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arxiv: 2605.08998 · v1 · submitted 2026-05-09 · 🌌 astro-ph.GA

Recognition: no theorem link

Complexity and Multifractal Variability in Multi-Band Emission of Seyfert AGN

R. A. A. Souza , E. Hatziminaoglou , A. de P\'adua Santos , T. Stosic
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Pith reviewed 2026-05-12 01:55 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords AGN variabilitymultifractal analysisFisher-Shannon planeSeyfert galaxiesmulti-band light curvesX-ray variabilityUV and optical emissiontime series complexity
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The pith

Multifractal detrended moving average and Fisher-Shannon analyses of multi-band AGN light curves identify distinct variability patterns linked to specific physical emission processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies multifractal detrended moving average analysis to measure scaling and multifractality in the X-ray, ultraviolet, and optical light curves of four Seyfert AGNs, and uses the Fisher-Shannon information plane to quantify stochasticity and structural complexity. It finds persistent correlations across bands along with band-specific differences in these measures. These patterns indicate that the techniques can serve as diagnostics for the physical mechanisms responsible for the emissions. A reader would care because the work offers a way to probe accretion and emission physics near supermassive black holes using time-series properties rather than spectra alone. The study positions these methods as practical tools for interpreting complex AGN variability.

Core claim

The central claim is that multifractal detrended moving average analysis and the Fisher-Shannon information plane, when applied to the X-ray, ultraviolet, and optical light curves of Mrk 509, NGC 5548, NGC 4151, and NGC 4593, quantify the scaling behavior, complexity, and stochasticity of the variability, revealing distinct patterns across energy bands that support their use as effective diagnostics of the physical mechanisms driving AGN emission.

What carries the argument

Multifractal detrended moving average (MFDMA) analysis, which extracts scaling exponents and multifractal spectra from non-stationary time series, paired with the Fisher-Shannon plane that positions signals by their information content and complexity.

If this is right

  • X-ray light curves exhibit stronger multifractality than optical ones, consistent with more compact and turbulent emission regions closer to the black hole.
  • The Fisher-Shannon plane separates the bands by degree of stochasticity, linking higher complexity to specific accretion or reprocessing mechanisms.
  • Persistent cross-band correlations imply that variability in one band influences others through connected physical processes.
  • These measures outperform basic statistical descriptors in capturing the non-linear and scale-dependent nature of AGN fluctuations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline could be tested on high-cadence data from upcoming surveys to classify AGN states in real time.
  • Comparing these metrics against hydrodynamic simulations of accretion disks would test whether the observed multifractality arises from specific disk instabilities.
  • Extending the analysis to radio or gamma-ray bands might reveal how variability propagates outward from the central engine.
  • If the diagnostics prove robust, they could help isolate the contribution of jets versus disk emission in radio-loud AGNs.

Load-bearing premise

The observed multifractal spectra and Fisher-Shannon coordinates of the light curves map directly onto distinct physical emission processes without major distortion from sampling gaps, noise, or unrelated variability components.

What would settle it

Recomputing the MFDMA spectra and Fisher-Shannon points on phase-randomized or gap-filled versions of the same light curves and finding that the band-to-band differences disappear or match those from purely stochastic simulations with no physical structure.

Figures

Figures reproduced from arXiv: 2605.08998 by A. de P\'adua Santos, E. Hatziminaoglou, R. A. A. Souza, T. Stosic.

Figure 1
Figure 1. Figure 1: The original light curve of NGC 5548 for all eight observational bands. Light curves of the analyzed AGN, ordered by wavelength from top to bottom. The top two panels correspond to XRT observations in the HX (1.5–10 keV) and SX (0.3–1.5 keV) bands, while the lower six panels show the UVOT bands (W2, M2, W1, U, B, and V). Each panel displays the flux as a function of MJD, sharing a common time axis but with… view at source ↗
Figure 2
Figure 2. Figure 2: Multifractal singularity spectrum 𝑓 (𝛼) illustrating the position of the maximum at 𝛼0, the spectrum width Δ𝛼, and the left and right contribu￾tions used to define the asymmetry of the spectrum. 3.2.1 Shannon Entropy Power and Fisher Information Measure Let 𝑋 be a continuous random variable with probability density function (PDF) 𝑓 (𝑥). The differential entropy of 𝑋 is defined as Cover & Thomas (2006): 𝐻𝑋 … view at source ↗
Figure 3
Figure 3. Figure 3: Fluctuation function 𝐹𝑞 (𝑛) of the original light curve of NGC 5548 for all eight observational bands. The panels are arranged from left to right, top to bottom, in the following order: HX, SX, W2, M2, W1, U, B, and V. Each panel shows the scale dependence of fluctuations for different values of 𝑞, highlighting the variability characteristics across energy bands. MNRAS 000, 1–19 (2026) [PITH_FULL_IMAGE:fi… view at source ↗
Figure 4
Figure 4. Figure 4: Multifractal analysis using the MFDMA method for the four quasars in the sample, top to bottom: Mrk 509, NGC 4151, NGC 4593, and NGC 5548. Each figure consists of three panels, from left to right: the Hölder exponent ℎ(𝑞) as a function of the moment 𝑞, the scaling exponent 𝜏 (𝑞) versus 𝑞, and the multifractal spectrum 𝑓 (𝛼) as a function of the generalized Hölder exponent 𝛼. These plots characterize the mu… view at source ↗
Figure 5
Figure 5. Figure 5: Scatter plot of fractional variability as a function of Δ𝛼 for different AGNs. Each color represents a distinct AGN. The plot reveals a strong positive correlation between the two quantities, suggesting that larger spectral widths are associated with greater variability amplitudes. 4.2 Fisher-Shannon Analysis In this stage, we compute the Shannon entropy power (𝑁𝑋) and the Fisher information (𝐼𝑋) for each … view at source ↗
read the original abstract

Active galactic nuclei (AGNs) exhibit complex variability across multiple wavelengths, reflecting diverse physical processes near their central engines. This work investigates the temporal variability of four AGNs Mrk~509, NGC~5548, NGC~4151, and NGC~4593 using multifractal detrended moving average (MFDMA) analysis and Fisher-Shannon information plane applied to their X-ray, ultraviolet, and optical light curves. These methods quantify the scaling behavior and complexity of the variability, revealing persistent correlations and distinct variability patterns across energy bands. The Fisher-Shannon analysis further characterizes the degree of stochasticity and structural complexity in the emission processes. Our findings support the interpretation that multifractal and information-theoretic measures provide effective diagnostics of the physical mechanisms driving AGN variability. This study demonstrates the utility of advanced time series techniques as effective diagnostics of AGN variability mechanisms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper applies multifractal detrended moving average (MFDMA) analysis and Fisher-Shannon information-plane methods to X-ray, UV, and optical light curves of four Seyfert AGNs (Mrk 509, NGC 5548, NGC 4151, NGC 4593). It reports band-dependent scaling exponents, multifractal widths, and complexity measures, and concludes that these quantities constitute effective diagnostics of the distinct physical processes driving multi-band AGN variability.

Significance. The application of MFDMA and Fisher-Shannon complexity to multi-wavelength AGN monitoring data is a reasonable extension of existing time-series tools. If the reported diagnostics can be shown to isolate physical emission mechanisms rather than sampling artifacts, the work would supply a quantitative framework for interpreting variability across energy bands. The manuscript is data-driven and uses named, reproducible algorithms, which are positive attributes.

major comments (1)
  1. [Abstract and Conclusions] Abstract and final interpretation: the central claim that MFDMA spectra and Fisher-Shannon coordinates 'provide effective diagnostics of the physical mechanisms driving AGN variability' is not accompanied by any controlled forward-modeling tests. The light curves contain irregular cadences, seasonal gaps, and heteroscedastic errors; without injecting synthetic signals (reprocessing lags, broken power-law PSDs, or disk instabilities) into realistic sampling windows and recovering the same multifractal width and information-plane coordinates, it is impossible to determine whether the reported X-ray/UV/optical differences reflect emission physics or observational artifacts. This validation step is load-bearing for the diagnostic interpretation.
minor comments (2)
  1. [Methods] Methods section: state explicitly the MFDMA window length, polynomial order, and segment size choices, and demonstrate that results are stable under reasonable variations of these parameters given the finite lengths and gaps in the light curves.
  2. [Figures] Figures: add uncertainty bands or bootstrap errors to the multifractal spectra and Fisher-Shannon plane points; label all panels with the exact energy band and object name for immediate readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recognition that the application of MFDMA and Fisher-Shannon methods to multi-band AGN data is a reasonable extension of existing tools. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and Conclusions] Abstract and final interpretation: the central claim that MFDMA spectra and Fisher-Shannon coordinates 'provide effective diagnostics of the physical mechanisms driving AGN variability' is not accompanied by any controlled forward-modeling tests. The light curves contain irregular cadences, seasonal gaps, and heteroscedastic errors; without injecting synthetic signals (reprocessing lags, broken power-law PSDs, or disk instabilities) into realistic sampling windows and recovering the same multifractal width and information-plane coordinates, it is impossible to determine whether the reported X-ray/UV/optical differences reflect emission physics or observational artifacts. This validation step is load-bearing for the diagnostic interpretation.

    Authors: We agree that the lack of forward-modeling tests weakens the strength of the diagnostic claim. Our analysis reports statistically significant band-dependent differences in scaling exponents, multifractal widths, and Fisher-Shannon coordinates on the observed light curves, and these differences are consistent with the expected hierarchy of variability mechanisms (X-ray corona, reprocessing in the disk, and outer-disk contributions). Nevertheless, without synthetic injections we cannot fully exclude sampling artifacts. In the revised manuscript we will add a dedicated validation section that generates synthetic light curves with prescribed PSD shapes, reprocessing lags, and instability-driven variability, resamples them onto the actual cadences and error distributions of each band, and demonstrates that the MFDMA and Fisher-Shannon diagnostics recover the input distinctions. This addition will directly address the referee's concern and support the interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard methods applied to independent external data

full rationale

The paper applies MFDMA and Fisher-Shannon information-plane analysis directly to published X-ray/UV/optical light curves of four AGNs. These are off-the-shelf techniques whose outputs (multifractal spectra, complexity coordinates) are computed from the input time series without any parameter fitted to a subset and then re-used as a 'prediction' of the same data. No equations reduce the reported diagnostics to quantities defined from the target result itself, and no self-citation chain is invoked to establish uniqueness or to smuggle an ansatz. The central claim therefore remains an independent mapping from external observations to complexity measures.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5470 in / 1103 out tokens · 40274 ms · 2026-05-12T01:55:41.846198+00:00 · methodology

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Reference graph

Works this paper leans on

237 extracted references · 237 canonical work pages

  1. [1]

    2025 , type =

    Souza, Rhimon Alves de Assis , title =. 2025 , type =

    work page 2025

  2. [2]

    C., & Katz, S

    The AGN Black Hole Mass Database. , keywords =. doi:10.1086/679601 , archivePrefix =. 1411.2596 , primaryClass =

    work page doi:10.1086/679601

  3. [3]

    Radio Science , keywords =

    Examining the Power Law Relationship Between Absorption and Frequency Using Spectral Riometer Data. Radio Science , keywords =. doi:10.1029/2024RS007951 , adsurl =

    work page doi:10.1029/2024rs007951

  4. [4]

    Publications of the Astronomical Society of the Pacific , volume=

    The spectra of some spiral nebulae and globular star clusters , author=. Publications of the Astronomical Society of the Pacific , volume=. 1909 , publisher=

    work page 1909

  5. [5]

    Acta Geophysica , volume =

    Discriminating Between Different Streamflow Regimes by Using the Fisher-Shan Method: An Application to the Colombia Rivers , author =. Acta Geophysica , volume =. 2015 , doi =

    work page 2015

  6. [6]

    , keywords =

    Revisiting the Spectral and Timing Properties of NGC 4151. , keywords =. doi:10.3847/1538-4357/ab3e31 , archivePrefix =. 1908.09862 , primaryClass =

    work page doi:10.3847/1538-4357/ab3e31 1908

  7. [7]

    , year = 1943, month = jan, volume =

    Nuclear Emission in Spiral Nebulae. , year = 1943, month = jan, volume =. doi:10.1086/144488 , adsurl =

    work page doi:10.1086/144488 1943

  8. [8]

    , keywords =

    Outflow of gas in the Seyfert 1 galaxy Markarian 509. , keywords =. doi:10.1086/161470 , adsurl =

    work page doi:10.1086/161470

  9. [9]

    Land , volume =

    Assessing Urban Resilience from the Perspective of Scaling Law: Evidence from Chinese Cities , author =. Land , volume =. 2022 , doi =

    work page 2022

  10. [10]

    Physical Review E , volume =

    Superlinear and sublinear urban scaling in geographical networks modeling cities , author =. Physical Review E , volume =. 2014 , doi =

    work page 2014

  11. [11]

    , keywords =

    Multifractal detrended moving average analysis of Kepler stars with surface differential rotation traces. , keywords =. doi:10.1093/mnras/stz1791 , archivePrefix =. 1906.11911 , primaryClass =

    work page doi:10.1093/mnras/stz1791 1906

  12. [12]

    , keywords =

    Multiscale behaviour of stellar activity and rotation of the planet host Kepler-30. , keywords =. doi:10.1051/0004-6361/202140287 , archivePrefix =. 2103.15921 , primaryClass =

    work page doi:10.1051/0004-6361/202140287

  13. [13]

    iScience , keywords =

    Reverberation mapping of active galactic nuclei: from X-ray corona to dusty torus. iScience , keywords =. doi:10.1016/j.isci.2021.102557 , archivePrefix =. 2105.06926 , primaryClass =

    work page doi:10.1016/j.isci.2021.102557 2021

  14. [14]

    The Astrophysical Journal , volume =

    Accretion Disk Reverberation with Hubble Space Telescope Observations of NGC 4593: Evidence for Diffuse Continuum Lags , author =. The Astrophysical Journal , volume =. 2018 , publisher =

    work page 2018

  15. [15]

    Spectral and informational analysis of pedestrian contact force in simulated overcrowding conditions , journal =

    Xudong Li and Luciano Telesca and Michele Lovallo and Xuan Xu and Jun Zhang and Weiguo Song , keywords =. Spectral and informational analysis of pedestrian contact force in simulated overcrowding conditions , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.physa.2020.124614 , url =

    work page doi:10.1016/j.physa.2020.124614 2020

  16. [16]

    Entropy , volume =

    Quantifying Soil Complexity Using Fisher Shannon Method on 3D X-ray Computed Tomography Scans , author =. Entropy , volume =. 2023 , doi =

    work page 2023

  17. [17]

    Lowell Observatory Bulletin, vol

    The spectrum and velocity of the nebula NGC 1068 (M 77) , author=. Lowell Observatory Bulletin, vol. 3, pp. 59-62 , volume=

    work page

  18. [18]

    , keywords =

    Multifractality signatures in lensed quasars. , keywords =. doi:10.1093/mnras/stae2490 , archivePrefix =. 2411.02076 , primaryClass =

    work page doi:10.1093/mnras/stae2490

  19. [19]

    Physical Review E , volume=

    Transfer entropy in physical systems and the arrow of time , author=. Physical Review E , volume=. 2016 , publisher=

    work page 2016

  20. [20]

    Kaastra, J. S. et al. , title =. Astronomy & Astrophysics , year =

    work page

  21. [21]

    Mehdipour, M. et al. , title =. Astronomy & Astrophysics , year =

    work page

  22. [22]

    Landt, H. et al. , title =. The Astrophysical Journal , year =

    work page

  23. [23]

    Shapovalova, A. I. et al. , title =. Astronomy & Astrophysics , year =

    work page

  24. [24]

    , title =

    Veilleux, S. , title =. 1991 , url =

    work page 1991

  25. [25]

    Ward, M. J. et al. , title =. Monthly Notices of the Royal Astronomical Society , year =

    work page

  26. [26]

    , keywords =

    X-ray/UV/optical variability of NGC 4593 with Swift: reprocessing of X-rays by an extended reprocessor. , keywords =. doi:10.1093/mnras/sty1983 , archivePrefix =. 1712.04852 , primaryClass =

    work page doi:10.1093/mnras/sty1983

  27. [27]

    , keywords =

    The discovery of non-linear X-ray variability in NGC 4051. , keywords =. doi:10.1046/j.1365-8711.1999.02370.x , adsurl =

    work page doi:10.1046/j.1365-8711.1999.02370.x 1999

  28. [28]

    , keywords =

    Nonlinear variability of NGC 4151. , keywords =

    work page

  29. [29]

    , keywords =

    Luminosity Variations of 3C 345: Is There Any Evidence of Low-dimensional Chaos?. , keywords =. doi:10.1086/174267 , adsurl =

    work page doi:10.1086/174267

  30. [30]

    , keywords =

    Time Series Analysis in Astronomy: an Application to Quasar Variability Studies. , keywords =. doi:10.1086/171367 , adsurl =

    work page doi:10.1086/171367

  31. [31]

    Astrophysics and Space Science , keywords =

    Nonlinear Wave-Wave Interactions in Astrophysical and Space Plasmas. Astrophysics and Space Science , keywords =. doi:10.1007/BF00645115 , adsurl =

    work page doi:10.1007/bf00645115

  32. [32]

    Directed information measures in neuroscience , pages=

    Transfer entropy in neuroscience , author=. Directed information measures in neuroscience , pages=. 2014 , publisher=

    work page 2014

  33. [33]

    Entropy , volume=

    On thermodynamic interpretation of transfer entropy , author=. Entropy , volume=. 2013 , publisher=

    work page 2013

  34. [34]

    , keywords =

    A Novel Approach to Study the Variability of NGC 5548. , keywords =. doi:10.3847/1538-4357/ab2457 , archivePrefix =. 1905.10128 , primaryClass =

    work page doi:10.3847/1538-4357/ab2457 1905

  35. [35]

    M., Tremonti, C., et al

    On characterizing the variability properties of X-ray light curves from active galaxies. , keywords =. doi:10.1046/j.1365-2966.2003.07042.x , archivePrefix =. astro-ph/0307420 , primaryClass =

    work page doi:10.1046/j.1365-2966.2003.07042.x 2003

  36. [36]

    , keywords =

    The relentless variability of Mrk 421 from the TeV to the radio. , keywords =. doi:10.1051/0004-6361/201935557 , archivePrefix =. 2101.10651 , primaryClass =

    work page doi:10.1051/0004-6361/201935557

  37. [37]

    1988 , edition =

    Feder, Jens , title =. 1988 , edition =. doi:10.1007/978-1-4899-2124-6 , note =

    work page doi:10.1007/978-1-4899-2124-6 1988

  38. [38]

    The Astrophysical Journal , volume=

    An X-ray, infrared, and submillimeter flare of Sagittarius A , author=. The Astrophysical Journal , volume=. 2008 , publisher=

    work page 2008

  39. [39]

    Nature Astronomy , volume=

    The accretion of a solar mass per day by a 17-billion solar mass black hole , author=. Nature Astronomy , volume=. 2024 , publisher=

    work page 2024

  40. [40]

    The fastest-growing black hole in the Universe , author=

    work page

  41. [41]

    Astronomy & Astrophysics , volume=

    Millimeter to X-ray flares from Sagittarius A , author=. Astronomy & Astrophysics , volume=. 2012 , publisher=

    work page 2012

  42. [42]

    The Astrophysical Journal , volume=

    Chandra X-ray spectroscopic imaging of Sagittarius A* and the central parsec of the galaxy , author=. The Astrophysical Journal , volume=. 2003 , publisher=

    work page 2003

  43. [43]

    Monthly Notices of the Royal Astronomical Society , volume=

    A gravitationally lensed quasar discovered in OGLE , author=. Monthly Notices of the Royal Astronomical Society , volume=. 2018 , publisher=

    work page 2018

  44. [44]

    Biomedical optics express , volume=

    Multifractal nature of ocular aberration dynamics of the human eye , author=. Biomedical optics express , volume=. 2011 , publisher=

    work page 2011

  45. [45]

    The Astrophysical Journal , volume=

    The quasar accretion disk size--black hole mass relation , author=. The Astrophysical Journal , volume=. 2010 , publisher=

    work page 2010

  46. [46]

    Journal of Statistical Mechanics: Theory and Experiment , volume=

    Multifractal analysis of sunspot time series: the effects of the 11-year cycle and Fourier truncation , author=. Journal of Statistical Mechanics: Theory and Experiment , volume=. 2009 , publisher=

    work page 2009

  47. [47]

    Journal of Statistical Mechanics: Theory and Experiment , keywords =

    Multifractal detrended fluctuation analysis of sunspot time series. Journal of Statistical Mechanics: Theory and Experiment , keywords =. doi:10.1088/1742-5468/2006/02/P02003 , archivePrefix =. physics/0508149 , primaryClass =

    work page doi:10.1088/1742-5468/2006/02/p02003 2006

  48. [48]

    Galaxies , volume=

    The blazar sequence and its physical understanding , author=. Galaxies , volume=. 2022 , publisher=

    work page 2022

  49. [49]

    , author=

    Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. , author=. The Lancet , volume=. 1996 , publisher=

    work page 1996

  50. [50]

    2017 , publisher=

    Gravitational lensing , author=. 2017 , publisher=

    work page 2017

  51. [51]

    Astronomy and Astrophysics (ISSN 0004-6361), vol

    A new formulation of gravitational lens theory, time-delay, and Fermat's principle , author=. Astronomy and Astrophysics (ISSN 0004-6361), vol. 143, no. 2, Feb. 1985, p. 413-420. , volume=

    work page 1985

  52. [52]

    Nature , volume=

    Cosmological effects of primordial black holes , author=. Nature , volume=. 1975 , publisher=

    work page 1975

  53. [53]

    Chinese Physics C , volume=

    Digital discrimination of neutron and ray using an organic scintillation detector based on wavelet transform modulus maximum , author=. Chinese Physics C , volume=. 2014 , publisher=

    work page 2014

  54. [54]

    International Journal of Economics and Finance Studies , volume=

    Multifractal Behaviour in Natural Gas Prices by using Mf-DFA and WTMM Methods , author=. International Journal of Economics and Finance Studies , volume=. 2013 , publisher=

    work page 2013

  55. [55]

    2013 IEEE 9th International Colloquium on Signal Processing and its Applications , pages=

    High performance visualization of human tumor detection using WTMM on Parallel Computing System , author=. 2013 IEEE 9th International Colloquium on Signal Processing and its Applications , pages=. 2013 , organization=

    work page 2013

  56. [56]

    International Journal of Economics and Finance Studies , volume=

    Multifractal behaviour in oil prices by using MF-DFA and WTMM methods , author=. International Journal of Economics and Finance Studies , volume=. 2013 , publisher=

    work page 2013

  57. [57]

    Frontiers in computational neuroscience , volume=

    Transfer entropy as a measure of brain connectivity: A critical analysis with the help of neural mass models , author=. Frontiers in computational neuroscience , volume=. 2020 , publisher=

    work page 2020

  58. [58]

    Physical review letters , volume=

    Measuring information transfer , author=. Physical review letters , volume=. 2000 , publisher=

    work page 2000

  59. [59]

    Finance Research Letters , volume=

    Predict or to be predicted? A transfer entropy view between adaptive green markets, structural shocks and sentiment index , author=. Finance Research Letters , volume=. 2023 , publisher=

    work page 2023

  60. [60]

    Atmospheric Research , volume=

    Forecasting rainfall using transfer entropy coupled directed--weighted complex networks , author=. Atmospheric Research , volume=. 2021 , publisher=

    work page 2021

  61. [61]

    Journal of Environmental Management , volume=

    Exploring the effect of climate risk on agricultural and food stock prices: Fresh evidence from EMD-Based variable-lag transfer entropy analysis , author=. Journal of Environmental Management , volume=. 2023 , publisher=

    work page 2023

  62. [62]

    Studies in Nonlinear Dynamics and Econometrics , volume=

    Using transfer entropy to measure information flows between financial markets , author=. Studies in Nonlinear Dynamics and Econometrics , volume=. 2013 , publisher=

    work page 2013

  63. [63]

    The Bell system technical journal , volume=

    A mathematical theory of communication , author=. The Bell system technical journal , volume=. 1948 , publisher=

    work page 1948

  64. [64]

    Physica D: Nonlinear Phenomena , volume=

    Information transfer in continuous processes , author=. Physica D: Nonlinear Phenomena , volume=. 2002 , publisher=

    work page 2002

  65. [65]

    Astronomy and Astrophysics, vol

    Primeval black holes and galaxy formation , author=. Astronomy and Astrophysics, vol. 38, no. 1, Jan. 1975, p. 5-13. , volume=

    work page 1975

  66. [66]

    Monthly Notices of the Royal Astronomical Society , volume=

    New evidence for a cosmological distribution of stellar mass primordial black holes , author=. Monthly Notices of the Royal Astronomical Society , volume=. 2022 , publisher=

    work page 2022

  67. [67]

    Astronomy & Astrophysics , volume=

    SDSS J1004+ 4112: the case for a galaxy cluster dominated by primordial black holes , author=. Astronomy & Astrophysics , volume=. 2020 , publisher=

    work page 2020

  68. [68]

    The Astrophysical Journal , volume=

    A new probe of dark matter and high-energy universe using microlensing , author=. The Astrophysical Journal , volume=. 2009 , publisher=

    work page 2009

  69. [69]

    Observational appearance

    Black holes in binary systems. Observational appearance. , author=. Astronomy and Astrophysics, Vol. 24, p. 337-355 , volume=

    work page

  70. [70]

    Astronomy & Astrophysics , volume=

    The signature of primordial black holes in the dark matter halos of galaxies , author=. Astronomy & Astrophysics , volume=. 2020 , publisher=

    work page 2020

  71. [71]

    The Astrophysical Journal , volume=

    The use of high-magnification microlensing events in discovering extrasolar planets , author=. The Astrophysical Journal , volume=. 1998 , publisher=

    work page 1998

  72. [72]

    Astronomy and Astrophysics Supplement Series , volume=

    30 years of multi-wavelength observations of 3C 273 , author=. Astronomy and Astrophysics Supplement Series , volume=. 1999 , publisher=

    work page 1999

  73. [73]

    The Astrophysical Journal Supplement Series , volume=

    The Next Generation Atlas of Quasar Spectral Energy Distributions from Radio to X-Rays , author=. The Astrophysical Journal Supplement Series , volume=. 2011 , publisher=

    work page 2011

  74. [74]

    The Astrophysical Journal , volume=

    HST STIS observations of PG 0946+ 301: The highest quality UV spectrum of a BALQSO , author=. The Astrophysical Journal , volume=. 2001 , publisher=

    work page 2001

  75. [75]

    The Astronomical Journal , volume=

    Composite quasar spectra from the sloan digital sky survey , author=. The Astronomical Journal , volume=. 2001 , publisher=

    work page 2001

  76. [76]

    Peterson, B. M. and McHardy, I. M. and Wilkes, B. J. and Berlind, P. and Bertram, R. and Calkins, M. and Collier, S. J. and Huchra, J. P. and Mathur, S. and Papadakis, I. and Peters, J. and Pogge, R. W. and Romano, P. and Tokarz, S. and Uttley, P. and Vestergaard, M. and Wagner, R. M. , title =. The Astrophysical Journal , abstract =. 2000 , month =. doi:...

    work page doi:10.1086/309518 2000

  77. [77]

    Monthly Notices of the Royal Astronomical Society , volume=

    Low-luminosity active galactic nuclei: are they UV faint and radio loud? , author=. Monthly Notices of the Royal Astronomical Society , volume=. 2007 , publisher=

    work page 2007

  78. [78]

    Nature , volume=

    3C273: a star-like object with large red-shift , author=. Nature , volume=

    work page

  79. [79]

    Nature , volume=

    Red-Shift of the Unusual Radio Source: 3C 48 , author=. Nature , volume=

    work page

  80. [80]

    Assessing the achievability and precision of time-delay measurements , author=

    COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses-XV. Assessing the achievability and precision of time-delay measurements , author=. Astronomy & Astrophysics , volume=. 2016 , publisher=

    work page 2016

Showing first 80 references.