diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfdtu" "b/data_all_eng_slimpj/shuffled/split2/finalzzfdtu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfdtu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nAlbregtsen \\& Maltby (\\cite{albregtsen1}, cf. Albregtsen \\& Maltby \n\\cite{albregtsen2}, Albregtsen et al. \\cite{albregtsen}) reported a dependence \nof umbral core brightness on the phase of the solar cycle based on 13 sunspots \nobserved at Oslo Solar Observatory. The umbral core is defined as the darkest \npart of the umbra. According to their findings, sunspots present in the early \nsolar cycle are the darkest, while as the cycle progresses spots have increasingly \nbrighter umbrae. Also, the authors did not find any dependence of this relation \non the size or the type of the sunspot. Following this discovery Maltby et al. \n(\\cite{maltby}) proposed three different semi-empirical model atmospheres \nfor the umbral core, corresponding to early, middle and late phases of the \nsolar cycle.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{6356fig1.eps} \n\\caption{Distribution of analysed sunspots over the ongoing solar cycle 23. \nThe grey solid line shows the International Sunspot Number (left scale). \nThe solid circles show the dates of observation and umbral radii of the analysed \nsunspots with umbral radii between 5 and 15 arc-sec, and the open circles show the \nsunspots with umbral radii less than 5 arc-sec or greater than 15 arc-sec (right-hand scale). \nNo sunspot in 1996 \\& 1997 fulfilled the selection criteria of $\\mu > 0.94$. \nThe hatched area marks the period when contact with SOHO was lost.}\n\\end{figure*}\n \nIn order to explain the umbral brightness variation with solar cycle \ntwo hypotheses have been put forward. Sch$\\mathrm{\\ddot{u}}$ssler \n(\\cite{schussler}) proposed that umbral brightness may be influenced \nby the age of the sub-photospheric flux tubes, whereas Yoshimura \n(\\cite{yoshimura}) suggested that the brightness of the umbra \ndepends on the depth in the convection zone at which the flux \ntube is formed. Confirmation of these results appears important for two reasons. \nFirstly, this is the only strong evidence for a dependence of local properties\nof the magnetic features on the global cycle. E. g., the facular contrast does not \ndepend on solar cycle phase (Ortiz et al. \\cite{ortiz}). Secondly, such a confirmation \nappears timely in the light of the recent paper by Norton \\& Gilman \n(\\cite{norton}), who reported a smooth decrease in umbral brightness from \nearly to mid phase in solar cycle 23, reaching a minimum intensity around \nsolar maximum, after which the umbral brightness increased again, based on the \nanalysis of more than 650 sunspots observed with the MDI instrument. \nThis decrease in brightness contradicts the results of Maltby et al. (\\cite {maltby}). \nAlso, the data used by Norton \\& Gilman (\\cite{norton}) were not corrected for stray \nlight and no sunspot size dependence of the brightness was discussed. \n\nIn this paper we investigate the dependence of umbral core brightness, \nas well as the mean umbral and penumbral brightness on the solar cycle and on the \nsize of the sunspot. In the following \nsection we describe the data selection. In the third \nsection we deal with the data correction for stray light and for the \ninfluence of Zeeman splitting of the nearby Ni~{\\scriptsize I} absorption line on \ncontinuum measurements. In sections 4 and 5 we present our results. We discuss our results \nand compare them with earlier findings in section 6.\n\\section{Data selection}\nContinuum full disk images recorded by the Michelson Doppler Imager (MDI; \nScherrer et al. \\cite{scherrer}) on board the SOHO spacecraft are used in \nthis analysis. The continuum images are obtained from five filtergrams \nobserved around the Ni~{\\scriptsize I} 6768 \\AA\\\/ mid-photospheric \nabsorption line with a spectral pass band of 94 m\\AA\\\/ each. The filtergrams \nare summed in such a way as to obtain the continuum intensity \nis free of Doppler cross talk at the 0.2\\% level. The \nadvantage of this data set is its homogeneity with no seeing fluctuations. \n\nWe selected 234 sunspots observed between March, 1998 and March, 2004. \nThe selected sunspots were located close to the \ndisk centre, i.e. for $\\mu > 0.94$, where $\\mu = \\cos \\theta $ and $\\theta$ is \nthe angle between the line-of-sight and the surface normal. Also, our analysis was \nmostly restricted to regular sunspots, this excluded complex sunspots having very \nirregular shape and multiple umbrae. By looking through the daily images and selecting \nthe sunspot when it was very close to the central meridian, we make sure that a \nparticular sunspot is included only once in our analysis during one solar rotation. \nOut of the selected sunspots, 164 sunspots have an umbral radius between 5\\arcsec\\\/ \nand 15\\arcsec. Even though all the 234 sunspots were used for the study of \nradius-brightness dependence, only the sunspots with umbral radius between 5\\arcsec\\\/ \nto 15\\arcsec\\\/ were used for the study of brightness dependence on solar cycle. \nThis is done in order to facilitate a direct comparison of our results with those \nof Maltby et al. (\\cite{maltby}). The data set covers most of solar cycle 23, \nalthough it does miss a few sunspots in the beginning of the cycle due to our \nselection criteria and the end of the cycle. \n\nFigure 1 shows the distribution of sunspots\\footnote{Sunspot numbers are \ncompiled by the Solar Influences Data Analysis Center \n(http:\/\/sidc.oma.be), Belgium} over cycle 23, \nused for the study of the solar cycle dependence of brightness (filled \ncircles) and those used only to determine the dependence on size (open circles). \nBy restricting the analysis to sunspots near the disk \ncentre, we avoid the suspected effect of centre-to-limb variation \non the umbral brightness (Albregtsen et al. \\cite{albregtsen}). \nThis is shown in Sect. 5. \n\nThe determination of umbral-penumbral and penumbral-quiet \nsun boundaries was carried out using the cumulative histogram \n(Pettauer \\& Brandt \\cite{pettauer}) \nof the intensity of the sunspot brightness and of the immediately surrounding quiet Sun \n(whose average is set to unity). This histogram was computed for 88 \nsymmetric sunspots scattered across the observational period and then \naveraged. This averaged cumulative histogram is shown in Fig. 2. Note that the \nhistogram is computed after stray light correction (see Sect. 3). The quiet \nSun corresponds to the steep rise around normalised intensity unity. The rise \nat around 0.6 corresponds to the penumbra, below that is the umbra. In order to\ndetermine the intensity threshold corresponding to the penumbra-photosphere and \numbra-penumbra boundaries linear fits to the flattest parts of the averaged histogram \nwere computed. The boundaries were chosen at the highest intensity \nat which the linear fit ceases to be a tangent to the histogram. \nThe reason why the penumbra is visible as a reasonably sharp drop, while the \numbra is not, is only partly due to the larger \nrange of intensity found in the umbra. It is mainly due to the large difference in \numbral brightness from spot to spot (see Sect. 4). From the average cumulative histogram\nit was found that values of 0.655 and 0.945 in normalised \nintensity correspond to umbral-penumbral, and penumbral-quiet sun boundaries, respectively. \nThese values were later used to determine the umbral and spot radius. All intensities are\nnormalised to quiet Sun values at roughly the same $\\mu$ value as the sunspot.\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig2.eps}} \n\\caption{Average cumulative intensity histogram used for obtaining the umbral-penumbral \nand penumbral-quiet Sun sunspot boundaries. Dotted lines are linear fits to the flattest \nparts of the histogram. Vertical dotted lines mark the values selected for umbral-penumbral \nand penumbral-quiet Sun boundaries.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig3.eps}} \n\\caption{Off-limb stray light profiles of SoHO\/MDI continuum images for 8 different years.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig4.eps}} \n\\caption{A typical stray-light fit to the observed limb profile. {\\bf Top:} the filled circles show the \nobserved average limb profile and the solid line shows the fit to the observed profile. \n{\\bf Bottom:} the residual after the fit.}\n\\end{figure}\n\\section{Data correction}\nBefore retrieving the brightness, we made a few corrections to the observed data.\nEven though the atmospheric seeing related blurring and distortions are absent, \nMDI continuum images are found to be contaminated by instrumental scattered light. \nBy checking the falloff of intensity just outside the solar limb, it was noticed \nthat the instrumental scattered light increased with the aging of the instrument, \nwhich we carefully correct for.\n\nAlso, continuum measurements in MDI are not carried out in a pure continuum spectral \nband. As described in the previous section, five filtergrams obtained around a \nNi~{\\scriptsize I} absorption line are used to compute the continuum intensities. \nWe investigate the effect of this absorption line in the vicinity of a filter \npass-band on observed continuum intensities. This is especially important in \nsunspots where the line profile changes due to the presence of a magnetic field. \nIn the following subsections we elaborate on these corrections.\n\n\\subsection{Stray light correction}\nIn order to remove the stray light, average radial profiles were obtained \nfrom the observed full-disk MDI continuum images. Figure 3 shows such intensity \nprofiles obtained from an MDI continuum image, averaged over the whole limb, \neach year just outside the solar disk. The gradual increase in scattered light \nwith time is clearly evident in the plot. These profiles were fitted to retrieve \nthe PSF (point-spread function) of the instrument \n(Mart\\' \\i nez Pillet \\cite{valentin}, Walton \\& Preminger \\cite {walton}). The \nradial profiles were generated using the spread function along with \nthe centre-to-limb variation (CLV). A fifth order polynomial is used to describe\nthe CLV. The initial values of the CLV coefficients were taken from \nPierce \\& Slaughter (\\cite {pierce}). The computed profiles were iteratively \nfitted to the observations by adjusting the coefficients of the PSF and CLV. \nA deconvolution of the observed image with the model PSF (generated from the \nfitted coefficients) is carried out to retrieve the original intensity. \n\nFigure 4 shows a typical fit to the observed radial profile and the difference \nbetween observed and fitted profiles. The spikes in the residual are due to the \nsharp change in intensity at the solar limb and are restricted to the points just \noutside and inside the Sun. Excluding these points, the residuals always lie \nbetween $\\pm 0.002$. \n\nWe tested our fitting procedure for the stray light correction \nusing MDI continuum images of the Mercury transit on 7th May, 2003. \nFigure 5 shows the observed and restored intensity for a \ncut across the solar disk through the Mercury image (at $\\mu=0.65$) whose expected full width \nat half maximum is 12\\arcsec, i.e. typical of the diameter of a sunspot umbra. \nIt is evident from the figure that while the intensity in the original cut \nthrough the Mercury image never drops below 16\\%, after the stray light removal \nthe intensity drops to very close to zero. More details of the stray light \ncorrection are given in Appendix A. \\\\\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig5.eps}} \n\\caption{Observed and restored intensity for a \ncut across the solar disk through the Mercury image.\nFilled circles show the observed intensity profile through the centre of the mercury image. \nThe solid line shows the profile after the stray light removal.}\n\\end{figure}\n\n\\subsection{Correction for the Zeeman splitting of the Ni~{\\scriptsize {\\rm \\it I}} line} \nIn MDI, the continuum is computed by measuring intensities at five filter \npositions (designated as F$_{0}$ through F$_{4}$, cf. Scherrer et al. \\cite{scherrer}) \non a spectral band which includes the Ni~{\\scriptsize I} absorption line. \nThe filter F$_{0}$ (whose profile is shown in Fig. 6) gives the main contribution to the\nmeasured continuum, while the intensities recorded through the other filters are used to\ncorrect for intrusions of the Ni~{\\scriptsize I} line into the continuum filter, \nmainly introduced by Doppler shifts. The claimed accuracy for such corrections is 0.2\\% \nof the continuum intensity (Scherrer et al. \\cite{scherrer}), which is perfectly \nadequate for our analysis. \n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig6.eps}} \n\\caption{Synthesised Ni~{\\scriptsize I} line profiles (solid lines). \nFilled circles represent the FTS quiet Sun (QS) spectrum for the same line. The dashed curve \nshows the position of the MDI-F$_{0}$ filter transmission profile.} \n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig7.eps}} \n\\caption{Brightness correction for the contribution of the Ni~{\\scriptsize I} \nline to the MDI-continuum measurements. The solid line shows the normalised continuum \nintensity computed including the influence of the spectral line plotted versus the \ncomputed true continuum for the same set of model atmospheres. Dashed lines are \nfor magnetic field strengths around 10\\% below or above the values listed in Table 1, \nfor a given value of temperature.} \n\\end{figure}\n\nIt is, however, not clear to what extent the Zeeman splitting of the line, as known to be\npresent in sunspots, affects the continuum intensity measurement. In order to quantify \nthis effect in sunspots, we carried out a series of calculations taking various values \nfor magnetic field strength and temperature, which approximately simulate the \nrelationship between the magnetic field and corresponding temperatures found in \nsunspots following Kopp \\& Rabin (\\cite {kopp}), Solanki et al. (\\cite{sol1993}), \nand Mathew et al. (\\cite{mathew}). The exact choice of the field \nstrength-temperature relation is not very critical, as we have found by considering \nalso other combinations (e.g. with lower or with higher field strength for a \ngiven temperature). The magnetic field strength and the corresponding temperature \nalong with other parameters used in the synthesis of Ni~{\\scriptsize I} line profiles \nare given in Table 1. The abundance is given on a logarithmic scale on which the \nhydrogen abundance is 12 and the oscillator strength implies the $\\log (gf)$ value. \n\\begin{table}\\caption{Parameters used for producing Ni~{\\scriptsize I} line profiles}\n\\begin{center}\n\\begin{tabular}{ll}\\hline \\hline\nCentre wavelength \t\t\t& \t6767.768 \\AA \\\\\nAbundance (Ni)\t\t\t\t&\t6.25\\\\\nOscillator strength \t\t\t&\t$-$1.84\\\\\nMacro turbulence \t\t\t&\t1.04 km s$^{-1}$\\\\\nMicro turbulence\t\t\t& 0.13 km s$^{-1}$\\\\\n\\hline\nTemperature (K) & Field strength (G) \\\\\n\\hline\n5750 \t \t&\t0 \\\\\n5500 \t\t&\t1100 \\\\\n5250 \t\t&\t1700 \\\\\n5000 \t\t&\t2250 \\\\\n4750 \t\t&\t2250 \\\\\n4500 \t\t&\t2500 \\\\\n4250 \t\t&\t3000 \\\\\n4000\t\t&\t3500 \\\\\n3750 \t\t&\t4000 \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fig8.eps}} \n\\caption{The change in full width (solid line; left scale), equivalent width (dotted line; left scale) and \nline depth (dashed line; right scale) \nwith effective temperature.} \n\\end{figure}\n\nFigure 6 shows the computed Ni~{\\scriptsize I} line profiles along with the position \nof the MDI-F$_{0}$ filter transmission profile. The solid circles are the quiet Sun \nFTS (Fourier Transform Spectrometer) spectrum for the same line \n(Kurucz et al. \\cite{kurucz_at}). Each plotted line profile was computed using a \nmodel atmosphere from Kurucz (\\cite{kurucz}) with effective temperature and \nheight independent vertical magnetic field of the strength listed in Table 1. \nThe Ni abundance is taken from Grevesse \\& Sauval (\\cite {grevesse}) and the atomic \nparameters are obtained from the Kurucz\/NIST\/VALD \\footnote {Kurucz data base - http:\/\/www.pmp.uni-hannover.de \\\\\nNIST - http:\/\/physics.nist.gov\/PhysRefData\\\\\nVALD - http:\/\/www.atro.uu.se\/ $\\tilde{}$~vald } atomic data bases. \nAs a first step, the Ni~{\\scriptsize I} line profile taken from the quiet Sun FTS spectrum \nis fitted using the Kurucz quiet Sun model atmosphere (T$_{\\rm eff} = $ 5750 K) keeping the \nmicro- and macro-turbulence and oscillator strength as free parameters. The values \nobtained for the oscillator strength and micro- and macro-turbulence from the fit \nare maintained when computing all the remaining line profiles with various magnetic \nfield strengths. MDI theoretical filter transmission profiles are created for the \nfive filter positions across the spectral line. The transmitted intensity through \neach filter is computed and combined following Scherrer et al. (\\cite{scherrer}) to \nderive the continuum intensities. In Fig. 7 we plot the `true continuum' intensities \n(i.e., the intensity which would have been measured through the filter if the line \nwere not present in the vicinity of the MDI filter) versus the intensities resulting \nfrom the MDI continuum measurements for different effective temperatures. \nClearly, the MDI continuum measurements in the presence of the Ni~{\\scriptsize I} absorption \nline provide a lower intensity than the real continuum in the sunspots, and the difference \nvaries with the changing field strength and temperature. Naively one would expect this\ndifference to increase for increasing magnetic field strengths, whereas the \nactually found behaviour is more complex. An explanation for this is given \nin Fig. 8. The decreasing temperature reduces the line depth (which is given in units \nrelative to the continuum intensity for the relevant line profile), while the \nequivalent width initially increases before decreasing again with decreasing temperature \nif the field is left unchanged. Increasing field strength leads to enhanced line \nbroadening and a slight increase in equivalent width. The combined influence of both \neffects is plotted in Fig. 8. Note that the width is measured as the wavelength \ndifference between the two outer parts of the line profile at which it drops to $1-d\/2$, \nwhere $d$ is the line depth in units of the continuum intensity. The behaviour seen in Fig. 7 therefore \npartly reflects the dependence of the equivalent width on temperature, but quite significantly also the \nfact that the total intensity absorbed by the line in terms of the {\\it continuum intensity of the quiet Sun}\ndecreases rapidly with decreasing temperature. This is the quality more relevant for our needs, rather than \ntotal absorbed intensity relative to sunspot continuum intensity, which corresponds to the equivalent width.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8.5cm]{6356fi9a.eps} \\includegraphics[width=8.5cm]{6356fi9b.eps}\n\\caption{Cut through two simple sunspots with different effective umbral radii of around \n{\\bf (a)} 15 arc-sec and \n{\\bf (b)} 6 arc-sec.}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5.6666cm]{6356f10a.eps} \n\\includegraphics[width=5.6666cm]{6356f10b.eps}\n\\includegraphics[width=5.6666cm]{6356f10c.eps}\n\\caption{Umbral core intensity versus umbral radius, {\\bf (a)} observed, {\\bf (b)} \ncorrected for stray light, and {\\bf (c)} corrected for stray light and the influence of \nthe Ni~{\\scriptsize I} \\\/line. Here sunspots with umbral radius less \nthan 5 arc-sec and greater than 15 arc-sec are also included.}\n\\end{figure*}\n\nThe two dashed lines in Fig. 7, which are hardly distinguishable from the solid line correspond to using \ndifferent field strength-temperature relations. The upper one is found if we decrease the field strength by around \n10\\% (i.e. the correction is minutely smaller), the lower one for around 10\\% higher field. We use the solid line in \nFig. 7 to correct the observed continuum intensities in sunspots. During the data reduction process \nall the resulting intensities after the stray light removal are replaced by reading \nout the corresponding value from the computed true continuum. \n\n\\section{Brightness-radius relationships} \nBefore discussing the brightness-radius relationship,in Fig. 9 we show cuts through two different sunspots. \nThose allow us to point out various intensity values used in our study. The big sunspot (Fig. 9(a)) has an \neffective umbral radius of around 15\\arcsec\\\/. The horizontal dashed lines indicate the umbral and penumbral \nboundaries, whereas the dotted lines represent mean and minimum umbral intensities as well as mean penumbral \nintensity in this particular sunspot. Similarly, Fig. 9(b) shows a cut through a small sunspot (umbral \nradius $\\approx 6$\\arcsec). \n\n\\subsection{Umbral core and mean intensity versus umbral radius}\nFigure 10 shows the relation between umbral core intensity and umbral \nradius. The umbral radius is computed as the radius of a circle with the \nsame area as the (irregularly shaped) umbra under study. \nThe umbral core intensity is the lowest intensity value found in the particular \numbra (see Fig. 9). \n\nAll intensities are normalised to the average local quiet Sun intensity. \nFigure 10(a) shows this relation for the observed intensity and (b) for the stray light \ncorrected intensities. The influence of the Ni~{\\scriptsize I} \nline on the continuum measurement is still present in this panel. Figure 10(c) is corrected for \nboth the stray light and the effect of the Ni~{\\scriptsize I} line \non the continuum measurement. In all \nthe figures the trend remains the same. It is clear from the figure \nthat the core intensity decreases very strongly with increasing umbral radius. \n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f11a.eps}\\includegraphics{6356f11b.eps}}\n\\caption{Mean umbral intensity versus umbral radius, {\\bf (a)} observed and, {\\bf (b)} \ncorrected for stray light and the influence of the Ni~{\\scriptsize I} \\\/line. \nHere sunspots with umbral radius less than 5 arc-sec and greater than 15 arc-sec are \nalso included.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f12a.eps}\\includegraphics{6356f12b.eps}} \n\\caption{Power law fit (solid line) and double linear fit (dash lines) to the {\\bf (a)} \numbral core intensity and, {\\bf (b)} mean umbral intensity. \nHere the filled circles represent bins of 10 spots each.} \n\\end{figure}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356f13a.eps}\\includegraphics{6356f13b.eps}}\n\\caption{{\\bf (a)} Mean penumbral intensity versus spot radius, corrected for stray light. \n{\\bf (b)} Linear fit to the binned mean penumbral intensity.}\n\\end{figure}\n\\begin{table*}\n\\caption{Fit parameters for radius-brightness relation}\n\\begin{tabular}{lcccccc} \n\\hline \n\\hline\n Dependence on umbral radius of,& Umbral radius\t& Constant & Exponent & Gradient & $\\sigma $\t& $\\chi^{2}$ \\\\\n\\hline\n\\bf{Power law fit} \t&\t\t&\t\t&\t &\t & & \\\\\numbral core intensity \t&(all) \t& 1.8598\t& $-1.0679$ &\t&0.063 &$3.5 \\times 10^{-3}$\\\\\numbral mean intensity \t&(all) \t\t& 0.8297 & $-0.3052$ & &0.013\t &$1.5 \\times 10^{-4}$ \\\\ \n\\bf {Double linear fit} \t&\t\t&\t\t&\t\t&\t\t& \\\\\numbral core intensity\t &$<$10$''$ \t&0.6515 &\t&$-0.0552$ \t&0.0029 \t&$6.7 \\times 10^{-4}$\\\\\n\t\t\t &$>$10$''$\t&0.2299 &\t&$-0.0094$ \t&0.0027 \t&$4.7 \\times 10^{-5}$\\\\\numbral mean intensity\t &$<$10$''$ \t&0.6536 &\t&$-0.0266$ \t&0.0013 \t&$1.2 \\times 10^{-4}$ \\\\\n\t\t\t &$>$10$''$\t&0.4858 &\t&$-0.0087$ \t&0.0026 \t&$4.3 \\times 10^{-5}$\\\\ \n\\hline \nDependence on spot radius of, \t&\t\t&\t\t&\t\t&\t\t&\\\\\n\\hline\n\\bf {Linear fit} \t\t&\t\t&\t\t&\t\t&\t\t&\\\\\npenumbral mean intensity \t&$>$10$''$\t&0.8561 &\t&$-0.0016$ \t&0.0001 \t&$1.1\\times 10^{-6}$\\\\ \n\\hline \n\\hline\n\\end{tabular}\n\\end{table*}\n\nA steeper decrease \nis found for spots with smaller umbral radius, while for the bigger \nspots a more gentle decrease in umbral core intensity with radius is observed \n(dictated by the fact that umbral intensity has to be positive). \nThis plot emphasises the need to take into account the dependence of the \numbral brightness on the size of the spot when looking for solar cycle variations.\n\nThe mean umbral intensity, plotted in Fig. 11, also shows a similar decrease \nwith increase in umbral radius. The difference between the umbral core \nand mean intensities is smallest for the smallest umbrae and increases with umbral size. \n\nIn order to obtain a relation between the umbral core and mean intensities with umbral radius,\nwe carried out two different fits to the respective corrected umbral intensities after binning \ntogether points with similar sunspot radius, such that each bin contains 10 samples. The dashed lines \nin Figs. 12(a) and (b) show the double linear \nfits to the umbral core and mean intensities, respectively. The individual linear fits are made \nto spots with umbral radii less than 10\\arcsec\\\/ and to those with radii above this limit, respectively.\nThe fit parameters along with errors and normalised $\\chi^{2}$ values are included in Table 2. \nThe solid lines in these figures show the power law fit. The power law fit to the umbral \ncore intensity seems to be a comparatively poor approximation, while the mean umbral brightness \nseems to obey the power law (i.e. it gives a very low $\\chi^2$ for half the number of free\nparameters as the double linear fit). The parameters for the power law fit are also included \nin Table 2. Figure 12 again demonstrates that the difference between the core and mean intensity \nincreases rapidly with umbral radius. It is equally clear that since the umbral core intensity \nvaries by a factor of nearly 6, the mean umbral intensity by a factor of nearly 2 between the smallest \nand the largest umbrae, employing a single value for umbral brightness of all spots is a very poor \napproximation. Such an approximation is often made e.g. for the reconstruction of solar irradiance \n(cf. Unruh et al. \\cite{unruh}, Krivova et al. \\cite{krivova}).\n\n\\subsection{Penumbral mean intensity and spot radius}\nFigure 13(a) shows the relation between mean penumbral intensity and spot radius. \nAn approximate linear relationship is evident for the spots with outer penumbral radius between \n10\\arcsec\\\/ and 30\\arcsec. The outer penumbral radius is the equivalent radius of the whole sunspot \n(including the umbra). The large scatter in mean penumbral intensities for spot \nsizes below 10\\arcsec\\\/ might result from the insufficient resolution of the \nfull disk images or may be due to the fact that the parameters for distinguishing \nbetween umbra and penumbra are possibly not appropriate for small spots. Note the \norder of magnitude smaller range of variation of penumbral contrast than of \numbral contrast. Figure 13(b) shows the linear fit to the mean penumbral brightness, \nafter binning 10 adjacent spots (taking only spots with radius greater than 10$''$). \nThe fit parameters are listed in Table 2. \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi14.eps}\n\\caption{Umbral core intensity versus solar cycle, {\\bf (a)} observed, {\\bf (b)} corrected for \nstray light, and {\\bf (c)} corrected for stray light and the influence of Ni~{\\scriptsize I} \\\/line. \nThe intensities are plotted for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. The solid line shows the linear regression \nand the dashed lines represent the $\\pm1 \\sigma$ deviation, due to the uncertainty in the regression \ngradient. The best linear fit is given in the upper left corner.}\n\\end{figure*}\n\n\\section{Solar cycle dependence of the brightness}\nIn Fig. 14 we plot the sunspot umbral core intensity versus time elapsed since the solar cycle \nminimum. September 1996 is taken as the minimum month, the upper axis shows the corresponding \nyear. This plot includes spots with umbral radius between 5\\arcsec\\\/ and 15\\arcsec\\\/ only, in order to be \nconsistent with the work of Albregtsen \\& Maltby (\\cite {albregtsen1}). Figure 14(a) shows the observed \nintensity, (b) the stray light corrected intensity, while Fig. 14(c) shows the intensity corrected for both \nstray light and the influence of the Ni~{\\scriptsize I} \\\/line on the continuum measurements. In all the figures \nthe trend remains the same. As the umbral radius and core brightness are related, the scatter in one quantity \nalso reflects the scatter in the other.\n\nThe solid line shows the linear regression to the brightness, whereas the dotted \nlines indicate the 1$\\sigma$ error in the gradient. A feeble trend of increasing umbral \nbrightness towards the later phase of the solar cycle is observed. This increase is \nwell within the 1$\\sigma$ error bars and statistically insignificant. Table 3 lists the fit \nparameters, including the errors and normalised $\\chi^{2}$ values. All the fit parameters \nare listed for the corrected intensity values. Also, in all the remaining figures we plot \nthe corrected intensities alone. \n\\begin{figure*}\n\\includegraphics[width=12cm]{6356f15a.eps}\\\\\n\\sidecaption\n\\includegraphics[width=12cm]{6356f15b.eps}\n\\caption{Umbral core intensity versus solar cycle plotted for {\\bf (a)} umbral radii \nranging from 5 to 10 arc-sec and, {\\bf (b)} from 10 to 15 arc-sec. \nThe solid lines show the regression fits and the dashed lines the $\\pm1 \\sigma$ deviations \ndue to the uncertainty in the regression gradient.}\n\\end{figure*}\n\\begin{table*}\\caption{Fit parameters for solar cycle dependence}\n\\begin{tabular}{lcccccc}\\hline \\hline\nSolar cycle dependence of, \t&Umbral radius\t\t&Constant \t& Gradient\t& $\\sigma $ \t& Gradient$\/ \\sigma $\t& $\\chi^{2} $ \\\\ \\hline\numbral core intensity \t\t& (all)\t\t\t&0.33605\t&$-0.00785$\t&0.00649\t&$-1.2095$\t\t&0.01901\\\\\n\t\t\t\t&5$''$ - 15$''$ \t&0.22188 \t&$+0.00358$ \t&0.00577 \t&$+0.6205$\t\t&0.01015\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.25001\t&$+0.00227$ \t&0.00582 \t&$+0.3900$\t\t&0.00841\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.12560 \t&$-0.00269$ \t&0.00547 \t&$-0.4918$\t\t&0.00158\\\\\n\t\t\t\t&(all, northern hemisphere)&0.33957\t&$-0.01071$\t&0.00622\t&$-1.7218$\t\t&0.00085\\\\\n\t\t\t\t&(all, southern hemisphere)&0.33062\t&$-0.00310$\t&0.00874\t&$-0.3547$\t\t&0.00164\\\\\\hline\numbral mean intensity \t\t&5$''$ - 15$''$ \t&0.44346 \t&$+0.00206$ \t&0.00312 \t&$+0.6603$\t\t&0.00296\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.45782\t&$+0.00168$ \t&0.00294 \t&$+0.5714$\t\t&0.00215\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.39609 \t&$-0.00333$ \t&0.00513 \t&$-0.6491$\t\t&0.00139\\\\\\hline\npenumbral mean intensity \t&5$''$ - 15$''$ \t&0.82974\t&$-0.00009$ \t&0.00049 \t&$-0.1837$\t\t&0.00007\\\\ \n\t\t\t\t&5$''$ - 10$''$ \t&0.83324\t&$-0.00034$ \t&0.00042 \t&$-0.8095$\t\t&0.00004\\\\\n\t\t\t\t&10$''$ - 15$''$\t&0.81723\t&$-0.00024$ \t&0.00074 \t&$-0.3243$\t\t&0.00003\\\\\\hline\numbral radius \t\t & (all)\t\t\t&5.79088\t&$+0.19029$\t&0.13549\t&$+1.4045$\t\t&8.29028\\\\\n\t\t\t\t&5$''$ - 15$''$ \t&8.06675\t&$-0.07457$ \t&0.12525 \t&$-0.5954$\t\t&4.77502\\\\\n\t\t\t\t&5$''$ - 10$''$\t \t&7.06682\t&$-0.01697$ \t&0.08311 \t&$-0.2042$\t\t&1.71632\\\\\n\t\t\t\t&10$''$ - 15$''$ \t&11.5536\t&$+0.06823$ \t&0.20443 \t&$+0.3338$\t\t&2.20379\\\\\n\t\t\t\t&(all, northern hemisphere)&5.75164\t&$+0.21357$\t&0.11716\t&$+1.8229$\t\t&0.30162\\\\\n\t\t\t\t&(all, southern hemisphere)&5.94171\t&$+0.12221$\t&0.19719\t&$+0.6198$\t\t&0.83724\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table*}\n\\begin{figure*}\n\\includegraphics[width=12cm]{6356f16a.eps}\\\\\n\\sidecaption\n\\includegraphics[width=12cm]{6356f16b.eps}\n\\caption{Solar cycle dependence of umbral radius for {\\bf (a)} sunspots with radii between \n5 and 10 arc-sec and {\\bf (b)} with radii between 10 and 15 arc-sec.}\n\\end{figure*} \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi17.eps}\n\\caption{$\\mu$ versus time since activity minimum for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. The solid line shows the linear regression \nand the dashed lines the $\\pm1 \\sigma$ deviation.}\n\\end{figure*}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fi18.eps}}\n\\caption{Umbral core intensity versus $\\mu$ for all spots with umbral radii between \n5 arc-sec and 15 arc-sec. \n}\n\\end{figure} \n\\begin{figure*}\n\\sidecaption\n\\includegraphics[width=12cm]{6356fi19.eps}\n\\caption{Umbral core intensity {\\bf (a)} and umbral radius {\\bf(b)} versus time since solar cycle minimum \nfor Northern (filled circles) and Southern (asterisks) hemispheres, for all observed sunspots. \nEach plotted symbol represents an average over 10 sunspots. The solid and dashed lines show the linear regression \nfits for Northern and Southern hemispheres, respectively. The best linear fits are given in the lower \nleft corner.}\n\\end{figure*}\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{6356fi20.eps}}\n\\caption{Umbral core intensity versus umbral radius. Different symbols represent spots observed \nin ascending, maximum and descending phases of solar cycle. The dates indicate the period of observation.}\n\\end{figure} \nIn Figs. 15(a) and (b) we display the umbral \ncore intensity for two different umbral size ranges, i.e. for spots with umbral \nradii in the range 5\\arcsec\\\/ - 10\\arcsec\\\/ and in 10\\arcsec\\\/ - 15\\arcsec, respectively. \nThe trend seen here is opposite for small and large spots, but is insignificant \nin both cases (see column Gradient\/$\\sigma$ in Table 3). If we plot all the sunspots, \nirrespective of radius, the results are not significantly different.\n \nIn Figs. 16(a) and (b) we plot the dependence on the time elapsed from \nactivity minimum, of the analysed sunspot umbral radius, separately for the \nsmall (5\\arcsec\\\/ - 10\\arcsec) and the large (10\\arcsec\\\/ - 15\\arcsec) spots, respectively. \nThe linear regression are overplotted. \nThe mean umbral radius of the analysed spots \nbetween 5\\arcsec\\\/ - 10\\arcsec\\\/ slightly decreases with time, whereas spots \nwith radius between 10\\arcsec\\\/ - 15\\arcsec\\\/ show the opposite trend. Similarly in \nFig. 1 the umbral radii of all studied spots are plotted. The regression parameters\nfor the full sample of spots (5\\arcsec\\\/ - 10\\arcsec) is given in Table 3. Although none of these\ntrends is statistically significant, they are opposite to the trends (also not \nstatistically significant) shown by the umbral brightness of these spots. This \nis completely consistent with the dependence of brightness on umbral radius shown \nin Figs. 10 - 12.\n\nAnother bias can be introduced by the fact that the sunspot latitude systematically \ndecrease over a solar cycle. We have to a certain extent reduced this effect by \nconsidering only sunspots at $\\mu > 0.94$. Figure 17 shows the average $\\mu$ of the \nanalysed sunspots, as expected, this displays an increase over the cycle. \nIn order to judge whether this introduces a bias into the cycle phase dependence of sunspot \nbrightness we plot in Fig. 18 umbral core brightness (i.e. contrast to local quiet Sun) \nversus $\\mu$ (for $0.94 < \\mu < 1$). We did not find any significant variation in the umbral \ncore brightness with $\\mu$.\n\nIn order to check for an asymmetry in umbral brightness between the northern and southern \nhemispheres as reported by Norton \\& Gilman (\\cite{norton}), in Fig. 19 we plot umbral core \nbrightness separately for northern and southern hemispheres. In this plot we used corrected \nintensities for all the observed spots. The intensities are binned and each bin contains 10 samples. \nThe linear regression fit provides a slightly higher gradient in the northern hemisphere. \nBut this can be well explained by the increase in umbral \nradius of the spots with the cycle phase (Fig. 19(b)). It should be noted that Norton \\& Gilman observed a significant \numbral brightness difference between the northern and southern hemisphere during the onset of cycle 23. \nDue to the restriction of $0.94 < \\mu $ in our selection criteria, we have analysed only few spots during \nthis period and hence cannot comment on that result.\n \n\nIn Table 3 we also list the parameters of the linear regressions to mean umbral and \npenumbral intensities versus time. None of the gradients is significant at even the \n1$\\sigma$ level. Also, the signs of the gradients of all umbral core and mean intensity \nsamples are opposite to those of the umbral radius of the corresponding sample, \nsuggesting that even any small gradient in the umbral brightness is due to a small \nbias in the umbral size with time. Hence we find no evidence at all for a change \nin sunspot brightness over the solar cycle. \n\nIn order to test whether the dependence of umbral core brightness on umbral radius in \nSect. 4.1 is itself dependent on solar cycle phase we plot in Fig. 20 the umbral core brightness \nversus radius but now for three different phases of the cycle. Asterisks, filled circles and \ndiamond symbols represent the spots observed in ascending, maximum and descending phases of solar \ncycle, respectively. As can be clearly seen there is no difference between the different phases. \nThis demonstrates that there is no cross-talk between cycle phase dependence and umbral radius \ndependence of umbral brightness. \n \n\\section{Discussion}\nWith a large sample of sunspots, we have tested \nif the umbral core brightness, the umbral average brightness, or the penumbral \nbrightness depend on solar cycle phase. In addition to this, we \nstudied the dependence of brightness on sunspot size.\n\nEarlier continuum observations suggested that large sunspots are \ndarker than smaller sunspots (Bray \\& Loughhead \\cite{bray}). But most \nof such observations were barely corrected for stray light (Zwaan \\cite{zwaan}). Subsequent \nobservations which were corrected for stray light showed no significant \ndependence of umbral core brightness on spot size (Albregtsen \\& Maltby \n\\cite{albregtsen2}). More recent observations however, reveal that even after stray-light \ncorrection a size dependence remains. Thus, Kopp \\& Rabin (\\cite{kopp}) \npresent observations at 1.56 ${\\rm {\\mu m}}$ that show clear evidence for the size dependence \nof umbral brightness. Also, results from two sunspots observed at the same \nwavelength combined with the Kopp \\& Rabin data confirm and strengthen the \nlinear dependence of brightness on sunspot umbral size (Solanki \\cite{solanki1997}, \nSolanki et al. \\cite{solanki1}, Kopp \\& Rabin \\cite{kopp}, R$\\mathrm{\\ddot{u}}$edi et al. \\cite{ruedi}).\n\nMart\\' \\i nez Pillet \\& V$\\acute{\\rm a}$zquez (\\cite{martinez}) confirmed the Kopp \\& Rabin result \nbased on the analysis of 7 sunspots. Collados et al. (\\cite{collados}) found from the inversion \nof Stokes profiles obtained in 3 sunspot umbrae that small umbrae are distinctly \nhotter than large umbrae. One disadvantage of all these studies is that each is restricted to a\nrelatively small number of sunspots. Another is that each sunspot was observed under \ndifferent seeing conditions, so that the level of stray light varied in an unsystematic\nmanner. Both shortcomings are addressed in the present paper. \n\nIn our analysis we found a clear dependence of umbral core brightness on \numbral size. Since we correct for the very slowly varying stray light and the MDI specific \nproblem of cross-talk of the spectral line into the continuum, as described \nin Sect. 3 and the appendix, our results are basically free from stray light contamination. \nThis is particularly true for umbral core intensities as we could show using the Mercury transit data.\nFor mean intensities some residual remains (see Appendix A).\nAlso, we have a relatively large sample of sunspots to support our results. \nWe carried out and compared two different fits to the umbral brightness-size \ndependence for the analysed spots, a double linear fit for two different umbral \nsize ranges and a power law fit to the entire data set. A similar analysis of \numbral brightness and diameter carried out at a wavelength of 1.56 $\\rm \\mu m$ \n(Solanki \\cite{solanki1997}) shows a smaller gradient than what we obtained in our \nwork. Umbral brightness for 9 spots, with umbral diameter ranging from 5\\arcsec\\\/ \nto 35\\arcsec\\\/ were included in the above study. From a linear fit, \na gradient of around $-0.012\/\\arcsec$ is obtained for the umbral brightness, \nwhereas in our case a much higher gradient of $-0.04\/\\arcsec$ is found. \nThis is not surprising since the brightness at 1.56 $\\rm \\mu m$ reacts much more weakly\nto a given temperature change than at 677 $\\rm {nm}$.\n \nAlbregtsen \\& Maltby (\\cite{albregtsen1}) reported a variation of umbral \ncore brightness with solar cycle by analysing 13 sunspots. The results \nwere mainly presented for the observations done at a wavelength of 1.67$\\rm {\\mu m}$. \nThey found an increase in umbral core intensity by as much as 0.15 from early to \nlate phase of the solar cycle. They also found no dependence of umbral brightness \non other sunspot parameters, such as size and type of the spot. In a later \npaper Maltby et al. (\\cite{maltby}) detailed this variation for a range of wavelengths \nstarting from 0.38 - 2.35 $\\rm {\\mu m}$. The nearest wavelength to our observations \n(i.e. 0.669 $\\rm {\\mu m}$) shows a variation of around 0.072 in umbral core intensity from \nearly to late phase, this corresponds to 0.0065 umbral intensity variation per year. \nBased on these findings they presented three different umbral \ncore model atmospheres for sunspots present in the early, mid and late phase of \nthe solar cycle. In a recent study carried out using MDI data Norton \\& Gilman \n(\\cite {norton}) find a relatively smooth decrease in the umbral brightness from activity minimum \nto maximum for Northern hemisphere and no distinct trend for the southern hemisphere. \nThe decrease in umbral brightness with solar cycle they found is opposite to the results \nof Maltby et al. (\\cite{maltby}). \n\nIn our analysis we found a very feeble, statistically insignificant dependence of \numbral brightness on solar cycle (i.e. any change remains well within the error bars).\nThe linear fit to umbral core brightness is given by the following \nequation,\n\\begin{eqnarray}\nI_{uc}=0.222+(0.004\\pm 0.006)\\times t\n\\end{eqnarray}\nwhere $t$ is the time elapsed from the minimum in units of years. \nAll sunspots within 5\\arcsec\\\/ - 15\\arcsec\\\/ umbral radius are included in the regression. \nIt is striking that the 1$\\sigma$ uncertainty in the gradient obtained in our \nanalysis is approximately equal to the trend found by Maltby et al. (\\cite{maltby}).\nTherefore, either the MDI wavelength is less suitable than the 1.56 $\\rm \\mu m$ wavelength \nband employed by Maltby and co-workers, or there is a selection bias affecting their results. \nIndeed, the change in umbral core intensity over the solar cycle reported by \nMaltby et al. (\\cite{maltby}), 0.072, is only $1\/3$ the umbral core intensity difference \nbetween spots with umbral radii of 5\\arcsec\\\/ and 15\\arcsec\\\/ found here and is 0.6 times the \nintensity difference between such spots at 1.56 $\\rm \\mu m$. Consequently, selection biases, \nwhich often afflict small samples, can introduce an artificial \ntrend of the correct magnitude over an activity cycle. \n\nIn order to reduce the effect of size dependence on the above relation, we grouped the sunspots \ninto two umbral radii bins. The linear regressions to these groups are given by the following \nequations,\n\\begin{eqnarray}\nI_{uc}= \\{\n\\begin{array}{ll}\n0.250+(0.002\\pm0.006)\\times t &{\\rm for} ~~5'' - 10''\\\\\n0.126-(0.003\\pm0.005)\\times t &{\\rm for} ~~10''- 15'' \n\\end{array}\n\\end{eqnarray}\nFor the group of spots with umbral radii between 5\\arcsec\\\/ - 10\\arcsec\\\/ the linear regression \nfit gives an increase in umbral brightness with increasing phase of the solar cycle, \nbut again the change is within the error bars. For the spots with umbral radii \n10\\arcsec\\\/ - 15\\arcsec\\\/ the opposite trend is found. We compared these results with the \nvariation of the sizes of the \nanalysed sunspots over the solar cycle. It turns out that for all three samples \n(5\\arcsec\\\/- 10\\arcsec\\\/, 10\\arcsec\\\/-15\\arcsec\\\/, 5\\arcsec\\\/-15\\arcsec\\\/) the average radii \nshow the opposite trend to the intensity. This suggests that at least part of any trend in \nbrightness over the solar cycle is due to a corresponding (opposite) trend in umbral radii.\n\nFrom these results it is evident that the \ndependence of the brightness of the spot on size is an important parameter to \nbe considered when the umbral brightness variation with solar cycle is studied. We believe that \nwithout showing the time dependence of the average area or size of the sunspot umbrae in the employed \nsamples, any studies of sunspot brightness (or even field strength) evolution over time are of limited \nvalue. Although there is no systematic variation in the relative distribution of \numbral areas with solar cycle (Bogdan et al. \\cite {bogdan}) in a smaller sunspot sample\na trend may be introduced by limited statistics. It would therefore be of great value to determine\nthe areas of the umbrae studied by Penn \\& Livingston (\\cite{penn}), who find a steady decrease of the \nmaximum umbral field strength over the last 7 years, since the field strength is related to brightness \n(Maltby \\cite{maltby2}, Kopp \\& Rabin \\cite{kopp}, Mart\\' \\i nez Pillet \\& V$\\acute{\\rm a}$zquez \\cite{martinez}, \nSolanki et al. \\cite{sol1993}, Mathew et al. \\cite{mathew2004}, Livingston \\cite{livingston}), \nwhich depends on size (Sect. 4.1). Also, our results imply that models of the umbral \natmosphere (e.g. Avrett \\cite{avrett}, Maltby et al. \\cite{maltby}, Caccin et al. \\cite{caccin}, \nSeverino et al. \\cite{severino}, Fontenla et al. \\cite{fontenla}) should always indicate the spot \nsize to which they refer.\n\n\nChapman et al. (\\cite{chapman})\nreported photometric observations of sunspot groups, that show considerable variation in their \nmean contrast. This could be due to umbra\/penumbra area ratio change, or due to intrinsic \nbrightness change. Our results suggest that, at least in part the later is the reason. This has \nimplications for irradiance reconstructions, particularly those using separate atmospheric \ncomponents for umbrae and penumbrae (e.g. Fligge et al. \\cite{fligge}, Krivova et al. \\cite{krivova}).\n\nThe result is also of importance for the physics of sunspots, since an explanation \nmust be found why a smaller heat flux is transported through the umbrae of larger sunspots. \nE. g. in Parker's (\\cite{parker}) spaghetti model it would imply that either the filamentation \nof the subsurface field is less efficient under larger umbrae, or that the energy flux transported \nbetween the filaments is smaller for larger filaments. One factor which probably plays a role is that\ndarker sunspots have higher field strengths (Maltby \\cite{maltby2}, Kopp \\& Rabin \\cite{kopp}, \nLivingston \\cite{livingston}) which are more efficient at blocking magnetoconvection. \nThis also implies that larger sunspots have stronger fields. \n \nThe only clear dependence we have found in our sample of sunspots is between \numbral intensity and radius. However, even this relationship shows considerable \nscatter, whose origin is not clear. We list some possibilities below:\n\\begin{enumerate}\n\\item dependence of the scattered light correction on the shape of the umbra \n(e.g. very elongated versus circular);\n\\item dependence of intrinsic brightness of umbrae on shape and complexity;\n\\item dependence of brightness on age;\n\\item (small) dependence of brightness on phase of the solar cycle;\n\\item CLV of umbral contrast (which is small according to Fig. 18).\n\\end{enumerate}\nSeparating clearly between these possibilities is beyond the scope of \nthe current paper. Our analysis is restricted to regular sunspots with single umbrae. \nComplex spots may in principle show a different behaviour. \n\\section{Concluding remarks}\nIn this paper we present the analysis of MDI continuum sunspot images aimed at \ndetecting umbral core brightness variation with solar cycle. We analysed a total of \n234 sunspots of which 164 sunspots have an umbral radius lying between 5\\arcsec\\\/ - 15\\arcsec. \nCareful corrections for stray light and the Zeeman splitting of the nearby \nNi~{\\scriptsize I} \\\/line on measured continuum intensities have been made.\nWe derive the following conclusions from our analysis. \n\\begin{itemize}\n\\item The umbral core and mean brightness decreases substantially with increasing umbral radius. \n\\item The mean penumbral intensity is also reduced with increased spot size, but by a small amount.\n\\item No significant variation in umbral core, umbral mean and mean penumbral intensities is \nfound with solar cycle.\n\\item The insignificant variation with solar cycle of the umbral intensity could be at least \npartly be explained by the dependence of the analysed spot size on \nsolar cycle.\n\\end{itemize}\n\\begin{acknowledgements}\nWe wish to express our thanks to SOHO\/MDI team for providing the full-disk continuum \nintensity images. Thanks are also due to Dr. Andreas Lagg for providing the updated \ncode for computing the line profiles and the referee Aimee Norton for her useful suggestions. \nThis work was partly supported by the Deutsche Forschungsgemeinschaft, DFG project number \nSO~711\/1-1. Funding by the Spanish National Space Program (PNE) under project ESP2003-07735 \nis gratefully acknowledged. \n\\end{acknowledgements}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\nState complexity is one of the fundamental topics in automata\ntheory. It is important from both theoretical aspect and\nimplications in automata applications, because the state complexity\nof an operation gives an upper bound of both time and space\ncomplexity of the operation. For example, programmers should know\nthe largest possible number of states that would be generated before\nthey perform an operation in an application, since they need to\nallocate enough space for the computation and make an estimate of\nthe time it takes.\n\nThe research on state complexity can be recalled to\n1950's~\\cite{RaSc59}. However, most results on state complexity came\nout after\n1990~\\cite{CCSY99,CaSaYu02,DaDoSa08,Domaratzki02,HoKu02,JiJiSz05,Jriaskova05,JiOk05,PiSh2002,SaWoYu04,Yu01,YuZhSa94}.\nTheir research focused on individual operations, e.g. union,\nintersection, star, catenation, reversal, etc, until A. Salomaa, K.\nSalomaa and S. Yu initiated the study of state complexities of\ncombined operations in 2007~\\cite{SaSaYu07}. In the following three\nyears, many papers were published on this\ntopic~\\cite{CGKY10-cat-sr,CGKY10-cat-ui,DoOk09,EsGaLiYu09,GaSaYu08,GaYu09,JiOk07,LiMaSaYu08}.\n\nPeople are interested in state complexities of combined operations\nnot only because it is a relatively new research direction but also\nbecause its importance in practice. For example, several operations\nare often applied in a certain order on languages in searching and\nlanguage processing. If we simply use the mathematical composition\nof the state complexities of individual participating operations, we\nmay get a very huge value which is far greater than the exact state\ncomplexity of the combined operation, because the resulting\nlanguages of the worst case of one operation may not be among the\nworst case input languages of the next\noperation~\\cite{GaSaYu08,JiOk07,LiMaSaYu08,SaSaYu07}. Although\ncomputer technology is developing fast, time and space should still\nbe used efficiently. Thus, state complexities of combined operations\nare at least as important as those of individual operations.\n\n\nIn~\\cite{SaSaYu07}, two combined operations were investigated:\n$(L(M)\\cup L(N))^*$ and $(L(M)\\cap L(N))^*$, where $M$ and $N$ are\n$m$-state and $n$-state DFAs, respectively. In~\\cite{LiMaSaYu08},\nBoolean operations combined with reversal were studied, including:\n$(L(M)\\cup L(N))^R$ and $(L(M)\\cap L(N))^R$. One natural question is\nwhat are the state complexities of these combined operations if we\nexchanged the orders of the composed individual operations. For\nexample, we perform star or reversal first and then perform union or\nintersection. Thus, in this paper, we investigate four particular\ncombined operations: $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$,\n$L(M)^R\\cup L(N)$ and $L(M)^R\\cap L(N)$.\n\nIt has been shown in~\\cite{YuZhSa94} that, (1) the state\ncomplexities of the union and intersection of an $m$-state DFA\nlanguage and an $n$-state DFA language are both $mn$, (2) the state\ncomplexity of star of a $k$-state DFA language is $\\frac{3}{4}2^k$,\nand (3), the state complexity of reversal of an $l$-state DFA\nlanguage is $2^l$. In this paper, we obtain the state complexities\nof $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$, $L(M)^R\\cup L(N)$ and\n$L(M)^R\\cap L(N)$ and show that they are all less than the\nmathematical compositions of individual state complexities for\n$m,n\\ge 2$.\n\nWe prove that the state complexity of $L(M)^*\\cup L(N)$ is\n$\\frac{3}{4}2^m\\cdot n-n+1$ for $m$, $n\\ge 2$ which is much less\nthan the known state complexity of $(L(M)\\cup L(N))^*$\n(\\cite{SaSaYu07}). We obtain that the state complexity of\n$L(M)^*\\cap L(N)$ is also $\\frac{3}{4}2^m\\cdot n-n+1$ for $m$, $n\\ge\n2$ whereas the state complexity of $(L(M)\\cap L(N))^*$ has been\nproved to be $\\frac{3}{4}2^{mn}$, the mathematical compositions of\nindividual state complexities (\\cite{SaSaYu07}). For $L(M)^R\\cup\nL(N)$ and $L(M)^R\\cap L(N)$, we prove both of their state\ncomplexities to be $2^m\\cdot n-n+1$ for $m$, $n\\ge 2$ while the\nstate complexities of $(L(M)\\cup L(N))^R$ and $(L(M)\\cap L(N))^R$\nare both $2^{m+n}-2^m-2^n+2$ (\\cite{LiMaSaYu08}).\n\nIn the next section, we introduce the basic notations and\ndefinitions used in this paper. In\nSections~\\ref{star-union},~\\ref{star-intersection},~\\ref{reversal-union}\nand~\\ref{reversal-intersection}, we investigate the state\ncomplexities of $L(M)^*\\cup L(N)$, $L(M)^*\\cap L(N)$, $L(M)^R\\cup\nL(N)$ and $L(M)^R\\cap L(N)$, respectively. In\nSection~\\ref{sec:conclusion}, we conclude the paper .\n\n\n\n\\section{Preliminaries}\nAn alphabet $\\Sigma$ is a finite set of letters. A word $w \\in\n\\Sigma^*$ is a sequence of letters in $\\Sigma$, and the empty word,\ndenoted by $\\varepsilon$, is the word of length 0.\n\nA {\\it deterministic finite automaton} (DFA) is usually denoted by a\n5-tuple $A = (Q, \\Sigma, \\delta, s, F)$, where $Q$ is the finite and\nnonempty set of states, $\\Sigma$ is the finite and nonempty set of\ninput symbols, $\\delta: Q\\times\\Sigma \\rightarrow Q$ is the state\ntransition function, $s\\in Q$ is the initial state, and $F\\subseteq\nQ$ is the set of final states. A DFA is said to be {\\it complete} if\n$\\delta$ is a total function. Complete DFAs are the basic model for\nconsidering state complexity. Without specific mentioning, all DFAs\nare assumed to be complete in this paper. We extend $\\delta$ to $Q\n\\times \\Sigma^* \\rightarrow Q$ in the usual way. Then this automaton\naccepts a word $w \\in \\Sigma^*$ if $\\delta(s,w) \\cap F \\neq\n\\emptyset$. Two states in a DFA are said to be {\\it equivalent} if\nand only if for every word $w \\in \\Sigma^*$, if $A$ is started in\neither state with $w$ as input, it either accepts in both cases or\nrejects in both cases. The language accepted by a DFA $A$ is denoted\nby $L(A)$. A language is accepted by many DFAs but there is only one\nessentially unique {\\it minimal} DFA for the language which has the\nminimum number of states.\n\nA {\\it non-deterministic finite automaton} (NFA) is also denoted by\na 5-tuple $B = (Q, \\Sigma, \\delta, s, F)$, where $Q$, $\\Sigma$, $s$,\nand $F$ are defined the same way as in a DFA and $\\delta:\nQ\\times\\Sigma\\rightarrow 2^Q$ maps a pair consisting of a state and\nan input symbol into a set of states rather than a single state. An\nNFA may have multiple initial states, in which case an NFA is\ndenoted $(Q, \\Sigma, \\delta, S, F)$ where $S$ is the set of initial\nstates. A language $L$ is accepted by an NFA if and only if $L$ is\naccepted by a DFA, and such a language is called a {\\it regular\nlanguage}. Two finite automata are said to be equivalent if they\naccepts the same regular language. An NFA can always be transformed\ninto an equivalent DFA by performing subset construction. The reader\nmay refer to~\\cite{HoMoUl01,Yu97} for more details about regular\nlanguages and automata theory.\n\nThe {\\it state complexity} of a regular language $L$ is the number\nof states of the minimal, complete DFA accepting $L$. The state\ncomplexity of a class of regular languages is the worst among the\nstate complexities of all the languages in the class. The state\ncomplexity of an operation on regular languages is the state\ncomplexity of the resulting languages from the operation. For\nexample, we say that the state complexity of union of an $m$-state\nDFA language and an $n$-state DFA language is $mn$. This implies\nthat the largest number of states of all the minimal, complete DFAs\nthat accept the union of an $m$-state DFA language and an $n$-state\nDFA language,\nis $mn$, and such languages exist. Thus, state complexity is a\nworst-case complexity.\n\n\n\n\\section{State complexity of $L_1^*\\cup L_2$}\\label{star-union}\nWe first consider the state complexity of $L_1^*\\cup L_2$, where\n$L_1$ and $L_2$ are regular languages accepted by $m$-state and\n$n$-state DFAs, respectively. It has been proved that the state\ncomplexity of $L_1^*$ is $\\frac{3}{4}2^m$ and the state complexity\nof $L_1\\cup L_2$ is $mn$~\\cite{Maslov70,YuZhSa94}. The mathematical\ncomposition of them is $\\frac{3}{4}2^m\\cdot n$. In the following, we\nshow that this upper bound can be lower.\n\n\\begin{theorem}\n\\label{star union upper bound}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ such that\n$|F_M-\\{ s_M \\}|=k\\geq 1$, $m\\geq 2$, $n\\geq 1$, there exists a DFA\nof at most $(2^{m-1}+2^{m-k-1})\\cdot n-n+1$ states that accepts\n$L(M)^*\\cup L(N)$.\n\\end{theorem}\n\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ be a\ncomplete DFA of $m$ states. Denote $|F_M-\\{ s_M \\}|$ by $F_0$. Then\n$F_0=k\\geq 1$ Let $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ be another\ncomplete DFA of $n$ states. Let DFA $M'=(Q_{M'},\\Sigma , \\delta_{M'}\n, s_{M'}, F_{M'})$ where\n\\begin{eqnarray*}\n& & s_{M'} \\notin Q_M\\mbox{ is a new start state,}\\\\\n& & Q_{M'} = \\{s_{M'}\\}\\cup \\{P\\mid P\\subseteq (Q_M-F_0)\\mbox{ \\& } P\\neq \\emptyset \\} \\\\\n& & \\qquad \\cup \\{R\\mid R\\subseteq Q_M \\mbox{ \\& } s_M\\in R \\mbox{ \\& }R\\cap F_0\\neq \\emptyset \\},\\\\\n& & \\delta_{M'}(s_{M'}, a)= \\{\\delta_M(s_M, a)\\mbox{ for any $a\\in \\Sigma$} \\},\\\\\n& & \\delta_{M'}(R, a)= \\{\\delta_M(R, a)\\}\\mbox{ for $R\\subseteq Q_M$ and $a\\in \\Sigma$ if $\\delta_M(R, a)\\cap F_0=\\emptyset$},\\\\\n& & \\delta_{M'}(R, a)= \\{ \\delta_M(R, a)\\}\\cup \\{s_M\\}\\mbox{ otherwise}, \\\\\n& & F_{M'}= \\{s_{M'}\\}\\cup\\{R\\mid R\\subseteq Q_M \\mbox{ \\& } R\\cap\nF_M\\neq \\emptyset \\}.\n\\end{eqnarray*}\n\nIt is clear that $M'$ accepts $L(M)^*$. In the second term of the\nunion for $Q_{M'}$ there are $2^{m-k}-1$ states. And in the third\nterm, there are $(2^k-1)2^{m-k-1}$ states. So $M'$ has\n$2^{m-1}+2^{m-k-1}$ states in total. Now we construct another DFA\n$A=(Q,\\Sigma , \\delta , s, F)$ where\n\\begin{eqnarray*}\n& & s=\\langle s_{M'},s_N \\rangle,\\\\\n& & Q = \\{\\langle i,j \\rangle \\mid i\\in Q_{M'}-\\{s_{M'}\\},j\\in Q_N\\}\\cup \\{s \\}, \\\\\n& & \\delta(\\langle i,j \\rangle, a)= \\langle \\delta_{M'}(i, a),\\delta_N(j, a) \\rangle \\mbox{, $\\langle i,j \\rangle \\in Q$, $a\\in \\Sigma$},\\\\\n& & F= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{ or }j\\in F_N \\}.\n\\end{eqnarray*}\nWe can see that $$L(A)=L(M')\\cup L(N)=L(M)^*\\cup L(N).$$ Note\n$\\langle s_{M'},j \\rangle \\notin Q$, for $j\\in Q_N-\\{s_N\\}$, because\nthere is no transition going into $s_{M'}$ in DFA $M'$. So there are\nat least $n-1$ states in $Q$ are not reachable. Thus, the number of states of minimal DFA accepting $L(M)^*\\cup L(N)$ is no more than\\\\\n$$|Q|=(2^{m-1}+2^{m-k-1})\\cdot n-n+1. \\ \\ \\Box$$\n\nIf $s_M$ is the only final state of $M$($k=0$), then $L(M)^*=L(M)$.\n\n\\begin{corollary}\n\\label{star union upper bound corollary}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$, $m>1$, $n>0$,\nthere exists a DFA $A$ of at most $\\frac{3}{4}2^m\\cdot n-n+1$ states\nsuch that $L(A)=L(M)^*\\cup L(N)$.\n\\end{corollary}\n{\\bf Proof.\\ \\ } Let $k$ be defined as in the above proof. There are\ntwo cases in the following.\n\\begin{itemize}\n\\item[{\\rm (I)}]$k=0$. In this case, $L(M)^*=L(M)$. Then $A$ simply needs at most $m\\cdot\nn$ states, which is less than $\\frac{3}{4}2^m\\cdot n-n+1$ when\n$m>1$.\n\\item[{\\rm (II)}]$k\\geq 1$. The claim is clearly true by Theorem~\\ref{star union upper bound}.\n$\\ \\ \\Box$\n\\end{itemize}\nNext, we show that the upper bound $\\frac{3}{4}2^m\\cdot n-n+1$ is\nreachable.\n\\begin{theorem}\n\\label{star union lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^*\\cup L(N)$ needs at least $\\frac{3}{4}2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , 0, \\{ m-1 \\})$ be a\nDFA, where $Q_M = \\{0,1,\\ldots ,m-1\\}$, $\\Sigma = \\{a,b,c\\}$ and the\ntransitions of $M$ are\n\\begin{eqnarray*}\n& & \\delta_M(i, a) = i+1 \\mbox{ mod $m$, } i=0,1, \\ldots , m-1,\\\\\n& & \\delta_M(0, b) = 0 \\mbox{, }\\delta_M(i, b) = i+1 \\mbox{ mod $m$, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(i, c) = i \\mbox{, } i=0,1, \\ldots , m-1.\n\\end{eqnarray*}\nThe transition diagram of $M$ is shown in\nFigure~\\ref{star-union-first}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{star-union-first.eps}\n \\caption{The transition diagram of the witness DFA $M$ of Theorems~\\ref{star union lower bound} and~\\ref{star intersection lower bound}}\n\\label{star-union-first}\n\\end{figure}\\\\\nLet $N=(Q_N,\\Sigma , \\delta_N , 0, \\{n-1\\})$ be another DFA, where\n$Q_N = \\{0,1,\\ldots ,n-1\\}$ and\n\\begin{eqnarray*}\n& & \\delta_N(i, a) = i \\mbox{, } i=0,1, \\ldots , n-1,\\\\\n& & \\delta_N(i, b) = i \\mbox{, } i=0,1, \\ldots , n-1,\\\\\n& & \\delta_N(i, c) = i+1 \\mbox{ mod $n$, } i=0,1, \\ldots , n-1.\n\\end{eqnarray*}\nThe transition diagram of $N$ is shown in\nFigure~\\ref{star-union-second}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{star-union-second.eps}\n \\caption{The transition diagram of the witness DFA $N$ of Theorems~\\ref{star union lower bound} and~\\ref{star intersection lower bound}}\n\\label{star-union-second}\n\\end{figure}\n\n\n\nIt has been proved in~\\cite{YuZhSa94} that the minimal DFA accepting\nthe star of an $m$-state DFA language has $\\frac{3}{4}2^m$ states in\nthe worst case. $M$ is a modification of worst case example given in\n~\\cite{YuZhSa94} by adding a $c$-loop to every state. So we design a\n$\\frac{3}{4}2^m$-state, minimal DFA $M'=(Q_{M'},\\Sigma , \\delta_{M'}\n, s_{M'}, F_{M'})$ that accepts $L(M)^*$, where\n\\begin{eqnarray*}\n& & s_{M'} \\notin Q_M\\mbox{ is a new start state,}\\\\\n& & Q_{M'} = \\{s_{M'}\\}\\cup \\{P\\mid P\\subseteq \\{0,1,\\ldots ,m-2\\}\\mbox{ \\& } P\\neq \\emptyset \\} \\\\\n& & \\qquad \\cup \\{R\\mid R\\subseteq \\{0,1,\\ldots ,m-1\\} \\mbox{ \\& } 0\\in R \\mbox{ \\& }m-1\\in R \\},\\\\\n& & \\delta_{M'}(s_{M'}, a)= \\{\\delta_M(0, a)\\mbox{ for any $a\\in \\Sigma$} \\},\\\\\n& & \\delta_{M'}(R, a)= \\{\\delta_M(R, a)\\}\\mbox{ for $R\\subseteq Q_M$ and $a\\in \\Sigma$ if $m-1\\notin \\delta_M(R, a)$},\\\\\n& & \\delta_{M'}(R, a)= \\{ \\delta_M(R, a)\\}\\cup \\{0\\}\\mbox{ otherwise}, \\\\\n& & F_{M'}= \\{s_{M'}\\}\\cup\\{R\\mid R\\subseteq \\{0,1,\\ldots ,m-1\\}\n\\mbox{ \\& } m-1\\in R\\}.\n\\end{eqnarray*}\n\nThen we construct a DFA $A=(Q,\\Sigma , \\delta , s, F)$ accepting\n$L(M)^*\\cup L(N)$ exactly as described in the proof of\nTheorem~\\ref{star union upper bound}, where\n\\begin{eqnarray*}\n& & s=\\langle s_{M'},0 \\rangle,\\\\\n& & Q = \\{\\langle i,j \\rangle \\mid i\\in Q_{M'}-\\{s_{M'}\\},j\\in Q_N\\}\\cup \\{s \\}, \\\\\n& & \\delta(\\langle i,j \\rangle, a)= \\langle \\delta_{M'}(i, a),\\delta_N(j, a) \\rangle \\mbox{, $\\langle i,j \\rangle \\in Q$, $a\\in \\Sigma$},\\\\\n& & F= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{ or }j=n-1 \\}.\n\\end{eqnarray*}\n\nNow we need to show that $A$ is a minimal DFA.\n\\begin{itemize}\n\\item[{\\rm (I)}]All the states in $Q$ are reachable.\\\\\nFor an arbitrary state $\\langle i,j\\rangle$ in $Q$, there always\nexists a string $w_1w_2$ such that $\\delta(\\langle s_M',0\\rangle,\nw_1w_2) = \\langle i,j\\rangle$, where\n\\begin{eqnarray*}\n& & \\delta_{M'}(s_{M'}, w_1)=i\\mbox{, }w_1\\in \\{a,b\\}^*,\\\\\n& & \\delta_N (0, w_2)=j\\mbox{, }w_2\\in \\{c\\}^*.\n\\end{eqnarray*}\n\\item[{\\rm (II)}]Any two different states $\\langle i_1,j_1\\rangle$ and $\\langle i_2,j_2\\rangle$ in $Q$ are\ndistinguishable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i_1\\neq i_2$, $j_2\\neq n-1$. We can find a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1)\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_M'(i_2, w_1) \\notin F_M'$.\n\n\\item[{\\rm 2.}]$i_1\\neq i_2$, $j_2= n-1$. There exists a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1c)\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1c) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_{M'}(i_2, w_1) \\notin F_{M'}$.\n\\item[{\\rm 3.}]$i_1= i_2\\notin F_{M'}$, $j_1\\neq j_2$. For this case, a string $c^{n-1-j_1}$ can distinguish the two states, since $\\delta(\\langle i_1,j_1\\rangle, c^{n-1-j_1})\\in\nF$ and $\\delta(\\langle i_2,j_2\\rangle, c^{n-1-j_1}) \\notin F$.\n\n\\item[{\\rm 4.}]$i_1= i_2\\in F_{M'}$, $j_1\\neq j_2$. A string $b^mc^{n-1-j_1}$ can distinguish them, because $\\delta(\\langle i_1,j_1\\rangle, b^mc^{n-1-j_1})\\in\nF$ and $\\delta(\\langle i_2,j_2\\rangle, b^mc^{n-1-j_1}) \\notin F$.\n\n\\end{itemize}\n\\end{itemize}\nSince all the states in $A$ are reachable and distinguishable, DFA\n$A$ is minimal. Thus, any DFA accepting $L(M)^*\\cup L(N)$ needs at\nleast $\\frac{3}{4}2^m\\cdot n-n+1$ states. $\\ \\ \\Box $\n\nThis result gives a lower bound for the state complexity of\n$L(M)^*\\cup L(N)$. It coincides with the upper bound in\nCorollary~\\ref{star union upper bound corollary}. So we have the\nfollowing Theorem~\\ref{Tight bound of star union}.\n\\begin{theorem}\n\\label{Tight bound of star union}\n\nFor any integer $m\\geq 2$, $n\\geq 2$, $\\frac{3}{4}2^m\\cdot n-n+1$\nstates are both sufficient and necessary in the worst case for a DFA\nto accept $L(M)^*\\cup L(N)$, where $M$ is an $m$-state DFA and $N$\nis an $n$-state DFA.\n\\end{theorem}\n\n\n\n\n\n\n\n\\section{State complexity of $L(M)^*\\cap L(N)$}\\label{star-intersection}\n\nSince the state complexity of intersection on regular languages is\nthe same as that of union~\\cite{YuZhSa94}, the mathematical\ncomposition of the state complexities of star and intersection is\nalso $\\frac{3}{4}2^m$. In this section, we show that the state\ncomplexity of $L(M)^*\\cap L(N)$ is $\\frac{3}{4}2^m\\cdot n-n+1$ which\nis the same as the state complexity of $L(M)^*\\cup L(N)$.\n\n\\begin{theorem}\n\\label{star intersection upper bound}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$ such that\n$|F_M-\\{ s_M \\}|=k\\geq 1$, $m>1$, $n>0$, there exists a DFA of at\nmost $(2^{m-1}+2^{m-k-1})\\cdot n-n+1$ states that accepts\n$L(M)^*\\cap L(N)$.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } We construct a DFA $A$ accepting $L(M)^*\\cap L(N)$\nthe same as in the proof of Theorem~\\ref{star union upper bound}\nexcept that its set of final states is\n\\[\nF= \\{\\langle i,j \\rangle \\mid i\\in F_{M'}\\mbox{, }j\\in F_N \\}.\n\\]\nThus, after reducing the $n-1$ unreachable states $\\langle s_{M'},j\n\\rangle \\notin Q$, for $j\\in Q_N-\\{s_N\\}$, the number of states of\n$A$ is sill no more than $(2^{m-1}+2^{m-k-1})\\cdot n-n+1. \\ \\ \\Box$\n\nSimilarly to the proof of Corollary~\\ref{star union upper bound\ncorollary}, we consider both the case that $M$ has no other final\nstate except $s_M$ ($L(M)^*=L(M)$) and the case that $M$ has some\nother final states (Theorem~\\ref{star intersection upper bound}).\nThen we obtain the following corollary. Detailed proof may be\nomitted.\n\n\\begin{corollary}\n\\label{star intersection upper bound corollary}\n\nFor any $m$-state DFA $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ and\n$n$-state DFA $N=(Q_N,\\Sigma , \\delta_N , s_N, F_N)$, $m>1$, $n>0$,\nthere exists a DFA $A$ of at most $\\frac{3}{4}2^m\\cdot n-n+1$ states\nsuch that $L(A)=L(M)^*\\cap L(N)$.\n\\end{corollary}\nNext, we show that this general upper bound of state complexity of\n$L(M)^*\\cap L(N)$ can be reached by some witness DFAs.\n\\begin{theorem} \\label{star intersection lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^*\\cap L(N)$ needs at least $\\frac{3}{4}2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } We use the same DFAs $M$ and $N$ as in the proof of\nTheorem~\\ref{star union lower bound}. Their transition diagrams are\nshown in Figure~\\ref{star-union-first} and\nFigure~\\ref{star-union-second}, respectively. Construct DFA\n$M'=(Q_{M'},\\Sigma , \\delta_{M'} , s_{M'}, F_{M'})$ that accepts\n$L(M)^*$ in the same way.\n\nThen we construct a DFA $A=(Q,\\Sigma , \\delta , s, F)$ accepting\n$L(M)^*\\cap L(N)$ exactly as described in the proof of\nTheorem~\\ref{star union lower bound} except that\n\\[\nF= \\{\\langle i,n-1 \\rangle \\mid i\\in F_{M'} \\}.\n\\]\n\nNow we prove that $A$ is minimal.\n\\begin{itemize}\n\\item[{\\rm (I)}]Every state of $A$ is reachable.\\\\\nLet $\\langle i,j\\rangle$ be an arbitrary state of $A$. Then there\nalways exists a string $w_1w_2$ such that $\\delta(\\langle\ns_{M'},0\\rangle, w_1w_2) = \\langle i,j\\rangle$, where\n\\begin{eqnarray*}\n& & \\delta_{M'}(s_{M'}, w_1)=i\\mbox{, }w_1\\in \\{a,b\\}^*,\\\\\n& & \\delta_N (0, w_2)=j\\mbox{, }w_2\\in \\{c\\}^*.\n\\end{eqnarray*}\n\\item[{\\rm (II)}]Any two different states $\\langle i_1,j_1\\rangle$ and $\\langle i_2,j_2\\rangle$ of $A$ are\ndistinguishable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i_1\\neq i_2$.\n\nWe can find a string $w_1$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_1c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_1c^{n-1-j_1}) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$, $\\delta_{M'}(i_1, w_1)\\in F_{M'}$ and\n$\\delta_{M'}(i_2, w_1) \\notin F_{M'}$.\n\n\\item[{\\rm 2.}]$i_1= i_2\\notin F_{M'}$, $j_1\\neq j_2$.\n\nThere exists a string $w_2$ such that\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, w_2c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, w_2c^{n-1-j_1}) \\notin F,\n\\end{eqnarray*}\nwhere $w_1\\in \\{a,b\\}^*$ and $\\delta_{M'}(i_1, w_2)\\in F_{M'}$.\n\\item[{\\rm 3.}]$i_1= i_2\\in F_{M'}$, $j_1\\neq j_2$.\n\\begin{eqnarray*}\n& & \\delta(\\langle i_1,j_1\\rangle, c^{n-1-j_1})\\in F,\\\\\n& & \\delta(\\langle i_2,j_2\\rangle, c^{n-1-j_1}) \\notin F.\n\\end{eqnarray*}\n\\end{itemize}\n\\end{itemize}\nDue to (I) and (II), $A$ is a minimal DFA with $\\frac{3}{4}2^m\\cdot\nn-n+1$ states which accepts $L(M)^*\\cap L(N)$. $\\ \\ \\Box $\n\nThis lower bound coincides with the upper bound in\nCorollary~\\ref{star intersection upper bound corollary}. Thus, the\nbounds are tight.\n\\begin{theorem}\n\\label{Tight bound of star intersection}\n\nFor any integer $m\\geq 2$, $n\\geq 2$, $\\frac{3}{4}2^m\\cdot n-n+1$\nstates are both sufficient and necessary in the worst case for a DFA\nto accept $L(M)^*\\cap L(N)$, where $M$ is an $m$-state DFA and $N$\nis an $n$-state DFA.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\\section{State complexity of $L_1^R\\cup L_2$}\\label{reversal-union}\nIn this section, we study the state complexity of $L_1^R\\cup L_2$,\nwhere $L_1$ and $L_2$ are regular languages. It has been proved that\nthe state complexity of $L_1^R$ is $2^m$ and the state complexity of\n$L_1\\cup L_2$ is $mn$~\\cite{Maslov70,YuZhSa94}. Thus, the\nmathematical composition of them is $2^m\\cdot n$. In this section we\nwill prove that this upper bound of state complexity of $L_1^R\\cup\nL_2$ can not be reached in any case. We will first try to lower the\nupper bound in the following.\n\n\n\\begin{theorem}\n\\label{reversal uion upper bound}\n\nLet $L_1$ and $L_2$ be two regular language accepted by an $m$-state\nand $n$-state DFAs, respectively. Then there exists a DFA of at most\n$2^m\\cdot n-n+1$ states that accepts $L_1^R\\cup L_2$.\n\\end{theorem}\n\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , s_M, F_M)$ be a\ncomplete DFA of $m$ states and $L_1=L(M)$. Let $N=(Q_N,\\Sigma ,\n\\delta_N , s_N, F_N)$ be another complete DFA of $n$ states and\n$L_2=L(N)$. Let $M'=(Q_M,\\Sigma , \\delta_{M'} , F_M, \\{s_M\\})$ be an\nNFA with multiple initial states. $\\delta_{M'}(p,a)=q$ if\n$\\delta_M(q,a)=p$ where $a\\in \\Sigma$ and $p,q\\in Q_M$. Clearly,\n$L(M')=L(M)^R=L_1^R$. After performing subset construction, we can\nget a $2^m$-state DFA $A=(Q_A,\\Sigma , \\delta_A , s_A, F_A)$ that is\nequivalent to $M'$. Since $A$ has $2^m$ states, one of its final\nstate must be $Q_M$. Now we construct a DFA $B=(Q_B,\\Sigma ,\n\\delta_B , s_B, F_B)$, where\n\\begin{eqnarray*}\n& & Q_B = \\{\\langle i,j \\rangle \\mid i\\in Q_A\\mbox{, } j\\in Q_N\\},\\\\\n& & s_B = \\langle s_A,s_N \\rangle,\\\\\n& & F_B = \\{\\langle i,j \\rangle\\in Q_B\\mid i\\in F_A\\mbox{ or } j\\in F_N\\},\\\\\n& & \\delta_B(\\langle i,j \\rangle, a) = \\langle i',j' \\rangle \\mbox{,\nif } \\delta_A(i,a)=i'\\mbox{ and }\\delta_N(j,a)=j'\\mbox{, }a\\in\n\\Sigma.\n\\end{eqnarray*}\nIt is easy to see that $\\delta_B(\\langle Q_M,j \\rangle, a) \\in F_B$\nfor any $j\\in Q_N$ and $a\\in \\Sigma$. This means all the states\n(two-tuples) starting with $Q_1$ are equivalent. There are $n$ such\nstates in total. Thus, the minimal DFA accepting $L_1^R\\cup L_2$ has\nno more than $2^m\\cdot n-n+1$ states.$\\ \\ \\Box $\n\n\n\nThis result gives an upper bound of state complexity of $L_1^R\\cup\nL_2$. Now let's see if this bound is reachable.\n\n\\begin{theorem}\n\\label{reversal uion lower bound}\n\nGiven two integers $m\\geq 2$, $n\\geq 2$, there exists a DFA $M$ of\n$m$ states and a DFA $N$ of $n$ states such that any DFA accepting\n$L(M)^R\\cup L(N)$ needs at least $2^m\\cdot n-n+1$ states.\n\\end{theorem}\n\n{\\bf Proof.\\ \\ } Let $M=(Q_M,\\Sigma , \\delta_M , 0, \\{0\\})$ be a\nDFA, where $Q_M = \\{0,1,\\ldots ,m-1\\}$, $\\Sigma = \\{a,b,c,d\\}$ and\nthe transitions are\n\\begin{eqnarray*}\n& & \\delta_M(0, a) = m-1 \\mbox{, }\\delta_M(i, a) = i-1 \\mbox{, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(0, b) = 1 \\mbox{, } \\delta_M(i, b) = i \\mbox{, } i=1, \\ldots , m-1,\\\\\n& & \\delta_M(0, c) = 1 \\mbox{, } \\delta_M(1, c) = 0 \\mbox{, }\\delta_M(j, c) = i \\mbox{, } j=2, \\ldots , m-1,\\\\\n& & \\delta_M(k, d) = k \\mbox{, } k=0, \\ldots , m-1.\n\\end{eqnarray*}\nThe transition diagram of $M$ is shown in\nFigure~\\ref{reversal-union-first}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{reversal.eps}\n \\caption{The transition diagram of the witness DFA $M$ of Theorems~\\ref{reversal uion lower bound} and~\\ref{reversal intersection lower bound}}\n\\label{reversal-union-first}\n\\end{figure}\nLet $N=(Q_N,\\Sigma , \\delta_N , 0, \\{0\\})$ be another DFA, where\n$Q_N = \\{0,1,\\ldots ,n-1\\}$, $\\Sigma = \\{a,b,c,d\\}$ and the\ntransitions are\n\\begin{eqnarray*}\n& & \\delta_N(i, a) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, b) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, c) = i \\mbox{, } i=0, \\ldots , n-1,\\\\\n& & \\delta_N(i, d) = i+1 \\mbox{ mod }n \\mbox{, } i=0, \\ldots , n-1.\n\\end{eqnarray*}\nThe transition diagram of $N$ is shown in\nFigure~\\ref{reversal-union-second}.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.60]{onecircle.eps}\n \\caption{The transition diagram of the witness DFA $N$ of Theorems~\\ref{reversal uion lower bound} and~\\ref{reversal intersection lower bound}}\n\\label{reversal-union-second}\n\\end{figure}\n\nNote that $M$ is a modification of worst case example given\nin~\\cite{YuZhSa94} for reversal, by adding a $d$-loop to every\nstate. Intuitively, the minimal DFA accepting $L(M)^R$ should also\nhave $2^m$ states. Before using this result, we will prove it first.\nLet $A=(Q_A,\\Sigma , \\delta_A , \\{0\\}, F_A)$ be a DFA, where\n\\begin{eqnarray*}\n& & Q_A = \\{q\\mid q\\subseteq Q_M\\},\\\\\n& & \\Sigma = \\{a,b,c,d\\},\\\\\n& & \\delta_A(p, e) = \\{j\\mid \\delta_M(i, e)=j\\mbox{, }i\\in p\\} \\mbox{, } p\\in Q_A\\mbox{, } e\\in \\Sigma,\\\\\n& & F_A = \\{q\\mid \\{0\\}\\in q \\mbox{, }q\\in Q_A\\}.\n\\end{eqnarray*}\nClearly, $A$ has $2^m$ states and it accepts $L(M)^R$. Now let's\nprove it is minimal.\n\\begin{itemize}\n\\item[{\\rm (i)}]Every state $i \\in Q_A$ is\nreachable.\\\\\n\\begin{itemize}\n\\item[{\\rm 1.}]$i=\\emptyset$.\\\\\n$|i|=0$ if and only if $i=\\emptyset$. $\\delta_A(\\{ 0 \\}, b) =\ni=\\emptyset .$\n\\item[{\\rm 2.}]$|i|=1$.\\\\\nAssume that $i=\\{ p \\}$, $0\\leq p\\leq m-1$. $\\delta_A(\\{ 0 \\}, a^p)\n=i.$\n\\item[{\\rm 3.}]$2\\leq |i|\\leq m$.\\\\\nAssume that $i=\\{ i_1, i_2, \\ldots ,i_k \\}$, $0\\leq i_10, \n\\eeq\nand showed that there is a maximum charge and size.\nTo construct large Q-balls, Anagnostopoulos {\\it et al.} \\cite{AAFT} introduced fermions with charge of the opposite sign.\nLi {\\it et al.} \\cite{LHL} assumed a different potential, a piecewise parabolic function, and Deshaies-Jacques and MacKenzie \\cite{DM} supposed the Maxwell-Chern-Simons theory with the $V_4$ potential (\\ref{V4}) in the 2+1 dimensional spacetime; it was shown that there is a maximum charge and size of Q-balls in both models.\n\nArod\\'z and Lis \\cite{Arodz} considered gauged Q-balls with the V-shaped potential,\n\\beq\\label{VV}\nV_{\\rm V}(\\phi):=\\lambda\\frac{|\\phi |}{\\sqrt{2}}\n~~~{\\rm with} ~~~ \\lambda >0, \n\\eeq\nBecause its three-dimensional plot has the form of a cone, it would be more appropriate to call it the cone-shaped potential.\nIn addition to normal Q-balls, which have a maximum charge, they found a new type of solutions, Q-shells.\nQ-shell solutions are obtained in such a way that the scalar field and the gauge field are assumed to be constant within a certain sphere $rr_0$. \nBecause the electric charge is concentrated on the shell, large Q-balls with any amount of charge can exist without additional fermions. \nThus this model overcomes the difficulty of the $V_4$ model.\nHowever, there is another drawback that it is so simplified and singular at $\\phi=0$. \n\nIn this paper we address the question whether such large gauged Q-balls can be formed in realistic or cosmologically-motivated theories without additional fermions nor a singular potential.\nOne of the physically-motivated theories is the AD mechanism \\cite{AD}, which includes two types of potentials,\ngravity-mediation type and gauge-mediation type. \nThe former is described by \n\\bea\\label{gravity}\n&&V_{\\rm grav.}(\\phi):=\\frac{m_{\\rm grav.}^2}{2}\\phi^2\\left[\n1+K\\ln \\left(\\frac{\\phi}{M}\\right)^2\n\\right]~~ \\nonumber \\\\\n&&{\\rm with} ~~ m_{\\rm grav.}^2,~M>0,\n\\eea\nwhile the latter by\n\\beq\\label{gauge}\nV_{\\rm gauge}(\\phi):=m_{\\rm gauge}^4 \\ln\\left(1+\\frac{\\phi^2}{m_{\\rm gauge}^2}\\right)~~~\n{\\rm with} ~~~ m_{\\rm gauge}^2>0\\ .\n\\eeq\nIf we take Maclaurin expansion of the two potentials in the vicinity of $\\phi=0$, the latter can be regarded as $V_{4}$ model, and is inappropriate for our purpose.\nThus, we concentrate on investigating gauged Q-balls in the former potential. \n\nThis paper is organized as follows.\nIn Sec. II, we show the basic equations of gauged Q-balls. \nIn Sec. III, we discuss general properties of ordinary and gauged Q-balls in words of Newtonian mechanics.\nIn Sec. IV, we review previous results of $V_{4}$ and $V_{\\rm V}$ models. \nIn Sec. V, we investigate equilibrium solutions in the $V_{\\rm grav.}$ model numerically.\nSection VI is devoted to concluding remarks.\n\n\\section{basic equations}\n\nConsider an SO(2) symmetric scalar field $\\bp=(\\phi_1,\\phi_2)$ coupled to a gauged field $A_\\mu$,\n\\beq\\label{S}\n{\\cal S}=\\int d^4x\\left[\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n-\\frac{1}{2}\\eta^{\\mu\\nu}D_{\\mu}\\phi_{a} D_{\\nu}\\phi_{a}-V(\\phi) \\right],\n\\eeq\nwhere \n\\bea\n&&\\phi:=\\sqrt{\\phi_{a}\\phi_{a}},~~~\nF_{\\mu\\nu}:=\\pa_\\mu A_\\nu-\\pa_\\nu A_\\mu,\\\\\n&&D_{\\mu}\\phi_{a}:=\\pa_{\\mu}\\phi_{a}+A_{\\mu}\\epsilon_{ab}\\phi_{b}~(a,b=1,2). \n\\eea\nTo find spherically symmetric and equilibrium solutions with vanishing magnetic fields, we assume\n\\beq\\label{qball-phase}\n\\bp=\\phi(r)(\\cos\\omega t,\\sin\\omega t),~~~\nA_{0}=A_{0}(r),~~~ A_i=0,\n\\eeq\nwhere the subscript $i$ denotes spatial components and runs 1 to 3.\nIntroducing a variable,\n\\beq\n\\Omega(r):=\\omega+qA_{0}(r),\n\\eeq\nwe obtain field equations, \n\\bea\\label{FEqball}\n&&\\frac{d^2\\phi}{dr^2}+\\frac{2}{r}\\frac{d\\phi}{dr}+\\Omega^2 \\phi=\\frac{dV}{d\\phi}, \\\\\n&&\\frac{d^2\\Omega}{dr^2}+\\frac{2}{r}\\frac{d\\Omega}{dr}=\\Omega (q\\phi)^{2}. \\label{FEqball2}\n\\eea\n\nThe boundary condition we assume is \n\\bea\n&&{d\\phi\\over dr}(r=0)=0,~~{d\\Omega\\over dr}(r=0)=0, \\label{BCqball} \\\\\n&&\\phi(r\\ra\\infty)=0,~~\\Omega(r\\ra\\infty)=\\omega+\\frac{C}{r}, \\label{BCqball2}\n\\eea\nwhere $C$ is a constant.\nIn numerical calculation we must choose $\\Omega$ and $\\phi$ at $\\tilde{r}=0$ to satisfy the asymptotic conditions \n(\\ref{BCqball2}). \nIn concrete, we seek for appropriate $\\phi (0)$ for a fixed $\\Omega (0)$. \n\nWe define the energy and the charge, respectively, as\n\\bea\\label{Edef}\nE&=&\\int d^3xT_{00}\\nn\n&=&2\\pi \\int_0^{\\infty}r^2 dr\n\\left\\{\\Omega^2\\phi^2+\\left({d\\phi\\over dr}\\right)^2+\\left({d\\Omega\\over dr}\\right)^2+2V\\right\\},\\nn\nQ&=&\\int d^3x(\\phi_1D_0\\phi_2-\\phi_2D_0\\phi_1)\\nn\n&=&4\\pi \\int_0^{\\infty}r^2\\Omega\\phi^2dr,\n\\label{Qdef}\\eea\nwhere $T_{00}$ is the time-time component of the energy momentum tensor, which is defined by\n\\bea\nT_{\\mu\\nu}&=&D_\\mu\\phi_aD_\\nu\\phi_a-\\eta_{\\mu\\nu}\\left[\\frac12(D_\\lambda\\phi_a)^2+V\\right]\\nn\n&&+F_{\\mu\\lambda}F_\\nu^\\lambda-\\frac14\\eta_{\\mu\\nu}(F_{\\lambda\\sigma})^2.\n\\eea\nEquations (\\ref{FEqball}), (\\ref{FEqball2}) and (\\ref{Qdef}) indicate that the sign transformation\n$\\Omega \\to -\\Omega$ changes nothing but $Q\\to -Q$ with keeping $E$ and $\\phi(r)$ unchanged.\nThus, we choose $\\Omega >0$ in this paper.\n\n\\section{General Properties of Ordinary and Gauged Q-ball Solutions. }\n\nTo begin with, to understand the effect of gauge fields on Q-balls, we review properties of ordinary Q-ball solutions.\nThe field equations are obtained by putting $\\Omega=\\omega$=constant in Eq.(\\ref{FEqball}),\n\\beq\\label{FEOQ}\n\\frac{d^2\\phi}{dr^2}+\\frac{2}{r}\\frac{d\\phi}{dr}=\\frac{dV_\\omega}{d\\phi},~~~\nV_\\omega :=V-\\frac12\\omega^2\\phi^2.\n\\eeq\nIf one regards the radius $r$ as \\lq time\\rq\\ and the scalar amplitude $\\phi(r)$ as \\lq the position of a particle\\rq,\none can understand solutions in words of Newtonian mechanics, as shown in Fig.\\ \\ref{f1}.\nEquation (\\ref{FEOQ}) describes a one-dimensional motion of a particle under the nonconserved force \ndue to the effective potential $-V_{\\omega}(\\phi)$ and the \\lq time\\rq-dependent friction $-(2\/r)d\\phi\/dr$.\nIf one chooses the \\lq initial position\\rq\\ $\\phi(0)$ appropriately, the static particle begins to roll down the potential slope, climbs up and approaches the origin over infinite time.\n\n\\begin{figure}[htbp]\n\\psfig{file=f1,width=3.2in}\n\\caption{\\label{f1}\nInterpretation of ordinary Q-balls by analogy with a particle motion in Newtonian mechanics.}\n\\end{figure}\n\nFrom the above picture, one can derive the existing conditions of equilibrium solutions of ordinary Q-balls as follows.\nThe first condition is that the \\lq initial altitude of the particle\\rq\\ $-V_\\omega(\\phi(0))$ is larger than the \\lq final altitude\\rq\\\n$-V_\\omega(\\phi(\\infty))=0$, which leads to\n\\beq\\label{cond1}\n{\\rm max}[-V_\\omega(\\phi)]>0,~~i.e.,~~\n{\\rm min}\\left[{2V\\over\\phi^2}\\right]<\\omega^2.\n\\eeq\nThe second condition is that the \\lq particle climbs up\\rq\\ at $r\\to\\infty$, which leads to\n\\beq\\label{cond2}\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(-{dV_\\omega\\over d\\phi}\\right)=\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(\\omega^2\\phi-{dV\\over d\\phi}\\right)<0\\ .\n\\eeq\nIf the lowest-order term of $V$ is quadratic, i.e., $\\ds V=\\frac12m^2\\phi^2+O(\\phi^3)$, the second condition (\\ref{cond2}) reduces to\n\\beq\\label{cond22}\n\\omega^20$, the condition (\\ref{cond2}) is satisfied regardless of $\\omega$. Similarly, in the case of $V_{\\rm grav.}$, if we take $K<0$, the condition (\\ref{cond2}) is satisfied regardless of $\\omega$. \n\nNow let us move on to gauged Q-balls. Without specifying a potential $V$, we can show that $\\Omega^2$ is a \nmonotonically increasing function of $r$~\\cite{Arodz}.\nUsing a variable $\\ds f:=r^2\\frac{d\\Omega}{dr}$, we can rewrite Eq. (\\ref{FEqball2}) as\n\\bea\\label{FEqball2-2}\n&&\\frac{df}{dr}=\\Omega (qr\\phi)^{2},~~~\n\\frac{d\\Omega}{dr}={f\\over r^2}\\ .\n\\eea\nThe Taylor expansion of $\\Omega$ and $f$ up to the first order is expressed as\n\\bea\nf(r+\\Delta r)&=&f(r_0)+(qr_0\\phi(r))^2\\Omega(r)\\Delta r+O(\\Delta r^2),\\nn\n\\Omega(r+\\Delta r)&=&\\Omega(r)+{f(r)\\over r^2}\\Delta r+O(\\Delta r^2).\n\\label{Taylor}\\eea\nBy definition $f(0)=0$. If $\\Omega (0)>0$, then $f(\\Delta r)>0$. Equation (\\ref{Taylor}) indicates that at every step $r\\to r+\\Delta r$ both $f$ and $\\Omega$ increases. \nSimilarly, if $\\Omega (0)<0$, then $f$ and $\\Omega$ decreases at every step.\nThus we can conclude that $\\Omega^2$ is a monotonically increasing function of $r$.\n\n\\begin{figure}[htbp]\n\\psfig{file=f2a,width=3.2in}\n\\psfig{file=f2b,width=3.2in}\n\\caption{\\label{Newton}\nInterpretation of gauged Q-balls by analogy with a particle motion in Newtonian mechanics. Examples of\n(a) monotonic solutions in $V_{4}$ model and (b) nonmonotonic solutions in $V_{\\rm V}$ model.}\n\\end{figure}\n\nWe can interpret their equilibrium solutions in words of Newtonian mechanics in the same fashion, except that the potential of a particle is \\lq time\\rq-dependent,\n\\beq\nV_{\\Omega}=V-\\frac12\\Omega^2\\phi^2.\n\\eeq\nBecause the \\lq potential energy of the particle\\rq\\ $-V_\\Omega$ increases as the \\lq time\\rq\\ $r$ increases,\nthe \\lq initial altitude\\rq\\ $-V_\\Omega(0)$ is not necessarily larger than the \\lq final altitude\\rq\\ $-V_\\Omega(\\infty)=0$, that is,\nthere is no condition which corresponds to (\\ref{cond1}).\nHowever, the condition that the \\lq particle climbs up\\rq\\ at $r\\to\\infty$ should hold, we find an existing condition, which corresponds to (\\ref{cond2}),\n\\beq\\label{gaugedQexist}\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(-{dV_\\Omega\\over d\\phi}\\right)=\n\\lim_{\\phi\\to +0}\\frac{1}{\\phi}\\left(\\Omega^2\\phi-{dV\\over d\\phi}\\right)<0\\ .\n\\eeq\n\nFigure \\ref{Newton} illustrates the \\lq time-dependent potential of a fictitious particle \\rq $-V_\\Omega$.\nAs $r$ increases, $\\Omega^2$ also increases; then $-V_\\Omega$ goes up as shown in the figure.\nThere are two types of solutions.\nOne is monotonic solutions as shown in (a): $\\phi$ decreases monotonically as $r$ increases.\nThe other is nonmonotonic solutions as shown in (b): $\\phi$ increase initially, but after the sign of $dV_\\Omega\/d\\phi$ changes, $\\phi$ turns to decreases.\nThe latter type exposes a characteristic of gauged Q-balls, which appears in the $V_{\\rm V}$ and $V_{\\rm grav.}$ models.\n\n\\begin{figure}[htbp]\n\\psfig{file=f3,width=3.2in}\n\\caption{\\label{V4m02Q09r-phi}\nThe field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ for the $V_{4}$ model with $\\tilde{m}^{2}=0.2$ \nand $\\tilde{Q}=9$. The dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.}\n\\end{figure}\n\\section{Review of previous results}\n\nIn this section we review gauged Q-ball solutions in the $V_{4}$ model \\cite{Lee} and in the $V_{\\rm V}$ model \\cite{Arodz}.\n\n\\subsection{$V_{4}$ model}\n\nFor the $V_4$ model (\\ref{V4}), the necessary condition of existing equilibrium solutions (\\ref{gaugedQexist}) is expressed as \n\\beq\\label{gaugedQexist2}\n\\lim_{r\\to\\infty}\\Omega^2 0$.\nContrary to the case of the $V_4$ model, this condition does not put any restriction on $\\Omega$. \nTherefore, large gauged Q-balls are expected in this model.\n\nUsing the normalized coupling $\\kappa:={q\\lambda}\/{\\sqrt{2}}$, we rescale the quantities as \n\\bea\n&&\\tp:=\\frac{q\\phi}{\\sqrt{\\kappa}},~~\\tilde{\\Omega}:=\\frac{\\Omega}{\\sqrt{\\kappa}},~~\n\\tilde{r}:= \\sqrt{\\kappa}r, \\nonumber \\\\\n&&\\tilde{Q}:= q^2 Q, ~~\\tilde{E}:= \\frac{q^2 E}{\\sqrt{\\kappa}}.\n\\label{rescale-VV}\n\\eea\nIn Fig.~\\ref{VVQ120r-phi}, we show the field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ \nwith $\\tilde{Q}=120$. \nThe dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively. \nIn the case of gauged Q-balls, $\\tilde{\\phi}$ initially increases as a function of $\\tilde{r}$ and \ntakes a maximum value at $\\tilde{r}=\\tilde{r}_{\\rm max}\\neq 0$; then it decreases due to the increase of $\\tilde{\\Omega}$.\nThis behavior can be understood by the effective potential shown in Fig.~\\ref{Newton} (b). \nHere we have defined $\\tilde{r}_{\\rm max}$ as the value of $\\tilde{r}$ where $\\tilde{\\phi}$ takes a maximum value.\nIn the case of ordinary Q-balls, by contrast, $\\tilde{r}_{\\rm max}$ is always zero.\n\n\\begin{figure}[htbp]\n\\psfig{file=f4,width=3.2in}\n\\caption{\\label{VVQ120r-phi}\nThe field configurations of $\\tilde{\\phi}$ and $\\tilde{\\Omega}$ for the $V_{\\rm V}$ model with $\\tilde{Q}=120$. \nThe dashed and solid lines correspond to the ordinary and gauged Q-balls, respectively.}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f5a,width=3.2in}\n\\psfig{file=f5b,width=3.2in}\n\\caption{\\label{Omega-phiVV}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for the $V_{\\rm V}$ model. \nThe dashed line corresponds to the ordinary Q-balls. \nThe dotted and black solid lines correspond to the gauged Q-balls with $\\tilde{r}_{\\rm max}= 0$ and those with \n$\\tilde{r}_{\\rm max}\\neq 0$, respectively.\nBlue solid line corresponds to the Q-shell solutions. }\n\\end{figure}\n\nWe show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ relations in Fig.~\\ref{Omega-phiVV} (a) and (b), respectively.\nThe dashed line corresponds to the ordinary Q-balls. \nThe dotted and black solid lines correspond to the gauged case with \n$\\tilde{r}_{\\rm max}= 0$ and that with $\\tilde{r}_{\\rm max}\\neq 0$, respectively. \nBlue solid line corresponds to the Q-shell solutions that will be explained below. \n\nIn the case of ordinary Q-balls ($\\tilde{\\Omega}=\\tilde{\\omega}$), the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation, \nwhich was represented by the dashed line in (a), can be understood as follows.\nIn the picture of a particle motion in Newtonian mechanics, which was shown in Fig.\\ \\ref{f1}, if we ignore the ``nonconserved force\" term, $(2\/r)d\\phi\/dr$, the maximum of $\\tilde{\\phi}$, $\\tilde{\\phi}_{\\rm max}=\\tilde{\\phi}(0)$ is determined by the nontrivial solution of $V_{\\Omega}=0$. Then we obtain\n\\bea\n\\tilde{\\phi}(0)=\\frac{2}{\\tilde{\\Omega}^2}, \n\\label{phimax-VV}\n\\eea\nwhich approximates the dashed line in (a).\n\nIn the case of gauged Q-balls, the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation for large \n$\\tilde{\\Omega}(0)$ (small $\\tilde{Q}$), which is represented by the dotted line in (a), almost coincides with that for ordinary Q-balls.\nFor small $\\tilde{\\Omega}(0)$ (large $\\tilde{Q}$), however, the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ relation for ordinary Q-balls and that for gauged Q-balls are qualitatively different.\nNevertheless, it is surprising that there is no qualitative difference in $\\tilde{Q}$-$\\tilde{E}$ relation between\nsolutions with $\\tilde{r}_{\\rm max}= 0$ and those with $\\tilde{r}_{\\rm max}\\neq 0$.\nBoth solutions are on the same quasi-linear relation across the point $A$. \n\n$Q$ reaches a maximum at the point $B$ where cusp structure appears in the $\\tilde{Q}$-$\\tilde{E}$ plane.\nQ-ball solutions with the boundary conditions (\\ref{BCqball}) disappear at the point $C$ where $\\tilde{\\phi}(0)\\to 0$. \nHowever, Arod\\'z and Lis \\cite{Arodz} found a new type of solutions with boundary conditions (\\ref{BCqball2}) and\n\\bea\n&&\\phi (r)={d\\phi\\over dr}(r)={d\\Omega\\over dr}(r)=0,~~{\\rm for}~00.\n\\eeq\nBecause the AD gravity mediation model (\\ref{gravity}) with $K<0$ satisfies this condition, we can expect that \nit allows for large $Q$ solutions. This special property is in common with the V-shaped model.\n\nWe rescale the quantities in (\\ref{gravity}) as \n\\bea\n&&\\tp:=\\frac{q\\phi}{M},~~\\tilde{\\Omega}:=\\frac{\\Omega}{M},~~ \\nonumber \\\\\n&&\\tilde{r}:= Mr, ~~\\tilde{m}_{\\rm grav.}:= \\frac{m_{\\rm grav.}}{M},\\nonumber \\\\\n&&\\tilde{Q}:= q^2 Q, ~~\\tilde{E}:= \\frac{q^2 E}{M}.\n\\label{rescale-gravity}\n\\eea\nWe fix $\\tilde{m}_{\\rm grav.}=q=1$ below. \n\n\\begin{figure}[htbp]\n\\psfig{file=f6,width=3.2in}\n\\caption{\\label{K-1fields}\nThe field configurations of $\\tilde{\\phi}$ for gauged Q-balls with $K =-1$ and $\\tilde{Q}\\simeq 1.7$, $11$ and $103$.\n}\n\\end{figure}\n\nWe show some solutions of gauged Q-balls in Fig.~\\ref{K-1fields}; we choose $K =-1$ and obtain \nsolutions with $\\tilde{Q}=1.7$ and $11$, in which case $r_{\\rm max}=0$, and that \nwith $\\tilde{Q}=103$, in which case $r_{\\rm max}\\ne0$. \nAs $\\tilde{Q}$ increases, the field configuration becomes shell-like and the location of the shell becomes farther from the center.\nThis behavior is explained by repulsive Coulomb force of electric charge.\nThese configurations are just like ``Q-shells,\" which were obtained by Arod\\'z and Lis for the V-shaped model \\cite{Arodz}.\nThe difference is that we use the boundary condition (\\ref{BCqball}) and (\\ref{BCqball2}) consistently and \ngive tiny but nonzero value for $\\tilde{\\phi}(0)$, while they adopted the special boundary condition (\\ref{BCqshell}).\n\n\\begin{figure}[htbp]\n\\psfig{file=f7a,width=3.2in}\n\\psfig{file=f7b,width=3.2in}\n\\caption{\\label{K-1}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1$.\nThe dashed lines correspond to ordinary Q-balls.\nThe dotted and solid lines correspond to gauged Q-balls with $\\tilde{r}_{\\rm max}= 0$ and those with \n$\\tilde{r}_{\\rm max}\\neq 0$, respectively. \n}\n\\end{figure}\n\nWe show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1$ \nin Fig.~\\ref{K-1}. For reference, we also plot the relations for ordinary Q-balls ($\\Omega=\\omega$), which \nare represented by the dashed lines.\nTheir extreme behavior in the thin-wall limit ($\\omega\\ra\\infty$) and in the thick-wall limit ($\\omega\\ra0$) can be discussed analytically as follows \\cite{TS2}.\nThe maximum of $\\phi$, $\\tilde{\\phi}_{\\rm max}=\\tilde{\\phi}(0)$, can be estimated by the nontrivial solution of $V_{\\Omega}=0$:\n\\bea\n\\tilde{\\phi}_{\\rm max}=e^{\\frac{1-\\tilde{\\omega}^2}{-2K}}. \n\\label{phimax-AD}\n\\eea\nBecause the energy and the charge are roughly estimated as\n\\beq\nE\\sim V(\\phi_{\\rm max})R^3,~~~\nQ\\sim\\omega\\phi_{\\rm max}^{~~~2}R^3,\n\\eeq\nwhere $R$ is the typical radius, we find\n\\bea\n\\omega\\ra0&:&\\phi_{\\rm max}\\ra{\\rm nzf},~~~E\\ra{\\rm nzf},~~~Q\\ra0,\\nn\n\\omega\\ra\\infty&:&\\phi_{\\rm max}\\ra0,~~~E\\ra0,~~~Q\\ra0,\n\\eea\nwhere nzf denotes nonzero finite. Therefore, there is an upper limit $Q_{\\rm max}$. \nThis analytic estimate agrees with the numerical results in Fig.\\ \\ref{K-1}. \nThere are two sequences of solutions which merge at the cusp. \nWe suppose by energetics that the sequences with high energy are unstable (unstable branch) while \nthose with low energy stable (stable branch). \n\n\n\\begin{figure}[htbp]\n\\psfig{file=f8a,width=3.2in}\n\\psfig{file=f8b,width=3.2in}\n\\caption{\\label{K-06}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-0.6$.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f9a,width=3.2in}\n\\psfig{file=f9b,width=3.2in}\n\\caption{\\label{K-04}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-0.4$. \n}\n\\end{figure}\n\nThe results for gauge Q-balls are represented by the dashed lines ($\\tilde{r}_{\\rm max}= 0$) and \nthe solid lines ($\\tilde{r}_{\\rm max}\\neq 0$). \nThe solutions denoted by red lines correspond to those with small $\\omega$ and unstable branch, \nwhile those by black lines large $\\omega$ and stable branch. \nFor dotted lines, the gauged Q-balls are similar to the ordinary Q-balls (dashed lines). In contrast, \ndue to the nonmonotonic behavior of $\\tilde{\\phi}(\\tilde{r})$ (i.e., $\\tilde{r}_{\\rm max}\\neq 0$), the \nproperties of gauged Q-balls with solid lines and ordinary Q-balls are quite different. \n\nAs for the stable solutions denoted by the black lines, both $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ \nand $\\tilde{Q}$-$\\tilde{E}$ relations of solutions are similar to those of the $V_{\\rm V}$ model, except that cusp structure does not appear in the $\\tilde{Q}$-$\\tilde{E}$ plane in \nFig.~\\ref{K-1}(b).\nBecause $\\tilde{E}$ is a monotonically increasing function of $\\tilde{Q}$ we judge that all equilibrium solutions by black lines are stable. We also suppose by energetics that the solutions denoted by red lines are unstable. \n\n\\begin{figure}[htbp]\n\\psfig{file=f10a,width=3.2in}\n\\psfig{file=f10b,width=3.2in}\n\\caption{\\label{K-106}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K =-1.06$. \nThe two sequences in red lines and in black lines are about to touch.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f11a,width=3.2in}\n\\psfig{file=f11b,width=3.2in}\n\\caption{\\label{K-107Omega-phi}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations with $K =-1.07$. \nThe ``recombination\" of the two sequences happens.\n}\n\\end{figure}\n\\begin{figure}[htbp]\n\\psfig{file=f12a,width=3.2in}\n\\psfig{file=f12b,width=3.2in}\n\\caption{\\label{k-107Q-E}\n(a) $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and (b) $\\tilde{Q}$-$\\tilde{E}$ relations for $K=-1.07$ and $\\tilde{Q}>500$.\nThe dotted lines extend from Fig.\\ \\ref{K-107Omega-phi}.}\n\\end{figure}\n\nFigures \\ref{K-06} and \\ref{K-04} show the $\\tilde{\\Omega}(0)$-$\\tilde{\\phi}(0)$ and $\\tilde{Q}$-$\\tilde{E}$ \nrelations for $K =-0.6$ and $-0.4$, respectively. \nWe find that, as $|K|$ decreases, the existing domain of the unstable solutions becomes small in the $\\tilde\\Omega(0)$-$\\tilde\\phi(0)$ plane and the two sequences leave away from each other.\n\nA drastic change occurs between $K=-1.06$ and $K=-1.07$, as shown in Figs.\\ \\ref{K-106} and \\ref{K-107Omega-phi}. \nAs $|K|$ increases, the two sequences approach further; eventually at some point in $-1.07500$) region for $K=-1.07$ in Fig.~\\ref{k-107Q-E}.\nComplicated structure appears along the sequence $C$ to $G$; there are several cusps about $C$-$D$-$E$-$F$. \nAs shown in Fig~\\ref{K-107r-phi}, field distributions in this region also have complicated structures. \nBeyond the point $F$, both $\\tilde{\\phi}_{\\rm max}$ and $\\tilde{r}_{\\rm max}$ monotonically increase. \nIt is interesting that small differences of boundary values $\\tilde{\\Omega}(0)$ and \n$\\tilde{\\phi}(0)$ result in such large differences in $\\tilde{Q}$ and $\\tilde{E}$. \n\n\n\\begin{figure}[htbp]\n\\psfig{file=f13a,width=3.2in}\n\\psfig{file=f13b,width=3.2in}\n\\caption{\\label{K-107r-phi}\nField distributions of $\\tilde{\\phi}$ with $K=-1.07$ for (a)solutions $C$-$D$-$E$ and (b)solutions $E$-$F$-$G$. }\n\\end{figure}\n\n\n\\section{Summary and Discussions}\n\nIn many models of gauged Q-balls, which were studied in the literature, there are upper limits for charge and size of Q-balls due to repulsive Coulomb force.\nAs a cosmologically-motivated model which could allow for gauged Q-balls with large charge and size, we have considered the gravity-mediation-type model in the Affleck-Dine mechanism.\nWe have found that stable Q-balls with any amount of charge and size exist in this model as long as $K<0$.\nAs the electric charge $Q$ increases, the field configuration of the scalar field becomes shell-like; \nbecause the charge is concentrated on the surface, the Coulomb force does not destroy the Q-ball configuration.\nThese properties are analogous to those in the V-shaped model, which was studied by Arod\\'z and Lis \\cite{Arodz}.\nBecause the V-shaped model is rather artificial, our results for the cosmologically-motivated model would be important if we consider gauged Q-balls as realistic dark matter model. \n\nWe have also found that for each $K$ there is another sequence of unstable solutions, which is separated from the other sequence of the stable solutions.\nAs $|K|$ increases, the two sequences approach; eventually at some point in $-1.07$ and their errors $\\left<\\sigma_{\\rm \\mu}\\right>$\nwould have to be weighted by the inverse variances, as:\n\\begin{equation}\n\\left< \\mu\\right> =\n{\\left(\\sum_{i=1}^{n}{\\mu_{\\rm i}\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}\n\/\n{\\left(\\sum_{i=1}^{n}{1\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}\\ .\n\\label{eqmu}\n\\end{equation}\n\\begin{equation}\n\\left< \\sigma_{\\rm \\mu}\\right> =\n{\\left(\\sum_{i=1}^{n}{1\/{\\sigma^{2}_{\\rm \\mu i}}}\\right)}^{-1\/2}\\ .\n\\label{eqsigmu}\n\\end{equation}\nWe know, however, that not all measurements are independent of each\nother, since some of the catalogues share the same plate material.\nFurthermore, though we checked as far as possible whether the star\nfound in the catalogue by the\nautomatic search procedure using its coordinates\nis indeed the white dwarf, in some cases misidentifications have occurred.\nComparing the proper motions of one star in different\nsources permits false detections to be eliminated.\n\nTo do this, we calculated the combined average and error, plus the\nquadratic deviation $\\Delta^{2}_{\\rm \\mu i}$\nof an individual measurement from this average:\n\\begin{equation}\n \\Delta^{2}_{\\rm \\mu i} =\n\\left( \\mu_{\\rm i}-\\left< \\mu\\right> \\right) ^{2}\\ .\n\\end{equation}\nIf each $\\Delta^{2}_{\\rm \\mu i}$ is divided by the corresponding\n$\\sigma_{\\rm \\mu i}^{2}$,\nthe sum over all $i$ is taken and divided by the number of measurements $n$.\nWe\nget a quantity $\\Delta_{\\rm check}$ that allows to check if the individual\nmeasurements are\nconsistent with each other and, if not, to eliminate the measurement\nwhich differs from the others:\n\\begin{equation}\n\\Delta_{\\rm check}=\n{1\\over n}\n\\left(\\sum_{i=1}^{n} \\left( \\Delta^{2}_{\\rm \\mu i}\/\\sigma^{2}_{\\mu i}\n\\right) \\right)\\ .\n\\end{equation}\nIf $\\Delta_{\\rm check}>1$, we checked the different catalogue values\nmanually in order to decide which values to choose and which to eliminate.\nHaving thus eliminated false detections the next step was to calculate\nthe quantity $\\left< \\Delta_{\\rm \\mu}\\right>$:\n\\begin{equation}\n\\left< \\Delta_{\\rm \\mu}\\right> =\n\\sqrt\n{{1\\over n} \\left(\\sum_{i=1}^{n} \\Delta^{2}_{\\rm \\mu i}\\right)}\\ .\n\\end{equation}\nWe adopted the weighted mean $\\left< \\mu\\right>$ from Eq.~(\\ref{eqmu}) \nand the maximum of $\\left< \\sigma_{\\rm \\mu}\\right>$\nand $\\left< \\Delta_{\\rm \\mu}\\right>$ as\nthe corresponding error.\nThis enabled us to obtain a realistic error estimate,\nwhich typically lies between $5\\,{\\rm mas~{yr}^{-1}}$ and\n$10\\,{\\rm mas~{yr}^{-1}}$.\n\nThe input parameters radial velocities, spectroscopic distances,\nand\nproper motion components together with their errors, are listed\nfor all white dwarfs in Table~8.\n\n\\section{Revised population classification scheme\\label{orbit}}\nIn Paper~I we presented\na new sophisticated\npopulation classification scheme based on the $U$\\\/-$V$-velocity diagram,\nthe $J_Z$-eccentricity-diagram, and the Galactic orbit.\nFor the computation of orbits and kinematic parameters, we used the code\nby \\citet{odenkirchen92} based on a Galactic potential by \\citet{allen91}.\nThe classification scheme was based on a calibration sample of\nmain-sequence stars.\nIn the meantime, new spectroscopic analyses have become available which \nallowed us to enlarge the calibration sample and to refine our\nclassification criteria.\n\n\\subsection{The calibration sample\\label{cal}}\nUnlike for main-sequence stars, the population membership of white dwarfs\ncannot be determined from spectroscopically measured metalicities.\nTherefore we have to rely on kinematic criteria.\nThose criteria have to be calibrated using a suitable calibration sample of\nmain-sequence stars.\nIn our case this sample consists of $291$ F and G main-sequence stars from\n\\citet{edvardsson93}, \\citet{fuhrmann98},\nFuhrmann (2000\\footnote{\\tt http:\/\/www.xray.mpe.mpg.de\/fuhrmann\/.},\n2004). It is important to note that the stars were \nselected from flux limited samples and not from proper motion surveys.\nThanks to the work of \\citet{fuhrmann04},\nthe number of calibration sample stars has been doubled, which makes it \nworthwhile revisiting the classification criteria outlined in Paper I.\n\nFor both samples a detailed abundance analysis was carried out.\n\\citet{fuhrmann98} combined abundances, ages, and 3D kinematics for\npopulation classification and found that the disk and halo populations\ncan be distinguished best\nin the [Mg\/Fe] versus [Fe\/H] diagram. Halo and thick-disk stars can be\nseparated by means of their [Fe\/H] abundances,\nas they possess a higher [Mg\/Fe] ratio than thin-disk stars\n\\citep[see also][]{bensby03}.\nIn Fig.~\\ref{met} the\n${\\rm [Mg\/Fe]}$ versus ${\\rm [Fe\/H]}$ abundances\nfor the $291$ main-sequence stars are shown.\nThese stars are divided into halo, thick disk, and\nthin disk according to their position in the diagram.\nThe halo stars have $[{\\rm Fe}\/{\\rm H}]<-1.05$,\nthe thick-disk stars\n$-1.05\\le[{\\rm Fe}\/{\\rm H}]\\le-0.3$ and\n$[{\\rm Mg}\/{\\rm Fe}]\\ge 0.3$, and the thin-disk stars \n$[{\\rm Fe}\/{\\rm H}]>-0.3$ and $[{\\rm Mg}\/{\\rm Fe}] \\le 0.2$.\nStars in the overlapping area between the thin and the thick disk (open\ntriangles in Fig.~\\ref{met}) were\nneglected in order to ensure a clear distinction between the\ntwo disk populations.\n\nThere are four stars left to the halo border, which according\nto \\citet{fuhrmann04} belong to the metal-weak thick disk\n(MWTD, open boxes).\nAs their kinematics are indeed incompatible with halo\nmembership, we omitted them\nfrom further analysis.\nAlso rejected was the star HD\\,148816, which though in\nthe thick-disk region in the abundance diagram, clearly shows\nhalo kinematics (not shown in the diagram).\n\nThis demonstrates that a clear distinction between halo and\nthick-disk stars by means of abundances is difficult,\nbut as will be shown later, halo and thick-disk stars\nshow very distinct kinematic\nproperties, so that they are unlikely to be confused.\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n\\begin{psfrags}\n\\psfrag{[Fe\/H]}{${\\rm [Fe\/H]}$}\n\\psfrag{[Mg\/Fe]}{${\\rm [Mg\/Fe]}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{trans}{trans}\n\\psfrag{metweak}{metweak}\n \\includegraphics[width=17cm]{2730fig1.eps}\n\\end{psfrags}}\n \\caption{${\\rm [Mg\/Fe] vs. [Fe\/H]}$ abundance diagram for the calibration \nsample (see text). \n}\n\\label{met}\n\\end{figure}\n\\subsection{The $U$\\\/-$V$-velocity diagram\\label{uv}}\nA classical tool for kinematic investigations is the\n$U$\\\/-$V$-velocity diagram. In Fig.~\\ref{uvms}, $U$ is plotted\nversus $V$ for the main-sequence stars.\nFor the thin-disk and the thick-disk stars, the mean values and standard\ndeviations of the two velocity components were calculated.\nThe values for the thin disk are:\n$\\left=3\\,\\rm{km~s^{-1}}$,\n$\\left=215\\,\\rm{km~s^{-1}}$,\n$\\sigma_{U_{\\rm ms}}=35\\,\\rm{km~s^{-1}}$, and\n$\\sigma_{V{\\rm ms}}=24\\,\\rm{km~s^{-1}}$.\nThe corresponding values for the thick disk are:\n$\\left=-32\\,\\rm{km~s^{-1}}$,\n$\\left=160\\,\\rm{km~s^{-1}}$,\n$\\sigma_{U_{\\rm ms}}=56\\,\\rm{km~s^{-1}}$, and\n$\\sigma_{V{\\rm ms}}=45\\,\\rm{km~s^{-1}}$.\nThe negative value of $\\left$ \nis explained\nin \\citet{fuhrmann04} as an effect of the Galactic bar.\nIndeed, nearly all thin-disk stars stay inside the\n$3\\sigma_{\\rm thin}$-limit, and all halo stars lie outside\nthe $3\\sigma_{\\rm thick}$-limit, as can be seen from Fig.~\\ref{uvms}.\nIn our previous paper, we used the $2\\sigma-$limit of the thin and\nthick-disk stars for finding thick-disk stars and\n$\\sqrt{U^2+(V-195)^2}\\ge 150\\,\\rm{km\\,s^{-1}}$ for finding halo stars.\nWe replaced these by the \nmore stringent \n$3\\sigma-$limits of the thin and thick-disk\nstars to obtain a clear-cut separation. \n\\begin{figure*}\n \\centering\n\\begin{psfrags}\n\\psfrag{V(kms^-1)}{$V\/{\\rm km~s^{-1}}$}\n\\psfrag{U(kms^-1)}{$U\/{\\rm km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{3 sig thin}{$3\\sigma_{\\rm thin}$-limit}\n\\psfrag{3 sig thick}{$3\\sigma-{\\rm thick}$-limit}\n \\includegraphics[width=17cm]{2730fig2.eps}\n\\end{psfrags}\n \\caption{$U$\\\/-$V$-velocity diagram for the calibration sample of \n main-sequence stars with $3\\sigma_{\\rm thin}$-, $3\\sigma_{\\rm\nthick}$-contours.}\n\\label{uvms}\n \\centering\n\\begin{psfrags}\n\\psfrag{e}{$e$}\n\\psfrag{Jz(kpc km s^-1)}{$J_Z\/{\\rm kpc\\,km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{RegionA}{Region~A}\n\\psfrag{RegionB}{Region~B}\n\\psfrag{RegionC}{Region~C}\n \\includegraphics[width=17cm]{2730fig3.eps}\n\\end{psfrags}\n \\caption{$J_Z$-$e$-diagram for the calibration sample of main-sequence stars}\n \\label{ecc1}\n\\end{figure*}\n\\subsection{The $Jz$-$e$-diagram \\label{jze}}\nThe $U$\\\/-$V$-plot is not the only source of information about population\nmembership.\nTwo important orbital parameters\nare the $z$-component of the angular momentum $J_Z$ and the\neccentricity of the orbit $e$. Both are plotted against each other\nfor the main-sequence stars in Fig.~\\ref{ecc1}.\nThe different populations can be distinguished well in this diagram.\nThe thin-disk stars cluster in a V-shaped area of\nlow eccentricity and $J_Z$ around $1\\,800\\,\\rm{kpc \\, km~s^{-1}}$,\nwhich we denote as region~A.\n\nIn general, the thick-disk stars possess higher\neccentricities $e>0.27$ and lower angular momenta.\nThey can be found in region~B.\nThere is also a clump of thick-disk\nstars with lower eccentricity around $0.2$ and higher $J_Z$.\nRegion~B is defined such that it excludes as many thin-disk\nstars as possible. The price that has to be paid for this is the\nloss of some thick-disk stars. But this way there is a high probability\nof identifying only those stars as thick-disk members that really\nbelong to the thick disk.\nIt should be noted that region~3 in our previous paper, which seemed\nto be different from the thin-disk and the thick-disk regions A and B, has\nproven to be just an extension of the thin-disk region to higher\neccentricities. Therefore it does not appear as an additional region in\nthis revised classification scheme.\n\nThe halo stars with very high eccentricity and smaller\n$J_Z$ can be found in Region~C, separated well from all other stars.\n\\subsection{Galactic orbits\\label{orb}}\nThe eccentricity was extracted from the Galactic orbit of the stars.\nThe classification can be confirmed by checking\nthe orbits themselves.\nTypical orbits for thin-disk, thick-disk and halo main-sequence stars\ncan be found \nin Paper~I and will not be repeated here.\n\\subsection{Population classification scheme \\label{class}}\n\nOur classification scheme (developed in Paper~I) combines three\ndifferent classification criteria:\ni) the position in $U$\\\/-$V$ diagram,\nii) the position in $J_Z$-$e$ diagram, and iii) the Galactic orbit.\n\nWe repeat some details here of the population classification scheme presented \nin Paper~I and then describe the new refinements and changes.\nWe classified white dwarfs as halo members if they had a value of\n$\\sqrt{U^2+(V-195)^2}\\ge 150\\,\\rm{km\\,s^{-1}}$ and \nlay in region~4 in the $J_Z$-$e$-diagram (see Paper~I).\n\nTo detect thick-disk white dwarfs, first all stars either\nsituated outside the $2\\sigma$-limit\nin the $U$\\\/-$V$-diagram or in region~2 or 3 in the $J_Z$-$e$-diagram\nwere selected as thick-disk candidates.\nIn a second step, each candidate was assigned a classification value\n$c$. $c$ was defined as the sum of the individual values $c_{\\rm UV}$,\n$c_{\\rm J_{Z}e}$ and $c_{\\rm orb}$ corresponding to the three\ndifferent criteria: position in $U$\\\/-$V$-diagram,\nposition in $J_Z$-$e$-diagram, and Galactic orbit.\n\nWe assigned $c_{\\rm UV}=+1$ to a star outside the $2\\sigma$-limit\nin the $U$\\\/-$V$-diagram, whereas one inside the $2\\sigma$-limit\ngot $c_{\\rm UV}=-1$.\nThe different regions in the $J_Z$-$e$-diagram are characterised by\n$c_{\\rm J_{Z}e}=-1$ for region~1, $0$ for region~3, and $+1$ for region~2.\nThe third classification value $c_{\\rm orb}$ described the orbits:\n$c=-1$ for orbits of thin-disk type and $c=+1$ for orbits of\nthick-disk type.\nThen the sum $c=c_{\\rm UV}+c_{\\rm J_{Z}e}+c_{\\rm orb}$ was computed.\nStars with $c=+3$ or $c=+2$ were considered as bona fide\nthick-disk members, and those with $c=+1$ as probable thick-disk members.\nIf $c \\le 0$, the star was classified as belonging to the thin disk.\n\nThe new classification scheme is more concise due to the elimination of\nregion~3. \nAs described in Sect.~\\ref{uv}, we also sharpened the selection criterion \nfor the $U$\\\/-$V$ plane by replacing the $2\\sigma$ by a $3\\sigma$ limit.\nA star is classified as a halo candidate if it lies either\noutside the $3\\sigma_{\\rm thick}$-limit in the $U$\\\/-$V$ diagram\nor in region~C in the $J_Z$-$e$ diagram.\nThen classification values $c_{\\rm UV}$, $c_{\\rm J_{Z}e}$, and $c_{\\rm orb}$\nare assigned to all halo candidates\nwhich take the value of $+1$ if the criterion favors a halo\nmembership and $-1$ if not.\nMore precisely:\n$c_{\\rm UV}=+1$ if the star lies outside the $3\\sigma_{\\rm thick}$-limit,\n$c_{\\rm J_{Z}e}=+1$ if the star lies in region~C,\nand $c_{\\rm orb}=+1$ if the star has a halo orbit.\nThen the sum $c=c_{\\rm UV}+c_{\\rm J_{Z}e}+c_{\\rm orb}$ is calculated.\nAll of the halo candidates with $c \\ge +1$\nare classified as halo members, the rest as thick-disk members.\n\nAll the remaining stars (not found to belong to the halo), \neither outside the $3\\sigma_{\\rm thin}$-limit in the $U$\\\/-$V$ diagram\nor in region~B in the $J_Z$-$e$ diagram, are classified\nas thick-disk candidates.\nThen the analogous procedure to the halo classification is applied:\n$c_{\\rm UV}=+1$ if the star lies outside the $3\\sigma_{\\rm thin}$-limit,\n$c_{\\rm J_{Z}e}=+1$ if the star lies in region~B,\nand $c_{\\rm orb}=+1$ if the star has a thick-disk orbit.\nIn contrast to Paper~I due to the elimination of region~3,\nthere is no longer a value 0 to be assigned\nto $c_{\\rm J_{Z}e}$; hence, we expect the number of thick-disk\ncandidates to decrease.\nAll of the thick-disk candidates with $c \\ge +1$\nare assigned to the thick-disk population, the rest to the thin-disk\npopulation.\n\\subsection{Consistency check for the kinematical classification \ncriteria \\label{consist}}\nIn this section a consistency check of our classification\nscheme is performed.\nThis is done by applying our kinematic classification criteria to\nour calibration main-sequence sample.\n\nThirty-three main-sequence stars are known to belong to the thick disk because \nof their abundance patterns\n(For reasons mentioned above we have excluded here the metal-weak thick-disk\nstars), and\n22 of them have a kinematical classification value $c \\geq +1$ and\nare classified as thick-disk stars.\nOnly one of them has $c=0$ and is thus misclassified as a thin-disk star.\nThis corresponds to a detection efficiency of about $67\\%$\nfor thick-disk members.\nIn addition to those 22 stars, six thin-disk main-sequence stars\nwith $c \\geq +1$ are misclassified as thick-disk stars, so that\nthe total number of stars classified as thick disk is 28 \nindicating a contamination with thin-disk stars of about $21\\%$.\n\n\n\\subsection{Application to the white dwarf sample of Paper~I\n\\label{application}}\n\n\nFurthermore, in order to be able to compare the results of Paper~I with\nthis paper, we applied the new classification scheme to the\n$107$ white dwarfs analysed in Paper~I.\nThe fraction of halo stars is not changed by this new scheme.\nDue to the elimination of region~3 in the $J_Z$-$e$ diagram, four stars\nlose their thick-disk candidate status, and we end up with a\ntotal number of eight thick-disk stars compared to twelve previously.\nThis reduces the local fraction of thick-disk white dwarfs from\n$11\\%$ to $7.5\\%$, and\ndemonstrates the uncertainty of kinematic population classification.\nEven higher errors are to be expected when the population separation\nis based on a single criterion such as the position in $U$\\\/-$V$ diagram alone,\nwhich is the case for most other kinematical studies of white dwarfs in the\nliterature.\n\n\\section{Kinematic population classification of the SPY white \ndwarfs\\label{popuclasswd}}\n\nWe calculated orbits and kinematic parameters for all $398$ white\ndwarfs (see Table~9\n).\nThe errors of $e$, $J_Z$, $U$, $V$, $W$ were\ncomputed with the Monte Carlo error propagation code\ndescribed in Paper~I.\nThey can be found in Table~9\nas well.\n\\begin{figure*}\n \\centering\n\\begin{psfrags}\n\\psfrag{V(kms^-1)}{$V\/{\\rm km~s^{-1}}$}\n\\psfrag{U(kms^-1)}{$U\/{\\rm km~s^{-1}}$}\n\\psfrag{HE0201}{\\footnotesize HE\\,0201}\n\\psfrag{HS1527}{\\footnotesize HS\\,1527}\n\\psfrag{WD0252}{\\footnotesize WD\\,0252}\n\\psfrag{WD1448}{\\footnotesize WD\\,1448}\n\\psfrag{WD1524}{\\footnotesize WD\\,1524}\n\\psfrag{WD2351}{\\footnotesize WD\\,2351}\n\\psfrag{WD2359}{\\footnotesize WD\\,2359}\n\\psfrag{WD2029}{\\footnotesize WD\\,2029}\n\\psfrag{3 sig thin}{$3\\sigma-{\\rm thin}$-limit}\n\\psfrag{3 sig thick}{$3\\sigma-{\\rm thick}$-limit}\n \\includegraphics[width=17cm]{2730fig4.eps}\n\\end{psfrags}\n \\caption{$U$\\\/-$V$-velocity diagram for the white dwarfs with\n$3\\sigma-{\\rm thin}$ and $3\\sigma-{\\rm thick}$ -- contours from \nFig.~\\ref{uvms}, Symbols with numbers are the white dwarfs \nmentioned in the text}\n \\label{uvwd}\n \\centering\n\\begin{psfrags}\n\\psfrag{e}{$e$}\n\\psfrag{Jz(kpc km s^-1)}{$J_Z\/{\\rm kpc\\,km~s^{-1}}$}\n\\psfrag{thin disk}{thin disk}\n\\psfrag{thick disk}{thick disk}\n\\psfrag{halo}{halo}\n\\psfrag{RegionA}{Region~A}\n\\psfrag{RegionB}{Region~B}\n\\psfrag{RegionC}{Region~C}\n\\psfrag{HE0201}{\\footnotesize HE\\,0201}\n\\psfrag{HS1527}{\\footnotesize HS\\,1527}\n\\psfrag{WD0252}{\\footnotesize WD\\,0252}\n\\psfrag{WD1448}{\\footnotesize WD\\,1448}\n\\psfrag{WD1524}{\\footnotesize WD\\,1524}\n\\psfrag{WD2351}{\\footnotesize WD\\,2351}\n\\psfrag{WD2359}{\\footnotesize WD\\,2359}\n\\psfrag{WD2029}{\\footnotesize WD\\,2029}\n \\includegraphics[width=17cm]{2730fig5.eps}\n\\end{psfrags}\n \\caption{$Jz$-$e$-diagram of the white dwarfs}\n \\label{ecc2}\n\\end{figure*}\n\\subsection{The $U$\\\/-$V$-velocity diagram\\label{uv_wd}}\nIn Fig.~\\ref{uvwd}, the $U$\\\/-$V$-velocity diagram for the\nwhite dwarfs is shown together with the $3\\sigma$-limits\nof the thin and thick-disk stars from the calibration sample.\nThe white dwarfs can be divided into two main groups that appear\nto be separated from each other:\none group that is clustered mainly within the $3\\sigma_{\\rm thin}$-limit\nwith some stars just outside the $3\\sigma_{\\rm thin}$-border\nand another second group with smaller $V$ that lies outside or just\ninside the $3\\sigma_{\\rm thick}$-border.\nAll the white dwarfs belonging to the second group are marked with\nthe first letters of their names in Fig.~\\ref{uvwd}.\n\nThe second group comprises five stars outside the\n$3\\sigma_{\\rm thick}$-limit (which qualify as halo candidates according to\nSect.~\\ref{class}) HS\\,1527+0614, WD\\,0252$-$350,\nWD\\,1448+077, WD\\,1524$-$749, and WD\\,2351$-$365.\n Exceptional are WD\\,1448+077 and WD\\,1524$-$749, which have\na negative value of $V$; i.e. they move on retrograde orbits.\nThis behaviour is incompatible with disk membership and\nstrongly suggests that they belong to the halo.\n\nThe other three white dwarfs of the second group are HE\\,0201$-$0513,\nWD\\,2029+183 and WD\\,2359$-$324. Situated inside the\n$3\\sigma_{\\rm thick}$, they do not qualify as halo candidates but\nwe must check if they belong to the halo or to the thick disk by means of \nthe $J_Z$-eccentricity diagram and the orbits .\n\\subsection{The $J_z$-$e$-diagram \\label{jze_wd}}\nWe now move on to the $J_Z$-eccentricity diagram of the SPY\nwhite dwarfs (Fig.~\\ref{ecc2}).\nAgain, two groups of stars can be detected:\none first group starting in Region~A with a high-eccentricity tail in\nRegion~B, which represents the disk population, and a second\ngroup in the right part of Region~B and in Region~C.\nContrary to the main-sequence stars there is a gap in Region~B\nthat is not populated at all by white dwarfs.\nIf this is real or just due to selection effects cannot be said at this point.\n\nThe second group contains all the stars discussed individually in the previous\nsection and labeled by name in Fig.~\\ref{ecc2}.\nHE\\,0201$-$0513, since situated in Region~C, is\nadded to the list of halo candidates.\nThe two retrograde stars, WD\\,1448+077 and WD\\,1524$-$749,\ncan be distinguished easily by their negative value of $J_{\\rm Z}$.\n\\subsection{Galactic orbits}\n\nNext we inspect the Galactic orbits of the SPY white dwarfs.\nWe display some meridional plots of white dwarfs with thin-disk, thick-disk, \nor halo like orbits, respectively, in Figs. \\ref{wd0310} to \\ref{hs1527}.\n\n\nMost white dwarfs have thin-disk-like orbits, an example is\nWD\\,0310$-$688 (Figure~\\ref{wd0310}).\nSome orbits, like the one of WD\\,1013$-$010 (Fig.~\\ref{wd1013}),\nshow thick-disk characteristics.\nThe star WD\\,2029+183 mentioned earlier has a thick-disk orbit.\nFive stars (HS\\,1527+0614,\nHE\\,0201$-$0513, WD\\,0252$-$350, WD\\,2351$-$365, and WD\\,2359$-$324)\nhave chaotic halo orbits, as can be seen from \nFig.~\\ref{hs1527} in the case of \nHS\\,1527+0614.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig6.eps}\n \\end{psfrags}\n }\n \\caption{WD\\,0310$-$688: a white dwarf with a thin-disk orbit}\n \\label{wd0310}\n\\end{figure}\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig7.eps}\n \\end{psfrags}\n }\n \\caption{WD\\,1013$-$010: a white dwarf with a thick-disk orbit}\n \\label{wd1013}\n\\end{figure}\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\n \\begin{psfrags}\n \\psfrag{rho\/kpc}{$\\rho\/{\\rm kpc}$}\n \\psfrag{Z\/kpc}{$Z\/{\\rm kpc}$}\n \\includegraphics{2730fig8.eps}\n \\end{psfrags}\n }\n \\caption{HS\\,1527+0614: a white dwarf with a (chaotic) halo orbit.} \n \\label{hs1527}\n\\end{figure}\n\n\n\n\\subsection{Classification}\n\nWe used the population classification scheme presented\nin Sect.~\\ref{class} to divide the SPY white dwarfs into\nthe three different populations. We start with the halo candidates, \ne.g. with all white dwarfs that are either\nsituated outside the $3\\sigma$-limit of the thick disk in the\n$U$\\\/-$V$-velocity diagram or that lie in Region~C in\nthe $J_Z$-eccentricity diagram.\nSix white dwarfs fulfill these conditions:\nall but one lie outside the $3\\sigma_{\\rm thick}$-limit, \nand all lie in Region~C.\nTwo white dwarfs, WD\\,1448+077 and WD\\,1524$-$749,\nare on retrograde orbits characterised by a negative value\nof $V$ and $J_{\\rm Z}$.\nWhen the classification values of the halo white\ndwarf candidates are added, it is found that all of them have $c>1$ and\ntherefore belong to the halo population.\nWe have mentioned before that the star WD\\,2359$-$324, though it\ndoes not fulfill the criteria for a halo candidate, has an\norbit typical for a halo object.\nAs its error-bar places it near Region~C in the\n$J_Z$-$e$ diagram, we therefore decided to classify it as a halo object.\nThis leaves us with seven halo white dwarfs.\nDetails can be found in Table~\\ref{ha_class}.\n\n\nWe now move on to the remaining $32$ white dwarfs that lie either\noutside the $3\\sigma$-limit of the thin disk in the\n$U$\\\/-$V$-velocity diagram or that lie in Region~B in\nthe $J_Z$-eccentricity diagram.\nTwenty-seven of them have a classification value of $c>1$\nand are classified as thick-disk members, the remaining\nfive are assigned a thin-disk membership (see Table~\\ref{di_class}).\nAll the remaining white dwarfs are assumed to belong to the thin disk,\nleaving us with seven halo, $27$ thick-disk, and\n$364$ thin-disk out of the $398$ SPY white dwarfs.\n\n\\begin{table}\n\\caption[]\n{Classification values for the halo candidates. Note that WD2359$-$324 is \nclassified as a halo star despite having $c=-1$; see text \\label{ha_class}}\n\\begin{tabular}{lrrrrl}\n\\\\\n\\hline\nstar & $c_{\\rm UV}$ & $c_{\\rm J_Z-e}$ & $c_{\\rm orb}$ & $c$ & classification\\\\ \n\\hline\nHE\\,0201$-$0513 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.45cm} \nhalo \\\\\nHS\\,1527+0614 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,0252$-$350 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,1448+077 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.45cm} halo \\\\\nWD\\,1524$-$749 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.45cm} halo \\\\\nWD\\,2351$-$368 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.45cm} halo \\\\\nWD\\,2359$-$324 & \\hspace*{.3cm} $-$1 & $-$1 & +1 & $-$1 & \\hspace*{.45cm} \nhalo \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Classification values for the thick-disk candidates. \n\\label{di_class}}\n\\begin{tabular}{lrrrrl}\n\\\\\n\\hline\nstar & $c_{\\rm UV}$ & $c_{\\rm J_Z-e}$ & $c_{\\rm orb}$ & $c$ & class.\\\\ \n\\hline\nHE\\,0409$-$5154 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0416$-$1034 & \\hspace*{.3cm} +1 & $-$1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0452$-$3444 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,0508$-$2343 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHE\\,1124+0144 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick disk \\\\\nHS\\,0820+2503 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick disk \\\\\nHS\\,1338+0807 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nHS\\,1432+1441 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0204$-$233 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0255$-$705 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0352+052 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0548+000 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,0732$-$427 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,0956+045 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,1013$-$010 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1152$-$287 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1323$-$514 & \\hspace*{.3cm} +1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1327$-$083 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1334$-$678 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1410+168 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1426$-$276 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1507+021 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1531+184 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,1614$-$128 & \\hspace*{.3cm} +1 & +1 & $-$1 & +1 & \\hspace*{.0cm} \nthick disk \\\\\nWD\\,1716+020 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1834$-$781 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,1952$-$206 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2029+183 & \\hspace*{.3cm} +1 & +1 & +1 & +3 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2136+229 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,2253$-$081 & \\hspace*{.3cm} $-$1 & +1 & $-$1 & $-$1 & \\hspace*{.0cm} \n\\it{thin disk} \\\\\nWD\\,2322$-$181 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\nWD\\,2350$-$083 & \\hspace*{.3cm} $-$1 & +1 & +1 & +1 & \\hspace*{.0cm} thick \ndisk \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\section{Age estimates \\label{chap_ages}}\n\nThe seven halo and $27$ thick-disk white dwarfs were assigned\nto the respective populations by means of purely kinematic\ncriteria. Accordingly they must be old stars; therefore, we attempted to \nestimate their ages.\nA check to see whether their physical parameters, mass and effective\ntemperature, are compatible with their belonging to an old population\nmust now be made. Masses $M$ for the white dwarfs were derived\nfrom ${\\rm log}~g$ and the mass-radius relation by \\citet{wood95}.\n\nThe halo is older than $10\\,{\\rm Gyr}$.\n\\cite{bensby03} determined a mean age for the thick disk as\n$11.2 \\pm 4.3\\,{\\rm Gyr}$.\nIt is very probable that stars that are younger than $7\\,{\\rm Gyr}$\ndo not belong to the thick disk.\nThus the main-sequence life-time (plus about $20\\%$ for time spent \nduring the giant phases and the horizontal branch),\nplus the time the white dwarf has cooled down\nuntil it reaches its actual $T_{\\rm eff}$, has to be greater than\nthe age of the youngest stars of the respective populations.\nThe main-sequence life-time $\\tau_{\\rm ms}$ depends on the mass of the\nwhite dwarf progenitor and is approximately proportional to\n$\\tau_{\\rm ms} \\propto M^{-2.5}$ \\citep{kippenhahn94}.\nThe main-sequence life-time is $\\tau_{\\rm ms}=10\\,{\\rm Gyr}$ for the Sun,\n$7.9\\,{\\rm Gyr}$ for a $1.1\\,\\Msolar$ mass star,\n$4.3\\,{\\rm Gyr}$ for a $1.4\\,\\Msolar$ mass star, and\n$1.8\\,{\\rm Gyr}$ for a $2\\,\\Msolar$ mass star.\nAdding the $20\\%$ horizontal branch plus giant phase lifetime, the total\npre-white dwarf lifetimes would be\n$12\\,{\\rm Gyr}$, $9.5\\,{\\rm Gyr}$, $5.2\\,{\\rm Gyr}$,\nand $2.2\\,{\\rm Gyr}$, respectively.\n\nThe mass of the white dwarf is related to the mass of its\nprogenitor by the initial-to-final mass relation.\nUntil now, no definitive initial-to-final mass relation\nhas been established;\nhowever, different estimates exist from\ndifferent groups derived from theoretical considerations\nand from observational investigations of open clusters; \nsee e.g. \\citet{weidemann00} and \\citet{schroeder01}.\nUnfortunately no initial-to-final mass relation for the\nhalo and the thick disk has been derived yet, so we have to work with what \nis available for the\nthin disk and keep in mind that our age \nestimates are crude.\nAccording to \\citet{weidemann00}, stars with initial masses\nof $1\\,\\Msolar$, $1.1\\,\\Msolar$, $1.4\\,\\Msolar$, and $2\\,\\Msolar$ would\nevolve into white dwarfs with masses of\n$0.55\\,\\Msolar$, $0.555\\,\\Msolar$, $0.57\\,\\Msolar$, and $0.6\\,\\Msolar$, \nrespectively.\nThe initial-to-final mass relation of \\citet{schroeder01}, on\nthe other hand, yields white dwarf masses of\n$0.55\\,\\Msolar$, $0.565\\,\\Msolar$, $0.605\\,\\Msolar$, and $0.67\\,\\Msolar$.\n\nWe now estimate how long it takes for a C\/O core white dwarf to cool down to\n$20\\,000\\,{\\rm K}$, $10\\,000\\,{\\rm K}$, $8\\,000\\,{\\rm K}$, and\n$5\\,000\\,{\\rm K}$ using the cooling tracks of \\citet{wood95}.\nFor a $0.5\\,\\Msolar$ mass white dwarf, the respective cooling times\nwould be $0.05\\,{\\rm Gyr}$, $0.5\\,{\\rm Gyr}$, $0.9\\,{\\rm Gyr}$,\nand $4\\,{\\rm Gyr}$.\nFor a $0.6\\,\\Msolar$ mass white dwarf, the corresponding values are\n$0.08\\,{\\rm Gyr}$, $0.6\\,{\\rm Gyr}$, $1.1\\,{\\rm Gyr}$, and\n$6\\,{\\rm Gyr}$.\nHence, only for white dwarfs cooler than \n$8\\,000\\,{\\rm K}$ does\nthe cooling time contribute significantly to the total age.\n\n\nAll the halo white dwarfs we found have masses less than $0.55\\,\\Msolar$;\ni.e. their\nprogenitors had a pre-white dwarf life-time of more than $12\\,{\\rm Gyr}$.\nThey are all hotter than $14\\,000\\,{\\rm K}$, meaning\nthey have all cooled less than $0.5\\,{\\rm Gyr}$.\nDue to the large pre-white dwarf lifetime, their total age\nis perfectly compatible with halo membership.\nIt should be noted that the low mass of WD\\,0252$-$350 of\nonly $0.35\\,\\Msolar$ indicates that it probably does not possess a\nCO core but instead a He one.\n\n\nNow the masses and effective temperatures of the\nthick-disk white dwarfs detected in the SPY sample were likewise \nchecked. We found that four white dwarfs WD\\,0255$-$705,\nWD\\,0352+052, WD\\,1013$-$010, and WD\\,1334$-$678 have masses \nwhich imply ages of less than $7\\,{\\rm Gyr}$,\nwhich would \nmake them too young to belong to the thick disk. \n\nThese four stars are the coolest in our sample of thick-disk\ncandidates (see Table~\\ref{di_par}), with $T_{\\rm eff}$ ranging from 8800~K\nto 10600~K. \\cite{liebert05}\nderived the mass\ndistribution of 348 DA white dwarfs from the PG survey and found that the\naverage gravities and masses increase with decreasing effective \ntemperature for\n$T_{\\rm eff} < 12000$~K. A similar trend is found in the analysis of more\nthan 600 DA white dwarfs from the SPY survey (Voss et al., in prep.). \nThe physical reason is unknown, but two conjectures have been\npublished. The high masses inferred from\nspectroscopy below $\\approx$ 12000~K may actually be due to helium being\nbrought to the surface by the hydrogen convection zone \\citep{bergeron92,\nliebert05}. On the other hand, \n\\citet{koester05} suggest\nthat the treatment of non-ideal effects for the level population with the \nHummer-Mihalas \\citep{hummer88} occupation probability mechanism may \nbe insufficient for neutral perturbers that become important at lower \n$T_{\\rm eff}$. Since these effects are unaccounted for in the model \natmospheres, \nwe may have overestimated the masses\nof cool DA white dwarfs ($T_{\\rm eff} < 12000$~K).\n\nAs a result, the four cool \nwhite dwarf stars with thick-disk-like kinematics may have a \nlower mass and, therefore, a significantly larger age, one that is perhaps \neven \nconsistent with that of the thick disk. Therefore we regard them\nas very likely belonging to the thick disk as \nwell.\n\n\n\n\n\n\n\n\\begin{table}\n\\caption[]\n{Effective temperatures, surface gravities, and masses of the halo white\n dwarfs \\label{ha_par}}\n\\begin{tabular}{llll}\n\\\\\n\\hline\nstar & $T_{\\rm eff}$ & ${\\rm log}~g$ & $M$ \\\\ \n &${\\rm K}$ & ${\\rm cm \\,s^{-2}} $ & \\Msolar \\\\\n\\hline\n HS1527+0614 & 14015 & 7.80 & 0.50 \\\\\n WD1448+077 & 14459 & 7.66 & 0.44 \\\\\n WD2351$-$368 & 14567 & 7.81 & 0.51 \\\\\n WD0252$-$350 & 17056 & 7.42 & 0.35 \\\\\n WD2359$-$324 & 23267 & 7.65 & 0.47 \\\\\n WD1524$-$749 & 23414 & 7.61 & 0.45 \\\\\n HE0201$-$0513 & 24604 & 7.67 & 0.48 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption[]\n{Effective temperatures, surface gravities, and masses of the thick-disk white \ndwarfs. The four coolest stars have higher masses than the rest;\nhowever, the masses of the former may have been overestimated (see text).\n\\label{di_par}}\n\\begin{tabular}{llll}\n\\\\\n\\hline\nstar & $T_{\\rm eff}$ & ${\\rm log}~g$ & $M$ \\\\ \n &${\\rm K}$ & ${\\rm cm \\,s^{-2}} $ & \\Msolar \\\\\n\\hline\n WD1013$-$010 & 8786 & 8.19 & 0.71 \\\\\n WD1334$-$678 & 8958 & 8.11 & 0.66 \\\\\n WD0352+052 & 10234 & 8.00 & 0.60 \\\\\n WD0255$-$705 & 10574 & 8.09 & 0.65 \\\\\n\\hline\nWD1716+020 & 12795 & 7.66 & 0.43 \\\\\n WD2029+183 & 12976 & 7.73 & 0.47 \\\\\n WD0204$-$233 & 13176 & 7.75 & 0.47 \\\\\n WD1952$-$206 & 13742 & 7.78 & 0.49 \\\\\n WD0732$-$427 & 14070 & 7.96 & 0.58 \\\\\n WD1327$-$083 & 14141 & 7.79 & 0.50 \\\\\n WD1614$-$128 & 15313 & 7.74 & 0.48 \\\\\n HS1432+1441 & 15414 & 7.77 & 0.49 \\\\\n HE0508$-$2343 & 15835 & 7.71 & 0.47 \\\\\n HE1124+0144 & 15876 & 7.68 & 0.45 \\\\\n WD1426$-$276 & 17526 & 7.67 & 0.45 \\\\\n WD1834$-$781 & 17564 & 7.76 & 0.49 \\\\\n WD2350$-$083 & 17966 & 7.76 & 0.49 \\\\\n WD1323$-$514 & 18604 & 7.71 & 0.47 \\\\\n WD1507+021 & 19384 & 7.79 & 0.51 \\\\\n HE0452$-$3444 & 20035 & 7.82 & 0.53 \\\\\n WD1152$-$287 & 20185 & 7.64 & 0.45 \\\\\n WD1410+168 & 20757 & 7.74 & 0.49 \\\\\n WD2322$-$181 & 21478 & 7.88 & 0.56 \\\\\n HE0416$-$1034 & 23809 & 7.88 & 0.57 \\\\\n HS1338+0807 & 25057 & 7.73 & 0.50 \\\\\n HE0409$-$5154 & 26439 & 7.75 & 0.52 \\\\\n HS0820+2503 & 33330 & 7.69 & 0.51 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\nAn alternative explanation for the four cool DA stars discussed\nabove having \ngained thick-disk-like\norbits could be that\nthey might be run-away stars that were born in a binary system in the\nthin disk and were thereafter ejected from it.\nTwo ejection mechanisms have been suggested. The first one implies a close\nbinary system in which the primary undergoes a supernova explosion and\nreleases the secondary at high velocity \\citep{davies02}.\nThis study showed that, indeed, a large fraction of such\nbinaries are broken up when the primary explodes as a supernova.\nA large number of the secondaries receive kick\nvelocities of $100 \\mbox{-}200\\,{\\rm km s^{-1}}$ and travel on Galactic\norbits similar to those of thick-disk stars.\nThus a population of white dwarfs originating in\nthe thin disk may contribute significantly to the observed\npopulation of high-velocity white dwarfs.\n\nAnother possibility for explaining young white dwarfs with thick-disk-like \nkinematics was proposed by \\citet{kroupa02}, who suggests\na scenario for the thickening of galactic disks\nthrough clustered star formation. Massive star clusters may add kinematically\nhot components to galactic field populations.\n\n\n\nAs their masses may be overestimated, we think it is not required to\ninvoke such run-away scenarios to explain the origin of the four cool white \ndwarfs discussed above. A more natural explanation would be that we have simply\nunderestimated their ages. \n \nWe therefore classify \nthose $23$ white dwarfs where age and\nkinematics both indicate a thick-disk membership as bona fide \nthick-disk members. \nIn addition, the four cool \nwhite dwarfs are classified as probable thick-disk stars, i.e. \nall 27 stars are retained as thick-disk members. \nThis leaves us with a fraction of $2\\%$ halo and\n$7\\%$ thick-disk white dwarfs.\n\n\n\n\\section{Discussion\\label{dis}}\nWe have refined \nand sharpened \nthe population classification scheme\ndeveloped in Paper I and applied it to a kinematical\nanalysis of a sample of $398$ DA white dwarfs from the SPY project.\nCombining three kinematic criteria, i.e. the position in the $U$\\\/-$V$-diagram,\nthe position in the $J_Z$-$e$-diagram, and the Galactic orbit with age \nestimates, we found seven halo and $23$ thick-disk members.\n\nTo be able to discuss the kinematic parameters of the three different\npopulations white dwarfs, we calculated the mean value and standard\ndeviation of the three velocity components. Of interest are the\nasymmetric drift ($V_{\\rm lag}=220{\\rm km~s^{-1}}-$) for the thick-disk\nwhite dwarfs and the velocity dispersions of the white dwarfs of all\nthree populations (Tables~5--7).\nFor comparison, the corresponding values derived by \\citet{chiba00}\nand \\citet{soubiran03} for main-sequence stars are also shown.\n\nThe velocity dispersions that were found for the thin-disk\nwhite dwarfs are compatible with the ones of \\citet{soubiran03}.\nThe same is the case for the asymmetric drift and the velocity\ndispersions of the thick disk.\nHere agreement with the results of \\citet{soubiran03} is much\nbetter than with the earlier results of \\citet{chiba00}.\nThere, $\\sigma_{\\rm U}$ and $\\sigma_{\\rm V}$ of the halo white dwarfs\nare similar to the values of \\citet{chiba00}, while\nour $\\sigma_{\\rm W}$ is much smaller.\nThis is probably due to the fact that our local sample\ndoes not extend as far in the $Z$-direction as the\nsample of \\citet{chiba00} does.\nAlso with only seven halo white dwarfs, we have to account for\nsmall number statistics.\nIn general, the kinematic parameters\nof the white dwarfs of the three different populations\ndo not differ much from those of the main-sequence samples.\n\\begin{table}\n\\caption[]\n{Standard deviation of $U$, $V$,\n$W$ for the $361$ SPY thin-disk\nwhite dwarfs, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, and $\\sigma_{\\rm W}$\nfrom \\citet{soubiran03} are shown for comparison \\label{uvw_duetab}}\n\\begin{tabular}{lccc}\n\\hline\n &$\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n &${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nThin-disk WDs & & & \\\\\n(our sample) & $34$ & $24$ & $18$\\\\\n\\hline\nThin-disk stars & & &\\\\\n(Soubiran et al.)& $39$ & $20$ & $20$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Asymmetric drift $V_{\\rm lag}$ and standard deviation of $U$, $V$,\n$W$ for the $27$ SPY thick-disk\nwhite dwarfs, $V_{\\rm lag}$, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, \nand $\\sigma_{\\rm W}$\nfrom \\citet{soubiran03}, and \\citet{chiba00} are shown for \ncomparison \\label{uvw_ditab}}\n\\begin{tabular}{lcccc}\n\\hline\n &$V_{\\rm lag}$ & $\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n &${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nThick-disk WDs & & & \\\\\n(our sample)& $-51$ & $79$ & $36$ & $46$\\\\\n\\hline\nThick-disk stars & & & &\\\\\n(Soubiran et al.) & $-51$ & $63$ & $39$ & $39$\\\\\n\\hline\nThick-disk stars & & & &\\\\\n(Chiba \\& Beers) & $-20$ & $46$ & $50$ & $35$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption[]\n{Standard deviation of $U$, $V$,\n$W$ for the seven SPY halo\nwhite dwarfs, $V_{\\rm lag}$, $\\sigma_{\\rm U}$, $\\sigma_{\\rm V}$, and \n$\\sigma_{\\rm W}$\nfrom \\citet{chiba00} shown for comparison \\label{uvw_hatab}}\n\\begin{tabular}{lccc}\n\\hline\n & $\\sigma_{\\rm U}$ & $\\sigma_{\\rm V}$ & $\\sigma_{\\rm W}$\\\\\n & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$ & ${\\rm km~s^{-1}}$\\\\\n\\hline\nHalo white dwarfs & & &\\\\ \n(our sample)& $138$ & $95$ & $47$\\\\\n\\hline\nHalo stars & & &\\\\\n(Chiba \\& Beers) & $141$ & $106$ & $94$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe found seven halo white dwarfs in our sample, which\ncorresponds to a fraction of $2\\%$.\nIn Paper I we found $4\\%$ halo white dwarfs,\na deviation possibly due to small number statistics or to\na target selection effect.\nIn our first paper, we analysed stars from the early phase of\nSPY. This sample contained a relatively large fraction of white dwarfs\ndetected in proper motion surveys \n\\citep[][and references therein]{Luyten79, giclas78}. Therefore an\nover-representation of white dwarfs with high tangential velocities\nis not unexpected.\n\nOur value is lower than the one derived by\n\\citet{sion88},\nwho identified about $5$\\% of their sample as halo white dwarfs.\n\\citet{liebert89}, on the other hand, obtained a percentage of\n14\\% halo white dwarfs\nby classifying all stars that exceed a certain value of tangential velocity\nas halo members.\nWhen comparing those samples with ours, it has to be kept in mind that\nour selection criteria are \nsharper and allow us to separate thick-disk\nfrom halo stars. It is likely that a fraction of the white dwarfs classified \nas halo stars by\n\\citet{sion88} and \\citet{liebert89} actually belong to the thick disk.\nFurthermore, both samples suffer from the lack of radial velocity \ninformation.\n\nIt is difficult to compare our sample to the one of \\citet{oppenheimer01},\nbecause the inhomogeneous sky coverage of SPY does not allow us to\ncalculate a space density for halo white dwarfs.\nIt has to be taken into account that our sample is a magnitude\nlimited sample and thus biased towards\nhigh temperatures \\citep[mean temperature of $21\\,000\\,\\mathrm{K}$; see\nalso discussion in][]{schroeder04}, \nwhereas \\citet{oppenheimer01} analyse much cooler white\ndwarfs.\n\nClassically, halo white dwarfs are supposed to be cool stars\nthat originated from high mass progenitors.\nThe main contribution to the total ages of these white dwarfs is\nthe cooling time.\nThis work demonstrates that another\nclass of hot, low-mass halo white dwarfs exists\nwith low-mass progenitors that only recently have become white dwarfs so\nhave not had much time to cool down.\nThis makes this SPY sample\ncomplement to samples that focus on cool halo white dwarfs.\n\n\nThere are $27$ SPY white dwarfs classified as thick-disk members \nout of which \nfour are too cool to allow reliable ages to be derived.\nThis corresponds to a local fraction of thick-disk white dwarfs of\n$7\\%$ or $6\\%$, if we reject the four cool stars. \nThese values are somewhat lower than the $11\\%$ found \nby \\citet{silvestri02} but are much smaller than that of \nFuhrmann (2000)\\footnote{http:\/\/www.xray.mpe.mpg.de\/fuhrmann\/.},\nwho predicted a fraction of $17$\\% thick-disk white dwarfs.\nThe differences are possibly caused by the temperature bias mentioned above.\nAn over-representation of white dwarfs compared to\nlow mass main-sequence stars, which would require a truncated initial\nmass function as suggested by \\citet{favata97}, has not been found.\n\nThe question of whether thick-disk white dwarfs contribute\nsignificantly to the total mass of the Galaxy is very important for clarifying \nthe dark matter problem.\nThis contribution can be estimated from the results derived above.\nTo derive the densities of thin-disk and thick-disk white dwarfs,\nwe used the $1\/V_{\\rm max}$ method \\citep{schmidt68}.\nThe mass density of thick-disk over thin-disk white dwarfs\n${M_{\\rm thick}\\over M_{\\rm thin}}$ was calculated as described\nin Paper I.\nFor the thick disk we adopted the values of\n\\citet{ojha01}, scale length $l_{\\rm 0,thick}=3.7\\,{\\rm kpc}$, and\ntried two extreme values of the\nscale height, $h_{\\rm 0,thick}=0.8\\,{\\rm kpc}$ \\citep{ojha99} and\n$h_{\\rm 0,thick}=1.3\\,{\\rm kpc}$ \\citep{chen97}.\nFor the thin disk, we assumed\n$l_{\\rm 0,thin}=2.8\\,{\\rm kpc}$ \\citep{ojha01} and\n$h_{\\rm 0,thin}=0.25\\,{\\rm kpc}$, in between the values of\n\\citet{kroupa92} and \\citet{haywood97}.\nWe found ${M_{\\rm thick}\\over M_{\\rm thin}}=0.12 \\pm 0.36$ and\n${M_{\\rm thick}\\over M_{\\rm thin}}=0.19 \\pm 0.57$ for thick-disk scale \nheights of $0.8\\,{\\rm kpc}$\nand $1.3\\,{\\rm kpc}$, respectively.\nAccordingly, upper limits for ${M_{\\rm thick}\\over M_{\\rm thin}}$ are \n$0.48$ and $0.76$, respectively.\nOf course the errors are huge because of the poor statistics of the\nrelatively small thick-disk sample.\nNevertheless, it can be concluded\nthat the total mass of thick-disk white dwarfs is less than $48\\%$\n($76\\%$) of the total mass of thin-disk white dwarfs.\nTherefore the mass contribution of the thick-disk white dwarfs\nmust not be neglected, but it is not sufficient to account for the\nmissing dark matter.\n\n\\section{Conclusions\\label{con}}\nWe have demonstrated how a combination of sophisticated kinematic analysis\ntools can distinguish halo, thick-disk, and thin-disk white\ndwarfs. We identified a fraction of 2\\% halo and 7\\% thick-disk white\ndwarfs.\nMost of our thick-disk and halo white dwarfs\nare hot and possess low masses.\nOur results suggest that the mass present in halo and thick-disk\nwhite dwarfs is not sufficient for explaining the missing mass of the Galaxy.\nBut to draw definite conclusions, more data are needed.\nOur goal is to extend this kinematic analysis to all\n$1\\,000$ \ndegenerate stars\nfrom the SPY project,\nin order to have a large data base for\ndeciding on the population membership of white dwarfs\nand their implications for the mass and evolution of the Galaxy.\n\\acknowledgements { We thank D. Koester for providing us with the\nresults of his spectral analysis prior to publication and B. Voss for prolific\ndiscussions. \nE.-M. P. acknowledges support by the Deutsche\nForschungsgemeinschaft (grant Na\\,365\/2-1) and is grateful to the\nStudienstiftung des Deutschen Volkes for a grant. M. Altmann\nacknowledges support from the DLR~50~QD~0102 and from FONDAP~1501~0003. \nR.N.\\ is supported by a\nPPARC Advanced Fellowship. Thanks go to\nJ.~Pauli for interesting and fruitful discussions. \nThis research has made use of the SIMBAD database,\noperated at the CDS, Strasbourg, France \nand of DSS images based on photographic data obtained with the UK Schmidt\nTelescope. \n}\n\\bibliographystyle{aa}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSequential decison making is a common problem in many practical applications, broadly encompassing situations in which an agent must choose the best action to perform at each iteration while maximizing cumulative reward over some period of time \\cite{BouneffoufBG13,ChoromanskaCKLR19,RiemerKBF19,LinC0RR20,lin2020online,lin2020unified,NoothigattuBMCM19,lin2019split,lin2020story}.\nOne of the key challenges in sequential decision making is to achieve a good trade-off between the exploration of new actions and the exploitation of known actions. This exploration vs exploitation trade-off in sequential decision making is often formulated as the {\\em multi-armed bandit (MAB)} problem. In the MAB problem setting, given a set of bandit ``arms'' (actions), each associated with a fixed but unknown reward probability distribution ~\\cite {surveyDB,LR85,UCB,Bouneffouf0SW19,LinBCR18,DB2019,BalakrishnanBMR19ibm,BouneffoufLUFA14,RLbd2018,balakrishnan2020constrained,BouneffoufRCF17}, the agent selects an arm to play at each iteration, and receives a reward, drawn according to the selected arm's distribution, independently from the previous actions. \n\nA particularly useful version of MAB is the {\\em contextual multi-armed bandit (CMAB)}, or simply the {\\em contextual bandit} problem, where at each iteration, the agent observes a $N$-dimensional {\\em context}, or {\\em feature vector} prior to choosing an arm \\cite{AuerC98,AuerCFS02,BalakrishnanBMR18,BouneffoufBG12}.\nOver time, the goal is to learn the relationship between the context vectors and rewards, in order to make better action choices given the context \\cite{AgrawalG13}. Common sequential decision making problems with side information (context) that utilize the contextual bandit approach range from clinical trials \\cite{villar2015multi} to recommender systems \\cite{MaryGP15,Bouneffouf16,aaai0G20}, where the patient's information (medical history, etc.) or an online user profile provide a context for making better decisions about which treatment to propose or ad to show. The reward reflects the outcome of the selected action, such as success or failure of a particular treatment option, or whether an ad is clicked or not.\n\nIn this paper we consider a new problem setting referred to as {\\em contextual bandit with missing rewards}, where the agent can always observe the context but may not always observe the reward. \nThis setting is motivated by several real-life applications where the reward associated with a selected action can be missing, or unobservable by the agent, for various reasons. For instance, in medical decision making settings, a doctor can decide on a specific treatment option for a patient, but the patient may not come back for follow-up appointments; though the reward feedback regarding the treatment success is missing, the context, in this case the patient's medical record, is still available and can be potentially used to learn more about the patient's population. Missing rewards can also occur in information retrieval or online search settings where a user enters a search request, but, for various reasons, may not click on any of the suggested website links, and thus the reward feedback about those choices is missing. Yet another example is in online advertisement, where a user clicking on a proposed ad represents a positive reward, but the absence of a click can be negative reward (the user did not like the ad), or can be a consequence of a bug or connection loss.\n\nThe contextual bandit with missing rewards framework proposed here aims to capture the situations described above, and provide an approach to exploit all context information for future decision making, even if some rewards are missing. More specifically, we will combine unsupervised online clustering with the standard contextual bandit. Online clustering allows us to learn representations of all the context vectors, with or without the observed rewards. Utilizing the contextual bandit on top of clustering makes use of the reward information when it is available. We demonstrate on several real-life datasets that this approach consistently outperforms the standard contextual bandit approach when rewards are missing. \n \n\n\\section{Related Work}\n\\label{sec:related}\nThe multi-armed bandit problem provides a solution to the exploration versus exploitation trade-off \\cite {AllesiardoFB14,dj2020,Sohini2019}. This problem has been extensively studied. Optimal solutions have been provided using a stochastic formulation ~\\cite {LR85,UCB,BouneffoufF16}, a Bayesian formulation ~\\cite {T33}, and an adversarial formulation ~\\cite{AuerC98,AuerCFS02}. However, these approaches do not take into account the relationship between context and reward, potentially inhibiting overall performance.\nIn LINUCB ~\\cite{Li2010,ChuLRS11} and in Contextual Thompson Sampling (CTS)~\\cite{AgrawalG13}, the authors assume a linear dependency between the expected reward of an action and its context; the representation space is modeled using a set of linear predictors. However, these algorithms assume that the bandit can observe the reward at each iteration, which is not the case in many practical applications, including those discussed earlier in this paper. Authors in \\cite{bartok2014partial} considered a kind of incomplete feedback called \"Partial Monitoring (PM)\", developing a general framework for sequential decision making problems with incomplete feedback. The framework allows the learner to retrieve the expected value of actions through an analysis of the feedback matrix when possible, assuming both are known to the learner.\n\nIn \\cite{bouneffouf2020online}, authors study a variant of the stochastic multi-armed bandit (MAB) problem in which the context are corrupted. The new problem is motivated by certain online settings including clinical trial and ad recommendation applications. In order to address the corrupted-context setting, the author propose to combine the standard contextual bandit approach with a classical multi-armed bandit mechanism. Unlike standard contextual bandit methods, they were able to learn from all iteration, even those with corrupted context, by improving the computing of the expectation for each arm. Promising empirical results are obtained on several real-life datasets. \n\nIn this paper we focus on handling incomplete feedback in the bandit problem setting more generally, without assuming the existence of a systematic corruption process. Our work is somewhat comparable to online semi-supervised learning \\cite{Yver2009, ororbia2015online}, a field of machine learning that studies learning from both labeled and unlabeled examples in an online setting. However, in online semi-supervised learning, the true label is available at each iteration, whereas in the contextual bandit with missing rewards, only bandit feedback is available, and the true label, or best action, is unknown.\n\n\n\\section{Problem Setting} \n\n{ Algorithm \\ref{alg:CBP1} presents at a high-level the contextual bandit setting, where $x_t\\in C$ (we will assume here $C = \\mathbf{R}^N$) is a vector describing the context at time $t$, $r_{t,i} \\in [0,1]$ is the reward of the action $i$ at time $t$, and $r_t \\in [0,1]^K$ denotes a vector of rewards for all arms at time $t$. Also, $D_{c,r}$ denotes a joint probability distribution over $(x,r)$, $A$ denotes a set of $K$ actions, $A = \\{1,...,K\\}$, and $\\pi: C \\rightarrow A$ denotes a policy.\nWe operate under the linear realizability assumption; that is, there exists an unknown weight vector $\\theta^* \\in R$ with $ ||\\theta^*||\\leq 1$ so that,\n\n\\begin{equation*}\n\\forall k, t: \\; \\mathbb{E}[r_k(t) \\vert x_t] = \\theta_k^\\top x_t \n+ n_t .\\end{equation*}\n\nwhere $\\theta_k \\in \\mathbb{R}^d$ is an unknown coefficient vector associated with the arm $k$ which needs to be learned from data. Hence, we assume that the $r_{t,k}$ are independent random variables with expectation $x^\\top \\theta^*+ n_t$. with $n_t$ some measurement noise.\nWe also assume here that, the measurement noise $n_t$ is independent of everything and is $\\sigma$-sub-Gaussian for some $\\sigma >0$, i.e., $E[e^{\\phi\\, n_t} ] \\leq exp(\\frac{\\phi^2 \\sigma^2}{2})$ for all $ \\phi \\in R$.\n\\begin{definition}[Cumulative regret]\n{The regret of an algorithm accumulated during $T$ iterations is given as:\n\\begin{equation*}\nR(T) =\\sum ^{T}_{t=1} r_{t,k^*(t)} - \\sum^{T}_{t=1} r_{t,k(t)}.\n\\end{equation*}}\n\\end{definition}\n where $k^*(t)= \\text{argmax}_k x_{t}^\\top \\theta^*$is the best action at step $t$ according to $\\theta^*$.}\n\n\\begin{algorithm}[H]\n\t\\caption{ Contextual Bandit }\n\n\t\\label{alg:CBP1}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE {\\bfseries }\\textbf{Repeat}\n\t\t\\STATE {\\bfseries } $(x_t,r_t)$ is drawn according to $D_{x,r}$\n\t\t\\STATE {\\bfseries }$x_t$ is revealed to the player\n\t\t\\STATE {\\bfseries } The player chooses an action $k =\\pi_t(x_t)$\n\t\t\t\t\\STATE {\\bfseries } The reward $r_t$\n\t\t\\STATE {\\bfseries } The player updates its policy $\\pi_t$\n\t\t\t\\STATE {\\bfseries } $t=t+1$\n\t\t\\STATE {\\bfseries }\\textbf{Until} t=T\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{LINUCB with Missing Rewards (MLINUCB)}\n\\label{sec:banditDL}\nOne solution for the contextual bandit is the LINUCB algorithm~\\cite{13} where the key idea is to apply online ridge regression to incoming data to obtain an estimate of the coefficients $\\theta_k$. \nIn order to make use of the context even in the absence of the corresponding reward, we propose to use an unsupervised learning approach; specifically, we use an online clustering step to retrieve missing rewards from available rewards with similar contexts. At each time step, the context vectors $x(t)$ are clustered into $N$ clusters where $N$ is selected \\emph{a-priori}.\n\nWe adapt the LINUCB algorithm for our setting, proposing to use a clustering step for imputing the reward data when missing. At each time step, we perform a clustering step on the context vectors where the total number of clusters $N$ is a hyperparameter. For each cluster $j$ we define the average reward for each arm as below:\n\\begin{equation}\n\\overline{r}_j=\\frac{\\sum_{\\tau=1}^{n_j} r_{\\tau}}{n_j}\n\\label{waverage}\n\\end{equation}\nAssuming $d_j=dist(x_t,\\gamma_j)$ is the metric used for clustering where $\\gamma_j$ is the $j^{th}$ cluster centroid and $n_j$ is the number of data points in cluster $j$, we choose the $m$ smallest $d_j$ as the closest clusters to $x_t$ and compute a weighted average of the average cluster rewards as formulated below: \\begin{equation}g(x_t)=\\frac{\\sum_{j=1}^{m}\\frac{\\overline{r}_j}{d_j}}{\\sum_{j=1}^{m}\\frac{1}{d_j}}\\end{equation}\nWhen $r_t$ is missing we assign\n$r_t=g(x_t)$. Note that if $m=1$, $g(x_t)$ is simply the average rewards of all the points within the cluster that $x_t$ belongs to. \n \n\\begin{algorithm}[H]\n\\caption{MLINUCB}\n \\label{alg:LINUCB}\n \\begin{algorithmic}[1]\n \\STATE {\\bfseries Input:} value for $\\alpha$, $b_0$, $\\textbf{A}_0$, $N$, $m$\n \\FOR{t=1 {\\bfseries to} T} \n \\STATE cluster \\{$x_1$, ... , $x_t$\\} into $N$ clusters\n \\FOR{all $k \\in K$}\n \\STATE $\\theta_k \\leftarrow$ \\textbf{A}$_{k_t}^{-1}*b_{k_t}$ \n \\STATE $p_{t,k} \\leftarrow \\theta^{\\top}_k x_{t} +\n \\alpha \\sqrt{x^{\\top}_{t}\n \\textbf{A}_{k_t}^{-1} x_{t}}$\n \\ENDFOR\n \\STATE Choose arm $k_t = \\text{argmax}_{k\\in K} p_{t,k}$,\n and observe real-valued payoff $r_t$\n \\IF{ $r_t$ available}\n \\STATE retrieve $r_t$ from data\n \\ELSE \n \n \\STATE $r_t \\leftarrow g(x_t)$.\n \n \n \n \\ENDIF \n \\STATE \\textbf{A}$_{k_t} \\leftarrow$ \\textbf{A}$_{k_t} + x_{t,k_t} x^{\\top}_{t,k_t}$\n \\STATE $b_{k_t} \\leftarrow b_{k_t} + r_t x_{t,k_t}$ \n \\ENDFOR\n \\end{algorithmic}\n\\end{algorithm}\n\nWe now upper bound the regret of MLINUCB. Note that the general CBP setting \\cite{abbasi2011improved} takes one context per arm instead for our setting of the one context share by actions. To upper bound our algorithm for the general CBP setting, we simply cast our setting as theirs by the following steps.\nWe simply choose a global vector $\\theta$ as the concatenation of the $K$ vectors, so $\\theta =[\\theta_{1},...,\\theta_{K}]$. We define a context $x_{t,k}$ per action with $x_t$, where $x_{t,k} =[...0,x_{t}^{\\top},0,... ]^{\\top}$ and $x_t$ being the $k$-th vector within the concatenation. All $A_t$,$r_t$, $b_t$ can be similarly defined from $A_{k(t)}$, $r_{k(t)}$, $b_{k(t)}$.\n\\begin{thm} \\label{thm:hlinucb}\nWith probability $1-\\delta$, where $0 < \\delta < 1$, the upper bound on the R(T) for the MLINUCB in the contextual bandit problem, $K$ arms and $d$ features (context size) is given as follows:\n\\begin{eqnarray}\nR(T)\\leq \\sigma (\\sqrt{d \\;log (\\frac{det(A_{T})^{1\/2} }{\\delta\\;det(\\mathbf{S})^{1\/2}} )}+\\nonumber \\\\ \\frac{||\\theta||}{\\sqrt{\\phi}})\\sqrt{18\\; T log(\\frac{det(A_{T})}{det(\\mathbf{S})})} \n\\end{eqnarray}\nwith $||x_t||_2 \\leq L$, with $\\mathbf{S}=\\mathbf{I}+ \\sum_{t \\in s} x_tx_t^\\top $ with $s \\subset T$ contains the contexts with missing rewards and $\\phi \\in R$\n\\end{thm}\nTheorem \\ref{thm:hlinucb} shows that MLINUCB has better upper bound compared to the LINUCB \\cite{abbasi2011improved}, where in LINUCB upper bound has $log(det(A_{T}))$ under the square root where we have $log(\\frac{det(A_{T})}{det(\\mathbf{H})})$. We can see that the upper bound depends on $H$, so more context with missing rewards better is the bound.\n\n\n\\section{Regret analysis of BILINUCB}\nWe now upper bound the regret of BILINUCB. General Contextual Bandit Problem (CBP) setting \\cite{abbasi2011improved} assumes one context per arm instead for BILINUCB setting with same context shared across arms. To upper bound regret of BILINUCB we cast our setting as general CBP setting in the following way.\nWe choose a global vector $\\theta$ as the concatenation of the $K$ vectors, so $\\theta =[\\theta_{1},...,\\theta_{K}]$. Next define a context $x_{t,k}$ per arm as $x_{t,k} =[...,0,x_{t}^{\\top},0,... ]^{\\top}$ with $x_t$ being the $k$-th vector within the concatenation. Let $\\mathbf{S_T} = \\mathcal{I}_D + \\sum_{t \\in s} x_t x_t^\\top $, where $s \\subset \\{1,\\ldots,T\\}$ contains the contexts with missing rewards up to step $T$, and let $\\mathbf{A}_T = \\mathbf{S_T} + \\sum_{t \\not\\in s} x_t x_t^\\top $. We have the following theorem regarding the regret bound up to step $T$.\n\nTheorem 1 of the main text shows that BILINUCB has better upper bound compared to the LINUCB \\cite{abbasi2011improved}, where in LINUCB upper bound has $\\log(\\det(A_{t}))$ under the square root where we have $\\log(\\frac{\\det(A_{t})}{\\det(\\mathbf{S}_T)})$. \nThe matrix $\\mathbf{S}_T$ is the sum of identity matrix $\\mathcal{I}_D$ and covariance matrix $\\Sigma_s= \\sum_{t \\in s} x_t x_t^\\top $ constructed using the contexts with missing reward. Both $\\mathcal{I}_D$ and $\\Sigma_s$ are real symmetric and hence Hermitian matrices. Further, $\\Sigma_s$ is positive semi-definite as a covariance matrix.\nSince all the eigenvalues of $\\mathcal{I}_D$ equal $1$ and since all the eigenvalues of $\\Sigma_s$ are non-negative, by Weyl's inequality in matrix theory for perturbation of Hermitian matrices, the eigenvalues of $\\mathbf{S}_T$ are lower bounded by $1$. Hence $\\det(\\mathbf{S}_T)$ which is the product of the eigenvalues of $\\mathbf{S}_T$ is lower bounded by $1$. Hence, BILINUCB which involves the term $\\frac{\\det (\\mathbf{A}_T)}{\\det (\\mathbf{S}_T)}$ has a provably better guarantee than LINUCB which involves only the term $\\det(\\mathbf{A}_T)$ (without $\\det(\\mathbf{S}_T)$).\n\\subsection{Proof of Theorem 1}\n\\begin{proof}\n\nWe need the following assumption:\nwe assume that the noise introduced by the imputed reward is heteroscedastic. \n Formally, let $\\rho : X \\rightarrow R$\n be a continuous, positive function, such that $\\epsilon_t$ is conditionally\n $\\rho(x_t)$-subgaussian, that is for all $t \\geq 1$ and $\\rho_t = \\rho(x_t)$,\n\\begin{equation}\n\\forall\\; \\lambda \\in R, \\quad E[e^{\\lambda n_t} | F_{t-1}, x_t] \\leq \\exp\\left(\\frac{\\lambda^2 \\rho_t^2}{2}\\right) \\text{ .}\\label{eq: noise assumption}\n\\end{equation}\nNote that this condition implies that the noise has zero mean, and common examples include Gaussian, Rademacher, and uniform random variables\nWe need the following lemma, \n\n\\begin{lem} \\label{lem:ct} \nAssuming that, the measurement noise $\\epsilon_t$ issatisfies assumption \\eqref{eq: noise assumption}.\nWith probability $1-\\delta$, where $0 < \\delta < 1$ and $\\theta^*$ lies in the confidence ellipsoid.\n\\begin{eqnarray*}\nC_{t}=\\{ \\theta: \\|\\theta-\\hat{\\theta}_{t}\\|_{A_{t}} \\leq c_{t} :=\\nonumber \\\\ \\rho_t \\sqrt{D \\log \\frac{\\det(\\mathbf{A}_{t})^{1\/2} \\det(\\mathbf{S}_t)^{-1\/2}}{\\delta}}+ \\|\\theta^*\\|_2\\}\n\\end{eqnarray*}\n\\end{lem}\n\nThe lemma is adopted from theorem 2 in \\cite{abbasi2011improved} using the noise being heteroscedastic. We follow the same step of proof, the main difference is that they have $\\mathbf{A}_T=\\lambda \\mathcal{I}_D+ \\sum_{t=1}^T x_tx_t^\\top$ and we have $\\mathbf{A}_T=\\mathbf{S}_T+ \\sum_{t \\not\\in s} x_tx_t^\\top $ with $\\mathbf{S}_T=\\mathcal{I}_D+ \\sum_{t \\in s} x_tx_t^\\top $ with $s \\subset T$ contains the contexts with missing rewards .\n \n$ R(t) = [ x_t^{*\\top} \\theta^*-x_t^\\top \\theta_t] = [ x_t^{*\\top} \\theta^*- x_t^\\top \\theta^l_t ]+[x_t^\\top \\theta_t^l -x_t^\\top \\theta_t]$ \n\nwhere $ \\theta^l$ is the parameter of the classical LINUCB, and then\n\n$ R(t) \\leq \\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta^l_t \\|_2+\\|x_t^\\top \\theta^l_t -x_t^\\top \\theta_t\\|_2$ \n \nNow we investigate $\\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta_t^l \\|_2$ and $\\|x_t^\\top \\theta_t^l -x_t^\\top \\theta_t\\|_2$ separately. \n \n \nFollowing the same step as the proof of theorem 2 in \\cite{abbasi2011improved} we also have the following,\n\n \n\n\n\n\n\n$\\| x_t^{*\\top} \\theta^*- x_t^\\top \\theta_t^l \\|_2\\leq 2 c_t \\|x_t\\|_{\\mathbf{A}_{t}^{-1}}$, and using Cauchy-Schwarz with $\\|\\theta_t^l -\\theta_t\\|_2\\leq \\epsilon_t $, we get\n\n$\\|x_t^\\top \\theta^l_t -x_t^\\top \\theta_t\\|_2 \\leq \\epsilon_t \\|x_t\\|_{\\mathbf{A}_{t}^{-1}} $ and then,\n\n$R(t) \\leq (2 c_t+\\epsilon_t) \\|x_t\\|_{\\mathbf{A}_{t}^{-1}}$\n\nSince $x^{\\top}\\theta_{t}^* \\in [-1,1]$ for all $x \\in X_t $ then we have $R(t) \\leq 2$. Therefore,\n\n$R(t) \\leq \\text{min}\\{(2 c_t+\\epsilon_t)\\|x\\|_{\\mathbf{A}^{-1}_{t}},2\\} \\leq 2( c_t+\\epsilon_t\/2) \\; \\text{min}\\{\\|x\\|_{\\mathbf{A}^{-1}_{t}},1\\}$\n\nOur bound on the imputed reward assures $\\epsilon_t \\leq c_t$. Therefore,\n\n$[R(t)]^2 \\leq 9 c_t^2 \\text{min}\\{\\|x\\|^2_{\\mathbf{A}^{-1}_{t}},1\\}$\n\nwe have,\n\n$R(T) \\leq \\sqrt{T\\sum_{t=1}^{T}[R(t)]^2}= \\sqrt{ \\sum_{t=1}^T 9 c_t^2 \\text{min}\\{\\|x\\|^{2}_{\\mathbf{A}^{-1}_{t}},1\\}}$ \n\n$R(T)\\leq 3 c_T \\sqrt{ T} \\sqrt{ \\sum_{t=1}^T \\text{min}\\{\\|x\\|^{2}_{\\mathbf{A}^{-1}_{t}},1\\}}$, with $c_{T}$ monotonically increasing\n\nsince $x \\leq 2\\,\\log(1+x)$ for $x \\in [0,1]$, \n\nwe have $\\sum_{t=1}^{T} \\text{min}\\{\\|x_t\\|^2_{\\mathbf{A}_{t}^{-1}}, 1\\} \\leq 2 \\sum_{t=1}^{T} \\log(1+\\|x_t\\|^2_{\\mathbf{A}^{-1}_t})\\leq 2 (\\log\\det(\\mathbf{A}_{T})-\\log\\det(\\mathbf{S}_T))$, \n\nhere we also use the fact that we have $\\mathbf{A}_T=\\mathbf{S}_T+ \\sum_{s=1}^T x_sx_s^\\top $ to get the last inequality. \n\n$R(T)\\leq 3 c_T \\sqrt{2(\\log\\det(\\mathbf{A}_{T})-\\log\\det(\\mathbf{S}_T))}$\n \nby upper bounding $c_{T}$ using lemma \\ref{lem:ct} we get our result.\n\\end{proof}\n\n\n\n\n\\section{Experiments}\n\nIn order to verify the proposed MLINUCB methodology, we ran the LINUCB and MLINUCB algorithms on four different datasets, three derived from the UCI Machine Learning Repository \\footnote{https:\/\/archive.ics.uci.edu\/ml\/datasets.html}: Covertype, CNAE-9, and Internet Advertisements, and one external dataset : Warfarin. The Warfarin dataset concerns the dosage of the drug Warfarin, where each record consists of a context of patient information and the corresponding appropriate dosage or action. The reward is then defined as 1 if the correct action is chosen and 0 otherwise. The details for each of these datasets are summarized in the Table \\ref{table:Synthetic}. \n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{Datasets}\n\t\\resizebox{0.6\\columnwidth}{!}{\n\t\t\\begin{tabular}{ l | c | r | r }\n\t\t\tDatasets & Instances & Features & Classes \\\\ \\hline\n Covertype & 500 000 & 95 & 7\\\\\n CNAE-9 & 1080 & 856 & 9\\\\\n Internet Advertisements & 3279 & 1558 & 2\\\\\n Warfarin\t\t\t\t & 5528 & 93 & 3 \\\\\n\t\t\t\n\t\t\\end{tabular}\n\t}\n\t\\label{table:Synthetic}\n\\end{table}\nTo evaluate the performance of MLINUCB and LINUCB we utilize an accuracy metric that checks the equality of the selected action and the best action, which is revealed for the purposes of evaluation. Defined as such, accuracy is inversely proportional to regret. In the following experiments we fix $m=1, \\alpha=0.25$ and utilize the mini batch K-means algorithm for clustering. In Table \\ref{accuracy}, we report the total average accuracies of running LINUCB and MLINUCB with 2, 5, 10, 15, and 20 clusters on each dataset.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Total average accuracy}\n\\label{accuracy}\n\\begin{tabular}{l|l|l|l|l}\n \\multicolumn{5}{c}{10\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.884} & 0.644 & 0.866 & 0.643 \\\\ \\hline\nMLINUCB - $N=2$ & 0.869 & 0.643 & 0.898 & 0.643 \\\\ \nMLINUCB - $N=5$ & 0.874 & 0.626 & 0.895 & \\textbf{0.656} \\\\ \nMLINUCB - $N=10$ & 0.880 & 0.664 & 0.894 & 0.650 \\\\ \nMLINUCB - $N=15$ & 0.877 & \\textbf{0.678} & \\textbf{0.902} & 0.647 \\\\ \nMLINUCB - $N=20$ & 0.878 & 0.675 & 0.898 & 0.653 \\\\ \\hline\n\\end{tabular}\n\\\\\\vspace{\\baselineskip}\n\\begin{tabular}{l|l|l|l|l}\n \\multicolumn{5}{c}{50\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.884} & 0.566 & 0.824 & 0.615 \\\\ \\hline\nMLINUCB - $N=2$ & 0.838 & 0.578 & 0.888 & 0.630 \\\\ \nMLINUCB - $N=5$ & 0.847 \t\t & 0.546 & 0.896 & \\textbf{0.641} \\\\ \nMLINUCB - $N=10$ & 0.863 & 0.592 & 0.897 & 0.640 \\\\ \nMLINUCB - $N=15$ & 0.854 & \\textbf{0.608} & \\textbf{0.903} & 0.638 \\\\ \nMLINUCB - $N=20$ & 0.853 & 0.592 & 0.901 & 0.639 \\\\ \\hline\n\\end{tabular}\n\\\\\\vspace{\\baselineskip} \n\\begin{tabular}{l|l|l|l|l}\n\n \\multicolumn{5}{c}{75\\% Missing Rewards} \\\\ \\hline\n & Covertype & CNAE-9 & Internet Ads & Warfarin \\\\ \\hline\nLINUCB & \\textbf{0.880} & 0.483 & 0.786 & 0.610 \\\\ \\hline\nMLINUCB - $N=2$ & 0.784 & 0.461 & 0.881 & 0.594 \\\\ \nMLINUCB - $N=5$ & 0.797 & 0.494 & 0.890 & 0.612 \\\\ \nMLINUCB - $N=10$ & 0.837 & \\textbf{0.521} & 0.887 & \\textbf{0.624} \\\\ \nMLINUCB - $N=15$ & 0.824 & 0.500 & 0.891 & 0.600 \\\\ \nMLINUCB - $N=20$ & 0.819 & 0.493 & \\textbf{0.896} & 0.611 \\\\ \\hline \n\\end{tabular}\n\\vspace{\\baselineskip}\n\\end{table}\n\n\nAs the MLINUCB regret upper bound is lower than the LINUCB regret upper bound when $\\epsilon$ is small, minimizing clustering error is critical to performance. Accordingly, successful MLINUCB operates on the assumption that the context vectors live in a manifold that can be described by a set of clusters. Thus MLINUCB has the potential to outperform LINUCB when this manifold assumption holds, specifically when the number of clusters chosen adequately describes the structure of the context vector space. Visualizing the context vectors suggests that some of our test datasets violate this assumption, some respect this assumption, and when an appropriate number of clusters is chosen, MLINUCB performance aligns as expected. \n\nConsider the Internet Advertisements and Warfarin datasets, where 2D projections of the context vectors capture the majority of the variance in the context vector space, 100.0\\% and 98.2\\% respectively. In Figures \\ref{Advertisement_Acc} and \\ref{warfarin_acc} the projected context vector spaces appear clustered, not randomly scattered, and MLINUCB outperforms LINUCB for most choices of $N$, the number of clusters. The Internet Advertisements dataset yields the best results - when switching from LINUCB to MLINUCB algorithms, accuracy jumps from $86.6\\%$ to $90.2\\%$ when $25\\%$ of the reward data is missing, from $82.4\\%$ to $90.3\\%$ when $50\\%$ of the reward data is missing, and from $78.6\\%$ to $89.6\\%$ when $75\\%$ of the reward data is missing.\n\nAlthough the 2D projections of the Covertype and CNAE-9 context vectors in Figures \\ref{covtree1_acc} and \\ref{CNAE_acc} appear well clustered, both projections only capture a small amount of the variance in the context vector space, $29.7\\%$ in the Covertype dataset and $13.9\\%$ in the CNAE-9 dataset. MLINUCB results do not show improvement for the cases tried for Covertype dataset suggesting that the Covertype dataset violates the manifold assumption for the context space. However in the CNAE-9 dataset, we see that MLINUCB outperforms LINUCB for most choices of $N$, which supports the observation that the context space is clustered. \n\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 29.7\\% of the variance in the Covertype dataset.]{\\includegraphics[scale=0.2]{covtree1nc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{covtree1_avg_acc.png}}\n\\caption{Covertype}\n\\label{covtree1_acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 13.9\\% of the variance in the CNAE-9 dataset.]{\\includegraphics[scale=0.2]{CNAEnc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{CNAE_avg_acc.png}}\n\\caption{CNAE-9}\n\\label{CNAE_acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 13.9\\% of the variance in the CNAE-9 dataset.]{\\includegraphics[scale=0.19]{Advertisementnc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{Advertisement_avg_acc.png}}\n\\caption{Internet Advertisements}\n\\label{Advertisement_Acc}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[Context vector visualization with 5 clusters and 2D PCA. 2D PCA captures 98.2\\% of the variance in the Warfarin dataset.]{\\includegraphics[scale=0.2]{warfarin_datanc5.png}}\n\\subfigure[LINUCB and MLINUCB accuracy comparison]{\\includegraphics[scale=0.1]{warfarin_avg_acc.png}}\n\\caption{Warfarin}\n\\label{warfarin_acc}\n\\end{figure}\n\nTaking a more in depth look at the CNAE-9 dataset, in Figure \\ref{CNAE9_alpha}, we vary LINUCB and MLINUCB's common hyperparameter $\\alpha$, which controls the ratio of exploration to exploitation, and see that MLINUCB continues to result in higher accuracies than LINUCB for most $\\alpha$. \n\nNote that $N$, the number of clusters, is a hyperparameter of the algorithm and while initialized \\emph{a-priori}, it could be changed and optimized online as more context vectors are revealed. Alternatively, we could leverage clustering algorithms that do not initialize $N$ \\emph{a-priori} and learn the best $N$ from the available data.\n\n\\begin{figure}[h]\n\\label{CNAE9_alpha}\n\\centering\n{\\includegraphics[scale=0.11]{CNAE_alpha.png}}\n\\caption{LINUCB and MLINUCB cumulative accuracies on CNAE-9 across various $\\alpha$}\n\\end{figure}\n\n\n\\section{Conclusions and Future Work}\nIn this paper we studied the effect of data imputation in the case of missing rewards for multi-arm bandit problems. We prove an upper bound for the total regret in our algorithm following the CBP upper bound. Our MLINUCB algorithm shows improvements over LINUCB in terms of total average accuracy for most cases. The main observation here is that when the context vector space lives in a clustered manifold, we can take advantage of this structure and impute the missing reward at each step given similar context in previous events. \nA very obvious next step is to try using the weighted average introduced in equation \\ref{waverage} with $m$ greater than $1$. This would use more topological information from the context feature space and wouldn't rely on a single cluster. Additionally, the algorithm doesn't rely on a fixed value for $N$ so we could optimize the value of $N$ at each event using some clustering metric to find the best $N$ at each time. This work can also be extended by replacing the simple clustering step with more complex methodologies to learn a representation of the context vector space, for example sparse dictionary learning.\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}