explicitly given.

Eggleston's dichotomy for characterized subgroups and the role of ideals

“Eggleston's dichotomy” is a “one of a kind” unique observation which broadly tells us that the characterized subgroups of the circle group (characterized by a sequence of positive integers (an)) are either countable or of cardinality c depending on the asymptotic behavior of the sequence of the ratios anan−1. One should note that these subgroups are generated by using the notion of usual convergence which is nothing but a special case of the more general notion of ideal convergence for the ideal Fin. It has been recently established that “Eggleston's dichotomy” fails in the case of modified versions of characterized subgroups when the ideal Fin is replaced by the natural density ideal Id, or more generally, by ideals which are now known as simple density and modular simple density ideals. As all the ideals mentioned above are analytic P-ideals, a natural question arises as to whether one can isolate some appropriate property of ideals which enforces the dichotomy or the failure of it. In this article we are able to isolate that particular feature of an ideal and come out with a new class of ideals which we call, “strongly non-translation invariant ideals” (in short snt-ideals). In particular, we are able to establish that for a sequence of positive integers (an), be it arithmetic or arising from the continued fraction expansion of an irrational number:(i)For non-snt analytic P ideals, the size of the corresponding characterized subgroups is always c even if the sequence (an) is b-bounded (i.e. the sequence of the ratios anan−1 is bounded) which implies the breaking down of “Eggleston's dichotomy”.(ii)For snt analytic P ideals, the corresponding characterized subgroups are always countable if the sequence (an) is b-bounded which means “Eggleston's dichotomy” holds.

Shadows of rationals and irrationals: Supersymmetric continued fractions and the super modular group

This paper is an attempt to apply the tools of supergeometry to arithmetic. Supergeometric objects are defined over supercommutative rings of coefficients. We consider an integral ring with two odd variables. In this case the even quantities, such as numbers and continued fractions, are dual integers, having both a classical and a nilpotent part. We refer to the nilpotent part as the “shadow”. We investigate the notions of supersymmetric continued fractions and the orthosymplectic modular group, and make some initial steps toward studying their properties.

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Let ℒ(S 1 ) be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle S 1 by its rational points. We give a complete description of the structure of ℒ(S 1 ) below its smallest accumulation point. To this end, we use digit expansions of points on S 1 , which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of ℒ(S 1 ) is 2. Also we characterize the points on S 1 whose Lagrange numbers are less than 2 in terms of Romik’s digit expansions. Our theorem is the analogue of the celebrated theorem of Markoff on badly approximable real numbers.

Low first-trimester fetal fraction is associated with chronic inflammation in the placenta

Some data suggest an association between abnormal fetal fraction on noninvasive prenatal screening and adverse pregnancy outcomes, including low birthweight, preeclampsia, and preterm birth in the absence of aneuploidy. These findings suggest that abnormal fetal fraction may be associated with placental pathologic processes in early gestation. This study aimed to determine the independent association of fetal fraction on genetic noninvasive prenatal screening with histologic placental types. This was a retrospective cohort study at a single institution in the period between January 2017 and March 2021, including live births at ≥24 weeks for which noninvasive prenatal screening was performed and placental pathology results were available. Results were stratified by trimester of noninvasive prenatal screening. Clinical characteristics were compared by quartile of fetal fraction using chi-square tests. Linear regression was used to model continuous fetal fraction as a function of 3 histologic types representing chronic placental injury-chronic inflammation, maternal vascular malperfusion, and fetal vascular malperfusion. Inverse probability weighting was used to account for selection bias in characteristics of patients with placental pathology examination. A total of 1374 patients had noninvasive prenatal screening in the first trimester and 262 in the second trimester. Preterm birth and hypertensive disorders of pregnancy were most common in the lowest quartile of fetal fraction. Chronic inflammation was associated with a 0.56 percentage point reduction in fetal fraction (95% confidence interval, -0.95 to -0.16), and maternal vascular malperfusion was associated with a 0.48 percentage point reduction in fetal fraction (95% confidence interval, -0.91 to -0.04) in adjusted models. The association with maternal vascular malperfusion was no longer statistically significant after accounting for selection bias in placentas sent for pathologic examination. Second-trimester fetal fraction was not associated with placental pathology. Chronic inflammation is associated with lower first-trimester fetal fraction even after accounting for selection bias. Higher fetal fraction in the second trimester was associated with fetal vascular pathology, although this association was no longer statistically significant after inverse probability weighting to account for selection bias. First-trimester fetal fraction may be a biomarker of adverse outcomes associated with chronic inflammation.

On length of the period of the continued fraction of nd

For a given quadratic irrational [Formula: see text], let us denote by [Formula: see text] the length of the periodic part of the continued fraction expansion of [Formula: see text]. We prove that for every positive integer [Formula: see text], which is not a perfect square, there are infinitely many even integers [Formula: see text], for which the equality [Formula: see text] holds for infinitely many integers [Formula: see text].

Maxima and visibility in involutions

We enumerate involutions according to the joint distribution of left-to-right and right-to-left maxima. From this computation we deduce the distribution of the static “number of visible pairs”, namely, the number of non-adjacent columns in the bargraph representation of a given permutation that are mutually visible to each other. The corresponding generating functions are expressible in terms of formal continued fractions. We obtain the same distribution over the set of involutions avoiding either the pattern 3412 or 4321. The proofs reside on a well-known bijection between involutions and labeled Motzkin paths.

Quasinormal modes of a holonomy corrected Schwarzschild black hole

We analyze the quasinormal modes (QNMs) of a recently obtained solution of a Schwarzschild black hole (BH) with corrections motivated by Loop Quantum Gravity (LQG). This spacetime is regular everywhere and presents the global structure of a wormhole, with a minimal surface whose radius depends on a LQG parameter. We focus on the investigation of massless scalar field perturbations over the spacetime. We compute the QNMs with the WKB approximation, as well as the continued fraction method. The QNM frequency orbits, for $l=0$ and $n>0$, where $l$ and $n$ are the multipole and overtone numbers, respectively, are self-intersecting, spiraling curves in the complex plane. These orbits accumulate to a fixed complex value corresponding to the QNMs of the extremal case. We obtain that, for small values of the LQG parameter, the overall damping decreases as we increase the LQG parameter. Moreover the spectrum of the quantum corrected black hole exhibits an oscillatory pattern, which might imply in the existence of QNMs with vanishing real part. This pattern suggests that the limit $n\rightarrow \infty$ for the real part of the QNMs is not well-defined, what differs from Schwarzschild's case. We also analyze the time-domain profiles for the scalar perturbations, showing that the LQG correction does not alter the Schwarzschild power-law tail. We compute the fundamental mode from the time profile by means of the Prony method, obtaining excellent agreement with the two previously mentioned methods.