Continued Fraction Method Research Articles

Statistical analysis in enrichment of total whey protein by continuous foam fractionation method

The objective of the present study was to optimize the operating conditions in the separation of the total whey proteins from whey by continuous foam fractionation method using response surface methodology (RSM). The effects of the different process variables such as pH (X 1 ) of proteins in feed, gas flow rate, (X 2 ) of initial feed solution, protein: surfactant ratio (X 3 ) and volumetric flow rate (X 4 ) where investigated on the performance criteria of fractionation of raw processed whey. Four factors, three levels Box-Behnken design was used for the optimization procedure. Quadratic model regression equations and response surface plots correlate independent variables (X 1 , X 2 , X 3 and X 4 ) and dependent variables (response) such as concentration of Foamate (C f ), Enrichment ratio (E r ), and percentage Recovery (% R p ) of total whey proteins. All the four factors had significant effects on the response variables. The model predicted that the optimized values of the factors (X 1 , X 2 , X 3 and X 4 ) such as 5, 290, 1.5 and 14, respectively. The predicted responses were (concentration of foamate, enrichment ratio, and percentage recovery) such as 6647.32, 13.27, and 78.02, respectively. Experiments were performed with the predicted values of factors.

Fokker-Planck equation with arbitrary dc and ac fields: Continued fraction method

The continued fraction method (CFM) is used to solve the Fokker-Planck equation with arbitrary dc and ac fields. With an appropriate choice of basis functions, the Fokker-Planck equation is converted into a set of linear algebraic equations with short-ranged coupling and then CFM is implemented to obtain numerical solutions with high efficiency. Both a proposed perturbative CFM and the numerically exact matrix CFM are used to study the nonlinear response of driven systems, with their results compared to assess the validity regime of the perturbative approach. The proposed perturbative CFM approach needs scalar quantities only and hence is more efficient within its validity regime. Two nonlinear systems of different nature are used as examples: molecular dipole (rotational Brownian motion) and particle in a periodic potential (translational Brownian motion). The associated full dynamics is presented in the compact form of hysteresis loops. It is observed that as the strength of an AC driving field increases, pronounced nonlinear effects are manifested in the deformation of the hysteresis loops.

Scalar Spheroidal Harmonics in Five Dimensional Kerr-(A)dS

We derive expressions for the general five-dimensional metric for Kerr-(A)dS black holes. The Klein-Gordon equation is explicitly separated and we show that the angular part of the wave equation leads to just one spheroidal wave equation, which is also that for charged five-dimensional Kerr-(A)dS black holes. We present results for the perturbative expansion of the angular eigenvalue in powers of the rotation parameters up to 6th order and compare numerically with the continued fraction method.

Band structure and accumulation point in the spectrum of quantum collision operator in a one-dimensional molecular chain

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point, and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces.

Vibrational cross sections for positron scattering by nitrogen molecules

We present a systematic study of low-energy positron collision with nitrogen molecules. Vibrational elastic and excitation cross sections are calculated using the multichannel version of the continued fractions method in the close-coupling scheme for the positron incident energy up to $20$ eV. The interaction potential is treated within the static-correlation-polarization approximation. The comparison of our calculated data with existing theoretical and experimental results is encouraging.

Vincent’s theorem of 1836: overview and future research

In this paper, we present two different versions of Vincent’s theorem of 1836 and discuss various real root isolation methods derived from them: one using continued fractions and two using bisections, the former being the fastest real root isolation method. Regarding the continued fractions method, we first show how, using a recently developed quadratic complexity bound on the values of the positive roots of polynomials, its performance has been improved by an average of 40% over its initial implementation, and then we indicate directions for future research. Bibliography: 45 titles.

Kinks and waterfalls in the dispersion of ARPES features

In this paper we use a generic form for the Green function G( k, ω) in a correlated metal, already proven successful in describing ARPES line shapes [1]. The associated many body self-energy function has only a single pole. We now investigate, whether this generic model can be used all the way to the limit of strong correlations and, when applied to ARPES intensities, whether it is able to explain some of the ubiquitous dispersive crossover phenomena that have been attributed to dynamical, i.e.: ω-dependent effects. We argue that a quantitative interpretation of experimental data requires to calculate extrema not only in the momentum distribution curve but also in the energy distribution curve. In passing, we give a formula for the extrema in the latter distribution that is valid for the general G( k, ω) in a many body system. To our knowledge, this is a new formula, not found in the literature. The investigation of the generic model proceeds on two levels: on the one hand, we explore the rich variety of crossovers that can be predicted and linked to well defined features in the complex ω-plain. On the other hand, we show that the generic one-pole self-energy can be viewed as a projection on the low energy sector of a microscopic solution, belonging to a lattice model of interacting fermions. To obtain approximate microscopic solutions, we use our continued fraction method [2,3]. As an explicit example, we study the projection for the case of a hole doped Hubbard model in infinite dimension. A discussion section gives examples, how the generic model is able to cope with the ubiquity of the crossover phenomena, also in finite dimension and beyond the Hubbard model.

The extremal black hole bomb

We analyze the spectrum of massive scalar bound states in the background of extremal Kerr black holes, focusing on modes in the superradiant regime, which grow exponentially in time and quickly deplete the black hole's mass and spin. Previous analytical estimates for the growth rate of this instability were limited to the $\mu M\ll1$ and $\mu M\gg1$ regimes, where $\mu$ and $M$ denote the scalar field and black hole masses, respectively. In this work, we discuss an analytical method to compute the superradiant spectrum for generic values of these parameters, namely in the phenomenologically interesting regime $\mu M\sim 1$. To do this, we solve the radial mode equation in two overlapping regions and match the solutions in their common domain of validity. We show that matching the functional forms of these functions involves approximations that are not valid for the whole range of scalar masses, exhibiting unphysical poles that produce a large enhancement of the growth rate. Alternatively, we match the functions at a single point and show that, despite the uncertainty in the choice of the match point, this method eliminates the spurious poles and agrees with previous numerical computations of the spectrum using a continued-fraction method.

Quasinormal modes of Kerr–de Sitter black holes

We calculate the fundamental quasinormal modes of the Kerr--de Sitter black hole for the first time. In order to calculate the quasinormal modes, we employ the master equations derived by Suzuki, Takasugi, and Umetsu, who transform the Teukolsky equations for the Kerr--de Sitter black hole into the standard form of the Heun's equation. The transformed functions are expanded around the outer horizon of the black hole or the symmetric axis in the Fr\"obenius series whose coefficients satisfy a three-term recurrence relation. These three-term recurrence relations allow us to use Leaver's continued fraction method to calculate the angular separation constant and the quasinormal mode frequency. Any unstable fundamental quasinormal mode is not found in this paper. It is also observed that for some black holes characterized by a large mass parameter, some retrograde modes in the slow rotation limit become prograde as the black hole spin increases. This phenomenon does not occur for the fundamental modes of the Kerr black hole.

Arrow diagram theory for non-orthogonal electronic groups: the continued fractionsmethod

The group function theory by Tolpygo and McWeeny is a useful tool in treating quantumsystems that can be represented as a set of localized electronic groups (e.g. atoms, moleculesor bonds). It provides a general means of taking into account intra-correlation effects insidethe groups without assuming that the interaction between the groups is weak. Fornon-orthogonal group functions the arrow diagram (AD) technique provides a convenientprocedure for calculating matrix elements of arbitrary symmetrical operators which are needed, for example, for calculating the total energy of the system or itselectron density. The total wavefunction of the system is represented as an antisymmetrized product of non-orthogonal electron group functionsΦI of eachgroup I in the system. However, application of the AD theory to extended (e.g. infinite)systems (such as biological molecules or crystals) is not straightforward,since the calculation of the mean value of an operator requires that eachterm of the diagram expansion be divided by the normalization integralS = ⟨Ψ|Ψ⟩ which is given by an AD expansion as well. In our previous work, we cast the mean value of a symmetrical operator in the form of an AD expansion which is a linear combination of linked (connected) ADsmultiplied by numerical pre-factors. To obtain the pre-factors, a method based on powerseries expansion with respect to overlap was developed and tested for a simple 1DHartree–Fock (HF) ring model. In the present paper this method is first tested on a 2D HFmodel, and we find that the power series expansion for the pre-factors converges extremelyslowly to the exact solution. Instead, we suggest another, more powerful, methodbased on a continued fraction expansion of the pre-factors that approaches theexact solution much faster. The method is illustrated on the calculation of theelectron density for the 2D HF model. It provides a powerful technique for treatingextended systems consisting of a large number of strongly localized electronic groups.

Low energy scattering of positrons by [formula omitted

We present a theoretical investigation on elastic positron- H 2 O collisions. More specifically, differential and integral cross sections in the 0–10 eV energy range are reported. The calculations were performed using two theoretical approaches, namely, the Schwinger multichannel method and the method of continued fractions. The positron–molecule interaction dynamics is described by using a potential composed of static and correlation–polarization contributions. Comparison of our calculated results with the recent experimental of Zecca et al. [J. Phys. B 39 (2006) 1597] and theoretical results is encouraging.

Asymptotic iteration method for spheroidal harmonics of higher-dimensional Kerr-(A)dS black holes

In this work we calculate the angular eigenvalues of the ($n+4$)-dimensional simply rotating Kerr-(A)dS spheroidal harmonics using the asymptotic iteration method. We make some comparisons between this method and that of the continued fraction method and use the latter to check our results. We also present analytic expressions for the small rotation limit up to $O({c}^{3})$ with the coefficient of each power up to $O({\ensuremath{\alpha}}^{2})$, where $c=a\ensuremath{\omega}$ and $\ensuremath{\alpha}={a}^{2}\ensuremath{\Lambda}$ ($a$ is the angular velocity, $\ensuremath{\omega}$ the frequency, and $\ensuremath{\Lambda}$ the cosmological constant).

The massive Dirac field on a rotating black hole spacetime: angular solutions

The massive Dirac equation on a Kerr–Newman background may be solved by the method of separation of variables. The radial and angular equations are coupled via an angular eigenvalue, which is determined from the Chandrasekhar–Page (CP) equation. Obtaining accurate angular eigenvalues is a key step in studying scattering, absorption and emission of the fermionic field. Here we introduce a new method for finding solutions of the CP equation. First, we introduce a novel representation for the spin-half spherical harmonics. Next, we decompose the angular solutions of the CP equation (the mass-dependent spin-half spheroidal harmonics) in the spherical basis. The method yields a three-term recurrence relation which may be solved numerically via continued-fraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case μ = ±ω (where ω and μ are the frequency and mass of the fermion) we obtain eigenvalues and eigenfunctions in a closed form. We study the eigenvalue spectrum and the zeros of the maximally co-rotating mode. We compare our results with previous studies, and uncover and correct some errors in the literature. We provide series expansions, tables of eigenvalues and numerical fits across a wide parameter range and present plots of a selection of eigenfunctions. It is hoped that this study will be a useful resource for all researchers interested in the Dirac equation on a rotating black hole background.

Imprints of the nuclear symmetry energy on gravitational waves from the axialw-modes of neutron stars

The eigenfrequencies of the axial $w$-modes of oscillating neutron stars are studied using the continued fraction method with an equation of state (EOS) partially constrained by the recent terrestrial nuclear laboratory data. It is shown that the density dependence of the nuclear symmetry energy ${E}_{\mathrm{sym}}(\ensuremath{\rho})$ affects significantly both the frequencies and the damping times of these modes. Besides confirming the previously found universal behavior of the mass-scaled eigenfrequencies as functions of the compactness of neutron stars, we explored several alternative universal scaling functions. Moreover, the ${w}_{\mathit{II}}$-mode is found to exist only for neutron stars having a compactness of $M/R\ensuremath{\geqslant}0.1078$ independent of the EOS used.

Probing dense matter in neutron stars with axialwmodes

We study the problem of extracting information about composition and equation of state of dense matter in neutron star interior using axial $w$ modes. We determine complex frequencies of axial $w$ modes for a set of equations of state involving hyperons as well as Bose-Einstein condensates of antikaons adopting the continued fraction method. Hyperons and antikaon condensates result in softer equations of state leading to higher frequencies and lower damping times of first axial $w$ modes than those of the nuclear matter case. The presence of condensates may lead to the appearance of a new stable branch of superdense stars beyond the neutron star branch called the third family. The existence of the same mass compact stars in both branches is known as neutron star twins. Further investigation of twins reveals that first axial $w$-mode frequencies of superdense stars in the third family are higher than those of the corresponding twins in the neutron star branch.

Net particle current in an adiabatically driven unbiased inhomogeneous inertial system in aperiodic potential

We extend Risken’s matrix continued fraction method (MCFM) to solve the Fokker–Planckequation to calculate the particle current in an inertial symmetric (sinusoidal) periodicpotential under the action of a constant force. We consider particle motion ina medium with a small friction coefficient varying periodically in space as thepotential but with a finite phase difference . Though the frictional inhomogeneity is considered a less likely candidate to trigger theratchet effect than the temperature inhomogeneity, for the former leaves the staticequilibrium distribution unaffected as opposed to the latter, it results in a finite netalgebraic sum of currents with symmetrically time-distributed small applied forces ± |F|. The MCFM results, with very rich and interesting qualitative characteristics, aresupported by Langevin dynamic simulations. The effects of different uniformF, temperatureT and averagefriction coefficient γ0 on the performance characteristics of the inhomogeneous ratchet are presented.

Double-Diffusive Convection in a Porous Layer in the Presence of Vibration

The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal layer of a binary fluid mixture saturating a porous medium. A constant temperature and concentration distribution are assigned on the rigid boundaries, so that there exist vertical temperature and concentration gradients. The mathematical model is described by equations of filtration convection in the Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed using Floquet theory. Employing the method of continued fractions allows derivation of the dispersion equation for the Floquet exponent in an explicit form. The neutral curves of the Rayleigh number versus horizontal wave number are constructed for the three types of instability modes: synchronous, subharmonic, and complex conjugate. Asymptotic formulas for these curves are derived for large values of vibration frequency using the method of averaging. It is shown that, at some finite frequencies of vibration, there exist regions of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility to convection. In addition, even in the presence of vibration, the onset of convection in the system is affected by variations in the concentration of the solute in the mixture.

Continued Fraction Method for an Ancient Chinese Musical Equation

Continued Fraction Method for an Ancient Chinese Musical Equation

Motional dispersions and ratchet effect in inertial systems

We obtain ratchet effect in inertial structureless systems in symmetric periodic potentials where the asymmetry comes from the non-uniform friction offered by the medium and driven by symmetric periodic forces. In the adiabatic limit the calculations are done by extending the matrix continued fraction method and also by numerically solving the appropriate Langevin equation. For finite frequency field drive the ratchet effect is obtained only numerically. In the transient time scales the system shows dispersionless behaviour as reported earlier when a constant force is applied. In the periodic drive case the dispersion behaviour is more complex. In this brief communication we report some of the results of our work.

Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots

In this paper we compare four implementations of the Vincent-AkritasStrzebo´nski Continued Fractions (VAS-CF) real root isolation method using four different (two linear and two quadratic complexity) bounds on the values of the positive roots of polynomials. The quadratic complexity bounds were included to see if the quality of their estimates compensates for their quadratic complexity. Indeed, experimentation on various classes of special and random polynomials revealed that the VAS-CF implementation using LMQ, the Quadratic complexity variant of our Local Max bound, achieved an overall average speed-up of 40 % over the original implementation using Cauchy’s linear bound.