Continued Fraction Coefficients Research Articles

Weil–Petersson geodesics on the modular surface

We consider the Weil–Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface, and analyse geometric properties of a lift as a path in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow traveling to derive a correspondence between recurrent WP and hyperbolic rays from a base-point. We show that the correspondence can be promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we show that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. In terms of continued fraction coefficients, the singularity boils down to a comparison that we prove, namely, the average coefficient is bounded along a typical WP ray but unbounded along a typical hyperbolic ray.

On the asymptotic behavior of Sudler products along subsequences

Let α∈(0,1) and irrational. We investigate the asymptotic behavior of sequences of certain trigonometric products (Sudler products) (PN(α))N∈N withPN(α)=∏r=1N|2sin⁡(πrα)|.More precisely, we are interested in the asymptotic behavior of subsequences of the form (Pqn(α)(α))n∈N, where qn(α) is the nth best approximation denominator of α. Interesting upper and lower bounds for the growth of these subsequences are given, and convergence results, obtained by Mestel and Verschueren (see [24]) and Grepstad and Neumüller (see [11]), are generalized to the case of irrationals with bounded continued fraction coefficients.

A fast method to compute dispersion diagrams of three-dimensional photonic crystals with rectangular geometry

We propose a method and codes for fast computation of complex dispersion relations in three-dimensional photonic crystals (PCs) with rectangular geometry. The main idea of the method is to convert the eigenproblem to a nonlinear equation equivalent to the zero-determinant condition. This equation is then solved iteratively either by fixed-point iteration or by rational approximation method. Additional mathematical elements include fast-converging continued-fraction expansion to compute the interaction tensor (appearing in the above nonlinear equation) and efficient accounting for the rectangular geometry in matrix-vector multiplications, which are involved in computing the continued fraction coefficients. The method allows one to perform realistic three-dimensional computations on a typical laptop computer, including finding the Bloch wave vector in the band gaps and in evanescent mode bands. This paper is focused on the method and includes its detailed explanation and illustration with examples. The associated computational package contains a detailed user guide and a set of further demonstrations, which can be run with the help of provided scripts. Program summaryProgram title: RectDisp v.1-3CPC Library link to program files:https://doi.org/10.17632/k7nk86cfbs.1Code Ocean capsule:https://codeocean.com/capsule/7141738Licensing provisions: CC BY NC 3.0Programming language: Fortran 2003Nature of problem: The program computes the complex Bloch wave number q as a function of real frequency ω in photonic crystals (PCs) with rectangular geometry. The PC constituents are characterized by complex dispersive permittivities and are non-magnetic. Projection of q onto the XY-plane of the PC is fixed and can be specified by the user, and the projection onto the Z-axis is computed.Solution method: The method converts the eigenproblem of finding q to the nonlinear equation equivalent to the zero-determinant condition for, at most, a 3×3 matrix, and solves the latter by an iterative method (either fixed-point iteration or rational approximation or a combination of the two approaches). Additional features include continued-fraction expansion for computing the coefficients in the nonlinear equation and utilization of variable separability characteristic of the rectangular geometry for fast matrix-vector multiplication.Additional comments including restrictions and unusual features: The current version of the codes computes at most one value of q for each frequency. This excludes spurious modes arising due to the artificial band folding. Such modes are usually not coupled to incident radiation. However, some other modes can be missed. Under most circumstances, this can happen at higher frequencies, i.e., for ωh/c>π, where h is the PC period. Verification of mode coupling to external radiation and exhaustive search for all coupled modes are not currently implemented but we plan to add these features in future releases. If a solution is not found at some frequency, it can potentially be found by fine-tuning input parameters as described in the User Guide.

On eventually always hitting points

We consider dynamical systems $$(X,T,\mu )$$ which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls $$(B_n)_{n=1}^\infty $$ , we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a $$k < m$$ with $$T^k (x) \in B_m$$ . We also give an asymptotic estimate as $$m \rightarrow \infty $$ on the number of $$k < m$$ with $$T^k (x) \in B_m$$ . As an application, we prove for almost every point x an asymptotic estimate on the number of $$k \le m$$ such that $$a_k \ge m^t$$ , where $$t \in (0,1)$$ and $$a_k$$ are the continued fraction coefficients of x.

Effective estimation of some oscillatory integrals related to infinitely divisible distributions

We present a practical framework to prove, in a simple way, two-term asymptotic expansions for Fourier integrals $$\begin{aligned} {{\mathcal {I}}}(t) = \int _{{\mathbb {R}}}(\mathrm{e}^{it\phi (x)}-1) \mathop {}\!\mathrm {d}\mu (x), \end{aligned}$$where \(\mu \) is a probability measure on \({{\mathbb {R}}}\) and \(\phi \) is measurable. This applies to many basic cases, in link with Levy’s continuity theorem. We present applications to limit laws related to rational continued fraction coefficients.

Finding both, the continued fraction and the Laurent series expansion of golden ratio analogs in the field of formal power series

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Knowing both of them is for instance important for the study of Kronecker type sequences in the theory of uniform distribution.Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions. We discover that both matrices in the LU decomposition of these Hankel matrices can be described by a simple recursive algorithm based on the continued fraction coefficients of the golden ratio analogs.The upper triangular factor U possesses several nice properties. First, its entries in the special case where the golden ratio analog is [0;X‾] can be given by using the Catalan's triangular numbers. Nicely, this relation together with our findings on the Hankel matrices are used to derive combinatorial identities involving Catalan's triangular numbers. As a side product we obtain a description of the distribution of the Catalan numbers modulo a prime number.Second, the upper triangular factor U is the columnwise composition of the Zeckendorf-type representations of the powers of X. This Zeckendorf representation for polynomials is introduced in the style of the Zeckendorf representation of the natural numbers and it is based on the Fibonacci polynomials instead of Fibonacci numbers.Third, in the case of positive characteristic, for each purely periodic golden ratio analog we detect a self-similar pattern in the upper triangular factor U of its Hankel matrix. We derive this pattern from a binomial type formula for two non-commutative matrices.Finally, we exemplary show, how this self-similar pattern in the upper triangular factor U can be used to describe the Laurent series expansion of the golden ratio analog [0;X‾].

A positive lower bound for liminf_{𝑁→∞}∏ᵣ₌₁^{𝑁}|2sin𝜋𝑟𝜑|

Nearly 60 years ago, Erdős and Szekeres raised the question of whether lim inf N → ∞ ∏ r = 1 N | 2 sin ⁡ π r α | = 0 \begin{equation*} \liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \alpha \right | =0 \end{equation*} for all irrationals α \alpha . Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if α \alpha has unbounded continued fraction coefficients, and he suggested that the answer is yes in general. However, we show in this paper that for the golden ratio φ = ( 5 − 1 ) / 2 \varphi =(\sqrt {5}-1)/2 , lim inf N → ∞ ∏ r = 1 N | 2 sin ⁡ π r φ | &gt; 0 , \begin{equation*} \liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \varphi \right | &gt;0 , \end{equation*} providing a negative answer to this long-standing open problem.

Probing Schrodinger equation with a continued fraction potential

We suggest a new perturbed form of the quantum potential and investigate the possible solutions of Schrodinger equation. The new form can be written as a finite or infinite continued fraction. a comparison has been given between the continued fractional potential and the non-perturbed potential. We suggest the validity of this continued fractional quantum form in some quantum systems. As the order of the continued fraction increases the difference between the perturbed and the ordinary potentials decreases. The physically acceptable solutions critically depend on the values of the continued fraction coefficients αi.

Symplectic Rational Blow-down Along Seifert Fibered 3-Manifolds

The rational blow-down procedure (introduced by Fintushel and Stern [4] and generalized by Park [25]) turned out to be one of the most effective operations in constructing smooth 4-manifolds with interesting topological properties, cf. [5, 26–28]. Recall that when performing the rational blow-down operation we simply replace the tubular neighbourhood of a string of 2-spheres (intersecting each other according to the linear plumbing tree with framings specified by the continued fraction coefficients of − p 2 pq−1 for some p, q relatively prime) with a rational homology disk. The success of this operation might be explained by the fact that—as Symington showed [30, 31]—it can be performed symplectically. More precisely, if the ambient 4-manifold is symplectic and the spheres are symplectic submanifolds intersecting each other orthogonally then the neighbourhood

Resolution of Maxwell's equations by the spectral moments method. Global approach

We present a systematic analysis, within the scope of electromagnetic theory, of the spectral moments method (SMM) and develop several ways to compute the linear response of any type of system. We show that the method can be used in diffraction studies regardless of the number, nature and form of the diffracting objects. Multiple diffraction is naturally taken into account in the computation. The method can thus be applied to determine propagation through any type of media. It is based on the computation of Green functions, solutions of discretized Maxwell's equations. Fourier transforms of Green functions are developed in continued fraction. Two approaches will be presented. In the first “global” approach, all space is discretized, the coefficients of continued fractions are computed directly from the dynamic matrix obtained by the discretization of Maxwell's equations and from sources and receivers. In the second “local” approach, only the diffracting system is discretized. This paper is devoted to the global approach. We study two important problems in electromagnetism, i.e. propagation of a plane wave through a heterogeneous layer and scattering of an isolated object. We present two computation techniques for plane wave propagation: one uses a small grid, is very rapid but the results are approximate; the other uses a large grid, is less rapid but the results are exact. We show that computing the reflectivity and/or transmissivity of photonic lattices is now a very simple problem. For scattering, we mainly report a series of tests on some canonical systems, such as cylinders or spheres, showing that SMM results are in very good agreement with the analytical results. Several types of absorbing boundary conditions are tested. We report results on backscattering cross-sections and the impulsional response of different one-, two- and three-dimensional systems.

Trimmed sums for non-negative, mixing stationary processes

We consider the effect of “trimming” ergodic sums of their maximal values on the strong law of large numbers for non-negative, non-integrable, mixing stationary processes. The results obtained are used to show the failure of the strong law of large numbers for modified continued fraction coefficients, and to study the “cusp visits” of a certain interval map.

Bounded geometry for Kleinian groups

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Applications include an improvement to the bounded geometry versions of Thurston's ending lamination conjecture, and of Bers' density conjecture.

Kleinian groups and the complex of curves

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invariants are points in the closed unit disk and the coefficients are closely related to classical continued-fraction coefficients. The estimates obtained play an essential role in the solution of Thurston's ending lamination conjecture in that case.

Elastic wave‐propagation simulation in heterogeneous media by the spectral moments method

The second‐order elastic wave propagation equations are solved using the spectral moments method. This numerical method, previously developed in condensed matter physics, allows the computation of Green’s functions for very large systems. The elastic wave equations are transformed in the Fourier domain for time derivatives, and the partial derivatives in space are computed by second‐order finite differencing. The dynamic matrix of the discretized system is built from the medium parameters and the boundary conditions. The Green’s function, calculated for a given source‐receiver couple, is developed as a continued fraction whose coefficients are related to the moments and calculated from the dynamic matrix. The continued fraction coefficients and the moments are computed using a very simple algorithm. We show that the precise estimation of the waveform for the successive waves arriving at the receiver depends on the number of moments used. For long recording times, more moments are needed for an accurate solution. Efficiency and accuracy of the method is illustrated by modeling wave propagation in 1-D acoustic and 2-D elastic media and by comparing the results obtained by the spectral moments method to analytical solutions and classical finite‐difference methods.

The symmetric strong moment problem

Sequences of polynomials that occur as denominators in the two point Paté table for two series expansions are considered in the special case when the series coefficients are solutions of a strong symmetric Stieltjes moment problem. The continued fractions whose convergents generate these polynomials as denominators are presented, together with determinant representations for the polynomials and the continued fraction coefficients. The log-normal distribution is used as an example.

Plaquette expansion proof and interpretation

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is established up to the first two orders for an arbitrary system. This method employs an expansion of the Lanczos coefficients, the tridiagonal Hamiltonian matrix elements or equivalently the continued fraction coefficients of the resolvent, in a descending series in the size of the system. The coefficients of this series are formed from the low order cumulants or connected Hamiltonian moments. The lowest order approximation in the plaquette expansion corresponds to a gaussian model which is a consequence of the central limit theorem. The first nontrivial order yields a model with a spectrum on a bounded energy interval, becoming asymptotically uniform in the thermodynamic limit.

Ordering and fluctuations in the ground state of the one-dimensional and two-dimensional S=1/2 XXZ antiferromagnets: A study of dynamical properties based on the recursion method.

The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ t${sum}_{\mathbf{q}}$${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.

Continued fraction as a discrete nonlinear transform

The connection between a Taylor series and a continued fraction involves a nonlinear relation between the Taylor coefficients {an} and the continued fraction coefficients {bn}. In many instances it turns out that this nonlinear relation transforms a complicated sequence {an} into a very simple one {bn}. This simplification is illustrated in the context of graph combinatorics.

Spectral signature of quantum spin diffusion in dimensions d=1, 2, and 3.

The spectral densities of dynamical spin autocorrelation functions at infinite temperature are studied for the S=1/2XXZ model (with exchange couplings ${\mathit{J}}_{\mathit{x}}$=${\mathit{J}}_{\mathit{y}}$\ensuremath{\equiv}J,${\mathit{J}}_{\mathit{z}}$) on the linear chain, the square lattice, and the simple cubic lattice. The low-frequency behavior of a given spectral density is inferred from certain characteristic properties of its continued-fraction coefficients as determined from computed frequency moments. The analysis yields estimates for the ${\mathit{J}}_{\mathit{z}}$/J dependence of the infrared-singularity exponent. In the d=1 case, the exponent for spin fluctuations perpendicular to the O(2) symmetry axis responds sensitively as the anisotropy parameter sweeps across the O(3) symmetry point ${\mathit{J}}_{\mathit{z}}$/J=1, while the exponent for the parallel fluctuations shows little variation. In the cases d=2 and d=3 the same observations are made for autocorrelation functions of aggregate spins in chains and lattice planes, respectively.

Continued fraction expansion for the X-ray absorption cross section

The Haydock recursion scheme is for the first time applied to the matrix inversion required for calculations of X-ray absorption cross sections. A mapping of the multiple-scattering contributions into continued fraction coefficients which appears to improve the convergence of the multiple-scattering series is derived. Numerical applications to three different systems-Mn ion in water solution, a copper-imidazole molecule, and c-Si-are presented.