Continued Fraction Method Research Articles

Around the [formula omitted]-binomial-Eulerian polynomials

We find a combinatorial interpretation of Shareshian and Wachs’ q-binomial-Eulerian polynomials, which leads to an alternative proof of their q-γ-positivity using group actions. Motivated by the sign-balance identity of Désarménien–Foata–Loday for the (des,inv)-Eulerian polynomials, we further investigate the sign-balance of the q-binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the q-binomial-Eulerian polynomials. We finally use the method of continued fractions to derive a new (p,q)-extension of the γ-positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations.

Low-energy elastic scattering of positrons by O2

We present a theoretical investigation on positron scattering by ${\text{N}}_{2}\text{O}$. Elastic differential and integral cross sections at both the static and static plus correlation-polarization levels of approximation are calculated and reported in the $0.1--100$ eV. Calculations were performed using two theoretical methods, namely, the Schwinger multichannel method and the method of continued fractions. Also, two different schemes were used to treat correlation-polarization effects. The comparison between our calculated results and the existing experimental data is encouraging.

Gravitational lensing by black holes in Einsteinian cubic gravity

We investigate the predictions of Einsteinian Cubic Gravity (ECG) for the lensing effects due to supermassive black holes at the center of Milky Way and other galaxies. Working in the context of spherical symmetry, we obtain the metric function from a continued fraction method and find that both time delays and the angular positions of images considerably deviate from general relativity, as large as milliarcseconds. This suggests that observational tests of ECG are indeed feasible.

Hankel determinants and shifted periodic continued fractions

Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang–Hu–Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set {(1,1),(q,0),(ℓ−1,−1)} for integer parameters m,q,ℓ. Again shifted periodic continued fractions appear.

Quasinormal modes for nh-stu black holes

We compute and investigate the behavior of scalar quasinormal frequencies for static, asymptotically anti de Sitter nh-stu black hole solution of {mathcal {N}}=2, D=4 supergravity. We report their exact expression when the horizon topology is flat. The modes are purely imaginary unless they carry high angular momentum number. These modes for spherical horizon topology were numerically computed by using continued fraction method and found to be of two kinds. One such family consists of purely negative imaginary modes. The other family consists of complex modes with their imaginary part always negative, depicting stability of black holes against scalar perturbations. The complex modes show symmetry about imaginary axis, linear relationship with inverse temperature for large horizon radius and increased oscillation frequency for higher harmonics. Many of these properties are common to a family of black holes which asymptotes to anti de Sitter space, thus pointing to common features of thermalization dynamics of a class of dual holographic theories.

The effects of atom–cavity coupling constant on physical observables for different transitions

The aim of this work is to investigate the changing effects of the atom–cavity coupling constant on an atom–cavity system. A three-level atom in the Λ configuration with q-photon transition between levels 2 and 3 is confined in a single-mode Fabry–Pérot optical cavity. To solve the master equation of this system in the steady-state by using the appropriate physical quantities, the matrix continued fractions method for recurrence equations is applied. The behavior of physical observables including atom–field correlation, mean photon number, and second-order coherence function is discussed. The effect of altering the atom–cavity coupling constant for different transitions on these observables is fully considered. The results of calculations show that by increasing this coupling constant, the range of atom–cavity correlation becomes longer, the maximum value of the output mean photon number from the cavity remains almost constant, the broadening in the curves of the mean photon number increases and the lasing process is amplified in the system. Finally, the transformation of the three-level atom into a two-level one under several specific conditions in a four-photon transition case has been studied. The obtained results of the two-level atomic pattern are adequately confirmed by the simulations related to the three-level atom.

Instability for massive scalar fields in Kerr-Newman spacetime

It is known that a massive charged scalar field can trigger a superradiant instability in the background of a Kerr-Newman black hole. In this paper, we present a numerical study of such an instability by using the continued fraction method. It is shown that for given a black hole, the unstable scalar mode with a specific azimuthal index $m$ only occurs in a finite region in the parameter space of the scalar field. The maximum mass of the scalar cloud is exactly the upper bound of the mass of the unstable modes. We show that due to the electromagnetic interaction between the scalar field and the Kerr-Newman black hole, the growth rate of the instability can be $15.7\%$ larger than that of a scalar field in Kerr spacetime of the same rotation parameter. In addition, we find a maximum value of the growth rate $\tau^{-1}=1.788\times 10^{-7}M^{-1}$, which is about $4\%$ larger than that in the Kerr case.

Ergoregion instability of a rotating quantum system

Using the analogy between acoustic perturbations in an ideal fluid and the description of a Klein-Gordon scalar field in a curved spacetime, we study the quasinormal modes of a quantum system: the rotating Bose-Einstein condensate. To compute quasinormal frequencies, we use two different numerical techniques, namely the direct integration and the continued-fraction methods. We study in detail the ergoregion instability of this linearly perturbed system, comparing the results with different setup configurations.

Cryptanalysis of RSA-type cryptosystems based on Lucas sequences, Gaussian integers and elliptic curves

In this paper, we apply the continued fraction method to launch an attack on the three RSA-type cryptosystems when the private exponent d is sufficiently small. The first cryptosystem, proposed by Kuwakado, Koyama and Tsuruoka in 1995, is a scheme based on singular cubic curves y2=x3+bx2(modN) where N=pq is an RSA modulus. The second cryptosystem, proposed by Elkamchouchi, Elshenawy and Shaban in 2002, is an extension of the RSA scheme to the field of Gaussian integers using a modulus N=PQ where P and Q are Gaussian primes such that p=|P| and q=|Q| are ordinary primes. The third cryptosystem, proposed by Castagnos in 2007, is a scheme over quadratic field quotients with an RSA modulus N=pq based on Lucas sequences. In the three cryptosystems, the public exponent e is an integer satisfying the key equation ed−k(p2−1)(q2−1)=1. Our attack is applicable to primes p and q of arbitrary sizes and we do not require the usual assumption that p and q have the same bit size. Thus, this is an extension of our recent result presented at ACISP 2016 conference. Our experiments demonstrate that for a 513-bit prime p and 511-bit prime q, our method works for values of d of up to 520 bits.

Diffusion mechanism of non-interacting Brownian particles through a deformed substrate

We study the diffusion mechanism of non-interacting Brownian particles through a deformed substrate. The study is done at low temperature for different values of the friction. The deformed substrate is represented by a periodic Remoissenet–Peyrard potential with deformability parameter s. In this potential, the particles (impurity, adatoms…) can diffuse. We ignore the interactions between these mobile particles consider them merely as non-interacting Brownian particles and this system is described by a Fokker–Planck equation. We solve this equation numerically using the matrix continued fraction method to calculate the dynamic structure factor S(q,ω). From S(q,ω) some relevant correlation functions are also calculated. In particular, we determine the half-width line λ(q) of the peak of the quasi-elastic dynamic structure factor S(q,ω) and the diffusion coefficient D.Our numerical results show that the diffusion mechanism is described, depending on the structure of the potential, either by a simple jump diffusion process with jump length close to the lattice constant a or by a combination of a jump diffusion model with jump length close to lattice constant a and a liquid-like motion inside the unit cell. It shows also that, for different friction regimes and various potential shapes, the friction attenuates the diffusion mechanism. It is found that, in the high friction regime, the diffusion process is more important through a deformed substrate than through a non-deformed one.

Scalar field quasinormal frequencies of Reissner–Nordström black hole surrounded by quintessence by using the continued fraction method

We evaluate the quasinormal modes of massless scalar field around Reissner–Nordström black hole surrounded by a static and spherically symmetric quintessence using the continued fraction method. The appropriate Frobenius series for three special cases of the quintessence parameter [Formula: see text] and [Formula: see text] are derived successfully. We show the variation of quasinormal frequencies with charge of the black hole and the quintessential parameters. The numerical results show that quintessence field decreases oscillation frequencies of all angular momentum [Formula: see text] modes and increases the damping time of [Formula: see text] modes.

The improvement of lasing without inversion in the presence of multi-photon transition in a three-level atom confined in an optical cavity in the steady-state regime

A three-level atom in \( \Lambda\) configuration trapped in a single-mode Fabry-Perot optical cavity is investigated. It is assumed that the transition between levels 2 and 3 in the atom is accomplished through the number of q photons. To solve the steady-state master equation for the atom-cavity system, the matrix continued-fraction method for applied physical quantities is used. The results show that by raising the number of transitions, the curves of the population inversion finally overlap with one another and the output mean photon number from the optical cavity rises and moreover the laser effect appears in the system. On the other hand, the outcomes reveal that by increasing the number of transitions despite the negativity of the population inversion, the number of output photons from the cavity grows and lasing interval becomes larger. Eventually the steps of the transformation from the three-level atom to two-level one in the case of multi-photon transition under several specific conditions have been studied. The obtained results of the simulations confirm the accuracy of the used approximations in the two-level pattern appropriately. In summary, the lasing process without population inversion along with the multi-photon transition in the three-level atom is discussed.

Problems in Calculating Moments and Distribution Functions of Ladder Heights

The problem of approximate calculation of distribution functions of ladder heights is considered in the context of the finite number of its known moments. This problem is solved by means of the Chebyshev continued fractions method. The midpoint is finding the terms of fraction of the nth convergents of continued fractions. The moments are calculated by S. Nagaev’s formulas, including solution of the Frobenius equation and applying the Fa di Bruno formulas for higher derivatives. The problem of high precision calculations is studied.

A new analysis of the complex two-dimensional multilayered anisotropic soil in time domain

The dynamic analysis of two-dimensionalmultilayered anisotropic soilwith rigid bedrock is studied. An efficient numerical approach named the modified scaled boundary finite element method (SBFEM) is proposed in the time domain. Based on introducing the continued fraction method and auxiliary variables, the time domain solution is obtained. This solution can be applied to the transversely isotropic medium without any difficulty. For the modified SBFEM, the original scaling center is replaced by a scaling line. These characteristics enable the modified SBFEM to model the horizontal layered medium. Three significant technologies have been introduced in the formula derivation and solving process. First, the dual system is used to derive the displacement equation of the modified SBFEM, which is built on a Hamilton system. According to the principle of virtual work, the displacement equation is transformed to the dynamic stiffness equation. Second, the new continued fraction method for the unbounded domain resting on rigid bedrock is proposed. By introducing auxiliary variables, the displacement equation of motion of an unbounded domain is built. Third, it is an extremely important point that the accurate precise time-integration method is first employed to solve the global equation of motion of the modified SBFEM. This numerical integral method can achieve the machine precision. By using this method in solving the equation of motion of the modified SBFEM, an extremely accurate solution can be achieved. Finally, numerical examples validate the accuracy of the new proposed method, especially for the complex inclined model with anisotropic soil.

Erratum: Damped and zero-damped quasinormal modes of charged, nearly extremal black holes [Phys. Rev. D93, 044033 (2016)

Despite recent progress, the complete understanding of the perturbations of charged, rotating black holes as described by the Kerr-Newman metric remains an open and fundamental problem in relativity. In this study, we explore the existence of families of quasinormal modes of Kerr-Newman black holes whose decay rates limit to zero at extremality, called zero-damped modes in past studies. We review the nearly extremal and WKB approximation methods for spin-weighted scalar fields (governed by the Dudley-Finley equation) and give an accounting of the regimes where scalar zero-damped and damped modes exist. Using Leaver's continued fraction method, we verify that these approximations give accurate predictions for the frequencies in their regimes of validity. In the nonrotating limit, we argue that gravito-electromagnetic perturbations of nearly extremal Reissner-Nordstr\"{o}m black holes have zero-damped modes in addition to the well-known spectrum of damped modes. We provide an analytic formula for the frequencies of these modes, verify their existence using a numerical search, and demonstrate the accuracy of our formula. These results, along with recent numerical studies, point to the existence of a simple universal equation for the frequencies of zero-damped gravito-electromagnetic modes of Kerr-Newman black holes, whose precise form remains an open question.

Quasinormal modes of extremal black holes

The continued fraction method (also known as Leaver's method) is one of the most effective techniques used to determine the quasinormal modes of a black hole. For extremal black holes, however, the method does not work (since, in such a case, the event horizon is an irregular singular point of the associated wave equation). Fortunately, there exists a modified version of the method, devised by Onozawa et al. [Phys. Rev. D 53, 7033 (1996)], which works for neutral massless fields around an extremal Reissner-Nordstrom black hole. In this paper, we generalize the ideas of Onozawa et al. to charged massless perturbations around an extremal Reissner-Nordstrom black hole and to neutral massless perturbations around an extremal Kerr black hole. Similarities and differences between the results of the original continued fraction method for near extremal black holes and the results of the new continued fraction method for extremal black holes are discussed.

Brownian diffusion in a triple-well potential

ABSTRACTIn this work we present a general theory for diffusion mechanism of Brownian particle submitted to a symmetric periodic triple-well potential. The kinetics description is done by the Fokker-Planck equation, which is resolved numerically using the Matrix Continued Fraction Method, in order to calculate some important correlation functions. The half-width λ(q) at half maximum of the quasi-elastic peak of dynamic structure factor S(q, ω) and the diffusion coefficient D are studied in the high friction regime and low temperature for different form of triple-well potential. Our numerical results of half-width λ(q), show that the diffusion process in triple-well potential can be described by a superposition of both simples hopping and liquid-like motion when the ratio Δ of two potential barriers V1 and V2 is less than one (Δ < 1) and by the longs jumps when Δ tends towards one. For some values of ratio of potential barriers, the diffusion coefficient results show that the intermediates potential barriers accelerate the diffusion process.

Quasinormal modes of relativistic stars and interacting fields

The quasinormal modes of relativistic compact objects encode important information about the gravitational response associated with astrophysical phenomena. Detecting such oscillations would provide us with a unique understanding of the properties of compact stars, and may give definitive evidence for the existence of black holes. However, computing quasinormal modes in realistic astrophysical environments is challenging due to the complexity of the spacetime background and of the dynamics of the perturbations. We discuss two complementary methods for computing the quasinormal modes of spherically symmetric astrophysical systems, namely, the direct integration method and the continued-fraction method. We extend these techniques to dealing with generic coupled systems of linear equations, with the only assumption being the interaction between different fields is effectively localized within a finite region. In particular, we adapt the continued-fraction method to include cases where a series solution can be obtained only outside an effective region. As an application, we compute the polar quasinormal modes of boson stars by using the continued-fraction method for the first time. The methods discussed here can be applied to other situations in which the perturbations effectively couple only within a finite region of space.

Damped and zero-damped quasinormal modes of charged, nearly extremal black holes

Despite recent progress, the complete understanding of the perturbations of charged, rotating black holes as described by the Kerr-Newman metric remains an open and fundamental problem in relativity. In this study, we explore the existence of families of quasinormal modes of Kerr-Newman black holes whose decay rates limit to zero at extremality, called zero-damped modes in past studies. We review the nearly extremal and WKB approximation methods for spin-weighted scalar fields (governed by the Dudley-Finley equation) and give an accounting of the regimes where scalar zero-damped and damped modes exist. Using Leaver’s continued fraction method, we verify that these approximations give accurate predictions for the frequencies in their regimes of validity. In the nonrotating limit, we argue that gravito-electromagnetic perturbations of nearly extremal Reissner-Nordstrom black holes have zero-damped modes in addition to the well-known spectrum of damped modes. We provide an analytic formula for the frequencies of these modes, verify their existence using a numerical search, and demonstrate the accuracy of our formula. These results, along with recent numerical studies, point to the existence of a simple universal equation for the frequencies of zero-damped gravito-electromagnetic modes of Kerr-Newman black holes, whose precise form remains an open question.

Hamiltonian-Based Derivation of the High-Performance Scaled-Boundary Finite-Element Method Applied to the Complex Multilayered Soil Field in Time Domain

AbstractWave propagation in a two-dimensional unbounded domain with rigid bedrock was studied. The study was based on the scaled-boundary finite-element method (SBFEM) combing the continued fraction method. The SBFEM is a novel semianalytical technique, combining the advantages of the finite-element and the boundary-element methods. Meanwhile, SBFEM has unique properties of its own. In this paper, the modified SBFEM was introduced. The original scaling center was replaced by a scaling line, which is more suitable for analyzing the multilayered soil model. As a key point, the modified SBFEM and the dual system, which is used to solve a proposed time domain problem, were combined to derive the governing equations. The derivation process was based on the framework of the Hamiltonian system by introducing dual variables. By using the continued fraction solution and introducing auxiliary variables, the equation of motion was built. More significantly, the efficient, precise time-integration method was first us...