Theory Of Continued Fractions Research Articles

R-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices

AbstractWe study the $R_\beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature $\beta$ associated to a family of nearest-neighbor matrices $(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $\beta$ , and (ii) the existence of weak limits as $\beta \to \infty$ . Some application examples are revised to put in context the main results of this work.

Continued fractions related to a group of linear fractional transformations

Abstract There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=z+2 . This group is the unique subgroup of the modular group PSL ( 2 , Z ) {\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.

Large algebraic integers

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than [Formula: see text]. An integral ideal is said large if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.

An involution on restricted Laguerre histories and its applications

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Françon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement.

Model of money income diffusion in the European integration context

A model of money income diffusion is constructed taking into account the processes of social comparison and the development of the income formation spatial structure. To implement such model, the analytical theory of continued fractions is used and the corresponding differential equation solution is found in the form of a formal functional continued fraction. The values of the approach fractions of a continuous fraction describing the dynamics of changes in the level of money income give an approximation of the real values with almost predetermined, arbitrarily high accuracy. This allowed us to qualitatively describe the general dynamics of the money income diffusion process.

Application of Continued Fraction in Pell's Equation

This paper uses a continued fraction to explain various approaches to solving Diophantine equations. It first examines the fundamental characteristics of continued fractions, such as convergent and approximations to real numbers. Using continued fractions, we can solve the Pell's equation. Certain theorems have also been discussed for how to expand quadratic irrational integers into periodic continued fractions. Finally, the relationship between convergent and best approximations and use of continuous fraction in calendar construction has-been investigated. The analytical theory of continued fractions is a significant generalization of continued fractions and represents a large field for current and future research.

Perrin numbers that are concatenations of two repdigits

Let (P_n)_{nge 0} be the sequence of Perrin numbers defined by ternary relation P_0=3 , P_1=0 , P_2=2 , and P_{n+3}=P_{n+1}+P_n for all nge 0 . In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two repeated digit numbers.

Continued Fractions and Probability Estimations in Shor’s Algorithm: A Detailed and Self-Contained Treatise

Shor’s algorithm for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books), filling the gap to allow a complete comprehension of Shor’s algorithm. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor.

Density of Oscillating Sequences in the Real Line

Abstract In this paper we study the density in the real line of oscillating sequences of the form ( g ( k ) ⋅ F ( k α ) ) k ∈ ℕ , {\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}}, where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1. More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

DEH Scheme DGTD-Based Transient Modeling Approach for the Cole–Cole Dispersive Media Using Tustin’s Method

The Cole–Cole model can simulate the electromagnetic properties of biological tissues more accurately in a wide frequency band. The current simulation of wave propagation in the Cole–Cole dispersive media usually uses the finite-difference time-domain (FDTD) method, but its requirements in small grid size and time step interval for curved objects limit its application in calculating multiscale complex biological tissues. The discontinuous Galerkin time-domain (DGTD) method is more suitable for simulating multiscale complex targets because it can use unstructured grids and high-order basis functions. Therefore, this article proposes a transient modeling approach for the Cole–Cole dispersive media based on DGTD method, in which the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> transform is performed on the Cole–Cole model, and the fractional power terms are approximated to integer power polynomials according to the continued fraction theory. Moreover, in order to avoid the frequency-dependent permittivity in Maxwell’s equations, the DEH scheme is constructed and a weak form equation for the constitutive relationship between D and E is created. Maxwell’s equations and the constitutive relationship are updated by the leap-frog time-stepping method. The accuracy and ability of the new method are verified by simulating the plane wave propagation in various Cole–Cole media.

Growth in Chevalley groups relatively to parabolic subgroups and some applications

Given a Chevalley group ${\mathbf G}(q)$ and a parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is a certain growth of $A$ relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also, we study a question about the intersection of $A^n$ with parabolic subgroups $P$ for large $n$. We apply our method to obtain some results on a modular form of Zaremba’s conjecture from the theory of continued fractions, and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than $1/2$.

Multi-dimensional Kronecker sequences with a small number of gap lengths

Abstract Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension d ∈ {2, 3} by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely N ∈ ℕ. Our proof relies on simple arguments from the theory of continued fractions.

Metrical properties for continued fractions of formal Laurent series

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let An(x) be the nth partial quotient of the continued fraction expansion of x in the field of formal Laurent series. We consider the sets of x such that deg⁡An+1(x)+⋯+deg⁡An+k(x)≥Φ(n) holds for infinitely many n and for all n respectively, where k≥1 is an integer and Φ(n) is a positive function defined on N. We determine the size of these sets in terms of Haar measure and Hausdorff dimension.

Necessary and sufficient conditions for convergence of integer continued fractions

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.

Repdigits as Product of Terms of k-Bonacci Sequences

For any integer k≥2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F−(k−2)(k)=⋯=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

Sets of Dirichlet non-improvable numbers with certain order in the theory of continued fractions * *This work was supported by NSFC 11571127, 11871208.

Let [a 1(x), a 2(x), …, a n (x), …] be the continued fraction expansion of x ∈ [0, 1) and q n (x) be the denominator of the nth convergent. The study of relative growth rate of the product of partial quotients a n+1(x)a n (x) compared with q n (x) originated from the improvability of Dirichlet’s theorem. In this note, we prove that, for any 0 ⩽ α ⩽ β ⩽ +∞, the Hausdorff dimension of the following set is or according to α > 0 or α = 0, respectively. This result extends an earlier result of Huang and Wu as well as gives insights on the metric theory of Dirichlet non-improvable sets.

Fourier coefficients of level 1 Hecke eigenforms

Lehmer's 1947 conjecture on whether $\tau(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $\tau(n)$. For example, among the odd numbers $\alpha$ such that $|\alpha|<99$, we determine that $$\tau(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $\tau(n) \neq -|\alpha|$ and that $\tau(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $\Delta E_4 = \sum_{n = 1}^{\infty} \tau_{16}(n)q^n$, we show that \begin{align*}\tau_{16}(n) \notin &\{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} & \quad \quad \quad \quad \quad \cup \{-33,-55,-59,-67,-89,-91\}.\end{align*} Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.

Periodic continued fractions over $S$-integers in number fields and Skolem’s $p$-adic method

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic polynomial in ${\mathcal O}[x]$. We show that PCFs $P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({\mathcal O})$ are related to writing matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the ${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(\alpha_n)$, $\alpha_{n}:=2\cos(2\pi/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf Z}[\alpha_n]$. We want to find the PCFs of $\alpha_{n+1}$ over ${\mathcal O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on $V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{\alpha_{n+1}, -\alpha_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the ${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for $p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.

ANALYSIS AND VERIFICATION OF THE OPERATION OF THE STABILIZING ROBUST CONTROLLER'S ELECTRICAL CIRCUIT OF THE ROTOR FLUX-LINKAGE CONTROL SYSTEM

On the basis of the classical theory of robust control, the theory of stability and the theory of continued fractions, the linearized mathematical model, the procedure for calculating and designing an analog electrical circuit of the H∞-suboptimal robust controller of the flux-linkage control system of the rotor of an asynchronous electric drive, protected by Ukrainian Patent no.137157, were built. The circuit contains three operational amplifiers, several resistors and capacitors. The proposed mathematical model, the procedure for calculating and choosing the parameters of the resistors and capacitors of the circuit take into account the random variations of the object and the controller within the specified boundaries. An analytical relationship between the transfer function coefficients and the resistances of the resistors and the capacitances of the capacitors of the controller's electrical circuit was established. On a specific example, by calculation according to the developed algorithm, the limiting values of the tolerances of the circuit's resistances of the resistors and capacitances of the capacitors were identified and these values were selected with a margin from the standard series. Analysis of noise filtering by the electrical circuit of the controller in the Multisim package shows its insensitivity to noise spreads within fairly wide boundaries.

K−Fibonacci numbers close to a power of 2

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence whose first k terms are 0, . , 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, by using a lower bound to linear forms in logarithms of algebraic numbers due to Matveev and some argument of the theory of continued fractions, we find all the members of F (k) which are close to a power of 2. This paper continues and extends the previous work of Chern and Cui which investigated the Fibonacci numbers close to a power of 2.