Continued Fraction Method Research Articles

Diffusion of a massive particle in a periodic potential: Application to adiabatic ratchets.

We generalize a theory of diffusion of a massive particle by the way in which transport characteristics are described by analytical expressions that formally coincide with those for the overdamped massless case but contain a factor comprising the particle mass which can be calculated in terms of Risken's matrix continued fraction method (MCFM). Using this generalization, we aim to elucidate how large gradients of a periodic potential affect the current in a tilted periodic potential and the average current of adiabatically driven on-off flashing ratchets. For this reason, we perform calculations for a sawtooth potential of the period L with an arbitrary sawtooth length (l<L) instead of the smooth potentials typically considered in MCFM-solvable problems. We find nonanalytic behavior of the transport characteristics calculated for the sharp extremely asymmetric sawtooth potential at l→0 which appears due to the inertial effect. Analysis of the temperature dependences of the quantities under study reveals the dominant role of inertia in the high-temperature region. In particular, we show, by the analytical strong-inertia approach developed for this region, that the temperature-dependent contribution to the mobility at zero force and to the related effective diffusion coefficient are proportional to T(-3/2) and T(-1/2), respectively, and have a logarithmic singularity at l→0.

Duality system-based derivation of the modified scaled boundary finite element method in the time domain and its application to anisotropic soil

In this study, an efficient method is proposed for the dynamic analysis of a two-dimensional semi-infinite soil with rigid bedrock, which is applicable to the cross-isotropic and anisotropic soil models. The original scaling center is replaced by a scaling line, so the modified scaled boundary finite element method (SBFEM) is more suitable for analyzing the horizontal layered soil. For the first time, the dual system is employed to derive the displacement equation for the modified SBFEM. By introducing the dual variables, the governing equations are derived in the framework of a Hamilton system. Next, the dynamic stiffness equation is obtained according to the weighted residuals method. The displacement equation of motion for the far field is built by applying the continued fraction method and introducing auxiliary variables. Based on the sub-structure method, the far field can be seamlessly coupled with the near field. Importantly, the efficient and precise time-integration method is first employed to solve the global equation of motion. High computational precision can be achieved using the proposed method. An extremely efficient and accurate solution can be obtained by applying this method to solve the equation of motion for the modified SBFEM. Finally, the accuracy and high efficiency of the proposed method is demonstrated for the anisotropic soil model based on numerical examples.

Elastic and inelastic vibrational cross sections for positron scattering by carbon monoxide

The vibrational cross sections of the CO molecule induced by positron impact is the focus of this work. The positron–molecule interaction is represented by the static potential plus a model potential designed to take into account the positron–target correlations. To calculate the vibrational cross sections, we applied the multichannel version of the continued fractions method in the close-coupling scheme. We present vibrational excitation cross sections and elastic ones, for the ground and excited vibrational states. The results are interpreted in terms of the vibrational coupling-scheme used in the scattering model.

Analytical representation of the relations of inertial diffusion transport

A generalized approach has been proposed to describe the diffusive transport of inertial particles at which the known inertialess relations (in particular, between the concentration of particles and the corresponding flux) are supplemented by a factor allowing the calculation of inertial effects in terms of the matrix continued fraction method. The advantage of this approach, which is in the analytical representation of the results, is illustrated by solving the problem of the determination of the effective diffusion coefficient of an inertial particle in a sawtooth potential and the average velocity of the adiabatic inertial ratchet. The character of the nonanalytic behavior of these quantities in the presence of large gradients of the potential has been established.

Voltage noise, multiple phase-slips, and switching rates in moderately damped Josephson junctions

We study the voltage noise properties including the statistics of phase-slips and switching rates in moderately damped Josephson junctions by using a novel efficient numerical approach that combines the matrix continued-fraction method with the full counting statistics. By analyzing the noise results obtained for the resistively and capacitively shunted junction (RCSJ) model we identify different dominating components; namely, the thermal noise close to equilibrium (small-current-bias regime), the shot noise of (multiple) phase-slips in the intermediate range of biases, and the switching noise for yet higher bias currents. We extract thus far inaccessible characteristic rates of phase-slips in the shot-noise regime as well as the escape and retrapping rates in the switching regime as functions of various junction parameters. The method can be extended and applied to other experimentally relevant Josephson junction circuits as well as to optical trap setups.

High performance of the scaled boundary finite element method applied to the inclined soil field in time domain

An efficient method for modelling the wave propagation in semi-infinite domain is proposed. It is applicable to soil–structure interaction problems for complex inclined soil field. The scaled boundary finite element method is modified through the original scaling center substituting by a scaling line. Based on this scaling line, the dynamic stiffness equation is derived. Then, an accurate and efficient continued fraction method is firstly introduced for solving equation for the model with rigid bedrock. By using the continued fraction solution and introducing auxiliary variables, the equation of motion of unbounded domain is built. Coupling the far field modelling by modified scaled boundary finite element method with the near field modelling by the finite element method, the global time-domain equation is obtained, which is a standard equation of motion for the whole domain. As a key point, the precise time-integration method is firstly employed to solve global equation of motion. The advantages of this integration method are that the integral interval is divided into quite small piece. It makes sure the precision achieve to computer precision. Owning to adopting five terms Taylor expansion for each small integral interval, the computational precision is increased greatly. Applying precise time-integration method in this paper, it greatly improves the accuracy and computing speed of proposed method. By using the sub-structure method, the inclined soil field is modelled. Numerical examples demonstrate accuracy and high efficiency of the new method, especially for complex dip mediums.

Continued fraction real root isolation using the Hong root bound

An investigation of the codominance maximum computing time of the continued fractions method (CF) for isolation of the real roots of a squarefree integral polynomial when applied to the two-parameter family of polynomials Aa,n(x)=xn−2(ax2−(a+2)x+1)2, with n≥5 and a≥1. These polynomials have two roots, r1 and r2, in the interval (0,1), with |r1−r2|<a−n. It is proved that for these polynomials the maximum time required by CF to isolate those two close roots would be codominant with n5(ln⁡a)2 even if an “ideal” root bound were available and either the Horner method or the Budan method is used for translations. It is proved that if a power-of-two Hong root bound is used by CF to determine translation amounts then the time required to isolate the two close roots is dominated by n6(ln⁡a) if a multiplication-free Budan translation method is used. Computations reveal that the Hong root bound is surprisingly effective when applied to the transformed polynomials that arise, engendering a minimum efficiency conjecture. It is proved that if the conjecture is true then the time to isolate the two close roots is dominated by n5(ln⁡a)2. There is also evidence for a maximum efficiency conjecture. The two conjectures together, if true, make it likely that this time is codominant with n5(ln⁡a)2.

Quasinormal modes of charged fields around a Reissner-Nordström black hole

The quasinormal spectrum of a charged scalar field around a non-extremal Reissner-Nordstr\"om black hole, specially in the limit of large electromagnetic interaction, has been comprehensively studied only very recently. In this work, we extend the analysis to Dirac fields using the continued fraction method and compare the results with the scalar case. In particular, we study the behaviour of the fundamental quasinormal mode as a function of the black hole's charge and of the electromagnetic interaction parameter. We derive an analytical formula for the quasinormal frequencies in the limit of large electromagnetic interaction. As the extremal limit of black hole charge is approached, we show that, unlike the case of neutral fields, the imaginary part of the quasinormal frequencies approach zero for charged fields.

ON MULTISERVER RETRIAL QUEUES: HISTORY, OKUBO-TYPE HYPERGEOMETRIC SYSTEMS AND MATRIX CONTINUED-FRACTIONS

In this paper, we study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiserver M/M/∞ Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a"minimal/nondominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (a) and (b), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest.The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: (i) the irregular singularity at 0; (ii) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and (iii) the point 1, whose Taylor series coefficients are the factorial moments.

Continued Fraction Method for Approximation of Heat Conduction Dynamics in a Semi-Infinite Slab

Heat conduction dynamics are described by partial differential equations. Their approximations with a set of finite number of ordinary differential equations are often required for simpler computations and analyses. Rational approximations of the Laplace solutions such as the Pade approximation can be used for this purpose. For some heat conduction problems appearing in a semi-infinite slab, however, such rational approximations are not easy to obtain because the Laplace solutions are not analytic at the origin. In this article, a continued fraction method has been proposed to obtain rational approximations of such heat conduction dynamics in a semi-infinite slab.

QUASINORMAL FREQUENCIES OF BLACK BRANES WITH HYPERSCALING VIOLATION

We investigated quasinormal frequencies (QNFs) in black brane with hyperscaling violating by using the continued fractions method. We calculate QNF of massless scalar field and electromagnetic field both with zero spatial momentum, find that QNFs have negative imaginary frequency suggesting black brane with hyperscaling violation is stable under those perturbations.

Quantum-interference-enhanced deep sub-Doppler cooling of39K atoms in gray molasses

We report enhanced sub-Doppler cooling of the bosonic atoms of ${}^{39}$K facilitated by formation of dark states with the cooling and repumping lasers tuned to the Raman resonance in $\ensuremath{\Lambda}$ configuration near the ${D}_{1}$ transition. A temperature of about 12 $\ensuremath{\mu}$K and phase-space density $&gt;$$2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ is achieved in the two-stage ${D}_{2}$-${D}_{1}$ molasses and spans a very large parameter region where quantum interference persists robustly. We also present results on enhanced radiation heating with a subnatural linewidth (0.07$\ensuremath{\Gamma}$) and a signature Fano-like profile of a coherently driven three-level atomic system. The optical Bloch equations relevant for the three-level atom in a bichromatic light field are solved with the method of continued fractions to show that cooling occurs only for a small velocity class of atoms, emphasizing the need for precooling in the ${D}_{2}$ molasses stage.

Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies $$\omega $$ for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $$\mathcal{R }(\omega )$$ tends to the mass of the field and the imaginary part $$\mathcal{I }(\omega )$$ tends to zero, for any angular momentum quantum number $$\ell $$ . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.

Towards the assignment of the REMPI spectrum of Ph2O using CIS and TD-DFT methods

Tentative assignment of the low-lying vibrational features of the first electronic excited singlet-state S1 of diphenyl ether (Ph2O), obtained from resonance-enhanced multiphoton ionisation (REMPI) spectroscopy, is performed using CIS and time-dependent density functional theory (TD-DFT) methods. The potential energy surfaces, regarding the rotation of the phenyl rings relatively to the C–O–C plane, are obtained at the B3LYP/6-31G(d) level of theory, for the ground-state of neutral and cationic Ph2O and for its first excited singlet state. The torsional barriers of the ground state of diphenyl ether were studied by means of quantum-chemical perturbations of increasing accuracy and an extrapolation to the complete basis set limit and full configuration interaction (FCI) was performed through the use of correlation consistent basis sets and the continued fraction method. The first adiabatic ionisation energy (AIE) of the twist conformer is computed at 8.60 eV in the FCI limit, much higher than the experimental results of Terlouw et al. (8.09±0.03 eV) and Paiva et al. (7.8±0.1 eV). The B3LYP result of 7.82 eV is, however, in reasonable agreement with the result of Paiva et al.The first singlet excitation energy for the twist conformation is found to be 5.5 eV at the CIS/6-311++G(d,p) level of theory. Some features of the experimental REMPI spectrum, previously obtained by one of the authors, are explained and a new insight on the ionisation energy of diphenyl ether is presented.

Degenerate rotating black holes, chiral CFTs and Fermi surfaces I — Analytic results for quasinormal modes

In this work we discuss charged rotating black holes in $AdS_5 \times S^5$ that degenerate to extremal black holes with zero entropy. These black holes have scaling properties between charge and angular momentum similar to those of Fermi surface operators in a subsector of $\mathcal{N}=4$ SYM. We add a massless uncharged scalar to the five dimensional supergravity theory, such that it still forms a consistent truncation of the type IIB ten dimensional supergravity and analyze its quasinormal modes. Separating the equation of motion to a radial and angular part, we proceed to solve the radial equation using the asymptotic matching expansion method applied to a Heun equation with two nearby singularities. We use the continued fraction method for the angular Heun equation and obtain numerical results for the quasinormal modes. In the case of the supersymmetric black hole we present some analytic results for the decay rates of the scalar perturbations. The spectrum of quasinormal modes obtained is similar to that of a chiral 1+1 CFT, which is consistent with the conjectured field-theoretic dual. In addition, some of the modes can be found analytically.

On the computing time of the continued fractions method

The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.

Resonances of a rotating black hole analogue

Under certain conditions, sound waves in a fluid may be governed by a Klein-Gordon equation on an ``effective spacetime'' determined by the background flow properties. Here we consider the draining bathtub: a circulating, draining flow whose effective spacetime shares key features with the rotating black hole (Kerr) spacetime. We present a complete investigation of the role of quasinormal (QN) mode and Regge pole (RP) resonances of this system. First, we simulate a perturbation in the time domain by applying a finite-difference method, to demonstrate the ubiquity of ``QN ringing.'' Next, we solve the wave equation in the frequency domain with the continued-fraction method, to compute QN and RP spectra numerically. We then explore the geometric link between (prograde and retrograde) null geodesic orbits on the spacetime, and the properties of the QN/RP spectra. We develop a ``geodesic-expansion'' method which leads to asymptotic expressions (in inverse powers of mode number $m$) for the spectra, the radial functions, and the residues. Next, the role of the Regge poles in scattering and absorption processes is revealed through the application of the complex angular momentum method. We elucidate the link between the Regge poles and oscillations in the absorption cross section. Finally, we show that Regge poles provide a neat explanation for ``orbiting'' oscillations seen in the scattering cross section.

A New Approach to Black Hole Quasinormal Modes: A Review of the Asymptotic Iteration Method

We discuss how to obtain black hole quasinormal modes (QNMs) using the asymptotic iteration method (AIM), initially developed to solve second-order ordinary differential equations. We introduce the standard version of this method and present an improvement more suitable for numerical implementation. We demonstrate that the AIM can be used to find radial QNMs for Schwarzschild, Reissner-Nordström (RN), and Kerr black holes in a unified way. We discuss some advantages of the AIM over the continued fractions method (CFM). This paper presents for the first time the spin 0, 1/2 and 2 QNMs of a Kerr black hole and the gravitational and electromagnetic QNMs of the RN black hole calculated via the AIM and confirms results previously obtained using the CFM. We also present some new results comparing the AIM to the WKB method. Finally we emphasize that the AIM is well suited to higher-dimensional generalizations and we give an example of doubly rotating black holes.

Calculating far field of an arbitrarily oriented electric dipole located above layered half space using reciprocity theorem

Based on the reciprocity theorem, the far field formulation of an arbitrarily oriented electric dipole located at the interface of the layered half space is deduced. Then, considering the optical path difference between directed wave and reflected wave, the formula of the electric dipole located above the interface of the layered half space is discussed. The TE and the TM reflection coefficients of the layered half space can be calculated by the propagation matrix method or the continued fraction method. The presente method, in which the deduction process possesses concise physics concept, is suitable for calculating the radiated field of an electric dipole when the observation point is away from the interface. The numerical results show that using our method the far radiation field of an arbitrarily oriented electric dipole located above the interface of layered half space can be guickly analyzed.

Onset of Vibrational Convection in a Binary Fluid Saturated Non-Darcy Porous Layer Heated from Above

A linear stability analysis is used to investigate the influence of mechanical vibration on the onset of thermosolutal convection in a horizontal porous layer heated and salted from above. Vibrations are considered with arbitrary amplitude and frequency. The Brinkman extended Darcy model is used to describe the flow and the Oberbeck-Boussinesq approximation is employed. Continued fraction method and Floquet theory are used to determine the convective instability threshold. It is found that the solutal Rayleigh number has the stabilizing effect. The existence of a closed disconnected loop of synchronous mode is predicted in the marginal curve for moderate values of solutal Rayleigh number and vibration amplitude.