diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfwzf" "b/data_all_eng_slimpj/shuffled/split2/finalzzfwzf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfwzf" @@ -0,0 +1,5 @@ +{"text":"\\section{Methods}\n\n\\subsection{The Controversy \u2013 Massive Black Hole or Supernova Remnant:}\n\nIn recent years, evidence has been mounting for a massive black hole powering a low-luminosity active galactic nucleus (AGN) at the center of Henize 2-10 \\cite{reines2011actively_Henize,reines2012parsec,reines2016deep,riffel2020evidence}, although a supernova remnant has been proposed as an alternative by some authors \\cite{hebbar2019x,cresci2017muse}. As discussed in the main text, the radio and X-ray point source luminosities are consistent with both. A recent study \\cite{hebbar2019x} argues for a supernova remnant based on their findings that the X-ray spectrum is better fit by a hot plasma model (typically used for supernova remnants) than a power-law model (typically used for luminous AGNs). However, the soft X-ray spectrum of the nuclear source in Henize 2-10 does resemble massive black holes accreting at very low Eddington fractions \\cite{constantin2009probing} including Sagittarius A$^*$ in the Milky Way \\cite{baganoff2003chandra}. Another study using ground-based spectroscopy favored a supernova remnant origin for the central source based on a lack of any AGN ionization signatures \\cite{cresci2017muse}. However, the ground-based observations used in that work had an angular resolution of $\\sim$0.7\", which is not sufficient to cleanly isolate the weakly accreting black hole from nearby young ($<$5 Myr) massive (M$_* \\sim 10^5 M_{\\odot}$) star clusters that dominate the line ratios at this relatively course angular resolution.\nThere are other observational results to consider regarding the origin of the nuclear radio\/X-ray source in Henize 2-10. For example, a recent study using adaptive optics integral field spectroscopy provides evidence for gas excited by an AGN and an enhanced stellar velocity dispersion at the location of the nuclear source consistent with a $\\sim10^6 M_{\\odot}$ black hole, favoring the low-luminosity AGN interpretation \\cite{riffel2020evidence}. There is also evidence for moderately significant variability on hour-long timescales in the X-ray light curve, which is incompatible with a supernova remnant \\cite{reines2016deep,hebbar2019x}. Moreover, it is reasonable to expect that Henize 2-10 hosts a massive black hole since its overall structure resembles an early-type galaxy (albeit with a central starburst) and its stellar mass may be as high as M$_* \\sim 10^{10} M_{\\odot}$ \\cite{nguyen2014extended}, a regime where the black hole occupation fraction is near unity \\cite{greene2020intermediate}. The central starburst complicates the identification of the weakly accreting black hole, yet a variety of multiwavelength observational results taken collectively strongly support its presence. These results are summarized in Figure \\ref{tab:AGNSNR_tab} (Extended Data Table 1). A highly sub-Eddington massive black hole is consistent with all of the observations, including the new work presented here, while a supernova remnant is not. The present study not only adds to the evidence for a massive black hole in Henize 2-10, it also demonstrates that a bipolar outflow from the black hole is enhancing\/triggering star formation in its vicinity.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figs\/M2_Table_1.jpg}\n\\caption{\\textbf{(Extended Data Table 1) Summary of observational results regarding the nature of the nucleus in Henize 2-10.} A highly sub-Eddington massive black hole is consistent with all the available observations including new results presented here, while a supernova remnant is not.}\n\\label{tab:AGNSNR_tab}\n\\end{figure}\n\n\n\\subsection{STIS Observations and Data Reduction:}\n\nSpatially resolved spectroscopic observations of the nuclear regions of the dwarf starburst galaxy Henize 2-10 were obtained using the Space Telescope Imaging Spectrograph (STIS) instrument on the Hubble Space Telescope (HST). We obtained observations with two slit orientations. The first is aligned with the quasi-linear ionized gas structure identified by Reines et al.\\cite{reines2011actively_Henize} and covers the central radio\/X-ray source and the bright knot of ionized gas to the east. We refer to this as the East-West (EW) orientation. The second slit orientation was placed perpendicular to the EW observation at the location of the central radio\/X-ray source. We refer to this as the North-South (NS) orientation. The candidate AGN itself was too faint to acquire directly, therefore we used a target acquisition with an offset from a bright point source 7.9\" to the southeast.\nSpectra were taken with the G750M and G430M gratings providing medium spectral resolution (R $\\sim$ 5000-6000) coverage of key optical emission lines. The central wavelengths were set at 6581 \\AA \\ and 4961 \\AA \\ for the G750M and G430M gratings, respectively. At each slit orientation, two orbits were spent in G430M and one orbit in G750M. The observations were taken with a two-point dither pattern with CR-SPLITS (multiple exposures taken to aid in cosmic ray rejection) at each position to help eliminate cosmic rays. The calibrated dithered images were combined and have a spatial resolution of $\\sim$0.1\", which corresponds to a physical scale of $\\sim$4 pc at the distance of Henize 2-10 ($\\sim$9 Mpc).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/M2_Figure_4.pdf}\n\\caption{\\textbf{(Extended Data Figure 1) Raw 2D spectra showing the [OI]6300 emission line at the location of the nucleus in the EW slit orientation.} The location of the nucleus is indicated by white circles and the two images correspond to the two dithered sub-exposures.}\n\\label{fig:OIRAW}\n\\end{figure}\n\n\n\\subsection{Emission Line Fitting:}\n\n\nBefore we fit emission lines in the spectra, we first modeled and removed the continuum. The reduced spectra were continuum-subtracted by masking emission line regions and fitting a low order polynomial to the continuum in each row in the spatial dimension. A low order polynomial was used given the lack of absorption features in the spectra. We did, however, consider the potential impact of stellar absorption lines on our measurements and found that the absorption line strengths are negligible compared to the emission line strengths. Scaling a Starburst99 \\cite{starburst99} model for a 4 Myr stellar population (see Stellar Ages section below) to our observed spectra, we find that the flux of H$\\alpha$ absorption is smaller by a factor of 61 than the H$\\alpha$ emission and the flux of the H$\\beta$ absorption is smaller by a factor of 7 than the H$\\beta$ emission at the location of the central source. Accounting for this absorption has a negligible effect on the line ratios of the nuclear source and does not impact the classifications based on the diagnostic diagrams. \nOnce the spectra were continuum-subtracted, we fit each emission line of interest with a linear combination of Gaussian profiles to characterize the flux and estimate kinematic properties along the spatial dimension of each slit. The fitting was done using lmfit \\cite{newville2016lmfit}, a non-linear least squares curve fitting package in Python. We fit each emission line with up to two Gaussian components when needed. To determine if a second Gaussian component is warranted, we require that the flux of both components be greater than the 3$\\sigma$ error of the flux. This process is performed row by row in the spatial direction along each slit for emission lines of interest. During this process we fit the H$\\beta$, [OIII]5007, H$\\alpha$, [NII]6548\/6583 and [SII]6716\/6731 emission lines. We fit the H$\\alpha$ and [NII] lines simultaneously, fixing the spacing between [NII] Gaussian components to their corresponding component in the H$\\alpha$ emission line to laboratory values. Additionally, we tie the widths of [NII] components and fix the flux ratio of the [NII] lines to the laboratory value of 1:2.96. Similarly, the two [SII] emission lines are fit simultaneously with the spacing between Gaussian components of the two lines held fixed and the widths of the Gaussian components are tied together. \nWe also fit [OI]6300 in the spectra of the nuclear source and the eastern star-forming region, but the line is too weak to be detected all along the slits. Since the [OI]6300 line has a complex profile at the location of the black hole, with possible double peaked narrow lines and a much broader component than the other emission lines, we confirmed that this was not due to an artifact in the data. In Figure \\ref{fig:OIRAW}, we show close-up views of the raw 2D spectra along the EW slit position. The two images correspond to the two dithered sub-exposures offset by 7.5 pixels. The broad [OI] line at the location of the central source is seen in both images at different positions on the detector (indicated by white circles). Note that the locations of hot pixels do not change between the two images. In Figure \\ref{fig:OICOMB} we also show the final reduced 2D image with the sub-exposures combined. The broad, double peaked nature of the [OI] line is clearly visible. We also note that [OI] is similarly broadened in the nuclear spectrum extracted from the NS slit position, although there is not an obvious double-peaked narrow line component (see Figure 2 in the main paper). In any case, broadened [OI] is clearly detected at the location of the nuclear source in both slit positions indicating an outflow on the order of ~500 km s$^{-1}$. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figs\/M2_Figure_5.pdf}\n\\caption{\\textbf{(Extended Data Figure 2) Combined 2D spectra showing the [OI]6300 emission line at the location of the nucleus in the EW slit orientation.} Same as Figure \\ref{fig:OIRAW} but showing the reduced 2D image with the dithered sub-exposures combined. }\n\\label{fig:OICOMB}\n\\end{figure}\n\n\n\\subsection{Gas Density:}\n\nWe estimate the electron density, n$_e$, using the density sensitive line ratio [SII]6716\/6731 \\cite{osterbrock2006astrophysics}. This ratio is sensitive to electron densities in the range of $\\sim$10$^2$-10$^4$ cm$^{-3}$. Along the EW slit orientation, we find a range of n$_e \\sim 10^{2.5}-10^4 cm^{-3}$, indicating a relatively high-density gas (see Figure \\ref{fig:gas_density}). These density estimates are in general agreement with the gas densities predicted by the Allen et al. \\cite{allen2008mappings} shock\/shock+precursor models in the central regions of Henize 2-10 as described in the next section.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figs\/M2_Figure_6.pdf}\n\\caption{\\textbf{(Extended Data Figure 3) The electron density, n$_e$, along the EW slit orientation.} We measure the electron density along the EW slit from the ratio of [SII]6716\/[SII]6731 and find the electron density ranges from $\\sim10^{2.5}-10^4 cm^{-3}$ , which is within the range the [SII] ratio is sensitive to density. The high densities are consistent with those predicted by optical emission line diagnostics derived from the Allen et al. \\cite{allen2008mappings} shock models. }\n\\label{fig:gas_density}\n\\end{figure}\n\n\n\\subsection{Emissin Line Diagnostics - Photoionization and Shock Models:}\n\nTo understand the ionization mechanisms in the central regions of Henize 2-10 we compare our emission line measurements in various regions to photoionization and shock models. In addition to the central radio\/X-ray source, we identified 7 regions of interest that are shown in Figure \\ref{fig:extract_reg} and serve to provide a spatially resolved picture of the kinematics and ionization conditions in the central regions of Henize 2-10. The extraction regions taken along the EW slit orientation were chosen to correspond with emission features seen in the H$\\alpha$ and I-band imaging from HST (young star clusters, knots of ionized gas) as well as features seen in the STIS spectroscopy (broad emission, double peaks). \\par\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/M2_Figure_7.pdf}\n\\caption{\\textbf{(Extended Data Figure 4) The spatial extraction regions taken along the EW slit orientation.} We place these regions on optical emission line diagnostic diagrams (Figures \\ref{fig:BPT}-\\ref{fig:ShockPre2}). Top panel: the extraction regions are shown on the narrow band H$\\alpha$ + continuum image from HST to highlight the ionized gas features that several of the spatial extractions probe. Bottom panel: the extraction regions are shown on the archival 0.8 micron HST image, showing young star clusters that the EW slit orientation passes through.}\n\\label{fig:extract_reg}\n\\end{figure}\n\n\nWe first utilize the standard emission line diagnostic diagrams described by Baldwin et al. \\cite{baldwin1981classification} and Veilleux et al. \\cite{veilleux1987spectral} that have been expanded upon and summarized in Kewley et al. \\cite{kewley2006host}. An accreting BH will produce a much harder continuum than is emitted by hot stars, and these diagrams take advantage of this fact by comparing strong emission line ratios that are close together in frequency to mitigate reddening effects. In this study we employ widely used emission line diagnostic diagrams that take [OIII]\/H$\\beta$ versus [NII]\/H$\\alpha$, [SII]\/H$\\alpha$, and [OI]\/H$\\alpha$ (see Figure \\ref{fig:BPT}). In the [NII]\/H$\\alpha$ diagram, line-emitting galaxies separate into a V-shape \\cite{kewley2006host} with star forming galaxies occupying the left most plume while AGNs occupy the right branch of galaxies. These regions are quantified by an empirical division between HII regions and emission from AGNs developed by Kauffmann et al. \\cite{kauffmann2003host}. The \"composite\" region between this empirical division and the theoretical maximum starburst line from Kewley et al. \\cite{kewley2001theoretical} indicates there is likely significant emission form both HII regions and AGNs. Like the [NII]\/H$\\alpha$ diagram, the [SII]\/H$\\alpha$ and [OI]\/H$\\alpha$ diagnostics provide diagnostics for differentiating between emission from HII regions and AGNs. These two diagrams add a dividing line to distinguish between emission from Seyferts and Low Ionization Nuclear Emission Regions (LINERs). LINER emission can be generated both by shocks and very hard AGN spectra and determining the primary ionization mechanism can be complicated \\cite{baldwin1981classification}. It should be noted that while these diagnostic diagrams are useful for identifying regions dominated by luminous AGNs, they have limitations and can yield ambiguous results for (or completely miss) massive black holes accreting at very low Eddington ratios such as the one in Henize 2-10. Indeed, non-stellar ionization is clearly indicated in the [OI]\/H$\\alpha$ diagram at the location of the nuclear source, but not so for the other diagnostic diagrams (see discussion in the main paper). Figure \\ref{fig:BPT} shows the [OIII]\/H$\\beta$ versus [NII]\/H$\\alpha$, [SII]\/H$\\alpha$, and [OI]\/H$\\alpha$ diagnostic diagrams for the various extraction regions along the EW slit. \\par\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{figs\/M2_Figure_8.pdf}\n\\caption{\\textbf{(Extended Data Figure 5) Narrow emission line diagnostic diagrams showing various extraction regions along the EW slit orientation (see Figure \\ref{fig:extract_reg}).} The nucleus (yellow point) falls in the Seyfert region of the [OI]\/H$\\alpha$ diagram. The young star-forming region $\\sim$70 pc to the east of the low-luminosity AGN is depicted with a blue triangle and star for the primary emission line component and the blue-shifted secondary component, respectively. [OI] is not detected in all of the regions.}\n\\label{fig:BPT}\n\\end{figure}\n\n\nIn addition to the diagnostic diagrams discussed above, we also investigate whether the ionization conditions seen in the central regions of Henize 2-10 can be explained by mechanical excitation from shocks. To investigate this, we employ ionization models of shock and shock+precursor emission developed by Allen et al. \\cite{allen2008mappings}. These models provide emission line fluxes for ionization from a pure shock (possibly driven by an outflow from an AGN or by regions of intense star formation), where the gas is collisionally ionized, or a shock+precursor where ionizing photons produced in the shock-heated gas travel upstream and ionize the gas before the shock reaches it. We explore models with a variety of electron densities (0.01-1000 cm$^{-3}$), shock velocities (100-600 km s$^{-1}$) and transverse magnetic field strengths (0.01-32 $\\mu$G). These are shown in Figures \\ref{fig:ShockPre1} and \\ref{fig:ShockPre2}. \\par\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/M2_Figure_9.pdf}\n\\caption{\\textbf{(Extended Data Figure 6) Optical emission line diagnostics from the shock and shock+precursor models with varying gas density.} We place the spatial extractions from the EW slit orientation shown in Figure \\ref{fig:extract_reg} on a grid of shock excitation models (presented in Allen et al. \\cite{allen2008mappings}) with varying gas density (n = 0.01-1000 cm$^{-3}$) and shock velocity (v = 100-600 km\/s). We fix the transverse magnetic field to be b = 1$\\mu$G and the assume solar metallicity.}\n\\label{fig:ShockPre1}\n\\end{figure}\n\n\nThe emission line ratios from the nuclear source are best described by the shock+precursor models with a low shock velocity (100-250 km s$^{-1}$), a high-density gas (n = 1000 cm$^{-3}$), and a low transverse magnetic field parameter (b = 0.01 \u2013 1 $\\mu$G) (Figures \\ref{fig:ShockPre1} and \\ref{fig:ShockPre2}). Shock+precursor models are thought to be a good description for AGN+outflow emission. Along the filament (extraction regions 2-5), the line ratios are explained well by a low velocity shock or shock+precursor model ($\\sim$200 km s$^{-1}$) in a high density (n$_e \\sim 1000$ cm$^{-3}$) gas with a transverse magnetic field parameter in the range of 1-10 $\\mu$G. This is consistent with a scenario where the central black hole is driving a bipolar outflow that shocks the gas and dominates the ionization conditions along the filament. \\par\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{figs\/M2_Figure_10.pdf}\n\\caption{\\textbf{(Extended Data Figure 7) Optical emission line diagnostics from the shock and shock+precursor models with varying magnetic field.} The models (presented in Allen et al. \\cite{allen2008mappings}) are shown as a grid with dashed blue lines indicating constant shock velocity and dashed black lines indicating constant transverse magnetic field. For these models, the density is fixed to n = 1000 cm$^{-3}$ and the transverse magnetic field parameter is allowed to vary from b = 0.01-32 $\\mu$G. }\n\\label{fig:ShockPre2}\n\\end{figure}\n\n\nAt the region of intense star formation located $\\sim$70 pc to the east of the black hole, we observe strong emission lines including a secondary, blue-shifted kinematic component. To properly fit the emission lines in this region, an additional Gaussian component is added (see Figure \\ref{fig:H210_AGN_spec2} in the main paper). The separation of this component is determined using the [SII] emission line region where the secondary peak is most clearly resolved. We find that the secondary peak is offset from the primary peak by 154 km s$^{-1}$, and we use this value when fitting the other emission line regions where the secondary peak is not as well resolved. When comparing to shock(+ precursor) models, we find that the [NII]\/H$\\alpha$ diagram indicates a low shock velocity (100-250 km s$^{-1}$) in a high-density gas (n = 1000 cm$^{-3}$) with a low transverse magnetic field parameter (b = 0.01 \u2013 1 $\\mu$G) (see Figures \\ref{fig:ShockPre1} and \\ref{fig:ShockPre2}). The [SII]\/H$\\alpha$ diagram shows the primary emission peak is inconsistent with emission from shocks or shocks+precursor models. This indicates that the primary emission peak at this location is primarily due to star formation. The secondary emission peak is consistent with shock+precursor models for the low velocity, high density conditions, indicating that this kinematically distinct emission component is dominated by a shock+precursor from the AGN-driven outflow. \\par\n\nEmission from extraction regions 6 and 7 (i.e., the western star-forming region) is not consistent with any shock or shock+precursor models, which is in agreement with their location in the HII region of the Baldwin\u2013Phillips\u2013Terlevich (BPT) diagram. The line ratios are dominated by star formation in this region.\n\n\\subsection{Star Cluster Ages:}\n\nWe estimate the ages of the young stellar clusters that fall within the EW slit from their H$\\alpha$ and H$\\beta$ equivalent widths. To ascertain ages from these equivalent widths we employ simple stellar population (SSP) models from Starburst99 \\cite{starburst99}. We use models from Version 7.1 with solar metallicity (appropriate for the central regions of Henize 2-10 \\cite{martin2006high}), an instantaneous burst of 10$^4$ M$_\\odot$ with a Kroupa IMF (0.1 \u2013 100 M$_\\odot$), the Geneva evolutionary tracks with high mass loss and the Pauldrach-Hillier atmospheres. \\par\n\nAt the location of the young stellar clusters in the eastern star-forming region (region 1 in Figure \\ref{fig:extract_reg}) we find an equivalent width of 478 \\AA \\ and 70 \\AA \\ for H$\\alpha$ and H$\\beta$ respectively. These both give stellar population age estimates of $\\sim$4.3 Myr, which is in good agreement with previous estimates of the ages of other young star clusters in the region \\cite{chandar2003stellar}. The ages of these clusters are larger than the crossing time for the AGN-driven outflow ($\\sim$0.3 Myr), based on the minimum outflow velocity measured from emission line spectra ($\\sim$200 km s$^{-1}$) and the distance between the AGN and the eastern star-forming knot ($\\sim$70 pc). Therefore, the timescales allow for the AGN-driven outflow to have triggered\/enhanced the formation of star clusters in the Eastern star-forming knot. \\par\n\nThe EW slit orientation also passes through a young star cluster in region 3. At this location we find equivalent widths of 212 \\AA \\ and 41 \\AA \\ for H$\\alpha$ and H$\\beta$ respectively, both indicating an age of $\\sim$5.2 Myr for the stellar cluster. Finally, region 6 in the western star-forming region has H$\\alpha$ and H$\\beta$ equivalent widths of 1092 \\AA \\ and 196 \\AA, respectively. These equivalent widths indicate the stellar clusters have ages $\\leq$3 Myr. \n\n\n\\subsection{Bipolar Outflow Model:}\n\nHere we provide a derivation of the model used to describe a precessing bipolar outflow emanating from the central radio\/X-ray source, which can explain the coherent velocity structure seen in the central $\\sim$120 pc of the EW orientation observations. In this model we align the EW slit orientation with the $z$-axis and assume the outflow precesses about this axis with a small angle $\\theta$ and an angular precession frequency $\\omega$. If the gas being ejected by the outflow has velocity $v_0$, and we orient the $x$-axis to be in the direction of the observer, then the radial (Doppler shifted) velocity seen at the location of the AGN as a function of time will be given by\n\n\\begin{equation}\n v_r(t) = v_x(t) = v_0 \\sin{\\left(\\theta\\right)} \\sin{\\left(\\omega t + \\gamma\\right)},\n\\end{equation}\n\n\\noindent\nwhere $\\gamma$ represents a phase shift that accounts for small asymmetries in the outflow profile (see Figure \\ref{fig:outflow_model} for an illustration of our model).\n\nIn order to find the radial velocity as a function of distance ($z$) along the slit axis, we must consider what angle the outflow made with the (line-of-sight) $x$-axis when the gas at distance $z$ was emitted. Since this angle is time dependent as the outflow precesses, the line-of-sight velocity of the gas will depend on the orientation of the outflow at some time $t_0$ in the past. The time that has passed since the gas at distance $z$ was ejected by the outflow is determined by the $z$ velocity of the gas. Due to the symmetry of the model about the $z$-axis, the $z$ component of the gas velocity will be time independent and only depend on the angle of the outflow with the $z$-axis,\n\n\\begin{equation}\n v_z = v_0 \\cos{(\\theta)}.\n\\end{equation}\n\n\\noindent\nThe time, $t_0$, for gas to reach a distance $z$ along the slit is then given by\n\n\\begin{equation}\n t_0 = \\frac{z}{v_0\\cos(\\theta)}.\n\\end{equation}\n\n\\noindent\nWe are then able to find the an expression for $v_r(z)$ by evaluating the expression for $v_r(t)$ at the time $-t_0$:\n\n\\begin{equation}\n v_r(z) = v_0 \\sin{\\left(\\theta\\right)} \\sin{\\left( \\gamma - \\frac{\\omega}{v_0\\cos{(\\theta)}} z \\right)}\n\\end{equation}\n\n\nTo fit this model to the data we require a rough estimate of the bulk outflow velocity, $v_0$. We estimate this parameter using $W80$, the velocity interval containing 80\\% of the line flux, of the broad emission seen at the location of the candidate AGN. We find $W80 \\approx 200 - 500$ km s$^{-1}$ based on measurements of the [OIII]5007 and [OI]6300 lines at the location of the candidate AGN. This allows us to determine the best-fit angle of precession, $\\theta$, and the frequency of precession, $f = \\omega\/2\\pi$, to be\n\n\\begin{align}\n \\theta &= 2.4^{\\circ} - 6.1^{\\circ} \\\\\n f &= 3.0 - 7.5 \\ {\\rm revolutions \\ Myr}^{-1}\n\\end{align}\n\nwhere the larger angle of precession and smaller frequency of precession corresponds to lower outflow velocities. We find consistent results when using the Doppler shift profile of H$\\alpha$, H$\\beta$ and [OIII] emission lines to fit the model derived above (the results using the [OIII] emission line is shown in Figure \\ref{fig:H210_AGN_spec2}). The Doppler shift profile can be coherently traced out to 50-60 pc on either side of the candidate AGN, most definitively out to the bright eastern star forming region after which the Doppler shifts of the emission lines are influenced by the bright young stars and then shown no coherent pattern further along the slit. The coherent velocity structure seen on the scale of 100 pc is not consistent with young supernova remnant as the compact radio\/x-ray source in the central regions of Henize 2-10, which provides further motivation that a low luminosity AGN is driving the outflow. \n\nThese results are roughly consistent with jet parameters derived in other studies where precessing or reorienting jet models have been applied. The long precession period we observe ($\\sim$200,000 years) is shorter by a factor of a few than those seen predicted by Dunn et al. \\cite{dunn2006precession}, Nawaz et al. \\cite{nawaz2016jet} and Cielo et al. \\cite{cielo2018feedback} but longer by a factor of a few than those predicted by Gower et al. \\cite{gower1982precessing} when jet precession is invoked to explain the complex bending and knotting seen in large radio jets.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/M2_Figure_11.pdf}\n\\caption{\\textbf{(Extended Data Figure 8) A diagram of the toy model of the bipolar outflow generated by the low-luminosity AGN in Henize 2-10.} Our simple model depends on the outflow velocity of the ionized gas (v$_{outflow}$), the angle the outflow makes with its precession axis ($\\theta$) and the angular frequency with which the outflow precesses ($\\omega$). Similar models have been used to describe the bending seen in large radio jets \\cite{gower1982precessing,dunn2006precession}.}\n\\label{fig:outflow_model}\n\\end{figure}\n\n\n\\clearpage\n\n\\section{Acknowledgments}\n\nWe are grateful to Mallory Molina for useful discussions regarding shocks. We also thank Mark Whittle and Kelsey Johnson for their assistance with the HST\/STIS proposal while AER was a graduate student at the University of Virginia, as well as subsequent discussions. Support for Program number HST-GO-12584.006-A was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. AER also acknowledges support for this work provided by NASA through EPSCoR grant number 80NSSC20M0231. ZS acknowledges support for this project from the Montana Space Grant Consortium.\n\n\\section{Author contributions statement}\n\nZS reduced and analyzed the STIS data and compared the results to models. AER led the HST\/STIS proposal and helped with the data reduction. Both authors worked on the interpretation of the results and writing of the paper. \n\n\\section{Data Availability Statement}\nThe spectroscopic data analyzed in this study are available from the Mikulski Archive for Space Telescopes (MAST), https:\/\/archive.stsci.edu\/\n\n\\section{Competing Interest Statement}\nThe authors declare no competing interests.\n\n\n\n\\clearpage\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro} Over the last few years\nnonlocal models have increasingly impacted upon a number of important\nfields in science and technology.\nThe evidence of anomalous diffusion processes, for example, has been\nfound in several physical and social environments \\cite{Klafter,\n MetzlerKlafter}, and corresponding transport models have been\nproposed in various areas such as electrodiffusion in nerve\ncells~\\cite{AnomalousElectrodiffusion} and ground-water solute\ntransport~\\cite{BensonWheatcraft}. Non-local models have also been\nproposed in fields such as finance~\\cite{CarrHelyette,RamaTankov} and\nimage processing~\\cite{GattoHesthaven,GilboaOsher}.\nOne of the fundamental non-local operators is the Fractional Laplacian\n$(-\\Delta)^s$ ($0 < s < 1$) which, from a probabilistic point of view\ncorresponds to the infinitesimal generator of a stable L\\'evy process\n\\cite{Valdinoci}.\n\nThe present contribution addresses theoretical questions and puts\nforth numerical algorithms for the numerical solution of the Dirichlet\nproblem\n\\begin{equation}\n\\left\\lbrace\n \\begin{array}{rl}\n (-\\Delta)^s u = f & \\mbox{ in }\\Omega, \\\\\n u = 0 & \\mbox{ in }\\Omega^c \\\\\n \\end{array}\n \\right.\n\\label{eq:fraccionario_dirichlet}\n\\end{equation}\non a bounded one-dimensional domain $\\Omega$ consisting of a union of\na finite number of intervals (whose closures are assumed mutually\ndisjoint). This approach to enforcement of (nonlocal) boundary\nconditions in a bounded domain $\\Omega$ arises naturally in connection\nwith the long jump random walk approach to the Fractional\nLaplacian~\\cite{Valdinoci}. In such random walk processes, jumps of\narbitrarily long distances are allowed. Thus, the payoff of the\nprocess, which corresponds to the boundary datum of the Dirichlet\nproblem, needs to be prescribed in $\\Omega^c$.\n\n\nLetting $s$ and $n$ denote a real number ($00$\nthen the solution $u$ may be written as $w^s\\phi + \\chi$, where $\\phi\n\\in H^{r+s}(\\Omega)$ and $\\chi \\in H^{r+2s}_0(\\Omega)$. Interior regularity\nresults for the Fractional Laplacian and related operators have also\nbeen the object of recent studies~\\cite{Albanese2015,Cozzi2016}.\n\n\nThe sharp regularity results put forth in the present contribution, in\nturn, are related to but different from those mentioned above. Indeed\nthe present regularity theorems show that the fractional Laplacian in\nfact induces a {\\em bijection} between certain weighted Sobolev\nspaces. Using an appropriate version of the Sobolev lemma put forth in\nSection~\\ref{regularity}, these results imply, in particular, that the\nregular factors in the decompositions of fractional Laplacian solutions\nadmit $k$ continuous derivatives for a certain value of $k$ that\ndepends on the regularity of the right-hand side. Additionally, this\npaper establishes the operator regularity in spaces of analytic\nfunctions: denoting by $A_\\rho$ the space of analytic functions in the\nBernstein Ellipse $\\mathcal{E}_\\rho$, the weighted operator $K_s(\\phi)\n= (-\\Delta)^s(\\omega^s \\phi)$ maps $A_\\rho$ into itself bijectively. In\nother words, for a right-hand side which is analytic in a Bernstein\nEllipse, the solution is characterized as the product of an analytic\nfunction in the same Bernstein Ellipse times an explicit singular\nweight.\n\nThe theoretical treatment presented in this paper is essentially\nself-contained. This approach recasts the problem as an integral\nequation in a bounded domain, and it proceeds by computing certain\nsingular exponents $\\alpha$ that make $(-\\Delta)^s (\\omega^\\alpha\n\\phi(x))$ analytic near the boundary for every polynomial $\\phi$. As\nshown in Theorem~\\ref{teo1} a infinite sequence of such values of\n$\\alpha$ is given by $\\alpha_n = s + n$ for all $n\\geq 0$. Morever,\nSection~\\ref{two_edge_sing} shows that the weighted operator $K_s$\nmaps polynomials of degree $n$ into polynomials of degree $n$---and it\nprovides explicit closed-form expressions for the images of each\npolynomial $\\phi$.\n \nA certain hypersingular form we present for the operator $K_s$ leads\nto consideration of a weighted $L^2$ space wherein $K_s$ is\nself-adjoint. In view of the aforementioned polynomial-mapping\nproperties of the operator $K_s$ it follows that this operator is\ndiagonal in a basis of orthogonal polynomials with respect to a\ncorresponding inner product. A related diagonal form was obtained in\nthe recent independent contribution~\\cite{Dyda2016} by employing\narguments based on Mellin transforms. The diagonal\nform~\\cite{Dyda2016} provides, in particular, a family of explicit\nsolutions in the $n$ dimensional ball in ${\\mathbb {R}}^n$, which are given by\nproducts of the singular term $(1-|z|^2)^s$ and general Meijer\nG-Functions. The diagonalization approach proposed in this paper,\nwhich is restricted to the one-dimensional case, is elementary and is\nsuccinctly expressed: the eigenfunctions are precisely the Gegenbauer\npolynomials.\n\nThis paper is organized as follows: Section~\\ref{integraleq} casts the\nproblem as an integral equation, and Section~\\ref{diagonalform}\nanalyzes the boundary singularity and produces a diagonal form for the\nsingle-interval problem. Relying on the Gegenbauer eigenfunctions and\nassociated expansions found in Section~\\ref{diagonalform},\nSection~\\ref{regularity} presents the aforementioned Sobolev and\nanalytic regularity results for the solution $u$, and it includes a\nweighted-space version of the Sobolev lemma. Similarly, utilizing\nGegenbauer expansions in conjunction with Nystr\\\"om discretizations\nand taking into account the analytic structure of the edge\nsingularity, Section~\\ref{HONM} presents a highly accurate and\nefficient numerical solver for Fractional-Laplacian equations posed on\na union of finitely many one-dimensional intervals. The sharp error\nestimates presented in Section~\\ref{HONM} indicate that the proposed\nalgorithm is spectrally accurate, with convergence rates that only\ndepend on the smoothness of the right-hand side. In particular,\nconvergence is exponentially fast (resp. faster than any power of the\nmesh-size) for analytic (resp. infinitely smooth) right-hand sides. A\nvariety of numerical results presented in Section~\\ref{num_res}\ndemonstrate the character of the proposed solver: the new algorithm is\nsignificantly more accurate and efficient than those resulting from\nprevious approaches.\n\n\n\n\n\n\n\\section{Hypersingular Bounded-Domain Formulation\\label{integraleq}}\n\nIn this section the one-dimensional operator \n\\begin{equation}\n(-\\Delta)^s u(x) = C_1(s) \\mbox{ P.V.} \\int_{-\\infty}^\\infty \\left( u(x)-u(x-y) \\right) |y|^{-1-2s} dy\n\\label{frac1d}\n\\end{equation} \ntogether with Dirichlet boundary conditions outside the bounded domain\n$\\Omega$, is expressed as an integral over $\\Omega$. The Dirichlet\nproblem~\\eqref{eq:fraccionario_dirichlet} is then identified with a\nhypersingular version of Symm's integral equation; the precise\nstatement is provided in Lemma~\\ref{lemma_hypersingular} below. In\naccordance with Section~\\ref{sec:intro}, throughout this paper we\nassume the following definition holds.\n\\begin{definition}\\label{union_intervals_def}\n The domain $\\Omega$ equals a finite union\n\\begin{equation}\n\\label{union_intervals}\n\\Omega = \\bigcup_{i=1}^M (a_i, b_i)\n\\end{equation}\nof open intervals $(a_i,b_i)$ with disjoint closures. We denote\n$\\partial\\Omega =\\{a_1,b_1,\\dots, a_M,b_M\\}$.\n\\end{definition}\n\n\\begin{definition}\n $C^2_0(\\Omega)$ will denote, for a given open set $\\Omega \\subset\n \\mathbb{R}$, the space of all functions $u \\in C^2(\\Omega) \\cap\n C(\\mathbb{R})$ that vanish outside of $\\Omega$. For $\\Omega =(a,b)$ we will\n simply write $C^2_0((a,b)) = C^2_0(a,b)$.\n\\end{definition}\nThe following lemma provides a useful expression for the Fractional\nLaplacian operator in terms of a certain integro-differential\noperator. For clarity the result is first presented in the following\nlemma for the case $\\Omega=(a,b)$; the generalization to \ndomains $\\Omega$ of the form~\\eqref{union_intervals} then follows easily in\nCorollary~\\ref{coro_lemma_hypersingular}.\n\n\\begin{lemma}\n\\label{lemma_hypersingular}\nLet $s \\in (0,1)$, let $u \\in\nC^2_0(a,b)$ such that $|u'|$ is integrable in $(a,b)$, let $x \\in \\mathbb{R}, x\\not \\in \\partial \\Omega=\\{a,b\\}$,\nand define\n\\begin{equation}\\label{eq:c_s}\nC_s = \\frac{C_1(s)}{2s(1-2s)} = -\\Gamma(2s-1)\\sin(\\pi s) \/ \\pi \\quad (s\\ne 1\/2);\n\\end{equation}\nWe then have\n\\begin{itemize}\n\\item [---] Case $s \\ne \\frac{1}{2}$:\n\\begin{equation}\n\\label{eq:hypersingular}\n(-\\Delta)^s u (x) = C_s \\frac{d}{dx} \\int_{a}^b |x-y|^{1-2s} \\frac{d}{dy} u(y) dy.\n\\end{equation}\n\\item [---] Case $s=\\frac{1}{2}$:\n\\begin{equation}\n\\label{eq:hypersingular_s12}\n(-\\Delta)^{1\/2} u (x) = \\frac{1}{\\pi} \\frac{d}{dx} \\int_{a}^b \\ln |x-y| \\frac{d}{dy} u(y) dy.\n\\end{equation}\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n We note that, since the support of $u=u(x)$ is contained in $[a,b]$,\n for each $x \\in \\mathbb{R}$ the support of the translated function\n $u=u(x-y)$ as a function of $y$ is contained in the set\n $[x-b,x-a]$. Thus, using the decomposition ${\\mathbb {R}} = [x-b,x-a] \\cup\n (-\\infty,x-b) \\cup (x-a,\\infty)$ in~\\eqref{frac1d}, we obtain the\n following expression for $(-\\Delta)^s u (x)$:\n\\begin{equation}\\label{spliting}\n C_1(s) \\Bigg( \\mbox{P.V.} \\int_{x-b}^{x-a} ( u(x)-u(x-y) ) |y|^{-1-2s} dy + \n \\left[\\int_{-\\infty}^{x-b} dy + \\int_{x-a}^\\infty dy\\right] u(x) |y|^{-1-2s} \\Bigg). \n\\end{equation}\n\n\nWe consider first the case $x\\not\\in [a,b]$, for\nwhich~\\eqref{spliting} becomes\n\\begin{equation}\\label{spliting_2}\n -C_1(s)\\Bigg( \\mbox{P.V.} \\int_{x-b}^{x-a} u(x-y) |y|^{-1-2s} dy\\Bigg). \n\\end{equation}\nNoting that the integrand~\\eqref{spliting_2} is smooth, integration by\nparts yields\n\\begin{equation}\\label{parts_firstterm}\n \\frac{C_1(s)}{2s} \\int_{x-b}^{x-a} u'(x-y) \\operatorname{sgn}(y)|y|^{-2s} dy\n\\end{equation}\n(since $u(a)=u(b)=0$), and, thus, letting $z=x-y$ we obtain\n\\begin{equation}\\label{singleinterval_smooth}\n (-\\Delta)^s u(x) = \\frac{C_1(s)}{2s} \\int_{a}^{b} \\operatorname{sgn}(x-z)|x-z|^{-2s} u'(z) dz\\quad , \\quad x\\not\\in [a,b].\n\\end{equation}\nThen, letting\n\\begin{equation*}\\label{K_def}\n\\Phi_s(y) = \\left\\lbrace\n \\begin{array}{ll}\n |y|^{1-2s}\/(1-2s) & \\mbox{ for } s\\in (0,1), \\, s\\ne 1\/2 \\\\\n \\log|y| & \\mbox{ for } s = 1\/2 \n \\end{array}\n \\right. ,\n\\end{equation*}\nnoting that\n\\begin{equation}\n\\label{sgn_x_der}\n\\operatorname{sgn}(x-z)|x-z|^{-2s} = \n\\frac{\\partial}{\\partial x} \\Phi_s(x-z),\n\\end{equation}\nreplacing~\\eqref{sgn_x_der} in ~\\eqref{singleinterval_smooth} and\nexchanging the $x$-differentiation and $z$-integration yields the\ndesired expressions~\\eqref{eq:hypersingular}\nand~\\eqref{eq:hypersingular_s12}. This completes the proof in the case\n$x\\not\\in [a,b]$.\n\nLet us now consider the case $x\\in(a,b)$. The second term\nin~\\eqref{spliting} can be computed exactly; we clearly have\n\\begin{equation}\\label{eq:bnd_term}\n\\left[\\int_{-\\infty}^{x-b} dy + \\int_{x-a}^\\infty dy\\right] u(x) |y|^{-1-2s} = \n\\left[ \\frac{u(x)}{2s} \\operatorname{sgn}(y) |y|^{-2s} \\bigg|_{y=x-b}^{y=x-a} \\right] .\n\\end{equation}\nIn order to integrate by parts in the P.V. integral in~\\eqref{spliting} consider the set $$ D_\\varepsilon =\n[x-b,x-a] \\setminus (-\\varepsilon,\\varepsilon).$$\nThen, defining\n\\begin{equation*}\n\\label{parts_secondterm_0}\nQ_\\epsilon (x) = \\int_{D_\\epsilon } \\left( u(x)-u(x-y) \\right) |y|^{-1-2s} dy \n\\end{equation*}\nintegration by parts yields\n\\begin{equation*}\n\\label{parts_secondterm}\nQ_\\epsilon (x) = \n-\\frac{1}{2s}\\left ( g_{a}^b(x) - h_{a}^b(x) - \\frac{\\delta^2_\\varepsilon}{\\varepsilon^{2s}} - \\int_{D_\\epsilon} u'(x-y) \\operatorname{sgn}(y)|y|^{-2s} dy \\right)\n\\end{equation*}\nwhere $\\delta_\\varepsilon = u(x+\\varepsilon) + u(x-\\varepsilon) - 2 u(x)$, $g_{a}^b(x) =\nu(x)(|x-a|^{-2s} + |x-b|^{-2s})$ and $h_{a}^b(x) = u(a)|x-a|^{-2s} +\nu(b)|x-b|^{-2s}$. \n\nThe term $h_{a}^b(x)$ vanishes since $u(a)=u(b)=0$. The contribution\n$g_{a}^b(x)$, on the other hand, exactly cancels the boundary terms in\nequation~\\eqref{eq:bnd_term}.\nFor the values $x\\in (a,b)$ under\nconsideration, a Taylor expansion in $\\varepsilon$ around $\\varepsilon=0$\nadditionally tells us that the quotient\n$\\frac{\\delta^2_\\varepsilon}{\\varepsilon^{2s}}$ tends to $0$ as $\\varepsilon\\to\n0$. Therefore, using the change of variables $z=x-y$ and letting\n$\\varepsilon\\to 0$ we obtain a principal-value expression valid for $x \\ne a, x\\ne b$:\n\\begin{equation}\n\\label{eq:pv_singleinterval}\n(-\\Delta)^s u (x) = \\frac{C_1(s)}{2s} \\mbox{ P.V.} \\int_{a}^b \\operatorname{sgn}(x-z)|x-z|^{-2s} u'(z) dz.\n\\end{equation}\nReplacing~\\eqref{sgn_x_der} in ~\\eqref{eq:pv_singleinterval} then\nyields~\\eqref{eq:hypersingular} and~\\eqref{eq:hypersingular_s12},\nprovided that the derivative in $x$ can be interchanged with the\nP.V. integral. This interchange is indeed correct, as it follows from\nan application of the following Lemma to the function $v=u'$. The\nproof is thus complete.\n\\end{proof}\n\\begin{lemma}\n\\label{lemma_exchangePV}\nLet $\\Omega \\subset \\mathbb{R}$ be as indicated in\nDefinition~\\ref{union_intervals_def} and let $v\\in C^1(\\Omega)$ such\nthat $v$ is absolutely integrable over $\\Omega$, and let\n$x\\in\\Omega$. Then the following relation holds:\n\\begin{equation} \\label{der_pv}\n P.V. \\int_\\Omega \\frac{\\partial}{\\partial x} \\Phi_s(x-y) v(y) dy = \\frac{\\partial}{\\partial x} \\int_\\Omega \\Phi_s(x-y) v(y) dy\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSee Appendix~\\ref{app_exchangePV}.\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{coro_lemma_hypersingular}\nGiven a domain $\\Omega$ as in Definition~\\eqref{union_intervals_def}, and\nwith reference to equation~\\eqref{eq:c_s}, for $u\\in C_0^2(\\Omega)$ and\n$x\\not\\in \\partial \\Omega$ we have\n\\begin{itemize}\n\\item [---] Case $s \\ne \\frac{1}{2}$:\n\\begin{equation}\n\\label{eq:mutli_int}\n(-\\Delta)^s u (x) = C_s \\frac{d}{dx} \\sum_{i=1}^M \\int_{a_i}^{b_i} |x-y|^{1-2s} \\frac{d}{dy} u(y) dy\n\\end{equation}\n\\item [---] Case $s=\\frac{1}{2}$:\n\\begin{equation}\n\\label{eq:mutli_int_s12}\n (-\\Delta)^{1\/2} u (x) = \\frac{1}{\\pi} \\frac{d}{dx} \\sum_{i=1}^M \\int_{a_i}^{b_i} \\ln |x-y| \\frac{d}{dy} u(y) dy\n\\end{equation}\n\\end{itemize}\nfor all $x\\in\\mathbb{R}\\setminus \\partial \\Omega = \\cup_i^M \\{a_i, b_i\\}$.\n\\end{corollary}\n\\begin{proof}\n Given $u\\in C_0^2(\\Omega)$ we may write $u=\\sum_i^M u_i$ where, for\n $i=1,\\dots,M$ the function $u_i=u_i(x)$ equals $u(x)$ for $x \\in\n (a_i,b_i)$ and and it equals zero elsewhere. In view of\n Lemma~\\ref{lemma_hypersingular} the result is valid for each\n function $u_i$ and, by linearity, it is thus valid for the function\n $u$. The proof is complete.\n\\end{proof}\n\\begin{remark}\\label{bound_rem}\n A point of particular interest arises as we examine the character of\n $(-\\Delta)^s u$ with $u\\in C_0^2(\\Omega)$ for $x$ at or near $\\partial\n \\Omega$. Both Lemma~\\ref{lemma_hypersingular} and its\n corollary~\\ref{coro_lemma_hypersingular} are silent in these\n regards. For $\\Omega = (a,b)$, for example, inspection of\n equation~\\eqref{eq:pv_singleinterval} leads one to generally expect\n that $(-\\Delta)^s u(x)$ has an infinite limit as $x$ tends to each\n one of the endpoints $a$ or $b$. But this is not so for all\n functions $u\\in C_0^2(\\Omega)$. Indeed, as established in\n Section~\\ref{diag}, the subclass of functions in $C_0^2(\\Omega)$ for\n which there is a finite limit forms a dense subspace of a relevant\n weighted $L^2$ space. In fact, a dense subset of functions exists\n for which the image of the fractional Laplacian can be extended as\n an analytic function in the complete complex $x$ variable plane.\n But, even for such functions, definition~\\eqref{frac1d} still\n generically gives $(-\\Delta)^s u (x) =\\pm \\infty$ for $x=a$ and\n $x=b$. Results concerning functions whose Fractional Laplacian blows\n up at the boundary can be found in~\\cite{Abatangelo2015}.\n\\end{remark}\nThe next section concerns the single-interval case ($M=1$\nin~\\eqref{eq:mutli_int},~\\eqref{eq:mutli_int_s12}). Using\ntranslations and dilations the single interval problem in any given\ninterval $(a_1, b_1)$ can be recast as a corresponding problem in any\ndesired open interval $(a,b)$. For notational convenience two\ndifferent selections are made at various points in\nSection~\\ref{diagonalform}, namely $(a,b)=(0,1)$ in\nSections~\\ref{edge_sing_left} and~\\ref{two_edge_sing}, and $(a,\nb)=(-1,1)$ in Section~\\ref{diag}. The conclusions and results can\nthen be easily translated into corresponding results for general\nintervals; see for example Corollary~\\ref{diag_ab}.\n\n\\section{Boundary Singularity and Diagonal Form of the Single-Interval\n Operator \\label{diagonalform}}\n\n\nLemma~\\ref{lemma_hypersingular} expresses the action of the operator\n$(-\\Delta)^s$ on elements $u$ of the space $C_0^2(\\Omega)$ in terms of the\nintegro-differential operators on the right-hand side of\nequations~\\eqref{eq:hypersingular} and~\\eqref{eq:hypersingular_s12}. A\nbrief consideration of the proof of that lemma shows that for such\nrepresentations to be valid it is essential for the function $u$ to\nvanish on the boundary---as all functions in $C_0^2(a,b)$ do, by\ndefinition. Section~\\ref{edge_sing_left} considers, however, the\naction under the integral operators on the right-hand side of\nequations~\\eqref{eq:hypersingular} and~\\eqref{eq:hypersingular_s12} on\ncertain functions $u$ defined on $\\Omega = (a,b)$ {\\em which do not\n necessarily vanish at $a$ or $b$}. To do this\nwe study the closely related integral operators\n \\begin{align}\n S_s[u](x) &:= C_s \\int_a^b \\left( |x-y|^{1-2s} - (b-a)^{1-2s} \\right) u(y) dy \\;\\; ( s \\ne \\frac{1}{2} ), \\label{Tsdef_eq1}\\\\\n S_{\\frac{1}{2}}[u](x) &:= \\frac{1}{\\pi} \\int_a^b \\log\\left(\\frac{|x-y|}{b-a}\\right) u(y) dy, \\label{Tsdef_eq2}\\\\\n T_s[u](x) &:= \\frac{\\partial}{\\partial x} S_s\\left[\n \\frac{\\partial}{\\partial y} u(y) \\right](x) .\\label{Tsdef_eq3}\n\\end{align}\n\n\n\\begin{remark}\\label{const_term}\n The addition of the constant term $-(b-a)^{1-2s}$ in the\n integrand~\\eqref{Tsdef_eq1} does not have any effect in the\n definition of $T_s$: the constant $-(b-a)^{1-2s}$ only results in the addition\n of a constant term on the right-hand side of~\\eqref{Tsdef_eq1},\n which then yields zero upon the outer differentiation in\n equation~\\eqref{Tsdef_eq3}. The integrand~\\eqref{Tsdef_eq1} is\n selected, however, in order to insure that the kernel of $S_s$\n (namely, the function $C_s \\left( |x-y|^{1-2s} -(b-a)^{1-2s}\\right)$) tends to\n the kernel of $S_{\\frac{1}{2}}$ in~\\eqref{Tsdef_eq2} (the function\n $\\frac{1}{\\pi} \\log(|x-y|\/(b-a))$) in the limit as $s\\to \\frac{1}{2}$.\n\\end{remark}\n\\begin{remark}\\label{Ts_PV}\n In view of Remark~\\ref{const_term} and Lemma~\\ref{lemma_exchangePV},\n for $u\\in C^2(a,b)$ we additionally have\n\\begin{equation}\\label{Tsdef2}\n T_s[u](x) = \\frac{C_1(s)}{2s} \\mbox{ P.V.} \\int_{a}^b\n \\operatorname{sgn}(x-z)|x-z|^{-2s} u'(z) dz.\n\\end{equation}\n\\end{remark}\n\\begin{remark}\n\\label{remark_Ts}\n The operator $T_s$ coincides with $(-\\Delta)^s$ for functions $u$\n that satisfy the hypothesis of Lemma~\\ref{lemma_hypersingular}, but\n $T_s$ does not coincide with $(-\\Delta)^s$ for functions $u$ which,\n such as those we consider in Section~\\ref{edge_sing_left} below, do\n not vanish on $\\partial \\Omega = \\{a,b\\}$.\n\\end{remark}\n\n\\begin{remark}\\label{remark_openarcs}\n The operator $S_{\\frac{1}{2}}$ coincides with Symm's integral\n operator~\\cite{Symm}, which is important in the context of\n electrostatics and acoustics in cases where Dirichlet boundary\n conditions are posed on infinitely-thin open\n plates~\\cite{OpenArcsRadioScience,OpenArcsTheoretical,Symm,YanSloan}. The\n operator $T_{\\frac{1}{2}}$, on the other hand, which may be viewed\n as a {\\em hypersingular version} of the Symms operator\n $S_{\\frac{1}{2}}$, similarly relates to electrostatics and\n acoustics, in cases leading to Neumann boundary conditions posed on\n open-plate geometries. The operators $S_{s}$ and $T_{s}$ in the\n cases $s \\ne \\frac{1}{2}$ can thus be interpreted as generalizations\n to fractional powers of classical operators in potential theory,\n cf. also Remark~\\ref{remark_Ts}.\n\\end{remark}\nRestricting attention to $\\Omega = (a,b) = (0,1)$ for notational\nconvenience and without loss of generality,\nSection~\\ref{edge_sing_left} studies the image $T_s[u_\\alpha]$ of the\nfunction \n\\begin{equation}\\label{u_alpha}\n u_\\alpha(y) =y^\\alpha\n\\end{equation}\nwith $\\Re\\alpha > 0$---which is smooth in $(0,1)$, but which has an\nalgebraic singularity at the boundary point $y=0$. That section shows\nin particular that, whenever $\\alpha = s +n$ for some $n\\in\n\\mathbb{N}\\cup \\{ 0 \\}$, the function $T_s[u_\\alpha](x)$ can be\nextended analytically to a region containing the boundary point $x=0$.\nBuilding upon this result (and assuming once again $\\Omega = (a, b) =\n(0,1)$), Section~\\ref{two_edge_sing}, explicitly evaluates the images\nof functions of the form $v(y) = y^{s+n}(1-y)^s$ ($n\\in \\mathbb{N}\\cup\n\\{ 0 \\}$), which are singular (not smooth) at the two boundary points\n$y=0$ and $y=1$, under the integral operators $T_s$ and $S_s$. The\nresults in Section~\\ref{two_edge_sing} imply, in particular, that the\nimage $T_s[v]$ for such functions $v$ can be extended analytically to\na region containing the interval $[0,1]$. Reformulating all of these\nresults in the general interval $\\Omega = (a,b)$, Section~\\ref{diag} then\nderives the corresponding single-interval diagonal form for weighted\noperators naturally induced by $T_s$ and $S_s$.\n\\subsection{Single-edge singularity\\label{edge_sing_left}}\nWith reference to equations~\\eqref{Tsdef2} and \\eqref{eq:c_s}, and\nconsidering the aforementioned function $u_\\alpha(y) = y^\\alpha$ we\nclearly have\n\\begin{equation*}\\label{eq:TsNs}\n T_s[u_\\alpha](x) =\\alpha (1-2s) C_s N_\\alpha^s(x) \\;\\; \\mbox{, where}\n\\end{equation*}\n\\begin{equation}\n N_\\alpha^s(x) := P.V.\\int_{0}^{1} \\operatorname{sgn}(x-y)|x-y|^{-2s} y^{\\alpha-1} dy .\n\\label{eq:def_Ns}\n\\end{equation} \nAs shown in Theorem~\\ref{teo1} below (equation~\\eqref{eq:Ns_Betas}),\nthe functions $N_\\alpha^s$ and (thus) $ T_s[u_\\alpha]$ can be\nexpressed in terms of classical special functions whose singular\nstructure is well known. Leading to the proof of that theorem, in what\nfollows we present a sequence of two auxiliary lemmas.\n\\begin{lemma}\n\\label{lemma0_analytic}\nLet $ E = (a, b)\\subset\\mathbb{R}$, and let $C\\subseteq \\mathbb{C}$\ndenote an open subset of the complex plane. Further, let $f=f(t,c)$ be\na function defined in $E\\times C$, and assume 1)~$f$ is continuous in\n$E\\times C$, 2)~$f$ is analytic with respect to $c=c_1+ic_2\\in C$ for\neach fixed $t\\in E$, and 3)~$f$ is ``uniformly integrable over\ncompact subsets of $C$''---in the sense that for every compact set $K\n\\subset C$ the functions\n\\begin{equation}\n\\label{hab_eta}\nh_a(\\eta,c) = \\left| \\int_a^{a+\\eta} f(t,c) dt \\,\\right|\\quad \\mbox{and}\\quad\nh_b(\\eta,c) = \\left| \\int_{b-\\eta}^b f(t,c) dt \\,\\right|\n\\end{equation}\ntend to zero uniformly for $c\\in K$ as $\\eta\\to 0^+$. Then the\nfunction\n$$F(c) := \\int_E f(t,c) dt$$\nis analytic throughout $C$.\n\\end{lemma}\n\\begin{proof} \n Let $K$ denote a compact subset of $C$. For each $c\\in K$ and each\n $n\\in\\mathbb{N}$ we consider Riemann sums $R_n^h(c)$ for the\n integral of $f$ in the interval $[a+\\eta_n,b-\\eta_n]$, where\n $\\eta_n$ is selected in such a way that $h_a(\\eta_n,c) \\leq 1\/n$ and\n $h_b(\\eta_n,c) \\leq 1\/n$ for all $c\\in K$ (which is clearly possible\n in view of the hypothesis~\\eqref{hab_eta}). The Riemann sums are\n defined by $R_n^h(c)=h \\sum_{j=1}^Mf(t_j,c)$, with $ h = (b-a\n +2\\eta_n)\/M$ and $t_{j+1} - t_j =h $ for all $j$.\n\n Let $n\\in \\mathbb{N}$ be given. In view of the uniform continuity of\n $f(t,c)$ in the compact set $[a+\\eta_n,b-\\eta_n]\\times K$, the\n difference between the maximum and minimum of $f(t,c)$ in each\n integration subinterval $(t_j,t_{j+1})\\subset [a+\\eta_n,b-\\eta_n]$\n tends uniformly to zero for all $c\\in K$ as the integration meshsize\n tends to zero. It follows that a meshsize $h_n$ can be found for\n which the approximation error in the corresponding Riemann sum\n $R_n^h(c)$ is {\\em uniformly small for all $c\\in K$}:\n\\[\n\\left| \\int_{a+\\eta_n}^{b-\\eta_n} f(t,c) dt - R_n^h(c)\\right| < \\frac 1n\n\\quad \\mbox{for all $c\\in K$ and for all $n\\in \\mathbb{N}$}.\n\\]\nThus $F(c)$ equals a uniform limit of analytic functions over every\ncompact subset of $C$, and therefore $F(c)$ is itself analytic\nthroughout $C$, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma_analytic}\n Let $x\\in (0,1)$ and let $g(s,\\alpha) = N_\\alpha^s(x)$ be defined\n by~\\eqref{eq:def_Ns} for complex values of $s$ and $\\alpha$\n satisfying $ \\Re s < 1$ and $\\Re \\alpha > 0$. We then have:\n\\begin{itemize}\n\\item[(\\it{i})] For each fixed $\\alpha$ such that $\\Re \\alpha > 0$,\n $g(s,\\alpha)$ is an analytic function of $s$ for $\\Re s < 1$; and\n\\item[(\\it{ii})] For each fixed $s$ such that $\\Re s < 1$,\n $g(s,\\alpha)$ is an analytic of $\\alpha$ for $\\Re \\alpha > 0$.\n\\end{itemize}\nIn other words, for each fixed $x\\in (0,1)$ the function $\nN_\\alpha^s(x)$ is jointly analytic in the $(s,\\alpha)$ domain $D = \\{\n\\Re s < 1\\} \\times \\{\\Re \\alpha > 0\\} \\subset \\mathbb{C}^2$.\n\\end{lemma}\n\\begin{proof} \n We express the integral that defines $N_\\alpha^s$ as the sum $g_1(s,\\alpha) +\n g_2(s,\\alpha)$ of two integrals, each one of which contains only one of the\n two singular points of the integrand ($y=0$ and $y=x$):\n\\begin{equation*}\\label{ns_two}\n g_1 = \\int_{0}^{x\/2}\n \\operatorname{sgn}(x-y)|x-y|^{-2s} y^{\\alpha-1} dy \\, \\mbox{ and } g_2 = P.V. \\int_{x\/2}^1\n \\operatorname{sgn}(x-y)|x-y|^{-2s} y^{\\alpha-1} dy. \n\\end{equation*}\nLemma~\\ref{lemma0_analytic} tells us that $g_1$ is an analytic\nfunction of $s$ and $\\alpha$ for $(s,\\alpha)\\in D_1 = \\mathbb{C}\\times\n\\{\\Re \\alpha > 0\\}$. \n\nIntegration by parts in the $g_2$ term, in turn, yields\n\\begin{equation}\n\\label{g2_parts}\n(1-2s) g_2(s,\\alpha) = (1-x)^{1-2s} - \\left( \\frac{x}{2}\\right) ^{\\alpha-2s}-\n(\\alpha-1) \\int_{x\/2}^{1} |x-y|^{1-2s} y^{\\alpha-2} dy .\n\\end{equation}\nBut, writing the the integral on the right-hand side\nof~\\eqref{g2_parts} in the form $\\int_{x\/2}^1 = \\int_{x\/2}^x +\n\\int_{x}^1$ and applying Lemma~\\ref{lemma0_analytic} to each one of\nthe resulting integrals shows that the quantity $(1-2s) g_2(s,\\alpha)$\nis an analytic function of $s$ and $\\alpha$ for $(s,\\alpha)\\in D_2 =\n\\mathbb{C}\\times \\{ \\alpha > 0\\}$. In view of the $(1-2s)$ factor,\nhowever, it still remains to be shown that $g_2(s,\\alpha)$ is analytic\nat $s=1\/2$ as well.\n\nTo check that both $g_2(s,\\alpha)$ and $g(s,\\alpha)$ are analytic\naround $s=1\/2$ for any fixed $\\alpha \\in \\{ \\Re \\alpha>0 \\}$, we first\nnote that since $\\int_0^1 1\\cdot y^{\\alpha-1} dy$ is a constant\nfunction of $x$ we may write\n$$ g(s,\\alpha) = \\frac{1}{1-2s} \\frac{\\partial}{\\partial x} \\int_0^1 \\left(|x-y|^{1-2s} - 1 \\right) y^{\\alpha-1} dy. $$\nBut since we have the uniform limit\n\\[\n\\lim_{s\\to 1\/2}\\frac{|x-y|^{1-2s} - 1 }{1-2s} =\n\\left .\\frac{\\partial}{\\partial r} |x-y|^r\\right|_{r=0} = \\log|x-y|\n\\]\nas complex values of $s$ approach $s=1\/2$, we see that $g$ is in fact a\ncontinuous and therefore, by Riemann's theorem on removable\nsingularities, analytic at $s=1\/2$ as well. The proof is now complete.\n\\end{proof}\n\\begin{theorem}\\label{teo1} \n Let $s\\in (0,1)$ and $\\alpha >0$. Then $N_\\alpha^s(x)$ can be\n analytically continued to the unit disc $\\{x:|x|<1\\}\\subset\n \\mathbb{C}$ if and only if either $\\alpha = s + n$ or $\\alpha = 2s +\n n$ for some $n\\in \\mathbb{N}\\cup \\{ 0 \\}$. In the case $\\alpha = s +\n n$, further, we have\n \\begin{equation}\n \\label{eq:teo1}\n N_{s+n}^s(x) = \\sum_{k=0}^{\\infty} \\frac{(2s)_k}{s-n+k} \\frac{x^k}{k!} \n\\end{equation}\nwhere, for a given complex number $z$ and a given non-negative integer $k$\n\\begin{equation}\n\\label{def_Pochhamer}\n(z)_k:=\\frac{\\Gamma(z+k)}{\\Gamma(z)}\n \\end{equation}\ndenotes the Pochhamer symbol.\n\\end{theorem}\n\\begin{proof}\n We first assume $s<\\frac{1}{2}$ (for which the integrand\n in~\\eqref{eq:def_Ns} is an element of $L^1(0,1)$) and $\\alpha < 2s$\n (to enable some of the following manipulations); the result for the\n full range of $s$ and $\\alpha$ will subsequently be established by\n analytic continuation in these variables. Writing\n$$ N_\\alpha^s(x) = x^{-2s} \\int_{0}^{1} \\operatorname{sgn}(x-y) \\left|1-\\frac{y}{x}\\right|^{-2s} y^{\\alpha-1} dy ,$$\nafter a change of variables and some simple calculations for $x\\in\n(0,1)$ we obtain\n\\begin{equation}\n\\label{eq:Ns_secondterm}\nN_\\alpha^s(x) = x^{-2s+\\alpha} \\left[ \\int_{0}^{1} (1-r)^{-2s} r^{\\alpha-1} dr - \\int_{1}^{\\frac{1}{x}} (r-1)^{-2s} r^{\\alpha-1} dr \\right].\n\\end{equation}\nIt then follows that\n\\begin{equation}\n\\label{eq:Ns_Betas}\nN_\\alpha^s(x) = x^{-2s+\\alpha} \\left[\\mbox{B}(\\alpha, 1 - 2 s)\n -\\mbox{B}(1 - 2 s,2s-\\alpha) + \\mbox{B}_x(-\\alpha + 2 s, 1 - 2 s)\n\\right],\n\\end{equation}\nwhere \n\\begin{equation}\\label{Betas}\n \\begin{split}\n\\mbox{B}(a,b) & := \\int_0^1 t^{a-1}(1-t)^{b-1} dt = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)} \\;\\;\\; \\mbox{ and} \\\\\n\\mbox{B}_x(a,b) & := \\int_0^x t^{a-1}(1-t)^{b-1} dt = x^{a} \\sum_{k=0}^{\\infty} \\frac{(1-b)_k}{a+k} \\frac{x^k}{k!}\n \\end{split}\n\\end{equation}\ndenote the Beta Function~\\cite[eqns. 6.2.2]{AbramowitzStegun} and the\nIncomplete Beta function~\\cite[eqns. 6.6.8 and\n15.1.1]{AbramowitzStegun}, respectively. Indeed, the first integral\nin~\\eqref{eq:Ns_secondterm} equals the first Beta function on the\nright-hand side of~\\eqref{eq:Ns_Betas}, and, after the change of\nvariables $w= 1\/r$, the second integral is easily seen to equal the\ndifference $\\mbox{B}(1-2s,2s-\\alpha) - \\mbox{B}_x(-\\alpha + 2 s, 1 - 2\ns)$.\n\nIn view of~\\eqref{eq:Ns_Betas} and the right-hand expressions in\nequation~\\eqref{Betas} we can now write\n\\begin{equation}\n\\label{eq:caso_s_unmedio}\n N_\\alpha^s(x) = x^{-2s+\\alpha} \\left[ \\frac{ \\Gamma(\\alpha) \\Gamma(1 - 2 s) }{ \\Gamma(1+\\alpha-2s) } - \\frac{ \\Gamma(1 - 2 s) \\Gamma(-\\alpha + 2 s) }{\\Gamma(1-\\alpha)} \\right] + \\sum_{k=0}^{\\infty} \\frac{(2s)_k}{2s-\\alpha+k} \\frac{x^k}{k!}\n\\end{equation}\nfor all $x\\in(0,1)$, $00 \\} \\subset \\mathbb{C}^2$. To do\nthis we consider the following facts:\n\\begin{enumerate}\n\\item \\label{dos} Since $\\Gamma(z)$ is a never-vanishing function of\n $z$ whose only singularities are simple poles at the nonpositive\n integers $z=-n$ ($n\\in \\mathbb{N}\\cup \\{0\\}$), and since, as a\n consequence, $1\/\\Gamma(z)$ is an entire function of $z$ which only\n vanishes at non-positive integer values of $z$, the quotient\n $\\Gamma(\\alpha) \/ \\Gamma(1+\\alpha-2s)$ is analytic and non-zero for\n $(s,\\alpha)\\in D$.\n\\item \\label{tres} The function $G(s)$ that appears on the right hand \n side of~\\eqref{eq:Ns_euler2} ($s\\ne 1\/2$) can be continued analytically to the domain $\\Re s < 1$ with\n the value $G(1\/2)=\\pi$. Further, this function does not vanish for\n any $s$ with $0 < \\Re s < 1$.\n\\item \\label{cuatro} For fixed $s\\in \\mathbb{C}$ the quotient\n $\\sin(\\pi (\\alpha-s)) \/ \\sin(\\pi (\\alpha-2s)) = \\sin(\\pi (q+s)) \/\n \\sin(\\pi q)$ is a meromorphic function of $q$---whose singularities\n are simple poles at the integer values $q = n\\in \\mathbb{Z}$ with\n corresponding residues given by $(-1)^n \\sin(\\pi\n (q+s))\/\\pi$. Further, for $s\\not\\in\\mathbb{Z}$ the quotient vanishes\n if and only if $q=n-s$ (or equivalently, $\\alpha = s + n$) for some $n\\in\n \\mathbb{Z}$.\n\\item \\label{cinco} For each $x$ in the unit disc $\\{x\\in\\mathbb{C}:\n |x|<1\\}$ the infinite series on the right-hand side\n of~\\eqref{eq:Ns_euler} converges uniformly over compact subsets of $D\n \\setminus \\{ \\alpha=2s+n, n\\in \\mathbb{N}\\cup \\{0\\} \\}$. This is\n easily checked by using the asymptotic\n relation~\\cite[6.1.46]{AbramowitzStegun} $\\lim_{k\\to \\infty}\n k^{1-2s}(2s)_k \/ k! = 1\/\\Gamma(2s)$, and taking into account that\n the functions $s\\to (2s)_k$ and $s\\to 1\/\\Gamma(2s)$ are entire and,\n thus, finite-valued for each $s\\in \\mathbb{C}$ and each $k\\in\n \\mathbb{N}\\cup \\{0\\}$.\n\\item \\label{seis} For each fixed $s\\in \\mathbb{C}$ and each\n $x\\in\\mathbb{C}$ with $|x|<1$ the series on the right hand side\n of~\\eqref{eq:Ns_euler} is a meromorphic function of $q$ containing\n only simple polar singularities at $q = n\\in \\mathbb{N}\\cup \\{0\\}$,\n with corresponding residues given by $(2s)_n x^n\/n!$. Indeed,\n point~\\eqref{cinco} above tells us that the series is an analytic\n function of $q$ for $q \\not\\in \\mathbb{N}\\cup \\{0\\}$; the residue at\n the non-negative integer values of $q$ can be computed immediately\n by considering a single term of the series.\n\\item \\label{siete} The residue of the two terms under brackets on the\n right-hand side of~\\eqref{eq:Ns_euler2} are negatives of each\n other. This can be established easily by considering points\n \\eqref{cuatro} and~\\eqref{seis} as well as the identity $\\lim_{q\\to\n n} (-1)^n G(s)\\sin(\\pi(q+s))\/\\pi = 1\/\\Gamma(2s)$---which itself\n results from Euler's reflection formula and standard trigonometric\n identities.\n\\item \\label{ocho} The sum of the bracketed terms\n in~\\eqref{eq:Ns_euler2} is an analytic function of $q$ up to and\n including non-negative integer values of this variable, as it\n follows from point~\\eqref{siete}. Its limit as $q\\to n$, further, is\n easily seen to equal the product of an analytic function of $q$ and\n $s$ times the monomial $x^n$.\n\\end{enumerate}\n\nExpressions establishing the $x$-analyticity properties of\n$N_\\alpha^s(x)$ can now be obtained. On one hand, by\nLemma~\\ref{lemma_analytic} the function $N_\\alpha^s(x)$ is a jointly\nanalytic function of $(s,\\alpha)$ in the domain $D$. In view of\npoints~\\eqref{cuatro} through ~\\eqref{ocho}, on the other hand, we see\nthat the right-hand side expression in equation~\\eqref{eq:Ns_euler} is\nalso an analytic function throughout $D$. Since, as shown above in\nthis proof, these two functions coincide in the open set $U :=\n(0,\\frac{1}{2}) \\times ( 0, 2s )\\subset D$, it follows that they must\ncoincide throughout $D$. In other words, interpreting the right-hand\nsides in equations~\\eqref{eq:Ns_euler} and~\\eqref{eq:Ns_euler2} as\ntheir analytic continuation at all removable-singularity points\n(cf. points~\\eqref{tres} and~\\eqref{siete}) these two equations hold\nthroughout $D$.\n\nWe may now establish the $x$-analyticity of the function\n$N_\\alpha^s(x)$ for given $\\alpha$ and $s$ in $D$. We first do this\nin the case $\\alpha = s+n$ with $n\\in \\mathbb{N}\\cup \\{0\\}$ and\n$s\\in(0,1)$. Under these conditions the complete first term\nin~\\eqref{eq:Ns_euler} vanishes---even at $s=1\/2$---as it follows from\npoints~\\eqref{dos} through~\\eqref{cuatro}. The function\n$N_\\alpha^s(x)$ then equals the series on the right-hand side\nof~\\eqref{eq:Ns_euler}. In view of point~\\eqref{cinco} we thus see\nthat, at least in the case $\\alpha = s+n$, $N_\\alpha^s(x)$ is analytic\nwith respect to $x$ for $|x|<1$ and, further, that the desired\nrelation~\\eqref{eq:teo1} holds.\n\nIn order to establish the $x$-analyticity of $N_\\alpha^s(x)$ in the\ncase $\\alpha = 2s+n$ (or, equivalently, $q=n$) with $n\\in\n\\mathbb{N}\\cup \\{0\\}$ and $s\\in (0,1)$, in turn, we consider the limit\n$q\\to n$ of the right-hand side in\nequation~\\eqref{eq:Ns_euler2}. Evaluating this limit by means of\npoints~\\eqref{cinco} and~\\eqref{ocho} results in an expression which,\nin view of point~\\eqref{cinco}, exhibits the $x$-analyticity of the\nfunction $N_\\alpha^s$ for $|x|<1$ in the case under consideration.\n\nTo complete our description of the analytic character of\n$N_\\alpha^s(x)$ for $(\\alpha,s)\\in D$ it remains to show that this\nfunction is not $x$-analytic near zero whenever\n$(\\alpha - s)$ and $(\\alpha - 2s)$ are not elements of $\\mathbb{N}\\cup\n\\{0\\}$. But this follows directly by consideration\nof~\\eqref{eq:Ns_euler}---since, per points~\\eqref{dos}, ~\\eqref{tres}\nand ~\\eqref{cuatro}, for such values of $\\alpha$ and $s$ the\ncoefficient multiplying the non-analytic term $x^{-2s+\\alpha}$\nin~\\eqref{eq:Ns_euler} does not vanish. The proof is now complete.\n\\end{proof}\n\n\\subsection{Singularities on both edges\\label{two_edge_sing}}\nUtilizing Theorem~\\ref{teo1}, which in particular establishes that the\nimage of the function $u_\\alpha(y) = y^\\alpha$\n(equation~\\eqref{u_alpha}) under the operator $T_s$ is analytic for\n$\\alpha = s+n$, here we consider the image of the function\n\\begin{equation}\\label{udef}\n u(y) := y^s(1-y)^s y^n \n\\end{equation}\nunder the operator $T_s$ and we show that, in fact, $T_s[u]$ is a\npolynomial of degree $n$. This is a desirable result which, as we\nshall see, leads in particular to (i)~Diagonalization of weighted\nversion of the fractional Laplacian operator, as well as (ii)~Smoothness\nand even analyticity (up to a singular multiplicative weight) of\nsolutions of equation~\\eqref{eq:fraccionario_dirichlet} under suitable\nhypothesis on the right-hand side $f$.\n\n\\begin{remark}\\label{remark_idea_2s}\n Theorem~\\ref{teo1} states that the image of the aforementioned\n function $u_\\alpha$ under the operator $T_s$ is analytic not only\n for $\\alpha = s+n$ but also for $\\alpha = 2s+n$. But, as shown in\n Remark~\\ref{remark_2s}, the smoothness and analyticity theory\n mentioned in point~(ii) above, which applies in the case $\\alpha =\n s+n$, cannot be duplicated in the case $\\alpha = 2s+n$. Thus, except\n in Remark~\\ref{remark_2s}, the case $\\alpha = 2s+n$ will not be further\n considered in this paper.\n\\end{remark}\n\n\nIn view of Remark~\\ref{Ts_PV} and in order to obtain an explicit\nexpression for $T_s[u]$ we first express the derivative of $u$ in the\nform\n$$u'(y) = \\frac{d}{dy} \\left(y^{s} (1-y)^s y^n \\right) = y^{s-1}\n(1-y)^{s-1} \\left[ y^n (s+n-(2s+n)y) \\right] $$\nand (using~\\eqref{eq:c_s}) we thus obtain\n\\begin{equation}\n\\label{eq:Ts_useful}\n T_s[ u ] = (1-2s)C_s \\left( (s+n) L^s_n - (2s+n) L^s_{n+1} \\right) .\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:Ks_n}\nL^s_n := P.V. \\int_{0}^{1} \\operatorname{sgn}(x-y)|x-y|^{-2s} y^{s-1} (1-y)^{s-1} y^n dy\n\\end{equation}\nOn the other hand, in view of definitions~\\eqref{Tsdef_eq1}\nand~\\eqref{Tsdef_eq2} and Lemma~\\ref{lemma_exchangePV} it is easy to\ncheck that\n\\begin{equation}\n\\label{eq:Ss_useful}\n\\frac{\\partial}{\\partial x} S_{s}( y^{s-1} (1-y)^{s-1} y^n ) = (1-2s)C_s L^s_n .\n\\end{equation}\nIn order to characterize the image $T_s[u]$ of the function $u$\nin~\\eqref{udef} under the operator $T_s$, Lemma~\\ref{lemma_Lns} below\npresents an explicit expression for the closely related function\n$L^s_n$. In particular the lemma shows that $L^s_n$ is a polynomial of\ndegree $n-1$, which implies that $T_s[u]$ is a polynomial of degree\n$n$.\n\\begin{lemma}\n\\label{lemma_Lns}\n $L^s_n(x)$ is a polynomial of degree $n-1$. More precisely,\n \\begin{equation}\n\\label{eq:lemma_Lns}\nL^s_n(x) = \\Gamma(s) \\sum_{k=0}^{n-1} \\frac{(2s)_k}{k!} \\frac{\\Gamma( n - k - s+1)} { (s+ k - n ) \\Gamma(n - k)} x^k.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n We proceed by substituting $(1-y)^{s-1}$ in the\n integrand~\\eqref{eq:Ks_n} by its Taylor expansion around $y=0$,\n\\begin{equation}\\label{taylor_one_bnd}\n (1-y)^{s-1} = \\sum_{j=0}^{\\infty} q_j y^j, \\mbox{ with } q_j = \\frac{(1-s)_j}{j!}, \n\\end{equation}\nand subsequently exchanging the principal value integration with the\ninfinite sum (a step that is justified in\nAppendix~\\ref{appendix_pvseries}). The result is\n\\begin{equation}\n\\label{eq:series_Ks}\nL^s_n(x) = \\sum_{j=0}^{\\infty} \\left( \\mbox{P.V.} \\int_0^1 \\operatorname{sgn}(x-y)|x-y|^{-2s} q_j y^{s-1+n+j} dy \\right)\n\\end{equation}\nor, in terms of the functions $N_\\alpha^s$ defined in\nequation~\\eqref{eq:def_Ns},\n\\begin{equation}\\label{lns2}\nL^s_n(x) = \\sum_{j=0}^{\\infty} q_j N^s_{{s+n+j}} .\n\\end{equation}\n\nIn view of~\\eqref{eq:teo1}, equation~\\eqref{lns2} can also be made to\nread\n\\begin{equation}\n \\label{eq:series_ajk}\n L^s_n(x) = \\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} \\frac{(1-s)_j}{j!} \\frac{(2s)_k}{k!} \\frac{1}{s-n-j+k} \\, x^k,\n\\end{equation}\nor, interchanging of the order of summation in this expression (which\nis justified in Appendix~\\ref{appendix_sumorder}),\n\\begin{equation}\n\\label{eq:Kz_series}\n L^s_n(x) = \\sum_{k=0}^{\\infty} \\frac{(2s)_k}{k!} a_{k}^n x^k, \\mbox{ where } a_{k}^n = \\sum_{j=0}^{\\infty} \\frac{(1-s)_j}{j!} \\frac{1}{s-n-j+k}.\n\\end{equation}\nThe proof will be completed by evaluating explicitly the coefficients\n$a_k^n$ for all pairs of integers $k$ and $n$.\n\nIn order to evaluate $a_k^n$ we consider the Hypergeometric function\n\\begin{equation}\\label{hyper_geom}\n _2F_1(a,b;c;z)=\\sum_{j=0}^\\infty \\frac{(a)_j\n (b)_j}{(c)_j} \\frac{z^j}{j!}.\n\\end{equation}\nComparing the $a_k^n$ expression in~\\eqref{eq:Kz_series}\nto~\\eqref{hyper_geom} and taking into account the relation\n$$\\frac{1}{s-n-j+k} = \\frac{(n-k-s)_j}{(n-k-s+1)_j} \\, \\frac{1}{s+k-n}$$\n(which follows easily from the recursion $(z+1)_j= (z)_j (z+j)\/z$ for\nthe Pochhamer symbol defined in equation~\\eqref{def_Pochhamer}),\nwe see that $a_k^n$ can be expressed in terms of the Hypergeometric\nfunction $_2F_1$ evaluated at $z=1$:\n$$a_k^n = 2F_1(1-s,n-k-s;n-k-s+1;1)\/(s+k-n). $$ \nThis expression can be simplified further: in view of Gauss' formula\n$_2F_1(a,b;c;1) =\n\\frac{\\Gamma(c)\\Gamma(c-a-b)}{\\Gamma(c-a)\\Gamma(c-b)}$ (see\ne.g.~\\cite[p. 2]{Bailey}) we obtain the concise expression\n\\begin{equation}\n\\label{eq:ak_finitos}\n a_{k}^n = \\frac{\\Gamma(n-k-s+1)\\Gamma(s)} { (s+k-n) \\Gamma(n - k) } .\n\\end{equation}\nIt then clearly follows that $a_{k}^n = 0$ for $k \\ge n$---since the\nterm $\\Gamma(n - k)$ in the denominator of this expression is infinite\nfor all integers $k\\geq n$. The series in~\\eqref{eq:Kz_series} is\ntherefore a finite sum up to $k=n-1$ which, in view\nof~\\eqref{eq:ak_finitos}, coincides with the desired\nexpression~\\eqref{eq:lemma_Lns}. The proof is now complete.\n\\end{proof}\n\\begin{corollary}\\label{cor_poly_w}\n Let $w(y) = u(y)\\chi_{(0,1)}(y)$ where $u= y^s(1-y)^s y^n$\n (equation~\\eqref{udef}) and where $\\chi_{(0,1)}$ denotes the\n characteristic function of the interval $(0,1)$. Then, defining the\n $n$-th degree polynomial $p(x) = (1-2s)C_s \\left( (s+n) L^s_n -\n (2s+n) L^s_{n+1} \\right)$ with $L^s_n$ given\n by~\\eqref{eq:lemma_Lns}, for all $x\\in \\mathbb{R}$ such that $x\\ne\n 0$ and $x\\ne 1$ (cf. Remark~\\ref{bound_rem}) we have\n \\begin{equation}\\label{Ts_pol}\nT_s[u] (x) = p(x)\n\\end{equation}\nand, consequently,\n\\begin{equation}\\label{frac_pol}\n(-\\Delta)^s w(x) = p(x).\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\n In view of equation~\\eqref{eq:Ts_useful} and Lemma~\\ref{lemma_Lns}\n we obtain~\\eqref{Ts_pol}. The relation~\\eqref{frac_pol} then follows\n from Remark~\\ref{remark_Ts}.\n\\end{proof}\n\n\n\n\nIn view of equation~\\ref{eq:Ss_useful} and Lemma~\\ref{lemma_Lns}, the\nresults obtained for the image of $u(y) = y^{s}(1-y)^{s} y^n$ under\nthe operator $T_s$ can be easily adapted to obtain analogous\npolynomial expressions of degree exactly $n$ for the image of the\nfunction $\\tilde{u}(y) = y^{s-1}(1-y)^{s-1} y^n$ under the operator\n$S_{s}$. And, indeed, both of these results can be expressed in terms\nof isomorphisms in the space $\\mathbb{P}_n$ of polynomials of degree\nless or equal than $n$, as indicated in the following corollary,\n\\begin{corollary}\n\\label{coro_diagonal}\nLet $s\\in (0,1)$, $m\\in \\mathbb{N}$, and consider the linear mappings\n$P:\\mathbb{P}_m \\to \\mathbb{P}_m$ and $Q:\\mathbb{P}_m \\to\n\\mathbb{P}_m$ defined by\n\\begin{equation}\\label{eq_PQ}\n \\begin{split}\n P : p & \\to T_s[y^s(1-y)^s p(y)] \\;\\;\\; \\mbox{and} \\\\\n Q : p & \\to S_{s}[y^{s-1}(1-y)^{s-1} p(y)].\n \\end{split}\n\\end{equation}\nThen the matrices $[P]$ and $[Q]$ of the linear mappings $P$ and $Q$\nin the basis $\\{ y^n:n=0,\\dots,m\\}$ are upper-triangular and their\ndiagonal entries are given by\n\\begin{equation*}\n \\begin{split}\n P_{nn} = & \\frac{\\Gamma(2s+n+1) }{n!}\\;\\;\\; \\mbox{and} \\\\\n Q_{nn} = & -\\frac{\\Gamma(2s+n-1) }{n!},\n \\end{split}\n\\end{equation*}\nrespectively. In particular, for $s = \\frac 12$ we have\n\\begin{equation}\\label{diagonal_entries}\n \\begin{split}\n P_{nn} = & \\;\\; 2 n \\\\\n Q_{nn} = & - \\frac{2}{n} \\;\\;\\; \\mbox{for } \\;\\;\\; n\\ne 0 \\;\\;\\; \\mbox{and }\\;\\;\\; Q_{00} = -2\\log(2). \\\\\n \\end{split} \n\\end{equation}\n\\end{corollary}\n\\begin{proof}\n The expressions for $n\\ne 0$ and for $P_{00}$ follow directly from\n equations \\eqref{eq:Ts_useful}, \\eqref{eq:Ss_useful}\n and~\\eqref{eq:lemma_Lns}. In order to obtain $Q_{00}$, in turn, we\n note from~\\eqref{eq:Ss_useful} that for $n=0$ we have\n $\\frac{\\partial}{\\partial x} S_{s}( y^{s-1} (1-y)^{s-1} y^n )=0$.\n In particular, $S_{s}( y^{s-1} (1-y)^{s-1} )$ does not depend on $x$\n and we therefore obtain\n\\begin{equation*}\n \\begin{split}\n Q_{00} = S_{s}( y^{s-1} (1-y)^{s-1} ) &= C_s \\int_0^1 ( y^{2s-1} - 1 ) y^{s-1} (1-y)^{s-1} dy \\\\\n &= C_s \\left( \\mbox{B}(3s-1,s) - \\mbox{B}(s,s) \\right).\n \\end{split}\n\\end{equation*} \nIn the limit as $s \\to 1\/2$, employing l'H\\^opital's rule together with\nwell known values\\cite[6.1.8, 6.3.2, 6.3.3]{AbramowitzStegun} for the\nGamma function and it's derivative at $z=1\/2$ and $z=1$, we obtain\n$S_{\\frac 12}( y^{-1\/2} (1-y)^{-1\/2} ) = -2\\log(2)$\n\\end{proof}\n\n\\subsection{Diagonal Form of the Weighted Fractional Laplacian\\label{diag}}\nIn view of the form of the mapping $P$ in equation~\\eqref{eq_PQ} and\nusing the ``weight function''\n$$\\omega^s(y) = (y-a)^s(b-y)^s,$$\nfor $\\phi \\in C^2(a,b)\\cap C^1[a,b]$ (that is, $\\phi$ smooth up to the\nboundary but it does not necessarily vanish on the boundary) we\nintroduce the weighted version\n\\begin{equation}\\label{eq:weighted_hypersingular}\nK_s(\\phi) = C_s \\frac{d}{dx} \\int_{a}^{b} |x-y|^{1-2s} \\frac{d}{dy} \\left( \\omega^s \\phi(y) \\right) dy \\quad (s\\ne 1\/2),\n\\end{equation}\nof the operator $T_s$ in equation~\\eqref{Tsdef_eq3}. In view of Lemma~\\ref{lemma_hypersingular}, $K_s$ can also be viewed as a weighted version of the Fractional Laplacian operator, and we therefore define\n\\begin{equation}\\label{eq:weighted_fractional}\n(-\\Delta)_\\omega^s[ \\phi] = K_s(\\phi) \\;\\; \\mbox{for} \\;\\; \\phi \\in C^2(a,b)\\cap C^1[a,b].\n\\end{equation}\n\n\\begin{remark}\\label{rem_connection_u}\n Clearly, given a solution $\\phi$ of the equation \n \\begin{equation}\\label{eqn_weighted}\n (-\\Delta)_\\omega^s[\\phi] = f\n\\end{equation}\nin the domain $\\Omega = (a,b)$, the function $u= \\omega^s \\phi$ extended\nby zero outside $(a,b)$ solves the Dirichlet problem for the\nFractional Laplacian~\\eqref{eq:fraccionario_dirichlet}\n(cf. Lemma~\\ref{lemma_hypersingular}).\n\\end{remark}\n\n\nIn order to study the spectral properties of the operator\n$(-\\Delta)^s_\\omega,$ consider the weighted $L^2$ space\n\\begin{equation}\\label{weighted_L2}\n L^2_s(a,b) = \\left\\{ \\phi:(a,b) \\to {\\mathbb {R}} \\ \\colon \\int_a^b |\\phi|^2 \\omega^s < \\infty \\right\\},\n\\end{equation}\nwhich, together with the inner product\n\\begin{equation}\\label{scalarproduct_L2}\n (\\phi,\\psi)^s_{a,b} = \\int_a^b \\phi \\, \\psi \\, \\omega^s \n\\end{equation}\nand associated norm is a Hilbert space. We can now establish the\nfollowing lemma.\n\\begin{lemma}\n\\label{teo:autoadj}\nThe operator $(-\\Delta)_\\omega^s$ maps $\\mathbb{P}_n$ into itself.\nThe restriction of $(-\\Delta)_\\omega^s$ to $\\mathbb{P}_n$ is a self\nadjoint operator with respect to the inner product\n$(\\cdot,\\cdot)^s_{a,b}$.\n\\end{lemma}\n\\begin{proof}\n Using the notation $K_s=(-\\Delta)_\\omega^s$, we first establish the\n relation $(K_s[p],q) = (p,K_s[q])$ for $p,q \\in \\mathbb{P}_n$. But\n this follows directly from application of integration by parts and\n Fubini's theorem followed by an additional instance of integration\n by parts in \\eqref{eq:weighted_hypersingular}, and noting that the\n the boundary terms vanish by virtue of the weight $\\omega^s$.\n\\end{proof}\nThe orthogonal polynomials with respect to the inner product under\nconsideration are the well known Gegenbauer\npolynomials~\\cite{AbramowitzStegun}. These are defined on the interval $(-1,1)$ by\nthe recurrence\n\\begin{equation}\\label{eq:recurrencia}\n\\begin{split}\n C_{0}^{(\\alpha)}(x) & = 1, \\\\\n C_{1}^{(\\alpha)}(x) & = 2 \\alpha x, \\\\\n C_{n}^{(\\alpha)}(x) & = \\frac{1}{n}\n \\left[2x(n+\\alpha-1)C_{n-1}^{(\\alpha)}(x) -\n (n+2\\alpha-2)C_{n-2}^{(\\alpha)}(x) \\right];\n\\end{split}\\end{equation}\nfor an arbitrary interval $(a,b)$, the corresponding orthogonal\npolynomials can be easily obtained by means of a suitable affine\nchange of variables. Using this orthogonal basis we can now produce an\nexplicit diagonalization of the operator $(-\\Delta)^s_\\omega$. We first consider the\ninterval $(0,1)$; the corresponding result for a general interval\n$(a,b)$ is presented in Corollary~\\ref{diag_ab}.\n\\begin{theorem} \\label{teo:diagonalform} Given $s\\in(0,1)$ and $n \\in\n \\mathbb{N}\\cup \\{ 0 \\} $, consider the Gegenbauer polynomial\n $C^{(s+1\/2)}_n$, and let $p_n(x) = C^{(s+1\/2)}_n(2x-1)$. Then the\n weighted operator $(-\\Delta)^s_\\omega$ in the interval $(0,1)$ satisfies\n the identity \n\\begin{equation} \\label{eq_diagform}\n(-\\Delta)^s_\\omega( p_n ) =\n\\frac{\\Gamma(2s+n+1)}{n!} \\, p_n .\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\n By Lemma~\\ref{teo:autoadj} the restriction of the operator\n $(-\\Delta)^s_\\omega$ to the subspace $\\mathbb{P}_m$ is self-adjoint and\n thus diagonalizable. We may therefore select polynomials $q_0,\n q_1,\\dots,q_m\\in\\mathbb{P}_m$ (where, for $0\\leq n\\leq m$, $q_n$ is\n a polynomial eigenfunction of $(-\\Delta)^s_\\omega$ of degree exactly\n $n$) which form an orthogonal basis of the space\n $\\mathbb{P}_m$. Clearly, the eigenfunctions $q_n$ are orthogonal and,\n therefore, up to constant factors, the polynomials $q_n$ must\n coincide with $p_n$ for all $n$, $0\\leq n\\leq m$. The corresponding\n eigenvalues can be extracted from the diagonal elements, displayed\n in equation~\\eqref{diagonal_entries}, of the upper-triangular matrix $[P]$\n considered in Corollary~\\ref{coro_diagonal}. These entries coincide\n with the constant term in~\\eqref{eq_diagform}, and the proof is\n thus complete.\n\\end{proof}\n\n\\begin{corollary}\\label{diag_ab}\n The weighted operator $(-\\Delta)^s_\\omega$ in the interval $(-1,1)$\n satisfies the identity\n\\[\n(-\\Delta)^s_\\omega(C_n^{(s+1\/2)}) = \\lambda_n^s \\, C_n^{(s+1\/2)} ,\n\\]\t\nwhere\n\\begin{equation}\n\\label{Eigenvalues}\n\\lambda_n^s = \n\\frac{\\Gamma(2s+n+1)}{n!}.\n\\end{equation}\nMoreover in the interval $(a,b)$, we have\t\n\\begin{equation}\\label{eigenfuncts} (-\\Delta)^s_\\omega(p_n) = \n\\lambda_n^s \\, p_n ,\n\\end{equation}\nwhere $p_n(x) = C_n^{(s+1\/2)}\\left(\\frac{2(x-a)}{b-a} - 1 \\right)$. \n\\end{corollary}\n\\begin{proof}\n The formula is obtained by employing the change of variables\n $\\tilde x=(x-a)\/(b-a) $ and $\\tilde y =(y-a)\/(b-a)$ in equation~\\eqref{eq:weighted_hypersingular}\n to map the weighted operator in $(a,b)$ to the corresponding operator in $(0,1)$, \n\tand observing that $\\omega^s(y) = (b-a)^{2s} \\tilde \\omega^s (\\tilde y)$, where \n\t$\\tilde \\omega^s (\\tilde y) = \\tilde{y}^s (1 - \\tilde y)^s.$\n\\end{proof}\n\n\\begin{remark}\\label{eigenvalue_asymptotics}\n It is useful to note that, in view of the formula $\\lim_{n\\to\\infty}\n n^{\\beta-\\alpha}\\Gamma(n+\\alpha)\/\\Gamma(n+\\beta) = 1$ (see\n e.g.~\\cite[6.1.46]{AbramowitzStegun}) we have the asymptotic\n relation $\\lambda_n^s \\approx O\\left( n^{2s} \\right)$ for the\n eigenvalues~\\eqref{Eigenvalues}. This fact will be exploited in the\n following sections in order to obtain sharp Sobolev regularity\n results as well as regularity results in spaces of analytic\n functions.\n\\end{remark}\n\nAs indicated in the following corollary, the background developed in\nthe present section can additionally be used to obtain the\ndiagonal form of the operator $S_s$ for all $s\\in (0,1)$. This\ncorollary generalizes a corresponding existing result\nfor the case $s=1\/2$---for which, as indicated in\nRemark~\\ref{remark_openarcs}, the operator $S_{s}$ coincides with the\nsingle-layer potential for the solution of the two-dimensional Laplace\nequation outside a straight arc or ``crack''.\n\\begin{corollary} The weighted operator $\\phi \\to\n S_s[\\omega^{s-1} \\phi]$ can be diagonalized in terms of the Gegenbauer\n polynomials $C_n^{(s-1\/2)}$\n\\begin{equation*}\n S_{s} \\left[\\omega^{s-1} C_n^{(s-1\/2)} \\right] = \\mu_n^s C^{(s-1\/2)}_n ,\n\\end{equation*}\nwhere in this case the eigenvalues are given by\n$$ \\mu_n^s =\n- \\frac{\\Gamma(2s+n-1) } {n!} .$$\n\\end{corollary}\n\\begin{proof}\nThe proof for the interval $[0,1]$ is analogous to that of Theorem \\ref{teo:diagonalform}. In this case, the eigenvalues are extracted from the diagonal entries of the upper triangular matrix $[Q]$ in equation \\eqref{diagonal_entries}. A linear change of variables allows to obtain the desired formula for an arbitrary interval.\n\\end{proof}\n\n\\begin{corollary}\n In the particular case $s=1\/2$ on the interval $(-1,1)$, the\n previous results amount, on one hand, to the known\n result~\\cite[eq. 9.27]{Handscomb} (cf also~\\cite{YanSloan}),\n \n \n\\begin{equation*}\n\\int_{-1}^1 \\log|x-y| T_n(y) (1-y^2)^{-1\/2} dy = \n\\left\\lbrace\n \\begin{array}{rl}\n -\\frac{\\pi}{n} T_n & \\mbox{ for } n \\ne 0 \\\\\n -2\\log(2) & \\mbox{ for } n = 0 \\\\\n \\end{array}\n \\right. \n\\end{equation*}\n(where $T_n$ denotes the Tchevyshev polynomial of the first kind),\nand, on the other hand, to the relation\n\\begin{equation*}\n\\frac{\\partial}{\\partial x} \\int_{-1}^1 \\log|x-y| \\frac{\\partial}{\\partial y} \\left( U_n(y) (1-y^2)^{1\/2} \\right) dy = (n+1) \\pi U_n \n\\end{equation*}\n(where $U_n$ denotes the Tchevyshev polynomial of the second kind).\n\\end{corollary}\n\n\n\n\n\n\n\\section{Regularity Theory}\n\\label{regularity}\nThis section studies the regularity of solutions of the fractional\nLaplacian equation~\\eqref{eq:fraccionario_dirichlet} under various\nsmoothness assumptions on the right-hand side $f$--including\ntreatments in both Sobolev and analytic function spaces, and for\nmulti-interval domains $\\Omega$ as in\nDefinition~\\ref{union_intervals_def}. In particular,\nSection~\\ref{Sobolev} introduces certain weighted Sobolev spaces\n$H^r_{s}(\\Omega)$ (which are defined by means of expansions in Gegenbauer\npolynomials together with an associated norm). The space $A_\\rho$ of\nanalytic functions in a certain ``Bernstein Ellipse''\n$\\mathcal{B}_\\rho$ is then considered in\nSection~\\ref{single_interval_analytic}. The main result in\nSection~\\ref{Sobolev} (resp. Section~\\ref{single_interval_analytic})\nestablishes that for right-hand sides $f$ in the space $H^r_{s}(\\Omega)$\nwith $r\\geq 0$ (resp. the space $A_\\rho(\\Omega)$ with $\\rho >0$) the\nsolution $u$ of equation~\\eqref{eq:fraccionario_dirichlet} can be\nexpressed in the form $u(x) = \\omega^s(x) \\phi(x)$, where $\\phi$ belongs\nto $H^{r+2s}_s(\\Omega)$ (resp. to $A_\\rho(\\Omega)$). Sections~\\ref{Sobolev}\nand~\\ref{single_interval_analytic} consider the single-interval case;\ngeneralizations of all results to the multi-interval context are\npresented in Section~\\ref{regularity_multi_int}. The theoretical\nbackground developed in the present Section~\\ref{regularity} is\nexploited in Section~\\ref{HONM} to develop and analyze a class of\neffective algorithms for the numerical solution of\nequation~\\eqref{eq:fraccionario_dirichlet} in multi-interval domains\n$\\Omega$.\n\n\\subsection{Sobolev Regularity, single interval case}\\label{Sobolev}\n\nIn this section we define certain weighted Sobolev spaces, which\nprovide a sharp regularity result for the weighted Fractional\nLaplacian $(-\\Delta)^s_\\omega$ (Theorem \\ref{teo_extended}) as well as a\nnatural framework for the analysis of the high order numerical methods\nproposed in Section~\\ref{HONM}. It is noted that these spaces\ncoincide with the non-uniformly weighted Sobolev spaces introduced\nin~\\cite{BabuskaGuo}; Theorem~\\ref{gegen_embedding_Hr} below provides\nan embedding of these spaces into spaces of continuously\ndifferentiable functions. For notational convenience, in the present\ndiscussion leading to the definition~\\ref{def:sobolev} of the Sobolev\nspace $H^r_s(\\Omega)$, we restrict our attention to the domain $\\Omega =\n(-1,1)$; the corresponding definition for general multi-interval\ndomains then follows easily.\n\n\nIn order to introduce the weighted Sobolev spaces we note that the set\nof Gegenbauer polynomials $C^{(s + 1\/2)}_n$ constitutes an orthogonal\nbasis of $L^2_s(-1,1)$ (cf. ~\\eqref{weighted_L2}). The $L^2_s$ norm of \na Gegenbauer polynomial (see ~\\cite[eq 22.2.3]{AbramowitzStegun}), is given by \n\\begin{equation} \n\\label{eq:norm_gegen}\nh_j^{(s+1\/2)} = \\left\\| C^{(s + 1\/2)}_j \\right\\|_{L^2_s(-1,1)} = \\sqrt{\\frac{2^{-2s}\\pi}{\\Gamma^2(s + 1\/2)} \\frac{\\Gamma(j+2s+1)}{\\Gamma(j+1)(j+s+1\/2)}}.\n\\end{equation}\n\\begin{definition}\\label{normal}\n Throughout this paper $\\tilde{C}^{(s + 1\/2)}_j$ denotes the\n normalized polynomial $C^{(s + 1\/2)}_j \/ h_j^{(s+1\/2)}$.\n\\end{definition}\nGiven a function $v\\in L^2_s(-1,1)$, we have the following expansion\n\\begin{equation}\\label{exp_gegen}\nv (x) = \\sum_{j=0}^\\infty v_{j,s} \\tilde{C}^{(s + 1\/2)}_j (x), \n\\end{equation}\nwhich converges in $L^2_s(-1,1)$, and where\n\\begin{equation}\\label{gegen_coef}\n v_{j,s} = \\int_{-1}^1 v(x) \\tilde{C}^{(s + 1\/2)}_j (x) (1-x^2)^s dx.\n\\end{equation}\n\n\nIn view of the expression\n \\begin{equation}\\label{gegen_poly_der}\n \\frac{d}{dx} C^{(\\alpha)}_j (x) = 2\\alpha C_{j-1}^{(\\alpha + 1)}(x), \\ j \\geq 1,\n \\end{equation}\n for the derivative of a Gegenbauer polynomial (see e.g.~\\cite[eq. 4.7.14]{szego}), we have\n\\begin{equation}\n \\frac{d}{dx} \\tilde{C}^{(s+1\/2)}_j (x) = (2s+1) \\frac{h^{(s+3\/2)}_{j-1}}{h^{(s+1\/2)}_j} \\tilde{C}^{s+3\/2}_{j-1}.\n\\end{equation}\nThus, using term-wise differentiation in~\\eqref{exp_gegen} we may\nconjecture that, for sufficiently smooth functions $v$, we have\n\\begin{equation}\\label{k-der}\n v^{(k)}(x) =\\sum_{j=k}^\\infty v_{j-k,s+k}^{(k)} \\tilde {C}_{j-k}^{(s+k+1\/2)}(x)\n\\end{equation}\nwhere $ v^{(k)}(x)$ denotes the $k$-th derivative of the function\n$v(x)$ and where, calling\n\\begin{equation}\\label{A_nk_def}\n A_j^{k} = \\prod_{r=0}^{k-1} \\frac{h^{(s+3\/2+r)}_{j-1-r}}{h^{(s+1\/2+r)}_{j-r}} (2(s+r)+1) = 2^k \\frac{h^{(s+1\/2+k)}_{j-k}}{h^{(s+1\/2)}_{j}} \\frac{\\Gamma(s+1\/2+k)}{\\Gamma(s+1\/2)},\n\\end{equation}\nthe coefficients in~\\eqref{k-der} are given by\n\\begin{equation}\\label{k-der-coeffs}\n v_{j-k,s+k}^{(k)} = A_j^{k}\\, v_{j,s}.\n\\end{equation}\n\n\n\nLemma~\\ref{lemma_gegen_parts} below provides, in particular, a\nrigorous proof of~\\eqref{k-der} under minimal hypothesis. Further,\nthe integration by parts formula established in that lemma together\nwith the asymptotic estimates on the factors $B_j^{k}$ provided in\nLemma~\\ref{A_jk_lemma}, then allow us to relate the smoothness of a\nfunction $v$ and the decay of its Gegenbauer coefficients; see\nCorolary~\\ref{gegen_decay_coro}.\n\\begin{lemma}[Integration by parts]\\label{lemma_gegen_parts}\n Let $k\\in \\mathbb{N}$ and let $v \\in C^{k-2}[-1,1]$ such that for a\n certain decomposition $[-1,1] = \\bigcup_{i=1}^n\n [\\alpha_i,\\alpha_{i+1}]$ ($-1=\\alpha_1<\\alpha_i<\\alpha_{i+1}\n <\\alpha_n=1$) and for certain functions $\\tilde v_i \\in\n C^{k}[\\alpha_i,\\alpha_{i+1}]$ we have $v(x)=\\tilde v_i(x)$ for all\n $x\\in (\\alpha_i,\\alpha_{i+1})$ and $1\\leq i\\leq n$. Then for $j\\geq\n k$ the $s$-weighted Gegenbauer coefficients $v_{j,s}$ defined in\n equation~\\eqref{gegen_coef} satisfy\n\\begin{equation}\\label{gegen_coef_parts_k}\n\\begin{split}\nv_{j,s} = B_j^{k} \\int_{-1}^1 & \\tilde v^{(k)}(x) \\tilde{C}_{j-k}^{(s+k+1\/2)}(x) (1-x^2)^{s+k} dx \\\\\n &-B_j^{k} \\sum_{i=1}^n \\left[ \\tilde v^{(k-1)}(x) \\tilde{C}_{j-k}^{(s+k+1\/2)}(x) (1-x^2)^{s+k} \\right]_{\\alpha_i}^{\\alpha_{i+1}},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\\label{B_jk_def}\n B_j^k = \\frac{h_{j-k}^{(s+k+1\/2)}}{h_j^{(s+1\/2)}} \\prod_{r=0}^{k-1} \\frac{(2(s+r)+1)}{(j-r)(2s+r+j+1)}.\n\\end{equation}\nWith reference to equation~\\eqref{A_nk_def}, further, we have\n$A_j^k=\\frac{1}{B_j^k}$. In particular, under the additional\nassumption that $v \\in C^{k-1}[-1,1]$ the\nrelation~\\eqref{k-der-coeffs} holds.\n\\end{lemma}\n\\begin{proof}\n Equation~\\eqref{gegen_coef_parts_k} results from iterated\n applications of integration by parts together with the\n relation~\\cite[eq. 22.13.2]{AbramowitzStegun}\n$$ \\frac{\\ell(2t+\\ell+1)}{2t+1} \\int (1-y^2)^{t} C_\\ell^{(t+1\/2)}(y) dy = \n- (1-x^2)^{t+1} C_{\\ell-1}^{(t+3\/2)}(x). $$ and subsequent\nnormalization according to Definition~\\ref{normal}. The validity of\nthe relation $A_j^k=\\frac{1}{B_j^k}$ can be checked easily.\n\\end{proof}\n\n\\begin{lemma}\\label{A_jk_lemma} There exist constants $C_1$ and $C_2$ such that the\n factors $B_j^k$ in equation~\\eqref{A_nk_def} satisfy\n\\begin{equation*}\\label{A_jk_inequality}\n C_1 j^{-k} < |B_j^k| < C_2 j^{-k}\n\\end{equation*}\n\\end{lemma} \n\\begin{proof}\n In view of the relation $\\lim_{j\\to\\infty}\n j^{b-a}\\Gamma(j+a)\/\\Gamma(j+b) = 1$\n (see~\\cite[6.1.46]{AbramowitzStegun}) it follows that\n $h_j^{(s+1\/2)}$ in equation~\\eqref{eq:norm_gegen} satisfies\n \\begin{equation}\\label{gegen_norm_est}\n \\lim_{j\\to\\infty} j^{1\/2-s} h_j^{(s+1\/2)} \\ne 0\n \\end{equation}\n and, thus, letting\n\\begin{equation}\\label{gegen_ratio}\n q_j^k=\\frac{h_{j-k}^{(s+k+1\/2)}}{h_j^{(s+1\/2)}},\n\\end{equation}\nwe obtain \n\\begin{equation}\\label{q_bound}\n \\lim_{j\\to\\infty} q_j^k\/j^{k} \\ne 0.\n\\end{equation}\nThe lemma now follows by estimating the asymptotics of the product\nterm on the right-hand side of~\\eqref{B_jk_def} as $j\\to\\infty$.\n\\end{proof}\n\\begin{corollary}\\label{gegen_decay_coro}\n Let $k\\in \\mathbb{N}$ and let $v$ satisfy the hypothesis of\n Lemma~\\ref{lemma_gegen_parts}. Then the Gegenbauer coefficients\n $v_{j,s}$ in equation~\\eqref{gegen_coef} are quantities of order\n $O(j^{-k})$ as $j\\to\\infty$:\n$$|v_{j,s}| < C j^{-k}$$\nfor a constant $C$ that depends on $v$ and $k$.\n\\end{corollary}\n\\begin{proof}\n The proof of the corollary proceeds by noting that the factor\n $B_j^{k}$ in equation~\\eqref{gegen_coef_parts_k} is a quantity of\n order $j^{-k}$ (Lemma~\\ref{A_jk_lemma}), and obtaining bounds for\n the remaining factors in that equation. These bounds can be\n produced by (i)~applying the Cauchy-Schwartz inequality in the space\n $L^2_{s+k}(-1,1)$ to the $(s+k)$-weighted scalar\n product~\\eqref{scalarproduct_L2} that occurs in\n equation~\\eqref{gegen_coef_parts_k}; and\n (ii)~using~\\cite[eq. 7.33.6]{szego} to estimate the boundary terms\n in equation~\\eqref{gegen_coef_parts_k}. The derivation of the bound\n per point~(i) is straightforward. From~\\cite[eq. 7.33.6]{szego}, on\n the other hand, it follows directly that for each $\\lambda >0$ there\n is a constant $C$ such that\n$$ |\\sin(\\theta)^{2\\lambda-1} C^{\\lambda}_j(\\cos(\\theta))| \\le C j^{\\lambda-1}. $$\nLetting $x=\\cos(\\theta)$, $\\lambda =s+k+1\/2$ and dividing by the\nnormalization constant $h_{j}^{(s+k+1\/2)}$ we then obtain\n$$ \\left |\\tilde{C}^{s+k+1\/2}_j(x)(1-x^2)^{s+k}\\right | < Cj^{s+k-1\/2} \/ h_{j}^{(s+k+1\/2)}.$$ \nIn view of~\\eqref{gegen_norm_est}, the right hand side in this equation\nis bounded for all $j\\geq 0$. The proof now follows from\nLemma~\\ref{A_jk_lemma}.\n\\end{proof}\n\n\n\n\nWe now define a class of Sobolev spaces $H^r_s$ \nthat, as shown in Theorem~\\ref{teo_extended}, completely characterizes\nthe Sobolev regularity of the weighted fractional Laplacian operator\n$(-\\Delta)^s_\\omega$. \n\\begin{remark}\\label{no-s}\n In what follows, and when clear from the context, we drop the\n subindex $s$ in the notation for Gegenbauer coefficients such as\n $v_{j,s}$ in~\\eqref{gegen_coef}, and we write e.g. $v_j=v_{j,s}$,\n $w_j=w_{j,s}$, $f_j=f_{j,s}$, etc.\n\\end{remark}\n\n\\begin{definition}\\label{def:sobolev} Let $r,s\\in\\mathbb{R}$, \n $r\\geq 0$, $s> -1\/2$ and, for $v \\in L^2_s(-1,1)$ call $v_j$ the\n corresponding Gegenbauer coefficient~\\eqref{gegen_coef} (see\n Remark~\\ref{no-s}). Then the complex vector space $H^r_s(-1,1) =\n \\left\\{ v \\in L^2_s(-1,1) \\colon \\sum_{j=0}^\\infty (1+j^{2})^r\n |v_j|^2 < \\infty \\right\\}$ will be called the $s$-weighted Sobolev\n space of order $r$.\n\\end{definition}\n\\begin{lemma}\\label{lemma_hilbert_space}\n Let $r,s\\in\\mathbb{R}$, $r\\geq 0$, $s> -1\/2$. Then the\n space $H^r_s(-1,1)$ endowed with the inner product $\\langle v, w\n \\rangle_s^r = \\sum_{j=0}^\\infty v_j w_j (1 + j^{2})^r$ and\n associated norm \n\\begin{equation}\\label{H_r_norm}\n \\| v \\|_{H_s^r} = \\sum_{j=0}^\\infty |v_j|^2 (1 + j^{2}\n )^r\n\\end{equation}\n is a Hilbert space.\n\\end{lemma}\n\\begin{proof}\n The proof is completely analogous to that of \\cite[Theorem\n 8.2]{Kress}.\n\\end{proof}\n\\begin{remark}\\label{gegen_aprox_Hs} \n By definition it is immediately checked that for every function\n $v\\in H^r_s(-1,1)$ the Gegenbauer expansion~\\eqref{exp_gegen} with\n expansion coefficients~\\eqref{gegen_coef} is convergent in\n $H_s^r(-1,1)$.\n\\end{remark}\n\\begin{remark}\\label{sobolev_remark}\n In view of the Parseval identity $\\|v\\|_{L^2_s(-1,1)}^2 =\n \\sum_{n=0}^\\infty |v_n|^2$ it follows that the Hilbert spaces\n $H^0_s(-1,1)$ and $L^2_s(-1,1)$ coincide. Further, we have the dense\n compact embedding $H^t_s(-1,1)\\subset H^r_s(-1,1)$ whenever $r0$,\n $H^r_s(-1,1)$ constitutes an interpolation space between $H^{\\lfloor\n r \\rfloor}_s(-1,1)$ and $H^{\\lceil r \\rceil}_s(-1,1)$ in the sense\n defined by \\cite[Chapter 2]{BerghLofstrom}.\n\\end{remark}\n\nClosely related ``Jacobi-weighted Sobolev spaces'' $\\mathcal{H}^k_s$\n(Definition~\\ref{Guo_def}) were introduced\npreviously~\\cite{BabuskaGuo} in connection with Jacobi approximation\nproblems in the $p$-version of the finite element method. As shown in\nLemma~\\ref{sobolev_equivalence} below, in fact, the spaces\n$\\mathcal{H}^k_s$ coincide with the spaces $H^k_s$ defined above, and\nthe respective norms are equivalent.\n\\begin{definition}[Babu\\v{s}ka and Guo~\\cite{BabuskaGuo}]\\label{Guo_def}\n Let $k\\in\\mathbb{N}\\cup \\{0\\}$ and $r>0$. The $k$-th order\n non-uniformly weighted Sobolev space $\\mathcal{H}^k_s(a,b)$ is\n defined as the completion of the set $C^\\infty(a,b)$ under the norm\n\\begin{equation*}\\label{gou_norm}\n \\|v \\|_{\\mathcal{H}^k_s} = \\left( \\sum_{j=0}^k \\int_{a}^b |v^{(j)}(x)|^2 \\omega^{s+j} dx \\right)^{1\/2} = \\left( \\sum_{j=0}^k \\| v^{(j)} \\|_{L^2_{s+j}}^2 \\right)^{1\/2}.\n\\end{equation*} \nThe $r$-th order space $\\mathcal{H}^r_s(a,b)$, in turn, is defined by\ninterpolation of the spaces $\\mathcal{H}^k_s(a,b)$\n($k\\in\\mathbb{N}\\cup \\{0\\}$) by the $K$-method (see ~\\cite[Section\n3.1]{BerghLofstrom}).\n\\end{definition}\n\n\\begin{lemma}\\label{sobolev_equivalence}\n Let $r>0$. The spaces $H^r_s(a,b)$ and $\\mathcal{H}^r_s(a,b)$\n coincide, and their corresponding norms $\\| \\cdot \\|_{H^r_s}$ and $\\|\n \\cdot \\|_{\\mathcal{H}^r_s}$ are equivalent.\n\\end{lemma}\n\\begin{proof}\n A proof of this lemma for all $r>0$ can be found in \\cite[Theorem\n 2.1 and Remark 2.3]{BabuskaGuo}. In what follows we present an\n alternative proof for non-negative integer values of $r$:\n $r=k\\in\\mathbb{N}\\cup \\{0\\}$. In this case it suffices to show that\n the norms $\\| \\cdot \\|_{H^k_s}$ and $\\| \\cdot \\|_{\\mathcal{H}^k_s}$\n are equivalent on the dense subset $C^\\infty[a,b]$ of both\n $H^k_s(a,b)$ (Remark~\\ref{gegen_aprox_Hs}) and\n $\\mathcal{H}^k_s(a,b)$. But, for $v\\in C^\\infty[a,b]$,\n using~\\eqref{k-der}, Parseval's identity in $L^2_{s+k}$ and\n Lemma~\\ref{lemma_gegen_parts} we see that for every integer $k\\geq\n 0$ we have $\\| v^{(k)} \\|_{L^2_{s+k}} = \\sum_{j=k}^\\infty\n |v_{j-k,s+k}^{(k)}|^2 = \\sum_{j=k}^\\infty |v_{j,s}|^2 \/\n |B_j^k|^2$. From Lemma~\\ref{A_jk_lemma} we then obtain\n$$ D_1 \\sum_{j=k}^\\infty |v_{j,s}|^2 j^{2k} \\le \\| v^{(k)} \\|_{L^2_{s+k}}^2 \\le D_2 \\sum_{j=k}^\\infty |v_{j,s}|^2 j^{2k} $$\nfor certain constants $D_1$ and $D_2$, where $v_{j-k,s+k}^{(k)}$. In\nview of the inequalities\n $$ (1+j^{2k}) \\le (1+j^2)^k \\le (2j^2)^k \\le 2^k(1+j^{2k}) $$ \n the claimed norm equivalence for $r=k\\in\\mathbb{N}\\cup \\{0\\}$ and\n $v\\in C^\\infty[a,b]$ follows. \n\\end{proof}\n\nSharp regularity results for the Fractional Laplacian in the Sobolev space\n$H^r_s(a,b)$ can now be obtained easily.\n\\begin{theorem}\\label{teo_extended}\n Let $r\\geq 0$. Then the weighted fractional Laplacian\n operator~\\eqref{eq:weighted_fractional} can be extended uniquely to\n a continuous linear map $(-\\Delta)^s_\\omega$ from $H_s^{r+2s}(a,b)$ into\n $H_s^{r}(a,b)$. The extended operator is bijective and bicontinuous.\n\\end{theorem}\n\\begin{proof} Without loss of generality, we assume $(a,b)=(-1,1)$.\n Let $\\phi\\in H^{r+2s}_s(-1,1)$, and let $\\phi^n = \\sum_{j=0}^n\n \\phi_j \\tilde{C}_j^{(s+1\/2)} $ where $\\phi_j$ denotes the Gegenbauer\n coefficient of $\\phi$ as given by equation~\\eqref{gegen_coef} with\n $v=\\phi$. According to Corollary~\\ref{diag_ab} we have\n $(-\\Delta)^s_\\omega \\phi^n = \\sum_{j=0}^n \\lambda_j^s \\phi_j\n \\tilde{C}_j^{(s+1\/2)}$. In view of Remarks~\\ref{gegen_aprox_Hs}\n and~\\ref{eigenvalue_asymptotics} it is clear that $(-\\Delta)^s_\\omega\n \\phi^n$ is a Cauchy sequence (and thus a convergent sequence) in\n $H_s^{r}(-1,1)$. We may thus define \n$$(-\\Delta)^s_\\omega \\phi = \\lim_{n\\to\\infty}\n(-\\Delta)^s_\\omega \\phi^n = \\sum_{j=0}^\\infty\\lambda_j^s \\phi_j\n\\tilde{C}_j^{(s+1\/2)}\\in H_s^{r}(-1,1).$$ \nThe bijectivity and bicontinuity of the\nextended mapping follows easily, in view of Remark~\\ref{eigenvalue_asymptotics}, \nas does the uniqueness of continuous extension. The proof is complete.\n\\end{proof}\n\\begin{corollary} \\label{coro_u_sobolev} The solution $u$\n of~\\eqref{eq:fraccionario_dirichlet} with right-hand side $f\\in\n H^r_s(a,b)$ ($r\\geq 0$) can be expressed in the form $u=\\omega^s\\phi$ for\n some $\\phi \\in H^{r+2s}_s(a,b)$.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem~\\ref{teo_extended} and\n Remark~\\ref{rem_connection_u}.\n\\end{proof}\n\nThe classical smoothness of solutions of\nequation~\\eqref{eq:fraccionario_dirichlet} for sufficiently smooth\nright-hand sides results from the following version of the ``Sobolev\nembedding'' theorem.\n\\begin{theorem}[Sobolev's Lemma for weighted\n spaces]\\label{gegen_embedding_Hr}\n Let $s \\ge 0$, $k\\in \\mathbb{N}\\cup \\{0\\}$ and $r > 2k + s +\n 1$. Then we have a continuous embedding $H^r_s(a,b)\\subset C^k[a,b]$\n of $H^r_s(a,b)$ into the Banach space $C^k[a,b]$ of $k$-continuously\n differentiable functions in $[a,b]$ with the usual norm $\\| v \\|_k$\n (given by the sum of the $L^\\infty$ norms of the function and the\n $k$-th derivative): $\\| v \\|_k = \\| v \\|_\\infty + \\| v^{(k)}\n \\|_\\infty$.\n\\end{theorem}\n\\begin{proof} Without loss of generality we restrict attention to\n $(a,b)=(-1,1)$. Let $0\\le \\ell \\le k$ and let $v\\in H^r_s(-1,1)$ be\n given. Using the expansion~\\eqref{exp_gegen} and in view of the\n relation~\\eqref{gegen_poly_der} for the derivative of a Gegenbauer polynomial, we\n consider the partial sums\n \\begin{equation}\\label{exp_gegen_der}\n v^{(\\ell)}_n(x) = 2^\\ell \\prod_{i=1}^\\ell (s + i - 1\/2)\n \\sum_{j=\\ell}^n \\frac{v_j}{h_j^{(s+1\/2)}} C^{(s+\\ell+1\/2)}_{j-\\ell}(x)\n \\end{equation}\n that result as the partial sums corresponding to~\\eqref{exp_gegen} up to $j=n$ are differentiated $\\ell$ times.\n But we have the estimate\n \\begin{equation}\\label{bound_gegen}\n \\|C^{(s+1\/2)}_{n}\\|_\\infty \\sim O(n^{2s}).\n \\end{equation}\n which is an immediate consequence of~\\cite[Theorem 7.33.1]{szego}. Thus, taking into account~\\eqref{gegen_norm_est}, we obtain \n $$ |v_n^{(\\ell)}(x)| \\le C(\\ell) \\sum_{j=0}^{n-\\ell} \\frac{| v_{j+\\ell}|} {h_{j+\\ell}^{(s+1\/2)}} |C^{(s+\\ell+1\/2)}_{j}(x)| \n \\le C(\\ell) \\sum_{j=0}^{n-\\ell} (1+j^2)^{(s+2\\ell)\/2 + 1\/4} | v_{j+\\ell}| ,$$\nfor some constant $C(\\ell)$. Multiplying and dividing by $(1+j^2)^{r\/2}$ and\napplying the Cauchy-Schwartz inequality in the space of square\nsummable sequences it follows that\n\\begin{equation}\\label{g_der}\n| v_n^{(\\ell)}(x) | \\le C(\\ell) \\left(\\sum_{j=0}^{n-\\ell} \\frac{1}{(1+j^2)^{r - (s+2\\ell+1\/2) }}\\right)^{1\/2} \\left(\\sum_{j=0}^{n-\\ell} (1+j^2)^r |v_{j+\\ell}|^2\\right)^{1\/2}. \n\\end{equation}\nWe thus see that, provided $r - (s + 2\\ell+1\/2)>1\/2$ (or equivalently,\n$r>2\\ell+s+1$), $v_n^{(\\ell)}$ converges uniformly to\n$\\frac{\\partial^\\ell}{\\partial x^\\ell} v(x)$ (cf. ~\\cite[Th. 7.17]{baby_rudin}) for all \n$\\ell$ with\n$0\\leq \\ell \\leq k$. It follows that $v\\in C^k[-1,1]$, and, in view\nof~\\eqref{g_der}, it is easily checked that there exists a constant\n$M(\\ell)$ such that $\\| \\frac{\\partial^{(\\ell)}}{\\partial x^k} v(x)\n\\|_\\infty \\le M(\\ell) \\| v \\|_s^r$ for all $0\\leq \\ell \\leq k.$ The\nproof is complete.\n\\end{proof}\n\n\\begin{remark}\nIn order to check that the previous result is sharp we consider an example in the case $k=0$: \nthe function $v(x)=|\\log(x)|^{\\beta}$ with $0<\\beta<1\/2$ is not bounded, but a straightforward computation \nshows that, for $s \\in {\\mathbb {N}}$, $v\\in \\mathcal{H}_s^{s+1}(0,1)$, or equivalently (see Lemma \\ref{sobolev_equivalence}),\n$v\\in H_s^{s+1}(0,1)$.\n\\end{remark}\n\n\n\\begin{corollary}\\label{cinfty}\n The weighted fractional Laplacian\n operator~\\eqref{eq:weighted_fractional} maps bijectively the\n space $C^\\infty[a,b]$ into itself.\n\\end{corollary}\n\\begin{proof}\n Follows directly from Theorem~\\ref{teo_extended} together \n with lemmas~\\ref{lemma_gegen_parts},~\\ref{A_jk_lemma} and~\\ref{gegen_embedding_Hr}.\n\\end{proof}\n\\subsection{Analytic Regularity, single interval case}\n\\label{single_interval_analytic}\nLet $f$ denote an analytic function defined in the closed interval\n$[-1,1]$. Our analytic regularity results for the solution of\nequation~\\eqref{eq:fraccionario_dirichlet} relies on consideration of\nanalytic extensions of the function $f$ to relevant neighborhoods of\nthe interval $[-1,1]$ in the complex plane. We thus consider the {\\em\n Bernstein ellipse} $\\mathcal{E}_\\rho$, that is, the ellipse with\nfoci $\\pm 1$ whose minor and major semiaxial lengths add up to\n$\\rho\\geq 1$. We also consider the closed set $\\mathcal{B}_\\rho$ in\nthe complex plane which is bounded by $\\mathcal{E}_\\rho$ (and which\nincludes $\\mathcal{E}_\\rho$, of course). Clearly, any analytic\nfunction $f$ over the interval $[-1,1]$ can be extended analytically\nto $\\mathcal{B}_\\rho$ for some $\\rho > 1$. We thus consider the\nfollowing set of analytic functions.\n\\begin{definition}\\label{A_rho_def}\n For each $\\rho >1$ let $A_\\rho$ denote the normed space of analytic\n functions $A_\\rho = \\{ f \\colon f \\text{ is analytic on }\n \\mathcal{B}_\\rho\\} $ endowed with the $L^\\infty$ norm\n $\\|\\cdot\\|_{L^\\infty\\left (\\mathcal{B}_{\\rho}\\right)}$.\n\\end{definition}\n\\begin{theorem}\n\\label{teo_analyticity}\nFor each $f \\in A_\\rho$ we have $((-\\Delta)^s_\\omega)^{-1} f \\in\nA_\\rho$. Further, the mapping $((-\\Delta)^s_\\omega)^{-1}: A_\\rho \\to\nA_\\rho$ is continuous.\n\\end{theorem}\n\\begin{proof}\n Let $f \\in A_\\rho$ and let us consider the Gegenbauer expansions\n\\begin{equation}\\label{expansions}\n f=\\sum_{j=0}^\\infty f_j \\tilde{C}_j^{(s+1\/2)} \\quad\\mbox{and}\\quad ((-\\Delta)^s_\\omega)^{-1} f=\\sum_{j=0}^\\infty (\\lambda_j^s)^{-1} f_j \\tilde{C}_j^{(s+1\/2)}.\n\\end{equation}\nIn order to show that $((-\\Delta)^s_\\omega)^{-1} f\\in A_\\rho$ it suffices\nto show that the right-hand series in this equation converges\nuniformly within $\\mathcal{B}_{\\rho_1}$ for some $\\rho_1 > \\rho$. To\ndo this we utilize bounds on both the Gegenbauer coefficients and the\nGegenbauer polynomials themselves.\n\nIn order to obtain suitable coefficient bounds, we note that, since $f\n\\in A_\\rho$, there indeed exists $\\rho_2 > \\rho$ such that $f \\in\nA_{\\rho_2}$. It follows~\\cite{ZhaoWangXie} that the Gegenbauer\ncoefficients decay exponentially. More precisely, for a certain\nconstant $C$ we have the estimate\n\\begin{equation} \\label{eq:cota_an} |f_j| \\le C\n \\max_{z\\in\\mathcal{B}_{\\rho_2}} |f(z)| \\rho_2^{-j}\n j^{-s}\\quad\\mbox{for some}\\quad \\rho_2>\\rho,\n\\end{equation} \nwhich follows directly from corresponding bounds~\\cite[eqns 2.28, 2.8,\n1.1, 2.27]{ZhaoWangXie} on Jacobi coefficients. (Here we have used\nthe relation\n$$ C_j^{(s+1\/2)} = r^s_j P_j^{(s,s)} \\quad \\mbox{with} \\quad r^s_j = \\frac{(2s+1)_j}{(s+1)_j} = O(j^{s}) $$\nthat expresses Gegenbauer polynomials $C_j^{(s+1\/2)}$ in terms of\nJacobi polynomials $P_j^{(s,s)}$.)\n\n\nIn order to the adequately account for the growth of the Gegenbauer\npolynomials, on the other hand, we consider the estimate\n\\begin{equation}\n\\label{eq:crecimiento_gegenbauer}\n\\frac{\\| C_j^{(s+1\/2)} \\|_{L^\\infty(\\mathcal{B}_{\\rho_1})}}{h_j^{(s+1\/2)}} \\le \nD \\rho_1^j\\quad\\mbox{for all}\\quad \\rho_1>1,\n\\end{equation}\nwhich follows directly from~\\cite[Theorem 3.2]{XieWangZhao} and\nequation~\\eqref{gegen_norm_est}, where $D=D(\\rho_1)$ is a constant\nwhich depends on $\\rho_1$.\n\nLet now $\\rho_1\\in [\\rho,\\rho_2)$. In view of~\\eqref{eq:cota_an}\nand~\\eqref{eq:crecimiento_gegenbauer} we see that the $j$-th term of\nthe right-hand series in equation~\\eqref{expansions} satisfies\n\\begin{equation}\\label{series_est}\n\\left | \\frac{\\lambda_j^s f_j C_j^{(s+1\/2)}(x)}{h_j^{(s+1\/2)}}\\right |\\leq C D(\\rho_1)\n\\left( \\frac{\\rho_1}{\\rho_2} \\right)^j j^{-s} (\\lambda_j^s)^{-1} \\max_{z\\in\\mathcal{B}_{\\rho_1}} |f(z)|\n\\end{equation} \nthroughout $\\mathcal{B}_{\\rho_1}$. Taking $\\rho_1\\in (\\rho,\\rho_2)$\nwe conclude that the series converges uniformly in\n$\\mathcal{B}_{\\rho_1}$, and that the limit is therefore analytic\nthroughout $\\mathcal{B}_{\\rho}$, as desired. Finally, taking\n$\\rho_1=\\rho$ in~\\eqref{series_est} we obtain the estimates\n\\begin{equation*}\n \\| ((-\\Delta)^s_\\omega)^{-1} f \\|_{L^\\infty(\\mathcal{B}_{\\rho})} \\le C D(\\rho) \\sum_{j=0}^\\infty \\left( \\frac{\\rho}{\\rho_2} \\right)^j j^{-s} (\\lambda_j^s)^{-1} \\max_{z\\in\\mathcal{E}_{\\rho}} |f(z)| \n \\le E \\|f \\|_{L^\\infty(\\mathcal{B}_{\\rho})}\n\\end{equation*} \nwhich establish the stated continuity condition. The proof is thus\ncomplete.\n\\end{proof}\n\n\\begin{corollary}\n Let $f \\in A_\\rho$. Then the solution $u$ of\n \\eqref{eq:fraccionario_dirichlet} can be expressed in the form\n $u=\\omega^s\\phi$ for a certain $\\phi \\in A_\\rho$.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem~\\ref{teo_analyticity} and\n Remark~\\ref{rem_connection_u}.\n\\end{proof}\n\n\\begin{remark}\\label{remark_2s}\n We can now see that, as indicated in Remark~\\ref{remark_idea_2s},\n the smoothness and analyticity theory presented throughout\n Section~\\ref{regularity} cannot be duplicated with weights of\n exponent $2s$, in spite of the ``local'' regularity result of\n Theorem~\\ref{teo1}---that establishes analyticity of\n $T[y^{\\alpha}](x)$ around $x=0$ for both cases, $\\alpha = s+n$ and\n $\\alpha = 2s+n$. Indeed, we can easily verify that\n $T(y^{2s}(1-y)^{2s} y^n)$ cannot be extended analytically to an open\n set containing $[0,1]$. If it could, Theorem~\\ref{teo_analyticity}\n would imply that $y^{s}(1-y)^{s}$ is an analytic function around\n $y=0$ and $y=1$.\n \n \n \n\\end{remark}\n\n\n\n\\subsection{Sobolev and Analytic Regularity on Multi-interval\n Domains}\\label{regularity_multi_int}\nThis section concerns multi-interval domains $\\Omega$ of the\nform~\\eqref{union_intervals}. Using the characteristic functions\n$\\chi_{(a_i,b_i)}$ of the individual component interval, letting\n$\\omega^s(x)=\\sum_{i=1}^M(x-a_i)^s(b_i-x)^s \\chi_{(a_i,b_i)}(x)$ and\nrelying on Corollary~\\ref{coro_lemma_hypersingular}, we define the\nmulti-interval weighted fractional Laplacian operator on $\\Omega$ by\n$(-\\Delta)^s_\\omega \\phi = (-\\Delta)^s[\\omega^s \\phi]$, where $\\phi: \\mathbb{R} \\to\n\\mathbb{R}$.\nIn view of the various\nresults in previous sections it is natural to use the decomposition\n$(-\\Delta)^s_\\omega = \\mathcal{K}_s + \\mathcal{R}_s$, where $\\mathcal{K}_s[\\phi] =\n\\sum_{i=1}^M\\chi_{(a_i,b_i)} K_s\\chi_{(a_i,b_i)}\\phi$ is a\nblock-diagonal operator and where $\\mathcal{R}_s$ is the associated off-diagonal\nremainder. Using integration by parts is easy to check that\n\n\\begin{equation}\\label{other_intervals}\n \\mathcal{R}_s\\phi(x) = C_1(s) \\int_{\\Omega\\setminus (a_j,b_j)} |x-y|^{-1-2s} \\omega^s(y) \\phi(y) dy\\quad\\mbox{for}\\quad x\\in (a_j,b_j).\n\\end{equation}\n\n\\begin{theorem}\n Let $\\Omega$ be given as in\n Definition~\\ref{union_intervals_def}. Then, given $f \\in L^2_s(\\Omega)$,\n there exists a unique $\\phi \\in L^2_s(\\Omega)$ such that\n $ (-\\Delta)^s_\\omega \\phi = f$. Moreover, for $f \\in H^r_s(\\Omega)$\n (resp. $f \\in A_\\rho(\\Omega)$) we have $\\phi \\in H^{r+2s}_s(\\Omega)$\n (resp. $\\phi \\in A_\\nu(\\Omega)$ for some $\\nu >1$).\n\\end{theorem} \n\\begin{proof}\n Since $(-\\Delta)^s_\\omega=(\\mathcal{K}_s+\\mathcal{R}_s)$,\n left-multiplying the equation $ (-\\Delta)^s_\\omega \\phi = f$ by\n $\\mathcal{K}_s^{-1}$ yields\n\\begin{equation}\n\\label{eq:Preconditioner}\n\\left( I + \\mathcal{K}_s^{-1}\\mathcal{R}_s \\right) \\phi = \\mathcal{K}_s^{-1} f .\n\\end{equation}\nThe operator $\\mathcal{K}_s^{-1}$ is clearly compact in $L^2_s(\\Omega)$\nsince the eigenvalues $\\lambda_j^s$ tend to infinity as $j\\to\\infty$\n(cf. \\eqref{Eigenvalues}). On the other hand, the kernel of the\noperator $\\mathcal{R}_s$ is analytic, and therefore $\\mathcal{R}_s$ is\ncontinuous (and, indeed, also compact) in $L^2_s(\\Omega)$. It follows\nthat the operator $\\mathcal{K}_s^{-1}\\mathcal{R}_s$ is compact in\n$L^2_s(\\Omega)$, and, thus, the Fredholm alternative tells us that\nequation~\\eqref{eq:Preconditioner} is uniquely solvable in $L^2_s(\\Omega)$\nprovided the left-hand side operator is injective.\n\nTo establish the injectivity of this operator, assume $\\phi \\in L^2_s$\nsolves the homogeneous problem. Then\n$\\mathcal{K}_s(\\phi) = - \\mathcal{R}_s(\\phi)$, and since\n$\\mathcal{R}_s(\\phi)$ is an analytic function of $x$, in view of the\nmapping properties established in Theorem~\\ref{teo_analyticity} for\nthe self operator $K_s$ (which coincides with the\nsingle-interval version of the operator $(-\\Delta)^s_\\omega)$), we\nconclude the solution $\\phi$ to this problem is again analytic. Thus,\na solution to~\\eqref{eq:fraccionario_dirichlet} for a null right-hand\nside $f$ can be expressed in the form be $u = \\omega^s\\phi$ for a certain\nfunction $\\phi$ which is analytic throughout $\\Omega$. But this\nimplies that the function $u = \\omega^s\\phi$ belongs to the classical\nSobolev space $H^s(\\Omega)$. (To check this fact we consider that\n(a)~$\\omega^s\\in H^s(\\Omega)$, since, by definition, the Fourier transform of\n$\\omega^s$ coincides (up to a constant factor) with the confluent\nhypergeometric function $M(s+1,2s+2,\\xi)$ whose\nasymptotics~\\cite[eq. 13.5.1]{AbramowitzStegun} show that $\\omega^s$ in\nfact belongs to the classical Sobolev space $H^{s+1\/2-\\varepsilon}(\\Omega)$ for\nall $\\varepsilon>0$; and (b)~the product $fg$ of functions $f$, $g$ in\n$H^s(\\Omega)\\cap L^\\infty(\\Omega)$ is necessarily an element of $H^s(\\Omega)$---as\nthe Aronszajn-Gagliardo-Slobodeckij semi-norm~\\cite{Hitchhikers} of\n$fg$ can easily be shown to be finite for such functions $f$ and $g$,\nwhich implies $fg \\in H^s(\\Omega)$~\\cite[Prop 3.4]{Hitchhikers}). Having\nestablished that $u = \\omega^s\\phi\\in H^s(\\Omega)$, the injectivity of the\noperator in~\\eqref{eq:Preconditioner} in $L^2_s(\\Omega)$ follows from the\nuniqueness of $H^s$ solutions, which is established for example\nin~\\cite{AcostaBorthagaray}. As indicated above, this injectivity\nresult suffices to establish the claimed existence of an $L^2_s(\\Omega)$\nsolution for each $L^2_s(\\Omega)$ right-hand side.\n\nAssuming $f$ is analytic (resp. belongs to $H^r_s(\\Omega)$), finally, the\nregularity claims now follow directly from the single-interval results\nof Sections~\\ref{Sobolev} and~\\ref{single_interval_analytic}, since a\nsolution $\\phi$ of $(-\\Delta)^s_\\omega \\phi = f$ satisfies\n\\begin{equation}\\label{multi_single}\n\\mathcal{K}_s(\\phi) = f - \\mathcal{R}_s(\\phi).\n\\end{equation}\nThe proof is now complete.\n\\end{proof}\n\n\n\n\\section{High Order Numerical Methods\\label{HONM}}\nThis section presents rapidly-convergent numerical methods for single-\nand multi-interval fractional Laplacian problems. In particular, this\nsection establishes that the proposed methods, which are based on the\ntheoretical framework introduced above in this paper, converge\n(i)~exponentially fast for analytic right-hand sides $f$,\n(ii)~superalgebraically fast for smooth $f$, and (iii)~with\nconvergence order $r$ for $f \\in H_s^r(\\Omega)$.\n\n\\subsection{Single-Interval Method: Gegenbauer Expansions\\label{num_single}}\nIn view of Corollary~\\ref{diag_ab}, a spectrally accurate algorithm\nfor solution of the single-interval equation~\\eqref{eqn_weighted} (and\nthus equation~\\eqref{eq:fraccionario_dirichlet} for $\\Omega=(a,b)$)\ncan be obtained from use of Gauss-Jacobi quadratures. Assuming\n$(a,b)=(-1,1)$ for notational simplicity, the method proceeds as\nfollows: 1)~The continuous scalar product~\\eqref{gegen_coef} with\n$v=f$ is approximated with spectral accuracy (and, in fact, exactly\nwhenever $f$ is a polynomial of degree less or equal to $n+1$) by\nmeans of the discrete inner product\n\\begin{equation}\n\\label{eq:disc_inner}\nf_j^{(n)}:= \\frac{1}{h_j^{(s+1\/2)}} \\sum_{i=0}^n f(x_i) C^{(s+1\/2)}_j(x_i) w_i, \n\\end{equation}\nwhere $(x_i)_{i=0}^n$ and $(w_i)_{i=0}^n$ denote the nodes and weights\nof the Gauss-Jacobi quadrature rule of order $2n+1$. (As is well\nknown~\\cite{HaleTowsend}, these quadrature nodes and weights can be\ncomputed with full accuracy at a cost of $O(n)$ operations.) 2)~For\neach $i$, the necessary values $C^{(s+1\/2)}_j(x_i)$ can be obtained\nfor all $j$ via the three-term recurrence\nrelation~\\eqref{eq:recurrencia}, at an overall cost of $O(n^2)$\noperations. The algorithm is then completed by 3)~Explicit evaluation\nof the spectrally accurate approximation\n\\begin{equation}\n\\label{eq:geg_rec}\n\\phi_n := K_{s,n}^{-1} f = \\sum_{j=0}^n \\frac{f_j^{(n)}}{\\lambda_j^s h_j^{(s+1\/2)}} C^{(s+1\/2)}_j\n\\end{equation}\nthat results by using the expansion~\\eqref{exp_gegen} with $v=f$\nfollowed by an application of equation~\\eqref{eigenfuncts} and\nsubsequent truncation of the resulting series up to $j=n$. The\nalgorithm requires accurate evaluation of certain ratios of Gamma\nfunctions of large arguments; see equations~\\eqref{Eigenvalues}\nand~\\eqref{eq:norm_gegen}, for which we use the Stirling's series as\nin~\\cite[Sec 3.3.1]{HaleTowsend}. The overall cost of the algorithm is\n$O(n^2)$ operations. The accuracy of this algorithm, in turn, is\nstudied in section~\\ref{error_estimates}.\n\n\\subsection{Multiple Intervals: An iterative Nystr\\\"om Method}\\label{num_multi}\nThis section pre\\-sents a spectrally accurate iterative Nystr\\\"om method\nfor the numerical solution of\nequation~\\eqref{eq:fraccionario_dirichlet} with $\\Omega$ as\nin~\\eqref{union_intervals}. This solver, which is based on use of the\nequivalent second-kind Fredholm equation~\\eqref{eq:Preconditioner},\nrequires (a)~Numerical approximation of $\\mathcal{K}_s^{-1}f$,\n(b)~Numerical evaluation of the ``forward-map''\n$(I+\\mathcal{K}_s^{-1}\\mathcal{R}_s)[\\tilde \\phi]$ for each given\nfunction $\\tilde \\phi$, and (c)~Use of the iterative linear-algebra\nsolver GMRES~\\cite{GMRES}. Clearly, the algorithm in\nSection~\\ref{num_single} provides a numerical method for the\nevaluation of each block in the block-diagonal inverse operator\n$\\mathcal{K}_s^{-1}$. Thus, in order to evaluate the aforementioned\nforward map it now suffices to evaluate numerically the off-diagonal\noperator $\\mathcal{R}_s$ in equation~\\eqref{other_intervals}.\n\nAn algorithm for evaluation of $\\mathcal{R}_s[\\tilde \\phi](x)$ for $x\\in\n(a_j,b_j)$ can be constructed on the basis of the Gauss-Jacobi\nquadrature rule for integration over the interval $(a_\\ell,b_\\ell)$\nwith $\\ell\\ne j$, in a manner entirely analogous to that described in\nSection~\\ref{num_single}. Thus, using Gauss-Jacobi nodes and weights\n$y_i^{(\\ell)}$ and $w_i^{(\\ell)}$ ($i = 1,\\dots, n_\\ell$) for each\ninterval $(a_\\ell,b_\\ell)$ with $\\ell\\ne j$ we may construct a\ndiscrete operator $R_n$ that can be used to approximate $\\mathcal{R}_s[\\tilde\n\\phi](x)$ for each given function $\\tilde\\phi$ and for all observation\npoints $x$ in the set of Gauss-Jacobi nodes used for integration in\nthe interval $(a_j,b_j)$ (or, in other words, for $x = y_k^{(j)}$ with\n$k=1,\\dots,n_j$). Indeed, consideration of the numerical\napproximation\n$$ R[\\tilde\\phi](y_k^{(j)}) \\approx \\sum_{\\ell \\ne j} \\sum_{i=0}^{n_\\ell} |y_k^{(j)}-y_i^{(\\ell)}|^{-2s-1} \\tilde\\phi(y_i^{(\\ell)}) w_i^{(\\ell)} $$\nsuggests the following definition. Using a suitable ordering to define\na vector $Y$ that contains all unknowns corresponding to\n$\\tilde\\phi(y_i^{(\\ell)})$, and, similarly, a vector $F$ that contains all\nof the values $f(y_i^{(\\ell)})$, the discrete equation to be solved\ntakes the form\n\\begin{equation*}\n\\label{eq:disc_precond}\n(I + K_{s,n}^{-1} R_{s,n}) Y = K_{s,n}^{-1} [F]\n\\end{equation*}\nwhere $R_n$ and $K_{s,n}^{-1}$ are the discrete operator that\nincorporate the aforementioned ordering and quadrature rules.\n\nWith the forward map $(I + K_{s,n}^{-1} R_{s,n})$ in hand, the multi-interval\nalgorithm is completed by means of an application of a suitable\niterative linear algebra solver; our implementations are based on the\nKrylov-subspace iterative solver GMRES~\\cite{GMRES}. Thus, the overall\ncost of the algorithm is $O(m\\cdot n^2)$ operations, where $m$ is the\nnumber of required iterations. (Note that the use of an iterative\nsolver allows us to avoid the actual construction and inversion of the\nmatrices associated with the discrete operators in\nequation~\\eqref{eq:disc_precond}, which would lead to an overall cost of\nthe order of $O(n^3)$ operations.) As the equation to be solved\noriginates from a second kind equation, it is not unreasonable to\nanticipate that, as we have observed without exception (and as\nillustrated in Section~\\ref{num_res}), a small number of GMRES\niterations suffices to meet a given error tolerance.\n\n\n\\subsection{Error estimates}\\label{error_estimates}\nThe convergence rates of the algorithms proposed in\nSections~\\ref{num_single} and~\\ref{num_multi} are studied in what\nfollows. In particular, as shown in Theorems~\\ref{teo_sobolev_error}\nand~\\ref{teo_analytic_error}, the algorithm's errors are exponentially\nsmall for analytic $f$, they decay superalgebraically fast (faster\nthan any power of meshsize) for infinitely smooth right-hand sides,\nand with a fixed algebraic order of accuracy $O(n^{-r})$ whenever $f$\nbelongs to the Sobolev space $H^r_s(\\Omega)$ (cf. Section~\\ref{Sobolev}).\nFor conciseness, fully-detailed proofs are presented in the\nsingle-interval case only. A sketch of the proofs for the\nmulti-interval cases is presented in\nCorollary~\\ref{multi_interval_error}.\n\n\\begin{theorem}\\label{teo_sobolev_error}\n Let $r > 0$, $00$). Indeed, it suffices to show that, for a\n given bounded sequence $\\{\\phi_{n}\\}\\subset H^r_s(\\Omega)$, the\n sequence $R_{s,n}[\\phi_{n}]$ admits a convergent subsequence in\n $H^r_s(\\Omega)$. But, selecting $00$ and any $0\\le s\n\\le 1$) are displayed in Fig.~\\ref{fig_nonsmooth}. The errors decay\nwith the order predicted by Theorem~\\ref{teo_sobolev_error} in the\n$H^{2s}_s(-1,1)$ norm, and with a slightly better order than predicted\nby that theorem for the $L_s^2(-1,1)$ error norm, although the\nobserved orders tend to the predicted order as $s\\to 0$ (cf.\nRemark~\\ref{remark_negative_norm}).\n\n\n \n\\begin{center}\n\\begin{figure}[htbp]\n \\begin{minipage}[t]{0.59\\linewidth}\n \\vspace{0pt}\n \\centering\n \\includegraphics[scale=0.27]{.\/sol_multi.png}\n \\end{minipage}\n \\begin{minipage}[t]{0.4\\linewidth}\n \\vspace{0pt}\n \\centering\n \\begin{tabular}{ c | c }\n $N$ \t& rel. err. \\\\\n \\hline\n \\hline\n\t8 & 9.3134e-05 \\\\ \n\t12 & 1.6865e-06 \\\\ \n\t16 & 3.1795e-08 \\\\ \n\t20 & 6.1375e-10 \\\\ \n\t24 & 1.1857e-11 \\\\ \n\t28 & 1.4699e-13 \\\\\n \\hline\n \\end{tabular} \n\\end{minipage}\n\\caption{Multiple (upper curves) vs. independent single-intervals\n solutions (lower curves) with $f=1$. A total of five GMRES\n iterations sufficed to achieve the errors shown on the right table\n for each one of the discretizations considered.}\n\\label{fig:multi}\n\\end{figure}\n\\end{center}\n \nA solution for a multi-interval (two-interval) test problem with right\nhand side $f=1$ is displayed in Figure~\\ref{fig:multi}. A total of\nfive GMRES iterations sufficed to reach the errors displayed for each\none of the discretizations considered on the right-hand table in\nFigure~\\ref{fig:multi}. The computational times required for each one\nof the discretizations listed on the right-hand table are of the order\nof a few hundredths of a second.\n\n\\section{Acknowledgments}\n\nThe authors thankfully acknowledge support from various agencies. GA\nwork was partially supported by CONICET, Argentina, under grant PIP\n2014--2016 11220130100184CO. JPB's and MM's efforts were made possible\nby graduate fellowship from CONICET, Argentina. OB efforts were\nsupported by the US NSF and AFOSR through contracts DMS-1411876 and\nFA9550-15-1- 0043, and by the NSSEFF Vannevar Bush Fellowship under\ncontract number N00014-16-1-2808.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n2M1207b\\ is a planet-mass companion orbiting the young brown dwarf 2M1207A at a\nprojected separation of $\\sim 46$ AU \\cite[]{Chauvin2004, Chauvin2005}. As a\nmember of the $\\sim 8$ Myr old TW Hydra association \\cite[]{Gizis2002}, 2M1207b\\\nis one of the youngest companions of its mass ($\\sim$ 2 -- 5 $M_{\\rm Jup}$) ever\nimaged. Regardless of how the system formed, 2M1207b\\ provides a unique look\ninto the atmospheric properties of giant planets and brown dwarfs in the very\nearly stages of their evolution.\n\nSoon after 2M1207b\\ was discovered, it was recognized that its low luminosity and\nred near-IR colors were seemingly inconsistent. With m$_K$ = 16.93 $\\pm 0.11$\n\\cite[]{Chauvin2004}, a $K$-band bolometric correction of 3.25 $\\pm 0.14$\n\\cite[]{Mamajek2005}, and a distance of 52.4 $\\pm 1.1$ pc\n\\cite[]{Ducourant2008}, 2M1207b\\ has a luminosity of $\\log L\/L_\\odot = -4.73 \\pm\n0.12$. With this luminosity and an age of 8$^{+4}_{-3}$ Myr\n\\cite[]{Chauvin2004}, brown dwarf cooling tracks \\cite[]{Baraffe2003}\npredict\\footnote{The ranges in predicted values come from the luminosity and\nage uncertainties.} $T_{\\rm eff}$\\ = 936 -- 1090 K, $\\log(g)$\\ = 3.5 -- 3.8 (cgs units),\nradius = 1.34 -- 1.43 $R_{\\rm Jup}$, and a mass of 2.3 -- 4.8 $M_{\\rm Jup}$. The red near-IR\ncolors (e.g., $H-K = 1.16 \\pm 0.24$; Chauvin et al. 2004\\nocite{Chauvin2004})\nand near-IR spectra of 2M1207b, on the other hand, are indicative of a mid to\nlate L spectral type and effective temperature of $\\sim 1600$K\n\\cite[]{Mohanty2007, Patience2010}. Such a combination of high effective\ntemperature and low luminosity would require a radius of $\\sim 0.7 $$R_{\\rm Jup}$,\nabout a factor of two below evolution model predictions. Under the assumption\nthat the higher $T_{\\rm eff}$\\ is correct, two possible explanations for the apparent\nunderluminosity of 2M1207b\\ have been put forward. An edge-on disk, albeit with\nunusual characteristics, could provide the necessary 2.5 mags of gray\nextinction to accommodate a larger and more reasonable radius\n\\cite[]{Mohanty2007}. A more provocative suggestion requires 2M1207b\\ to be the\nhot afterglow of a recent protoplanetary collision allowing it to be\nsimultaneously hot and small \\cite[]{Mamajek2007}. \n\nBoth of these early explanations for the observed properties of 2M1207b\\ hinge\nupon the effective temperature being as high as 1600 K. Earlier estimates of\n$T_{\\rm eff}$\\ were likely lead astray by the lack of color-$T_{\\rm eff}$\\ relationships\nproperly calibrated for young, planet-mass objects and the lack of model\natmospheres spanning a broad enough range of cloud and chemical parameters to\nencompass objects like 2M1207b. The discovery of HR8799b \\cite[]{Marois2008}, a\nsecond planet-mass object with similar near-IR colors and luminosity as 2M1207b,\nmotivates this new model atmosphere study of 2M1207b, as it is highly unlikely\nthat either the edge-on disk or recent-collision explanation applies to both\nobjects.\n\nIn this Letter, an atmosphere-only explanation for the observed properties of\n2M1207b\\ is presented. A combination of clouds of modest thickness and\nnon-equilibrium CO\/CH$_4$ ratio is shown to simultaneously reproduce both the\nobserved photometric and spectroscopic properties of 2M1207b, with bulk\nproperties consistent with evolution model predictions. Such an explanation\nwas touched upon in several recent papers \\cite[]{Currie2011, Skemer2011,\nBarman2011}, but here the atmosphere of 2M1207b\\ is explored in more detail. \n\n\\section{Clouds}\n\nThe properties of brown dwarfs and giant planets are known to be influenced by\natmospheric cloud opacity and it is well established that clouds play a\ncentral role in the transition from spectral types L to T \\cite[]{Allard2001}.\nEarly work on brown dwarf atmospheres approached cloud modeling\nphenomenologically, parameterizing the problem with an emphasis on vertical\nmixing \\cite[]{Ackerman2001}. Cloudy and cloud-free limits provide useful\ninsight into the expected spectroscopic and photometric trends but often fail,\nunsurprisingly, to match individual brown dwarfs, especially in the L-to-T\ntransition region \\cite[]{Burrows2006}.\n\nFigure \\ref{fig1} compares field brown dwarfs to 2M1207b\\ and the HR8799 planets\nin a color-magnitude diagram (CMD). 2M1207b\\ is located between the cloudy and\ncloud-free limits and, consequently, one should not expect either of these\nlimiting cases to be appropriate when modeling its photometric and\nspectroscopic properties. Figure \\ref{fig1} shows the path brown dwarfs or\ngiant planets of various effective temperatures would follow in a near-IR CMD\nif the vertical thickness of clouds is allowed to continuously increase from\nzero (cloud-free) to well above the photosphere (pure equilibrium clouds). The\nradius for these tracks, 1.4 $R_{\\rm Jup}$, was specifically chosen to match the radius\npredicted for 2M1207b\\ as discussed above; however, the paths traced by these\ntracks are otherwise independent of evolutionary models. 2M1207b\\ is intersected\nby the $T_{\\rm eff}$ = 1000 K cloud-track, which is the expected value from evolution\nmodels, demonstrating that low-temperature cloudy atmospheres can achieve very\nred near-IR colors, even with clouds significantly thinner than the extreme\nlimits.\n\n\\begin{figure}[t]\n\\plotone{figure1.eps}\n\\caption{Absolute $H$-band magnitude vs. $H$-$K$ near-IR color-magnitude\ndiagram for field brown dwarfs \\cite[]{Leggett2002, Knapp2004}. The planets\n2M1207b\\ and HR8799bcd are indicated by symbols with 1-$\\sigma$ error bars (see\nlegend). Dotted lines show color-magnitude tracks for chemical equilibrium\nmodels with radius = 1.4 $R_{\\rm Jup}$, $\\log(g) = 3.5$, mean particle size equal to 5\n$\\mu$m and T$_{\\rm eff}$ equal to 700 -- 1200 K in steps of 100 K, from bottom to\ntop. Cloud thickness increases from left to right. The arrows indicate the\napproximate locations of cloud free (left) and extremely cloudy (right) models,\nwith arrow-direction pointing toward decreasing $T_{\\rm eff}$. The ``transition\"\nregion, between cloudy and cloud-free atmospheres covers a broad range in the\ncolor-magnitude diagram, beyond what is currently occupied by field brown dwarfs.\n\\label{fig1}}\n\\end{figure}\n\n\\section{Non-local Equilibrium Chemistry}\n\nIf the methane-rich atmospheres of mid to late T dwarfs were in a pure chemical\nequilibrium state, CO mole fractions would be too small to have a major\nimpact on their spectra. However, CO has been detected in many T dwarfs,\nsuggesting that their atmospheres are out of equilibrium \\cite[]{Noll1997,\nSaumon2000, Saumon2006, Geballe2009}. The most likely mechanism for CO\nenhancement is vertical mixing from deep layers, where the temperatures and\npressures are higher and CO is in ready supply. At photospheric depths and\nabove, the chemical timescale ($\\tau_{\\rm chem}$) to reestablish an equilibrium\nCO\/CH$_4$ ratio becomes far greater than the mixing timescale ($\\tau_{\\rm mix}$),\nthereby allowing larger CO mole fractions to exist at otherwise\nmethane-dominated pressures. Through the same mixing process, N$_2$\/NH$_3$ can\nalso depart from local chemical equilibrium (LCE; Saumon et al. 2006;\nHubeny \\& Burrows 2007) \\nocite{Saumon2006, Hubeny2007}. The standard non-LCE\nmodel quenches the CO and CH$_4$ mole fractions at the atmospheric pressure\n($P_{q}$) where $\\tau_{\\rm mix}$ = $\\tau_{\\rm chem}$, with $\\tau_{\\rm mix}$\\ computed following\n\\cite{Smith1998}. Below the quenching depth ($P >$ $P_{q}$) the mole fractions for\nCO and CH$_4$ are in chemical equilibrium while above ($P \\le P_{q}$) they are\nset to the values at $P_{q}$.\n\nThe mole fractions for CO and CH$_4$, N(CO) and N(CH$_4$), at and above $P_{q}$\\\nare very sensitive to the underlying temperature-pressure (T-P)\nprofile and, thus, are sensitive to gravity, cloud opacity, and metallicity\n\\cite[]{Hubeny2007, Fortney2008, Barman2011}. Certain combinations of low\ngravity and clouds can result in $P_{q}$\\ {\\em below} the CO\/CH$_4$\nequilibrium chemistry crossing point ($P_{eq}$). When $P_{q}$\\ is sufficiently deep\nand $P_{q}$\\ $>$ $P_{eq}$, N(CO) can be quenched at its maximum value while N(CH$_4$)\nis quenched near its minimum. When this situation occurs, the non-LCE\nCO\/CH$_4$ ratio becomes fairly insensitive to the mixing timescale in the\nradiative zone (determined by the adopted coefficient of eddy diffusion, $K_{\\rm zz}$).\nThis situation is similar to the N$_2$\/NH$_3$ chemistry where the NH$_3$ mole\nfractions can also be nearly independent of $K_{\\rm zz}$, even in high-gravity\ncloud-free atmospheres \\cite[]{Saumon2006, Hubeny2007}. \n\nThis atmospheric behavior is highly significant to 2M1207b\\ as it allows for \na photosphere with much higher N(CO) and much lower N(CH$_4$) mole fractions at the\nlow $T_{\\rm eff}$\\ predicted by evolution models. Previous studies, focused primarily\non field brown dwarfs, present far less severe situations with non-LCE only\naltering spectra at $\\lambda > 4$ $\\mu$m where the strongest absorption\nbands of CO and NH$_3$ occur \\cite[]{Hubeny2007}. At lower surface gravities,\nhowever, strong non-LCE effects have been shown to extend well into the near-IR\n\\cite[]{Fortney2008, Barman2011}. \n\n\\section{Model Comparisons}\n\n2M1207b\\ has been observed extensively from the ground and space, with\nphotometric coverage between 0.9 and $\\sim$ 9 \\micron\\ \\cite[]{Chauvin2004,\nSong2006, Mohanty2007, Skemer2011}. High signal-to-noise $J$, $H$, and $K$\nspectra are also available \\cite[]{Patience2010}. With the goal of finding a\nmodel atmosphere that agrees with these observations and that has $T_{\\rm eff}$\\ and $\\log(g)$\\ in the\nrange predicted by evolution models, a sequence of atmosphere models was\ncomputed covering $T_{\\rm eff}$\\ = 900 -- 1200 K and $\\log(g)$ from 3.0 to 4.5 (cgs\nunits). The same intermediate cloud (ICM) and non-LCE prescriptions from\n\\cite{Barman2011} were used. Synthetic photometry was generated by convolving\nsynthetic spectra with filter response curves. $J$, $H$, and $K$ spectra were\nproduced by convolving the model spectra with a Gaussian filter matching the\nobserved spectral resolution, then interpolated onto the same wavelength\npoints. The best-fit was determined by least-squares minimization.\n\nFigure \\ref{fig2} illustrates the basic structure of the model that best fits the\ndata ($T_{\\rm eff}$ = 1000 K and $\\log(g)$ = 4.0). The atmospheric cloud, composed mostly\nof Fe and Mg-Si grains, has a base at $\\sim 3$ bar and extends upward before\ndropping off in number-density at $\\sim 1$ bar. Despite the rapid drop in\nnumber-density, the cloud extends across the photospheric depths. Also, \n$P_{q}$\\ ($\\sim 3$ bar) is well below the N(CO)$_{eq}$ = N(CH$_4$)$_{eq}$\npoint ($P \\sim 0.3$ bar), with the CO mole fractions set to the maximum value and\nCH$_4$ is close to its minimum value.\n\nWhile the model cloud and non-LCE properties are determined by free-parameters, they\nare likely supported by low gravity and efficient vertical mixing. In the\nconvection zone $\\tau_{\\rm mix}$ $\\propto$ $H_{\\rm P}$\/$V_{\\rm c}$, where $V_{\\rm c}$\\ is the convective velocity\nand $H_{\\rm P}$\\ is the local pressure scale-height. With $K_{\\rm zz}$\\ $\\propto$\n$H_p^2\/$$\\tau_{\\rm mix}$\\ $\\propto$ $V_{\\rm c}$$H_{\\rm P}$, $K_{\\rm zz}$\\ increases with decreasing\ngravity in the convection zone as velocity and scale-height increase. The\nradiative-convective boundary also shifts toward the photosphere as surface\ngravity decreases, further suggesting that vertical mixing in the radiative\nzone near this boundary will also be enhanced in low gravity atmospheres. In\nthe radiative zone it is unclear whether the vertical mixing is predominantly\ndriven by convective overshoot or gravity waves, but the picture emerging from\nmulti-dimensional hydrodynamical simulations in M dwarfs and brown dwarfs \nindicates that $K_{\\rm zz}$\\ is both depth dependent and easily achieves values $>\n10^8$ cm$^2$ s$^{-1}$ \\cite[]{Ludwig2006, Freytag2010}. \n \nFigure \\ref{fig3} compares the best-fitting model photometry and spectrum to\nthe observations. The model photometry agrees very well with the observations\nin nearly ever bandpass, with most bands agreeing at 1$\\sigma$. The model\n$J$-band spectrum has about the right slope and nicely reproduces the water\nabsorption starting at 1.33 $\\mu$m. At $H$ band, the model also has roughly\nthe correct shape but is slightly too linear across the central wavelengths.\nThe CO band in the $K$ band is very well reproduced by the model but the\ncentral wavelength region is again too flat. \\cite{Skemer2011} also compare\nmodels \\cite[]{Burrows2006, Madu2011} with similar $T_{\\rm eff}$\\ and $\\log(g)$\\ to the\nsame observations; however, these models likely underestimate the non-LCE, as\nthey do not adequately reproduce the near-IR spectrum, especially the CO band\nat 2.3 \\micron. An LCE version of the best-fit non-LCE model is shown in Figure\n\\ref{fig3} and demonstrates the impact non-LCE has on the near-IR spectrum.\n\n\\begin{figure}[t]\n\\plotone{figure2.eps}\n\\caption{\nAtmospheric properties for the best fitting model for 2M1207b. {\\em Top:}\ntemperature-pressure structure compared to condensation curves for two abundant\ncloud species. {\\em Middle:} CO and CH$_4$ mole fractions for equilibrium\n(dashed, red) and non-equilibrium (solid, with $K_{\\rm zz}$\\ = 10$^8$ cm$^2$ s$^{-1}$)\nchemistry. Chemical and mixing timescales are also plotted (dotted lines).\n{\\em Bottom:} dust-to-gas ratio for the intermediate cloud model (ICM) and the\npure equilibrium cloud model.\n\\label{fig2}}\n\\end{figure}\n \nAlso shown in Figure \\ref{fig3} is the 1600 K ``Dusty\" \\cite[]{Allard2001}\nequilibrium cloud model often selected as the best match to the near-IR spectra\n\\cite[]{Mohanty2007, Patience2010}. The 1600 K model photometry does not agree\nwell with the observations across the full wavelength range. The 1600 K model\nspectrum, however, is very similar to the best-fit 1000 K model across the $K$\nband, but only when the near-IR bands are scaled to match individually (to\naccount for the photometric disagreement for the 1600 K model). In both $J$ and\n$H$ bands, the 1600 K model overpredicts the peak flux and is noticeably more\ntriangular at $H$ than the observations. At $K$ band, the 1600 K model provides\nonly slightly better agreement with the observations than the 1000 K model. \n \nThe remaining differences between the 1000 K model and near-IR spectra can be\nattributed to an incorrect proportion of dust opacity relative to molecular\nopacity. Given the simplicity of the cloud model used here such discrepancies\nare not surprising. Without a doubt, a more parameter-rich cloud model could\nbe used to fine tune the comparison, but it is unlikely that such an exercise\nwill lead to significantly greater insights into the physical properties of the\ncloud. The primary lesson from this comparison is that atmospheric clouds and\nchemistry can dramatically alter the spectral shape and potentially lead to\nerrors in effective temperature as great as 50\\%.\n\nThe model comparisons to the photometry provide a new estimate for the\nbolometric luminosity. The mean luminosity, found by comparing to pure\nequilibrium models, ICM\/non-LCE models and black bodies, is $\\log L\/L_\\odot =\n-4.68$ with rms of 0.05. This luminosity is consistent with the earlier value\nmentioned above (based on $K$-band bolometric corrections). \n\n\\begin{figure*}[t]\n\\epsscale{1.04}\n\\plotone{figure3.eps}\n\\caption{\nPhotometric (top) and spectroscopic (bottom) comparison between 2M1207b\\\nobservations (shown in black, see the text for details) and best-fitting model\n(blue). For comparison, synthetic photometry and spectra from a 1600 K model\n(red) with an equilibrium cloud model (aka ``Dusty\") along with an LCE model (green) with the\nsame parameters as the non-LCE model. Surface gravities, radii, and effective\ntemperatures are indicated in the legend. All fluxes have been scaled to 10\npc.\n\\label{fig3}}\n\\end{figure*}\n\n\\section{Discussion and Conclusions}\n\nThe best fitting $T_{\\rm eff}$, $\\log(g)$, and radius (1000 K, 10$^4$ cm s$^{-2}$, and 1.5 $R_{\\rm Jup}$)\nfor 2M1207b\\ are consistent with the cooling track predictions discussed above. This\nmodel demonstrates that including typical cloud thickness and non-LCE are all\nthat is required to reproduce the current observations of 2M1207b. Such a model\nreminds us that the spectra of brown dwarfs are not strictly a function of\ntemperature and, at young ages, can deviate significantly from expectations\nderived from older field brown dwarfs. The primary evidence supporting the\nedge-on-disk and protoplanetary collision hypotheses, was the previously\ndeduced 1600 K effective temperature. A model with this temperature is shown to\ninadequately reproduce the available photometry and compares no better to\nnear-IR spectra than a cooler cloudy model. Also, the disk-model comparison by\n\\cite{Skemer2011} further weakens the case for an edge-on disk. Consequently,\nno strong evidence remains for the previous disk or collision hypotheses. This\nconclusion is independent of the existence of HR8799b, but is certainly\nsupported by it.\n\nThe primary atmospheric contributors to the L-type appearance of 2M1207b, despite\nits low $T_{\\rm eff}$, are clouds extending across the photosphere, thereby reddening\nthe near-IR colors, and non-equilibrium chemistry, establishing a CO\/CH$_4$\nratio that is nearly the reciprocal of what is present in the photospheres of\nolder field T dwarfs. The cloud properties, in particular the thickness, are\nprobably not substantially different from those found in late L dwarfs and the\nrequired $K_{\\rm zz}$\\ ($\\sim 10^8$ cm$^2$ s$^{-1}$) is well within the range of\nfield dwarfs. \\cite{Skemer2011} stress clouds as the primary explanation for\nthe photometric and spectroscopic properties of 2M1207b. However, non-LCE plays\nan equally important role. Probably the most important underlying distinction\nbetween 2M1207b\\ and field brown dwarfs is low surface gravity, provided by its\nyouth and low mass. Without low surface gravity, it is unlikely that clouds or\nnon-LCE would be sufficient to give 2M1207b\\ its current appearance. Such\nobjects, therefore, should not be considered members of a new class, but rather\nrepresent the natural extension of substellar atmospheres to low gravity.\n\nGiven their similar masses ($\\sim 5$$M_{\\rm Jup}$), it is possible that 2M1207b\\ and\nHR8799b represent two distinct states in the evolution of substellar\natmospheric properties (despite potentially different formation scenarios).\nAfter $\\sim 20$ Myr of cooling, perhaps 2M1207b\\ will spectroscopically evolve\ninto something resembling HR8799b and, eventually, into a traditional looking\nmethane-rich T-dwarf. The very distinct spectra of the two objects (see Figure\n15, \\nocite{Barman2011}Barman et al. 2011) would suggest rapid spectral\nevolution in the first 50 Myr, post formation. It is worth noting that the\nbest-fit models for 2M1207b\\ and HR8799b, though similar in atmospheric\nparameters, differ in one significant respect -- the radius derived from the\neffective temperature and luminosity. Our model here gives a radius of 1.5 $R_{\\rm Jup}$\\\nfor 2M1207b, while even the coldest fit to HR8799b in \\cite{Barman2011} gives a\nradius of only 1 $R_{\\rm Jup}$. There are several possible explanations for this.\nFirst, the recovered radius is of course very sensitive to the effective\ntemperature -- a 100 K increase in 2M1207b\\ and a 100 K decrease in HR8799b would\nremove the discrepancy, though neither would be a good fit to the spectra.\nSecond, this may represent a fundamental difference in their internal state due\nto different formation scenarios. 2M1207b\\ almost certainly did not form through\na core-accretion process as it is highly unlikely that a disk surrounding a\nlow-mass brown dwarf would have had enough material to accrete a giant planet,\nespecially at large separations and in a short time. It is more likely that\n2M1207b\\ represents the tail end of binary star formation and, thus, might be\nexpected to follow a similar cooling evolution as brown dwarfs. The formation\nof the HR8799 planets is less clear but could potentially have involved an\naccretion period. The ``cold start\" accretion models of \\cite{Marley2007} predict\nsignificantly smaller radii for a given age and mass. Although the temperature\nand luminosity of HR8799b are too high for the extreme cold start models, the\nsmaller radius may be pointing toward a formation process that involved at\nleast some loss of entropy.\n\nFinally, one can speculate on the implications that 2M1207b\\ and HR8799b might\nhave on the spectral properties for the broader young brown dwarf and planet\npopulation. If one adopts, $\\sim$ 1500 K as the upper $T_{\\rm eff}$\\ limit for\nT-dwarfs, then all T dwarfs younger than $\\sim 100$ Myr should be in the planet\nmass regime ($\\lesssim 13$ $M_{\\rm Jup}$) and should have very low gravity ($\\log(g)\n\\lesssim 4.5$). However, if 2M1207b\\ and HR8799b are representative of cool, low\ngravity, substellar atmospheres, then non-LCE (and possibly clouds) will\ndiminish the strength of CH$_4$ absorption across the $H$ and $K$ bands, making\nvery young methane dwarfs rare. This prediction, however, is at odds with the\ntentative discoveries of $\\sim 1$ Myr-old T dwarfs \\cite[]{Zap2002,\nBurgess2009, Marsh2010}. While at least one of these discoveries has been\ndrawn into question \\cite[]{Burgasser2004}, if others with strong near-IR\nCH$_4$ absorption are confirmed, then it must be explained why substellar\nobjects of similar age and gravity have atmospheres with wildly different cloud\nand non-LCE properties.\n\n\\vspace{-0.5cm}\n\\acknowledgements\nWe thank the anonymous referee for their review. This Letter benefited from\nmany useful discussions with Brad Hansen, Mark Marley, and Didier Saumon. This\nresearch was supported by NASA through Origins grants to Lowell Observatory and\nLLNL along with support from the HST GO program. Support was also provided by\nthe NASA High-End Computing (HEC) Program through the NASA Advanced\nSupercomputing (NAS) Division at Ames Research Center. Portions of this work\nwere performed under the auspices of the U.S. Department of Energy by Lawrence\nLivermore National Laboratory under Contract DE-AC52-07NA27344\n(LLNL-JRNL-485291).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAtomistic simulations are indispensable in materials science and require a robust description of the interatomic interaction. \nThe application of highly accurate density functional theory (DFT) calculations is often limited by the computationally involved description of the electronic structure. \nA systematic coarse-graining of the electronic structure~\\cite{Drautz2006,Drautz2011} leads to the tight-binding (TB) bond model~\\cite{Sutton-88} and to analytic bond-order potentials (BOP)~\\cite{Hammerschmidt-09-IJMR,Drautz2015}.\nThese TB\/BOP models can be evaluated at significantly reduced numerical effort and provide a transparent and intuitive model of the interatomic bond for the prediction of materials properties, see e.g. Refs.~\\cite{Mrovec2004,Mrovec2007,Mrovec2011,Seiser-11-2,Cak2014,Ford-14,Ford2015,Wang-19}. \n\nThe TB\/BOP models employ adjustable parameters that need to be optimized for a particular material.\nThis optimization is in principle comparable to the parameterization of classical potentials which can be performed with various existing software packages~\\cite{Brommer2007,Duff2015,Barrett2016,Stukowski2017}.\nThe parameterization of TB\/BOP models, however, requires sophisticated successive optimization steps, computationally-efficient handling of large data sets and an interface to a TB\/BOP calculator~\\cite{Horsfield-96,Hammerschmidt2019}.\n\nHere, we present BOPcat (\\textbf{B}ond-\\textbf{O}rder \\textbf{P}otential \\textbf{c}onstruction \\textbf{a}nd \\textbf{t}esting), a software to parameterize TB\/BOP models as implemented in the BOPfox software~\\cite{Hammerschmidt2019}. \nThe parameters of the models are optimized to reproduce target properties like energies, forces, stresses, eigenvalues, defect formation energies, elastic constants, etc. \nWith the interface of BOPfox to LAMMPS~\\cite{Plimpton-95} and ASE~\\cite{Larsen2017}, the list of target properties can in principle be extended to include dynamical properties. \nWe illustrate the capability of the BOPcat software by constructing and testing an analytic BOP with collinear magnetism for Fe. \nExtensive tests show the good transferability of the BOP to properties which were not included in the parameterization, particularly to elastic constants, point defects, $\\gamma$ surfaces, phonon spectra and deformation paths of the ferromagnetic bcc groundstate and to other crystal structures.\nThe structures and target properties used here are taken from DFT calculations but could also include experimental data or other data sources. \n\nWe first provide a brief introduction of bond-order potentials in Sec.~\\ref{sec:BOPformalism}. \nIn Sec.~\\ref{sec:programflow}, the BOPcat program is outlined and the implemented modules are described.\nIn Sec.~\\ref{sec:application}, we discuss the construction of an analytic BOP for Fe and its testing. In the appendix we provide\nfurther examples of parameterization protocols for BOPcat.\n\n\\section{Bond-order potential formalism}\\label{sec:BOPformalism}\n\nAnalytic BOPs provide a robust and transparent description of the interatomic interaction that is based on a coarse-graining of the electronic structure~\\cite{Drautz2006,Drautz2011,Drautz2015} from DFT to TB to BOP.\nIn the following we compile the central equations and functions of the TB\/BOP formalism that are parameterized in BOPcat. \n\nIn the TB\/BOP models, the total binding energy is given by\n\\begin{equation}\n \\label{eq:U_B_general}\n U_B = U_{\\rm{bond}} + U_{\\rm{prom}} + U_{\\rm{ion}} + U_{\\rm{es}} + U_{\\rm{rep}} + U_{\\rm{X}}\n\\end{equation}\nwith the bond energy $U_{\\rm{bond}}$ due to the formation of chemical bonds, the promotion energy $U_{\\rm{prom}}$ from redistribution of electrons across orbitals upon bond formation, the onsite ionic energy $ U_{\\rm{ion}}$ to charge an atom, the intersite electrostatic energy $U_{\\rm{es}}$, \nthe exchange energy $U_{\\rm{X}}$ due to magnetism and the repulsive energy $U_{\\rm{rep}}$ that includes all further terms of the second-order expansion of DFT. \nThe individual energy and force contributions are described in detail in Ref.~\\cite{Hammerschmidt2019}, their functional forms and according parameters for the exemplary construction of a magnetic BOP for Fe are given in Sec.~\\ref{sec:application}.\n\nThe bond energy is obtained by integration of the electronic density of states (DOS) $n_{i\\alpha}$,\n\\begin{equation}\n U_\\mathrm{bond} = 2\\sum_{i\\alpha}\\int^{E_F} (E-E_{i\\alpha})n_{i\\alpha}(E)dE\n\\end{equation}\nwith atomic onsite levels $E_{i\\alpha}$ for orbital $\\alpha$ of atom $i$.\nIn TB calculations, the DOS is obtained by diagonalization of the Hamiltonian $\\hat{H}$ for a given structure.\nIn analytic BOPs~\\cite{Drautz2006,Drautz2011,Drautz2015}, the DOS is determined analytically from the moments \n\\begin{eqnarray}\n \\label{eq:mu}\n \\mu_{i\\alpha}^{(n)} &=& \\braopket{i\\alpha}{\\hat{H}^n}{i\\alpha} \\nonumber \\\\ \n &=&H_{i\\alpha j\\beta}H_{j\\beta k\\gamma}H_{k\\gamma\\ldots}\\dotso H_{\\ldots i\\alpha} \\nonumber \\\\\n &=&\\int E^{n} n_{i\\alpha}(E)dE\n\\end{eqnarray}\nthat are computed from the matrix elements $H_{i\\alpha j\\beta}$ of the Hamiltonian between pairs of atoms.\nThe BOP DOS provides an approximation to the TB DOS at a computational effort for energy and force calculations that scales linearly with the number of atoms~\\cite{Teijeiro-16-2}.\nThe quality of the BOP approximation can be improved systematically by including higher moments with a power-law scaling for the increase in computational effort~\\cite{Teijeiro-16-1}.\nA detailed introduction to TB\/BOP calculations in BOPfox is given in Ref.~\\cite{Hammerschmidt2019}.\n\n\\section{Program Flow}\\label{sec:programflow}\n\nBOPcat is a collection of Python modules and tools for the construction and testing of TB\/BOP models.\nThe individual steps are specified by the user as a protocol in terms of a Python script. \nThe overall construction and testing proceeds as follows:\n\\begin{itemize}\n \\item define and initialize input controls (CATControls)\n \\item generate initial model (CATParam)\n \\item read reference data (CATData) \n \\item initialize calculator interface (CATCalc)\n \\item build and run optimization kernel (CATKernel)\n\\end{itemize}\nThe individual tasks are modularized in the modules CATControls, CATParam, CATData, CATCalc and CATKernel that interact as illustrated in Fig.~\\ref{fig:workflow}.\nDue to its modular structure, one can also use only a subset of the modules, e.g., for successive optimizations.\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=1.9\\columnwidth]{workflow}\n \\caption{Workflow of the core optimization process in BOPcat. The input controls are specified and initialized.\n The reference data is read from a database and the reference and\n testing sets are generated. The initial model is read from a model database.\n The structures and initial model are assigned to the calculator interface.\n The calculator, reference properties and weights are then used to build the \n optimization kernel. The kernel can be run as standalone process or be sent to \n the process management of a queuing system.}\n \\label{fig:workflow} \n\\end{figure*}\nIn the following, the individual modules are described in detail with snippets of the python protocol for a basic construction process of a magnetic BOP for Fe.\nFurther example protocols are given in the appendix.\n\n\\subsection{Input controls: CATControls}\n\\label{sec:CATControls}\n\nThe variables required for running BOPcat are defined in the module CATControls. \nThese include the list of chemical symbols for which the model is constructed, calculator settings, filters for the reference data, optimization variables, specifications of the model, etc.. \nThe consistency of the input variables is checked within the module to catch inconsistent settings at an early stage. \nThe CATControls object is then passed on to succeeding modules from which the relevant variables are assigned. \nAs an example initialization of the CATControls object we define the functions of the TB\/BOP model and the initial guess for the according parameters by providing an existing model from a file (\\verb+models.bx+).\n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.catcontrols import CATcontrols\n ctr = CATControls()\n ctr.elements = ['Fe']\n ctr.model_pathtomodels = 'models.bx'\n ctr.calculator_nproc = 4\n ctr.calculator_parallel = 'multiprocessing'\n ctr.calculator_settings = {'scfsteps':500}\n ctr.data_filename = 'Fe.fit'\n ctr.data_free_atom_energies = {'Fe':-0.689}\n ctr.data_system_parameters = {'spin':[1,2],\n 'system_type':[0]}\n ctr.opt_optimizer = 'leastsq'\n ctr.opt_variables = [{'bond':['Fe','Fe'],\n 'rep1':[True,True]}]\n \\end{verbatim}\n}\n\nAlternatively, BOPcat can generate an initial guess from raw bond integrals~\\cite{Jenke2019} with additional user-specified information on the functions to be used for the individual energy contributions, the valence-type ($s$\/$p$\/$d$) and number of valence electrons of the elements, the cut-off radii for determining pairs of interacting atoms, etc. (see ~\\ref{sec:app2}) \n\n\\subsection{Reference data: CATData}\n\\label{sec:CATData}\n\nThe set of reference data of structures and properties is read from a text file by the CATData module.\nThe entries for reference data have the following format:\n\n{\\footnotesize\n\\begin{verbatim}\ndata_type = 0\ncode = 7\nbasis_set = 0\npseudopotential = 30\nstrucname = bcc\na1 = -1.3587 1.3587 1.3587\na2 = 1.3587 -1.3587 1.3587\na3 = 1.3587 1.3587 -1.3587\ncoord = cartesian\nFe 0.0000 0.0000 0.0000\nenergy = -7.9561\nforces 0.0000 0.0000 0.0000\nstress = -0.08423 -0.0843 -0.0843 -0. -0. -0.\nsystem_type = 0\nspace_group = 229\ncalculation_type = 1\nstoichiometry = Fe\ncalculation_order = 10.033\nweight = 1.0\nspin = 1\n\\end{verbatim}\n}\n\nIn contrast to optimization procedures for classical potentials, the optimization of TB\/BOP models with BOPcat can be carried out not only for atomic properties but also at the electronic-structure level by providing eigenvalues and corresponding $k$-points as reference data. \nThe optimization can be further extended to other properties of TB\/BOP calculations (e.g. band width, magnetic moment) by supplying the according keyword and the data type of the BOPfox software in the reference data module of BOPcat.\n\nCATData stores the reference data in a set of \\verb+Atoms+ objects of the ASE python framework for atomistic simulations~\\cite{Larsen2017}. \nEach of the ASE \\verb+Atoms+ objects can have calculator-specific identifiers such as the settings of the DFT calculation (\\verb+code+, \\verb+basis_set+, \\verb+pseudopotential+) as well as structure-related identifiers such as calculation type (\\verb+calculation_type+) and calculation order (\\verb+calculation_order+), which are used as filters to use only a subset of the available reference data.\nA complete description of the implemented identifiers and filters is given in the manual of BOPcat. \nThe corresponding reference data can include basic quantities like energies, forces, stresses as well as derived quantities such as defect energies. \nExtracting the reference structures and their corresponding properties is illustrated in the following: \n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.catdata import CATData\n data = CATData(controls=controls)\n ref_atoms = data.get_ref_atoms(\n structures=['bcc*','fcc*','hcp*'],\n quantities=['energy'],\n sort_by='energy')\n ref_data = data.get_ref_data()\n\\end{verbatim}\n}\n\nThe user-specified reference data, i.e. structures and properties, are passed to the calculator and optimization kernel, respectively.\n\n\\subsection{Initial model: CATParam}\n\\label{sec:CATParam}\n\nThe construction of TB\/BOP models in BOPcat relies on iterative optimization steps that require an initial guess that can be provided by the user or generated by the CATParam module. \nIn the latter case the functions and parameters are initialized from a recently developed database of TB bond integrals from downfolded DFT eigenspectra~\\cite{Jenke2019}. \nIn the following snippet, the initial model is read from the filename specified in the input controls.\n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.catparam import CATParam\n param = CATParam(controls=controls)\n ini_model = param.models[0]\n\\end{verbatim}\n}\n\nThe CATParam module also provides meta-operations on sets of models such as identifying the optimum model for a given set of reference data.\n\n\\subsection{Calculator: CATCalc}\n\\label{sec:CATCalc}\n\nAfter the reference structures are read and the initial model is generated, the CATCalc module sets up the calculator for the computation of the specified properties for the reference structures by the initial and then optimized TB\/BOP model. \nTo this end, a list of BOPfox-ASE calculators is constructed which can be run in serial or in multiprocessing mode. \nThe latter is optimized by a load balancing scheme that is based on the size of the structures. \nAdditional properties for the model optimization can easily be included by extending this module accordingly.\nIn the following snippet, an initial model is provided and an ASE calculator is initialized for each of the reference structures.\n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.catcalc import CATCalc\n calc = CATCalc(controls=ctr,\n model=ini_model,\n atoms=ref_atoms)\n\\end{verbatim}\n}\n\nHere, the input controls (\\verb+ctr+), the initial guess for the TB\/BOP model (\\verb+ini_model+) and the reference structures (\\verb+ref_atoms+) are specified in the code snippets given in Sec.~\\ref{sec:CATControls},~\\ref{sec:CATParam} and~\\ref{sec:CATData}, respectively.\n\n\\subsection{Construction and testing kernel: CATKernel}\n\\label{sec:CATKernel}\n\nThe reference data and associated calculators are then used to set up the CATKernel module.\nThis module provides an interface to the objective function for the construction and testing of TB\/BOP models. \nThe default objective function for $N_p$ properties of $N_s$ structures is given by\n\\begin{equation}\n\\chi^2 = \\sum_p \\frac{w_p}{N_p} \\sum_s \\frac{\\tilde{w_{ps}}}{N_s^{(p)}} \\frac{X_{ps}^\\mathrm{model}-X_{ps}^\\mathrm{ref}}{\\bar{X}^\\mathrm{ref}_p} .\n\\end{equation}\nThe user can choose other definitions of the objective function that are implemented in BOPcat or provide external implementations.\nThe weights for the individual structures \n\\begin{equation}\n \\tilde{w_{ps}} = \\frac{1}{\\gamma_pN_\\mathrm{atoms}^{(i)}}\n\\end{equation}\nare specified by the user or determined by the module. \nThe dimensionality factor $\\gamma_{p}$ balances the relative weights of energies, forces and stress by taking values of $1,3,6$, respectively. \nThe objective function can also be normalized by the variance of the properties as in Ref.~\\cite{Krishnapriyan2017}.\nFor construction purposes, this module provides an interface between objective function and optimizer algorithms that are either readily available in the Python modules Scipy~\\cite{Scipy} and NLopt~\\cite{Nlopt} or provided by the user as external module. \n\nIn the following example we illustrate the initialization and running of the CATkernel object for the input controls (\\verb+ctr+), calculator (\\verb+calc+) and reference properties (\\verb+ref_data+) defined in Sec.~\\ref{sec:CATControls},~\\ref{sec:CATCalc} and~\\ref{sec:CATParam}.\n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.catkernel import CATKernel\n kernel = CATKernel(controls=ctr,\n calc=calc,\n ref_data)\n kernel.run() \n\\end{verbatim}\n}\n\n\\subsection{Parallel execution of BOPcat}\n\nThe optimization kernel can be executed efficiently in parallel by distributing the required property calculations with Python subprocesses and the message passing interface. \nFor this purpose, BOPcat also features a process management module to interact with the queuing system of compute clusters.\nThis allows high-throughput optimizations of TB\/BOP models with, e.g., different initial models or different sets of reference data.\nWe illustrate this in the following snippet:\n\n{\\footnotesize\n\\begin{verbatim}\n from bopcat.process_management\\\n import Process_catkernels\n from bopcat.process_management\\\n import Process_catkernel\n from bopcat.process_management import queue\n models = [ini_model.rattle(kernel.variables) \n for i in range(5)]\n subprocs = []\n for i in range(len(kernels)):\n ckern = kernel.copy()\n ckern.calc.set_model(models[i])\n proc = Process_catkernel(catkernel=ckern,\n queue=queue)\n subprocs.append(proc)\n proc = Process_catkernels(procs=subprocs)\n proc.run()\n\\end{verbatim}\n}\n\nA list of random models (\\verb+models+) is first generated by applying noise to the parameters of the initial model (\\verb+ini_model+). \nA copy of the CATKernel object is then generated (\\verb+ckern+) and a model from the list is assigned. \nThe kernel is then serialized by the \\verb+Process_catkernel+ object which contains the details on how to execute the kernel. \nFinally, the individual subprocesses are wrapped in the main object \\verb+Process_catkernels+ which submits them to a specified queue (\\verb+queue+) of the compute cluster.\n\n\\subsection{Graphical user interface}\n\nThe setup of a construction or testing process with BOPcat discussed so far is based on writing and editing Python scripts. \nAlternatively, BOPcat can be driven with a graphical user interface (GUI), see snapshots in Fig.~\\ref{fig:gui}. \n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=1.75\\columnwidth]{gui}\n \\caption{Snapshots of the graphical user interface for (a) constructing a TB\/BOP model, (b) generating reference structures,\n (c) setting optimization variables and (d) defining a parameterization protocol.}\n \\label{fig:gui} \n\\end{figure*}\nBasic information of a construction such as the current value of the parameters and objective function can be visualized. \nThe reference data can be inspected visually and corresponding attributes such as the weights can be set directly.\nFor testing purposes, a given TB\/BOP model can be evaluated and visualized for a given set of reference data. \nThe GUI also comes with a parameterization builder which allows one to design and execute a parameterization protocol. \nIn particular, one can add a number of optimization layers that consist of several optimization kernels with communication of intermediate TB\/BOP models between subsequent layers. \nThis provides the technical prerequisite for automatizing the construction of TB\/BOP models by sophisticated protocols.\n\n\\section{Construction and testing of simple Fe BOP}\\label{sec:application}\n\n\\subsection{Functional form}\n\nAs an example application of BOPcat we construct a simple BOP model for Fe including magnetism.\nWe note that this Fe BOP is constructed to demonstrate BOPcat and that capturing the reference data more precisely would require a more complex functional form.\nThe atoms are assumed to be charge-neutral so that the binding energy given in Eq.~\\ref{eq:U_B_general} reduces to\n\\begin{equation}\n\\label{eq:U_B}\n U_\\mathrm{binding} = U_\\mathrm{bond} + U_\\mathrm{rep} + U_\\mathrm{X} + U_\\mathrm{emb}.\n\\end{equation}\nWe choose a $d$-valent model including only two-center contributions, similar to previous TB\/BOP models for Fe~\\cite{Mrovec2011,Ford2015,Madsen2011}, with an initial number of $d$-electrons of $N_d=6.8$. \nWe use a BOP expansion up to the 9-th moment (Eq.~\\ref{eq:mu}) as a good compromise between performance and accuracy of the DOS.\n\nThe Hamiltonian matrix elements $H_{i\\alpha j\\beta}$ which enter the calculation of the moments are constructed from Slater-Koster bond integrals $\\beta_{ij}^\\sigma$ with the angular character $\\sigma=dd\\sigma, dd\\pi, dd\\delta$. \nThe initial guess of the bond integral parameters is taken from projections of the DFT eigenspectrum on a TB minimal basis for the Fe-Fe dimer~\\cite{Jenke2019}. \nThe distance dependence of the bond integrals is represented by a simple exponential function\n\\begin{equation}\n\\beta_{ij}^\\sigma = a\\exp{-bR_{ij}^c} .\n\\end{equation}\nFor this simple BOP model, the repulsive energy is given as a simple pairwise contribution.\n\\begin{equation}\n U_\\mathrm{rep} = \\frac{1}{2}\\sum_{i,i\\neq j}a_\\mathrm{rep}\\exp\\left(-b_\\mathrm{rep}R_{ij}^{c_\\mathrm{rep}}\\right).\n\\end{equation}\nThe magnetic contribution to the energy $U_X$ is evaluated using\n\\begin{equation}\\label{eq:Stoner}\n U_\\mathrm{X} = -\\frac{1}{4}\\sum_i I m_i^2\n\\end{equation}\nwith $m_i=N_i^\\uparrow-N_i^\\downarrow$ the magnetic moment of atom $i$ and $I$ the Stoner exchange integral that is initially set to 0.80~eV similar to Ref.~\\cite{Mrovec2011}.\nIn extension to Eq.~\\ref{eq:U_B_general} we use an additional empirical embedding contribution in Eq.~\\ref{eq:U_B}\n\\begin{equation}\n U_\\mathrm{emb} = -\\sum_i\\sqrt{\\sum_{j,j\\neq i} a_\\mathrm{emb}\\exp\\left(-b_\\mathrm{emb}(R_{ij}-c_\\mathrm{emb})^2\\right)}\n\\end{equation}\nwhich may be understood as providing contributions due to the missing $s$ electrons and non-linear exchange correlation.\n\nThe bond integrals, repulsive and embedding functions are multiplied with a cut-off function\n\\begin{equation}\n f_\\mathrm{cut} = \\frac{1}{2}\\left[ \\cos\\left(\\pi \\frac{R_{ij}-(r_\\mathrm{cut}-d_\\mathrm{cut})}{d_\\mathrm{cut}}\\right) + 1 \\right]\n\\end{equation}\nin the range of $[r_\\mathrm{cut} - d_\\mathrm{cut}$, $r_\\mathrm{cut}]$ in order to restrict the range of the interatomic interaction.\nFor the Fe BOP constructed here we used $r_\\mathrm{cut}=3.8~\\AA$, $d_\\mathrm{cut}=0.5~\\AA$ for the bond integrals and $r_\\mathrm{cut}=6.0~\\AA$, $d_\\mathrm{cut}=1.0~\\AA$ for the repulsive and embedding energy terms.\n\n\\subsection{Reference data}\n\nThe reference data for constructing and testing the Fe BOP is obtained by DFT calculations using the \\texttt{VASP} software~\\cite{Kresse-96-1,Kresse-96-2} with a high-throughput environment~\\cite{Hammerschmidt-13}.\nWe used the PBE exchange-correlation functional~\\cite{Perdew-96} and PAW pseudopotentials~\\cite{Bloechl-94} with $p$ semicore states. \nA planewave cut-off energy of 450~eV and Monkhorst-Pack $k$-point meshes~\\cite{Monkhorst-76} with a density of 6~\/\\AA$^{-1}$ were sufficient to converge the total energies to 1~meV\/atom. \nThe reference data for constructing the Fe BOP comprised the energy-volume curves of the bulk structures bcc, fcc, hcp, and the topologically close-packed (TCP) phases A15, $\\sigma$, $\\chi$, $\\mu$, C14, C15 and C36 that are relevant for Fe compounds~\\cite{Ladines2015}. \n\nFor testing the Fe BOP, several additional properties were determined that are related to the ferromagnetic bcc groundstate structure. \nIn particular, the elastic constants at the equilibrium volume were determined by fitting the energies as function of the relevant strains~\\cite{Golesorkhtabar2013}. \nThe formation energies of point defects were calculated with the supercell approach with fixed volume corresponding to the bulk equilibrium volume. \nA $6\\times 6\\times 6$ supercell was found to be sufficient to converge the formation energies to an uncertainty of 0.1~eV.\nThe energy barriers for vacancy migration were calculated with the nudged elastic band method.\nThe phonon spectra are computed with the Phonopy software~\\cite{Togo2015}. \nThe transformation paths from bcc to other crystal structures were determined according to Ref.~\\cite{Paidar1999}. \n\n\\subsection{Construction}\n\nThe parameters of the Fe BOP with the functions and reference data given above were optimized with the following BOPcat protocol: \n\\begin{enumerate}\n \\item The magnitudes of the repulsive and embedding terms $a_\\mathrm{rep}$ and $a_\\mathrm{emb}$ are adjusted to reproduce the energies of hydrostatically deformed bcc, fcc and hcp with fixed values for the exponents taken from Ref.~\\cite{Madsen2011}.\n \\item The prefactors of the bond integrals are adjusted including the energies of the TCP phases.\n \\item The exponents of the repulsive functions are optimized including the randomly deformed structures in the reference set.\n \\item The exponents of the bond integrals are optimized by increasing the size of the reference data. \n \\item The other parameters are optimized further while increasing the number of reference structures.\n\\end{enumerate}\n\nThe resulting parameters of the model are compiled in Tab.~\\ref{tab:parameters}.\n\\begin{table}[t]\n \\centering\n \\caption{Parameters of the BOP model for Fe. The unit of $b_\\mathrm{emb}$ is $\\AA^{-2}$ and $c_\\mathrm{emb}$ is $\\AA$.}\n \\scalebox{1.00}{\n \\begin{tabular}{l c c c}\n \\hline\n \\hline\n & $a$ (eV) & $b$ ($\\AA^{-1}$) & c \\\\\n \\hline \n $dd\\sigma$ & -24.9657 & 1.4762 & 0.9253 \\\\\n $dd\\pi$ & 21.7965 & 1.4101 & 1.0621 \\\\\n $dd\\delta$ & -2.3536 & 0.7706 & 1.3217 \\\\\n \\hline\n $U_\\mathrm{rep}$ & 1797.4946 & 3.2809 & 1.0067 \\\\\n \\hline\n $U_\\mathrm{emb}$ & -1.3225 & 1.3374 & 2.1572 \\\\\n \\hline\n $N_d$ & 6.8876 & & \\\\\n $I (eV)$ & 0.9994 & & \\\\\n \\hline\n \\hline\n \\end{tabular}}\n \\label{tab:parameters}\n\\end{table}\nThe initial and optimized bond integrals, and the empirical potentials are plotted in Fig.~\\ref{fig:bondint_fe}. \nThere is no substantial change in the bond integrals except that the bond integrals become longer-ranged after optimization. \nAt shorter distance the bond integrals become weaker which can be rationalized by the screening influence of the neighboring atoms in the bulk structure. \nThe effective number of $d$ electrons and the Stoner integral increased during optimization.\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.99\\columnwidth]{bondint_fe}\n \\caption{Distance dependence of the $d$ bond integrals as obtained by downfolding from DFT eigenspectra for the Fe-Fe dimer (dashed)~\\cite{Jenke2019} and after optimization to Fe bulk structures (solid).}\n \\label{fig:bondint_fe} \n\\end{figure}\nThe optimized pairwise repulsive term $U_{\\mathrm{rep}}$ shown in Fig.~\\ref{fig:U_rep} is repulsive for all distances. \nThe embedding term $U_{\\mathrm{emb}}$, in contrast, is negative for all distances.\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.99\\columnwidth]{uemp_fe}\n \\caption{Distance dependence of the pairwise repulsive potential $U_{\\mathrm {rep}}$ (black) and the embedding term $U_{\\mathrm {emb}}$ (violet) after optimization to Fe bulk structures.}\n \\label{fig:U_rep} \n\\end{figure}\n\n\\subsection{Testing}\n\n\\subsubsection{Properties related to bcc-Fe}\n\nAn important basic test of the Fe BOP is the performance for properties that are closely related to the ferromagnetic bcc groundstate structure at 0~K. \nIn Tab.~\\ref{tab:bulk_properties} we compare elastic constants as well as point defect and surface properties as predicted by the Fe BOP to available DFT and experimental values.\nThe overall good agreement demonstrates the robust transferability of the Fe BOP to properties not included in the training set. \n\\begin{table}[htb]\n \\centering\n \\caption{Bulk properties of ferromagnetic bcc-Fe. The experimental values for the elastic constants, vacancy formation energy and vacancy migration energy are taken from Refs.~\\cite{Rayne1961}, ~\\cite{DeSchepper1983} and ~\\cite{Vehanen1982}, respectively. The DFT values for the vacancy migration energy and surface energies are taken from Refs. ~\\cite{Domain2001} and ~\\cite{Blonski2007}, respectively.}\n \\scalebox{1.00}{\n \\begin{tabular}{l c c c}\n \\hline\n \\hline\n & BOP & DFT & experiment \\\\\n \\hline \n V\/atom (\\AA$^3$) & 11.48 & 11.46 & 11.70 \\\\\n $B$ (GPa) & 171 & 176 & 168 \\\\\n $C_{11}$ (GPa) & 265 & 257 & 243 \\\\\n $C_{12}$ (GPa) & 125 & 154 & 138 \\\\\n $C_{44}$ (GPa) & 87 & 85 & 122 \\\\\n $E_f^\\mathrm{vac}$ (eV) & 2.03 & 2.20 & 2.00 \\\\\n $E_\\mathrm{mig}^\\mathrm{vac}$ (eV) & 1.33 & 0.65 & 0.55 \\\\\n $E_f^\\mathrm{100}$ (eV) & 3.65 & 4.64 & \\\\\n $E_f^\\mathrm{110}$ (eV) & 3.13 & 3.64 & \\\\\n $E_f^\\mathrm{111}$ (eV) & 3.59 & 4.34 & \\\\ \n $\\gamma_{(100)}$ (J\/m$^2$) & 1.44 & 2.47 & \\\\\n $\\gamma_{(110)}$ (J\/m$^2$) & 1.27 & 2.37 & \\\\\n $\\gamma_{(111)}$ (J\/m$^2$) & 2.04 & 2.58 & \\\\\n $\\gamma_{(211)}$ (J\/m$^2$) & 1.50 & 2.50 & \\\\\n \\hline\n \\hline\n \\end{tabular}}\n \\label{tab:bulk_properties}\n\\end{table}\n\nThe predicted elastic constants $C_{11}$, $C_{12}$ and $C_{44}$ are in good agreement with DFT although the specific deformed structures were not included in the reference set for constructing the BOP. \nThe prediction for the vacancy formation energy $E_f^\\mathrm{vac}$ is also in good agreement with DFT while the vacancy migration energy $E_\\mathrm{mig}^\\mathrm{vac}$ is overestimated as in previous TB\/BOP models for Fe~\\cite{Madsen2011, Mrovec2011}.\nThe formation energies of the vacancy and self-interstitial atoms calculated by the Fe BOP exhibit the correct energetic ordering although they were not included in the parameterization. \nThe absolute values are slightly underestimated as compared to DFT, similar to previous models~\\cite{Madsen2011, Mrovec2011}.\nThe relative stability of the low-index surfaces are also satisfactorily reproduced by the present Fe BOP. However, similar to the previous BOP model of Mrovec\\cite{Mrovec2011}, the energy difference of the other surfaces relative to $(110)$ is overestimated.\nThe deviations for point defects and surfaces are attributed to the missing $s$ electrons in the model and the lack of screening in the orthogonal TB model.\n\nAs a test towards finite-temperature applications we determine the phonon bandstructure for ferromagnetic bcc Fe shown in Fig.~\\ref{fig:phonon_fe}.\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=0.99\\columnwidth]{phonon_fe}\n \\caption{Phonon bandstructure of ferromagnetic bcc-Fe. The solid (dashed) lines are the BOP (DFT) results\n while the symbols correspond to experimental data taken from Ref~\\cite{Klotz2000}.}\n \\label{fig:phonon_fe} \n\\end{figure}\nThe prediction of the Fe BOP is in good overall agreement with DFT and experimental results. \nThe close match near the $\\Gamma$ point is expected from the good agreement of the elastic constants. \nThe experimental data were obtained at 300~K which can explain the deviation of the experimental from the\nDFT\/BOP data for the $T2$ branch along $T-N$. \n\nThe transferability of the model in the case of large deformations of the bcc groundstate structure is tested using deformation paths that connect the high symmetry structures bcc, fcc, hcp, sc and bct.\nThe energy profile versus the deformation parameter for the tetragonal, hexagonal, orthogonal and trigonal deformation paths is shown in Fig.~\\ref{fig:trans_fe}. \n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.99\\columnwidth]{trans_fe}\n \\caption{Energy profile along the transformation path connecting the ferromagnetic bcc Fe with other high-symmetry structures as obtained by the Fe BOP (red) and DFT (black).}\n \\label{fig:trans_fe} \n\\end{figure}\nThe energy profiles for all transformation paths are predicted very well by the Fe BOP model. \nThe energies at the high symmetry points are in good agreement with DFT except for the sc structure which is slightly underestimated by BOP. \n\n\\subsubsection{Transferability to other crystal structures}\n\nIn order to verify the transferability of the Fe BOP model to other crystal structures, we determined the equilibrium binding energy, volume and bulk modulus for a broader set of crystal structures as shown in Fig.~\\ref{fig:eos_fe}. \n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=0.99\\columnwidth]{eos_fe}\n \\caption{Relative binding energy, bulk modulus and equilibrium volume of bulk structures including TCP phases. The difference in background shading indicates structures that belong to the reference set used for construction (green) and for testing (blue).}\n \\label{fig:eos_fe} \n\\end{figure}\nThe properties of the bcc, fcc, hcp, A15, $\\sigma$, $\\chi$, $\\mu$, C14, C15 and C36 structures that were included in the parameterization are reproduced with excellent agreement.\nThis shows that the chosen functional form of the Fe BOP is sufficiently flexible to adapt to this set of reference data.\nThe Fe BOP model also shows robust predictions of equilibrium binding energy, volume and bulk modulus across the set of tested structures.\nThe open crystal structures (e.g. A4) show comparably larger errors which can be expected as the reference set used in the parameterization covers mostly close-packed local atomic environments~\\cite{Jenke2018} while local atomic environments of open structures were not part of the training set.\n\n\\section{Conclusions}\n\nWe present the BOPcat software for the construction and testing of TB\/BOP models as implemented in the BOPfox code. \nTB\/BOP models are parameterized to reproduce reference data from DFT calculations including energies, forces, and stresses as well as derived properties like defect formation energies.\nThe modular framework of BOPcat allows one to implement complex parameterization protocols by flexible python scripts. \nBOPcat provides a graphical user interface and a highly-parallelized optimization kernel for fast and efficient handling of large data sets. \n\nWe illustrate the key features of the BOPcat software by constructing and testing a simple $d$-valent BOP model for Fe including magnetism. \nThe resulting BOP predicts a variety of properties of the groundstate structure with good accuracy and shows good quantitative transferability to other crystal structures.\n\n\\section*{Acknowledgements}\n\nThe authors are grateful to Aparna Subramanyam, Malte Schr{\\\"o}der, Alberto Ferrari, Jan Jenke, Yury Lysogorskiy, Miroslav Cak, and Matous Mrovec for discussions and feedback using BOPcat.\nWe acknowledge financial support by the German Research Foundation (DFG) through research grant HA 6047\/4-1 and project C1 of the collaborative research centre SFB\/TR 103.\nPart of the work was carried out in the framework of the International Max-Planck Research School SurMat.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nBelief change\\footnote{\\, In some literature, the term ``belief revision\" is used as a synonym for belief change. In what follows, we use belief revision to refer to a particular kind of belief change.} theory studies how a rational agent changes her belief state when she is exposed to new information. Studies in this field have traditionally had a strong focus on two types of change: \\emph{contraction} in which a specified sentence has to be removed from the original belief state, and \\emph{revision} in which a specified sentence has instead to be consistently added. This paper is mainly concerned with the latter.\n\nAlchourr\\'on, G\\\"ardenfors and Makinson (AGM) performed the pioneering formal study on these two types of change in their seminal paper \\cite{alchourron_logic_1985}. In the AGM theory of belief change, the agent's belief state is represented by a set of sentences from some formal language $\\mathcal{L}$, usually denoted by $K$. The new information is represented by a single sentence in $\\mathcal{L}$. Belief revision and contraction on $K$ are formally represented by two operations $\\ast$ and $\\div$, mapping from a sentence $\\varphi$ to a new set of sentences $K \\ast \\varphi$ and $K \\div \\varphi$ respectively. \\cite{alchourron_logic_1985} postulated some conditions that a rational revision or contraction operation should satisfy, which are called AGM postulates on revision and contraction. \n\nFurthermore, \\cite{alchourron_logic_1985} showed that contraction and revision satisfying AGM postulates could be precisely constructed from a model based on partial meet functions on remainder sets. After that, many alternative models \\cite[etc.]{alchourron_logic_1985_safe,grove_two_1988,gardenfors_revisions_1988,hansson_kernel_1994} have been proposed to construct the contraction and revision operations satisfying AGM postulates. Although these models look entirely different on the surface, most of them employ the same select-and-intersect strategy \\cite[p. 19]{hansson_descriptor_2017}. For example, in partial meet construction for contraction \\cite{alchourron_logic_1985}, a selection is made among remainders and in sphere modelling for revision \\cite{grove_two_1988}, a selection is made among possible worlds. The intersection of the selected objects is taken as the outcome of the operation in both cases.\n\nAlthough the AGM theory has virtually become a standard model of theory change, many researchers are unsatisfied with its settings in several aspects and have proposed several modifications and generalizations to that framework (see \\cite{ferme_agm_2011} for a survey). Here we only point out two inadequatenesses of the AGM theory.\n\nOn the one hand, in the original AGM model, the input is represented by a single sentence. This is unrealistic since agents often receive more than one piece of information at the same time. In order to cover these cases, we can generalize sentential revision to multiple revision, where the input is a finite or infinite set of sentences. On the other hand, in AGM revision, new information has priority over original beliefs. This is represented by the success postulate: $\\varphi \\in K \\ast \\varphi$ for all $\\varphi$. The priority means that the new information will always be entirely incorporated, whereas previous beliefs will be discarded whenever the agent need do so in order to incorporate the new information consistently. This is a limitation of AGM theory since in real life it is a common phenomenon that agents do not accept the new information that they receive or only accept it partially. As a modification, we can drop the success postulate and generalize prioritized revision to non-prioritized belief revision. \n\nIn this contribution, we will put these two generalizations together and consider the so called multiple non-prioritized belief revision. In \\cite{falappa_prioritized_2012}, two different kinds of such generalized revision are distinguished:\n\n\\begin{enumerate}\n\\item Merge: $K$ and $A$ are symmetrically treated, i.e., sentences of $K$ and $A$ could be accepted or rejected. \n\\item Choice revision\\footnote{\\, Here we use the term ``choice revision'', introduced by \\cite{Fuhrmann_phd}, to replace the term``selective change'' used in \\cite{falappa_prioritized_2012}, for it is easier for us to distinguish it from the ``selective revision'' introduced in \\cite{ferme_selective_1999}, which is a sort of non-prioritized revision with a single sentence as input. It should be noted that generally choice revision by a finite set $A$ cannot be reduced to selective revision by the conjunction $\\& A$ of all elements in $A$. To see this, let $\\ast_{s}$ be some selective revision. It is assumed that $\\ast_{s}$ satisfies extensionality, i.e. if $\\varphi $ is logically equivalent to $ \\psi$, then $K \\ast_s \\varphi = K \\ast_s \\psi$. So, $K \\ast^\\prime \\& \\{\\varphi, \\neg \\varphi\\} = K \\ast^\\prime \\& \\{\\psi, \\neg \\psi\\}$ for all $\\varphi$ and $\\psi$. However, it is clearly unreasonable for choice revision $\\ast_c$ that $K \\ast_c \\{\\varphi, \\neg \\varphi\\} = K \\ast_c \\{\\psi, \\neg \\psi\\}$ should hold for all $\\varphi$ and $\\psi$.}: some sentences of A could be accepted, some others could be rejected.\n\\end{enumerate}\n\nWe use $\\ast_c$ to denote a choice revision operation. \\cite{falappa_prioritized_2012} investigated the formal properties of merge but left the study on choice revision as future work. As far as we know, little work has been done on this kind of revision in the literature. This fact can be partly explained by that the operation $\\ast_c$ has the unusual characteristic that the standard select-and-intersect approach is not in general applicable. To see why, let the set $K$ of original beliefs not contain any element of $A =\\{ \\varphi, \\neg \\varphi \\}$. We are going to construct a set $K \\ast_c A$ which incorporates $\\varphi$ or its negation. Suppose that we do that by first selecting a collection $\\mb{X} =\\{X_1, X_2, X_3, \\cdots \\}$ of sets of beliefs, each of which satisfies the success condition for choice revision with $A$, i.e. $X_i \\cap A \\neq \\emptyset$ for each $X_i$. Then it may be the case that $\\varphi \\in X_1$ and $\\neg \\varphi \\in X_2$. Given that $X_1$ and $X_2$ are consistent, it follows that the intersection $\\cap \\mb{X}$ cannot satisfy the success condition, i.e. it contains neither $\\varphi$ or $\\neg \\varphi$.\n\nTherefore, to develop a modelling for choice revision, we need to choose another strategy than the select-and-intersect method. \\cite{hansson_descriptor_2013} introduced a new approach of belief change named ``descriptor revision'', which employs a ``select-direct'' methodology: It assumes that there is a set of belief sets as potential outcomes of belief change, and the belief change is performed by a direct choice among these potential outcomes. Furthermore, this is a very powerful framework for constructing belief change operations since success conditions for various types of belief changes are described in a general way with the help of a metalinguistic belief operator $\\mathfrak{B}$. For instance, the success condition of contraction by $\\varphi$ is $\\neg \\mathfrak{B} \\varphi$, that of revision by $\\varphi$ is $\\mathfrak{B} \\varphi$. Descriptor revision on a belief set $K$ is performed with a unified operator $\\circ$ which applies to any success condition that is expressible with $\\mathfrak{B}$. Hence, choice revision $\\ast_c$ with a finite input set can be constructed from descriptor revision in the way of $K \\ast_c \\{\\varphi_1, \\varphi_2, \\cdots, \\varphi_n\\} = K \\circ \\{\\mathfrak{B} \\varphi_1 \\vee \\mathfrak{B} \\varphi_2 \\vee \\cdots \\vee \\mathfrak{B} \\varphi_n \\}$ \\cite[p. 130]{hansson_descriptor_2017}. \n\nAlthough the construction of choice revision in the framework of descriptor revision has been introduced in \\cite{hansson_descriptor_2017}, the formal properties of this type of belief change are still in need of investigation. The main purpose of this contribution is to conduct such an investigation. After providing some formal preliminaries in Section \\ref{Preliminaries}, we will review how to construct choice revision in the framework of descriptor revision in Section \\ref{section choice revison based on descriptor revision}. More importantly, in this section, we will investigate the postulates on choice revision which could axiomatically characterize these constructions. In Section \\ref{section on an alternative modelling for choice revision} we will propose an alternative modelling for choice revision, which is based on \\textit{multiple believability relations}, a generalized version of \\textit{believability relation} introduced in \\cite{hansson_relations_2014} and further studied in \\cite{zhang_believability_2017}. We will investigate the formal properties of this modelling and prove the associated representation theorems. Section \\ref{section conclusion} concludes and indicates some directions for future work. All proofs of the formal results are placed in the appendix.\n\n\n\\section{Preliminaries}\\label{Preliminaries}\n\nThe object language $\\mathcal{L}$ is defined inductively by a set $v$ of propositional variables $\\{p_0 ,\\, p_1, \\, \\cdots, \\, p_n, \\, \\cdots \\}$ and the truth-functional operations $\\neg, \\wedge, \\vee$ and $\\rightarrow$ in the usual way. $\\taut$ is a tautology and ${\\scriptstyle \\perp}$ a contradiction. $\\mathcal{L}$ is called finite if $v$ is finite. Sentences in $\\mathcal{L}$ will be denoted by lower-case Greek letters and sets of such sentences by upper-case Roman letters. \n\n$\\cn$ is a consequence operation for $\\mathcal{L}$ satisfying supraclassicality (if $\\varphi$ can be derived from $A$ by classical truth-functional logic, then $\\varphi \\in \\cons{A}$), compactness (if $\\varphi \\in \\cons{A}$, then there exists some finite $B \\subseteq A$ such that $\\varphi \\in \\cons{B}$) and the deduction property ($\\varphi \\in \\cn(A \\cup \\{\\psi\\})$ if and only if $\\psi \\rightarrow \\varphi \\in \\cn(A)$). $X\\vdash \\varphi$ and $X \\nvdash \\varphi$ are alternative notations for $\\varphi \\in \\cn(X)$ and $\\varphi \\notin \\cn(X)$ respectively. $\\{ \\varphi \\} \\vdash \\psi$ is rewritten as $\\varphi \\vdash \\psi$ for simplicity. And $\\varphi \\dashv \\Vdash \\psi$ means $\\varphi \\vdash \\psi$ and $\\psi \\vdash \\varphi$. $A \\equiv B$ holds iff for every $\\varphi \\in A$, there exists some $\\psi \\in B$ such that $\\varphi \\dashv \\Vdash \\psi$ and vice versa. \n\nFor all finite $A$, let $\\& A$ denote the conjunction of all elements in $A$. For any $A$ and $B$, $A \\owedge B = \\{\\varphi \\wedge \\psi \\mid \\varphi \\in A \\mbox{ and } \\psi \\in B\\}$. We write $\\varphi \\owedge \\psi$ and $A_0 \\owedge A_1 \\owedge \\cdots \\owedge A_n$ instead of $\\{ \\varphi\\} \\owedge \\{\\psi\\}$ and $( \\cdots ( A_0 \\owedge A_1) \\owedge \\cdots) \\owedge A_n$ for simplicity.\n\nThe set of beliefs an agent holds is represented by a \\emph{belief set}, which is a set $X \\subseteq \\mathcal{L}$ such that $X = \\cn(X)$. $K$ is fixed to denote the original beliefs of the agent. We assume that $K$ is consistent unless stated otherwise.\n\n\n\n\\section{Choice revision based on descriptor revision}\\label{section choice revison based on descriptor revision}\n\nBefore investigating the properties of choice revision constructed in the framework of descriptor revision, we first present some formal basics of this framework, which is mainly based on \\cite{hansson_descriptor_2013}. \n\n\\subsection{Basics of descriptor revision}\n\nAn atomic belief descriptor is a sentence $\\mathfrak{B}\\varphi$ with $\\varphi\\in \\mathcal{L}$. Note that the symbol $\\mathfrak{B}$ is not part of the object language $\\mathcal{L}$. A \\emph{molecular belief descriptor} is a truth-functional combination of atomic descriptors. A composite belief descriptor (henceforth: descriptor; denoted by upper-case Greek letters) is a set of molecular descriptors. \n\n$\\mathfrak{B}\\varphi$ is \\emph{satisfied} by a belief set $X$, if and only if $\\varphi \\in X$. Conditions of satisfaction for molecular descriptors are defined inductively, hence, provided that $\\varphi$ and $\\psi$ stand for molecular descriptors, $X$ satisfies $\\neg \\varphi$ if and only if it does not satisfy $\\varphi$, it satisfies $\\varphi \\wedge \\psi$ if and only if it satisfies both $\\varphi$ and $\\psi$, etc. It satisfies a composite descriptor $\\Phi$ if and only if it satisfies all its elements. $X \\Vdash \\Phi$ denotes that set $X$ satisfies descriptor $\\Phi$.\n\n\nDescriptor revision on a belief set $K$ is performed with a unified operator $\\circ$ such that $K \\circ \\Phi$ is an operation with $\\Phi$ as its success condition. \\cite{hansson_descriptor_2013} introduces several constructions for descriptor revision operations, of which the relational model defined as follows has a canonical status.\n\n\\begin{DEF}[\\cite{hansson_descriptor_2013}]\\label{definition of relational model}\n\n\n$(\\mb{X}, \\leqq)$ is a relational select-direct model (in short: relational model) with respect to $K$ if and only if it satisfies:\\footnote{\\, We will drop the phrase ``with respect to $K$'' if this does not affect the understanding, and write $\\set{\\varphi}$ and $\\mini{\\varphi}$ instead of $\\set{\\{\\mathfrak{B} \\varphi\\}}$ and $\\mini{\\{\\mathfrak{B} \\varphi\\}}$ for simplicity.}\n\\begin{enumerate}\n\\item[]$(\\mb{X} 1)$ $\\mb{X}$ is a set of belief sets.\n\\item[]$(\\mb{X} 2)$ $K \\in \\mb{X}$.\n\\item[]$(\\leqq1)$ $K \\leqq X$ for every $X \\in \\mb{X}$.\n\\item[]$(\\leqq2)$ For any $\\Phi$, if $\\{X \\in \\mb{X} \\mid X \\Vdash \\Phi \\}$ (we denote it as $ \\set{\\Phi}$) is not empty, then it has a unique $\\leqq$-minimal element denoted by $\\mini{\\Phi}$.\n\\end{enumerate}\n\nA descriptor revision $\\circ$ on $K$ is \\emph{based on} (or \\emph{determined by}) some relational model $(\\mathbb{X},\\leqq)$ with respect to $K$ if and only if for any $\\Phi$,\n\n\\begin{eqnarray}\\nonumber\n\\mbox{$\\langle \\leqq$ $\\mr{to}$ $\\circ \\rangle$}\\footnotemark \\,\\,\\,\\,\\,\\,\\, K \\circ \\Phi=\n \\begin{cases}\n \\mini{\\Phi} &\\mbox{if $\\set{\\Phi}$ is not empty,}\\\\\n K &\\mbox{otherwise.}\n \\end{cases}\n\\end{eqnarray}\n\n\\footnotetext{\\, Provided that $(\\mathbb{X},\\leqq)$ is a relational model, $\\mb{X} $ is equivalent to the domain of $\\leqq$ since $K \\in \\mb{X}$ and $K \\leqq X$ for all $X \\in \\mb{X}$. So $\\leqq$ in itself can represent the $(\\mathbb{X},\\leqq)$ faithfully.}\n\n\\end{DEF}\n\n\n$\\mb{X}$ could be seen as an \\textit{outcome set} which includes all the potential outcomes under various belief change patterns. The ordering $\\leqq$ (with the strict part $<$) brings out a direct-selection mechanism, which selects the final outcome among candidates satisfying a specific success condition. Given condition $(\\leqq2)$, this sort of selection is achievable for any success condition satisfiable in $\\mb{X}$. We call descriptor revision constructed in this way \\emph{relational} descriptor revision.\n\nAs \\cite{zhang_believability_2017} pointed out, in so far as the selection mechanism is concerned, descriptor revision is at a more abstract level comparing with the AGM revision. In the construction of descriptor revision $\\circ$, ``it assumes that there exists an outcome set which contains all the potential outcomes of the operation $\\circ$, but it says little about what these outcomes should be like'' \\cite[p. 41]{zhang_believability_2017}. In contrast, in the AGM framework, the belief change is supposed to satisfy the principle of consistency preservation and the principle of the informational\neconomy \\cite{gardenfors_knowledge_1988}. Therefore, the intersection step in the construction of belief change in the AGM framework becomes dispensable in the context of descriptor revision. This may explain why the descriptor revision model could be a select-direct approach.\n\n\n\\subsection{Choice revision constructed from descriptor revision}\n\nThe success condition for choice revision $\\ast_c$ with a finite input could be easily expressed by descriptor $\\{ \\mathfrak{B} \\varphi_0 \\vee \\cdots \\vee \\mathfrak{B} \\varphi_n \\}$. So, it is straightforward to construct choice revision through descriptor revision as follows.\n\n\\begin{DEF}[\\cite{hansson_descriptor_2017}]\nLet $\\circ$ be some descriptor revision. A choice revision $\\ast_c$ on $K$ is \\emph{based on} (or \\emph{determined by}) $\\circ$ if and only if for any finite set $A$, \n\n\\begin{eqnarray}\\nonumber\n\\mbox{$\\langle \\circ$ $\\mr{to}$ $\\cro \\rangle$} \\,\\,\\,\\,\\,\\,\\, K \\ast_c A= \n\\begin{cases}\nK \\circ \\{ \\mathfrak{B} \\varphi_0 \\vee \\cdots \\vee \\mathfrak{B} \\varphi_n \\} & \\mbox{if $A = \\{\\varphi_0 , \\cdots , \\varphi_n \\} \\neq \\emptyset$},\\\\\nK & \\mbox{otherwise}.\n\\end{cases}\n\\end{eqnarray}\n\n\\end{DEF}\n\nHenceforth, we say $\\ast_c$ is \\emph{based on} (or \\emph{determined by}) some relational model if it is based on the descriptor revision determined by the same model. The main purpose of this section is to investigate the formal properties of choice revision based on such models.\n\n\\subsection{Postulates and representation theorem}\n\nIt is observed that the choice revision determined by relational models should satisfy a set of arguably plausible postulates on choice revision. \n\n\\begin{OBS}\\label{Observation on the postulates satisfied by the choice revision constructed from relational model}\nLet $\\ast_c$ be a choice revision determined by any relational descriptor revision $(\\mathbb{X},\\leqq)$. Then it satisfies the following postulates: \n\\begin{enumerate}\n\\item[] $\\mathrm{(\\ast_c 1)}$ $\\cons{K \\ast_c A} = K \\ast_c A$. (\\textbf{$\\ast_c$-closure})\n\\item[] $\\mathrm{(\\ast_c 2)}$ $K \\ast_c A = K$ or $A \\cap (K \\ast_c A) \\neq \\emptyset$. (\\textbf{$\\ast_c$-relative success})\n\\item[] $\\mathrm{(\\ast_c 3)}$ If $A \\cap (K \\ast_c B) \\neq \\emptyset$, then $A \\cap (K \\ast_c A) \\neq \\emptyset$. (\\textbf{$\\ast_c$-regularity}) \n\\item[] $\\mathrm{(\\ast_c 4)}$ If $A \\cap K \\neq \\emptyset$, then $K \\ast_c A = K$. (\\textbf{$\\ast_c$-confirmation})\n\\item[] $\\mathrm{(\\ast_c 5)}$ If $(K \\ast_c A) \\cap B \\neq \\emptyset$ and $(K \\ast_c B) \\cap A \\neq \\emptyset$, then $K \\ast_c A = K \\ast_c B$. (\\textbf{$\\ast_c$-reciprocity})\n\\end{enumerate}\n\\end{OBS}\n\nMoreover, another plausible condition on choice revision follows from this set of postulates. \n\n\\begin{OBS}\\label{Observation on cro-syntax irrelevance}\nIf $\\ast_c$ satisfies $\\ast_c$-closure, relative success, regularity and reciprocity, then $\\ast_c$ satisfies:\n\\begin{enumerate}\n\\item[] If $A \\equiv B$, then $K \\ast_c A = K \\ast_c B$. (\\textbf{$\\ast_c$-syntax irrelevance})\n\\end{enumerate}\n\\end{OBS}\n\nIt is easy to see that the postulates in above are natural generalizations of the following postulates on sentential revision:\n\n\\begin{enumerate}\n\\item[] $\\mathrm{(\\ast 1)}$ $\\cn(K \\ast \\varphi)=K \\ast \\varphi$ \\textit{\\textbf{($\\ast$-closure)}}\n\\item[] $\\mathrm{(\\ast 2)}$ If $K \\ast \\varphi \\neq K $, then $\\varphi \\in K \\ast \\varphi$ \\textit{\\textbf{($\\ast$-relative success)}}\n\\item[] $\\mathrm{(\\ast 3)}$ If $\\varphi \\in K $, then $K \\ast \\varphi = K$ \\textit{\\textbf{($\\ast$-confirmation)}}\n\\item[] $\\mathrm{(\\ast 4)}$ If $\\psi \\in K \\ast \\varphi$, then $\\psi \\in K \\ast \\psi$ \\textit{\\textbf{($\\ast$-regularity)}}\n\\item[] $\\mathrm{(\\ast 5)}$ If $\\psi \\in K \\ast \\varphi$ and $\\varphi \\in K \\ast \\psi$, then $K\\ast \\varphi = K \\ast \\psi$ \\textit{\\textbf{($\\ast$-reciprocity)\\footnote{\\, This postulate is first discussed in \\cite{alchourron1982logic} in the context of maxichoice revision.}}}\n\\end{enumerate}\nand \n\\begin{enumerate}\n\\item[] If $\\varphi \\dashv \\Vdash \\psi$, then $K\\ast \\varphi = K \\ast \\psi$. \\footnote{\\, It is easy to check that $\\ast$-extensionality is derivable from $(\\ast 1)$, $(\\ast 2)$, $(\\ast 3)$ and ($\\ast 5$).} \\textit{\\textbf{($\\ast$-extensionality)}}\n\\end{enumerate}\n\n\nThe above postulates on choice revision are as intuitively plausible as their correspondents on sentential revision, except that the meaning of $\\ast_c$-reciprocity seems not so transparent as that of $\\ast$-reciprocity. However, given some weak conditions, we can show that the $\\ast_c$-reciprocity postulate is equivalent to a more understandable condition as follows.\n\n\\begin{OBS}\\label{Observation that reciprocity is equivalent to cautiousness}\nLet choice operation $\\ast_c$ satisfy $\\ast_c$-relative success and \\linebreak $\\ast_c$-regularity. Then it satisfies $\\ast_c$-reciprocity iff it satisfies:\n\\begin{enumerate}\n\\item[] If $A \\subseteq B$ and $(K \\ast_c B) \\cap A \\neq \\emptyset$, then $K \\ast_c A =K \\ast_c B$. (\\textbf{$\\ast_c$-cautiousness})\n\\end{enumerate} \n\\end{OBS}\n\n\\noindent The postulate $\\ast_c$-cautiousness reflects a distinctive characteristic of choice revision modelled by relational models: The agent who performs this sort of belief change is cautious in the sense of only adding the smallest possible part of the new information to her original beliefs. It follows immediately from $\\ast_c$-relative success and $\\ast_c$-cautiousness that if $A \\cap (K \\ast_c A) \\neq \\emptyset$, then $K \\ast_c A = K \\ast_c \\{\\varphi\\}$ for some $\\varphi \\in A$. Thus, it is not surprising that the following postulate follows.\n\n\\begin{OBS}\\label{Observation that dichotomy can be derived from reciprocity}\nIf $\\ast_c$ satisfies $\\ast_c$-relative success, regularity and reciprocity, then $\\ast_c$ satisfies:\n\\begin{enumerate}\n\\item[] $K \\ast_c (A \\cup B) = K \\ast_c A$ or $K \\ast_c (A \\cup B) = K \\ast_c B$. (\\textbf{$\\ast_c$-dichotomy})\n\\end{enumerate}\n\\end{OBS}\n\n\n\n\\noindent In contrast to $(\\ast_c 1)$ through $(\\ast_c 5)$, postulates $\\ast_c$-cautiousness and $\\ast_c$-dichotomy do not have directly corresponding postulates in the context of sentential revision. This suggests that though $(\\ast_c 1)$ through $(\\ast_c 5)$ naturally generalize $(\\ast 1)$ through $(\\ast 5)$, this sort of generalization is not so trivial as we may think of. As another evidence for this, the following observation shows that the properties of $(\\ast 1)$ through $(\\ast 5)$ and those of their generalizations are not always paralleled.\n\n\\begin{OBS}\\label{Observation that reciprocity is equivalent to strong reciprocity}\nLet $\\ast_c$ satisfy $\\ast_c$-regularity. Then it satisfies $\\ast_c$-reciprocity iff it satisfies \n\\begin{enumerate}\n\\item[] For any $n \\geq 1 $, if $(K \\ast_c A_1 ) \\cap A_0 \\neq \\emptyset$, $\\cdots$, $(K \\ast_c A_{n} ) \\cap A_{n-1} \\neq \\emptyset$, $(K \\ast_c A_{0} ) \\cap A_{n} \\neq \\emptyset$, then $K \\ast_c A_0 = K \\ast_c A_1 = \\cdots = K \\ast_c A_n$. (\\textbf{$\\ast_c$-strong reciprocity})\n\\end{enumerate}\n\\end{OBS}\n\n\\noindent $\\ast$-strong reciprocity is a generalization of the following postulate on sentential revision:\n\n\\begin{enumerate}\n\\item[] For any $n \\geq 1 $, if $\\varphi_0 \\in K \\star \\varphi_{1}$, $\\cdots$, $\\varphi_{n-1} \\in K \\ast \\varphi_n$ and $\\varphi_n \\in K \\star \\varphi_0 $, then $K \\star \\varphi_0 = K \\star \\varphi_2= \\cdots = K \\star \\varphi_n$. (\\textbf{$\\ast$-strong reciprocity} )\\footnote{\\, $\\ast$-strong reciprocity \nis closely related to a non-monotonic reasoning rule named ``loop'' which is first introduced in \\cite{kraus_nonmonotonic_1990}. For more discussion on this, see \\cite{makinson_relations_1991}.}\n\\end{enumerate}\n\n\\noindent However, in contrast to the result in Observation \\ref{Observation that reciprocity is equivalent to strong reciprocity}, $\\ast$-strong reciprocity is not derivable from $(\\ast 1)$ through $(\\ast 5)$.\\footnote{\\, To see this, let $K = \\conp{\\taut}$ and revision operation $\\ast$ on $K$ defined as: (i) if $p_0 \\wedge p_1 \\vdash \\varphi$ and $\\varphi \\vdash p_0$, then $K \\ast \\varphi = \\conp{p_0 \\wedge p_1}$; (ii) if $p_1 \\wedge p_2 \\vdash \\varphi$ and $\\varphi \\vdash p_1$, then $K \\ast \\varphi = \\conp{p_1 \\wedge p_2}$; (iii) if $p_0 \\wedge p_2 \\vdash \\varphi$ and $\\varphi \\vdash p_2$, then $K \\ast \\varphi = \\conp{p_0 \\wedge p_2}$; (iv) otherwise, $K \\ast \\varphi = \\conp{\\varphi}$. It is easy to check that $\\ast$ satisfies $(\\ast 1)$ through $(\\ast 5)$ but not $\\ast$-strong reciprocity.}\n\nAfter an investigation on the postulates $(\\ast_c 1)$ through $(\\ast_c 5)$ satisfied by choice revision based on rational models, the question raises naturally whether the choice revision could be axiomatically characterized by this set of postulates. We get a partial answer to this question: a representation theorem is obtainable when $\\mathcal{L}$ is finite.\n\n\\begin{THE}\\label{Representation theorem for choice revision derived from descriptor revision of finite language}\nLet $\\mathcal{L}$ be a finite language. Then, $\\ast_c$ satisfies $(\\ast_c 1)$ through $(\\ast_c 5)$ iff it is a choice revision based on some relational model.\n\\end{THE}\n\n\\subsection{More properties of choice revision}\n\nIn this subsection, we will study additional properties of choice revision from the point of view of postulates. The postulates introduced in the previous subsection do not necessarily cover all the reasonable properties of this operation. In what follows we are going to investigate some additional ones, in particular, the following:\n\n\\begin{enumerate}\n\\item[] If $A \\neq \\emptyset$, then$A \\cap (K \\ast_c A) \\neq \\emptyset$. (\\emph{\\textbf{$\\ast_c$-success}})\n\\item[] If $A \\not \\equiv \\{{\\scriptstyle \\perp}\\}$, then $K \\ast_c A \\nvdash {\\scriptstyle \\perp}$. (\\emph{\\textbf{$\\ast_c$-consistency}})\n\\end{enumerate}\n\nTo some extent, $\\ast_c$-success is a strengthening of $\\ast_c$-relative success and $\\ast_c$-regularity, but it does not say anything about the limiting case in which the input is empty. To cover this limiting case, we need the following postulate:\n\n\\begin{enumerate}\n\\item[] If $A = \\emptyset$, then $K \\ast_c A = K$. (\\textbf{$\\ast_c$-vacuity})\n\\end{enumerate}\n\nThe interrelations among $\\ast_c$-success, $\\ast_c$-relative success and \\linebreak $\\ast_c$-regularity are summarized as follows.\n\n\\begin{OBS}\\label{Observation on the inter-derivability among success, relative , vacuity and regularity }\nLet $\\ast_c$ be some choice revision on $K$.\n\\begin{enumerate}\n\\item If $\\ast_c$ satisfies relative success, then it satisfies vacuity.\n\\item If $\\ast_c$ satisfies success and vacuity, then it satisfies relative success.\n\\item If $\\ast_c$ satisfies success, then it satisfies regularity.\n\\end{enumerate}\n\\end{OBS}\n\n$\\ast_c$-consistency is a plausible constraint on a rational agent. While accepting $\\ast_c$-success and $\\ast_c$-consistency as ``supplementary'' postulates for choice revision $\\ast_c$, the corresponding relational model on which $\\ast_c$ is based will also need to satisfy some additional properties. We use the following representation theorem to conclude this subsection.\n\n\\begin{THE}\\label{Representation thoerem for choice revision additionally satisfying success and consistency }\nLet $\\mathcal{L}$ be a finite language and $\\ast_c$ some revision operation on $K \\subseteq \\mathcal{L}$. Then, $\\ast_c$ satisfies $\\ast_c$-closure, $\\ast_c$- success, $\\ast_c$-vacuity, $\\ast_c$-confirmation, $\\ast_c$-reciprocity and $\\ast_c$-consistency iff \n it is a choice revision determined by some relational model which satisfies the following two condition:\n\\begin{enumerate}\n\\item[] $(\\mb{X} 3)$ $\\conp{{\\scriptstyle \\perp}} \\in \\mb{X}$; \n\\item[] $(\\leqq 3)$ $\\set{\\mathfrak{B} \\varphi} \\neq \\emptyset$ and $\\mini{\\mathfrak{B} \\varphi } < \\conp{{\\scriptstyle \\perp}}$ for every $\\varphi \\not \\vdash {\\scriptstyle \\perp}$.\n\\end{enumerate}\n\\end{THE}\n\n\\section{An alternative modelling for choice revision}\\label{section on an alternative modelling for choice revision}\n\nIn this section, we propose an alternative modelling for choice revision, which is based on so-called \\textit{multiple believability relations}. A believability relation $\\preceq$ is a binary relation on sentences of $\\mathcal{L}$. Intuitively, $\\varphi \\preceq \\psi$ means that the subject is at least as prone to believing $\\varphi$ as to believing $\\psi$.\\footnote{\\, For more detailed investigation on believability relations, including its relationship with the epistemic entrenchment relation introduced in \\cite{gardenfors_revisions_1988}, see \\cite{hansson_relations_2014} and \\cite{zhang_believability_2017}.} We can generalize $\\preceq$ to a multiple believability relation $\\preceq_{\\ast}$ which is a binary relation on the set of all finite subsets of $\\mathcal{L}$ satisfying:\n\n\\begin{enumerate}\n\\item[] \\mbox{$\\langle \\preceq_{\\ast}$ $\\mr{to}$ $\\preceq \\rangle$} \\,\\,\\,\\,\\,\\,$\\varphi \\preceq \\psi$ iff $\\{\\varphi\\} \\preceq_{\\ast} \\{\\psi\\}$.\n\\end{enumerate}\n\n\\noindent This kind of generalization can be done in different ways, and at least two distinct relations can be obtained, namely \\textit{package multiple believability relations}, denoted by $\\preceq_{p}$, and \\textit{choice multiple believability relations}, denoted by $\\preceq_{c}$ (with symmetric part $\\simeq_{c}$ and strict part $\\prec_{c}$). Intuitively, $A \\preceq_{p} B$ means that it is easier for the subject to believe all propositions in $A$ than to believe all propositions in $B$ and $A \\preceq_{c} B$ means that it is easier for the subject to believe some proposition in A than to believe some proposition in $B$. \n\n\\noindent $\\preceq_{p}$ is of little interest since $A \\preceq_{p} B$ can be immediately reduced to $\\& A \\preceq \\& B$, given that $A$ and $B$ are finite. In what follows, multiple believability relations (or multi-believability relations for short) only refer to choice multiple believability relations $\\preceq_{c}$. ($\\{\\varphi\\} \\preceq_{c} A$ will be written as $\\varphi \\preceq_{c} A$ for simplicity.) \n\n\\subsection{Postulates on multi-believability relations}\n\nRecall the following postulates on believability relations $\\preceq$ introduced in \\cite{zhang_believability_2017}:\n\\begin{enumerate}\n\\item[] \\emph{\\textbf{$\\preceq$-transitivity}}: If $\\varphi \\preceq \\psi$ and $\\psi \\preceq \\lambda$, then $\\varphi \\preceq \\lambda$. \n\\item[] \\emph{\\textbf{$\\preceq$-weak coupling}}: If $\\varphi \\simeq \\varphi \\wedge \\psi $ and $\\varphi \\simeq \\varphi \\wedge \\lambda$, then $\\varphi \\simeq \\varphi \\wedge (\\psi \\wedge \\lambda)$.\n\\item[] \\emph{\\textbf{$\\preceq$-coupling}}: If $\\varphi \\simeq \\psi$, then $\\varphi \\simeq \\varphi \\wedge \\psi$.\n\\item[] \\emph{\\textbf{$\\preceq$-counter dominance}}: If $\\varphi \\vdash \\psi$, then $\\psi \\preceq \\varphi$.\n\\item[] \\emph{\\textbf{$\\preceq$-minimality}}: $\\varphi \\in K$ if and only if $\\varphi \\preceq \\psi$ for all $\\psi$.\n\\item[] \\emph{\\textbf{$\\preceq$-maximality}}: If $\\psi \\preceq \\varphi$ for all $\\psi$, then $\\varphi \\equiv {\\scriptstyle \\perp}$.\n\\item[] \\emph{\\textbf{$\\preceq$-completeness}}: $\\varphi \\preceq \\psi$ or $\\psi \\preceq \\varphi$\n\\end{enumerate} \n\nTransitivity is assumed for almost all orderings. In virtue of the intuitive meaning of believability relation, $\\varphi \\simeq \\varphi \\wedge \\psi$ represents that the agent will accept $\\psi$ in the condition of accepting $\\varphi$. Thus, the rationale for $\\preceq$-weak coupling is that if the agent will consequently add $\\psi$ and $\\lambda$ to her beliefs when accepting $\\varphi$, then she also adds the conjunction of them to her beliefs in this case. This is reasonable if we assume that the beliefs of the agent are closed under the consequence operation. The justification of $\\preceq$-counter dominance is that if $\\varphi$ logically entails $\\psi$, then it will be a smaller change and hence easier for the agent to accept $\\psi$ rather than to accept $\\varphi$, because then $\\psi$ must be added too, if we assume that the beliefs of the agent are represented by a belief set. $\\preceq$-coupling is a strengthening of $\\preceq$-weak coupling.\\footnote{\\, It is easy to see that $\\preceq$-coupling implies $\\preceq$-weak coupling, provided that $\\preceq$-transitivity and $\\preceq$-counter dominance hold. } It says that if $\\varphi$ is equivalent to $\\psi$ in believability, then the agent will consequently add $\\psi$ to her beliefs in case of accepting $\\varphi$ and vice versa. $\\preceq_{c}$-minimality is justifiable since nothing needs to be done to add $\\varphi$ to $K$ if it is already in $K$. $\\preceq$-maximality is justifiable since it is reasonable to assume that it is strictly more difficult for a rational agent to accept ${\\scriptstyle \\perp}$ than to accept any non-falsum. $\\preceq$-completeness seems a little bit strong. It says that all pairs of sentences are comparable in believability. In accordance with \\cite{zhang_believability_2017}, we call relations satisfying all these postulates \\textit{quasi-linear believability relations}.\n\nWe can generalize these postulates on believability relations in a natural way to postulates multi-believability relations as follows:\\footnote{\\, In what follows, it is always assumed that all sets $A$ and $B$ and $C$ mentioned in postulates on multi-believability relations are finite sets.}\n\n\\begin{enumerate}\n\\item[] \\emph{\\textbf{$\\preceq_{c}$-transitivity}}: If $A \\preceq_{c} B$ and $B \\preceq_{c} C$, then $A \\preceq_{c} C$.\n\\item[] \\emph{\\textbf{$\\preceq_{c}$-weak coupling}}: If \\( {A \\simeq_{c} A \\owedge B} \\) and \\( {A \\simeq_{c} A \\owedge C} \\), then $A \\simeq_{c} A \\owedge B \\owedge C$.\n\\item[] \\emph{\\textbf{$\\preceq_{c}$-coupling}}: If $A \\simeq_{c} B$, then $A \\simeq_{c} A \\owedge B$. \n\\item[] \\emph{\\textbf{$\\preceq_{c}$-counter dominance}}: If for every $\\varphi \\in B$ there exists $\\psi \\in A$ such that $\\varphi \\vdash \\psi$, then $A \\preceq_{c} B$. \n\\item[] \\emph{\\textbf{$\\preceq_{c}$-minimality}}: $A \\preceq_{c} B$ for all $B$ if and only if $A \\cap K \\neq \\emptyset$. \n\\item[] \\emph{\\textbf{$\\preceq_{c}$-maximality}}: If $B$ is not empty and $A \\preceq_{c} B$ for all non-empty $A$, then $B \\equiv \\{{\\scriptstyle \\perp}\\}$. \n\\item[] \\emph{\\textbf{$\\preceq_{c}$-completeness}}: $A \\preceq_{c} B$ or $B \\preceq_{c} A$.\n\\end{enumerate}\n\n\\noindent These postulates on multi-believability relations can be understood in a similar way that their correspondents on believability relations are understood. \n\nFurthermore, we propose the following two additional postulates on multi-believability relations:\n\n\\begin{enumerate}\n\\item[] \\emph{\\textbf{$\\preceq_{c}$-determination}}: $A \\prec_{c} \\emptyset$ for every non-empty $A$.\n\\item[] \\emph{\\textbf{$\\preceq_{c}$-union}}: $A \\preceq A \\cup B$ or $B \\preceq A \\cup B$.\n\\end{enumerate}\n\n\\noindent At least on the surface, these two could not be generalizations of any postulate on believability relation. In some sense the meaning of \\linebreak $\\preceq_{c}$-determination is correspondent to that of $\\ast_c$-success, since if it is a \\linebreak strictly smaller change for the agent to accept some sentences from a non-empty $A$ rather than to take some sentences from the empty set, which is obviously impossible, then it seems to follow that the agent will successfully add some sentences in $A$ to her original beliefs when exposed to the new information represented by $A$, and vice versa. Similarly, there is an obvious correspondence between the forms and meanings of $\\preceq_{c}$-union and $\\ast_c$-dichotomy. They both suggest that to partially accept a non-empty $A$ is equivalent to accept some single sentence in $A$. This is plausible if we assume that the agent is extremely cautious to the new information.\n\n\n\\begin{OBS}\\label{Observation on the interderivability among postulates on multiple believability relations}\nLet $\\preceq_{c}$ be some multi-believability relation satisfying $\\preceq_{c}$-transitivity and $\\preceq_{c}$-counter dominance. If it satisfies $\\preceq_{c}$-union in addition, then\n\\begin{enumerate}\n\\item It satisfies $\\preceq_{c}$-completeness.\n\\item It satisfies $\\preceq_{c}$-weak coupling iff it $\\preceq_{c}$-satisfies coupling.\n\\end{enumerate} \n\\end{OBS}\n\n\\noindent Observation \\ref{Observation on the interderivability among postulates on multiple believability relations} indicates that $\\preceq_{c}$-union is strong. It should be noted that for a believability relation, neither $\\preceq$-completeness nor $\\preceq$-coupling can be derived from $\\preceq$-transitivity, $\\preceq$-counter dominance and $\\preceq$-weak coupling.\n\nIn what follows, we name multi-believability relations satisfying all the above postulates \\textit{standard multi-believability relations}. \n\n\\subsection{Translations between believability relations and multiple believability relations}\nIn this subsection, we will show that although it is impossible to find a postulate on believability relations that corresponds to $\\preceq_{c}$-determination or $\\preceq_{c}$-union, there exists a translation between quasi-linear believability relations and standard multi-believability relations.\n\n\\begin{OBS}\\label{observation for reduction to single sentence}\nLet $\\preceq_{c}$ satisfy $\\preceq_{c}$-determination, $\\preceq_{c}$-transitivity and $\\preceq_{c}$-counter dominance. Then, for any non-empty finite sets $A$ and $B$,\n\\begin{enumerate}\n\\item $A \\preceq_{c} B$ if and only if there exists $\\varphi \\in A$ such that $\\varphi \\preceq_{c} B$.\n\\item $A \\preceq_{c} B$ if and only if $A \\preceq_{c} \\varphi$ for all $\\varphi \\in B$.\n\\end{enumerate}\n\\end{OBS}\n\n\\noindent This observation suggests that $\\preceq$ and $\\preceq_{c}$ can be linked through the following two transitions:\n\\begin{enumerate}\n\\item[] \\mbox{$\\langle \\preceqc$ $\\mr{to}$ $\\preceq \\rangle$} \\,\\,\\,\\,\\,\\,$\\varphi \\preceq \\psi$ iff $\\{\\varphi\\} \\preceq_{c} \\{\\psi\\}$.\n\\item[] \\mbox{$\\langle \\preceq$ $\\mr{to}$ $\\preceqc \\rangle$} \\,\\,\\,\\,\\,\\,$A \\preceq_{c} B$ iff $B = \\emptyset$ or there exists $\\varphi \\in A$ such that $\\varphi \\preceq \\psi$ for every $\\psi \\in B$.\n\\end{enumerate}\n\n\\noindent This is confirmed by the following theorem.\n\n\\begin{THE}\\label{Theorem on the translation between single and multi believability relations}\n\\begin{enumerate}\n\\item If $\\preceq$ is a quasi-linear believability relation and $\\preceq_{c}$ is constructed from $\\preceq$ through the way of \\mbox{$\\langle \\preceq$ $\\mr{to}$ $\\preceqc \\rangle$}, then $\\preceq_{c}$ is a standard multi-believability relation and $\\preceq$ can be retrieved from $\\preceq_{c}$ in the way of \\mbox{$\\langle \\preceqc$ $\\mr{to}$ $\\preceq \\rangle$}.\n\\item If $\\preceq_{c}$ is a standard multi-believability relation and $\\preceq$ is constructed from $\\preceq_{c}$ through \\mbox{$\\langle \\preceqc$ $\\mr{to}$ $\\preceq \\rangle$}, then $\\preceq$ is a quasi-linear believability relation and $\\preceq_{c}$ can be retrieved from $\\preceq$ through \\mbox{$\\langle \\preceq$ $\\mr{to}$ $\\preceqc \\rangle$}.\n\\end{enumerate}\n\\end{THE}\n\n\n\\subsection{Choice revision constructed from multi-believability relations}\n\nNow we turn to the construction of choice revision through \\linebreak multi-believability relations. Recall that a sentential revision $\\ast$ can be constructed from a believability relation $\\preceq$ in this way \\cite{zhang_believability_2017}:\n\\begin{eqnarray}\\nonumber\n\\mbox{$\\langle \\preceq$ $\\mr{to}$ $\\ast \\rangle$} \\,\\,\\,\\,\\,\\,\\,K \\ast \\varphi= \\{ \\psi \\mid \\varphi \\simeq \\varphi \\wedge \\psi \\} \n\\end{eqnarray} \n\n\\noindent As we have explained, $\\varphi \\simeq \\varphi \\wedge \\psi$ could be understood as that the agent will consequently accept $\\psi$ in case of accepting $\\varphi$. So, the set $\\{ \\psi \\mid \\varphi \\simeq \\varphi \\wedge \\psi \\} $ is just the agent's new set of beliefs after she performed belief revision with input $\\varphi$. Thus, we can similarly construct choice revision from multi-believability relations in the following way:\n\n\\begin{DEF}\\label{definition of choice revision based on multi-believability relations}\nLet $\\preceq_{c}$ be some multi-believability relation. A choice revision $\\ast_c$ on $K$ is based on (or determined by) $\\preceq_{c}$ if and only if: for any finite $A$, \n\\begin{eqnarray}\\nonumber\n\\mbox{$\\langle \\preceqc$ $\\mr{to}$ $\\cro \\rangle$} \\,\\,\\,\\,\\,\\,\\, K \\ast_c A= \n\\begin{cases}\n\\{\\varphi \\mid A \\simeq_{c} A \\owedge \\varphi \\} & \\mbox{If $A \\prec_{c} \\emptyset$},\\\\\nK & \\mbox{otherwise}.\n\\end{cases}\n\\end{eqnarray}\n\\end{DEF}\n\nThe primary results of this section are the following two representation theorems. Comparing with Theorems \\ref{Representation theorem for choice revision derived from descriptor revision of finite language} and \\ref{Representation thoerem for choice revision additionally satisfying success and consistency }, these two theorems are applicable to more general cases since they do not assume that the language $\\mathcal{L}$ is finite. These two theorems demonstrate that multi-believability relations provide a fair modelling for choice revision characterized by the set of postulates mentioned in Section \\ref{section choice revison based on descriptor revision}.\n\n\\begin{THE}\\label{Representation theorem for choice revision based on weak multiple believability relation}\nLet $\\ast_c$ be some choice revision on $K$. Then, $\\ast_c$ satisfies $(\\ast_c 1)$ through $(\\ast_c 5)$ iff it is determined by some multi-believability relation $\\preceq_{c}$ satisfying $\\preceq_{c}$-transitivity, $\\preceq_{c}$-weak coupling, $\\preceq_{c}$-counter-dominance, $\\preceq_{c}$-minimality \\linebreak and $\\preceq_{c}$-union.\n\\end{THE}\n\n\\begin{THE}\\label{Representation theorem for choice revision based on standard multiple believability relation}\nLet $\\ast_c$ be some choice revision on $K$. Then, $\\ast_c$ satisfies $\\ast_c$-closure, $\\ast_c$- success, $\\ast_c$-vacuity, $\\ast_c$-confirmation, $\\ast_c$-reciprocity and $\\ast_c$-consistency iff it is determined by some standard multi-believability relation.\n\\end{THE}\n\nConsidering the translation between multi-believability relations and believability relations (Theorem \\ref{Theorem on the translation between single and multi believability relations}), it seems that these results can be easily transferred to the context of believability relations. However, if we drop some postulates on multi-believability relation such as $\\preceq_{c}$-determination, the translation between multi-believability relation and believability relation will not be so transparent, at least it will not be so straightforward as \\mbox{$\\langle \\preceq$ $\\mr{to}$ $\\preceqc \\rangle$}{ }and \\mbox{$\\langle \\preceqc$ $\\mr{to}$ $\\preceq \\rangle$}. As a consequence, the result in Theorem \\ref{Representation theorem for choice revision based on weak multiple believability relation} may not be possible to transfer to believability relations in a straightforward way. Moreover, comparing with postulates on believability relations, postulates on multi-believability relations such as $\\preceq_{c}$-determination and $\\preceq_{c}$-union can present our intuitions on choice revision in a more direct way. Thus, the multi-believability relation is still worth to be studied in its own right.\n\n\n\\section{Conclusion and future work}\\label{section conclusion}\n\nAs a generalization of traditional belief revision, choice revision has more realistic characteristics. The new information is represented by a set of sentences and the agent could partially accept these sentences as well as reject the others. From the point of technical view, choice revision is interesting since the standard ``select-and-intersect'' methodology in modellings for belief change is not suitable for it. But instead, it can be modelled by a newly developed framework of descriptor revision, which employs a ``select-direct'' approach. After reviewing the construction of choice revision in the framework of descriptor revision, under the assumption that the language is finite, we provided two sets of postulates as the axiomatic characterizations for two variants of choice revision based on such constructions (in Theorem \\ref{Representation theorem for choice revision derived from descriptor revision of finite language} and \\ref{Representation thoerem for choice revision additionally satisfying success and consistency }). These postulates, in particular, \\linebreak $\\ast_c$-cautiousness and $\\ast_c$-dichotomy, point out that choice revision modelled by descriptor revision has the special characteristic that the agent who performs this sort of belief change is cautious in the sense that she only accepts the new information to the smallest possible extent.\n\nFor AGM revision and contraction, there are various independently motivated modellings which are equivalent in terms of expressive power. In this contribution, we also propose an alternative modelling for choice revision. We showed that multi-believability relations can also construct the choice revision axiomatically characterized by the sets of postulates proposed for choice revision based on descriptor revision (Theorem \\ref{Representation theorem for choice revision based on weak multiple believability relation} and \\ref{Representation theorem for choice revision based on standard multiple believability relation}). Moreover, these results are obtainable without assuming that the language is finite. This may indicate that multi-believability relations are an even more suitable modelling for choice revision.\n\nThe study in this contribution can be developed in at least three directions. First, the cautiousness constraint on choice revision, reflected by $\\ast_c$-cautiousness, certainly could be loosened. We think it is an interesting topic for future work to investigate the modeling and axiomatic characterization of more ``reckless'' variants of choice revision. Secondly, as it was showed in \\cite{zhang_believability_2017} that AGM revision could be reconstructed from believability relations satisfying certain conditions, it is interesting to ask which conditions a multi-believability relation should satisfy so that its generated choice revision coincides with an AGM revision when the inputs are limited to singletons. Finally, it is technically interesting to investigate choice revisions with an infinite input set, though they are epistemologically unrealistic.\n\n\\section*{Appendix: Proofs}\\label{appendix}\n\n\\begin{LEM}\\label{Lemma on the representation element}\nLet $\\preceq_{c}$ be some multiple believability relation which satisfies \\linebreak $\\preceq_{c}$-counter~dominance and $\\preceq_{c}$-transitivity. Then, \n\\begin{enumerate}\n\\item If $\\preceq_{c}$ satisfies $\\preceq_{c}$-union, then for every non-empty $A$, there exists some $\\varphi \\in A$ such that $\\varphi \\simeq_{c} A$.\n\\item For every $\\varphi \\in A$, $A \\simeq_{c} A \\owedge \\varphi$ if and only if $\\varphi \\simeq_{c} A$.\n\\end{enumerate}\n\\end{LEM}\n\n\\begin{proof}[Proof for Lemma \\ref{Lemma on the representation element}:]\n\\textit{1.} We prove this by mathematical induction on the size $n$ ($n \\geq 1$) of $A$. Let $n = 1$, then it follows immediately. Suppose hypothetically that it holds for $n = k$ ($k \\geq 1$). Let $n = k+1$. Since $k \\geq 1$, there exists a non-empty set $B$ containing $k$ elements and a sentence $\\varphi$ such that $A = B \\cup \\{\\varphi\\}$. By $\\preceq_{c}$-counter dominance and $\\preceq_{c}$-union, (i) $A \\simeq_{c} \\{\\varphi\\}$ or (ii) $A \\simeq_{c} B$. The case of (i) is trivial. In the case of (ii), by the hypothetical supposition, there exists some $\\psi \\in B \\subseteq A$ such that $A \\simeq_{c} B \\simeq_{c} \\psi$. So, by $\\preceq_{c}$-transitivity, $A \\simeq_{c} \\varphi$. To sum up (i) and (ii), there always exists some $\\varphi \\in A$ such that $\\varphi \\simeq_{c} A$.\\\\\n\\textit{2. From left to right:} Let $\\varphi \\in A $ and $A \\owedge \\varphi \\simeq_{c} A$. By $\\preceq_{c}$-counter dominance, $A \\preceq_{c} \\varphi$ and $ \\varphi \\preceq_{c} A \\owedge \\varphi$. And it follows from $\\varphi \\preceq_{c} A \\owedge \\varphi$ and $A \\owedge \\varphi \\simeq_{c} A$ that $\\varphi \\preceq_{c} A$ by $\\preceq_{c}$-transitivity. Thus, $\\varphi \\simeq_{c} A$. \\textit{From right to left:} Let $\\varphi \\in A$ and $\\varphi \\simeq A$. By $\\preceq_{c}$-counter-dominance, $A \\owedge \\varphi \\preceq_{c} \\varphi$. So $A \\owedge \\varphi \\preceq_{c} A$ by $\\preceq_{c}$-transitivity. Moreover, $A \\preceq_{c} A \\owedge \\varphi$ by $\\preceq_{c}$-counter-dominance. Thus, $A \\owedge \\varphi \\simeq_{c} A$.\n\\end{proof}\n\n\\begin{proof}[Proof for Observation \\ref{Observation on the postulates satisfied by the choice revision constructed from relational model}:]\nIt is easy to see that $\\ast_c$ satisfies $\\ast_c$-closure and $\\ast_c$-relative success. We only check the remaining three postulates. We let $\\Belsome{A}$ denote the descriptor $ \\{ \\mathfrak{B} \\varphi_0 \\vee \\cdots \\vee \\mathfrak{B} \\varphi_n\\}$ when $A = \\{ \\varphi_0 , \\cdots , \\varphi_n\\} \\neq \\emptyset$.\\\\\n\\textit{$\\ast_c$-regularity:} Let $(K \\ast_c B) \\cap A \\neq \\emptyset$. It follows that $A \\neq \\emptyset$ and $\\set{\\Belsome{A}} \\neq \\emptyset$. So $K \\ast_c A = \\mini{\\Belsome{A}}$ by the definition of $\\ast_c$. Thus, $(K \\ast_c A) \\cap A \\neq \\emptyset$.\\\\\n\\textit{$\\ast_c$-confirmation:} Let $A \\cap K \\neq \\emptyset$. Then $A \\neq \\emptyset $ and $K \\in \\set{\\Belsome{A}}$. It follows from $(\\leqq 1)$ and $(\\leqq 2)$ that $K$ is the unique $\\leqq$-minimal element in $\\set{\\Belsome{A}}$. Thus, $K \\ast_c A = \\mini{\\Belsome{A}} = K$.\\\\ \n\\textit{Reciprocity:} Let $(K \\ast_c A_0 ) \\cap A_1 \\neq \\emptyset$ and $(K \\ast_c A_{1} ) \\cap A_{0} \\neq \\emptyset$. Let $i \\in \\{0,1\\}$. It follows that $A_i \\neq \\emptyset $ and $\\set{\\Belsome(A_i)} \\neq \\emptyset$ and hence $K \\ast_c A_i = \\mini{\\Belsome(A_i)}$ by the definition of $\\ast_c$. So it follows from $\\mini{\\Belsome{A_0}} \\cap A_1 \\neq \\emptyset$ and $\\mini{\\Belsome{A_1}} \\cap A_{0} \\neq \\emptyset$ that $\\mini{\\Belsome{A_0}} \\in \\set{\\Belsome{A_1}}$ and $\\mini{\\Belsome{A_1}} \\in \\set{\\Belsome{A_0}}$ and hence $\\mini{\\Belsome{A_0}} \\leqq \\mini{\\Belsome{A_1}} \\leqq \\mini{\\Belsome{A_0}}$ by $(\\leqq 2)$. Since the minimal element in $\\set{\\Belsome(A_0)}$ is unique by $(\\leqq 2)$, it follows that $\\mini{\\Belsome{A_0}} = \\mini{\\Belsome{A_1}}$, i.e. $ K \\ast_c A_{0} = K \\ast_c A_{1}$.\n\\end{proof}\n\n\\begin{proof}[Proof for Observation \\ref{Observation on cro-syntax irrelevance}:]\nLet $A \\equiv B$. Suppose $A \\cap (K \\ast_c A) = \\emptyset$, then $A \\cap (K \\ast_c B) = \\emptyset$ due to $\\ast_c$-regularity. Hence, $B \\cap (K \\ast_c B) = \\emptyset$ by $\\ast_c$-closure. It follows that $K \\ast_c A = K \\ast_c B = K$ by $\\ast_c$-relative success. Suppose $A \\cap (K \\ast_c A) \\neq \\emptyset$, then $B \\cap (K \\ast_c A) \\neq \\emptyset$ by $\\ast_c$-closure, so $B \\cap (K \\ast_c B) \\neq \\emptyset$ by $\\ast_c$-regularity, and hence $A \\cap (K \\ast_c B) \\neq \\emptyset$ by $\\ast_c$-closure. It follows that $K \\ast_c A = K \\ast_c B$ by $\\ast_c$-reciprocity. Thus, $\\ast_c$ satisfies syntax irrelevance in any case.\n\\end{proof}\n\n\n\\begin{proof}[Proof for Observation \\ref{Observation that reciprocity is equivalent to cautiousness}:]\n\\textit{From left to right:} Let $A \\subseteq B$ and $(K \\ast_c B) \\cap A \\neq \\emptyset$. Then, $A \\neq \\emptyset $ and hence $ (K \\ast_c A) \\cap A \\neq \\emptyset$ by $\\ast_c$-regularity. Since $A \\subseteq B$, it follows that $(K \\ast_c A) \\cap B \\neq \\emptyset$. Thus, $K \\ast_c A =K \\ast_c B$ by $\\ast_c$-reciprocity.\\\\ \n\\textit{From right to left:} Let $A \\cap (K \\ast_c B) \\neq \\emptyset$ and $B \\cap (K \\ast_c A) \\neq \\emptyset$. It follows that $(A \\cup B) \\cap (K \\ast_c B) \\neq \\emptyset$. By $\\ast_c$-regularity, it follows that $(A \\cup B) \\cap (K \\ast_c (A \\cup B)) \\neq \\emptyset$. So $A \\cap (K \\ast_c (A \\cup B)) \\neq \\emptyset$ or $B \\cap (K \\ast_c (A \\cup B)) \\neq \\emptyset$. Without loss of generality, let $A \\cap (K \\ast_c (A \\cup B)) \\neq \\emptyset$, then $K \\ast_c A = K \\ast_c (A \\cup B)$ by $\\ast_c$-cautiousness. It follows that $B \\cap (K \\ast_c (A \\cup B)) = B \\cap (K \\ast_c A) \\neq \\emptyset$ and hence $K \\ast_c B = K \\ast_c (A \\cup B)$ by $\\ast_c$-cautiousness. So $K \\ast_c A = K \\ast_c (A \\cup B) = K \\ast_c B$.\n\\end{proof}\n\n\\begin{proof}[Proof for Observation \\ref{Observation that dichotomy can be derived from reciprocity}:]\nSuppose $(A \\cup B) \\cap (K \\ast_c (A\\cup B)) = \\emptyset$, then $(A \\cup B) \\cap (K \\ast_c A) = (A \\cup B) \\cap (K \\ast_c B)= \\emptyset$ by $\\ast_c$-regularity. So $A \\cap (K \\ast_c A) = B \\cap (K \\ast_c B) = \\emptyset$ and hence $K \\ast_c (A \\cup B) = K \\ast_c A = K \\ast_c B = K$ by $\\ast_c$-relative success. Suppose $(A \\cup B) \\cap (K \\ast_c (A\\cup B)) \\neq \\emptyset$, then $A \\cap (K \\ast_c (A\\cup B)) \\neq \\emptyset $ or $B \\cap (K \\ast_c (A\\cup B)) \\neq \\emptyset$. Let $A \\cap (K \\ast_c (A\\cup B)) \\neq \\emptyset $, then $A \\cap (K \\ast_c A) \\neq \\emptyset$ by $\\ast_c$-regularity and hence $(A \\cup B) \\cap (K \\ast_c A) \\neq \\emptyset$. It follows that $K \\ast_c (A \\cup B) = K \\ast_c A$ by $\\ast_c$-reciprocity. Similarly, we can show that $K \\ast_c (A \\cup B) = K \\ast_c B$ holds in the case of $B \\cap (K \\ast_c (A\\cup B)) \\neq \\emptyset$. Thus, $\\ast_c$ satisfies $\\ast_c$-dichotomy in any case.\n\\end{proof}\n\n\\begin{proof}[Proof for Observation \\ref{Observation that reciprocity is equivalent to strong reciprocity}:]\n\\textit{From right to left:} It follows immediately.\\\\\n\\textit{From left to right:} Assume $(\\star)$ that $(K \\ast_c A_1 ) \\cap A_0 \\neq \\emptyset$, $\\cdots$, $(K \\ast_c A_n ) \\cap A_{n-1} \\neq \\emptyset$ and $(K \\ast_c A_{0} ) \\cap A_{n} \\neq \\emptyset$ for some $n \\geq 1$. We prove that $K \\ast_c A_0 = K \\ast_c A_1 = \\cdots = K \\ast_c A_n$ by mathematical induction on $n$. For $n =1$, this follows immediately from $\\ast_c$-reciprocity. Let us hypothetically suppose that it holds for $n = k$ ($k \\geq 1$), then we should show that it also holds for $n = k+1$.\\\\\nLet $A = \\bigcup_{0 \\leq i \\leq k+1} A_{i}$. It follows from $(\\star)$ that $A \\cap (K \\ast_c A_i) \\neq \\emptyset$ for every $0 \\leq i \\leq k+1$. So $A \\cap (K \\ast_c A) \\neq \\emptyset$ by $\\ast_c$-regularity. It follows that there exists some $j$ with $0 \\leq j \\leq k+1$ such that $A_j \\cap (K \\ast_c A) \\neq \\emptyset$. Moreover, according to $(\\star)$, if $j = 0$ then $A_{k+1} \\cap (K \\ast_c A_j) \\neq \\emptyset$ else $A_{j-1} \\cap (K \\ast_c A_j) \\neq \\emptyset$. It follows that $A \\cap (K \\ast_c A_j) \\neq \\emptyset$ in any case. So $K \\ast_c A_j = K \\ast_c A$ by $\\ast_c$-reciprocity. Hence, as $A \\cap (K \\ast_c A_i) \\neq \\emptyset$ for every $0 \\leq i \\leq k+1$, it follows from $(\\star)$ and $\\ast_c$-reciprocity that if $0