Error Bounds of Continued Fractions for Complex Transport Coefficients and Spectral Functions

1978 ◽  
Vol 33 (11) ◽  
pp. 1380-1382 ◽  
Author(s):  
P. Hänggi
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Truncation Error ◽  
Transport Coefficients ◽  
Power Spectra ◽  
A Posteriori ◽  
Spectral Functions ◽  
Dynamical Equilibrium ◽  
Complex Continued Fractions

We study the calculation of complex transport coefficients x (ω) and power spectra in terms of complex continued fractions. In particular, we establish classes of dynamical equilibrium and non-equilibrium systems for which we can obtain a posteriori bounds for the truncation error | x (ω) - x(n)(ω)| ≦ c (ω) | x(n)(ω) - x(n-1)(ω)| when the transport coefficient is approximated by its n-th continued fraction approximant x(n)(ω).

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

A Posteriori Bounds for the Truncation Error of Continued Fractions

1971 ◽  
Vol 8 (4) ◽  
pp. 693-705 ◽  
Author(s):  
William B. Jones ◽  
W. J. Thron
Keyword(s):  
Continued Fractions ◽  
Truncation Error ◽  
A Posteriori

Truncation Error Bounds for Continued Fractions $K({{a_n } / 1})$ with Parabolic Element Regions

1983 ◽  
Vol 20 (6) ◽  
pp. 1219-1230 ◽  
Author(s):  
William B. Jones ◽  
W. J. Thron ◽  
Haakon Waadeland
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Truncation Error ◽  
Parabolic Element

scholarly journals Truncation error bounds for branched continued fraction whose partial denominators are equal to unity

2020 ◽  
Vol 54 (1) ◽  
pp. 3-14
Author(s):  
R. I. Dmytryshyn ◽  
T. M. Antonova
Keyword(s):  
Complex Plane ◽  
Error Bounds ◽  
Continued Fraction ◽  
Truncation Error ◽  
Independent Variables ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of obtaining error bounds for branched continued fraction of the form $\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{a_{i(2)}}{1}{\atop+}\sum_{i_3=1}^{i_2}\frac{a_{i(3)}}{1}{\atop+}\ldots$. By means of fundamental inequalities method the truncation error bounds are obtained for the above mentioned branched continued fraction providing its elements belong to some rectangular sets ofa complex plane. Applications are considered for several classes of branched continued fraction expansions including the multidimensional \emph{S}-, \emph{A}-, \emph{J}-fractions with independent variables.

Truncation Error Bounds for Continued Fractions

1969 ◽  
Vol 6 (2) ◽  
pp. 210-221 ◽  
Author(s):  
William B. Jones ◽  
R. I. Snell
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Truncation Error

scholarly journals Nonperturbative effects on radiative energy loss of heavy quarks

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Shuai Y.F. Liu ◽  
Ralf Rapp
Keyword(s):  
Energy Loss ◽  
Scattering Amplitude ◽  
Transport Coefficients ◽  
Power Spectra ◽  
Heavy Quarks ◽  
Radiative Energy ◽  
Spectral Functions ◽  
Fixed Temperature ◽  
Strong Suppression ◽  
Radiative Energy Loss

Abstract The radiative energy loss of fast partons traveling through the quark-gluon plasma (QGP) is commonly studied within perturbative QCD (pQCD). Nonperturbative (NP) effects, which are expected to become important near the critical temperature, have been much less investigated. Here, we utilize a recently developed T -matrix approach to incorporate NP effects for gluon emission off heavy quarks propagating through the QGP. We set up four cases that contain, starting from a Born diagram calculation with color- Coulomb interaction, an increasing level of NP components, by subsequently including (remnants of ) confining interactions, resummation in the heavy-light scattering amplitude, and off-shell spectral functions for both heavy and light partons. For each case we compute the power spectra of the emitted gluons, heavy-quark transport coefficients (drag and transverse-momentum broadening, $$ \hat{q} $$ q ̂ ), and the path-length dependent energy loss within a “QGP brick” at fixed temperature. Investigating the differences in these quantities between the four cases illustrates how NP mechanisms affect gluon radiation processes. While the baseline perturbative processes experience a strong suppression of soft radiation due to thermal masses of the emitted gluons, confining interactions, ladder resummations and broad spectral functions (re-)generate a large enhancement toward low momenta and low temperatures. For example, for a 10 GeV charm quark at 200 MeV temperature, they enhance the transport coefficients by up to a factor of 10, while the results smoothly converge to perturbative results at sufficiently hard scales.

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