Sequence Of Quotients Research Articles

A conjecture of Zagier and the value distribution of quantum modular forms

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when J_{K,0} denotes the colored Jones polynomial of a knot K , Zagier’s modularity conjecture describes the asymptotics of the quotient J_{K,0}(e^{2 \pi i \gamma(x)}) / J_{K,0}(e^{2 \pi i x}) as x \to \infty along rationals with bounded denominators, where \gamma \in \mathrm{SL}(2,\mathbb{Z}) . This problem is most accessible for the figure-eight knot 4_{1} , where the colored Jones polynomial has a simple explicit expression in terms of the q -Pochhammer symbol. Zagier also conjectured that the function h(x) = \log (J_{4_1,0}(e^{2 \pi i x}) / J_{4_1,0}(e^{2 \pi i /x})) can be extended to a function on \mathbb{R} which is continuous at irrationals. In the present paper, we prove Zagier’s continuity conjecture for all irrationals for which the sequence of partial quotients in the continued fraction expansion is unbounded. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of h , uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of \log J_{4_1,0}(e^{2 \pi i x}) , when x ranges over all reduced rationals in (0,1) with denominator at most N , as N \to \infty , thereby confirming a conjecture of Bettin and Drappeau.

Doubly sequenceable groups

AbstractGiven a sequence in a finite group with , let be the sequence of consecutive quotients of defined by and for . We say that is doubly sequenceable if there exists a sequence in such that every element of appears exactly twice in each of and . We show that if a group is abelian, odd, sequenceable, R‐sequenceable, or terraceable, then it is doubly sequenceable. We also show that if is an odd or sequenceable group and is an abelian group, then is doubly sequenceable.

The Thue–Morse continued fractions in characteristic 2 are algebraic

Let $a, b$ be distinct, non-constant polynomials in $\mathbb F_2 [z]$. Let $\xi_{a, b}$ be the power series in $\mathbb F_2((z^{-1}))$ whose sequence of partial quotients is the Thue–Morse sequence over $\{a, b\}$. We establish that $\xi_{a,b}$ is algebra

Reflexive group topologies on the integers generated by sequences

We establish reflexivity of a family of group topologies on Z generated by sequences, extending results of Gabriyelyan [21]. More precisely, for a T-sequence b=(bn)n∈N of integers and the associated topology Tb on Z (in the sense of [28]), we prove that (Z,Tb) is reflexive whenever the ratios qn=bn+1bn are integers and diverge to ∞ (whereas the same conclusion was obtained in [21] under the more stringent condition ∑n≥11qn<∞). The character group of (Z,Tb) is the subgroup ttb(T):={x+Z∈T:bnx+Z→0} of the torus T. If the ratios qn are integers and for some ℓ∈N the sequence of quotients (bn+ℓbn) diverges to ∞, then ttb(T) with the compact-open topology is reflexive.

On the distribution of partial quotients of reduced fractions with fixed denominator

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions a / N a/N , where N N is fixed and a a runs through the set of mod N N residue classes which are coprime with N N . Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators N N . We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauß–Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient.

On the asymptotic behaviour of Sudler products for badly approximable numbers

Given a badly approximable number α, we study the asymptotic behaviour of the Sudler product defined by PN(α)=∏r=1N2|sin⁡πrα|. We show that liminfN→∞PN(α)=0 and limsupN→∞PN(α)/N=∞ whenever the sequence of partial quotients in the continued fraction expansion of α exceeds 7 infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value 7 is optimal, even when restricting α to be a quadratic irrational, which gives a negative answer to a question of the latter authors.

On the Lévy constants of Sturmian continued fractions

The L\'evy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a L\'evy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a L\'evy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of L\'evy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval $[\log ((1+\sqrt 5)/2 ), + \infty)$.

Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the viscosity corresponding to the 2D incompressible Navier–Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier–Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will be well behaved. Furthermore, our analysis can be extended to analyze the sensitivity of the 2D Euler equations to a viscous regularization. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier–Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the viscosity.

ON THE LIMITS OF SOME p-ADIC SCHNEIDER CONTINUED FRACTIONS

In the present paper, we first generalize some convergence results for continued fractions given in real domain and p-adic domain. However, we prove the transcendence of a p-adic number given by it's Schneider continued fractions, such that the sequence of partial quotients is a Thue-Morse sequence.

Mini-Workshop: Superexpanders and Their Coarse Geometry

It is a deep open problem whether all expanders are superexpanders. In fact, it was already a major challenge to prove the mere existence of superexpanders. However, by now, some classes of examples are known: Lafforgue’s expanders constructed as sequences of finite quotients of groups with strong Banach property (T), the examples coming from zigzag products due to Mendel and Naor, and the recent examples coming from group actions on compact manifolds. The methods which are used to construct superexpanders are typically functional analytic in nature, but also rely on arguments from geometry and combinatorics. Another important aspect of the study of superexpanders is their (coarse) geometry, in particular in order to distinguish them from each other. The aim of this workshop was to bring together researchers working on superexpanders and their coarse geometry from different perspectives, with the aim of sharing expertise and stimulating new research.

Credibility Evaluation of Operational Test Simulation Data under Small Sample Circumstance

It is highly necessary to study how to analyze the reliability of simulation data under small sample circumstance when the number of times live operational test is strictly limited. Based on the analysis of existing test ideas and methods, combined with the characteristics of sequence statistics of uniform distribution, a new method of consistency verification is proposed by constructing the variable-scale differential quotient sequence statistic. The research shows that this method is not limited by the sample size, and the credibility of the simulation data can be quickly judged by MATLAB programming.

Limit theorems for sub-sums of partial quotients of continued fractions

This paper studies the limit behaviour of sums of the form Tn(x)=∑1≤j≤nckj(x),(n=1,2,...)where (cj(x))j≥1 is the sequence of partial quotients in the regular continued fraction expansion of the real number x and (kj)j≥1 is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational α, the sequence (kjα)j≥1 is uniformly distributed modulo one and (kj)j≥1 is good universal. It was observed by the second author, for this class of sequences (kj)j≥1 that we have limn→∞Tn(x)n=+∞ almost everywhere with respect to Lebesgue measure. The case kj=j(j=1,2,…) is classical and due to A. Ya. Khinchin. Building on work of H. Diamond, Khinchin, W. Philipp, L. Heinrich, J. Vaaler and others, in the special case where kj=j(j=1,2,…,) we examine the asymptotic behaviour of the sequence (Tn(x))n≥1 in more detail.

New algorithms for modular inversion and representation by the form x2+ 3xy + y2

We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.

Slow increasing functions and the largest partial quotients in continued fraction expansions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

Transcendence of Thue–Morse p-Adic Continued Fractions

It has been proved that real numbers defined as a limit of continued fractions with a Thue–Morse sequence as the sequence of positive integer partial quotients are transcendental. The same holds for more general sequences of partial quotients. Moreover, transcendence results have been proved for p-adic numbers defined by Hensel development. Here, we tackle the transcendence question in case of p-adic numbers defined as a limit of continued fractions by proving a similar result as the first one.

ParQuoSCI: Pseudorandom Partial Quotient Sequences for Content based Image Authentication

The application of a number theory based pseudorandom sequence called Partial Quotient (PQ) sequence generated from the continued fraction expansion of irrational numbers in a semi fragile watermarking scheme for content based image authentication is considered in this paper. The generated pseudorandom PQ sequence is used to create sub vectors to be used at various instances in the watermarking process. The watermark is derived from the image and is embedded in the higher textured blocks. The watermarked images demonstrate high imperceptibility and make it suitable for use in artistic, medical and military applications where high quality and minimal distortion of the watermarked images is required.

Statistical characteristics of selected elements in vegetables from Kosovo.

Zinc, copper, iron, chromium and cobalt are essential elements for human health, showing toxicity only in high concentrations, while lead and cadmium are extremely toxic even as traces. Therefore, it is important to monitor the contents of toxic metals in vegetables. Large number of vegetables is grown and used in nutrition, in Kosovo. The concentrations of selected elements in vegetables (radish, onion, garlic and spinach) from Kosovo were determined using ICP-OES method. Oral intake of metals and health risk index were calculated. Statistical analysis indicated numerous positive correlations between concentrations of selected elements in vegetables. As a result of principal component analysis, 15 new variables were obtained which were characterized by eigenvalues. The sequence of health quotients for the heavy metals followed the decreasing order Zn = Mn > Pb > Cu > Ni > Fe > Cd > Co > Cr. The health quotients for all investigated heavy metals were below 1 (one), which is considered safe. The vegetables from Kosovo are mainly safe for use in everyday diet.

On the complexity of a putative counterexample to the -adic Littlewood conjecture

Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.

Quadratic approximation to automatic continued fractions

We study the sets of values taken by the exponents of quadratic approximation w 2 and w 2 * evaluated at real numbers whose sequence of partial quotients is generated by a finite automaton. Among other results, we show that these sets contain every sufficiently large rational number and also some transcendental numbers.

Continued Fractions and Transcendence of Formal Power Series Over a Finite Field

The aim of this paper was to establish a new transcendence criterion for a class of quasi-periodic continued fractions of formal power series over a finite field depending only on the length of specific blocks appearing in the sequence of partial quotients.