Farey Sequence Research Articles

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.

Ubiquity and the Farey graph

We construct a countable planar graph which, for any two vertices u,v and any integer k≥1, contains k edge-disjoint order-compatible u–v paths but not infinitely many. This graph has applications in Ramsey theory, in the study of connectivity and in the characterisation of the Farey graph.

A sequence of algebraic integer relation numbers which converges to 4

Let α∈R and letA=[1101]andBα=[10α1]. The subgroup Gα of SL2(R) is a group generated by the matrices A and Bα. In this paper, we investigate the property of the group Gα. We construct a generalization of the Farey graph for the subgroup Gα. This graph determines whether the group Gα is a free group of rank 2. More precisely, the group Gα is a free group of rank 2 if and only if the graph is tree. In particular, we show that if 1/2 is a vertex of the graph, then Gα is not a free group of rank 2. Using this, we construct a sequence of real numbers so that the sequence converges to 4 and each number has the corresponding group that is not a free group of rank 2. It turns out that the real numbers are algebraic integers.

105.02 Some essential properties of Farey series in one Pic(k)ture

105.02 Some essential properties of Farey series in one Pic(k)ture

Farey sequence and Graham's conjectures

Let Fn be the Farey sequence of order n. For S⊆Fn we let Q(S)={x/y:x,y∈S,x≤y and y≠0}. We show that if Q(S)⊆Fn, then |S|≤n+1. Moreover, we prove that in any of the following cases: (1) Q(S)=Fn; (2) Q(S)⊆Fn and |S|=n+1, we must have S={0,1,12,…,1n} or S={0,1,1n,…,n−1n} except for n=4, where we have an additional set {0,1,12,13,23} for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan.

Computational analysis of a 9D model for a small DRG neuron.

Small dorsal root ganglion (DRG) neurons are primary nociceptors which are responsible for sensing pain. Elucidation of their dynamics is essential for understanding and controlling pain. To this end, we present a numerical bifurcation analysis of a small DRG neuron model in this paper. The model is of Hodgkin-Huxley type and has 9 state variables. It consists of a Nav1.7 and a Nav1.8 sodium channel, a leak channel, a delayed rectifier potassium, and an A-type transient potassium channel. The dynamics of this model strongly depend on the maximal conductances of the voltage-gated ion channels and the external current, which can be adjusted experimentally. We show that the neuron dynamics are most sensitive to the Nav1.8 channel maximal conductance ([Formula: see text]). Numerical bifurcation analysis shows that depending on [Formula: see text] and the external current, different parameter regions can be identified with stable steady states, periodic firing of action potentials, mixed-mode oscillations (MMOs), and bistability between stable steady states and stable periodic firing of action potentials. We illustrate and discuss the transitions between these different regimes. We further analyze the behavior of MMOs. As the external current is decreased, we find that MMOs appear after a cyclic limit point. Within this region, bifurcation analysis shows a sequence of isolated periodic solution branches with one large action potential and a number of small amplitude peaks per period. For decreasing external current, the number of small amplitude peaks is increasing and the distance between the large amplitude action potentials is growing, finally tending to infinity and thereby leading to a stable steady state. A closer inspection reveals more complex concatenated MMOs in between these periodic MMO branches, forming Farey sequences. Lastly, we also find small solution windows with aperiodic oscillations which seem to be chaotic. The dynamical patterns found here-as consequences of bifurcation points regulated by different parameters-have potential translational significance as repetitive firing of action potentials imply pain of some form and intensity; manipulating these patterns by regulating the different parameters could aid in investigating pain dynamics.

Optimal networks for exact controllability

The exact controllability can be mapped to the problem of maximum algebraic multiplicity of all eigenvalues. In this paper, we focus on the exact controllability of deterministic complex networks. First, we explore the eigenvalues of two famous networks, i.e. the comb-of-comb network and the Farey graph. Due to their special structure, we find that the eigenvalues of each network are mutually distinct, showing that these two networks are optimal networks with respect to exact controllability. Second, we study how to optimize the exact controllability of a deterministic network. Based on the spectral graph theory, we find that reducing the order of duplicate sets or co-duplicate sets which are two special vertex subsets can decrease greatly the exact controllability. This result provides an answer to an open problem of Li et al. [X. F. Li, Z. M. Lu and H. Li, Int. J. Mod. Phys. C 26, 1550028 (2015)]. Finally, we discuss the relation between the topological structure and the multiplicity of two special eigenvalues and the computational complexity of our method.

On tunnel numbers of a cable knot and its companion

Let K be a nontrivial knot in S3 and t(K) its tunnel number. For any (p≥2,q)-slope in the torus boundary of a closed regular neighborhood of K in S3, denoted by K⋆, it is a nontrivial cable knot in S3. Though t(K⋆)≤t(K)+1, Example 1.1 in Section 1 shows that in some case, t(K⋆)≤t(K). So it is interesting to know when t(K⋆)=t(K)+1.After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K⋆ and its companion K, t(K⋆)≥t(K); (2) if either K admits a high distance Heegaard splitting or p/q is far away from a fixed subset in the Farey graph, then t(K⋆)=t(K)+1. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

Canonical threefold singularities with a torus action of complexity one and $k$-empty polytopes

We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold singularities. Our classification relies on lattice point emptiness of certain polytopes with rational vertices. We show that in dimension two, such polytopes are sporadic or are given by Farey sequences. We finally present the Cox ring iteration tree of the classified singularities.

Counting spanning trees of a type of generalized Farey graphs

The Farey graph Fn is derived from the famous Farey sequence and it is a small-world network with a connectivity distribution decaying exponentially. By using the Matrix-Tree theorem, Zhang et al. (2012) obtained the exact formula of the number of spanning trees of Fn. In this paper, by using the electrical network method, we consider a type of generalized Farey graphs and give the exact solution for the number of spanning trees of these generalized Farey graphs, which generalizes some previous results about the Farey graphs.

On sidon sequences of farey sequences, square roots and reciprocals

Abstract In this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that Sidon sequences over them have cardinality of between c 1 N 3 / 4 log N $\begin{array}{} \displaystyle c_1\frac{N^{3/4}} {\log{N}} \end{array}$ and c 2 N 3/4, c 3 N, and c 4 N log log N log N . $\begin{array}{} \displaystyle c_4 \frac{N\log{\log{N}}}{\log{N}}. \end{array}$

Limiting distribution of eigenvalues in the large sieve matrix

The large sieve inequality is equivalent to the bound \lambda_1 \leq N + Q^2-1 for the largest eigenvalue \lambda_1 of the N \times N matrix A^*A , naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is N \asymp Q^2 . Based on his numerical data Ramaré conjectured that when N \sim \alpha Q^2 as Q \to \infty for some finite positive constant \alpha , the limiting distribution of the eigenvalues of A^*A , scaled by 1/N , exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of A^*A as Q\to\infty . Previously only the second moment was known, due to Ramaré. Furthermore, we obtain an explicit description of the moments of the limiting distribution, and establish that they vary continuously with \alpha . Some of the main ingredients in our proof include the large-sieve inequality and results on n -correlations of Farey fractions.

Интервалы малой длины, содержащие алгебраическое число заданной высоты

Рациональные числа распределены равномерно, хотя расстояния между соседними рациональными числами в последовательности Фарея могут сильно разниться. Для алгебраических чисел это свойство не выполняется. В 2013 г. Д. Коледа [6, 7] нашел функцию плотности распределения действительных алгебраических чисел любой степени при их естественном упорядочивании.Можно доказать, что количество действительных алгебраических чисел $ \alpha $ степени $n$ и высоты $H(\alpha ) \le Q$ асимптотически равно $c_{1}(n)Q^{n+1}$. Недавно было доказано, что существуют интервалы длины $Q^{- \gamma }, \gamma >1$, свободные от алгебраических чисел $ \alpha , H( \alpha ) \le Q$, однако уже при $0 \le \gamma <1$ их не менее чем $c_{2}(n)Q^{n+1- \gamma }$.В работе показано, что специальные интервалы длины $Q^{- \gamma }$ и при больших $ \gamma $ могут содержать алгебраические числа, однако их количество не превосходит $c_{3}Q^{n+1- \gamma }$. Ранее аналогичный результат был получен А. Гусаковой \cite{Gus15} лишь для случая $\gamma = \frac{3}{2}$.

Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle

This paper shows that the inverse chirp z-transform (ICZT), which generalizes the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, can also be used with chirp contours that perform partial or multiple revolutions on the unit circle. This is done as a special case of the ICZT, which in algorithmic form has the same computational complexity as the IFFT, i.e., O(n log n). Here we evaluate the ICZT algorithm for chirp contours on the unit circle and show that it is numerically accurate for large areas of the parameter space. The numerical error in this case depends on the polar angle between two adjacent contour points. More specifically, the error profile for a transform of size n is determined by the elements of the Farey sequence of order n − 1. Furthermore, this generalization allows the use of non-orthogonal frequency components, thus lifting one of the main restrictions of the IFFT.

A note on friezes of type [formula omitted

A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and Jørgenson introduced friezes of type Λp which correspond to p-angulations of a polygon. In this paper we explore the connection between these two types of friezes; in particular we show that the friezes of type Λp for p=4 and p=6 contain integral friezes within them. We also consider the relationships with Farey graphs.

Extraction of Farey Sequence from Stern-Brocot Sequence

In this paper we define and discuss the properties of Farey Sequence and Stern - Brocot Tree and the principal aim of this paper is to extract Farey sequence from Stern - Brocot Tree .

Combinatorics of $k$-Farey graphs

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the k-Farey graphs ℱk and ℱ≤k, two natural variants of the Farey graph ℱ in which we relax the edge condition to indicate intersection number =k or ≤k, respectively. The former, ℱk, is disconnected when k>1. In fact, we find that the number of connected components is infinite if and only if k is not a prime power. Moreover, we find that each component of ℱk is a quasitree (in fact, a tree when k is even) and Aut(ℱk) is uncountable for k>1. As for ℱ≤k, Agol obtained an upper bound of 1+ min{p:p is a prime>k} for both chromatic and clique numbers, and observed that this is an equality when k is either one or two less than a prime. We add to this list the values of k that are three less than a prime equivalent to 11(mod12), and we show computer-assisted computations of many values of k for which equality fails.

-DEFORMED RATIONALS AND -CONTINUED FRACTIONS

We introduce a notion of$q$-deformed rational numbers and$q$-deformed continued fractions. A$q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the$q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the$q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties,$q$-deformation of the Farey graph, matrix presentations and$q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

Geodesic Gaussian Integer Continued Fractions

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph F, whose vertices are reduced Gaussian rationals in Q∞(i) and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in H3 where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of H3 by Farey octahedrons. A natural distance ϱ on Q∞(i) is also recalled, where ϱ(∞,w)=n is the least number of edges in F from ∞ to w∈Q(i), where n is called the generation of w and a relevant path in F is called a geodesic expansion for w. The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation n−1, n, and n+1. Subsequently, it is seen that there can be at most four Farey neighbors of generation n−1 in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any w is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

FAREY SEQUENCES MAP PLAYABLE NODES ON A STRING

AbstractNatural harmonics, i.e. partials and their harmonic series, may be isolated on a vibrating string by lightly touching specific points along its length. In addition to the two endpoints, stationary nodes for a given partial n present themselves at n − 1 locations along the string, dividing it into n parts of equal length. It is not the case, however, that touching any one of these nodes will necessarily isolate the nth partial and its integer multiples. The subset of nodes that will activate the nth partial (termed playable nodes by the authors) may be derived by following a mathematically predictable pattern described by so-called Farey sequences. The authors derive properties of these sequences and connect them to physical phenomena. This article describes various musical applications: locating single natural harmonics, forming melodies of neighbouring harmonics, sounding multiphonic aggregates, as well as predicting the relative tuneability of just intervals.