The sequence formed by the fractional tune resonances up to order n exactly coincides with the Farey sequence, F n . In circular accelerators, it is important to place the tunes in regions free of resonances. From numerical evaluations, it is well known that the largest resonance-free gaps happen next to low order resonances. In this report, we provide a mathematical demonstration of this statement for any order n ∈ N showing that the resonance gaps are strictly ordered according to the order of the adjacent lowest order resonance up to some given value. We present a connection between the resonance sequence and one of the most important unsolved problems in mathematics, the Riemann hypothesis (RH). If RH is true, it implies that the resonance sequence is significantly more regularly spaced than if it were built at random; however, this has not yet been demonstrated. A new analytical estimate of the number of resonance lines up to order n in the two-dimensional tune diagram is also derived and compared to previous estimates.

On the stable norm of slit tori and the Farey sequence

This paper concerns the pair correlation of Farey fractions within an interval I⊂[0,1], specifically focusing on the fractions whose denominators are square-free numbers in an arithmetic sequence. We prove that the limiting pair correlation function of the sequence of such Farey fractions exists and is independent of the interval I.

ABSTRACTIn this paper, with inspiration of the definition of Bernstein basis functions and their recurrence relation, we give construction of a new word family that we refer Bernstein‐based words. By classifying these special words as the first and second kinds, we investigate their some fundamental properties involving periodicity and symmetricity. Providing schematic algorithms based on tree diagrams, we also illustrate the construction of the Bernstein‐based words. For their symbolic computation, we also give computational implementations of Bernstein‐based words in the Wolfram Language. By executing these implementations, we present some tables of Bernstein‐based words and their decimal equivalents. In addition, we present black–white and four‐colored patterns arising from the Bernstein‐based words with their potential applications in computational science and engineering. We also give some finite sums and generating functions for the lengths of the Bernstein‐based words. We show that these functions are of relationships with the Catalan numbers, the centered ‐gonal numbers, the Laguerre polynomials, certain finite sums, and hypergeometric functions. We also raise some open questions and provide some comments on our results. Finally, we investigate relationships between the slopes of the Bernstein‐based words and the Farey fractions.

Given an imaginary quadratic number field K with ring of integers [Formula: see text], we are interested in the asymptotic distance to nearest neighbor (or gap) statistic of complex fractions [Formula: see text], with [Formula: see text] and [Formula: see text], as [Formula: see text]. Reformulating this problem in a homogeneous dynamical setting, we follow the approach of J. Marklof for real Farey fractions with several variables and adapt an equidistribution result in the real 3-dimensional hyperbolic space of J. Parkkonen and F. Paulin to derive the existence of a probability measure describing this asymptotic gap statistic. We obtain an integral formula for the associated cumulative distribution function, and use geometric arguments to find an explicit estimate for its tail distribution in the cases of Gaussian and Eisenstein fractions.

This paper presents and examines a discrete-time predator–prey model of the Leslie type, integrating a Holling type IV functional response for analysis. The mathematical analysis succinctly identifies fixed points and evaluates their local stability within the model. The study employs the normal form approach and bifurcation theory to explore codimension-one and two bifurcation behaviors for this model. The primary conclusions are substantiated by a combination of rigorous theoretical analysis and meticulous computational simulations. Additionally, utilizing fractal basin boundaries, periodicity variations, and Lyapunov exponent distributions within two-parameter spaces, we observe a mode-locking structure akin to Arnold tongues. These periods are arranged in a Farey tree sequence and embedded within quasi-periodic/chaotic regions. These findings enhance comprehension of bifurcation cascade emergence and structural patterns in diverse biological systems with discrete dynamics.

Abstract Semiconductor quantum-well mode-locked lasers with time-delayed optical feedback are versatile dynamical systems for applications but also paramount examples to explore the universal properties of non-linear systems with two competing frequencies. For rational ratios between the feedback delay-time and the laser-round-trip time, second harmonic, third harmonic, and fourth harmonic combs occur. With this contribution, we explore the parameter dependence of different dynamic regimes which range from frequency-locking at Farey fractions to complex quasi-periodic pulse trains with a focus on the impact of feedback phase, strength, and delay length. We characterize the emission in the temporal, spectral, and radio-frequency domain. Our numerical approach is based on traveling-wave equations and is able to explain experimental data over a large parameter range. We show that the predictive power of the model allows for on demand sub-THz harmonic comb generation. We achieve a fine-tuning range of 450 MHz around the fundamental beat note of 13.6 GHz and a continuous pulse-width tuning from 7 to 26 ps solely by delay-time control.

Algorithm applications on graphs are intensively researched. Graph theory systematizes complex and difficult problems and algorithms provide fast and clear solutions, which increases interest in the discipline. The Floyd-Warshall algorithm determines the shortest paths between all the vertices in a graph. In this paper, we consider the Floyd-Warshall algorithm on the Farey graph defined in a non-Euclidean hyperbolic space. A Farey graph with 15 edges and 9 vertices is constructed and the shortest paths from all vertices to other vertices are detected. By defining the weight between consecutive vertices, the shortest paths between the vertices are measured in terms of the number of steps.

Metric properties of Farey series and connections to the Riemann hypothesis

Abstract We prove extreme value laws for cusp excursions of the horocycle flow in the case of surfaces of constant negative curvature. The key idea of our approach is to study the hitting time distribution for shrinking Poincaré sections that have a particularly simple scaling property under the action of the geodesic flow. This extends the extreme value law of Kirsebom and Mallahi-Karai (2022 arXiv:2209.07283) for cusp excursions for the modular surface. Here we show that the limit law can be expressed in terms of Hall’s formula for the gap distribution of the Farey sequence.

Frequency combs show various applications in molecular fingerprinting, imaging, communications, and so on. In the terahertz frequency range, semiconductor-based quantum cascade lasers (QCLs) are ideal platforms for realizing the frequency comb operation. Although self-started frequency comb operation can be obtained in free-running terahertz QCLs due to the four-wave mixing locking effects, resonant/off-resonant microwave injection, phase locking, and femtosecond laser based locking techniques have been widely used to broaden and stabilize terahertz QCL combs. These active locking methods indeed show significant effects on the frequency stabilization of terahertz QCL combs, but they simultaneously have drawbacks, such as introducing large phase noise and requiring complex optical coupling and/or electrical circuits. Here, we demonstrate Farey tree locking of terahertz QCL frequency combs under microwave injection. The frequency competition between the Farey fraction frequency and the cavity round-trip frequency results in the frequency locking of terahertz QCL combs, and the Farey fraction frequencies can be accurately anticipated based on the downward trend of the Farey tree hierarchy. Furthermore, dual-comb experimental results show that the phase noise of the dual-comb spectral lines is significantly reduced by employing the Farey tree locking method. These results pave the way to deploying compact and low phase noise terahertz frequency comb sources.

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to an apparently new problem on oscillations of characters on differences between Farey fractions is given. Other applications include a modular analogue of a multiplicative hybrid problem of Iwaniec and Sárközy (1987) and the solvability of some prime type equations with constraints.

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner’s lambda lengths and SL2-tilings. In particular, we prove a three-dimensional version of the Ptolemy relation, and generalise results of Short to classify tame SL2-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

A discrete SIS epidemic system with disease incubation period and saturated contact rate is formulated and studied. The qualitative properties of both the disease-free and endemic equilibria in hyperbolic and non-hyperbolic cases are analysed. All potential local bifurcations of the two equilibria are explored. The direction, stability, and explicit approximate expressions for each type of bifurcation are derived. All possible rational rotation numbers on the family of closed-invariant curves generated through the Neimark–Sacker bifurcation are derived and arranged based on the five-level Farey sequence tree. The 1:5 weak resonance on the invariant curve is analysed. By the study of contact bifurcation, the global dynamic behaviour is discussed. Based on the epidemiological background, the significance of each bifurcation is explained. A large number of numerical simulation and numerical continuation examples support the theoretical analyses. At last, the model is used to fit the number of Hepatitis A infections in Guangdong Province, China, from 2018 to 2024. After analysing the residuals, conducting goodness-of-fit test, Student's t-test and Overfitting test, short-term forecasting for the number of infections is carried out.

New analytical formulas are derived for the rank and the local discrepancy of Farey fractions. The new rank formula is applicable to all Farey fractions and involves sums of a lower order compared to the searched one. This serves to establish a new unconditional estimate for the local discrepancy of Farey fractions that decrease with the order of the Farey sequence. This estimate improves the currently known estimates. A new recursive expression for the local discrepancy of Farey fractions is also given. A second new unconditional estimate of the local discrepancy of any Farey fraction is derived from a sum of the Mertens function, again, improving the currently known estimates.

Abstract We study the relative $\mathrm {SU}(2,1)$ -character varieties of the one-holed torus, and the action of the mapping class group on them. We use an explicit description of the character variety of the free group of rank two in $\mathrm {SU}(2,1)$ in terms of traces, which allow us to describe the topology of the character variety. We then combine this description with a generalization of the Farey graph adapted to this new combinatorial setting, using ideas introduced by Bowditch. Using these tools, we can describe an open domain of discontinuity for the action of the mapping class group which strictly contains the set of convex cocompact characters, and we give several characterizations of representations in this domain.

In this work, we investigate the dynamics of a discrete-time prey-predator model considering a prey reproductive response as a function of the predation risk, with the prey population growth factor governed by two parameters. The system can evolve toward scenarios of mutual or only of predators extinction, or species coexistence. We analytically show all different types of equilibrium points depending on the ranges of growth parameters. By numerical study, we find the occurrence of quasiperiodic, chaotic, and hyperchaotic behaviors. Our analytical results are corroborated by the numerical ones. We highlight Arnold tongue-like periodic structures organized according to the Farey sequence, as well as pairs of twin shrimps connected by two links. The mathematical model captures two possible prey responsive strategies, decreasing or increasing the reproduction rate under predatory threat. Our results support that both strategies are compatible with the populations coexistence and present rich dynamics.

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from 1 to b – 1 is equal to the sum of the numbers from b + 1 to b + r for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

Tutte polynomial for a small world connected copies of Farey graphs