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\section{Introduction}
The extended nature of disk galaxies coupled with the brightness of the night sky makes the study of the properties of large samples of these objects difficult in optical wavelengths. Any sample chosen from an optical catalog will inevitably be biased toward higher surface brightness objects, as the surface brightness of the night sky in the B-band, even at a dark site, is $\sim$23 mag arcsecond$^{-2}$. The surface brightness bias can be avoided, however, by choosing a sample from an H{\sc i} survey since sky emission is virtually nonexistent at 21 cm. Galaxies contained within such a catalog are certain to contain a significant amount of hydrogen gas and are therefore likely to be detected in H$\alpha$ emission.\par
Among the structures typically found in the ISM, H{\sc ii} regions are of particular interest as they trace well the amount of recent star formation and the initial star cluster mass function of a galaxy. The number of luminous H{\sc ii} regions, the luminosity of the brightest H{\sc ii} regions \citep{ken88}, and the slope of the H{\sc ii} region luminosity function (LF) \citep{ken89, ban93, cal91} have all been found to depend slightly on morphological type. However, these results come from optical studies; the effect of the exclusion of low surface brightness (LSB) disks is therefore not well constrained.\par
A sample of galaxies was chosen from the H{\sc i} Parkes All Sky Survey (H{\sc i}PASS) \citep{bar01} to be imaged in broad band B and R and narrow band H$\alpha$ to examine star formation properties free of the surface brightness bias. The overall properties of this sample are presented in paper I \citep{hel04}. This paper will focus on the characteristics of the H{\sc ii} regions by presenting the results of automated H{\sc ii} region photometry.\par
\section{HII Region Photometry}
\subsection{The Sample}
A sample of 132 galaxies with declinations less than -65$^{\circ}$ and radial velocities less than 2500 km s$^{-1}$ was chosen from H{\sc i}PASS; due mainly to weather constraints, 68 of these 132 were imaged through broad-band B and R and narrow-band H$\alpha$ filters. Because of extinction issues, lower priority was given during the observing run to galaxies that appeared nearly edge-on on their Digitized Sky Survey images (available through the NASA/IPAC Extra Galactic Database (NED)\footnote{NED is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.}). We also include M83 as a nearby example of a typical spiral galaxy and obtained images in all three bands for it. There was no significant detection of integrated H$\alpha$ emission for only one of these galaxies. Slightly more than half of the sample galaxies have morphological types that are Sd or later. The sample contains a higher fraction of relatively blue, LSB galaxies than is typically seen in optical catalogs. A comparison of observed and model values for the H$\alpha$ equivalent widths, star formation rates (SFRs) per unit H{\sc i} mass, and B-R colors shows that the star formation histories of these objects are best characterized by SFRs that began $\sim$4 Gyr ago and were higher in the past. For a detailed description of the sample properties as well as the data acquisition, reduction, calibration, and measurements, see paper I.\par
\subsection{HII Region Fluxes and Diffuse Fractions}
Automated H{\sc ii} region photometry was performed on the H$\alpha$ continuum subtracted images utilizing the IDL program HII{\it phot} developed by \citet{thi00}. The images were corrected for $[$NII$]$ contamination using the following relation derived from a linear least squares fit to the photometric and spectroscopic data of \citet{jan00}
\begin{equation}
\mbox{log }\frac{[NII]}{H\alpha}=(-0.057\pm0.0093)M_{R}+(0.78\pm0.069)(B-R)_{e}+(-2.62\pm0.17)
\end{equation}
where $M_{R}$ is the absolute R-band magnitude and $(B-R)_{e}$ is the color index measured within the half-light radius. The scatter about this relation is $\sim$0.1 dex smaller than that used in paper I which did not utilize the galaxies' color indexes. For our sample, the values for $[$NII$]$/H$\alpha$ computed with equation (1) range from about 5\%-6\% for the faintest, bluest galaxies to about 50\%-60\% for the brightest, reddest galaxies.\par
The basic procedure for the HII{\it phot} program is as follows. On each image, a region of interest is selected around the galaxy which is convolved with a Gaussian kernel to remove any background fluctuations. The region is then convolved with several other kernels of various sizes to identify possible H{\sc ii} regions which are then fit with models of different morphologies. The models have the following functional form\par
\begin{equation}
I_{H\alpha}=I_{H\alpha,p}\mbox{ exp} \left [-\frac{(r-r_{o})^2}{2\sigma^2} \right ]
\label{eqn1}
\end{equation}
where $I_{H\alpha,p}$ is the peak H$\alpha$ surface brightness and the ratio of the inner ring radius, $r_{o}$, to the Gaussian width, $\sigma$, is varied among six different values, 0, 0.5, 1, 2, 4, and 8. For each region, these six models are stretched and rotated to match the shape of the region; the model that produces the best fit is used to obtain values for the axis ratio, position angle, and {\sc fwhm} along the major and minor axes, {\sc fwhm}$_{maj}$ and {\sc fwhm}$_{min}$.\par
``Footprints'' of the H{\sc ii} regions are then constructed by altering the boundaries of the objects so that any pixels that were shared by more than one object become associated with the object that is best fit by its model morphology. These model boundaries are then trimmed to create H{\sc ii} region ``seeds'' by rejecting any pixels that are less than half of the median value for the object. These seeds are then grown iteratively by lowering the threshold for the isophotal boundary by 0.02 dex at a time until the slope of the surface brightness profile reaches some minimum or until no more acceptable pixels (i.e. pixels that do not belong to other seeds or pixels that are below some maximum value) are available. Seeds that are best fit by their model morphologies are grown first. For this paper, a limiting surface brightness slope of 1.5 EM pc$^{-1}$ was used in the growth process where EM is the emission measure\footnote{On each image, the H$\alpha$ surface brightness in each pixel was converted to emission measure assuming a temperature of 10$^{4}$K and case B recombination (i.e. EM=5$\times$10$^{17} \frac{I_{H\alpha}}{1 \mbox{\scriptsize{erg s}}^{-1} \mbox{\scriptsize{ cm}}^{-2} \mbox{\scriptsize{ }} \Box ''}$ cm$^{-6}$ pc)} in standard units of cm$^{-6}$ pc. An example of this process is illustrated by the images of one of the sample galaxies displayed in Fig. 1.\par
After the H{\sc ii} region boundaries are determined, the flux within these boundaries is automatically measured. However, these fluxes may contain a contribution from any diffuse ionized gas (DIG) in the galaxy along the line of sight. HII{\it phot} estimates this contribution for each region using the flux from background pixels defined to be all of the pixels outside the region's boundary within a projected distance of 250 pc from its boundary. Any background pixel that is at least 75\% surrounded by other background pixels within a circular region of 250 pc is then identified as a control point. The median values for the circular regions surrounding the control points are then computed and a surface is fit to them to estimate the flux within the boundary of the H{\sc ii} region that is emitted by the DIG. This is then subtracted off the total flux within the boundary to give a corrected flux that will be used in the construction of the H{\sc ii} region luminosity function (LF). Following this, HII{\it phot} produces a "background" image where the pixels belonging to H{\sc ii} regions are replaced with the background surface fits.\par
For each galaxy, the flux from the galaxy on the background image was measured and compared to the total H$\alpha$ flux to determine the diffuse fraction, $f_{d}$, or the fraction of the total H$\alpha$ flux that originates from the DIG. We also formally compute the error in each value of $f_{d}$ using the counts measured from the galaxy on the H$\alpha$ and R-band images, a measure of the uncertainty in the value used for the background subtraction for the H$\alpha$ continuum subtracted image, and an estimate of the uncertainty in the continuum subtraction. The individual values for $f_{d}$ as well as their 1$\sigma$ errors are listed in Table 1. It should be noted that for some of the galaxies, the errors in the diffuse fractions are large ($>$50\%). The error in $f_{d}$ is influenced heavily by the uncerties in the continuum and background subtraction processes, causing the error for some of the more LSB galaxies and galaxies with relatively bright stellar continua to be larger. In some cases, the counts from the DIG are low which also causes the relative error in $f_{d}$ to be high.\par
\placefigure{fig1}
\placetable{tab1}
\subsection{The Brightest HII Regions}
For each galaxy, the mean H$\alpha$ luminosity and diameter of the three brightest H{\sc ii} regions were calculated; we will refer to these mean quantities hereafter as L$_{3}$ and D$_{3}$. We chose to compute a weighted mean where the weights used for the two quantities for each H{\sc ii} region were $\left(\frac{\mbox{\sc snr}}{L}\right)^{2}$ and $\left(\frac{\mbox{\sc snr}}{D}\right)^{2}$ respectively where {\sc snr} is the signal to noise ratio computed by HII{\it phot}. This weighting scheme was chosen because many galaxies contained few if any large H{\sc ii} regions implying that the {\sc snr} for any one of the three brightest regions may be relatively low. For each H{\sc ii} region, the diameter was taken to be the diameter of a circle that occupies the same area on the image as the H{\sc ii} region. To correct for the effects of the {\sc psf}, we first measure the median {\sc fwhm} of Gaussian fits to the radial profiles of ten stars on the H$\alpha$ image of each galaxy. We then used the effective circular {\sc fwhm} from the Gaussian model fit for each region (equal to $\sqrt{\mbox{\sc fwhm}_{maj} \; \mbox{\sc fwhm}_{min}}$) to effectively deconvolve the measured diameter for each region according to
\begin{equation}
\mbox{D}_{corr} = \mbox{D } \frac{\sqrt{\mbox{\sc fwhm}_{mod}^{2} - \mbox{\sc fwhm}_{psf}^{2}}}{\mbox{\sc fwhm}_{mod}}
\end{equation}
were {\sc fwhm}$_{mod}$ is the effective circular {\sc fwhm} from the Gaussian model fit. The weighted mean diameter for the three brightest H{\sc ii} regions computed using these corrected diameters is referred to as D$_{3,corr}$ and is listed for each galaxy in Table 1.\par
To compare our results with those of \citet{ken88}, we plot in Fig. 2 L$_{3}$ as a function of total B-band absolute magnitude corrected for internal extinction according to the correction used to produce the $B_{T}^{0}$ magnitudes in the Revised Shapley-Ames Catalog \citep{san81} used by \citet{ken88}, which we refer to as $M_{B}(0)$. We fit a line to this data for Sbc/Sc galaxies only and find that the trend for our galaxies is somewhat less steep than the fit obtained for the Sbc/Sc galaxies from \citet{ken88}; both lines are plotted with the data in Fig. 2. We also plot the fit to the Sbc/Sc galaxies in our sample $\pm 1 \sigma$ where $\sigma$ is the rms scatter about the fitted line. These lines demonstrate that while the slope for the fit to our data is formally less steep, the two fits agree within 1$\sigma$. We find, as \citet{ken88} did, that a significant number of Sm and Im galaxies in our sample lie above the line fit to the Sbc/Sc galaxies. We also find that a small but significant fraction of these extremely late-type disk galaxies lie near or below both the line fit to our data and the line fit to the data of \citet{ken88}. In contrast, all but one of the Sm and Im galaxies within the sample of \citet{ken88} lie above the fit to the data for the Sbc/Sc galaxies. It should be noted that \citet{ken88} used isophotal H{\sc ii} region luminosities and computed the average of the three brightest regions without using any weighting. To examine the effect of this on our measured trend, we also computed the luminosity of the H{\sc ii} regions for each galaxy using the same limiting isophote as \citet{ken88}, 100 cm$^{-6}$ pc. The median difference between the value for L$_{3}$ and the unweighted mean of the isophotal luminosities for the three brightest H{\sc ii} regions is 0.07 dex, implying that the difference between the two techniques used for measuring luminosities contributes weakly, if at all, to the difference in the two observed trends.\par
\placefigure{fig2}
\subsection{Measured Properties and Biases}
For H$_{\circ}$=70 km s$^{-1}$ Mpc$^{-1}$, one arcsecond corresponds to about 170 pc at the radial velocity limit of our sample. This implies that the ability of HII{\it phot} to find smaller, lower luminosity H{\sc ii} regions is limited for the most distant galaxies in the sample. To explore the effect this has on the measured values of L$_{3}$, D$_{3}$, and $f_{d}$, we first compute a minimum detectable H{\sc ii} region H$\alpha$ luminosity, L$_{lim}$, for each galaxy. We do this by fitting a line to log L$_{H\alpha}$ as a function of log {\sc snr} for all the regions in the galaxy. This fit was used to solve for the luminosity at a signal to noise ratio of five, the H{\sc ii} region detection limit recommended by \citet{thi00}. The value for L$_{lim}$ was then taken to be this luminosity plus 1$\sigma$, where $\sigma$ is the rms deviation of the log L$_{H\alpha}$ values from the fitted line. All but five of the sample galaxies have L$_{lim} > 10^{38}$ ergs s$^{-1}$ cm$^{-2}$; the largest value is 10$^{38.5}$ ergs s$^{-1}$ cm$^{-2}$. Because of this, we list in Table 1 the number of regions found by HII{\it phot} with log L$_{H\alpha} >$38 rather than the total number of regions detected so that a comparison of the number of regions found in one galaxy to that found in another can be made without any bias caused by the completeness limit increasing with distance.\par
In Fig. 3, we plot radial velocity relative to the Local Group and corrected for in-fall into Virgo, $V_{LG}$, versus L$_{lim}$. As expected, L$_{lim}$ increases with $V_{LG}$ but with a significant amount of scatter ($\sim$0.3 dex about a line fit to the data). In the lower panel of Fig. 3, we plot $V_{LG}$ versus L$_{3}$. We also plot the median value of L$_{3}$ in five bins for V$_{LG}$; we display these values as shaded rectangles with widths equal to the widths of the V$_{LG}$ bins and with lower and upper boundaries that correspond to the lower and upper quartiles for L$_{3}$ for the bins. It is evident from this plot that there is a bias toward higher values of L$_{3}$ for the more distant galaxies. However, the largest values of L$_{3}$ for the most distant galaxies are not significantly higher than the largest values for the most nearby galaxies, implying that this bias is due mostly to the increase in typical detection limit with $V_{LG}$ than to a problem with artificial blending of luminous (log L$_{H\alpha} \gtrsim$38) H{\sc ii} regions for more distant objects. This is consistent with tests run by \citet{thi00} on M51 which implied that blending had no significant effect on the number of H{\sc ii} regions with log L$_{H\alpha} \geq$38.6 detected by HII{\it phot} for distances up to 45 Mpc \citep[see Fig. 11 of][]{thi00}.\par
The same is not true, however, for D$_{3}$. We plot $V_{LG}$ versus D$_{3}$ in Fig. 4 along with median values for five $V_{LG}$ bins and find that there is an apparent positive correlation between the two quantities. This is what one may expect if artificial blending of H{\sc ii} regions were a significant problem at higher redshift. However, the absence of a similar trend between $V_{LG}$ and L$_{3}$ indicates that this is most likely not the case. It is then more likely that this trend is caused by the increase in the physical size that corresponds to the size of the seeing disk for the more distant galaxies. The median {\sc fwhm} of the {\sc psf} for our H$\alpha$ images was about 1{\huge \H{.}}6, or about 280 pc for $V_{LG}$=2500 km s$^{-1}$. To estimate the magnitude of the effect the size of the seeing disk has on the trend in Fig. 4, we plot D$_{3,corr}$ as a function of $V_{LG}$ in the lower panel of Fig. 4. From this plot, it can be seen that the {\sc psf} corrections from equation (3) significantly reduce the magnitude of the trend seen in the upper panel of Fig. 4 while retaining a bias toward larger values of D$_{3}$ at larger distances similar to the bias observed in Fig. 3 for L$_{3}$. This implies again that blending of H{\sc ii} regions is not a significant problem for our sample for the brightest regions. It also implies that the measured H{\sc ii} region sizes are significantly affected by the width of the {\sc psf} and are not reliable indicators for how the typical region diameter may depend on various galaxy properties.\par
We plot $f_{d}$ as a function of $V_{LG}$ and median values for $f_{d}$ in five $V_{LG}$ bins in Fig. 5 to explore the possibility that the higher detection limit at larger distances causes the amount of DIG to be overestimated. If this is the case, one would expect the diffuse fractions to be significantly higher at larger redshifts. From Fig. 5, it is evident that the median value for $f_{d}$ for the lowest $V_{LG}$ bin is smaller than that for the other bins. However, the values for the remaining four bins are roughly constant, implying that there is not a significant bias toward higher values of $f_{d}$ at larger distances.
\placefigure{fig3}
\placefigure{fig4}
\placefigure{fig5}
\subsection{Surface Brightness and Color}
In paper I, it was discovered that the H{\sc i}PASS sample contains a larger fraction of galaxies that are relatively bluer and lower surface brightness than the optically selected Nearby Field Galaxy Survey (NFGS) of \citet{jan00}. To explore the possibility that the properties of the H{\sc ii} regions and DIG in the bluest and most LSB galaxies which appear in smaller numbers in similar optical samples may be significantly different, we first plot the R-band surface brightness at the half-light radius, $\mu_{e,R}$, and the color within the half-light radius, $(B-R)_{e}$, as functions of $V_{LG}$ in Fig. 5. We again plot median values within five $V_{LG}$ bins and display the results as shaded rectangles in the same way as was done for L$_{3}$ in Fig. 3 (see \S 2.4). From these plots, there appears to be no significant bias toward higher or lower surface brightness galaxies at higher redshift. Similarly, there is no apparent bias toward bluer or redder galaxies at larger distances. Because of this, any trend observed between L$_{3}$ and either surface brightness or color cannot be the result of the bias toward larger values of L$_{3}$ at higher redshift that is evident in Fig. 3. To explore possible relationships among these quantities, we plot $\mu_{e,R}$ and $(B-R)_{e}$ versus L$_{3}$ and $f_{d}$ for all of our sample galaxies in Fig. 6. We do not include similar plots using D$_{3}$ because of inaccuracies caused by the relative size of the seeing disk as described in the previous section.\par
Within four bins in surface brightness and color, each containing roughly the same number of galaxies, we compute the median values for L$_{3}$ and $f_{d}$ and display them in the appropriate plots in Fig. 6 as shaded rectangles with widths equal to the width of the bins in $\mu_{e,R}$ and $(B-R)_{e}$ and with upper and lower boundaries that correspond to the median values for L$_{3}$ and $f_{d}$ $\pm 1 \sigma$ where $\sigma$ is the error in the median value (i.e. not the difference between the upper and lower quartiles as was used in previous figures). We also plot median values for L$_{3}$ and $f_{d}$ for all galaxies as horizontal dashed lines. From these median values, it can be seen that the lowest surface brightness galaxies have higher diffuse fractions and lower values for L$_{3}$; the same is true to a lesser degree for relatively bluer galaxies. The fact that lower surface brightness galaxies are found to have lower values of L$_{3}$ may be heavily influenced by small number statistics as the number of detected H{\sc ii} regions is correlated with surface brightness. The degree to which this may effect the observed trend will be addressed later in \S 3.4. The trend found between diffuse fraction and surface brightness is similar to the trend found by \citet{gre98} who demonstrated that within individual galaxies, areas with higher SFR surface densities had lower diffuse fractions. Despite the instances of highly uncertain diffuse fractions alluded to in \S 2.2, the errors in the median values for the four surface brightness bins are small enough to conclude that the trend observed between $\mu_{e,R}$ and $f_{d}$ is real.
\placefigure{fig6}
\section{HII Region LFs}
\subsection{Individual LFs}
H{\sc ii} region LFs were constructed for all galaxies within our H{\sc i}PASS sample with $V_{LG}>$840 km s$^{-1}$. This velocity requirement was used because the major sources of uncertainty in the corrected velocities come from the errors in the assumed values for the velocity of the sun relative to the Local Group and the distance to the Virgo cluster. These uncertainties produce errors in the corrected radial velocities that are on the order of 50 km s$^{-1}$ \citep{yah77} and 80 km s$^{-1}$ \citep{lu94} respectively yielding a total uncertainty of about 94 km s$^{-1}$. Given that the luminosity bins for the LFs were chosen to be 0.2 dex, the error in the luminosity originating from the error in the distance determination must be less than 0.1 dex for the computed LF to accurately reflect the true LF for a given galaxy. Therefore, the largest allowed error in $V_{LG}$ is 11.5\%, and the minimum allowable corrected radial velocity is 817.4 km s$^{-1}$. A more conservative limit of 840 km s$^{-1}$ was used to ensure that the shapes of the LFs would not be significantly affected by distance errors and because this velocity corresponds to a distance of 12 Mpc for H$_{\circ}$=70 km s$^{-1}$ Mpc$^{-1}$. For each galaxy, only H{\sc ii} regions with signal to noise ratios greater than five were considered to be detected regions \citep[i.e. the detection limit recommended by][]{thi00}. Example LFs are plotted in Fig. 7.\par
Previous studies of H{\sc ii} region LFs \citep[e.g.][]{ham01} have used power law fits to the bright end of the LFs to characterize their shapes, assuming the LF follows the form dN$\propto$L$^{\alpha}$dL. Fits are typically confined to the largest values of L$_{H \alpha}$ because for a \citet{sal55} initial mass function (IMF) (i.e. power law slope of 2.35) and an upper limit of 100 M$_{\odot}$, the largest H$\alpha$ luminosity one would expect from a region ionized by a single star is about 3$\times 10^{38}$ ergs s$^{-1}$. \citet{oey98} demonstrated that for a distribution of regions ionized by poor or "unsaturated" clusters, the LF drops steeply at an H$\alpha$ luminosity of about 10$^{39}$ ergs s$^{-1}$; beyond this, the shape of the LF traces well the distribution of masses for rich clusters. Because of this, for any galaxy that forms a significant number of poor clusters, the shape of the LF can be markedly different below and above the limit of 10$^{39}$ ergs s$^{-1}$. Therefore, to characterize the shapes of the LFs, we fit a power law for log L$_{H\alpha}>$39 for all LFs where at least five luminosity bins with log L$_{H\alpha}>$39 are not empty. The results of these fits are plotted with the appropriate LFs in Fig. 7; values for the power law slope, $\alpha$, for individual galaxies are listed in Table 1. No significant trend was found between $\alpha$ and either $\mu_{e}$ or $(B-R)_{e}$. It is possible that this is due to the large uncertainty in the determination of $\alpha$ (the median error is $\sim$20\%) or to the fact that more LSB, fainter galaxies tend to form few if any regions with log L$_{H\alpha}>$39, implying that these power law fits do not provide any information regarding the difference between the shapes of the LFs for these galaxies and those for more HSB, brighter galaxies.\par
\placefigure{fig7}
\subsection{Co-added LFs}
To explore how the shapes of the H{\sc ii} region LFs for the lowest surface brightness and bluest galaxies in the H{\sc i}PASS sample may differ from those for galaxies that are more typically contained within optically selected samples such as the NFGS, we have constructed co-added H{\sc ii} region LFs. We have opted for this approach because, as stated above, the most LSB and bluest galaxies tend to be less luminous and to have fewer detectable H{\sc ii} regions. The co-adding process used was as follows. For a particular co-added group, within each luminosity bin, a weighted mean number of regions was computed for all galaxies that had L$_{lim}$ (see \S 2.4) less than the lower boundary of the bin. Since larger galaxies tend to form more H{\sc ii} regions and since the H{\sc i}PASS sample was selected from an H{\sc i} catalog, for this computation, the number of regions within each bin from each galaxy was weighted by $M_{HI}^{-1}$. This was done to ensure that no one galaxy dominated the shape of the resulting co-added LF. Because of this weighting scheme, only galaxies with $M_{HI}$ greater than the H{\sc i}PASS 3$\sigma$ detection limit at a radial velocity of 2500 km s$^{-1}$ (1.6$\times10^{8}$ M$_{\odot}$ for H$_{\circ}$=70 km s$^{-1}$ Mpc$^{-1}$) were included. Again, galaxies with corrected radial velocities less than 840 km s$^{-1}$ were excluded. These two requirements qualify 58 of the 69 H{\sc i}PASS galaxies for the co-adding process.\par
To asses how distance effects influence this co-adding process, all galaxies eligible to be co-added were sorted by radial velocity and then placed in three roughly equal sized groups designated near, intermediate, and far. The ranges in values of $V_{LG}$ for the three groups are roughly 880$<V_{LG}<$1380 km s$^{-1}$ for the near group, 1380$<V_{LG}<$1730 km s$^{-1}$ for the intermediate group, and 1730$<V_{LG}<$2500 km s$^{-1}$ for the far group. The co-added LFs for these groups are displayed in Fig. 8 along with the ratio of the near group LF to the intermediate group LF and the ratio of the near group LF to the far group LF as functions of luminosity. For log L$_{H\alpha}>$38, the LF for the intermediate group appears to have fewer regions at all luminosities than the nearby group; the opposite is true for the LF for the far group. However, the residuals between the near group LF and both the intermediate and far group LFs are relatively flat as functions of H$\alpha$ luminosity for 38$\leq \mbox{log } L_{H\alpha} \leq$40, implying that the shape of the co-added LFs do not change significantly with distance.\par
\citet{thi00} used images of M51 altered to make the galaxy appear more distant to asses the effects of sensitivity and blending on the shape of the LF as a function of distance. The altered images were created to simulate the appearance of M51 if it were at distances of 15, 30, and 45 Mpc (the actual distance to M51 was assumed to be 9.6 Mpc). The results from this analysis predict that for more distant galaxies, HII{\it phot} will measure an LF with a slope that is artificially flattened by blending for intermediate luminosity regions while maintaining the same shape for higher luminosity regions (log L$_{H\alpha} \geq$38.6 for the case of M51 for a distance of up to 45 Mpc). Since the most distant galaxies in our sample are closer than 40 Mpc and the shapes of the LFs for all three distance groups agree reasonably well, flattening of the LFs from distance effects does not appear to be a significant problem for our co-adding process.\par
While these results imply that the effects of blending do not depend on distance within the volume occupied by our sample, they may be worse for locations where H{\sc ii} regions are more closely spaced \citep[e.g.][]{sco01} such as spiral arms or higher surface brightness galaxy disks. This may cause the shapes of the measured LFs for such locations to artificially appear less steep when compared to LFs for locations where the typical separation between H{\sc ii} regions is larger such as interarm regions and LSB galaxies. The possible impact of this on the analysis to follow will be discussed in \S 3.3.\par
To examine how the LFs for the most LSB and bluest galaxies may differ from those for typical spirals, six co-added groups were selected. First, LFs for two comparison groups were constructed. The comparison groups were chosen so that they spanned the ranges in surface brightness and color typical of optical catalogs and so that they contained enough galaxies that the resulting LFs would be well determined (i.e. less than 10\% error for the majority of the log L$_{H\alpha}$ bins). The surface brightness comparison group was chosen to consist of all galaxies from our sample that qualified to be co-added according to the criteria described above that were within $\pm 2 \sigma$ of the mean surface brightness for the NFGS (24.4$<\mu_{e,B}<$20.4), 50 galaxies in all. The color comparison group was chosen in a similar way by selecting all qualified galaxies from our sample within $\pm 2 \sigma$ of the mean color for the NFGS (0.60$<$(B-R)$_{e}<$1.60), resulting in a group of 56 galaxies.\par
To examine how the shapes of the LFs for galaxies that are at the extreme ends of the color and surface brightness distributions may differ from the typical shapes represented by the LFs computed for the comparison groups, the remaining four co-added groups were constructed as follows. Among the 58 galaxies that qualify to be co-added, the ten lowest and ten highest surface brightness galaxies were co-added in two separate groups; similarly, the ten bluest and ten reddest galaxies were co-added in two other groups. These groups are hereafter referred to as the LSB, HSB, blue and red groups respectively. The number of galaxies within these groups was chosen to be ten as that was the number required to keep the errors within the majority of the log L$_{H\alpha}$ bins to less than 30\% for the LFs for all four groups. The resulting co-added LFs for all six groups are plotted in the two left panels of Fig. 9.\par
At the lowest detectable H{\sc ii} region luminosities, all the LFs appear to roughly agree. However, the shape of the LSB group LF is markedly different from that of the comparison LF; the LSB group LF is approximated well by a power law down to log L$_{H\alpha}\sim$38. The slope of the LF for the blue group appears to be flatter than that for the comparison and red groups for intermediate luminosities (37.5$\leq$log L$_{H\alpha} \leq$38.5); the slope for the red group LF appears to be steeper than that of the color comparison LF. To quantify these differences in shape, we compute the reduced $\chi^{2}$ (referred to hereafter as $\chi^{2}_{red}$) between the LSB, HSB, blue, and red group LFs and their comparison LFs. We do this by assuming that the LFs for the four extreme groups each differ from the corresponding comparison LF by a scale factor. We then determine the scale factor that minimizes $\chi^{2}$ over the range 37.7$\leq$log L$_{H\alpha}<$40.5 and divide the minimized $\chi^{2}$ by the number of nonempty log L$_{H\alpha}$ bins over this range minus one. This lower limit for the range in log L$_{H\alpha}$ was chosen because less than half of the galaxies in the comparison groups have log L$_{lim}<$37.7, implying that below log L$_{H\alpha}$=37.7, the completeness corrections made during the construction of the comparison LFs are significant and that this portion of each LF is not as representative of the entire co-added group as the rest of the LF. The upper limit was chosen because beyond log L$_{H\alpha}$=40.5, only one or two galaxy contribute to each LF as evident by the large errorbars seen in Fig. 9. The resulting values for $\chi^{2}_{red}$ are printed in the appropriate panels in Fig. 9. From these values, it can be seen that the shapes of the HSB and blue group LFs agree with those of the their comparison LFs within roughly 1$\sigma$. The shapes of the LSB and red group LFs do not match those of their comparison LFs as well with $\chi^{2}_{red}$ being about 2.5 and 1.7 for the LSB and red groups respectively.
\placefigure{fig8}
\placefigure{fig9}
\subsection{Surface Brightness and Blending}
Using narrow-band images obtained with HST, \citet{sco01} found that in ground-based images of galaxies as nearby as M51 ($d \approx$10 Mpc), a significant fraction of H{\sc ii} regions will be blended, causing the measured H{\sc ii} region LF to be artificially flattened. They argue that blending will have a larger effect on LFs measured for locations where the number density of H{\sc ii} regions is higher. For example, they assert that this could explain why the LFs for interarm regions appear steeper than those for regions found in in spiral arms. While Fig. 8 indicates that the effects of blending on the measured LFs do not change with distance within the relatively small volume covered by our sample, the effects of blending may be worse in higher surface brightness galaxies where the density of H{\sc ii} regions is likely to be higher, just as the density of H{\sc ii} regions is higher in spiral arms. To explore this, we computed the projected distance to the nearest neighboring region for each H{\sc ii} region with {\sc snr}$>$5 within each galaxy which we will refer to as $\Delta_{min}$. For the LSB, comparison, and HSB co-added groups, the median values for $\Delta_{min}$ are 420, 380, and 300 pc respectively, confirming that on average, the density of H{\sc ii} regions is higher in higher surface brightness galaxies. Since we are limited by the resolution of our ground-based images and the lack of higher resolution images for the galaxies in our sample, we cannot explicitly determine the effect blending may have on the shape of the LF for each co-added group. However, the shapes of the LFs for the HSB and comparison groups agree quite well even though the typical densities of H{\sc ii} regions for the two groups are significantly different. It is therefore unlikely that the relatively lower densities of regions within the galaxies of the LSB group contribute significantly to the comparatively steeper shape of the LSB group LF.
\subsection{Dust Extinction}
The discrepancy in LF shapes may be influenced by the fact that no correction was made for internal extinction, especially in the case of the red group LF. To explore this, we applied a correction for internal extinction at H$\alpha$, $A(H\alpha)_{int}$, derived from a linear least squares fit to the data of \citet{jan00} given by
\begin{equation}
\mbox{log } A(H\alpha)_{int} = (-0.063\pm0.021)M_{R} + (1.33\pm0.16)(B-R)_{e} + (-3.08\pm0.40)
\end{equation}
where $A(H\alpha)_{int}$ was derived using the measured Balmer decrements taken from the \citet{jan00} data, an intrinsic ratio of H$\alpha$ to H$\beta$ flux of 2.85 \citep{ost89}, the extinction curve of \citet{odo94}, and $R_{V}$=3.1. As with the $[$NII$]$ correction given by equation (1), fitting a plane to the $A(H\alpha)_{int}$ data as a function of both $M_{R}$ and $(B-R)_{e}$ yielded a relation with $\sim$0.1 dex less scatter than the relation between $A(H\alpha)_{int}$ and $M_{R}$ used in paper I. Such a correction may be inappropriate for studying the properties of the brightest H{\sc ii} regions as the extinction can vary significantly from one location to the next within a galaxy \citep{ken88}. However, for the purpose of constructing co-added LFs, such a statistical correction provides a reasonable estimate of the effect that attenuation by dust has on the measured LFs.\par
HII{\it phot} was rerun on the images after the extinction corrections were applied. We chose to rerun HII{\sc phot} rather than simply apply the extinction corrections to the measured H{\sc ii} region luminosities because the region boundaries are defined by the slope of their surface brightness profiles (see \S 2.2) which will be altered by the multiplicative extinction corrections. However, we note that this is only relevant for relatively isolated regions. The value for the limiting H{\sc ii} region luminosity was redetermined for each galaxy using the new HII{\it phot} output, and the co-added LFs were reconstructed following the procedure described above. These LFs are plotted in the two right panels in Fig. 9. For the new co-added LFs, the values for $\chi^{2}_{red}$ were recomputed as described above and are printed in the appropriate panels in Fig. 9. From these plots and the $\chi^{2}_{red}$ values, it can be seen that using the extinction corrections causes the number of H{\sc ii} regions per galaxy to increase for both the HSB and red groups and that the shapes of the LFs for all of the color groups agree within less than 1$\sigma$. For both the LSB and HSB group LFs, the values for $\chi^{2}_{red}$ are slightly smaller than they were for the LFs that did not use extinction corrections. However, the LSB group LF has a relatively irregular shape and its value for $\chi^{2}_{red}$ is still relatively large at about 2.3.
\subsection{Small Number Statistics}
Since the number of regions with {\sc snr}$>$5 is less than 60 for all but one of the ten galaxies in the LSB group and is as low as four, the fact that the co-added LF for that group differs in shape from the LFs for the other groups may be the result of small number statistics. To explore this possibility, we reconstruct the LF for each of the galaxies in the LSB group in the following way. We first assume that the LF for the comparison surface brightness group is the ``true'' LF over the interval 37.7$\leq$log L$_{H\alpha} \leq$40.5 (see \S 3.2.2). For galaxies where the limiting H{\sc ii} region luminosity is more than 10$^{37.7}$ ergs s$^{-1}$, we consider only the interval log $L_{lim} \leq \mbox{log } L_{H\alpha} \leq$40.5. For each galaxy, we counted the number of H{\sc ii} regions found within this interval, $N_{R}$, and generated a set of $N_{R}$ random numbers which we used to sample the assumed true LF. Following this, we co-added the new LFs in the same manner as described above and computed a value of $\chi^{2}_{red}$ between each fake co-added LF and the measured LSB group LF. We repeated this procedure 1000 times to explore the range in shapes that could be produced by randomly sampling the assumed true LF with small numbers of regions. As a control, we also perform the same procedure for the galaxies in the HSB group which have a median number of regions with {\sc snr}$>$5 of $\sim$150.\par
The first five fake co-added LFs created with this procedure using the data without internal extinction corrections are plotted in the upper panels of Fig. 10 with the actual LFs plotted in the upper left panel. From these plots, it appears that the power law shape of the LSB group LF is most likely not the result of small number statistics while the shape of the HSB group LF is recovered in each instance. We repeated the procedure using data with the internal extinction correction given in equation (4) applied to it; the first five fake co-added LFs are plotted in the lower panels of Fig. 10. In this case, the LFs generated by randomly sampling the assumed true LF do appear to resemble the somewhat irregular shape observed for the LSB group.\par
To examine the results in a more quantitative manner, we plot the distribution of $\chi^{2}_{red}$ values in Fig. 11 for the LSB and HSB groups. The results for the fake co-added LFs generated using data with no extinction corrections are plotted in the upper panel; the results for the fake co-added LFs generated using data with extinction corrections are plotted in the lower panel. The modes for the $\chi^{2}_{red}$ distributions for the HSB group closely match the values printed in the upper panels of Fig. 9 which is what one would expect if the LFs for the galaxies in the HSB group are approximately as well sampled as the LFs for the galaxies in the comparison group. The mode for the $\chi^{2}_{red}$ distribution for the LSB group without internal extinction corrections is about 1.9. This is significantly lower that the value printed in Fig. 9 but is still not as low as that computed for the HSB group. The mode for the $\chi^{2}_{red}$ distribution for the LSB group with extinction corrections is about 1.1 and the shape of the distribution is similar to that for the HSB group. However, the mode is still larger than that found for the HSB group.\par
We also note that while the sample of \citet{jan00} that was used to derive the extinction corrections does span the same range in R-band luminosity as our sample, it contains a substantially lower fraction of LSB galaxies. Since LSB galaxies have been found to have on average relatively low metallicities \citep[about 1/3 solar;][]{mcg94} and typically contain little if any molecular gas \citep{one03}, it is likely that equation (4) overestimates the amout of internal extinction for the LSB galaxies in our sample. Taking this into account, the results displayed in Fig. 11 imply that while the effects of dust extinction and small number statistics contribute significantly to the shapes of the co-added LFs, they cannot fully explain the discrepancy between the shapes of the LSB and comparison group co-added LFs displayed in Fig. 9. These results also imply that while small number statistics may strongly influence the trend found between $\mu_{e,R}$ and L$_{3}$ discussed in \S 2.5, there is also a real correlation between surface brightness and the luminosity of the brightest H{\sc ii} regions.
\placefigure{fig10}
\placefigure{fig11}
\section{Discussion and Conclusions}
The trends displayed in Fig. 6 as well as the co-added LFs plotted in Fig. 9 suggest that the conditions under which star formation takes place in lower surface brightness galaxies may be different than in higher surface brightness systems. This possibility is of particular interest since optically selected catalogs tend to be biased against these galaxies. This observational bias is particularly relevant in regards to the shape of the H{\sc ii} region LF since eight out of the ten galaxies contained within the LSB co-added group have values for $\mu_{e,B}$ that are more than 2$\sigma$ fainter than the mean for the NFGS. Since the extinction corrections derived from the \citet{jan00} data most likely overestimate the amount of dust in our LSB galaxies, the results in Fig. 11 imply that the difference in LF shapes can only be partially explained with a combination of the effects of dust extinction and small number statistics. The effects of blending which may be stronger for higher surface galaxies do not appear to significantly contribute to the difference between the shapes of the comparison and LSB group LFs. Therefore, this difference most likely reflects a real discrepancy between the distribution of H{\sc ii} region luminosities in LSB galaxies and that for relatively higher surface brightness spiral galaxies.\par
In terms of gas density and surface brightness, the disks of LSB galaxies are more similar to the extreme outer disks of more typical spiral galaxies than they are to the entire disk components of those galaxies. It is therefore reasonable to compare the properties of the H{\sc ii} regions that have been found in the outer portions of spiral disks \citep[e.g][]{fer98, lel00, thi05} to those that we have observed for the LSB galaxies in our sample. The luminosities found for the star forming regions in the outer disks of NGC 628 \citep{lel00} and M83 \citep{thi05} imply that the H$\alpha$ luminosities for the vast majority of such regions are less than $\sim10^{38}$ ergs s$^{-1}$. The LF for the LSB group indicates that LSB galaxies are capable of forming a significant number of regions with L$_{H\alpha}>10^{38}$ ergs s$^{-1}$ and in some cases, a few regions with L$_{H\alpha}>10^{40}$ ergs s$^{-1}$. This implies that while H{\sc ii} regions in LSB galaxies are less luminous than those in higher surface brightness spiral galaxies, LSB galaxies may be capable of forming more luminous regions than those found in the outer portions of nearby disk galaxies. However, we note that the number of galaxies for which outer disk H{\sc ii} regions have been study in detail is small and that future observations may reveal that it is possible for larger H{\sc ii} regions to form in these locations as well.\par
The difference in shape between the LSB group and comparison LFs displayed in the upper left panel of Fig. 9 resembles the observed difference between arm and interarm H{\sc ii} region LFs \citep[e.g.][]{ken89, ban93, oey98, thi00} with the LSB group LF more closely resembling the typical interarm H{\sc ii} region LF. The fact that interarm H{\sc ii} regions are on average less luminous that those contained in spiral arms is consistent with the idea that the star clusters that are ionizing the regions within spiral arms are younger that those that are ionizing interarm regions \citep{oey98}. \citet{sco01} have argued that the difference in LF shapes may result from the effects of blending being less severe for interarm regions where the typical spacing between H{\sc ii} regions is relatively larger. However, as discussed in \S 3.2, the difference in shape between the LF for the LSB group, where the spacing between H{\sc ii} regions is typically larger, and that for the comparison group LF is most likely not the result of the comparison group LF being flattened more by blending than the LSB group LF.\par
The fact that the LSB group LF is similar to interarm region LFs may then imply that the star formation in LSB galaxies is somewhat episodic so that on average, the relatively new star clusters found in these galaxies will be slightly older than those found in typical spirals that are continually forming stars. Indeed, the relatively blue B-H colors of many gas rich, LSB galaxies \citep[see ][]{bot84, deb95} cannot be reconciled with an exponentially declining or constant star formation history (over the last few billion years). Such star formation histories would cause the giant branch to be more populated yielding redder colors. To simultaneously explain the blue broad band colors and low abundances in systems which do form massive stars as evidenced by the LSB group co-added H{\sc ii} region LF, it seems likely that episodic star formation events define the recent star formation history of the typical LSB disk.\par
The difference in LF shape along with the trend between surface brightness and L$_{3}$ seen in Fig. 6 may also result from LSB galaxies forming a higher fraction of stars in lower mass star clusters or outside of clusters altogether. Indeed, an interesting observation yet to be performed is to examine whether the frequency of stellar clusters in LSB disks is similar to that seen in normal disk galaxies. The toy model of \citet{one98} argues that stellar cluster formation in LSB disk galaxies is suppressed (because of the very large Jeans length in these low density disks) and that the formation of individual massive stars occurs stochastically within the disk and not exclusively within stellar clusters. While there is yet no direct observational evidence for a paucity of stellar clusters in LSB disks, our data is at least indirectly consistent with this view. The trend between surface brightness and diffuse fraction may support this notion as \citet{hoo00} estimates that main sequence O and B stars that reside in the field rather than in H{\sc ii} regions account for $\sim$40\% of the ionization of the DIG in typical spiral galaxies. If a relatively larger fraction of the O and B stars in LSB galaxies are not formed in dense, massive molecular clouds that would produce H{\sc ii} regions, they would be expected to have larger diffuse fractions as observed. In any case, the results imply that the manner in which star formation proceeds in LSB galaxies is considerably different than that for more HSB spiral galaxies.\par
The authors would like to thank the NOAO TAC for allocation of observing time and the CTIO staff for expert assistance at the telescope. We would also like to thank D. Thilker for assisting with the usage of HII{\it phot} and the referee for useful comments and suggestions.
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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import pdb
import json as mod_json
import logging as mod_logging
import base64 as mod_base64
import http as mod_http
import models as mod_models
import object as mod_object
import utils as mod_utils
DEFAULT_BASE_URL = 'https://oneapi.infobip.com'
class AbstractOneApiClient:
VERSION = '0.02'
"""
Note that this is *not* a http session. This class is just a utility class
holding authorization data and a few utility methods for http requests.
"""
def __init__(self, username, password, base_url=None):
self.base_url = base_url if base_url else DEFAULT_BASE_URL
self.username = username
self.password = password
self.oneapi_authentication = None
if not self.base_url.endswith('/'):
self.base_url += '/'
# If true -- an exception will be thrown on error, otherwise, you have
# to check the is_success and exception methods on resulting objects.
self.raise_exception = True
def login(self):
params = {
'username': self.username,
'password': self.password,
}
is_success, result = self.execute_POST('/1/customerProfile/login', params)
return self.fill_oneapi_authentication(result, is_success)
def fill_oneapi_authentication(self, content, is_success):
self.oneapi_authentication = self.create_from_json(mod_models.OneApiAuthentication,
content, not is_success)
self.oneapi_authentication.username = self.username
self.oneapi_authentication.password = self.password
self.oneapi_authentication.authenticated = len(self.oneapi_authentication.ibsso_token) > 0
return self.oneapi_authentication
def get_client_correlator(self, client_correlator=None):
if client_correlator:
return client_correlator;
return mod_utils.get_random_alphanumeric_string()
def get_rest_url(self, rest_path):
if not rest_path:
return self.base_url
if rest_path.startswith('/'):
return self.base_url + rest_path[1:]
return self.base_url + rest_path
def is_valid(self):
""" Check if the authorization (username/password) is valid. """
is_success, result = self.execute_GET('/1/customerProfile')
return is_success
def get_headers(self, headers=None):
assert headers is None or isinstance(headers, dict)
result = headers
if result is None:
result = {}
result["User-Agent"] = "OneApi-python-{0}".format(self.VERSION)
if self.oneapi_authentication and self.oneapi_authentication.ibsso_token:
result['Authorization'] = 'IBSSO {0}'.format(self.oneapi_authentication.ibsso_token)
else:
auth_string = '%s:%s' % (self.username, self.password)
auth_string = mod_base64.encodestring(auth_string)
result['Authorization'] = 'Basic {0}'.format(auth_string).strip()
return result
def execute_GET(self, rest_path, params=None, leave_undecoded=False, headers=None):
response = mod_http.execute_GET(self.get_rest_url(rest_path), data=params,
headers=self.get_headers(headers))
mod_logging.debug('status code:{0}'.format(response.status_code))
mod_logging.debug('content:{0}'.format(response.content))
is_success = 200 <= response.status_code <= 299
if leave_undecoded or not is_success:
return is_success, response.content
return is_success, mod_json.loads(response.content)
def execute_POST(self, rest_path, params=None, leave_undecoded=False, headers=None, data_format=None):
response = mod_http.execute_POST(self.get_rest_url(rest_path), data=params,
headers=self.get_headers(headers), data_format=data_format)
mod_logging.debug('status code:{0}'.format(response.status_code))
mod_logging.debug('params: {0}'.format(params))
mod_logging.debug('content:{0}'.format(response.content))
is_success = 200 <= response.status_code <= 299
if leave_undecoded or not is_success:
return is_success, response.content
return is_success, mod_json.loads(response.content)
def execute_PUT(self, rest_path, params=None, leave_undecoded=False, headers=None):
response = mod_http.execute_PUT(self.get_rest_url(rest_path), data=params,
headers=self.get_headers(headers))
mod_logging.debug('status code:{0}'.format(response.status_code))
mod_logging.debug('params: {0}'.format(params))
mod_logging.debug('content:{0}'.format(response.content))
is_success = 200 <= response.status_code <= 299
if leave_undecoded or not is_success:
return is_success, response.content
return is_success, mod_json.loads(response.content)
def execute_DELETE(self, rest_path, params=None, leave_undecoded=False, headers=None, use_absolute_path=None):
if use_absolute_path:
response = mod_http.execute_DELETE(rest_path, data=params,
headers=self.get_headers(headers))
else:
response = mod_http.execute_DELETE(self.get_rest_url(rest_path), data=params,
headers=self.get_headers(headers))
mod_logging.debug('status code:{0}'.format(response.status_code))
mod_logging.debug('content:{0}'.format(response.content))
is_success = 200 <= response.status_code <= 299
if leave_undecoded or not is_success or response.status_code == 204:
return is_success, response.content
return is_success, mod_json.loads(response.content)
def create_from_json(self, classs, json, is_error):
""" Converti API result from json to model. """
result = mod_object.Conversions.from_json(classs, json, is_error);
if self.raise_exception and not result.is_success():
message = "{0}: {1} [{2}]".format(result.exception.message_id, result.exception.text, result.exception.variables)
raise Exception(message)
return result
def create_to_json(self, json):
result = mod_object.Conversions.to_json(json);
return result
class OneApiClient(AbstractOneApiClient):
""" Generic OneApi client. May be used for direct rest requests. """
def __init__(self, username, password, base_url=None):
AbstractOneApiClient.__init__(self, username, password, base_url=base_url)
class SmsClient(AbstractOneApiClient):
def __init__(self, username, password, base_url=None):
AbstractOneApiClient.__init__(self, username, password, base_url=base_url)
def send_sms(self, sms, header=None, data_format=None, is_legacy=None):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
client_correlator = sms.client_correlator
if not client_correlator:
client_correlator = mod_utils.get_random_alphanumeric_string()
if data_format == "json":
params = {
'address' : [
'tel:{0}'.format(sms.address)
],
'clientCorrelator': client_correlator,
'senderAddress': sms.sender_address,
'outboundSMSTextMessage' : {
'message' : sms.message
},
'senderName': 'tel:{0}'.format(sms.sender_address),
'receiptRequest' : {
'callbackData': sms.callback_data,
'notifyURL': sms.notify_url
}
}
elif data_format == "url":
params = {
'senderAddress': sms.sender_address,
'address': sms.address,
'message': sms.message,
'clientCorrelator': client_correlator,
'senderName': 'tel:{0}'.format(sms.sender_address),
}
if sms.mo_response_key:
params['moResponseKey'] = sms.mo_response_key
if sms.notify_url:
params['notifyURL'] = sms.notify_url
if sms.callback_data:
params['callbackData'] = sms.callback_data
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/requests'.format(sms.sender_address)
else:
tmp='/smsmessaging/v1/outbound/{0}/requests'.format(sms.sender_address)
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.ResourceReference, result, not is_success)
def send_flash_sms(self, sms, header=None, data_format=None, is_legacy=None):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
client_correlator = sms.client_correlator
if not client_correlator:
client_correlator = mod_utils.get_random_alphanumeric_string()
if data_format == "json":
params = {
'address' : [
'tel:{0}'.format(sms.address)
],
'clientCorrelator': client_correlator,
'senderAddress': sms.sender_address,
'outboundSMSFlashMessage' : {
'flashMessage' : sms.message
},
'senderName': 'tel:{0}'.format(sms.sender_address),
'receiptRequest' : {
'callbackData': sms.callback_data,
'notifyURL': sms.notify_url
}
}
elif data_format == "url":
params = {
'senderAddress': sms.sender_address,
'address': sms.address,
'flashMessage': sms.message,
'clientCorrelator': client_correlator,
'senderName': 'tel:{0}'.format(sms.sender_address),
}
if sms.mo_response_key:
params['moResponseKey'] = sms.mo_response_key
if sms.notify_url:
params['notifyURL'] = sms.notify_url
if sms.callback_data:
params['callbackData'] = sms.callback_data
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/requests'.format(sms.sender_address)
else:
tmp='/smsmessaging/v1/outbound/{0}/requests'.format(sms.sender_address)
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.ResourceReference, result, not is_success)
def send_ringtone_sms(self, sms, header=None, data_format=None, is_legacy=None, sms_format='Ems'):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
client_correlator = sms.client_correlator
if not client_correlator:
client_correlator = mod_utils.get_random_alphanumeric_string()
if data_format == "json":
params = {
'address' : [
'tel:{0}'.format(sms.address)
],
'clientCorrelator': client_correlator,
'senderAddress': sms.sender_address,
'outboundSMSRingToneMessage' : {
'ringTone' : sms.message,
'smsFormat': sms_format,
},
'senderName': 'tel:{0}'.format(sms.sender_address),
'receiptRequest' : {
'callbackData': sms.callback_data,
'notifyURL': sms.notify_url
}
}
elif data_format == "url":
params = {
'senderAddress': sms.sender_address,
'address': sms.address,
'ringTone': sms.message,
'smsFormat': sms_format,
'clientCorrelator': client_correlator,
'senderName': 'tel:{0}'.format(sms.sender_address),
}
if sms.mo_response_key:
params['moResponseKey'] = sms.mo_response_key
if sms.notify_url:
params['notifyURL'] = sms.notify_url
if sms.callback_data:
params['callbackData'] = sms.callback_data
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/requests'.format(sms.sender_address)
else:
tmp='/smsmessaging/v1/outbound/{0}/requests'.format(sms.sender_address)
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.ResourceReference, result, not is_success)
def send_logo_sms(self, sms, header=None, data_format=None, is_legacy=None, sms_format='Ems'):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
client_correlator = sms.client_correlator
if not client_correlator:
client_correlator = mod_utils.get_random_alphanumeric_string()
if data_format == "json":
params = {
'address' : [
'tel:{0}'.format(sms.address)
],
'clientCorrelator': client_correlator,
'senderAddress': sms.sender_address,
'outboundSMSLogoMessage' : {
'image' : sms.message,
'smsFormat': sms_format,
},
'senderName': 'tel:{0}'.format(sms.sender_address),
'receiptRequest' : {
'callbackData': sms.callback_data,
'notifyURL': sms.notify_url
}
}
elif data_format == "url":
params = {
'senderAddress': sms.sender_address,
'address': sms.address,
'image': sms.message,
'smsFormat': sms_format,
'clientCorrelator': client_correlator,
'senderName': 'tel:{0}'.format(sms.sender_address),
}
if sms.mo_response_key:
params['moResponseKey'] = sms.mo_response_key
if sms.notify_url:
params['notifyURL'] = sms.notify_url
if sms.callback_data:
params['callbackData'] = sms.callback_data
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/requests'.format(sms.sender_address)
else:
tmp='/smsmessaging/v1/outbound/{0}/requests'.format(sms.sender_address)
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.ResourceReference, result, not is_success)
def query_delivery_status(self, client_correlator_or_resource_reference, sender, is_legacy=None):
if hasattr(client_correlator_or_resource_reference, 'client_correlator'):
client_correlator = client_correlator_or_resource_reference.client_correlator
else:
client_correlator = client_correlator_or_resource_reference
if is_legacy == None: is_legacy=True
client_correlator = self.get_client_correlator(client_correlator)
params = {
'clientCorrelator': client_correlator,
}
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/requests/{1}/deliveryInfos'.format(sender, client_correlator)
else:
tmp='/smsmessaging/v1/outbound/{0}/requests/{1}/deliveryInfos'.format(sender, client_correlator)
is_success, result = self.execute_GET(
tmp,
params = params
)
if not is_success:
return is_success
# TODO: Simplify the resulting object
return self.create_from_json(mod_models.DeliveryInfoList, result, not is_success)
def retrieve_inbound_messages(self, max_number=None, is_legacy=None):
if not max_number or max_number < 0:
max_number = 100
if is_legacy == None: is_legacy=True
params = {
'maxBatchSize': max_number,
}
if is_legacy == True:
tmp='/1/smsmessaging/inbound/registrations/INBOUND/messages'
else:
tmp='/smsmessaging/v1/inbound/registrations/INBOUND/messages'
is_success, result = self.execute_GET(
tmp,
params
)
if not is_success:
return is_success
return self.create_from_json(mod_models.InboundSmsMessages, result, not is_success)
def subscribe_delivery_status(self, sms, header=None, data_format=None, is_legacy=None):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
if data_format == "json":
params = {
'deliveryReceiptSubscription': {
'callbackReference' : {
'callbackData' : sms.callback_data,
'notifyURL' : sms.notify_url
},
'filterCriteria' : sms.filter_criteria
}
}
elif data_format == "url":
params = {
'callbackData' : sms.callback_data,
'notifyURL' : sms.notify_url,
'filterCriteria' : sms.filter_criteria
}
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/outbound/{0}/subscriptions'.format(sms.sender_address)
else:
tmp='/smsmessaging/v1/outbound/{0}/subscriptions'.format(sms.sender_address),
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.DeliveryReceiptSubscription, result, not is_success)
# TODO (pd) only subscriptionID should be passed into this method
def delete_delivery_status_subscription(self, resource_url):
is_success = self.execute_DELETE(
resource_url, use_absolute_path=True
)
return is_success
def subscribe_messages_sent_notification(self, sms, header=None, data_format=None, is_legacy=None):
if not data_format: data_format='json'
if is_legacy == None: is_legacy=True
if data_format == "json":
params = {
'subscription' : {
'callbackReference' : {
'callbackData' : sms.callback_data,
'notifyURL' : sms.notify_url
},
'criteria' : sms.filter_criteria,
'destinationAddress' : sms.address,
'clientCorrelator' : sms.client_correlator
}
}
elif data_format == "url":
params = {
'callbackData' : sms.callback_data,
'notifyURL' : sms.notify_url,
'destinationAddress' : sms.address
}
if sms.filter_criteria:
params['criteria'] = sms.filter_criteria
if sms.client_correlator:
params['client_correlator'] = sms.client_correlator
#resourceURL
else:
raise Exception("invalid asked data format (supported url or json")
if is_legacy == True:
tmp='/1/smsmessaging/inbound/subscriptions'
else:
tmp='/smsmessaging/v1/inbound/subscriptions'
is_success, result = self.execute_POST(
tmp,
params = params,
headers = header,
data_format = data_format
)
if not is_success:
return is_success
return self.create_from_json(mod_models.InboundSMSMessageReceiptSubscription, result, not is_success)
# TODO (pd) only subscriptionID should be passed into this method
def delete_messages_sent_subscription(self, resource_url):
is_success = self.execute_DELETE(
resource_url, use_absolute_path=True
)
return is_success
# ----------------------------------------------------------------------------------------------------
# Static methods used for http push events from the server:
# ----------------------------------------------------------------------------------------------------
@staticmethod
def unserialize_inbound_messages(json):
return mod_object.Conversions.from_json(mod_models.InboundSmsMessages, json, False)
@staticmethod
def unserialize_delivery_status(json):
return mod_object.Conversions.from_json(mod_models.DeliveryInfoNotification, json, False)
class UssdClient(AbstractOneApiClient):
"""
Warning, this is an experimental feature. The API may change!
"""
def __init__(self, username, password, base_url=None):
AbstractOneApiClient.__init__(self, username, password, base_url=base_url)
def send_message(self, address, message):
params = {
'address': address,
'message': message,
}
is_success, json = self.execute_POST(
'/1/ussd/outbound',
params = params
)
return self.create_from_json(mod_models.InboundSmsMessage, json, not is_success)
def close_session(self, address, message):
params = {
'address': address,
'message': message,
'stopSession': 'true',
}
is_success, json = self.execute_POST(
'/1/ussd/outbound',
params = params,
leave_undecoded = True
)
return True
class DataConnectionProfileClient(AbstractOneApiClient):
def __init__(self, username, password, base_url=None):
AbstractOneApiClient.__init__(self, username, password, base_url=base_url)
def retrieve_roaming_status(self, destination_address, notify_url=None):
"""
Retrieve asynchronously the customer's roaming status for a single network-connected mobile device (HLR)
"""
params = {
'address': destination_address,
}
if notify_url:
params['notifyURL'] = notify_url
# TODO(TK) Add these includeExtendedData, clientCorrelator, callbackData
is_success, result = self.execute_GET('/1/terminalstatus/queries/roamingStatus', params,
leave_undecoded=True)
if notify_url:
return self.create_from_json(mod_models.GenericObject, {}, not is_success);
else:
assert result
json = mod_json.loads(result)
assert json.has_key('roaming')
return self.create_from_json(mod_models.TerminalRoamingStatus, json['roaming'], not is_success);
# ----------------------------------------------------------------------------------------------------
# Static methods used for http push events from the server:
# ----------------------------------------------------------------------------------------------------
@staticmethod
def unserialize_roaming_status(json):
return mod_object.Conversions.from_json(mod_models.TerminalRoamingStatusNotification, json, False)
class CustomerProfileClient(AbstractOneApiClient):
def __init__(self, username, password, base_url=None):
AbstractOneApiClient.__init__(self, username, password, base_url=base_url)
def get_account_balance(self):
is_success, result = self.execute_GET('/1/customerProfile/balance')
return self.create_from_json(mod_models.AccountBalance, result, not is_success)
def get_customer_profile(self):
is_success, result = self.execute_GET('/1/customerProfile')
return self.create_from_json(mod_models.CustomerProfile, result, not is_success)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,902
|
Fall of TPP Under Trump Signals New Era for American Trade
Trump's move received support from some opponents, including Bernie Sanders.
ByMICHAEL EDISON HAYDEN
Trump Signs Order to Move Forward Keystone, Dakota Access Pipelines
President Trump used his fifth day in office to move forward the Keystone and Dakota Access pipelines, as well as to streamline regulations for infrastructure and manufacturing.
— -- President Donald Trump formally withdrew the United States from the Trans-Pacific Partnership on Monday, and critics of the doomed deal say the move will protect American jobs.
The withdrawal, which threatens to distance the U.S. from some of its Asian allies, fulfilled a campaign pledge by Trump to end U.S. involvement in the 2015 pact and likely marks the beginning of a new chapter in Washington's approach to global trade — one more focused on shielding domestic industries from foreign competition.
Unlikely Allies
Trump, who hopes to boost U.S. manufacturing jobs during his term, signed the executive memorandum authorizing the withdrawal in the Oval Office. Later, as he met with union leaders in the White House's Roosevelt Room, he said, "We're going to stop the ridiculous trade deals that have taken everybody out of our country and taken companies out of our country."
The move received support from some of his opponents, including Rust Belt Democrats.
Sen. Bernie Sanders, I-Vt., who denounced the TPP in last year's Democratic presidential primaries, arguably pushing Hillary Clinton to retract her support for the pact, praised Trump yesterday and said he was glad the deal was "dead and gone."
Democratic presidential candidate Senator Bernie Sanders (I-VT) speaks during an event on the Trans Pacific Partnership on Capitol Hill June 3, 2015 in Washington, DC.
End of a Long Fight
Although the agreement was already considered dead because of its lack of support from either major party's presidential candidate, Trump's signing of the memo nevertheless ended what Thea Lee, the policy director and chief international economist at the AFL-CIO, described to ABC News as a "six year fight" to stop the pact.
Union leaders like AFL-CIO President Richard Trumka and Teamsters General President James P. Hoffa, praised the move in separate statements, suggesting it would protect workers.
Union households — long considered a key component of the Democratic base — helped Trump win the 2016 presidential election. He matched Ronald Reagan's support in that cohort in 1984 — the highest level in that group for a Republican since then, according to ABC News exit polls.
Lee, however, was reluctant to praise Trump so easily for signing the directive.
"The TPP was dead already," she said. "Trump just disposed of the body."
Lee said her organization saw the accord, which President Obama made the centerpiece of his foreign policy pivot toward Asia, as "deeply flawed." She called it a "corporate empowerment agreement," part of a trend of deals that made it easier for companies to ship jobs overseas.
Despite her criticisms of the deal, she said she is reserving judgment on whether Trump's promise to fight for American workers can be believed.
"It's hard to say right now what we can expect from his administration," she said, since some of his proposed appointees — who she said are from the "Goldman Sachs wing of the Republican Party" — appear more likely to support future similar trade deals than his rhetoric would suggest.
Lee also said that it's still unclear what Trump will seek to do with other trade deals, as with his promise to renegotiate the North American Free Trade Agreement.
"Will he label China a currency manipulator?" Lee asked, referring to one of his promises. "That still remains to be seen."
Questions About Cause of Job Losses
Mireya Solis, the Philip Knight chair in Japan studies and a senior fellow at the Brookings Institution, argued in favor of the TPP in the past, praising the agreement for its potential to "enhance American influence in Asia."
She told ABC News that the deal would have had a small but positive effect on American jobs and cited a study by the United States International Trade Commission to support that claim.
Solis said opponents of the TPP "are misjudging the problem," referring to the much discussed decline in American manufacturing jobs. "We're very bad at training American workers to reinvent themselves."
She said that advances in technology were hurting workers more than trade deals like the TPP and that it could have boosted exports.
Robert E. Scott, the senior economist and director of trade and manufacturing policy research at the Economic Policy Institute, who argued against the deal, said the notion that it would have strengthened exports is "simply not coherent," based on the degree to which it would have empowered corporations.
Scott, who voiced skepticism about how effective the TPP would have been in strengthening U.S. influence in Asia, said that the framework of the deal was designed to help "really big companies" and implied that what would have happened to workers as a result of the deal was an afterthought.
"It would have redistributed wealth from the poor to the ultrarich," he said. "This is a win for workers, in my view."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,833
|
Q: how to display to a dbgrid my query in this code? hello is it possible for this code to display to a tdbgrid the search results in a list like style? (e.g. if i searched for john, all the data conataining john on a certain field will be displayed to the tdbgrid)
procedure Tspcb.dccolbtnClick(Sender: TObject);
begin
zdctable.First;
while not zdctable.EOF do
begin
if (zdctable.FieldByName('Collector').AsString = dcedit.Text)
then begin
cn.Caption := zdctable.FieldByName('Client_Name').AsString;
col.Caption := zdctable.FieldByName('Collector').AsString;
pay.Caption := zdctable.FieldByName('Daily_Payment').AsString;
date.Caption := zdctable.FieldByName('Date').AsString;
ddate.Caption := zdctable.FieldByName('Due_Date').AsString;
id.Caption := zdctable.FieldByName('ID').AsString;
la.Caption := zdctable.FieldByName('Loan').AsString;
tc.Caption := zdctable.FieldByName('Total_Collectibles').AsString;
end;
ShowMessage('click ok for next profile');
zdctable.Next;
end;
end;
A: Just add a datasource, set property dataset to your dataset zdctable, add a DBgrid to your form and set the property datasource to the datasource.
The only piece of code you will need is in the OnchangeEvent of dcedit
procedure TForm3.dceditChange(Sender: TObject);
begin
zdctable.FilterOptions:=[foCaseInsensitive]; // if wished
zdctable.Filtered := Length(dcEdit.Text) > 0;
if zdctable.Filtered then
// zdctable.Filter := 'Collector like ' + QuotedStr('%' + dcEdit.Text + '%')
zdctable.Filter := 'Collector like ' + QuotedStr('*' + dcEdit.Text + '*') // Zeos- Syntax
else zdctable.Filter := '';
end;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,843
|
\section{Introduction}
\medskip
Given a topological group ${\cal G}$, the dual space
${{\luc}(\G)^*}$ of the space of bounded complex-valued left
uniformly continuous functions has a natural
structure of Banach algebra which has been studied
since the 1970's. In particular, the problem of
describing its topological
centre ${\Lambda}({\luc}(\G)^*)$ has been studied by many authors.
The question was first considered by A. Zappa in
\cite{zappa} for abelian groups and for locally compact groups
was completely solved by A. T.-M. Lau in
\cite{lau}, where it was proved that for any such
group ${\cal G}$ the topological centre ${\Lambda}({\luc}(\G)^*)$ equals the measure
algebra ${\mathsf{M}}({\cal G})$.
The second author of this note in \cite{unified} considered the
topological centre problem, in the non-compact case, for $\mathrm{L}_1({\cal G})^{**}$ with the first
Arens product and for its quotient ${{\luc}(\G)^*}$ and,
using a factorization theorem similar to those appearing in
\cite[Satz 3.6.2]{neufangthesis}
and in \cite{equi}, proved that
${\Lambda}(\mathrm{L}_1({\cal G})^{**})=\mathrm{L}_1({\cal G})$ and that ${\Lambda}({\luc}(\G)^*)={\mathsf{M}}({\cal G})$.
A similar factorization will be also used in this work.
The same problem has been studied for general
(not necessarily locally compact) groups in \cite{ferrineufang}
and in \cite{ambitable} where it was proved
that for a large class of groups --- which includes all the
locally compact groups as well as many non-locally
compact ones --- the topological centre ${\Lambda}({\luc}(\G)^*)$ is the
space of uniform measures on ${\cal G}$.
The definition of ${\Lambda}({\luc}(\G)^*)$ apparently involves a
requirement on the continuity of certain maps at
every point of ${{\luc}(\G)^*}$.
However, it was proved
in \cite{budak} and in \cite{dales} that for locally
compact groups it is possible to determine whether
an element of ${{\luc}(\G)^*}$ belongs to ${\Lambda}({\luc}(\G)^*)$ by testing
the same type of continuity at just a few specific
points of ${{\luc}(\G)^*}$. This led to the definition of a DTC-set
(meaning a set Determining the Topological Centre),
i.e., a set with the property that
it is sufficient to test continuity only at the points of this set
in order to decide whether a given element belongs to ${\Lambda}({\luc}(\G)^*)$.
Several flavours of DTC-sets have been considered in the literature,
differing in the type of continuity assumed at the points of the set.
Our main result deals with the DTC-sets in the sense of Definition~\ref{def:DTC}.
We shall prove a one-point DTC-set
theorem for the topological groups that
satisfy a fairly weak cardinality condition
(property ($\dagger$) in Theorem~\ref{th:mainresult}).
This answers a question posed in \cite{dales1}.
From this theorem we shall obtain as
a corollary a result about the topological centre of
any subsemigroup of
${{\luc}(\G)^*}$ that contains
the uniform compactification ${\cal G}^{{\mathsf{LUC}}}$
of ${\cal G}$; in fact, any subsemigroup that contains
${\cal G}^{{\mathsf{LUC}}}\setminus {\cal G}$.
In particular, we shall
prove that there are one-point DTC-sets for the
topological centre of the uniform compactification
${\cal G}^{{\mathsf{LUC}}}$ of ${\cal G}$ itself, extending a similar
result given in \cite{budak}.
\par
We start by introducing the basic notation and
terminology used throughout the note.
\par
Let ${\cal G}$ be a topological group, here always
assumed to be Hausdorff, with identity $e$.
Given a function $f:{\cal G}\longrightarrow X$ (where $X$
can be any range) and $x\!\in\!{\cal G}$,
the left translate of $f$ by $x$ is the function ${\mathsf{L}}_xf$
defined by ${\mathsf{L}}_xf(y) = f(xy)$ for $y\!\in\!{\cal G}$.
We denote by $\mathsf{RP}({\cal G})$ the set of all
right-invariant continuous pseudometrics on ${\cal G}$.
From now on we denote by ${\cal G}$ not only the group with its topology but
also the uniform space on the set ${\cal G}$ induced by $\mathsf{RP}({\cal G})$;
since we do not consider here any other uniform
structures on ${\cal G}$, this notation will not lead to
any ambiguity.
Then ${\mathsf{LUC}}({\cal G})$ is the space of bounded
complex-valued uniformly continuous functions on ${\cal G}$
with the sup norm.
\par
Given $\nu\in{{\luc}(\G)^*}$ and $f\in{\mathsf{LUC}}({\cal G})$ the function
$\nu\bullet f$, defined by
$$(\nu\bullet f)(x):=\langle
\nu,{\mathsf{L}}_xf\rangle\qquad(x\in{\cal G}),$$
is in ${\mathsf{LUC}}({\cal G})$ (see for example~\cite{redbook}), i.e. ${\mathsf{LUC}}({\cal G})$
is left introverted.
This operation induces the convolution operation on
${{\luc}(\G)^*}$, defined by
$$\langle \mu\star \nu,f\rangle:=\langle
\mu,\nu\bullet f\rangle\qquad
(\mu,\nu\in{{\luc}(\G)^*},\;f\in{\mathsf{LUC}}({\cal G})),$$
which turns ${{\luc}(\G)^*}$ into a Banach algebra and
${\mathsf{LUC}}({\cal G})$ into a left ${{\luc}(\G)^*}$-module.
If we denote by $\delta(x)$ the point evaluation at $x$ ($x\in{\cal G}$)
and consider the $w^*$-closure of $\delta[{\cal G}]$ in ${{\luc}(\G)^*}$,
then it can be proved (see for example \cite{redbook}) that this set with
the induced product
is a semigroup compactification of ${\cal G}$ which
topologically coincides with the uniform
compactification of the uniform space ${\cal G}$.
This compactification, denoted here by ${\cal G}^{{\mathsf{LUC}}}$,
coincides with the spectrum of the C$^*$-algebra ${\mathsf{LUC}}({\cal G})$.
In the locally compact case this is the largest semigroup compactification of ${\cal G}$,
meaning that every other compactification is its quotient.
For a general topological group ${\cal G}$ the space ${\cal G}^{{\mathsf{LUC}}}$ is the greatest ambit ${\cal G}$,
i.e. the greatest ${\cal G}$-flow with a point whose orbit is dense.
In the sequel we identify ${\cal G}$ with its image
$\delta[{\cal G}]$ in ${\cal G}^{{\mathsf{LUC}}}$,
so that ${\cal G}\subseteq{\cal G}^{{\mathsf{LUC}}}\subseteq{{\luc}(\G)^*}$.
More on the subject can be found in
\cite{redbook} and in \cite{pestovbook}.
\begin{definition}
\label{def:DTC}
Let $S$ be a subsemigroup of ${{\luc}(\G)^*}$ with the convolution operation
and the $w^*$-topology.
For $D\subseteq S$ write
\[
\mathsf{Cont}(S,D):=\{\,\mu\in S:
\forall \,\nu_0\!\in\! D
\text{ the mapping } \nu\mapsto \mu\star\nu \text{ on } S
\text{ is continuous at } \nu_0 \}
\]
The \emph{topological centre of $S$} is
$
\Lambda(S):= \mathsf{Cont}(S,S)
$.
The set $D$ is said to be a \emph{DTC-set for $S$} iff
$\Lambda(S)=\mathsf{Cont}(S,D)$.
\end{definition}
In the literature $\Lambda({{\luc}(\G)^*})$ is often denoted by $Z_t({{\luc}(\G)^*})$.
Note that DTC-sets are interesting only when the group ${\cal G}$ is not precompact:
If ${\cal G}$ is precompact then $\Lambda(S)=S$
for every subsemigroup of ${{\luc}(\G)^*}$, and thus every subset of $S$
is a DTC-set.
In this paper we deal only with the DTC-sets of Definition~\ref{def:DTC}.
However, other variants of the DTC-set notion are also of interest.
In particular, for $D\subseteq S\cap{\cal G}^{{\mathsf{LUC}}}$, if we let
\begin{align*}
\mathsf{Cont}_{\cal G}(S,D):= \{&\mu\!\in\! S: \\
& \forall \,\nu_0\!\in\! D
\text{ the mapping } \nu\mapsto \mu\star\nu \text{ on } (S\cap{\cal G}) \cup\{\nu_0\}
\text{ is continuous at } \nu_0 \} ,
\end{align*}
then the condition $\Lambda(S)=\mathsf{Cont}_{\cal G}(S,D)$ is stronger (more restrictive)
than the condition $\Lambda(S)=\mathsf{Cont}(S,D)$
in Definition~\ref{def:DTC}.
As is explained in~\cite[sec.2]{budak} and~\cite[Ch.12]{dales},
if ${\cal G}$ is any non-compact locally compact abelian group then
${\Lambda}({\luc}(\G)^*)\neq\mathsf{Cont}_{\cal G}({{\luc}(\G)^*},\{\nu\})$ for every $\nu\!\in\!{\cal G}^{{\mathsf{LUC}}}$
but there exists $\nu_0\!\in\!{\cal G}^{{\mathsf{LUC}}}$ such that
${\Lambda}({\luc}(\G)^*)=\mathsf{Cont}({{\luc}(\G)^*},\{\nu_0\})$.
For any $\Delta\in\mathsf{RP}({\cal G})$ we define
$$\mathsf{BLip_b}^+(\Delta):=\{f:{\cal G}\longrightarrow
[0,1]:|f(x)-f(y)|\le\Delta(x,y)\text{ for all
}x,y\in{\cal G}\},$$
$$\mathsf{B}(\Delta):=\{x\in{\cal G}:\Delta(e,x)<1\}.$$
\par
It is known that for a large class of topological
groups ${\Lambda}({\luc}(\G)^*)$ coincides with the space
$\mathsf{M_u}({\cal G})$ of uniform measures on
the uniform space ${\cal G}$. One
of several equivalent definitions of
$\mathsf{M_u}({\cal G})$ is that a functional
$\mu\in{{\luc}(\G)^*}$ is in $\mathsf{M_u}({\cal G})$ if and
only if it is ${\cal G}$-pointwise
continuous on $\mathsf{BLip_b}^+(\Delta)$ for every
$\Delta\in\mathsf{RP}({\cal G})$. When ${\cal G}$ is locally compact,
$\mathsf{M_u}({\cal G})$ can be identified with the space
of finite Radon measures on ${\cal G}$ (\cite[sec.7.3]{uniformmeasures}).
More about uniform measures,
with references to original sources, may be found
in \cite{uniformmeasures}.
\par
The key step in proving our results will be
a factorization theorem which has its roots in
\cite[Satz 3.6.2]{neufangthesis} and that was later
generalized in a number of ways by many authors
considering problems related to topological centres
(see for example \cite{budak}, \cite{ferrineufang}, \cite{equi}
and \cite{ambitable}). The next
section is devoted just to proving the appropriate
version of the theorem which we shall need in order
to prove our main result.
\section{The factorization theorem}
We start the section by defining some cardinal
numbers associated to
topological groups which will be needed in order to
state our results.
\begin{definition}
{\rm We say that a topological group ${\cal G}$ is
\emph{$\kappa$-bounded\/}, where $\kappa$ is an
infinite cardinal, if and only if for every neighbourhood {$U$} of the identity in ${\cal G}$
there is a set $A\subseteq{\cal G}$ with $|A|\le\kappa$ such that ${\cal G}=UA$.
This is equivalent to saying that for every $\Delta\in\mathsf{RP}({\cal G})$ there is a set
$A\subseteq{\cal G}$ with $|A|\le\kappa$ such that ${\cal G}=\mathsf{B}(\Delta)A$.
We denote by $\mathfrak{B}_{{\cal G}}$ the least infinite cardinal for
which ${\cal G}$ is $\kappa$-bounded.
Given $\Delta\in\mathsf{RP}({\cal G})$ we denote by
$\eta^{\sharp}(\Delta)$ the least (finite or
infinite) cardinal $\kappa$ for which there exists
a subset $A\subseteq{\cal G}$ of cardinality $\kappa$ and
such that ${\cal G}=\mathsf{B}(\Delta)A$
and we denote by
$\eta(\Delta)$ the least cardinal number
$\kappa$ such that there exists a set
$A\subseteq{\cal G}$ with $|A|\le\kappa$ and a finite set
$K\subseteq{\cal G}$ such that ${\cal G}=KB(\Delta)A$.\/}
\end{definition}
We are ready to state and prove the factorization
theorem.
\begin{theorem}\label{th:factorization}
Let ${\cal G}$ be a topological group and let
$\Delta_1\in\mathsf{RP}({\cal G})$ with
$\eta(\Delta_1)={\mathfrak B}_{{\cal G}}$, then there
exists a family
$$\{\nu_{\psi}\in{\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}:\psi\in{\mathfrak
B}_{{\cal G}}\}$$
such that for every $\Delta\in\mathsf{RP}({\cal G})$ such that
$\Delta\ge\Delta_1$ and for every family
$$\{h_{\psi}\in\mathsf{BLip_b}^+(\Delta):\psi\in{\mathfrak
B}_{{\cal G}}\}$$
there is $h\in\mathsf{BLip_b}^+(2\Delta)$ for which
$$h_{\psi}=\nu_{\psi}\bullet h\qquad\text{for all
}\psi\in{\mathfrak B}_{{\cal G}}.$$
\end{theorem}
\begin{proof}
By \cite[Lemma 7]{ambitable}, ${\cal G}$ has a
$\Delta_1$-dense subset $D$ of cardinality
${\mathfrak B}_{{\cal G}}$. Denote by ${\cal P}_f(D)$ the
set of all finite subsets of $D$ and set
$A:={\mathfrak B}_{{\cal G}}\times{\cal P}_f(D)$.
\par
By \cite[Lemma 8]{ambitable} there are
$x_{(\psi,K)}\!\in\!{\cal G}$ for $(\psi,K)\!\in\! A$ such that
$\Delta_1(Kx_{(\psi,K)},Lx_{(\varphi,L}))>1$ whenever
$(\psi,K)\neq(\varphi,L)$ are elements of
$A$.
\par
For every $\psi\in{\mathfrak B}_{{\cal G}}$ let
$\nu_{\psi}$ be a cluster point of the net
$(x_{(\psi,K)})_{K\in{\cal P}_f(D)}$, where ${\cal
P}_f(D)$ is ordered by reversed inclusion.
\par
Define a real-valued function $u_K$ in the
variable $x\in{\cal G}$ for every $K\in{\cal P}_f(D)$ by:
$$u_K(x):=(1-2\Delta_1(x,K))^+.$$
Then we have that $u_K\in\mathsf{BLip_b}^+(2\Delta_1)$ and
that $\displaystyle{\lim_K u_K=1}$ pointwise.
\par
Now take any $\Delta\in\mathsf{RP}({\cal G})$ with
$\Delta\ge\Delta_1$ and a family of functions
$\{h_{\psi}\in\mathsf{BLip_b}^+(\Delta):\psi\in{\mathfrak
B}_{{\cal G}}\}$. For every $x\in{\cal G}$ there is at most
one $(\psi,K)\in A$ such that
$u_K(xx_{(\psi,K)}^{-1})\neq 0$. We can then define
$$h(x):=\sup\{h_{\psi}(xx_{(\psi,K)}^{-1})\wedge
u_K(xx_{(\psi,K)}^{-1}):(\psi,K)\in A\}$$
and we have that $h\in\mathsf{BLip_b}^+(2\Delta)$. Take
$x\in{\cal G}$ and $\psi\in{\mathfrak B}_{{\cal G}}$. By
density, there is $y\in D$ with
$\Delta_1(x,y)\le\frac{1}{2}$. For every
$K\in{\cal P}_f(D)$ with
$y\in K$ we have that $h_{\psi}(x)\wedge
u_K(x)=h(xx_{(\psi,K)})=x_{(\psi,K)}\bullet h(x)$,
hence
$$h_{\psi}(x)=\lim_Kh_{\psi}(x)\wedge
u_K(x)=\lim_Kx_{(\psi,K)}\bullet h(x).$$
By \cite[Lemma 19]{ambitable} the mapping
$\nu\mapsto\nu\bullet h$ is continuous from
${\cal G}^{{\mathsf{LUC}}}$ to $\mathsf{BLip_b}^+(\Delta)$ with the
${\cal G}$-pointwise topology, hence
$h_{\psi}(x)=\nu_{\psi}\bullet h(x)$.
Finally, $\nu_{\psi}\notin{\cal G}$ because
from $\nu_{\psi}\in{\cal G}$ we would get a contradiction by
choosing $h_{\psi}=0$ and $h_{\varphi}=1$ for
$\psi\neq\varphi$.
\end{proof}
\section{DTC-sets for ${\Lambda}({\luc}(\G)^*)$ and $\Lambda({\cal G}^{{\mathsf{LUC}}})$}
We are finally ready to give the main results of
this note. We begin with a technical lemma.
\begin{lemma}\label{lemma:technical}
Let ${\cal G}$ be a $\kappa$-bounded topological group
(where $\kappa$ is an infinite cardinal). The
following properties of a functional
$\mu\in{{\luc}(\G)^*}$ are equivalent:
\begin{itemize}
\item[{\rm (i)\/}] $\mu\in\mathsf{M_u}({\cal G})$.
\item[{\rm (ii)\/}] If $\Delta\in\mathsf{RP}({\cal G})$ and
$(h_{\psi})_{\psi\in\Psi(\kappa)}$ is a net
in $\mathsf{BLip_b}^+(\Delta)$ indexed by the
set $\Psi(\kappa):={\cal P}_f(\kappa)\times\omega$
(ordered by $(K,i)\le(L,j)$ if and only if
$K\subseteq L$ and $i\le j$) which converges
pointwise to $0$, then $0$ is a cluster point of
the net $(\mu(h_{\psi}))_{\psi}$.
\item[{\rm (iii)\/}] The restriction of $\mu$ to
$\mathsf{BLip_b}^+(\Delta)$ is ${\cal G}$-pointwise continuous at
$0\in\mathsf{BLip_b}^+(\Delta)$ for every
$\Delta\in\mathsf{RP}({\cal G})$.
\end{itemize}
\end{lemma}
\begin{proof}
Evidently, (i) implies (ii).
\par
To prove that (ii) implies (iii), take any $\mu$
that does not have
the property stated in (iii). There is
$\Delta\in\mathsf{RP}({\cal G})$ for which the restriction of
$\mu$ to $\mathsf{BLip_b}^+(\Delta)$ is not ${\cal G}$-pointwise
continuous at $0$. By \cite[Lemma 7]{ambitable},
${\cal G}$ has a $\Delta$-dense subset $D$ with
cardinality smaller or equal than $\kappa$. Fix a
surjection $\alpha:\kappa\longrightarrow D$. For
every $\psi=(K,i)\in\Psi(\kappa)$ let $U_{\psi}$
be the $D$-pointwise neighbourhood
$$\{f\in\mathsf{BLip_b}^+(\Delta):f(\alpha(x))<\frac{1}{i+1}\text{
for every }x\in K\}$$
of $0$ in $\mathsf{BLip_b}^+(\Delta)$. There is
$\varepsilon>0$ such that for every
$\psi\in\Psi(\kappa)$ there is $h_{\psi}\in
U_{\psi}$ for which $|\mu(h_{\psi})|>\varepsilon$.
Since the ${\cal G}$-pointwise and the $D$-pointwise
topology coincide on $\mathsf{BLip_b}^+(\Delta)$, the net
$(h_{\psi})_{\psi}$ converges ${\cal G}$-pointwise to
$0$. Hence $\mu$ does not have property (ii).
\par
In order to prove that (iii) implies (i) take any
net $(f_{\gamma})_{\gamma}$ in $\mathsf{BLip_b}^+(\Delta)$
converging ${\cal G}$-pointwise to a function
$f\in\mathsf{BLip_b}^+(\Delta)$. Then the functions
$(f_{\gamma}-f)^+$ and $(f_{\gamma}-f)^-$ are in
$2\mathsf{BLip_b}^+(\Delta)$ and converge ${\cal G}$-pointwise to
$0$.
\end{proof}
\par
We are ready to state the main result of this
paper, answering the question raised in \cite{dales1}.
\begin{theorem}\label{th:mainresult} Let ${\cal G}$ be a topological group
with the following property:
\begin{itemize}
\item[{\rm ($\dagger$)}] There exists $\Delta_0\in\mathsf{RP}({\cal G})$
such that $\eta^{\sharp}(\Delta_0)=\mathfrak{B}_{{\cal G}}$.
\end{itemize}
Then there are $\nu\in{\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}$ and a
net $(\nu_{\gamma})_{\gamma\in\Gamma}$ in
${\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}$ such that:
\begin{itemize}
\item[{\rm ({\bf 1}.)}]
$\displaystyle{\lim_{\gamma\in\Gamma}\nu_{\gamma}=\nu}$,
with the limit taken in ${\cal G}^{{\mathsf{LUC}}}$; and
\item[{\rm ({\bf 2}.)}] if $\mu\in{{\luc}(\G)^*}$ and
$\displaystyle{w^*\hskip-2mm
-\hskip-1mm\lim_{\gamma\in\Gamma}\mu\star
\nu_{\gamma}=\mu\star\nu}$ then
$\mu\in\mathsf{M_u}({\cal G})$.
\end{itemize}
\end{theorem}
\begin{proof}
Since ${\cal G}$ has property ($\dagger$), by \cite[Theorem 5]{ambitable} there is
$\Delta_1\in\mathsf{RP}({\cal G})$ such that
$\eta(\Delta_1)={\mathfrak B}_{{\cal G}}$.
Write $\Psi:={\cal P}_f({\mathfrak B}_{{\cal G}})\times\omega$
and note that $|\Psi|=|{\mathfrak B}_{{\cal G}}|$.
Let
$\{\nu_{\psi}\in{\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}:\psi\in\Psi \}$ be as
in Theorem \ref{th:factorization}
with $\Psi$ in place of ${\mathfrak B}_{{\cal G}}$.
The net
$(\nu_{\psi})_{\psi}$ has a subnet
$(\nu_{\gamma})_{\gamma}$ converging to a limit
$\nu\in{\cal G}^{{\mathsf{LUC}}}$.
\par
Take any $\mu\in{{\luc}(\G)^*}$ such that
$w^*\hskip-1mm-\lim_{\gamma}\mu\star\nu_{\gamma}=\mu\star\nu$
in ${{\luc}(\G)^*}$. Take any $\Delta\in\mathsf{RP}({\cal G})$ and a
net $(h_{\psi})_{\psi}$ in $\mathsf{BLip_b}^+(\Delta)$
indexed by $\Psi$ and
converging pointwise to $0$. By Theorem
\ref{th:factorization} there is
$h\in\mathsf{BLip_b}^+(2\Delta)$ such that
$h_{\psi}=\nu_{\psi}\bullet h$ for all
$\psi\in\Psi$. By \cite[Lemma 19]{ambitable}, the
mapping $\nu\mapsto\nu\bullet h$ is continuous from
${\cal G}^{{\mathsf{LUC}}}$ to $\mathsf{BLip_b}^+(2\Delta)$ with the
${\cal G}$-pointwise topology, hence $\nu\bullet h=0$.
Since
$$\lim_{\gamma}\mu(\nu_{\gamma}\bullet
h)=\lim_{\gamma}\mu\star\nu_{\gamma}(h)=\mu\star\nu(h)=\mu(\nu\bullet
h)=0$$
and $(\mu(\nu_{\gamma}\bullet h)\}_{\gamma}$ is a
subnet of $(\mu(h_{\psi}))_{\psi}$, we have that
$0$ is a cluster point of this last net and so
$\mu\in\mathsf{M_u}({\cal G})$ by Lemma
\ref{lemma:technical}.
\end{proof}
Note that if ${\cal G}$ is precompact then
$\eta^{\sharp}(\Delta)$ is finite for every
$\Delta\in\mathsf{RP}({\cal G})$ and it is known that in this
case ${\Lambda}({\luc}(\G)^*)=\mathsf{M_u}({\cal G})$. Thus property
($\dagger$) implies that ${\cal G}$ is not precompact.
\par
As a direct corollary to Theorem
\ref{th:mainresult} we obtain the following result.
\begin{corollary}
Let ${\cal G}$ be a topological group
with property \emph{($\dagger$)},
and let $S$ be a subsemigroup of ${{\luc}(\G)^*}$ such that
$S\supseteq{\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}$.
Then there exists $\nu_0\in S$ such that
$$
\Lambda(S)= \mathsf{M_u}({\cal G}) \cap S
= \{\mu\in S:
\text{ the mapping } \nu\mapsto \mu\star\nu \text{ on } S
\text{ is continuous} \text{ at } \nu_0 \}.
$$
\end{corollary}
\begin{proof}
By~\cite[Prop. 4.2]{ferrineufang} or~\cite[Cor. 9.36]{uniformmeasures} we have
$\mathsf{M_u}({\cal G}) \cap S \subseteq \Lambda(S)$.
By Theorem~\ref{th:mainresult} there exists $\nu_0\in S$ such that
$\mathsf{Cont}(S,\{\nu_0\})\subseteq\mathsf{M_u}({\cal G}) \cap S$,
and obviously $\Lambda(S)\subseteq\mathsf{Cont}(S,\{\nu_0\})$.
\end{proof}
Thus, for any ${\cal G}$ with property ($\dagger$), any subsemigroup $S$ of ${{\luc}(\G)^*}$ containing
${\cal G}^{{\mathsf{LUC}}}\setminus{\cal G}$ has a one-point DTC-set.
It is easy to see that every non-compact
locally compact group has property ($\dagger$):
Simply take any $\Delta_0\in\mathsf{RP}({\cal G})$ such that
$\mathsf{B}(\Delta_0)$ is relatively compact.
Thus Theorem \ref{th:mainresult} and its corollary
generalize the recent
results of Budak, I\c sik and Pym \cite{budak}, who
proved the same for non-compact locally compact
groups, and therefore the existence of one-point DTC-sets for
${{\luc}(\G)^*}$ and for ${\cal G}^{{\mathsf{LUC}}}$ for such groups.
Many other (not necessarily locally compact)
groups also have property ($\dagger$):
From the definition,
if ${\mathfrak B}_{{\cal G}}=\aleph_0$ and ${\cal G}$ is not precompact
then ${\cal G}$ has property ($\dagger$);
and if ${\mathfrak B}_{{\cal G}}$ is a successor cardinal
then ${\cal G}$ has property ($\dagger$).
\par
It is an open problem whether the property ($\dagger$)
may be omitted in Theorem~\ref{th:mainresult} or in its corollary
(for non-precompact topological groups).
\par
\par
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 483
|
\section*{Introduction}
$\mathcal{N}=1$ flux compactifications of eleven-dimensional supergravity on
eight-manifolds $M$ down to $\mathrm{AdS}_3$ spaces \cite{MartelliSparks,
Tsimpis} provide a vast extension of the better studied class of
compactifications down to 3-dimensional Minkowski space \cite{Becker1,
Becker2, Constantin}, having the advantage that they are already
consistent at the classical level \cite{MartelliSparks}. They form a
useful testing ground for various proposals aimed at providing unified
descriptions of flux backgrounds \cite{Grana} and may be relevant to
recent attempts to gain a better understanding of F-theory
\cite{Bonetti}. When the internal part $\xi$ of the supersymmetry
generator is everywhere non-chiral, such backgrounds can be studied
\cite{g2} using foliations endowed with longitudinal $G_2$ structures,
an approach which permits a geometric description of the supersymmetry
conditions while providing powerful tools for studying the topology of
such backgrounds.
In this paper, we extend the results of \cite{g2} to the general case
when the internal part $\xi$ of the supersymmetry generator is allowed
to become chiral on some locus $\mathcal{W}\subset M$. Assuming that $\mathcal{W}\neq
M$, i.e. that $\xi$ is not everywhere chiral, we show that, at the
classical level, the Einstein equations imply that the chiral locus
$\mathcal{W}$ must be a set with empty interior, which therefore is negligible
with respect to the Lebesgue measure of the internal space. As a
consequence, the behavior of geometric data along this locus can be
obtained from the non-chiral locus $\mathcal{U}\stackrel{{\rm def.}}{=} M\setminus \mathcal{W}$ through
a limiting process. The geometric information along the non-chiral
locus $\mathcal{U}$ is encoded \cite{g2} by a regular foliation $\mathcal{F}$ which
carries a longitudinal $G_2$ structure and whose geometry is
determined by the supersymmetry conditions in terms of the
supergravity four-form field strength. When $\emptyset \neq
\mathcal{W}\subsetneq M$, we show that $\mathcal{F}$ extends to a singular foliation
$\bar{\mathcal{F}}$ of the whole manifold $M$ by adding leaves which are
allowed to have singularities at points belonging to $\mathcal{W}$. This
singular foliation ``integrates'' a cosmooth\footnote{Note that
$\mathcal{D}$ is {\em not} a singular distribution in the sense of
Stefan-Sussmann \cite{Stefan, Sussmann} (it is cosmooth rather than
smooth). See Appendix \ref{app:gendist}.} \cite{BulloLewis, Drager, Michor, Ratiu}
singular distribution $\mathcal{D}$ (a.k.a. generalized sub-bundle of $TM$),
defined as the kernel distribution of a closed one-form ${\boldsymbol{\omega}}$
which belongs to a cohomology class $\mathfrak{f}\in H^1(M,\mathbb{R})$ determined by
the supergravity four-form field strength. The set of zeroes of
${\boldsymbol{\omega}}$ coincides with the chiral locus $\mathcal{W}$. In the most general
case, $\bar{\mathcal{F}}$ can be viewed as a Haefliger structure
\cite{Haefliger} on $M$. The singular foliation $\bar{\mathcal{F}}$ carries a
longitudinal $G_2$ structure, which is allowed to degenerate at the
singular points of singular leaves. On the non-chiral locus $\mathcal{U}$,
the problem can be studied using the approach of \cite{g2} or the
approach advocated in \cite{Tsimpis}, which makes use of two
$\mathrm{Spin}(7)_\pm$ structures. We show explicitly how one can translate
between these two approaches and prove that the results of \cite{g2}
agree with those of \cite{Tsimpis} along this locus.
While the topology of singular foliations defined by a closed one-form
can be extremely complicated in general, the situation is better
understood in the case when ${\boldsymbol{\omega}}$ is a Morse one-form. The Morse
case is generic in the sense that such 1-forms constitute an open and
dense subset of the set of all closed one-forms belonging to the
cohomology class $\mathfrak{f}$. In the Morse case, the singular foliation
$\bar{\mathcal{F}}$ can be described using the {\em foliation graph}
\cite{MelnikovaThesis, MelnikovaGraph, FKL} associated to the
corresponding decomposition of $M$ (see \cite{Melnikova2, Melnikova3,
FKL} and \cite{Gelbukh1}--\cite{Gelbukh9}), which provides a
combinatorial way to encode some important aspects of the foliation's
topology --- up to neglecting the information contained in the
so-called {\em minimal components} of the decomposition, components
which should possess an as yet unexplored non-commutative geometric
description. This provides a far-reaching extension of the picture
found in \cite{g2} for the everywhere non-chiral case $\mathcal{U}=M$, a case
which corresponds to the situation when the foliation graph is reduced
to either a circle (when $\mathcal{F}$ has compact leaves, being a fibration
over $S^1$) or to a single so-called exceptional vertex (when $\mathcal{F}$
has non-compact dense leaves, being a minimal foliation). In the
minimal case of the backgrounds considered \cite{g2}, the exceptional
vertex corresponds to a noncommutative torus which encodes the
noncommutative geometry \cite{ConnesFol, ConnesNG} of the leaf space.
The paper is organized as follows. Section 1 gives a brief review of
the class of compactifications we consider, in order to fix notations
and conventions. Section 2 discusses a geometric characterization of
Majorana spinors $\xi$ on $M$ which is inspired by the rigorous
approach developed in \cite{ga1,ga2, gf} for the method of bilinears
\cite{Tod}, in the case when the spinor $\xi$ is allowed to be chiral
at some loci. It also gives the K\"{a}hler-Atiyah~ parameterizations of this spinor
which correspond to the approach of \cite{g2} and to that of
\cite{Tsimpis} and describes the relevant $G$-structures using both
spinors and idempotents in the K\"{a}hler-Atiyah~ algebra of $M$. In the same
section, we give the general description of the singular foliation
$\bar{\mathcal{F}}$ as the Haefliger structure defined by the closed one-form
${\boldsymbol{\omega}}$. Section 3 describes the relation between the $G_2$ and
$\mathrm{Spin}(7)_\pm$ parameterizations of the fluxes as well as the relation
between the torsion classes of the leafwise $G_2$ structure and the
Lee form and characteristic torsion of the $\mathrm{Spin}(7)_\pm$ structures
defined on the non-chiral locus. The same section gives the comparison
of the approach of \cite{g2} with that of \cite{Tsimpis} along that
locus. Section 4 discusses the topology of the singular foliation
$\bar{\mathcal{F}}$ in the Morse case while Section 5 concludes. The
appendices contain various technical details.
\paragraph{Notations and conventions. }
Throughout this paper, $M$ denotes an oriented, connected and compact
smooth manifold (which will mostly be of dimension eight), whose
unital commutative $\mathbb{R}$-algebra of smooth real-valued functions we
denote by $\Omega^0(M)={{\cal \mathcal{C}}^\infty(M,\mathbb{R})}$. Given a subset $A$ of $M$, we let
${\bar A}$ denote the closure of $A$ in $M$ (taken with respect to the
manifold topology of $M$). The {\em large topological frontier} (also
called {\em topological boundary}) of $A$ is defined as $\mathrm{Fr}(A)\stackrel{{\rm def.}}{=}
{\bar A}\setminus \mathrm{Int}(A)$, where $\mathrm{Int}(A)$ denotes the interior of
$A$. The {\em small topological frontier} is $\mathrm{fr}(A)\stackrel{{\rm def.}}{=} {\bar
A}\setminus A$. Notice that $\mathrm{fr}(A)\subseteq\mathrm{Fr}(A)$ and that
$\mathrm{fr}(A)=\mathrm{Fr}(A)$ when $A$ is open, in which case we speak simply of the
{\em frontier} of $A$. All fiber bundles we consider are
smooth\footnote{The ``generalized bundles''\cite{Drager, BulloLewis}
considered occasionally in this paper are {\em not} fiber
bundles.}. We use freely the results and notations of
\cite{ga1,ga2,gf,g2}, with the same conventions as there. To simplify
notation, we write the geometric product $\diamond$ of
\cite{ga1,ga2,gf} simply as juxtaposition while indicating the wedge
product of differential forms through $\wedge$. If $\mathcal{D}\subset TM$ is
a singular (a.k.a. generalized) distribution on $M$ and $\mathcal{U}$ is an
open subset of $M$ such that $\mathcal{D}|_\mathcal{U}$ is a regular Frobenius
distribution (see Appendix \ref{app:gendist}), we let
$\Omega_\mathcal{U}(\mathcal{D})=\Gamma(\mathcal{U},\wedge (\mathcal{D}|_\mathcal{U})^\ast)$ denote the
$\mathcal{C}^\infty(\mathcal{U},\mathbb{R})$-module of $\mathcal{D}|_\mathcal{U}$-longitudinal differential
forms defined on $\mathcal{U}$. When $\dim M=8$, then for any 4-form
$\omega\in \Omega^4(M)$ we let $\omega^\pm\stackrel{{\rm def.}}{=} \frac{1}{2}(\omega\pm
\ast \omega)$ denote the selfdual and anti-selfdual parts of $\omega$
(namely, $\ast \omega^\pm=\pm \omega^\pm$). When $M$ is
eight-dimensional, we let $\Omega^{4\pm}(M)$ denote the spaces of
selfdual and anti-selfdual four-forms, respectively. We use the
``{\rm Det}'' convention for the wedge product and the corresponding
``{\rm Perm}'' convention for the symmetric product. Hence given a
local coframe $e^a$ of $M$, we have:
\begin{equation}
\label{DetPerm}
\begin{split}
&e^{a_1}\wedge \ldots \wedge e^{a_k}\stackrel{{\rm def.}}{=} \sum_{\sigma\in S_k}\epsilon(\sigma) e^{a_{\sigma(1)}}\otimes \ldots \otimes e^{a_{\sigma(k)}}~~,\\
&e^{a_1}\odot \ldots \odot e^{a_k}\stackrel{{\rm def.}}{=} \sum_{\sigma\in S_k}e^{a_{\sigma(1)}}\otimes \ldots \otimes e^{a_{\sigma(k)}}~~,
\end{split}
\end{equation}
{\em without} prefactors of $\frac{1}{k!}$ in the right hand side,
where $S_k$ is the symmetric group on $k$ letters and
$\epsilon(\sigma)$ denotes the signature of a permutation $\sigma$.
This is the convention used, for example, in \cite{Spivak}. We use
$\mathrm{Sym}^2_0(T^\ast M)$ to denote the space of traceless symmetric covariant
2-tensors on $M$ and $\mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ to denote the space of
traceless symmetric covariant 2-tensors defined on $\mathcal{U}$ and which are
longitudinal to the Frobenius distribution $\mathcal{D}|_\mathcal{U}$, when $\mathcal{D}$ is
as above. By definition, a $\mathrm{Spin}(7)_+$ structure on $M$ is a
$\mathrm{Spin}(7)$ structure with respect to the orientation chosen for $M$
while a $\mathrm{Spin}(7)_-$ structure is a $\mathrm{Spin}(7)$ structure with respect
to the opposite orientation.
\section{Basics}
\label{sec:basics}
We start with a brief review of the set-up, in order to fix notation.
As in \cite{MartelliSparks, Tsimpis}, we consider 11-dimensional
supergravity \cite{sugra11} on an eleven-dimensional connected and paracompact
spin manifold $\mathbf{M}$ with Lorentzian metric $\mathbf{g}$ (of `mostly plus'
signature). Besides the metric, the classical action of the theory contains the
three-form potential with four-form field strength
$\mathbf{G}\in\Omega^4(\mathbf{M})$ and the gravitino
$\mathbf{\Psi}$, which is a Majorana spinor of spin $3/2$. The bosonic part of the action takes the form:
\begin{equation*}
S_{\rm bos}[\mathbf{g}, \mathbf{C}]=
\frac{1}{2\mathbf{\kappa}_{11}^2}\int_{\mathbf M}R\boldsymbol{\nu}-
\frac{1}{4\mathbf{\kappa}_{11}^2}\int_{\mathbf M}\big(\mathbf{G}\wedge \star \mathbf{G}+\frac{1}{3}\mathbf{C}\wedge \mathbf{G}\wedge \mathbf{G}\big)~~,
\end{equation*}
where $\mathbf{\kappa}_{11}$ is the gravitational coupling constant in eleven dimensions, $\boldsymbol{\nu}$ and $R$ are the volume form and the scalar curvature of $\mathbf{g}$ and
$\mathbf{G}=\mathrm{d} \mathbf{C}$.
For supersymmetric bosonic classical backgrounds, both the gravitino and
its supersymmetry variation must vanish, which requires that there
exist at least one solution $\boldsymbol{\eta}$ to the equation:
\begin{equation}
\label{susy}
\delta_{\boldsymbol{\eta}} \mathbf{\Psi} \stackrel{{\rm def.}}{=} \mathfrak{D} \boldsymbol{\eta} = 0~~,
\end{equation}
where $\mathfrak{D}$ denotes the supercovariant connection. The
eleven-dimensional supersymmetry generator $\boldsymbol{\eta}$ is a
Majorana spinor (real pinor) of spin $1/2$ on $\mathbf{M}$.
As in \cite{MartelliSparks, Tsimpis}, consider compactification down
to an $\mathrm{AdS}_3$ space of cosmological constant $\Lambda=-8\kappa^2$,
where $\kappa$ is a positive real parameter --- this includes the
Minkowski case as the limit $\kappa\rightarrow 0$. Thus $\mathbf{
M}=N\times M$, where $N$ is an oriented 3-manifold diffeomorphic
to $\mathbb{R}^3$ and carrying the $\mathrm{AdS}_3$ metric $g_3$ while $M$ is an oriented,
compact and connected Riemannian eight-manifold whose metric we denote by $g$. The metric on
$\mathbf{M}$ is a warped product:
\begin{eqnarray}
\label{warpedprod}
\mathrm{d} \mathbf{s}^2 & = & e^{2\Delta} \mathrm{d} s^2~~~{\rm where}~~~\mathrm{d} s^2=\mathrm{d} s^2_3+ g_{mn} \mathrm{d} x^m \mathrm{d} x^n~~.
\end{eqnarray}
The warp factor $\Delta$ is a smooth real-valued function defined on $M$ while $\mathrm{d} s_3^2$ is the
squared length element of the $\mathrm{AdS}_3$ metric $g_3$. For the field strength $\mathbf{G}$, we use the ansatz:
\begin{equation}
\label{Gansatz}
\mathbf{ G} = \nu_3\wedge \mathbf{f}+\mathbf{F}~~,~~~~\mathrm{with}~~
\mathbf{F}\stackrel{{\rm def.}}{=} e^{3\Delta}F~~,~~\mathbf{f}\stackrel{{\rm def.}}{=} e^{3\Delta} f~~,
\end{equation}
where $f\in \Omega^1(M)$, $F\in \Omega^4(M)$ and $\nu_3$ is the
volume form of $(N,g_3)$. For $\boldsymbol{\eta}$, we use the ansatz:
\begin{equation*}
\boldsymbol{\eta}=e^{\frac{\Delta}{2}}(\zeta\otimes \xi)~~,
\end{equation*}
where $\xi$ is a Majorana spinor of spin $1/2$ on the internal space
$(M,g)$ (a section of the rank 16 real vector bundle $S$ of indefinite chirality real
pinors) and $\zeta$ is a Majorana spinor on $(N,g_3)$.
Assuming that $\zeta$ is a Killing spinor on the $\mathrm{AdS}_3$ space $(N,g_3)$, the
supersymmetry condition \eqref{susy} is equivalent with the following
system for $\xi$:
\begin{equation}
\label{par_eq}
\boxed{\mathbb{D}\xi = 0~~,~~Q\xi = 0}~~,
\end{equation}
where
\begin{equation*}
\mathbb{D}_X=\nabla_X^S+\frac{1}{4}\gamma(X\lrcorner F)+\frac{1}{4}\gamma((X_\sharp\wedge f) \nu) +\kappa \gamma(X\lrcorner \nu)~~,~~X\in \Gamma(M,TM)
\end{equation*}
is a linear connection on $S$ (here $\nabla^S$ is the connection induced on $S$ by the Levi-Civita
connection of $(M,g)$, while $\nu$ is the volume form of $(M,g)$) and
\begin{equation*}
Q=\frac{1}{2}\gamma(\mathrm{d} \Delta)-\frac{1}{6}\gamma(\iota_f\nu)-\frac{1}{12}\gamma(F)-\kappa\gamma(\nu)
\end{equation*}
is a globally-defined endomorphism of $S$. As in \cite{MartelliSparks,
Tsimpis}, {\em we do not require that $\xi$ has definite chirality}.
The set of solutions of \eqref{par_eq} is a finite-dimensional
$\mathbb{R}$-linear subspace $\mathrm{\cal K}(\mathbb{D},{Q})$ of the infinite-dimensional
vector space $\Gamma(M,S)$ of smooth sections of $S$. Up to rescalings by
smooth nowhere-vanishing real-valued functions defined on $M$, the
vector bundle $S$ has two admissible pairings $\Scr B_\pm$ (see \cite{gf,
AC1, AC2}), both of which are symmetric but which are distinguished
by their types $\epsilon_{\Scr B_\pm}=\pm 1$. Without loss of generality,
we choose to work with $\Scr B\stackrel{{\rm def.}}{=} \Scr B_+$. We can in fact take $\Scr B$
to be a scalar product on $S$ and denote the corresponding norm by
$||~||$ (see \cite{ga1,ga2} for details). Requiring that the
background preserves exactly $\mathcal{N}=1$ supersymmetry amounts to asking
that $\dim \mathrm{\cal K}(\mathbb{D},Q)=1$. It is not hard to check \cite{ga1} that
$\Scr B$ is $\mathbb{D}$-flat:
\begin{equation}
\label{flatness}
\mathrm{d} \Scr B(\xi',\xi'')=\Scr B(\mathbb{D}\xi', \xi'')+\Scr B(\xi',\mathbb{D}\xi'')~~,
~~\forall \xi',\xi''\in \Gamma(M,S)~~.
\end{equation}
Hence any solution of \eqref{par_eq} which has unit $\Scr B$-norm at a
point will have unit $\Scr B$-norm at every point of $M$ and we can take
the internal part $\xi$ of the supersymmetry generator to be
everywhere of norm one.
\section{Parameterizing a Majorana spinor on $M$}
\label{sec:fierz}
\subsection{Globally valid parameterization}
Fixing a Majorana spinor $\xi\in \Gamma(M,S)$ which is everywhere of
$\Scr B$-norm one, consider the inhomogeneus differential form:
\begin{equation}
\label{checkE}
\check{E}_{\xi,\xi}=\frac{1}{16} \sum_{k=0}^8 \boldsymbol{\check{E}}^{(k)}_{\xi,\xi}\in \Omega(M)~~,
\end{equation}
whose rescaled rank components have the following expansions in any
local orthonormal coframe $(e^a)_{a=1\ldots 8}$ of $M$ defined on some
open subset $U$:
\begin{equation*}
\boldsymbol{\check{E}}^{(k)}_{\xi,\xi}=_U\frac{1}{k!}\Scr B(\xi,\gamma_{a_1...a_k}\xi)e^{a_1...a_k} \in \Omega^k(M)~~.
\end{equation*}
The conditions:
\begin{equation}
\label{Esquare}
\check{E}^2=\check{E}~~,~~~\mathcal{S}(\check{E})=1~~,~~\tau(\check{E})=\check{E}~~
\end{equation}
encode the fact that an inhomogeneous form $\check{E}\stackrel{{\rm def.}}{=}\check{E}_{\xi,\xi} $ is of the type \eqref{checkE} for some Majorana spinor $\xi$ which is everywhere of norm one.
As a result of the last condition in \eqref{Esquare}, the non-zero components of $\check{E}$ have ranks $k=0,1,4,5$ and we have
$\mathcal{S}(\check{E}_{\xi,\xi})=\boldsymbol{\check{E}}^{(0)}_{\xi,\xi}=||\xi||^2=1$, where
$\mathcal{S}$ is the canonical trace of the K\"{a}hler-Atiyah~ algebra. Hence:
\begin{equation}
\label{Eexp}
\boxed{\check{E}=\frac{1}{16}(1+V+Y+Z+b\nu)}~~,
\end{equation}
where we introduced the notations:
\begin{equation}
\label{forms8}
V\stackrel{{\rm def.}}{=} \boldsymbol{\check{E}}^{(1)} ~~,~~ Y\stackrel{{\rm def.}}{=}\boldsymbol{\check{E}}^{(4)} ~,~~ Z\stackrel{{\rm def.}}{=} \boldsymbol{\check{E}}^{(5)}~~,~~ b\nu\stackrel{{\rm def.}}{=} \boldsymbol{\check{E}}^{(8)}~~.
\end{equation}
Here, $b$ is a smooth real valued function defined on $M$ and $\nu$ is
the volume form of $(M,g)$, which satisfies $||\nu||=1$; notice the
relation $\mathcal{S}(\nu\check{E}_{\xi,\xi})=b$. On a small enough open subset
$U\subset M$ supporting a local coframe $(e^a)$ of $M$, one has the
expansions:
\begin{eqnarray}
\label{forms8alt}
&&V=_U\Scr B(\xi,\gamma_a\xi)e^a ~~,~~ Y=_U\frac{1}{4!}\Scr B(\xi,\gamma_{a_1\ldots a_4}\xi) e^{a_1\ldots a_4}~,\nonumber\\
&& Z=_U\frac{1}{5!} \Scr B(\xi,\gamma_{a_1\ldots a_5}\xi) e^{a_1\ldots a_5}~~,
~~ b=_U\Scr B(\xi, \gamma(\nu)\xi)~~.
\end{eqnarray}
One finds \cite{ga1} that \eqref{Esquare} is equivalent with the following relations which hold globally on $M$:
\begin{equation}
\label{SolMS}
\boxed{
\begin{split}
& ||V||^2=1-b^2\geq0~~,~~||Y^\pm||^2=\frac{7}{2}(1\pm b)^2~~,\\
& \iota_V(\ast Z)=0~~,~~\iota_V Z=Y-b\ast Y~~,\\
&(\iota_\alpha (\ast Z)) \wedge (\iota_\beta (\ast Z)) \wedge (\ast Z)
= - 6 \langle \alpha\wedge V, \beta\wedge V\rangle \iota_V \nu~~,~~\forall \alpha,\beta\in \Omega^1(M)~~.
\end{split}}
\end{equation}
Notice that the first relation in the second row is equivalent with $V\wedge Z=0$,
which means that $V$ and $Z$ commute in the K\"{a}hler-Atiyah~ algebra of $(M,g)$.
\paragraph{Remark.}
Let (R) denote the second relation (namely $\iota_V Z=Y-b\ast Y$) on
the second row of \eqref{SolMS}. Separating the selfdual and
anti-selfdual parts shows that (R) is {\em equivalent} with the
following two conditions:
\begin{equation}
\label{iVZpm}
(\iota_V Z)^\pm=(1\mp b)Y^\pm~~.
\end{equation}
\paragraph{Proposition.} Relations \eqref{SolMS} imply that the following normalization
conditions hold globally on $M$:
\begin{equation}
\label{Ynorms}
||Y||^2=7(1+b^2)~~,~~||Z||^2=7 (1-b^2)~~.
\end{equation}
\noindent{\bf Proof.} The first equation in \eqref{Ynorms} follows
from the last relations on the first row of \eqref{SolMS} by noticing
that $||Y||^2=||Y^+||^2+||Y^-||^2$ (since $\langle Y^+,Y^-\rangle=0$).
We have:
\begin{equation}
\label{intmd}
||\iota_VZ||^2=||\ast \iota_V Z||^2=||V\wedge (\ast
Z)||^2=||V||^2||\ast Z||^2=||V||^2||Z||^2~~,
\end{equation}
where in the middle equality we used the first equation on the second
row of \eqref{SolMS}, which tells us that $\ast Z$ is orthogonal on
$V$. The second equation in \eqref{Ynorms} now follows from
\eqref{intmd} and from the identity:
\begin{equation*}
||\iota_VZ||^2=(1-b)^2||Y^+||^2+(1+b)^2||Y^-||^2=7(1-b^2)=7||V||^2~~,
\end{equation*}
where we used \eqref{iVZpm} and both relations in the first row of
\eqref{SolMS}. $\blacksquare$
\paragraph{The twisted selfdual and twisted anti-selfdual parts of $\check{E}$.}
The identity $\nu^2=1$ implies that the elements:
\begin{equation*}
R^\pm\stackrel{{\rm def.}}{=} \frac{1}{2}(1\pm \nu)~~
\end{equation*}
are complementary idempotents in the K\"{a}hler-Atiyah~ algebra:
\begin{equation}
\label{pirels}
(R^\pm)^2=R^\pm~~,~~R^\pm R^\mp=0~~,~~R^++R^-=\mathrm{id}_{\Omega(M)}~~.
\end{equation}
The (anti)selfdual part of a four-form $\omega\in \Omega^4(M)$ can be expressed as:
\begin{equation*}
\omega_\pm=R^\pm\omega~~.
\end{equation*}
Notice that this relation also gives the twisted (anti)selfdual parts \cite{ga1} of an inhomogeneous form $\omega\in \Omega(M)$.
The identities:
\begin{equation*}
YR^\pm=R^\pm Y=Y^\pm~~,~~(1+b\nu)R^\pm=(1\pm b)R^\pm~~
\end{equation*}
allow us to compute the twisted selfdual part $\check{E}^+$ and twisted anti-selfdual part $\check{E}^-$ of $\check{E}$:
\begin{equation}
\label{Epm}
\check{E}^\pm=\check{E}R^\pm=\frac{1}{16}\left[(1\pm b+V+Z)R_\pm + Y^\pm\right]\in \Omega(M)~~.
\end{equation}
The following decomposition holds globally on $M$:
\begin{equation*}
\check{E}=\check{E}^++\check{E}^-~~.
\end{equation*}
\subsection{The chirality decomposition of $M$}
Let $S^\pm\subset S$ be the rank eight subbundles of $S$ consisting of
positive and negative chirality spinors (the eigen-subbundles of
$\gamma(\nu)$ corresponding to the eigenvalues $+1$ and $-1$). Since
$\gamma(\nu)$ is $\Scr B$-symmetric, $S^+$ and $S^-$ give a
$\Scr B$-orthogonal decomposition $S=S^+\oplus S^-$. Decomposing a
normalized spinor as $\xi=\xi^++\xi^-$ with $\xi^\pm\stackrel{{\rm def.}}{=}
\frac{1}{2}(\mathrm{id}_S\pm \gamma(\nu))\xi\in \Gamma(M,S^\pm)$, we have:
\begin{equation*}
||\xi||^2=||\xi^+||^2+||\xi^-||^2=1
\end{equation*}
and:
\begin{equation*}
b=\Scr B(\xi,\gamma(\nu)\xi)=||\xi^+||^2-||\xi^-||^2~~.
\end{equation*}
These two relations give:
\begin{equation}
\label{xipmnorms}
\boxed{||\xi^\pm||^2=\frac{1}{2}(1\pm b)}~~.
\end{equation}
Notice that $b$ equals $\pm 1$ at a point $p\in M$ iff $\xi_p\in
S^\pm_p$. Since $||V||^2=1-b^2$, the one-form $V$ vanishes at $p$
iff $\xi_p$ is chiral i.e. iff $\xi_p\in S_p^+\cup S_p^-$. Consider
the {\em non-chiral locus} (an open subset of $M$):
\begin{equation*}
\mathcal{U} \stackrel{{\rm def.}}{=} \{p\in M|\xi\not \in S_p^+\cup S_p^-\}=\{p\in M|\xi^+_p\neq 0~\mathrm{and}~~\xi_p^-\neq 0\}=\{p\in M|V_p\neq 0\}=\{p\in M||b(p)| < 1\}~~,
\end{equation*}
and its closed complement, the {\em chiral locus}:
\begin{equation*}
\mathcal{W}\stackrel{{\rm def.}}{=} M\setminus \mathcal{U}=\{p\in M|\xi_p\in S_p^+\cup S^-_p\}=\{p\in M|\xi^+_p=0~\mathrm{or}~\xi^-_p=0\}=\{p\in M|V_p=0\}=\{p\in M| |b(p)|=1\}~~.
\end{equation*}
The chiral locus $\mathcal{W}$ decomposes further as a disjoint union of two closed
subsets, the {\em positive and negative chirality loci}:
\begin{equation*}
\mathcal{W}=\mathcal{W}^+\sqcup \mathcal{W}^-~~,
\end{equation*}
where:
\begin{eqnarray*}
\mathcal{W}^\pm\stackrel{{\rm def.}}{=} \{p\in M|\xi_p\in S^\pm_p\}=\{p\in M| b(p)=\pm 1\}=\{p\in M|\xi_p^\mp=0\}~~.
\end{eqnarray*}
The extreme cases $\mathcal{W}^+=M$ or $\mathcal{W}^-=M$, as well as $\mathcal{W}^+=\mathcal{W}^-=\emptyset$
are allowed. However, the case $\mathcal{U}=\emptyset$ with both $\mathcal{W}^+$ and
$\mathcal{W}^-$ nonempty (then $M=\mathcal{W}^+\sqcup \mathcal{W}^-$) is forbidden (recall that $b$ is
smooth and hence continuous while $M$ is connected). Since $\xi$ does not
vanish on $M$, we have:
\begin{equation*}
\mathcal{U}^\pm\stackrel{{\rm def.}}{=} \mathcal{U}\cup \mathcal{W}^\pm=\{p\in M|\xi^\pm_p\neq 0\}~~.
\end{equation*}
\paragraph{Remark.}
Since $|b|\leq 1$ on $M$, the sets $\mathcal{W}^\pm$ (when non-empty) consist
of critical points of $b$, namely the absolute maxima and minima of
$b$ on $M$. Hence the differential of $b$ vanishes at every point of
$\mathcal{W}$. In general $\mathcal{W}^\pm$ can be quite `wild' (they can
be very far from being immersed submanifolds of $M$).
\subsection{A topological no-go theorem}
\label{subsec:nogo}
Recall that $M$ is compact. The following result clarifies the kind of
topologies of the chiral loci which are of physical interest.
\paragraph{Theorem}
Assume that the supersymmetry conditions, the Bianchi identity and
equations of motion for $G$ as well as the Einstein equations are
satisfied. There exist only the following four possibilities:
\begin{enumerate}
\item The set $\mathcal{W}^+$ coincides with $M$ and hence $\mathcal{W}^-$ and $\mathcal{U}$
are empty. In this case, $\xi$ is a chiral spinor of positive
chirality which is covariantly constant on $M$ and we have
$\kappa=f=F=0$ while $\Delta$ is constant on $M$.
\item The set $\mathcal{W}^-$ coincides with $M$ and hence $\mathcal{W}^+$ and $\mathcal{U}$
are empty. In this case, $\xi$ is a chiral spinor of negative
chirality which is covariantly constant on $M$ and we have
$\kappa=f=F=0$ while $\Delta$ is constant on $M$.
\item The set $\mathcal{U}$ coincides with $M$ and hence $\mathcal{W}^+$ and $\mathcal{W}^-$
are empty.
\item At least one of the sets $\mathcal{W}^+$ or $\mathcal{W}^-$ is non-empty but
both of these sets have empty interior. In this case, $\mathcal{U}$ is dense
in $M$ and the union $\mathcal{W}=\mathcal{W}^+\cup\mathcal{W}^-$ coincides with the
topological frontier $\mathrm{Fr}(\mathcal{U})=\mathrm{fr}(\mathcal{U})={\bar \mathcal{U}}\setminus \mathcal{U}$ of $\mathcal{U}$.
\end{enumerate}
\noindent The proof of the theorem is given in Appendix \ref{app:nogo}.
\paragraph{Remarks.}
\begin{itemize}
\itemsep 0.0em
\item The theorem is a strengthening of an observation originally made in \cite{MartelliSparks} in the case when $\xi$ is nowhere-chiral.
\item The theorem holds in classical supergravity only.
One may be able to avoid its conclusions by considering quantum corrections.
\item Cases 1 and 2 correspond to the classical limit of the compactifications studied in \cite{Becker1, Becker2, Constantin}. Case 3 was studied in
\cite{MartelliSparks,g2}.
\end{itemize}
The study of Case 4 is the focus of the present paper. Due to the
theorem, we shall from now on assume that we are in this case,
i.e. that $\mathcal{W}$ is non-empty and that it coincides with the frontier
of $\mathcal{U}$; in particular, we can assume that the closure of $\mathcal{U}$
coincides with $M$:
\begin{equation*}
M={\bar \mathcal{U}}=\mathcal{U}\sqcup \mathcal{W}~~,~~\mathcal{W}=\mathrm{Fr} \mathcal{U}~~.
\end{equation*}
In Figure \ref{fig:loci}, we sketch the chirality decomposition in two
sub-cases of Case 4, which correspond to the assumptions that the
one-form ${\boldsymbol{\omega}}\stackrel{{\rm def.}}{=} 4\kappa e^{3\Delta}V$ is of Morse and
Bott-Morse type, respectively.
\!\!\!\!\!\begin{figure}
\centering
\!\!\!\!\!\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{morse.eps}
\
\caption{Sketch of the chiral loci in the Morse sub-case of Case 4 of
the Theorem. In this case, each of $\mathcal{W}^+$ and $\mathcal{W}^-$ is a finite
set of points, with the points of $\mathcal{W}^+$ indicated in red and those
of $\mathcal{W}^-$ indicated in blue.}
\end{subfigure}~~~~~~~~~
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{bott.eps}
\caption{Sketch of $\mathcal{W}^\pm$ in the Bott-Morse sub-case of Case 4 of
the Theorem. The connected components of $\mathcal{W}$ are submanifolds of
various dimensions, shown respectively in red and blue for
$\mathcal{W}^+$ and $\mathcal{W}^-$.}
\end{subfigure}
\caption{Sketch of chiral loci in two sub-cases of Case 4 of the
Theorem, for the case of a two-dimensional manifold $M$. The
non-chiral locus $\mathcal{U}$ is the complement of $\mathcal{W}$ in $M$ and is
indicated by white space, after performing appropriate cuts which
allow one to map $M$ to some region of the plane which is not
indicated explicitly. The figures should be interpreted with care in
our case $\dim M=8$.}
\label{fig:loci}
\end{figure}
\subsection{The singular distribution $\mathcal{D}$}
The one-form $V$ determines a singular (a.k.a. generalized) distribution $\mathcal{D}$ (generalized sub-bundle of $TM$) which is defined through:
\begin{equation*}
\mathcal{D}_p\stackrel{{\rm def.}}{=}{\mathrm{ker} V_p}~~,~~\forall p\in M~~.
\end{equation*}
This singular distribution is {\em cosmooth} (rather than smooth) in
the sense of \cite{Drager} (see Appendix \ref{app:gendist}). Notice
that $\mathcal{D}$ is smooth iff $\xi$ is everywhere non-chiral --- i.e. iff
$\mathcal{W}=\emptyset$, which is the case studied in \cite{g2}; in that case,
$\mathcal{D}$ is a regular Frobenius distribution. Since in this paper we
assume $\mathcal{W}\neq \emptyset$, it follows that $\mathcal{D}$ is {\em not} a
singular distribution in the sense of Stefan-Sussmann
\cite{Stefan,Sussmann}. The set of regular points of $\mathcal{D}$ equals the
non-chiral locus $\mathcal{U}$ and we have:
\begin{eqnarray*}
{\rm rk}\mathcal{D}_p&=&7~~\mathrm{when}~~p\in\mathcal{U}~~,\\
{\rm rk}\mathcal{D}_p&=&8~~\mathrm{when}~~p\in \mathcal{W}~~.
\end{eqnarray*}
In particular, the restriction $\mathcal{D}|_\mathcal{U}$ is a regular Frobenius
distribution on the non-chiral locus $\mathcal{U}$. As in \cite{g2}, we endow
$\mathcal{D}|_\mathcal{U}$ with the orientation induced by that of $M$ using the unit
norm vector field $n\stackrel{{\rm def.}}{=} {\hat V}^\sharp=\frac{1}{||V||}V^\sharp$,
which corresponds to the $\mathcal{D}|_\mathcal{U}$-longitudinal volume form:
\begin{equation*}
\nu_\top\stackrel{{\rm def.}}{=} \iota_{\hat V}\nu|_\mathcal{U}=n\lrcorner \nu|_\mathcal{U}\in \Omega^7_{\mathcal{U}}(\mathcal{D})~~.
\end{equation*}
Let $\ast_\perp:\Omega_\mathcal{U}(\mathcal{D})\rightarrow \Omega_\mathcal{U}(\mathcal{D})$ denote the
corresponding Hodge operator along the Frobenius distribution
$\mathcal{D}|_\mathcal{U}$:
\begin{equation}
\label{AstV}
\ast_\perp \omega=\ast({\hat V}\wedge \omega)=-\iota_{\hat V}(\ast \omega)=\tau(\omega)\nu_\top~~,~~\forall \omega\in \Omega_\mathcal{U}(\mathcal{D})~~.
\end{equation}
\subsection{Spinor parameterization and $G_2$ structure on the non-chiral locus}
\paragraph{Proposition \cite{g2}.}
Relations \eqref{Esquare} are equivalent on $\mathcal{U}$ with the following conditions:
\begin{equation}
\label{fsol}
V^2|_\mathcal{U}=1-b^2~~,~~Y|_\mathcal{U}=(1+b\nu)|_\mathcal{U} \psi~~,~~Z|_\mathcal{U}=V|_\mathcal{U} \psi~~,
\end{equation}
where $\psi\in \Omega^4_\mathcal{U}(\mathcal{D})$ is the canonically normalized
coassociative form of a $G_2$ structure on the Frobenius distribution $\mathcal{D}|_\mathcal{U}$
which is compatible with the metric $g|_\mathcal{D}$ induced by $g$ and with the
orientation of $\mathcal{D}|_\mathcal{U}$.
\
\noindent Let $\varphi\stackrel{{\rm def.}}{=} \ast_\perp \psi\in \Omega^3_\mathcal{U}(\mathcal{D})$ be
the associative form of the $G_2$ structure on $\mathcal{D}|_\mathcal{U}$ mentioned in the
proposition. We have \cite{g2}:
\begin{eqnarray}
\psi&=&\frac{1}{1-b^2} VZ=\frac{1}{1-b^2}(1-b\nu)Y\in \Omega^4_\mathcal{U}(\mathcal{D})~~, \label{psidef}\\
\varphi&=&\frac{1}{||V||}\ast Z=\frac{1}{\sqrt{1-b^2}} Z\nu \in \Omega^3_\mathcal{U}(\mathcal{D})~~.\label{phidef}
\end{eqnarray}
On the non-chiral locus, one can parameterize $\check{E}$ as \cite{g2}:
\begin{equation}
\label{Enr}
\check{E}|_\mathcal{U}=\frac{1}{16}(1+V+b\nu)(1+\psi)=P|_\mathcal{U}\Pi~~,
\end{equation}
where:
\begin{equation*}
P\stackrel{{\rm def.}}{=}\frac{1}{2}(1+V+b\nu)\in \Omega(M)~~,~~\Pi\stackrel{{\rm def.}}{=}
\frac{1}{8}(1+\psi)\in \Omega_\mathcal{U}(\mathcal{D})
\end{equation*}
and where $P|_\mathcal{U}$ and $\Pi$ are commuting idempotents in the K\"{a}hler-Atiyah~
algebra of $\mathcal{U}$. Notice the relations:
\begin{equation}
\varphi=\ast_\perp\psi=\ast ({\hat V}\wedge \psi)~~,~~
\ast\varphi=-{\hat V}\wedge \psi~~,~~\ast\psi={\hat V}\wedge \varphi~~
\end{equation}
and:
\begin{equation}
\label{phipsirels}
V\varphi=-\varphi V =V\wedge \varphi~~,~~V\psi=\psi V=V\wedge \psi~~.
\end{equation}
\paragraph{The selfdual and anti-selfdual parts of $\psi$.}
We have:
\begin{equation}
\label{psipm}
\psi^\pm=\frac{1}{2}(\psi\pm \ast \psi)=\frac{1}{2}(\psi\pm {\hat V}\wedge \varphi)\in \Omega(\mathcal{U})~~.
\end{equation}
\paragraph{Lemma.} The four-forms $\psi^\pm\in \Omega(\mathcal{U})$ satisfy the relations:
\begin{eqnarray}
&&{\hat V}\psi^\pm{\hat V}=\psi^\mp~~,\label{psipmrel}~\\
&&\psi^+\psi^-=\psi^-\psi^+=0~~,\label{psipmprod}~\\
&& \psi^\pm=\frac{Y^\pm}{1\pm b}|_\mathcal{U}~~,\label{psiY}~\\
&& ||\psi^+||^2=||\psi^-||^2=\frac{7}{2}~~.\label{psipmnorms}~
\end{eqnarray}
\paragraph{Proof.}
Using $\psi^\pm=R^\pm \psi$, relation \eqref{psipmprod} follows
immediately from the fact that $\nu$ commutes with $\psi$. The last
relation in \eqref{phipsirels} gives:
\begin{equation}
\label{psirel}
{\hat V}\psi{\hat V}=\psi~~\mathrm{on}~~\mathcal{U}~~.
\end{equation}
Using the fact that ${\hat V}$ and $\nu$ anti-commute in the K\"{a}hler-Atiyah~
algebra while $\psi$ and $\nu$ commute (because $\nu$ is twisted
central), relation \eqref{psirel} implies \eqref{psipmrel}.
Separating $Y$ into its selfdual and anti-selfdual parts and using
the fact that $\nu Y=Y\nu=\ast Y$, the last equality in \eqref{psidef}
implies \eqref{psiY}, which implies \eqref{psipmnorms} when combined
with the first relation in \eqref{Ynorms}. $\blacksquare$
\paragraph{Proposition.}
The inhomogeneous differential forms:
\begin{equation*}
\Pi^\pm\stackrel{{\rm def.}}{=} R^\pm|_\mathcal{U}\Pi=\Pi R^\pm|_\mathcal{U}=\frac{1}{8}(R^\pm|_\mathcal{U}+\psi^\pm)=\frac{1}{16}(1\pm \nu|_\mathcal{U}+2\psi^\pm)\in \Omega(\mathcal{U})
\end{equation*}
satisfy $\Pi=\Pi^++\Pi^-$ and ${\hat V}\Pi^\pm{\hat V}=\Pi^\mp$ and are orthogonal
idempotents in the K\"{a}hler-Atiyah~ algebra of $\mathcal{U}$:
\begin{equation*}
(\Pi^\pm)^2=\Pi^\pm~~,~~\Pi^\pm\Pi^\mp=0~~.
\end{equation*}
Furthermore, we have:
\begin{equation}
\label{EpmDec}
\check{E}^\pm|_{\mathcal{U}}=P|_\mathcal{U}\Pi^\pm~~.
\end{equation}
Notice that $\Pi^\pm$ are twisted (anti-)selfdual:
\begin{equation*}
\Pi^\pm \nu=\pm \Pi^\pm~~.
\end{equation*}
\noindent{\bf Proof.} Notice that $\psi$ and $R^\pm$ commute since
$\psi$ and $\nu$ commute. The conclusion now follows immediately using
the properties of $\Pi$ and $R^\pm$.
$\blacksquare$
\subsection{Spinor parameterization and $\mathrm{Spin}(7)_\pm$ structures on the loci $\mathcal{U}^\pm$}
\label{subsec:L}
\paragraph{Extending $\psi^\pm$ to $\mathcal{U}^\pm$.}
Notice that $P\in \Omega(M)$ is globally defined on $M$ while $\Pi\in
\Omega(\mathcal{U})$ is only defined on the non-chiral locus.
\paragraph{Proposition.}
The four-form $\psi^\pm$ has a continuous extension to the locus
$\mathcal{U}^\pm$, which we denote through
$\bar{\psi}^\pm\in\Omega^4(\mathcal{U}^\pm)$. Namely:
\begin{equation*}
\bar{\psi}^\pm\stackrel{{\rm def.}}{=} \frac{1}{1\pm b}(Y^\pm|_{\mathcal{U}^\pm})\in \Omega^4(\mathcal{U}^\pm)~~.
\end{equation*}
Furthermore, the idempotents $\Pi^\pm\in \Omega(\mathcal{U})$ have continuous
extensions to idempotents $\bar{\Pi}^\pm\in \Omega(\mathcal{U}^\pm)$, which
are given by:
\begin{equation}
\label{barPi}
{\bar \Pi}^\pm\stackrel{{\rm def.}}{=} \frac{1}{8}(R^\pm|_{\mathcal{U}^\pm}+\bar{\psi}^\pm)=\frac{1}{16}(1+2\psi^\pm\pm \nu)\in \Omega(\mathcal{U}^\pm)~~
\end{equation}
and which are twisted (anti-)selfdual:
\begin{equation*}
{\bar \Pi}^\pm R^\pm|_{\mathcal{U}^\pm}={\bar \Pi}^\pm~~,~~{\bar \Pi}^\pm R^\mp|_{\mathcal{U}^\pm}=0~~.
\end{equation*}
\paragraph{Remarks.}
\begin{enumerate}
\itemsep 0.0em
\item Notice that \eqref{psiY} does not provide any information about
the limit of $\psi^\mp$ along $\mathcal{W}^\pm$, so $\psi^\mp$ (and hence
also $\Pi^\mp$) will not generally have an extension to
$\mathcal{U}^\pm$. However, \eqref{psipmnorms} tells us that $\psi^\mp$ is
bounded on $M$. In particular, we have:
\begin{equation}
\label{limpsi}
\lim_{b\rightarrow \pm 1}(V\psi^\mp)= \lim_{b\rightarrow \pm 1}(\psi^\mp V)=0~~.
\end{equation}
\item On the locus $\mathcal{W}^\pm$ we have:
\begin{equation}
\label{WpmRes}
b|_{\mathcal{W}^\pm}=\pm 1~~,~~V|_{\mathcal{W}^\pm}=Z|_{\mathcal{W}^\pm}=Y^\mp|_{\mathcal{W}^\pm}=0~~,
\end{equation}
where the last relations follow from the last equation in
\eqref{SolMS} and from \eqref{psiY}. The remaining conditions in
\eqref{SolMS} are automatically satisfied.
\item Notice the relation:
\begin{equation*}
Y^\pm|_{\mathcal{W}^\pm}=2{\bar \psi}^\pm|_{\mathcal{W}^\pm}~~,
\end{equation*}
which follows from the fact that $b|_{\mathcal{W}^\pm}=\pm 1$.
\end{enumerate}
\noindent{\bf Proof.} Since $Y^\pm\in \Omega(M)$ is well-defined on
$M$, the conclusion follows immediately from relation \eqref{psiY} and
from the fact that $1\pm b$ does not vanish on $\mathcal{U}^\pm$. The
relations satisfied by $\bar{\Pi}^\pm$ on $\mathcal{U}^\pm$ follow by
continuity from the similar relations satisfied by $\Pi^\pm$ on $\mathcal{U}$.
$\blacksquare$
\
\noindent While $\Pi^\mp$ does not generally have an extension to
$\mathcal{W}^\pm$, the product $P\Pi^\mp$ has zero limit on $\mathcal{W}^\pm$:
\paragraph{Proposition.}
We have $P|_{\mathcal{W}^\pm}=R^\pm$ as well as:
\begin{equation}
\label{EW}
\exists \lim_{b\rightarrow \pm 1} P\Pi^\mp=\check{E}^\mp|_{\mathcal{W}^\pm}=0~~,~~\check{E}^\pm|_{\mathcal{W}^\pm}={\bar \Pi}^\pm|_{\mathcal{W}^\pm}=\frac{1}{8}(R^\pm+\bar{\psi}^\pm)|_{\mathcal{W}^\pm}=\frac{1}{16}(1\pm \nu +2\bar{\psi}^\pm)|_{\mathcal{W}^\pm}~~.
\end{equation}
\noindent{\bf Proof.}
The relation $P|_{\mathcal{W}^\pm}=R^\pm$ is obvious. The other statements
follow from \eqref{Epm} and \eqref{EpmDec} using
\eqref{WpmRes}. $\blacksquare$
\paragraph{The $\mathrm{Spin}(7)_\pm$ structures on $\mathcal{U}^\pm$.}
\paragraph{Lemma.}
Let $(e^a)_{a=1\ldots 8}$ be a local coframe defined over an open
subset $U\subset M$ and let $\eta\in \Gamma(U,S)$. Then:
\begin{equation*}
\Scr B(\gamma^a\eta,\gamma^b\eta)=g^{ab}||\eta||^2~~,
\end{equation*}
where $\gamma^a=\gamma(e^a)$ and $g^{ab}=\langle e^a,e^b\rangle$.
\
\noindent{\bf Proof.}
Using the property $(\gamma^a)^t=\gamma^a$ and the fact that
$(\gamma^a\gamma^b)^t=\gamma^b\gamma^a$, compute:
\begin{equation*}
\Scr B(\gamma^a\eta,\gamma^b\eta)=\Scr B(\eta,\gamma^a\gamma^b\eta)=
\Scr B(\eta,\gamma^b\gamma^a\eta)=\frac{1}{2}\Scr B(\eta,\{\gamma^a,\gamma^b\}\eta)=g^{ab}\Scr B(\eta,\eta)=g^{ab}||\eta||^2~~.
\end{equation*}
$\blacksquare$
\
\noindent When $\eta$ is non-vanishing everywhere on $U$, the
proposition implies that the spinors $\gamma^a\eta$ form a
linearly-independent set of sections of $S$ above $U$. Taking $\eta$
to have chirality $\pm 1$ and recalling that $\gamma^a$ map $S^\pm$
into $S^\mp$ and that ${\rm rk} S^+={\rm rk} S^-=8$, this gives:
\paragraph{Corollary.} Let $(e^a)_{a=1\ldots 8}$ be a local orthonormal coframe defined over an open subset
$U\subset M$ and $\eta\in \Gamma(U,S^\pm)$ be a spinor of chirality
$\pm 1$ which is nowhere vanishing on $U$. Then
$(\gamma^a\eta)_{a=1\ldots 8}$ is a $\Scr B$-orthogonal local frame of
$S^\mp$ above $U$. Every local section $\xi\in \Gamma(U,S^\mp)$
expands in this frame as:
\begin{equation*}
\xi=\frac{1}{||\eta||^2}\sum_{a=1}^8 \Scr B(\xi,\gamma_a\eta) \gamma^a\eta~~.
\end{equation*}
\paragraph{Proposition.} Let $U$ be an open subset of $M$ which supports
an orthonormal coframe $e^a$ of $(M,g)$. Then:
\
\noindent 1. If $\xi^+$ is everywhere non-vanishing on $U$, then
$\xi^-$ expands above $U$ as $\xi^-= \sum_{a=1}^8
L^+_a\gamma^a\xi^+=\gamma(L^+)\xi^+$, where $L^+_a$ are the
coefficients of the one-form $L^+=L_a^+\mathrm{d} x^a=\frac{1}{1+b}V$.
\
\noindent 2. If $\xi^-$ is everywhere non-vanishing on $U$, then
$\xi^+$ expands above $U$ as $\xi^+=\sum_{a=1}^8
L^-_a\gamma^a\xi^-=\gamma(L^-)\xi^-$, where $L^-_a$ are the
coefficients of the one-form $L^-=L_a^-\mathrm{d} x^a=\frac{1}{1-b}V$.
\
\noindent {\bf Proof.} Assume that $\xi^+$ (respectively $\xi^-$)
vanishes nowhere on $U$. The corollary shows that $\xi^\mp$ expands
as $\xi^\mp=\sum_{a=1}^8 L^\pm_a\gamma^a\xi^\pm$ where:
\begin{equation}
\label{Ld}
L^\pm_a=\frac{1}{||\xi^\pm||^2}\Scr B(\xi^\mp,\gamma_a\xi^\pm)~~.
\end{equation}
Recalling that $S^+$ and $S^-$ are $\Scr B$-orthonormal while $\gamma^a$
are $\Scr B$-symmetric, we find:
\begin{equation*}
\Scr B(\xi^+,\gamma_a\xi^-)=\Scr B(\xi^-,\gamma_a\xi^+)=\frac{1}{2}\Scr B(\xi,\gamma_a\xi)=\frac{1}{2}V_a~~.
\end{equation*}
Using this and \eqref{xipmnorms}, equation \eqref{Ld} becomes
$L^\pm_a=\frac{1}{1\pm b} V_a$.~~$\blacksquare$
\paragraph{Remarks.}
\begin{enumerate}
\itemsep 0.0em
\item The ``+'' case of \eqref{Ld} was used in \cite{Tsimpis}, where
no explicit expression for $L^+$ (which is denoted by $L$ in
loc. cit.) was given\footnote{Notice that $L^+$ is not a quadratic
function of $\xi$, since it involves the denominator $1+b$ and
thus it is not homogeneous under rescalings $\xi\rightarrow
\lambda\xi$ with $\lambda\neq 0$.}.
\item Notice that $L^+$ and $L^-$ are not independent (they are
proportional to each other) and that each of them contains the same
information as $V$ and $b$.
\end{enumerate}
\noindent Recalling \eqref{xipmnorms}, consider the unit norm spinors (of chirality $\pm 1$):
\begin{equation}
\label{etadef}
\boxed{\eta^\pm=\sqrt{1+||L^\pm||^2}\xi^\pm=\sqrt{\frac{2}{1\pm b}}\xi^\pm\in \Gamma(\mathcal{U}^\pm,S^\pm)}~.
\end{equation}
Using the fact that $||\eta^\pm||=1$ while $\Scr B(\eta^\pm,\gamma_{a_1\ldots a_k}\eta^\pm)$ vanishes unless $k\equiv_4 0$, we find:
\begin{equation}
\label{Eeta}
\check{E}_{\eta^\pm,\eta^\pm}=\frac{1}{16}(1+\Phi^\pm\pm \nu)\in \Omega(\mathcal{U}^\pm)~~,
\end{equation}
where:
\begin{equation}
\label{PhiDef}
\Phi^\pm\stackrel{{\rm def.}}{=} \frac{1}{4!}\Scr B(\eta^\pm,\gamma_{a_1\ldots a_4}\eta^\pm)e^{a_1\ldots a_4}
=\boldsymbol{\check{E}}^{(4)}_{\eta^\pm,\eta^\pm}=\frac{2}{1\pm b}\boldsymbol{\check{E}}^{(4)}_{\xi^\pm,\xi^\pm}\in \Omega^4(\mathcal{U}^\pm)~~
\end{equation}
and where we noticed that $\Scr B(\eta^\pm,\gamma(\nu)\eta^\pm)=\pm 1$.
\paragraph{Proposition.}
The four-form $\Phi^+$ is selfdual while the four-form $\Phi^-$ is
anti-selfdual. They satisfy the following relations on the locus $\mathcal{U}^\pm$:
\begin{equation}
\boxed{\Phi^\pm= 2\bar{\psi}^\pm}~.~\label{PhiT}\\
\end{equation}
In particular, the inhomogeneous form \eqref{Eeta} coincides with the
extension \eqref{barPi} of $\Pi^\pm$ to this locus:
\begin{equation*}
\check{E}_{\eta^\pm,\eta^\pm}=\bar{\Pi}^\pm
\end{equation*}
and we have:
\begin{equation}
||\Phi^\pm||^2 = 14~~\label{PhiNorms}~~.
\end{equation}
Moreover, the restriction of $\Phi^+$ is the canonically-normalized
calibration defining a $\mathrm{Spin}(7)$ structure on the open submanifold
$\mathcal{U}$ of $M$ while the restriction of $\Phi^-$ is the
canonically-normalized calibration defining a $\mathrm{Spin}(7)$ structure on
the orientation reversal of $\mathcal{U}$.
\paragraph{Proof.}
Recalling that $\xi^\pm=\frac{1}{2}(1\pm \gamma(\nu))\xi$, the
identities $\check{E}_{\xi,\gamma(\nu)\xi}=\check{E}_{\xi,\xi}\nu$ and
$\check{E}_{\gamma(\nu)\xi,\xi}=\nu \check{E}_{\xi,\xi}$ of \cite{ga1}
and the fact that $\nu$ is involutive and twisted central give:
\begin{equation*}
\check{E}_{\xi^\pm,\xi^\pm}=\frac{1}{4}(\check{E}_{\xi,\xi}\pm \nu \check{E}_{\xi,\xi}\pm \check{E}_{\xi,\xi}\nu+\nu\check{E}_{\xi,\xi}\nu)=
\frac{1}{4}(\check{E}_{\xi,\xi}+\pi(\check{E}_{\xi,\xi}))(1\pm \nu)=\frac{1}{2}\check{E}_{\xi,\xi}^\mathrm{ev}(1\pm\nu)=
\frac{1}{2}(\check{E}^\mathrm{ev}_{\xi,\xi}\pm \ast \tau(\check{E}_{\xi,\xi}^\mathrm{ev}))~~
\end{equation*}
Since the Hodge operator preserves $\Omega^4(M)$ and since the
reversion $\tau$ of the K\"{a}hler-Atiyah~ algebra restricts to the identity on the
space of four-forms, this implies:
\begin{equation*}
\boldsymbol{\check{E}}^{(4)}_{\xi^\pm,\xi^\pm}=\frac{1}{2}(\boldsymbol{\check{E}}^{(4)}_{\xi,\xi}\pm \ast \boldsymbol{\check{E}}^{(4)}_{\xi,\xi})=\frac{1}{2}(Y\pm \ast Y)=Y^\pm~~,
\end{equation*}
where the superscript $\pm$ indicates the selfdual/anti-selfdual part. Substituting this
into \eqref{PhiDef} gives relation \eqref{PhiT}. The statements of the
proposition regarding the restrictions of $\Phi^\pm$ to the open
submanifold $\mathcal{U}$ follow from the fact that $\eta_\pm$ is a
Majorana-Weyl spinor of norm one and of chirality $\pm 1$; it is
well-known \cite{Joyce} that giving such a spinor on an eight-manifold
$\mathcal{U}$ induces $\mathrm{Spin}(7)$ structures on the underlying manifold or on
its orientation reversal, whose normalized calibrations are given by
\eqref{PhiDef}. In particular, \eqref{PhiNorms} holds on $\mathcal{U}$ since
there it amounts to the condition that $\Phi^\pm$ are canonically
normalized. By continuity, this implies that \eqref{PhiNorms} also
holds on $\mathcal{W}^\pm$. $\blacksquare$
\paragraph{Remarks.}
\begin{enumerate}
\itemsep 0.0em
\item The proposition implies that the following relation holds on the non-chiral locus:
\begin{equation*}
\check{E}_{\xi,\xi}|_\mathcal{U}=P|_{\mathcal{U}}(\check{E}_{\eta^+,\eta^+}+\check{E}_{\eta^-,\eta^-})~~.
\end{equation*}
This shows how the idempotent $\check{E}_{\xi,\xi}|_{\mathcal{U}}$ which
characterizes the normalized Majorana spinor $\xi$ on the locus $\mathcal{U}$
relates to the two idempotents
$\check{E}_{\eta^\pm,\eta^\pm}|_\mathcal{U}=\Pi^\pm$ which characterize the
Majorana-Weyl spinors $\eta^\pm$ and which encode the
$\mathrm{Spin}(7)_\pm$ structures through the K\"{a}hler-Atiyah~ algebra. While $\check{E}_{\eta^+,\eta^+}$
depends only on the positive chirality spinor $\eta^+$ and
$\check{E}_{\eta^-,\eta^-}$ depends only on the negative chirality
spinor $\eta^-$, the idempotent $P$ contains the quantities $b$ and
$V$, each of which involves both chirality components of the spinor
$\xi$:
\begin{equation*}
b=||\xi^+||^2-||\xi^-||^2~~,~~V=2 \Scr B(\xi^+,\gamma_m\xi^-)e^m=(1-b^2)\Scr B(\eta^+,\gamma_m\eta^-)e^m~~.
\end{equation*}
The object $P$ encodes in the K\"{a}hler-Atiyah~ algebra the $\mathrm{SO}(7)$ structure which
corresponds to the distribution $\mathcal{D}$ on $\mathcal{U}$. Finally, notice that the idempotent $\Pi$ encodes
the $G_2$ structure along the distribution $\mathcal{D}$. Notice that $P$ and $\Pi$ commute, while $P$ and
$\Pi_\pm$ do not commute.
\item Equation \eqref{PhiT} implies that $\Phi^\pm$ coincides with
$\pm Y^\pm$ on the locus $\mathcal{W}^\pm$ since $b=\pm 1$ there. Notice
that \eqref{PhiNorms} agrees via \eqref{PhiT} with the last
equations in \eqref{SolMS}.
\end{enumerate}
\paragraph{Spinor parameterization on the loci $\mathcal{U}^\pm$.}
On the locus $\mathcal{U}$, relations \eqref{fsol} and \eqref{PhiT} give:
\begin{equation}
\label{YZT}
\boxed{
\begin{split}
&Z|_\mathcal{U}=\frac{1}{2}V(\Phi^++\Phi^-)~~,\\
&Y|_\mathcal{U}=\frac{1}{2}[(1+b)\Phi^++(1-b)\Phi^-]~~.
\end{split}}
\end{equation}
In these relations, $\Phi^+$
and $\Phi^-$ are not independent but related through:
\begin{equation*}
\Phi^\mp={\hat V}\Phi^\pm{\hat V}
\end{equation*}
as a consequence of \eqref{psipmrel}. Hence on the non-chiral locus we
can eliminate $\Phi^\mp$ in terms of $\Phi^\pm$ to obtain the
following non-redundant parameterizations:
\begin{equation*}
Z|_\mathcal{U}=\frac{1}{2}\sqrt{1-b^2}({\hat V}\Phi^\pm+\Phi^\pm {\hat V})~~,~~
Y|_\mathcal{U}=\frac{1}{2}\left[(1\pm b)\Phi^\pm +(1\mp b){\hat V}\Phi^\pm{\hat V}\right]~~,
\end{equation*}
which give:
\begin{equation*}
16 \check{E}|_\mathcal{U}=P|_{\mathcal{U}}(\Pi_\pm+{\hat V}\Pi_\pm {\hat V})=1+V+\frac{1}{2}\left[(1\pm b)\Phi^\pm+(1\mp b){\hat V}\Phi^\pm {\hat V}\right]+
\frac{1}{2} \sqrt{1-b^2}({\hat V} \Phi^\pm+\Phi^\pm {\hat V})+b\nu~~.
\end{equation*}
This imply the following parameterizations on the loci $\mathcal{U}^\pm$:
\begin{equation*}
\boxed{16 \check{E}|_{\mathcal{U}^\pm}=1+V+\frac{1}{2}\left[(1\pm b)\Phi^\pm+\frac{1}{1\pm b}V\Phi^\pm V\right]+
\frac{1}{2} (V \Phi^\pm+\Phi^\pm V)+b\nu }~~,
\end{equation*}
where it is understood that (see \eqref{limpsi}):
\begin{equation*}
\lim_{b\rightarrow \pm 1}{V\Phi^\mp}=\lim_{b\rightarrow \pm 1}{\Phi^\mp V}=0~~
\end{equation*}
and hence (see \eqref{EW}):
\begin{equation*}
16 \check{E}|_{\mathcal{W}^\pm}={\bar \Pi}^\pm|_{\mathcal{W}^\pm}=\frac{1}{16}(1+\Phi^\pm \pm \nu)|_{\mathcal{W}^\pm}~~.
\end{equation*}
Up to expressing $V$ and $b$ through $L^\pm$, this is the parameterization which corresponds to the
approach of \cite{Tsimpis}.
\subsection{Comparing spinors and G structures on the non-chiral locus}
Equation \eqref{psipm} gives:
\begin{equation*}
\Phi^\pm|_\mathcal{U}=2\psi^\pm=\psi\pm {\hat V}\wedge \varphi~~,
\end{equation*}
i.e.:
\begin{equation}
\label{PhiDec}
\boxed{(\Phi^\pm|_\mathcal{U})_\top=\pm \varphi~~,~~(\Phi^\pm|_\mathcal{U})_\perp=\psi}~~.
\end{equation}
The relation $\xi^\mp=\gamma(L^\pm)\xi^\pm$ gives
$\eta^\mp=\gamma({\hat V})\eta^\pm$, which shows that the everywhere
normalized spinor:
\begin{equation}
\label{eta0}
\eta_0\stackrel{{\rm def.}}{=} \frac{1}{\sqrt{2}}(\eta^++\eta^-)\in \Gamma(\mathcal{U},S)
\end{equation}
is a Majorana spinor along $\mathcal{D}$ in the seven-dimensional sense,
i.e. we have $D(\eta_0)=\eta_0$ where $D\stackrel{{\rm def.}}{=} \gamma({\hat V})$ is
the real structure of $S$, when the latter is viewed as a complex
spinor bundle over $\mathcal{D}$ (see \cite{g2}). The identity
$\boldsymbol{\check{E}}^{(4)}_{\eta^\pm,\eta^\mp}=0$ implies the following
spinorial expression for $\psi$:
\begin{equation}
\label{psieta}
\psi=\boldsymbol{\check{E}}^{(4)}_{\eta_0,\eta_0}=\frac{1}{4!}\Scr B(\eta_0,\gamma_{a_1\ldots a_4}\eta_0)e^{a_1\ldots a_4}~~.
\end{equation}
The relation $\xi^\mp=\gamma(L^\pm)\xi^\pm$ gives $\eta^\mp=\gamma({\hat V})\eta^\pm$, which implies:
\begin{equation*}
\eta_0=\frac{1}{\sqrt{2}}(\mathrm{id}_S+\gamma({\hat V}))\eta^+= \frac{1}{\sqrt{2}}(\mathrm{id}_S+\gamma({\hat V}))\eta^-~~.
\end{equation*}
Notice that $\frac{1}{2}(\mathrm{id}_S+\gamma({\hat V}))$ is an idempotent
endomorphism of $S$. As explained in \cite{g2}, the spinor $\eta_0$
induces the $G_2$ structure of the distribution $\mathcal{D}$. The situation
is summarized in Table \ref{table:G}.
\!\!\!\!\!\!\!\begin{table}[tt]
\centering
\!\!\!\!\!\!\!\begin{tabular}{|c|c|c|c|c|}
\hline
G structure & $\mathrm{Spin}(7)_+$ & $\mathrm{Spin}(7)_{-}$ & $G_2$ (on $\mathcal{D}|_\mathcal{U}$) & $\mathrm{SO}(7)$ ($\mathcal{D}|_\mathcal{U}$)\\
\hline\hline
spinor & $\eta^+$ & $\eta^-$ & $\eta_0=\frac{1}{\sqrt{2}}(\eta^+\!\!+\eta^-)$ & ---\\
\hline
idempotent & \!\!$\Pi^+\!\!=\!\!\frac{1}{16}(1+\Phi^+\!\!+\nu)$ & \!\!$\Pi^-\!\!=\!\!\frac{1}{16}(1+\Phi^-\!\!-\nu)$ &
\!\!$\Pi=\Pi^+\!\!+\Pi^-\!\!=\frac{1}{8}(1+\psi)$& \!\!$P=\frac{1}{2}(1+V+b\nu)$ \\
\hline
forms & $\Phi^+=2\psi^+$ & $\Phi^-=2\psi^-$ & $\varphi$ and $\psi=\ast_\perp\varphi$ & $b$ and $V$ \\
\hline
extends to & $\mathcal{U}^+$ & $\mathcal{U}^-$ & $\mathcal{U}$ & $\mathcal{U}$ \\
\hline
\end{tabular}
\
\caption{Summary of various $G$ structures and of their reflections in the K\"{a}hler-Atiyah~ algebra.}
\label{table:G}
\end{table}~~~~~~~~~~~~~~
\paragraph{Remarks.}
\begin{enumerate}
\itemsep 0.0em
\item None of the $G$ structures in Table \ref{table:G} extends to $M$. In
fact, the structure group $\mathrm{SO}(8)$ of the frame bundle of $M$ {\em does not
globally reduce, in general, to any proper subgroup}. As pointed out in
\cite{Tsimpis}, this is due to the fact that the action of $\mathrm{Spin}(8)$
on the fibers $S_p\simeq \mathbb{R}^{16}$ of $S$ (which is the action of
$\mathrm{Spin}(8)$ on the direct sum ${\bf 8}_s\oplus {\bf 8}_c$ of the
positive and negative chirality spin $1/2$ representations) is not
transitive when restricted to the unit sphere $S^{15}\subset
\mathbb{R}^{16}$. As shown in loc. cit, one can in some sense ``cure'' this problem by
considering the manifold ${\hat M}\stackrel{{\rm def.}}{=} M\times S^1$, using the fact that $\mathrm{Spin}(9)$
acts transitively on $S^{15}$. However, such an approach does not
immediately provide useful information on the geometry of $M$, in
particular the geometry of the singular foliation $\bar{\mathcal{F}}$
discussed in the next subsection is not immediately visible in that
approach. It was also shown in loc. cit. that one can repackage the
information contained in the $\mathrm{Spin}(7)_\pm$ structures into a
generalized $\mathrm{Spin}(7)$ structure on ${\hat M}$ in the sense of
\cite{Witt}. In particular, it is easy to check that relations (4.8)
of \cite{Tsimpis} are equivalent with some of the exterior
differential constraints which can be obtained by expanding equation
(3.5) of \cite{g2} into its rank components --- exterior differential
constraints which were discussed at length in \cite{ga1} and in the
appendix of \cite{g2}. As shown in detail in \cite{g2}, those exterior
differential constraints do not suffice to encode the full supersymmetry
conditions for such backgrounds.
\item The fact that the structure group of $TM$ does not globally
reduce beyond $\mathrm{SO}(8)$ in this class of examples illustrates some
limits of the philosophy that flux compactifications can be
described using reductions of structure group. That philosophy is
based on the observation that a collection of (s)pinors defines a
{\em local} reduction of structure group over any open subset of the
compactification manifold $M$ along which the stabilizer of the
pointwise values of those spinors is fixed up to conjugacy in the
corresponding $\mathrm{Spin}$ or $\mathrm{Pin}$ group. However, such a reduction
does {\em not} generally hold globally on $M$, since the local
reductions thus obtained can ``jump'' --- in our class of examples,
the jump occurs at the points of the chiral locus $\mathcal{W}$. The
appropriate notion is instead that of {\em generalized} reduction of
structure group, of which the class of compactifications considered
here is an example. In this respect, we mention that the cosmooth
generalized distribution $\mathcal{D}$ can be viewed as providing a
generalized reduction of structure group of $M$, which is an
ordinary reduction from $\mathrm{SO}(8)$ to $\mathrm{SO}(7)$ only when restricted to
its regular subset $\mathcal{U}$, on which $\mathcal{D}|_\mathcal{U}$ provides \cite{g2} an
almost product structure. We also mention that the conditions
imposed by supersymmetry can be formulated globally by using an
extension of the language of Haefliger structures (see Section
\ref{subsec:Haefliger}), an approach which can in fact be used to
give a fully general approach to flux compactifications. It is such
concepts, rather than the classical concept of $G$ structures
\cite{Chern}, which provide the language appropriate for giving
{\em globally valid} descriptions of the most general flux
compactifications.
\end{enumerate}
\subsection{The singular foliation of $M$ defined by $\mathcal{D}$}
\label{subsec:Haefliger}
As in \cite{g2}, one can show that the one-form:
\begin{equation*}
{\boldsymbol{\omega}}\stackrel{{\rm def.}}{=} 4\kappa e^{3\Delta} V~~
\end{equation*}
satisfies the following relations which hold globally on $M$ as a
consequence of the supersymmetry conditions \eqref{par_eq}:
\begin{eqnarray}
\label{meq}
\mathrm{d}{\boldsymbol{\omega}}&=&0~~,~~\\
{\boldsymbol{\omega}}~&=&\mathbf{f}-\mathrm{d}\mathbf{b}~~,~~\mathrm{where}~~\mathbf{b}\stackrel{{\rm def.}}{=} e^{3\Delta}b~~.\nonumber
\end{eqnarray}
As a result of the first equation, the generalized distribution
$\mathcal{D}=\mathrm{ker} V=\mathrm{ker}{\boldsymbol{\omega}}$ determines a singular foliation $\bar{\mathcal{F}}$
of $M$, which degenerates along the chiral locus $\mathcal{W}$, since that locus
coincides with the set of zeroes of ${\boldsymbol{\omega}}$. The second
equation implies that ${\boldsymbol{\omega}}$ belongs to the cohomology class $\mathfrak{f}\in
H^1(M,\mathbb{R})$ of $\mathbf{f}$.
Since $\mathcal{D}$ is cosmooth rather than smooth, the notion of singular
foliation which is appropriate in our case\footnote{Notice that this
is not the notion of singular foliation considered in
\cite{Androulidakis1,Androulidakis2}, which is instead based on
Stefan-Sussmann (i.e. smooth, rather than cosmooth) distributions.}
is that of Haefliger structure \cite{Haefliger}. More precisely,
$\bar{\mathcal{F}}$ can be described as the Haefliger structure defined as
follows. Consider an open cover $(U_\alpha)_{\alpha\in I}$ of $M$ such
that each $U_\alpha$ is simply-connected and let ${\boldsymbol{\omega}}_\alpha\stackrel{{\rm def.}}{=}
{\boldsymbol{\omega}}|_{U_\alpha}\in \Omega^1(U_\alpha)$. We have
${\boldsymbol{\omega}}_\alpha=\mathrm{d} {\mathbf h}_\alpha$ for some ${\mathbf h}_\alpha\in
\Omega^0(U_\alpha)$, where ${\mathbf h}_\alpha$ are determined up to shifts:
\begin{equation}
\label{hshifts}
{\mathbf h}_\alpha\rightarrow \mathbf{h}'_\alpha+{\mathbf c}_\alpha~~,~~{\mathbf c}_\alpha\in \mathbb{R}~~.
\end{equation}
For any $\alpha,\beta\in I$ and any $p\in U_\alpha\cap U_\beta$,
consider the orientation-preserving diffeomorphism
$\boldsymbol{\phi}_{\alpha\beta}(p)\in \mathrm{Diff}_+(\mathbb{R})$ of the real line
given by the translation:
\begin{equation*}
\boldsymbol{\phi}_{\alpha\beta}(p)(x)\stackrel{{\rm def.}}{=} x+{\mathbf h}_\beta(p)-{\mathbf h}_\alpha(p)~~\forall x\in \mathbb{R}~~.
\end{equation*}
Then $\boldsymbol{\phi}_{\alpha\beta}(p)({\mathbf
h}_\alpha(p))={\mathbf h}_\beta(p)$. The germ
$\hat{\boldsymbol{\phi}}_{\alpha\beta}(p)$ of
$\boldsymbol{\phi}_{\alpha\beta}(p)$ at ${\mathbf h}_\alpha(p)$ is an
element of the Haefliger groupoid $\Gamma_1^\infty$ and
it is easy to check that
$\hat{\boldsymbol{\phi}}_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow
\Gamma_1^\infty$ is a Haefliger cocycle on $M$:
\begin{equation*}
\hat{\boldsymbol{\phi}}_{\beta\gamma}(p)\circ \hat{\boldsymbol{\phi}}_{\alpha\beta}(p)=\hat{\boldsymbol{\phi}}_{\alpha\gamma}(p)~~~~\forall \alpha,\beta,\gamma\in I~~,~~\forall p\in U_\alpha\cap U_\beta\cap U_\gamma~~.
\end{equation*}
Moreover, the shifts \eqref{hshifts} correspond to transformations:
\begin{equation*}
\hat{\boldsymbol{\phi}}_{\alpha\beta}\rightarrow \hat{\boldsymbol{\phi}}'_{\alpha\beta}=\hat{{\mathbf q}}_\beta\circ \hat{\boldsymbol{\phi}}_{\alpha\beta}\circ \hat{{\mathbf q}}_\alpha^{-1}~~,
\end{equation*}
where $\hat{\mathbf{q}}_\alpha:U_\alpha\rightarrow \Gamma_1^\infty$ are defined by
declaring that $\hat{\mathbf{q}}_{\alpha}(p)$ is the germ
at $p\in U_\alpha$ of the orientation-preserving diffeomorphism
$\mathbf{t}_\alpha\in \mathrm{Diff}_+(\mathbb{R})$ given by the following translation
of the real line:
\begin{equation*}
\mathbf{t}_\alpha(x)=x+{\mathbf c}_\alpha~~\forall x\in \mathbb{R}~~.
\end{equation*}
It follows that the closed one-form ${\boldsymbol{\omega}}$ determines a
well-defined element of the non-Abelian cohomology $\in
H^1(M,\Gamma_1^\infty)$, which is the Haefliger structure defined by
${\boldsymbol{\omega}}$. The singular foliation $\bar{\mathcal{F}}$ which ``integrates''
$\mathcal{D}$ can be identified with this element.
The approach through Haefliger structures allows one to define
rigorously the singular foliation $\bar{\mathcal{F}}$ in the most general
case, i.e. without making any supplementary assumptions on the closed
one-form ${\boldsymbol{\omega}}$. In general, such singular foliations can be
extremely complicated and little is known about their topology and
geometry. However, the description of $\bar{\mathcal{F}}$ simplifies when
${\boldsymbol{\omega}}$ is a closed one-form of Morse or Bott-Morse type. In
Section \ref{sec:Morse}, we discuss the Morse case, recalling some
results which apply to $\bar{\mathcal{F}}$ in that situation.
\section{Relating the $G_2$ and $\mathrm{Spin}(7)$ approaches on the non-chiral locus}
\label{sec:G2Spin7}
On the non-chiral locus $\mathcal{U}$, we have the regular foliation $\mathcal{F}$
which is endowed with a longitudinal $G_2$ structure having
associative and coassociative forms $\varphi$ and $\psi$. We also have
a $\mathrm{Spin}(7)_+$ and a $\mathrm{Spin}(7)_-$ structure, which are determined
respectively by the calibrations $\Phi^\pm=2\psi^\pm=\psi\pm {\hat
V}\wedge \varphi$. Given this data, one can relate various
quantities determined by $(\mathcal{D},\varphi)$ to quantities determined by
$\Phi^\pm$ as we explain below. We stress that the results of this
subsection are independent of the supersymmetry conditions
\eqref{par_eq} and hence they hold in the general situation described
above. We mention that the relation between the type of $G_2$
structure induced on an oriented submanifold of a $\mathrm{Spin}(7)$ structure
manifold and the intrinsic geometry of such submanifolds was studied
in \cite{Gray,Cabrera}.
\subsection{The $\G_2$ and $\mathrm{Spin}(7)_\pm$ decompositions of $\Omega^4(\mathcal{U})$}
The group $G_2$ has a natural fiberwise rank-preserving action on the
graded vector bundle $\wedge (\mathcal{D}|_\mathcal{U})^\ast$, which is given at every $p\in
\mathcal{U}$ by the local embedding of $G_2$ as the stabilizer $G_{2,p}$ in
$\mathrm{SO}(\mathcal{D}_p)$ of the 3-form $\varphi_p\in \wedge^3(\mathcal{D}_p^\ast)$. Since
$\mathrm{SO}(\mathcal{D}_p)$ embeds into $\mathrm{SO}(T_pM)$ as the stabilizer of the 1-form
$V_p\in T_p^\ast M$, this induces a rank-preserving action of
$G_{2,p}$ on $\wedge T_p^\ast\mathcal{U}$ which can be described as
follows. Decomposing any form $\omega\in \wedge T_p^\ast \mathcal{U}$ as
$\omega=\omega_\perp+{\hat V}\wedge \omega_\top$, the action of an
element of $g$ of $G_2$ on $\omega$ is given by the simultaneous
action of $g$ on the components $\omega_\perp$ and $\omega_\top$, both
of which belong to $\wedge \mathcal{D}_p^\ast$. The corresponding
representation of $G_2$ at $p$ is equivalent with the direct sum of
the representations in which the components $\omega_\top$ and
$\omega_\perp$ transform at $p$. In particular, $F_{\perp,p}$ and
$F_{\top,p}$ transform in a $G_2$ representation which is equivalent
with the direct sum $\wedge^3 \mathcal{D}^\ast_p\oplus \wedge^4 \mathcal{D}^\ast_p$.
The group $\mathrm{Spin}(7)$ is embedded inside $\mathrm{SO}(T_pM)$
in two ways, namely as the stabilizers $\mathrm{Spin}(7)_{\pm,p}$ of the
selfdual 4-forms $\Phi^{\pm}_p$. Then \eqref{PhiT} shows that
$G_{2,p}$ is the stabilizer of $V_p$ in $\mathrm{Spin}(7)_{\pm,p}$. The action
of $G_{2,p}$ on $\wedge T^\ast_p M$ is obtained from that of
$\mathrm{Spin}(7)_{\pm,p}$ by restriction. Hence the irreducible components of
the action of $\mathrm{Spin}(7)_{\pm,p}$ on $\wedge^k(T^\ast_p M)$ decompose
as direct sums of the irreducible components of the action of
$G_{2,p}$ on the same space. We have the following decompositions into
irreps. (see, for example, \cite{Fernandez, Kthesis}):
\begin{eqnarray}
\label{irreps}
\wedge^4 T^\ast_p M &=& \wedge^4_{{\bf 1},\pm}T^\ast_p M\oplus \wedge^4_{{\bf 7},\pm}T^\ast_p M\oplus \wedge^4_{{\bf 27},\pm}T^\ast_p M\oplus \wedge^4_{{\bf 35},\pm}T^\ast_p M~~~\mathrm{for}~~\mathrm{Spin}(7)_{\pm,p}~~,\nonumber\\
\wedge^4 T^\ast_p M &=& \wedge^4_{1}T^\ast_p M\oplus \wedge^4_{7}T^\ast_p M\oplus \wedge^4_{27}T^\ast_p M~~~\mathrm{for}~~G_{2,p}~~,\\
\wedge^3 T^\ast_p M &=& \wedge^3_{1}T^\ast_p M\oplus \wedge^3_{7}T^\ast_p M\oplus \wedge^3_{27}T^\ast_p M~~~\mathrm{for}~~G_{2,p}~~,\nonumber
\end{eqnarray}
where the numbers used as lower indices indicate the dimension of the
corresponding irrep. The last two of these decompositions imply
similar decompositions into irreps. of $G_{2,p}$ for the spaces of
selfdual and anti-selfdual three- and four-forms:
\begin{equation}
(\wedge^4 T^\ast_p M)^\pm = \wedge^4_{1}T^\ast_p M\oplus \wedge^4_{7}T^\ast_p M\oplus \wedge^4_{27}T^\ast_p M~~~\mathrm{for}~~G_{2,p}~~.
\end{equation}
Furthermore, we have:
\begin{eqnarray}
\label{Spin7ASD}
\begin{split}
& (\wedge^4 T^\ast_p M)^\pm=\wedge^4_{{\bf 1},\pm}T^\ast_p M\oplus \wedge^4_{{\bf 7},\pm}T^\ast_p M\oplus \wedge^4_{{\bf 27},\pm}T^\ast_p M~~~\mathrm{for}~~\mathrm{Spin}(7)_{\pm,p}~~,\\
& (\wedge^4 T^\ast_p M)^\mp = \wedge^4_{{\bf 35},\pm}T^\ast_p M~~~\mathrm{for}~~\mathrm{Spin}(7)_{\pm,p}~~,
\end{split}
\end{eqnarray}
where the $\pm$ superscripts indicate the subspaces of selfdual and
anti-selfdual forms while the $\pm$ subscripts indicate which of the
$\mathrm{Spin}(7)_p$ subgroups of $\mathrm{SO}(T_pM)$ we consider. Comparing these
two decompositions, one sees immediately that the irreps of
$\mathrm{Spin}(7)_{\pm,p}$ appearing in \eqref{Spin7ASD} decompose as follows
under the $G_2$ action on $\wedge^4 T^\ast_p M$ which was discussed
above:
\begin{eqnarray}
\label{G2Spin7Irreps}
\boxed{
\begin{split}
&\wedge^4_{{\bf k},\pm}T^\ast_p M = \wedge^4_{k}T^\ast_p M~~,~~\mathrm{for}~~k=1,7,27~~\\
&\wedge^4_{{\bf 35},\pm}T^\ast_p M = \wedge^4_{1}T^\ast_p M\oplus \wedge^4_{7}T^\ast_p M\oplus \wedge^4_{27}T^\ast_p M
\end{split}}~~.
\end{eqnarray}
Let $\omega^{(k)}\in \Omega_k(\mathcal{U})$ and $\omega^{{\bf [k]}}_\pm\in
\Omega_{{\bf k}}(\mathcal{U})$ denote the (pointwise) projections of a form
$\omega$ on the irreps of $G_2$ and $\mathrm{Spin}(7)_\pm$ respectively.
\subsection{The $G_2$ and $\mathrm{Spin}(7)_\pm$ parameterizations of $F$}
\label{subsec:Spin7Param}
\paragraph{$G_2$ parameterization.}
Recall from \cite{g2} that $F|_\mathcal{U}=F_\perp+{\hat V}\wedge F_\top$ and
$f|_\mathcal{U}=f_\perp+f_\top{\hat V} $, where $f_\top\in \Omega^0(\mathcal{U})$,
$f_\perp\in \Omega^1_\mathcal{U}(\mathcal{D})$, $F_\top\in \Omega^3_\mathcal{U}(\mathcal{D})$ and $F_\perp\in
\Omega^4_\mathcal{U}(\mathcal{D})$, with:
\begin{eqnarray}
\label{Fdecomp}
\boxed{
\begin{split}
&F_\perp=F_\perp^{(7)}+F_\perp^{(S)}~~\mathrm{where}~
~F_\perp^{(7)}=\alpha_1\wedge\varphi\in \Omega^4_7(\mathcal{D})~~,
~~F_\perp^{(S)}=-\hat{h}_{kl}e^k\wedge\iota_{e^l}\psi\in \Omega^4_{\mathcal{U},S}(\mathcal{D})~~~\\
&F_\top=F_\top^{(7)}+F_\top^{(S)}~~\mathrm{where}~
~F_\top^{(7)}=-\iota_{\alpha_2}\psi\in \Omega^3_{\mathcal{U},7}(\mathcal{D})~~,
~~F_\top^{(S)}=\chi_{kl}e^k\wedge\iota_{e^l}\varphi\in \Omega^3_{\mathcal{U},S}(\mathcal{D})~
\end{split}
}~~.~~~~~~~
\end{eqnarray}
Here $\alpha_1,\alpha_2\in\Omega^1_\mathcal{U}(\mathcal{D})$, while ${\hat h}=\frac{1}{2}{\hat
h}_{ij}e^i\odot e^j$ and $\chi=\frac{1}{2}\chi_{ij}e^i\odot e^j$ are
sections of the bundle $\mathrm{Sym}^2_\mathcal{U}(\mathcal{D}^\ast)$. We have
$F_\top^{(S)}=F_\top^{(1)}+F_\top^{(27)}$ with $F_\top^{(1)}\in
\Omega^3_1(\mathcal{D})~,~F_\top^{(27)}\in \Omega^3_{\mathcal{U},27}(\mathcal{D})$ and a similar
decomposition for $F_\perp^{(S)}$. The last relations correspond to
the decompositions of $\chi$ and ${\hat h}$ into their homothety parts
$\mathrm{tr}(\chi)g|_\mathcal{D}$, $\mathrm{tr}({\hat h})g|_\mathcal{D}$ and traceless parts:
\begin{equation*}
\chi^{(0)}\stackrel{{\rm def.}}{=} \chi-\frac{1}{7}\mathrm{tr}(\chi)g|_\mathcal{D}~~,~~h^{(0)}={\hat h}-\frac{1}{7}\mathrm{tr}({\hat h})g|_\mathcal{D}~~.
\end{equation*}
Let $h,{\hat \chi}\in \mathrm{Sym}^2_\mathcal{U}(\mathcal{D}^\ast)$ denote the symmetric
tensors defined through:
\begin{equation*}
h_{ij}\stackrel{{\rm def.}}{=} {\hat h}_{ij}-\frac{1}{3}\mathrm{tr}_g({\hat h})g_{ij}~~,~~{\hat \chi}_{ij}\stackrel{{\rm def.}}{=} \chi_{ij}-\frac{1}{4}\mathrm{tr}_g(\chi)g_{ij}~~,
\end{equation*}
where:
\begin{equation*}
\mathrm{tr}_g(\chi)=-\frac{4}{3}\mathrm{tr}_g({\hat \chi})~~,~~\mathrm{tr}_g({\hat h})=-\frac{3}{4}\mathrm{tr}_g( h)~~.
\end{equation*}
The situation is summarized in Table 2.
\begin{table}[tt]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$G_2$ representation & $1$ & $7$ & $27$ \\
\hline\hline
$F_\perp\in \Omega^4_\mathcal{U}(\mathcal{D})$ & $\mathrm{tr}_g({\hat h})$ & $\alpha_1 \in \Omega^1_\mathcal{U}(\mathcal{D})$ & $h^{(0)}\in \mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ \\
\hline
$F_\top\in \Omega^3_\mathcal{U}(\mathcal{D})$ & $\mathrm{tr}_g({\hat \chi})$ & $\alpha_2 \in \Omega^1_\mathcal{U}(\mathcal{D})$ & $\chi^{(0)} \in \mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ \\
\hline
\end{tabular}
\
\caption{The $G_2$ parameterization of $F$ on the non-chiral locus.}
\label{table:G2param}
\end{table}
\paragraph{$\mathrm{Spin}(7)_\pm$ parameterization.}
The discussion of the previous subsection gives the following decompositions of the selfdual and anti-selfdual parts of $F$:
\begin{equation*}
F^\pm=F^{[\bf{1}]}_\pm+F^{[\bf{7}]}_\pm+F^{[\bf{27}]}_\pm\in \Omega^{4\pm}(\mathcal{U})~~,~~F^\mp=F^{[\bf{35}]}_\pm\in \Omega^{4\mp}(\mathcal{U})~~.
\end{equation*}
Since the Hodge operator intertwines $\mathrm{Spin}(7)_\pm$ representations, we have:
\begin{eqnarray*}
&&(F^{[\bf{k}]}_\pm)_\perp=\pm \ast_\perp (F^{[\bf{k}]}_\pm)_\top~~~\mathrm{for}~~k=1,7,27~~,\\
&&(F^{[\bf{35}]}_\pm)_\perp=\mp \ast_\perp (F^{[\bf{35}]}_\pm)_\top~~.
\end{eqnarray*}
One can parameterize $F^{[{\bf k}]}_\pm$ through a zero-form
$\mathcal{F}^{[\bf{1}]}_\pm\in \Omega^0(\mathcal{U})$, a 2-form $\mathcal{F}^{[{\bf 7}]}_\pm\in
\Omega^2(\mathcal{U})$, a $\mathcal{D}$-longitudinal traceless symmetric covariant tensor $\mathcal{F}^{[{\bf
27}]}_\pm\in \mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ and a traceless symmetric
covariant tensor $\mathcal{F}^{[{\bf 35}]}_\pm\in \mathrm{Sym}^2_0(T^\ast \mathcal{U})$, which are
defined by:
\begin{eqnarray}
\label{TF1F7}
&&F^{[\bf{1}]}_\pm=\frac{1}{42}\mathcal{F}^{[\bf{1}]}_\pm \Phi^\pm~~,\nonumber\\
&&F^{[\bf{7}]}_\pm=\frac{1}{96}\Phi\bigtriangleup_1\mathcal{F}^{[\bf{7}]}_\pm~~,\nonumber\\
&&F^{[\bf{27}]}_\pm=\frac{1}{24}(\mathcal{F}^{[{\bf 27}]}_\pm)_{ij}e^i\wedge \iota_{e^j}\Phi^\mp~~,\\
&&F^{[\bf{35}]}_\pm=\frac{1}{24}(\mathcal{F}^{[\bf{35}]}_{\pm})_{ab}e^a\wedge \iota_{e^b}\Phi^\pm\nonumber~~.
\end{eqnarray}
The quantities $\mathcal{F}^{[{\bf k}]}$ with $k=1,7,35$ can be recovered
from $F$ through the relation:
\begin{equation}
\label{cFF}
6(\iota_{e^a}F)\bigtriangleup_3(\iota_{e^b}\Phi^\pm)=g_{ab}\mathcal{F}^{[{\bf 1}]}_\pm+(\mathcal{F}^{[{\bf 7}]}_\pm)_{ab}+(\mathcal{F}^{[{\bf 35}]}_\pm)_{ab}~~.
\end{equation}
Define:
\begin{equation}
\label{Tparam}
\boxed{
\begin{split}
& \beta_{1\pm}\stackrel{{\rm def.}}{=} (\mathcal{F}^{[\bf{7}]}_\pm)_\top\in \Omega^1_\mathcal{U}(\mathcal{D})~~,\\
& \beta_{2\pm}\stackrel{{\rm def.}}{=} n~\lrcorner~{\cal F}_\pm^{[{\bf 35}]}=(\mathcal{F}^{[{\bf 35}]}_\pm)_{1j}e^j\in \Omega^1_\mathcal{U}(\mathcal{D})~~,\\
& \sigma_\pm\stackrel{{\rm def.}}{=}\frac{1}{2}(\mathcal{F}^{[\bf{35}]}_\pm)_{ij}e^i\odot e^j\in \mathrm{Sym}^2_{\mathcal{U}}(\mathcal{D}^\ast)~~,
\end{split}
}~~
\end{equation}
where $e_a$ is a local orthonormal frame such that $e_1=n\stackrel{{\rm def.}}{=} {\hat
V}^\sharp$ and $j=2,\ldots, 8$. The fact that $F^{[{\bf 7}]}_\pm$ is
(anti-)selfdual implies:
\begin{equation}
\label{cF7perp}
(\mathcal{F}^{[\bf{7}]}_\pm)_\perp=\mp\iota_{\beta_{1\pm}}\varphi~~.
\end{equation}
\begin{table}[tt]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$\mathrm{Spin}(7)_\pm$ representation & $\mathbf{1}$ & $\mathbf{7}$ & $\mathbf{27}$ & $\mathbf{35}$\\
\hline\hline
component & $F^{[{\bf 1}]}_\pm\in \Omega^{4\mp}(\mathcal{U})$ & $F^{[{\bf 7}]}_\pm\in \Omega^{4\mp} (\mathcal{U})$ & $F^{[{\bf 27}]}_\pm\in \Omega^{4\mp}(\mathcal{U})$ & $F^{[{\bf 35}]}_\pm\in \Omega^{4\pm}(\mathcal{U})$\\
\hline
$\mathcal{U}$-tensors & $\mathcal{F}^{[{\bf 1}]}_\pm\in \Omega^0(\mathcal{U})$ & $\mathcal{F}^{[{\bf 7}]}_\pm\in \Omega^2(\mathcal{U})$ & $\mathcal{F}^{[{\bf 27}]}_\pm\in \mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ & $\mathcal{F}^{[{\bf 35}]}_\pm\in \mathrm{Sym}^2_0(T^\ast \mathcal{U})$ \\
\hline
$\mathcal{D}$-tensors & $\mathcal{F}^{[{\bf 1}]}_\pm\in \Omega^0(\mathcal{U})$ & $\beta_{1\pm}\in \Omega^1_\mathcal{U}(\mathcal{D})$ & $\mathcal{F}^{[{\bf 27}]}_\pm\in \mathrm{Sym}^2_{\mathcal{U},0}(\mathcal{D}^\ast)$ & $\begin{array}{c}\beta_{2\pm}\in \Omega^1_\mathcal{U}(\mathcal{D})~~ \\ \sigma\in \mathrm{Sym}^2_\mathcal{U}(\mathcal{D}^\ast)\end{array}$ \\
\hline
\end{tabular}
\
\caption{The $\mathrm{Spin}(7)_\pm$ parameterization of $F$ on the non-chiral locus and its $\mathcal{D}$-refined version.}
\label{table:Spin7param}
\end{table}
\noindent Choosing an orthonormal frame with $e_1=n={\hat V}^\sharp$
and recalling \eqref{PhiDec}, relations \eqref{TF1F7} and
\eqref{Tparam} give the following parameterization of $F$, which
refines the parameterization used in \cite{Tsimpis} by taking into
account the decomposition into directions parallel and perpendicular
to ${\hat V}$:
\begin{equation}
\label{FSpin7TParam}
\!\!\!\!\!\!\!\!\!\boxed{
\begin{split}
&(F^{[\bf{1}]}_\pm)_\top=\pm \frac{1}{42}\mathcal{F}^{[\bf{1}]}_\pm \varphi~~,~~~~~~~~~~~~~~~~~~~(F^{[\bf{1}]}_\pm)_\perp=\frac{1}{42}\mathcal{F}^{[\bf{1}]}_\pm\psi~~,\\
&(F^{[\bf{7}]}_\pm)_\top=\frac{1}{24}\iota_{\beta_{1\pm}}\psi~~,~~~~~~~~~~~~~~~~~~
~~~~(F^{[\bf{7}]}_\pm)_\perp=\mp \frac{1}{24}\beta_{1\pm}\wedge \varphi~~,\\
& (F^{[\bf{27}]}_\pm)_\top=\mp\frac{1}{24}(\mathcal{F}^{[{\bf 27}]}_\pm)_{ij}e^i\wedge \iota_{e^j}\varphi~~,~~~(F^{[\bf{27}]}_\pm)_\perp=\frac{1}{24}(\mathcal{F}^{[{\bf 27}]}_\pm)_{ij}e^i\wedge \iota_{e^j}\psi~~,\\
&(F^{[\bf{35}]}_\pm)_\top\!=\pm\frac{1}{24}\big[\iota_{\beta_{2\pm}} \psi-\frac{4}{7}(\mathrm{tr}\sigma_\pm)\varphi+(\sigma_\pm^{(0)})_{ij}e^i\wedge \iota_{e^j}\varphi\big]~~,\\
& (F^{[\bf{35}]}_\pm)_\perp\!=\frac{1}{24}\big[{\beta_{2\pm}}\wedge \varphi+\frac{4}{7}(\mathrm{tr}\sigma_\pm)\psi+(\sigma_\pm^{(0)})_{ij}e^i\wedge \iota_{e^j}\psi\big]~~.
\end{split}}
\end{equation}
To arrive at the above, we used the relations:
\begin{equation*}
\varphi\bigtriangleup_1 (\mathcal{F}^{[\bf{7}]}_\pm)_\perp=\mp 3\iota_{\beta_{1\pm}}\psi~~,~~\psi\bigtriangleup_1(\mathcal{F}^{[\bf{7}]}_\pm)_\perp =\mp 3\beta_{1\pm}\wedge \varphi~~,
\end{equation*}
which follow from \eqref{cF7perp} and the identities given in the
Appendix of \cite{Kflows}.
\paragraph{Relating the $G_2$ and $\mathrm{Spin}(7)_\pm$ parameterizations of $F$.}
Relation \eqref{G2Spin7Irreps} implies:
\begin{equation}
\label{FSpin7TopPerp}
\!\!\!\!\!\!\begin{split}
&(F_\pm^{\bf [k]})_\top=\frac{1}{2}(F_\top^{(k)}\pm \ast_\perp F_\perp^{(k)})~~~~,~~~(F_\pm^{\bf [k]})_\perp=\frac{1}{2}(F_\perp^{(k)}\pm \ast_\perp F_\top^{(k)}) ~~~\mathrm{for}~~k=1,7,27~~,\\
&(F_\pm^{\bf [35]})_\top=\frac{1}{2}(F_\top\mp \ast_\perp F_\perp)~~~~~,~~~(F_\pm^{\bf [35]})_\perp=\frac{1}{2}(F_\perp\mp \ast_\perp F_\top)~~.
\end{split}
\end{equation}
Comparing \eqref{FSpin7TParam} with \eqref{FSpin7TopPerp} and using
the $G_2$ parameterization of $F_\top$ and $F_\perp$ given in
\eqref{Fdecomp}, one can express the quantities in the last row of
Table \ref{table:Spin7param} in terms of $\alpha_1,\alpha_2$ and
${\hat h},{\hat \chi}$:
\begin{equation}
\label{Spin7G2Param}
\boxed{
\begin{split}
&\mathcal{F}^{[{\bf 1}]}_\pm=-12\mathrm{tr}_g({\hat h}\pm{\hat \chi})~~,\\
&\sigma_\pm~~~=-12({\hat h}\mp{\hat \chi})~~,\\
&\mathcal{F}^{[{\bf 27}]}_\pm=-12({\hat h}^{(0)}\pm{\hat \chi}^{(0)})~~,\\
&\beta_{1\pm}~~=-12(\alpha_2\pm\alpha_1)~~,\\
&\beta_{2\pm}~~=+12(\alpha_1\mp \alpha_2)~~.
\end{split}}
\end{equation}
These simple relations provide the connection between the $G_2$
parameterization \eqref{Fdecomp} and the refined $\mathrm{Spin}(7)_\pm$
parameterizations \eqref{FSpin7TParam}, thus allowing one to relate
the $G_2$ and $\mathrm{Spin}(7)_\pm$ decompositions of $F$.
\subsection{Relating the $G_2$ torsion classes to the Lee form and characteristic torsion of the $\mathrm{Spin}(7)_\pm$ structures}
Recall that the {\em Lee form} of the $\mathrm{Spin}(7)_\pm$ structure
determined by $\Phi^\pm$ on $\mathcal{U}$ is the one-form defined through:
\begin{equation}
\label{Lee}
\boxed{\theta_\pm\stackrel{{\rm def.}}{=} \pm \frac{1}{7}\ast (\Phi^\pm\wedge \updelta\Phi^\pm) =-\frac{1}{7}\ast[\Phi^\pm\wedge (\ast \mathrm{d}\Phi^\pm)]\in \Omega^1(\mathcal{U})~\Longrightarrow ~\Phi^\pm\wedge \updelta\Phi^\pm=\mp 7\ast \theta_\pm}~~,
\end{equation}
where we use the conventions of \cite{Ivanov} and the fact that $\ast
\Phi^\pm=\pm\Phi^\pm$. Also recall from loc. cit. that there exists a
unique $g$-compatible connection $\nabla^c$ with skew-symmetric
torsion such that $\nabla^c\Phi^\pm=0$. This connection is called the
{\em characteristic connection} of the $\mathrm{Spin}(7)_\pm$ structure. Its
torsion form (obtained by lowering the upper index of the torsion
tensor of $\nabla^c$) is given by:
\begin{equation}
\label{Spin7Torsion}
\boxed{T_\pm=-\updelta\Phi^\pm\mp \frac{7}{6}\ast(\theta_\pm\wedge\Phi^\pm)=-\updelta\Phi^\pm
-\frac{7}{6}\iota_{\theta_\pm}\Phi^\pm=\pm \ast (\mathrm{d}\Phi^\pm - \frac{7}{6}\theta_\pm\wedge\Phi^\pm)\in \Omega^3(\mathcal{U})}
\end{equation}
and is called the {\em characteristic torsion} of the $\mathrm{Spin}(7)_\pm$
structure. The normalization relation $||\Phi^\pm||^2=14$,
i.e. $\Phi^\pm\wedge \Phi^\pm=\pm 14\nu$ implies $\Phi^\pm\wedge
\iota_{\theta_\pm}\Phi^\pm=\pm 7\ast\theta_\pm$. Thus $\Phi^\pm\wedge
T_\pm=\mp\frac{7}{6}\ast \theta_\pm$, where we used \eqref{Lee} and
\eqref{Spin7Torsion}. It follows that the Lee form is determined by
the characteristic torsion through the equation:
\begin{equation}
\label{thetaT}
\boxed{\theta_\pm=\pm \frac{6}{7}\ast(\Phi^\pm\wedge T_\pm)}~~.
\end{equation}
Relation \eqref{Spin7Torsion} shows that the exterior derivative of $\Phi^\pm$ takes the form:
\begin{equation}
\label{Spin7TorsionEq}
\mathrm{d} \Phi^\pm= \frac{7}{6}\theta_\pm\wedge \Phi^\pm\mp \ast T_\pm=\pm[\ast(\Phi^\pm\wedge T_\pm)]\wedge \Phi^\pm \mp \ast T_\pm~~.
\end{equation}
Recall the relation (see \cite{g2}):
\begin{equation*}
D_n\psi=-3\vartheta\wedge \varphi~~,
\end{equation*}
where $\vartheta\in \Omega^1(\mathcal{D})$. Together with \eqref{PhiDec} and with the formula for the exterior
derivative of longitudinal forms (see Appendix C. of \cite{g2}), this gives:
\begin{eqnarray*}
(\mathrm{d} \Phi^\pm)_\top &=&\pm (H_\sharp \mp3\vartheta - 3\tau_1) \wedge \varphi -(\frac{4}{7}\mathrm{tr} A\pm\tau_0 )\psi
- A^{(0)}_{jk}e^j\wedge \iota_{e^k}\psi \mp \ast_\perp\tau_3~~, \\
(\mathrm{d} \Phi^\pm)_\perp &=& 4\tau_1\wedge \psi+\ast_\perp\tau_2~~,
\end{eqnarray*}
which implies:
\begin{eqnarray}
(\ast \mathrm{d} \Phi^\pm)_\top&=&-\tau_2- 4\iota_{\tau_1}\varphi~~,\nonumber\\
(\ast \mathrm{d} \Phi^\pm)_\perp&=&\mp ~\iota_{(H_\sharp \mp3\vartheta-3\tau_1)}\psi - (\frac{4}{7}\mathrm{tr} A\pm\tau_0 )\varphi + A^{(0)}_{jk}e^j\wedge \iota_{e^k}\varphi \mp\tau_3~~.
\end{eqnarray}
Using this relation and \eqref{PhiDec}, we can compute $\theta_\pm$ from
\eqref{Lee} and then determine $T_\pm$ from equation
\eqref{Spin7Torsion}. We find:
\begin{equation}
\label{Ttheta}
\!\!\!\!\boxed{
\begin{split}
&(\theta_\pm)_\top\!\!=-\frac{4}{7}\mathrm{tr} A \mp\tau_0~~~~~~~,
~~~~~~~(\theta_\pm)_\perp\!\!=- \frac{4}{7}(H_\sharp \mp 3\vartheta-6\tau_1)~~,\\
&(T_\pm)_\top\!\!=- \frac{2}{3}\iota_{(\pm H_\sharp-3\vartheta)}\varphi \mp \tau_2~~,
~~(T_\pm)_\perp\!\!=- \frac{1}{6}(\frac{4}{7}\mathrm{tr} A \pm \tau_0)\varphi
- \frac{1}{3}\iota_{(H_\sharp \mp 3\vartheta+ 3\tau_1)}\psi \pm A_{jk}^{(0)}e^j\wedge \iota_{e^k}\varphi - \tau_3~.
\end{split}}
\end{equation}
To arrive at the last two relations, we used the identities:
\begin{equation*}
\iota_{\tau_2}\varphi=\iota_{\tau_3}\psi=\langle \tau_3,\varphi\rangle=0~~,~~
\end{equation*}
which follow from relations (B.13) and (B.14) given in Appendix B
of \cite{g2} upon using the fact that $\tau_3\in \Omega^3_{\mathcal{U},27}(\mathcal{D})$.
\subsection{Relation to previous work}
The problem of determining the fluxes $f,F$ in terms of the geometry
along the locus $\mathcal{U}^+$ was considered in reference \cite{Tsimpis},
where the quantities denoted here by $L^+,\Phi^+$ were denoted simply
by $L,\Phi$. Using the results of the previous subsections, one can
show that the relations given in Theorem 3 of \cite{g2} are equivalent,
on the non-chiral locus $\mathcal{U}$, with equations (3.16), (3.17) and
(3.18) of \cite{Tsimpis}. This solves the problem of comparing the
approach of loc. cit. with that of \cite{MartelliSparks, g2}. The
major steps of the comparison with loc. cit. are given in Appendix
\ref{app:T}.
\section{Description of the singular foliation in the Morse case}
\label{sec:Morse}
In this section, we consider the case when the closed one-form
${\boldsymbol{\omega}}\in \Omega^1(M)$ is Morse. This case is generic in the sense
that Morse one-forms form a dense open subset of the set of all closed
one-forms belonging to the fixed cohomology class $\mathfrak{f}$ --- hence a form
which satisfies equations \eqref{meq} can be replaced by a Morse form
by infinitesimally perturbing $b$. Singular foliations defined by
Morse one-forms were studied in \cite{Gelbukh1}--\cite{Gelbukh9} and \cite{Levitt1}--\cite{Honda}.
Let $\Pi_f=\mathrm{im} (\mathrm{per}_\mathfrak{f})\subset \mathbb{R}$ be the period group of the
cohomology class $\mathfrak{f}$ and $\rho(\mathfrak{f})={\rm rk} \Pi_\mathfrak{f}$ be its irrationality
rank. The general results summarized in the following subsection hold
for any smooth, compact and connected manifold of dimension $d$ which
is strictly bigger than two, under the assumption that the set of
zeroes of ${\boldsymbol{\omega}}$ (which in Novikov theory \cite{Farber} is called
the set of {\em singular points}):
\begin{equation*}
\mathrm{Sing}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} \{p\in M|\omega_p=0\}
\end{equation*}
is non-empty. Notice that $\mathrm{Sing}({\boldsymbol{\omega}})$ is a finite set since $M$
is compact and since the zeroes of a Morse 1-form are isolated.
The complement:
\begin{equation*}
M^\ast\stackrel{{\rm def.}}{=} M\setminus \mathrm{Sing}({\boldsymbol{\omega}})
\end{equation*}
is a non-compact open submanifold of $M$. Below, we shall use the notations
$\mathcal{F}_{\boldsymbol{\omega}}$ for the regular foliation induced by ${\boldsymbol{\omega}}$ on $M^\ast$
and $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ for the singular foliation induced on $M$.
In our application we have $n=8$ and:
\begin{equation*}
\mathrm{Sing}({\boldsymbol{\omega}})=\mathcal{W}~~,~~M^\ast=\mathcal{U}~~,~~\mathcal{F}_{\boldsymbol{\omega}}=\mathcal{F}~~,~~\bar{\mathcal{F}}_{\boldsymbol{\omega}}=\bar{\mathcal{F}}~~.
\end{equation*}
\subsection{Types of singular points}
\label{subsec:singtypes}
Let $\mathrm{ind}_p({\boldsymbol{\omega}})$ denote the Morse index of a point $p\in \mathrm{Sing}
({\boldsymbol{\omega}})$, i.e. the Morse index at $p$ of a Morse function $h_p\in
\mathcal{C}^\infty(U_p,\mathbb{R})$ such that $\mathrm{d} h_p$ equals ${\boldsymbol{\omega}}|_{U_p}$, where
$U_p$ is some vicinity of $p$. This index does not depend on the
choice of $U_p$ and $h_p$. Let:
\begin{eqnarray*}
&&\mathrm{Sing}_k({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} \{p\in \mathrm{Sing}({\boldsymbol{\omega}})|\mathrm{ind}_p({\boldsymbol{\omega}})=k\}~~,~~k=1,\ldots, d\\
&&\Sigma_k({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} \{p\in \mathrm{Sing}({\boldsymbol{\omega}})|\mathrm{ind}_p({\boldsymbol{\omega}})=k~\mathrm{or}~\mathrm{ind}_p({\boldsymbol{\omega}})=d-k\}~~,~~k=1,\ldots,\left[\frac{d}{2}\right]~~.
\end{eqnarray*}
Thus $\Sigma_k({\boldsymbol{\omega}})=\mathrm{Sing}_k({\boldsymbol{\omega}})\cup\mathrm{Sing}_{n-k}({\boldsymbol{\omega}})$ for
$k<\frac{d}{2}$ and $\Sigma_{d_0}({\boldsymbol{\omega}})=\mathrm{Sing}_{d_0}({\boldsymbol{\omega}})$ when $d=2d_0$
is even. In a small enough vicinity of $p\in
\mathrm{Sing}_k({\boldsymbol{\omega}})$ (which we can assume to equal $U_p$ by shrinking the
latter if necessary), the Morse lemma applied to $h_p$ implies that
there exists a local coordinate system $(x_1,\ldots,x_d)$ such that:
\begin{equation*}
h_p=-\sum_{j=1}^k{x_j^2}+\sum_{j=k+1}^{d} x_j^2~~.
\end{equation*}
\paragraph{Definition.} The elements of $\Sigma_0({\boldsymbol{\omega}})$ are called {\em centers}
while all other singularities of ${\boldsymbol{\omega}}$ are called {\em saddle
points}. The elements of $\Sigma_1({\boldsymbol{\omega}})$ are called {\em strong
saddle points}, while all other saddle points are called {\em weak}.
\paragraph{Remark.} Strong saddle points are sometimes called ``conical
points''. That terminology can lead to confusion, since all singular
points which are not centers are conical singularities of the singular
leaf to which they belong (see below), in the sense that the singular
leaf can be modeled by a cone (with one or two sheets) in a vicinity
of such a singular point. In other references, a ``conical point'' means
any singularity which is not a center, i.e. what we call a saddle point.
\subsection{The regular and singular foliations defined by a Morse 1-form}
\paragraph{The regular foliation $\mathcal{F}_{\boldsymbol{\omega}}$.}
The Morse 1-form ${\boldsymbol{\omega}}$ defines a regular foliation $\mathcal{F}_{\boldsymbol{\omega}}$
of the open submanifold $M^\ast$, namely the foliation which, by the
Frobenius theorem, integrates the regular Frobenius distribution
$\mathrm{ker}({\boldsymbol{\omega}})|_{M^\ast}$. Following \cite{Gelbukh7}, we say that a
singular point $p\in \mathrm{Sing}({\boldsymbol{\omega}})$ {\em adjoins} a leaf
$L$\footnote{This should not be confused with the quantity $L^\pm$
discussed in Subsection 3.4 (or with the quantity denoted by $L$ in
\cite{Tsimpis}).} of $\mathcal{F}_{\boldsymbol{\omega}}$ if the union $\{p\}\cup L$ is
connected; notice that a center cannot adjoin any leaf of
$\mathcal{F}_{\boldsymbol{\omega}}$. Let:
\begin{equation*}
s(L)\stackrel{{\rm def.}}{=} \{p\in \mathrm{Sing}({\boldsymbol{\omega}})|p~\mathrm{adjoins}~L\}\subset \mathrm{Sing}({\boldsymbol{\omega}})~~.
\end{equation*}
The set $s(L)$ is contained in the intersection of $\mathrm{Sing}({\boldsymbol{\omega}})$ with
the small topological frontier $\mathrm{fr}(L)$ of $L$:
\begin{equation}
\label{inclusion}
s(L)\subseteq \mathrm{fr}(L)\cap \mathrm{Sing}({\boldsymbol{\omega}})~~.
\end{equation}
Notice that this inclusion can be strict; a beautifully drawn example
illustrating this in the two-dimensional case can be found in
\cite{Gelbukh9} (see Figure 2(c) of loc. cit.). We have $s(L)\cap
\Sigma_0({\boldsymbol{\omega}})=\emptyset$ and hence
$s(L)=\sqcup_{k=1}^{\left[\frac{d}{2}\right]}s_k(L)$, where:
\begin{equation*}
s_k(L)\stackrel{{\rm def.}}{=} s(L)\cap \Sigma_k(L)~~.
\end{equation*}
\paragraph{Classification of the leaves of $\mathcal{F}_{\boldsymbol{\omega}}$.}
\begin{itemize}
\itemsep 0.0em
\item{\bf Compactifiable and non-compactifiable leaves.} We say that a
leaf $L$ of $\mathcal{F}_{\boldsymbol{\omega}}$ is {\em compactifiable} if the set $L\cup
\mathrm{Sing}({\boldsymbol{\omega}})$ is compact, which amounts to the condition that the
small topological frontier $\mathrm{fr}(L)\stackrel{{\rm def.}}{=} {\bar L}\setminus L$ of $L$
in $M$ is a (possibly void) subset of $\mathrm{Sing}({\boldsymbol{\omega}})$ and hence a
finite set. With this definition, compact leaves of $\mathcal{F}_{\boldsymbol{\omega}}$
are compactifiable, but not all compactifiable leaves are compact. A
{\em non-compactifiable leaf} of $\mathcal{F}_{\boldsymbol{\omega}}$ is a leaf which is
not compactifiable; obviously such a leaf is also non-compact. The
closure of a non-compactifiable leaf is a set with non-empty
interior \cite{ImanishiSing, Levitt1}, so the small frontier of such
a leaf is an infinite set.
\item{\bf Ordinary and special leaves.} The leaf $L$ is called {\em
ordinary} if $s(L)$ is empty and {\em special} if $s(L)$ is
non-empty. An ordinary leaf is either compact or
non-compactifiable. Any non-compact but compactifiable leaf is a
special leaf, but there also exist non-compactifiable special leaves
(see Table \ref{table:leaves}).
\end{itemize}
\noindent
The foliation $\mathcal{F}_{\boldsymbol{\omega}}$ has only a finite number of special leaves,
because the local form of leaves near the points of $\mathrm{Sing}({\boldsymbol{\omega}})$
(see below) shows that at most two special leaves can contain each
such point in their closures (recall that we assume $d\geq 3$) . We
shall see later that each non-compactifiable leaf (whether special or
not) covers densely some open and connected subset of $M^\ast$. Notice
that every singular point which is not a center adjoins some special leaf.
Hence:
\begin{equation}
\label{SigmakDec}
\Sigma_k({\boldsymbol{\omega}})=\cup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}s_k(L)~~,~~\forall k=1\ldots \left[\frac{d}{2}\right]~~.
\end{equation}
\begin{table}[tt]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\multirow{2}{*}{type of $L$} & \multicolumn{2}{c}{compactifiable} \vline & \multirow{2}{*}{non-compactifiable}\\
\cline{2-3}
& compact & non-compact & \\
\hline\hline
ordinary & Y & --- & Y \\
\hline
special & --- & Y & Y\\
\hline
$\mathrm{Card}(\mathrm{fr} L)$ &\multicolumn{2}{c}{finite} \vline & infinite\\
\hline
\end{tabular}
\caption{Classification of the leaves of $\mathcal{F}_{\boldsymbol{\omega}}$, where the
allowed combinations are indicated by the letter ``Y''. A
compactifiable leaf is ordinary iff it is compact and it is special
iff it is non-compact. A non-compactifiable leaf may be either
ordinary or special. Non-compactifiable leaves coincide
\cite{ImanishiSing,Levitt1} with those leaves whose small frontier is an
infinite set, while compactifiable leaves are those leaves whose
small frontier is finite.}
\label{table:leaves}
\end{table}
\paragraph{The singular foliation $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$.}
One can describe \cite{FKL, Farber} the singular
foliation $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ of $M$ defined by ${\boldsymbol{\omega}}$ as
the partition of $M$ induced by the equivalence relation $\sim$
defined as follows. We put $p\sim q$ if there exists a smooth curve
$\gamma:[0,1]\rightarrow M$ such that:
\begin{equation*}
\gamma(0)=p~~,~~\gamma(1)=q~~\mathrm{and}~~{\boldsymbol{\omega}}(\dot{\gamma}(t))=0~~\forall t\in [0,1]~~.
\end{equation*}
The {\em leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$} are the equivalence classes of this
relation; they are connected subsets of $M$ (which need not be
topological manifolds when endowed with the induced topology). Any such
leaf is either of the form $\{p\}$ where $p\in \Sigma_0({\boldsymbol{\omega}})$ is a center
or is a topological subspace of $M$ of Lebesgue covering dimension
equal to $n-1$.
\paragraph{Remark.}
We stress that $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ is not generally a foliation of $M$ in
the ordinary sense of foliation theory but (as explained in the
previous section) it should be viewed as a Haefliger structure. It is
not even a $\mathcal{C}^0$-foliation, i.e. a foliation in the category of
topological manifolds (locally Euclidean Hausdorff topological
spaces), because singular leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ which pass through
strong saddle points can be locally disconnected by removing those
points and hence are not topological manifolds.
\paragraph{Regular and singular leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$.}
A leaf $\mathcal{L}$ of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ is called {\em singular} if it
intersects $\mathrm{Sing}({\boldsymbol{\omega}})$ and {\em regular} otherwise. The regular
leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ coincide with the ordinary leaves of $\mathcal{F}_{\boldsymbol{\omega}}$; notice
that every center singularity is a singular leaf.
On the other hand, each singular leaf which is not a center is a
disjoint union of a finite number of special leaves of $\mathcal{F}_{\boldsymbol{\omega}}$
and of some subset $s(\mathcal{L})\stackrel{{\rm def.}}{=} \mathcal{L}\cap \mathrm{Sing}({\boldsymbol{\omega}})$ of
$\mathrm{Sing}({\boldsymbol{\omega}})$, which we shall call the {\em set of singular points
of $\mathcal{L}$}. We have:
\begin{equation*}
\mathcal{L}\setminus s(\mathcal{L})=L_1\sqcup \ldots \sqcup L_r\sqcup L'_1\sqcup \ldots \sqcup L'_t~~,
\end{equation*}
where $L_1,\ldots, L_r$ are compactifiable special leaves while
$L'_1,\ldots, L'_t$ are non-compactifiable special leaves of
$\mathcal{F}_{\boldsymbol{\omega}}$. We also have $s(\mathcal{L})=s^\mathrm{c}(\mathcal{L})\cup s^\mathrm{nc}(\mathcal{L})$ (generally
a non-disjoint union), with:
\begin{equation*}
s^c(\mathcal{L})\stackrel{{\rm def.}}{=} \cup_{i=1}^r s(L_i)~~,~~s^\mathrm{nc}(\mathcal{L})\stackrel{{\rm def.}}{=} \cup_{j=1}^t(L'_j)~~.
\end{equation*}
The singular leaf $\mathcal{L}$ decomposes as:
\begin{equation}
\label{cLdec}
\mathcal{L}=\mathcal{L}^{\rm c}\sqcup \mathcal{L}^\mathrm{nc}~~,
\end{equation}
where the {\em compact part} and {\em non-compact part} of $\mathcal{L}$ are defined through:
\begin{eqnarray*}
\mathcal{L}^c~&\stackrel{{\rm def.}}{=}& \bar{L}_1\cup \ldots \cup \bar{L}_r=L_1\sqcup \ldots \sqcup L_r\sqcup s^c(\mathcal{L})\nonumber\\
\mathcal{L}^\mathrm{nc}&\stackrel{{\rm def.}}{=}& \mathcal{L}\setminus \mathcal{L}^c=(L'_1\sqcup \ldots \sqcup L'_t)\sqcup (s(\mathcal{L})\setminus s^c(\mathcal{L}))~~.
\end{eqnarray*}
The set $s^c(\mathcal{L})$ consists of those singular points of $\mathcal{L}$ which
lie on the compact part $\mathcal{L}^c$. Notice that both the compact and
non-compact parts of $\mathcal{L}$ can be void and that a non-compactifiable
special leaf component $L'_j$ of $\mathcal{L}$ can adjoin points from
$s^c(\mathcal{L})$ as well as from $s(\mathcal{L})\setminus s^c(\mathcal{L})$ simultaneously;
furthermore, $\mathcal{L}^\mathrm{nc}$ may meet itself at certain points of
$s(\mathcal{L})\setminus s^c(\mathcal{L})$\footnote{We thank I. Gelbukh for drawing
our attention to these points.}. When $\mathcal{L}$ is a center leaf
$\{p\}$, we define $\mathcal{L}^{\rm c}\stackrel{{\rm def.}}{=} s(\mathcal{L})=\{p\}$ and $\mathcal{L}^\mathrm{nc}
\stackrel{{\rm def.}}{=} \emptyset$. Notice that any non-empty subset $A$ of $\mathcal{L}$
determines $\mathcal{L}$ as the saturation of $A$ with respect to the
equivalence relation $\sim$. If $S_{\boldsymbol{\omega}}$ denotes the union of
$\mathrm{Sing}({\boldsymbol{\omega}})$ with all special leaves of $\mathcal{F}_{\boldsymbol{\omega}}$, then the
singular leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ (including the centers) coincide
with the connected components of $S_{\boldsymbol{\omega}}$. Notice that
$\mathrm{fr}(L_i)=s(L_i)={\bar L}_i\cap \mathrm{Sing}({\boldsymbol{\omega}})$ for each
compactifiable leaf component $L_i$, $i=1\ldots r$. The compact sets
${\bar L}_i$ meet themselves or each other only in strong saddle
points. In particular, we have:
\begin{equation*}
{\bar L}_{i_1}\cap {\bar L}_{i_2}=s(L_{i_1})\cap s(L_{i_2})=s_1(L_{i_1})\cap s_1(L_{i_2})\subset \Sigma_1({\boldsymbol{\omega}})~~\mathrm{for}~~1\leq i_1<i_2\leq r~~.
\end{equation*}
\noindent The following definition generalizes the notion of generic
Morse function:
\paragraph{Definition.}
The Morse form ${\boldsymbol{\omega}}$ is called {\em generic} if every singular
leaf of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ contains exactly one singular point $p\in
\mathrm{Sing}({\boldsymbol{\omega}})$.
\subsection{Behavior of the singular leaves near singular points}
In a small enough vicinity of $p\in \mathrm{Sing}_k({\boldsymbol{\omega}})$, the singular
leaf $\mathcal{L}_p$ passing through $p$ is modeled by the locus $Q_k\subset
\mathbb{R}^n$ given by the equation $h_p=0$, where $p$ corresponds to the
origin of $\mathbb{R}^n$. One distinguishes the cases (see Tables
\ref{table:singtypes} and \ref{table:strongsaddles}):
\begin{itemize}
\itemsep 0.0em
\item $k\in \{0, n\}$, i.e. $p$ is a center. Then $\mathcal{L}_p=\{p\}$ and
the nearby leaves of $\mathcal{F}_p$ are diffeomorphic to $S^{n-1}$.
\item $2\leq k\leq n-2$, i.e. $p$ is a weak saddle point. Then $Q_k$
is diffeomorphic to a cone over $S^{k-1}\times S^{n-k-1}$ and
$\mathbb{R}^n\setminus Q_k$ has two connected components while
$Q_k\setminus\{p\}$ is connected. Removing $p$ does not {\em
locally} disconnect $\mathcal{L}_p$.
\item $k\in \{1,n-1\}$, i.e. $p$ is a strong saddle point. Then $Q_k$
is diffeomorphic to a cone over $\{-1,1\}\times S^{n-2}$ and
$\mathbb{R}^n\setminus Q_k$ has three connected components while
$Q_k\setminus\{0\}$ has two components. Removing $p$ {\em locally}
disconnects $\mathcal{L}_p$. A strong saddle point $p\in \Sigma_1({\boldsymbol{\omega}})$
is called {\em splitting} \cite{Gelbukh7} (or {\em blocking}
\cite{Levitt3}) if it adjoins two different (special) leaves of the regular
foliation $\mathcal{F}_{\boldsymbol{\omega}}$ and it is called {\em non-splitting} (or a
{\em transformation point} \cite{Gelbukh7}) if it adjoins a single (special)
leaf of $\mathcal{F}_{\boldsymbol{\omega}}$ (see Table \ref{table:strongsaddles}). If a
singular leaf $\mathcal{L}$ contains only one splitting point, then removing
it disconnects $\mathcal{L}$. If a singular leaf $\mathcal{L}$ contains more than
one splitting point, then removing it may not disconnect $\mathcal{L}$ (an
example of such behavior is given in \cite[Figure 7(b)]{Gelbukh7}).
\end{itemize}
\begin{table}[h!]
\centering
\begin{tabular}{ | c | c| c| c| }
\hline
Name & Morse index & Local form of $\mathcal{L}_p$ & Local form of regular leaves\\
\hline
Center & $0$ or $n$ & $\bullet=\{p\}$ &
\begin{minipage}{.3\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.25]{sphere.eps}
\vspace{0.2em}
\end{minipage}\\
\hline
Weak saddle & between $2$ and $n-2$ &
\begin{minipage}{.23\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.3]{onecone.eps}
\vspace{0.2em}
\end{minipage}
& \begin{minipage}{.23\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.3]{upperhyp.eps}
\vspace{0.2em}
\end{minipage}\\
\hline
Strong saddle & $1$ or $n-1$ &
\begin{minipage}{.23\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.3]{cone.eps}
\vspace{0.2em}
\end{minipage}
&
\begin{minipage}{.3\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.3]{hyperboloid.eps}
\vspace{0.2em}
\end{minipage}\\
\hline
\end{tabular}
\caption{Types of singular points $p$. The first and third figure on
the right depict the case $d=3$ for centers and strong saddles,
while the second figure attempts to depict the case $d>3$ for a weak
saddle (notice that weak saddles do not exist unless $d>3$). In
that case, the topology of the leaves does not change locally when
they ``pass through" the weak saddle point. $\mathcal{L}_p$ denotes the
singular leaf of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ which passes through $p$.}
\label{table:singtypes}
\end{table}
\noindent We have a decomposition
$\Sigma_1({\boldsymbol{\omega}})=\Sigma_1^\mathrm{sp}({\boldsymbol{\omega}})\sqcup\Sigma_1^\mathrm{nsp} ({\boldsymbol{\omega}})$
of the set of strong saddle points, where:
\begin{eqnarray*}
&&\Sigma_1^\mathrm{sp}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=}\{p\in \Sigma_1({\boldsymbol{\omega}})|p~~\mathrm{is~a~splitting~singularity}\}\\
&&\Sigma_1^\mathrm{nsp}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=}\{p\in \Sigma_1({\boldsymbol{\omega}})|p~~\mathrm{is~a~non-splitting~singularity}\}~~.
\end{eqnarray*}
Taking into account the local behavior of leaves near the various
types of singular points, we find that the decomposition
\eqref{SigmakDec} is disjoint for $k\neq 1$:
\begin{equation*}
\Sigma_k({\boldsymbol{\omega}})=\sqcup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}s_k(L)~~,~~\forall k=2\ldots \left[\frac{d}{2}\right]~~.
\end{equation*}
while the decomposition for $k=1$ may fail to be disjoint:
\begin{equation}
\label{Sigma1Dec}
\Sigma_1({\boldsymbol{\omega}})=\cup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}s_1(L)~~.
\end{equation}
More precisely:
\begin{eqnarray*}
\Sigma^\mathrm{nsp}_1({\boldsymbol{\omega}})&=&\sqcup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}s^\mathrm{nsp}_1(L)~~\\
\Sigma_1^\mathrm{sp}({\boldsymbol{\omega}})&=&\cup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}s_1^\mathrm{sp}(L)~~,
\end{eqnarray*}
where we defined:
\begin{equation*}
s^\mathrm{nsp}_1(L)\stackrel{{\rm def.}}{=} s_1(L)\cap \Sigma_1^\mathrm{nsp}({\boldsymbol{\omega}})~~,~~s^\mathrm{sp}_1(L)\stackrel{{\rm def.}}{=} s_1(L)\cap \Sigma_1^\mathrm{sp}({\boldsymbol{\omega}})
\end{equation*}
and where the second union may be non-disjoint. This is because two
distinct special leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ can meet each other only at a
strong saddle point which is a splitting singularity.
\begin{table}[h!]
\centering
\begin{tabular}{ | c | c | }
\hline
Singularity type & Example of global shape for $\mathcal{L}_p$ \\
\hline
Splitting &
\begin{minipage}{.3\textwidth}
\centering
\vspace{0.6em}
\includegraphics[scale=0.08, angle=90]{splitting.eps}
\vspace{0.6em}
\end{minipage}\\
\hline
Non-splitting &
\begin{minipage}{.3\textwidth}
\centering
\vspace{0.2em}
\includegraphics[scale=0.4]{nonsplitting.eps}
\vspace{0.2em}
\end{minipage}\\
\hline
\end{tabular}
\caption{Types of strong saddle points. The figures illustrate the two
types through two simple examples in the case $d=3$. The figure in
the first row uses different colors to indicate two different
special compactifiable leaves of $\mathcal{F}_\omega$ which are subsets of
the same singular leaf of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$, each of them adjoining the same splitting
singular point. The figure in the second row shows a single special
compactifiable leaf of $\mathcal{F}_{\boldsymbol{\omega}}$ which adjoins a single
non-splitting singular point.}
\label{table:strongsaddles}
\end{table}
\subsection{Combinatorics of singular leaves}
\
\paragraph{Definition.}
A singular leaf of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ which is not a center is called a
{\em strong singular leaf} if it contains at least one strong saddle
point and a {\em weak singular leaf} otherwise.
\
\noindent A weak singular leaf is obtained by adjoining weak saddle
points to a single special leaf of $\mathcal{F}_{\boldsymbol{\omega}}$. Such singular leaves
are mutually disjoint and their singular points determine a partition
of the set $\Sigma_{>1}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=}
\cup_{k=2}^{\left[\frac{d}{2}\right]}\Sigma_k({\boldsymbol{\omega}})$. The
situation is more complicated for strong singular leaves, as we now
describe.
At each $p\in \Sigma_1({\boldsymbol{\omega}})$, consider the strong singular leaf
$\mathcal{L}$ passing through $p$. The intersection of $\mathcal{L}\setminus\{p\}$
with a sufficiently small neighborhood of $p$ is a disconnected manifold
diffeomorphic to a union of two cones without apex, whose rays near $p$ determine a
connected cone $C_p\subset T_pM$ inside the tangent space to $M$ at
$p$ (see the last row of Table \ref{table:singtypes}). The set
$\overset{\bullet}{C}_p\stackrel{{\rm def.}}{=} C_p\setminus\{0_p\}$ (where $0_p$ is the zero vector of
$T_p M$) has two connected components, thus $\pi_0(\overset{\bullet}{C}_p)$ is a
two-element set. Hence the finite set:
\begin{equation*}
{\hat \Sigma_1}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} \sqcup_{p\in \Sigma_1(M)} \pi_0(\overset{\bullet}{C}_p)
\end{equation*} is a
double cover of $\Sigma_1({\boldsymbol{\omega}})$ through the projection $\sigma$
that takes $\pi_0(\overset{\bullet}{C}_p)$ to $\{p\}$. Consider the complete
unoriented graph having as vertices the elements of ${\hat
\Sigma_1}({\boldsymbol{\omega}})$. This graph has a dimer covering given by the
collection of edges:
\begin{equation*}
{\hat \mathcal{E}}=\{\pi_0(\overset{\bullet}{C}_p)|p\in \Sigma_1({\boldsymbol{\omega}})\}~~,
\end{equation*}
which connect vertically the vertices lying above the same point of
$\Sigma_1({\boldsymbol{\omega}})$ (see Figure \ref{fig:dimer}). If $L$ is a special
leaf of $\mathcal{F}_{\boldsymbol{\omega}}$ and $p\in \Sigma_1({\boldsymbol{\omega}})$ adjoins $L$, then
the connected components of the intersection of $L$ with a
sufficiently small vicinity of $p$ are locally approximated at $p$ by
one or two of the connected components of $\overset{\bullet}{C}_p$. The second case
occurs iff $p$ is a non-splitting strong saddle point (see Table
\ref{table:strongsaddles}). Hence $L$ determines a subset ${\hat
s}_1(L)$ of ${\hat \Sigma}_1({\boldsymbol{\omega}})$ such that $\sigma({\hat
s}_1(L))=s_1(L)$ and such that the fiber of ${\hat s}_1(L)$ above a
point $p\in s_1(L)$ has one element if $p$ is a splitting singularity
and two elements if $p$ is non-splitting. If $L'$ is a different
special leaf of $\mathcal{F}_{\boldsymbol{\omega}}$, then the sets ${\hat s}_1(L')$ and
${\hat s}_1(L)$ are disjoint, even though their projections $s_1(L)$
and $s_1(L')$ through $\sigma$ may intersect in
$\Sigma_1({\boldsymbol{\omega}})$. Hence the special leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ define
a partition of ${\hat \Sigma}_1({\boldsymbol{\omega}})$:
\begin{equation*}
{\hat \Sigma}_1({\boldsymbol{\omega}})=\sqcup_{L=\mathrm{special~leaf~of~}\mathcal{F}_{\boldsymbol{\omega}}}{\hat s}_1(L)~~,
\end{equation*}
which projects through $\sigma$ to the generally non-disjoint
decomposition \eqref{Sigma1Dec}. Viewing ${\hat \mathcal{E}}$ as a
disconnected graph on the vertex set ${\hat \Sigma}_1({\boldsymbol{\omega}})$, we
let $\mathcal{E}$ denote the (generally disconnected) graph obtained from
${\hat \mathcal{E}}$ upon identifying all vertices belonging to ${\hat
s}_1(L)$ for each special leaf $L$ of $\mathcal{F}_{\boldsymbol{\omega}}$, i.e. by
collapsing ${\hat s}_1(L)$ to a point for each special
leaf\footnote{If $s_1(L)$ is empty, this operation does nothing.} $L$.
Let $p:{\hat
\mathcal{E}}\rightarrow \mathcal{E}$ denote the corresponding projection. The graph
$\mathcal{E}$ has one vertex for each special leaf of $\mathcal{F}_{\boldsymbol{\omega}}$ which
adjoins some strong saddle point and an edge for each strong saddle
point. Notice that this edge is a loop when the strong saddle point
is a non-splitting singularity, since a non-splitting singularity
adjoins a single special leaf.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{\input{dimer.pstex_t}}
\caption{Example of the graphs ${\hat \mathcal{E}}$ and $\mathcal{E}$ for a Morse form
foliation $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ with two compact strong singular leaves.
The regular foliation $\mathcal{F}_{\boldsymbol{\omega}}$ of
$M^\ast$ has four special leaves, each of which is compactifiable;
they are depicted using four different colors. At the bottom of the
picture, we depict $\Sigma_1({\boldsymbol{\omega}})$ as well as the schematic
shape of the special leaves in the case $d=3$. The strong singular leaves
of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ correspond to the left and right parts of the
figure at the bottom; each of them is a union of two special leaves
of $\mathcal{F}_{\boldsymbol{\omega}}$ and of singular points. Each special leaf
corresponds to a vertex of $\mathcal{E}$.}
\label{fig:dimer}
\end{center}
\end{figure}
A strong singular leaf of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ can be written as:
\begin{equation}
\label{cLdec2}
\mathcal{L}=(\sqcup_{\alpha=1}^{r+t}{L''_\alpha})\sqcup s(\mathcal{L})~~,
\end{equation}
where $L''_\alpha$ are special leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ (compactifiable
or not). Its set of strong saddle singular points $s_1(\mathcal{L})=
\cup_{\alpha=1}^{r+t}s_1(L''_\alpha)$ is the projection through
$\sigma$ of the set ${\hat s}_1(\mathcal{L})\stackrel{{\rm def.}}{=}
\sqcup_{\alpha=1}^{r+t}{\hat s}_1(L''_\alpha)$. Let ${\hat \mathcal{E}}_\mathcal{L}$
be the (generally disconnected) subgraph of ${\hat \mathcal{E}}$ consisting of
those edges of ${\hat \mathcal{E}}$ which meet ${\hat s}_1(\mathcal{L})$. Then
$s_1(\mathcal{L})$ is obtained from ${\hat \mathcal{E}}_\mathcal{L}$ by contracting each edge
to a single point. If all special leaves $L$ of $\mathcal{F}_{\boldsymbol{\omega}}$ are
known, then ${\hat \mathcal{E}}_{\mathcal{L}}$ uniquely determines the strong singular leaf
$\mathcal{L}$. Indeed, ${\hat \mathcal{E}}_\mathcal{L}$ contains the information about how the
special leaves which form $\mathcal{L}$ meet themselves and each other at the strong saddle
points. Since $\mathcal{L}$ is connected and maximal with this property, the graph $\mathcal{E}_\mathcal{L}$
obtained from ${\hat \mathcal{E}}_\mathcal{L}$ by identifying to a single point the
vertices of each of the subsets ${\hat s}_1(L''_\alpha)$ is a connected
component of $\mathcal{E}$. It follows that the strong singular leaves of
$\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ are in one to one correspondence with the connected
components of the graph $\mathcal{E}$ --- namely, their subgraphs ${\hat
\mathcal{E}}_\mathcal{L}$ are the preimages through $p$ of those components.
\
\noindent In our application, the set
$\mathrm{Sing}({\boldsymbol{\omega}})=\mathcal{W}=\mathcal{W}^+\sqcup\mathcal{W}^-$ consists of positive and
negative chirality points of $\xi$, which are the points where $b$
attains the values $b=\pm 1$. Relation \eqref{meq} implies that $\mathbf{f}$
satisfies:
\begin{equation*}
\oint_{\gamma}\mathbf{f}=0~~
\end{equation*}
for any smooth closed curve $\gamma\in \mathcal{L}\setminus \mathcal{W}$ and hence
$\mathfrak{f}$ restricts to a trivial class in singular cohomology along each
leaf $\mathcal{L}$ of $\bar{\mathcal{F}}$:
\begin{equation*}
\iota^*(\mathfrak{f})=0\in H^1(\mathcal{L},\mathbb{R})~~,
\end{equation*}
where $\iota:\mathcal{L}\hookrightarrow M$ is the inclusion map while
$H^1(\mathcal{L},\mathbb{R})$ is the first singular cohomology group (which coincides
with the first de Rham cohomology group when $\mathcal{L}$ is non-singular).
The pull-back of $\mathbf{f}$ to $\mathcal{L}\setminus \mathcal{W}$ is given by:
\begin{equation*}
\mathbf{f}|_{\mathcal{L}\setminus\mathcal{W}}=\mathbf{f}_\perp=\mathrm{d}_\perp\mathbf{b}~~.
\end{equation*}
Notice that $f_\perp$ and $b$ have well-defined limits (equal to $f_p$
and $b(p)\in \{-1,1\}$) at each singular point $p\in \mathcal{L}\cap \mathcal{W}$ of a
singular leaf $\mathcal{L}$. If $p_1,p_2\in \mathcal{L}\cap \mathcal{W}$ are two singular
points lying on the same singular leaf $\mathcal{L}$ and
$\gamma:(0,1)\rightarrow \mathcal{L}\setminus \mathcal{W}$ is a smooth path which has
limits at $0,1$ given by $p_1$ and $p_2$, then the integral
$\int_\gamma \mathbf{f}$ is well-defined and given by:
\begin{equation*}
\int_\gamma\mathbf{f}=e^{3\Delta(p_2)}b(p_2)-e^{3\Delta(p_1)}b(p_1)~~,
\end{equation*}
where $b(p_i)\in \{-1,1\}$.
\subsection{Homology classes of compact leaves}
Let $H_{\boldsymbol{\omega}}$ be the (necessarily free) subgroup of $H_{n-1}(M,\mathbb{Z})$
generated by the compact leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ and let
$c({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} {\rm rk} H_{\boldsymbol{\omega}}$ denote the number of homologically
independent compact leaves. It was shown in \cite{Gelbukh1} that
$H_{\boldsymbol{\omega}}$ admits a basis consisting of homology classes $[L_i]$
$(i=1,\ldots, c({\boldsymbol{\omega}}))$ of compact leaves\footnote{Such a basis is
provided by the homology classes of the compact leaves corresponding
to the edges of any spanning tree of the foliation graph defined
below.} and that the homology class of any compact leaf $L$ of
$\mathcal{F}_{\boldsymbol{\omega}}$ expands in this basis as:
\begin{equation*}
[L]=\sum_{i=1}^{c({\boldsymbol{\omega}})}n_i[L_i]~~\mathrm{where}~~n_i\in \{-1,1\}~~.
\end{equation*}
Furthermore \cite{Gelbukh1,Gelbukh3}, there exists a system of
$\mathbb{Z}$-linearly independent one-cycles $\gamma_i\in H_1(M,\mathbb{Z})$
$(i=1,\ldots, c({\boldsymbol{\omega}}))$ such that $(\gamma_i,[L_j])=\delta_{ij}$
and such that $\gamma_i$ provide a direct sum decomposition:
\begin{equation*}
H_1(M,\mathbb{Z})=\langle \gamma_1,\ldots, \gamma_{c({\boldsymbol{\omega}})}\rangle \oplus \iota_\ast(H_1(\Delta))~~,
\end{equation*}
where $\iota:\Delta \hookrightarrow M$ is the inclusion map.
Let $\mathcal{H}_{\boldsymbol{\omega}}\stackrel{{\rm def.}}{=} H_{\boldsymbol{\omega}}\cap (\mathrm{ker}\mathrm{per}_{\boldsymbol{\omega}})^\perp$. Then
\cite{Gelbukh5} the subgroup $\mathcal{H}_{\boldsymbol{\omega}}$ is a direct summand in
$H_{\boldsymbol{\omega}}$ while $H_{\boldsymbol{\omega}}$ is a direct summand in
$H_{n-1}(M,\mathbb{Z})$. Furthermore, only the following
values are allowed for ${\rm rk}\mathcal{H}_{\boldsymbol{\omega}}$:
\begin{equation*}
{\rm rk}\mathcal{H}_{\boldsymbol{\omega}}\in \{0,\ldots \rho ({\boldsymbol{\omega}})-2\}\cup\{\rho({\boldsymbol{\omega}})\}~~.
\end{equation*}
\subsection{The Novikov decomposition of $M$}
What we shall call the ``Novikov decomposition'' is a generalization
of the Morse decomposition \cite{Milnor, Morse1, Morse2}, which was
introduced in \cite{Melnikova2, Melnikova3} (see also \cite{FKL,
Honda}) and used extensively in \cite{Gelbukh1}--\cite{Gelbukh9};
the name is motivated by analogy with ``Morse decomposition'', due to
the role which this decomposition plays in the modern study of the
topology of closed one-forms \cite{Farber}. Define $C^\mathrm{max}$ to be the
union of all compact leaves and $C^\mathrm{min}$ to be the union of all
non-compactifiable leaves of $\mathcal{F}_{\boldsymbol{\omega}}$; it is clear that these two
subsets of $M$ are disjoint. Then it was shown in
\cite{ImanishiSing,Levitt1} that both $C^\mathrm{max}$ and $C^\mathrm{min}$ are open
subsets of $M$ which have a common topological small frontier
$F$\footnote{This should not be confused with the internal part of the
flux which is denoted by the same letter.} given by the (disjoint)
union $F_0\cup \mathrm{Sing}({\boldsymbol{\omega}})$, where $F_0$ is the union of all those
leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ which are compactifiable but non-compact:
\begin{equation*}
\mathrm{fr} C^\mathrm{max}=\mathrm{fr} C^\mathrm{min}=F\stackrel{{\rm def.}}{=} F_0\sqcup \mathrm{Sing}({\boldsymbol{\omega}})~~.
\end{equation*}
Each of the open sets $C^\mathrm{max}$ and $C^\mathrm{min}$ has a finite number of
connected components, which are called the {\em maximal} and {\em
minimal} components of the set $M\setminus F=C^\mathrm{max}\sqcup C^\mathrm{min}$. We
let:
\begin{itemize}
\itemsep0em
\item $N_\mathrm{max}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} |\pi_0(C^\mathrm{max})|$ denote the number of maximal components
\item $N_\mathrm{min}({\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} |\pi_0(C^\mathrm{min})|$ denote the number of minimal components
\end{itemize}
Indexing these by $C^\mathrm{max}_j$ and $C^\mathrm{min}_a$ (where $j=1,\ldots,N_\mathrm{max}({\boldsymbol{\omega}})$
and $a=1,\ldots,N_\mathrm{min}({\boldsymbol{\omega}})$), we have:
\begin{equation}
\label{CMm}
C^\mathrm{max}=\sqcup_{j=1}^{N_\mathrm{max}({\boldsymbol{\omega}})}C_j^\mathrm{max}~~,~~C^\mathrm{min}=\sqcup_{a=1}^{N_\mathrm{min}({\boldsymbol{\omega}})}C_a^\mathrm{min}~~
\end{equation}
and hence (since \eqref{CMm} are {\em finite} and {\em disjoint}
unions) we also have:
\begin{eqnarray*}
&&\overline{C^\mathrm{max}}=\cup_{j=1}^{N_\mathrm{max}({\boldsymbol{\omega}})}\overline{C_j^\mathrm{max}}~~,
~~\overline{C^\mathrm{min}}=\cup_{a=1}^{N_\mathrm{min}({\boldsymbol{\omega}})}\overline{C_a^\mathrm{min}}~~,\\
&&F=\mathrm{fr} C^\mathrm{max}=\cup_{j=1}^{N_\mathrm{max}({\boldsymbol{\omega}})}\mathrm{fr} C_j^\mathrm{max}=\mathrm{fr} C^\mathrm{min}=\cup_{a=1}^{N_\mathrm{min}({\boldsymbol{\omega}})}\mathrm{fr} C_a^\mathrm{min}~~.
\end{eqnarray*}
Notice that the unions appearing in these equalities need not be
disjoint anymore, in particular the small frontiers of two distinct maximal
components can intersect each other and similarly for two distinct
minimal components. Let \footnote{$\Delta$ should not be confused with
the warp factor.}:
\begin{equation*}
\Delta\stackrel{{\rm def.}}{=} M\setminus C^\mathrm{max}=\overline{C^\mathrm{min}}=C^\mathrm{min}\sqcup F~~
\end{equation*}
be the union of all non-compact leaves and singularities. This subset
has a finite number (which we denote by $v({\boldsymbol{\omega}})$) of connected
components $\Delta_s$:
\begin{equation}
\label{DeltaDec}
\Delta=\sqcup_{s=1}^{v({\boldsymbol{\omega}})}\Delta_s~~.
\end{equation}
The connected components of $F$ (which are again in finite number) are
finite unions of singular points and of non-compact but compactifiable
leaves of $\mathcal{F}_{\boldsymbol{\omega}}$ which coincide with the `compact parts' of
the singular leaves of $\bar{\mathcal{F}}_{\boldsymbol{\omega}}$ (see \eqref{cLdec}).
One can show \cite{FKL, Levitt2} that each maximal component $C_j^\mathrm{max}$ is
diffeomorphic to the open unit cylinder over any of the (compact)
leaves $L_j$ of the restricted foliation $\mathcal{F}_{\boldsymbol{\omega}}|_{C_j^\mathrm{max}}$,
through a diffeomorphism which maps this restricted foliation to the
foliation of the cylinder given by its sections $L_j\times \{t\}$:
\begin{equation}
\label{maxcyl}
C_j^\mathrm{max}\simeq L_j\times (0,1)~~.
\end{equation}
In particular, we have:
\begin{equation*}
\rho({\boldsymbol{\omega}}|_{C_j^\mathrm{max}})=0~~.
\end{equation*}
Being connected, each non-compactifiable leaf $L$ of $\mathcal{F}_{\boldsymbol{\omega}}$ is
contained in exactly one minimal component. It was shown in
\cite{ImanishiSing} (see also Appendix of \cite{Levitt1}) that $L$ is
{\em dense} in that minimal component. Furthermore, one has
\cite{FKL, Levitt1}:
\begin{equation*}
\rho({\boldsymbol{\omega}}|_{C_a^\mathrm{min}})\geq 2~~,~~a=1,\ldots,N_\mathrm{min}({\boldsymbol{\omega}})~~.
\end{equation*}
In particular, any minimal component $C_a^\mathrm{min}$ must satisfy $b_1(C_a^\mathrm{min})\geq 2$.
\paragraph{Definition.}
The foliation $\mathcal{F}_{\boldsymbol{\omega}}$ is called {\em compactifiable} if each of its
leaves is compactifiable, i.e. if it has no minimal components.
\subsection{The foliation graph}
Since each maximal component $C_j^\mathrm{max}$ is a cylinder, its frontier
consists of either one or two connected components. When the frontier
of $C_j^\mathrm{max}$ is connected, there exists exactly one connected component
$\Delta_{s_j}$ of $\Delta$ such that $\mathrm{fr} C_j^\mathrm{max}\subset
\Delta_{s_j}$. When the frontier of $C_j^\mathrm{max}$ has two connected
components, there exist distinct indices $s'_1$ and $s''_j$ such that
these components are subsets of $\Delta_{s'_j}$ and $\Delta_{s''_j}$,
respectively. These observations allow one to define a graph as follows \cite{FKL, Honda}:
\paragraph{Definition.}
The {\em foliation graph} $\Gamma_{\boldsymbol{\omega}}$ of ${\boldsymbol{\omega}}$ is the unoriented
graph whose vertices are the connected components $\Delta_s$ of
$\Delta$ and whose edges are the maximal components $C_j^\mathrm{max}$. An edge
$C_j^\mathrm{max}$ is incident to a vertex $\Delta_s$ iff a connected component
of $\mathrm{fr} C_j^\mathrm{max}$ is contained in $\Delta_s$; it is a loop at $\Delta_s$ iff
$\mathrm{fr} C_j^\mathrm{max}$ is connected and contained in $\Delta_s$. A vertex $\Delta_s$ of
$\Gamma_{\boldsymbol{\omega}}$ is called {\em exceptional} (or of {\em type II}) if it contains at least one
minimal component; otherwise, it is called {\em regular} (or of {\em type I}).
\
\noindent The terminology {\em type I}, {\em type II} for vertices is
used in \cite{Gelbukh7}. Since $M$ is connected, it follows that $\Gamma_{\boldsymbol{\omega}}$ is a connected
graph. Notice that $\Gamma_{\boldsymbol{\omega}}$ can have loops and multiple edges as well
as terminal vertices.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.5]{genfolgraph.eps}
\caption{An example of foliation graph. Regular (a.k.a type I)
vertices are represented by black dots, while exceptional
(a.k.a. type II) vertices are represented by green blobs. All
terminal vertices are regular vertices and correspond to center
singularities. Notice that the graph can have multiple edges as
well as loops.}
\end{center}
\end{figure}
Let ${\rm deg} \Delta_s$ denote the degree (valency) of $\Delta_s$ as a
vertex of the foliation graph. A regular vertex $\Delta_s$ can be of two types:
\begin{itemize}
\itemsep 0.0em
\item A center singularity $\Delta_s=\{p\}$ (with $p\in \Sigma_0({\boldsymbol{\omega}})$), when ${\rm deg}
\Delta_s=1$. In this case, $\Delta_s$ is a terminal vertex of $\Gamma_{\boldsymbol{\omega}}$.
\item A compact singular leaf when ${\rm deg} \Delta_s \geq 2$.
\end{itemize}
Every exceptional vertex is a union of minimal components, singular points and compactifiable non-compact leaves of
$\mathcal{F}_{\boldsymbol{\omega}}$. For any vertex $\Delta_s$ of the foliation graph, we have \cite{Gelbukh7}:
\begin{equation*}
|\Delta_s \cap \Sigma_1^\mathrm{sp}({\boldsymbol{\omega}})|\geq {\rm deg} \Delta_s+2m_{\Delta_s}-2~~,
\end{equation*}
where $m_{\Delta_s}$ is the number of minimal components contained in
$\Delta_s$. In particular, a regular vertex with ${\rm deg} \Delta_s>2$ is a compact singular leaf which
contains at least one splitting strong saddle singularity. The number of edges $e(\Gamma_{\boldsymbol{\omega}})$ equals
$N_\mathrm{max}({\boldsymbol{\omega}})$ while the number of vertices $v(\Gamma_{\boldsymbol{\omega}})$ equals
$v({\boldsymbol{\omega}})$. Furthermore, it was shown in \cite{Gelbukh8} that the
cycle rank $b_1(\Gamma_{\boldsymbol{\omega}})$ equals $c({\boldsymbol{\omega}})$. Thus:
\begin{equation*}
e(\Gamma_{\boldsymbol{\omega}})=N_\mathrm{max}({\boldsymbol{\omega}})~~,~~ v(\Gamma_{\boldsymbol{\omega}})=v({\boldsymbol{\omega}})\leq |\mathrm{Sing}({\boldsymbol{\omega}})|~~,~~b_1(\Gamma_{\boldsymbol{\omega}})=c({\boldsymbol{\omega}})~~.
\end{equation*}
The graph Euler identity $e(\Gamma_{\boldsymbol{\omega}})=v(\Gamma_{\boldsymbol{\omega}})+b_1(\Gamma_{\boldsymbol{\omega}})-1$
implies:
\begin{equation*}
N_\mathrm{max}({\boldsymbol{\omega}})=c({\boldsymbol{\omega}})+v({\boldsymbol{\omega}})-1 \leq c({\boldsymbol{\omega}})+|\mathrm{Sing}({\boldsymbol{\omega}})|-1~~,
\end{equation*}
where we noticed that $v({\boldsymbol{\omega}})\leq |\mathrm{Sing}({\boldsymbol{\omega}})|$ since each $\Delta_s$ contains at least one
singular point. An example of foliation graph is depicted in Figure 3.
\paragraph{Constraints on the foliation graph from the irrationality rank of ${\boldsymbol{\omega}}$.}
When the chiral locus $\mathcal{W}$ is empty (i.e. when ${\boldsymbol{\omega}}$ is
nowhere-vanishing) we have $\mathrm{Sing}({\boldsymbol{\omega}})=\emptyset$ and ${\bar
\mathcal{F}}_{\boldsymbol{\omega}}=\mathcal{F}_{\boldsymbol{\omega}}$ is a regular foliation. Even though this
doesn't fit our assumption $\mathrm{Sing}{\boldsymbol{\omega}}\neq \emptyset$, one can
define a (degenerate) foliation graph also in this situation (which was
considered in \cite{g2}). In this case, knowledge of the irrationality
rank of $\mathfrak{f}$ determines the topology of the foliation $\mathcal{F}_{\boldsymbol{\omega}}$
for any ${\boldsymbol{\omega}}\in \mathfrak{f}$. Namely, one has only two possibilities (see
Figure \ref{fig:regfolgraph}):
\begin{itemize}
\itemsep 0.0em
\item $\rho(\mathfrak{f})=1$, i.e. $\mathfrak{f}$ is projectively rational. Then there
exists exactly one maximal component (which coincides with $M$) and
no minimal component. The foliation ``graph'' consists of one loop
and has no vertices; $M$ is a fibration over $S^1$ as a
consequence of Tischler's theorem \cite{Tischler}.
\item $\rho(\mathfrak{f})>1$, i.e. $\mathfrak{f}$ is projectively irrational. There exists
exactly one minimal component (which coincides with $M$) and no
maximal component, i.e. $\mathcal{F}_{\boldsymbol{\omega}}$ is a minimal foliation. Then
the foliation graph consists of a single exceptional vertex and
every leaf of $\mathcal{F}_{\boldsymbol{\omega}}$ is dense in $M$. As explained in \cite{g2}, the
noncommutative geometry of the leaf space is described by the $C^\ast$ algebra
$C(M/\mathcal{F}_{\boldsymbol{\omega}})$ of the foliation, which is a non-commutative torus of dimensions
$\rho({\boldsymbol{\omega}})$. Notice that this refined topological information is not reflected
by the foliation graph.
\end{itemize}
\!\!\!\!\!\begin{figure}
\centering
\!\!\!\!\!\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.2\linewidth]{regfol1.eps}
\
\caption{Foliation graph when $\mathcal{W}=\emptyset$ and $\rho({\boldsymbol{\omega}})=1$.}
\end{subfigure}~~~~~~
\begin{subfigure}{.5\textwidth}
\centering
\vspace{1.5em}
\includegraphics[width=0.05\linewidth]{regfol2.eps}
\vspace{2.6em}
\caption{Foliation graph when $\mathcal{W}=\emptyset$ and $\rho({\boldsymbol{\omega}})>1$.}
\end{subfigure}
\caption{Degenerate foliation graphs in the everywhere non-chiral case.}
\label{fig:regfolgraph}
\end{figure}
\
\noindent The situation is much more complicated when
$\mathrm{Sing}({\boldsymbol{\omega}})$ is non-empty, in that knowledge of
$\rho({\boldsymbol{\omega}})$ does not suffice to specify the topology of the
foliation. In this case, knowledge of $\rho(\mathfrak{f})$ allows one to say
only the following:
\begin{itemize}
\itemsep 0.0em
\item When $\rho(\mathfrak{f})=1$, then the foliation $\mathcal{F}_{\boldsymbol{\omega}}$ is
compactifiable for any ${\boldsymbol{\omega}}\in \mathfrak{f}$ \cite{FKL} and the inequality
\eqref{crank} below requires $c({\boldsymbol{\omega}})\geq 1$. Hence the foliation
graph $\Gamma_{\boldsymbol{\omega}}$ has only regular vertices and must have at
least one cycle. Except for this, nothing else can be
said about $\mathcal{F}_{\boldsymbol{\omega}}$ only by knowing that $\rho(\mathfrak{f})=1$. Indeed,
it was shown in \cite{Gelbukh8} that any compactifiable Morse form
foliation $\mathcal{F}_{{\boldsymbol{\omega}}'}$ with $c({\boldsymbol{\omega}}')\geq 1$ can be realized
as the foliation defined by a Morse form ${\boldsymbol{\omega}}$ belonging to a
projectively rational cohomology class. It was also shown in
loc. cit. that such a foliation can in fact be realized by a Morse
form of any irrationality rank lying between $1$ and
$c({\boldsymbol{\omega}}')$, inclusively.
\item When $\rho(\mathfrak{f})>1$, then $\mathcal{F}_{\boldsymbol{\omega}}$ may be
either compactifiable or non-compactifiable, hence the foliation
graph may or may not have exceptional vertices; when $\mathcal{F}_{\boldsymbol{\omega}}$
is compactifiable, then $\Gamma_{\boldsymbol{\omega}}$ has no exceptional vertices
and has a number of cycles at least equal to
$\rho({\boldsymbol{\omega}})$. Criteria for compactifiability of $\mathcal{F}_{\boldsymbol{\omega}}$ can
be found in \cite{FKL, Gelbukh1,Gelbukh5} and are given below.
\end{itemize}
\paragraph{Theorem \cite{FKL, Gelbukh1,Gelbukh5}.}The following statements are equivalent:
\begin{enumerate}[(a)]
\itemsep0em
\item $\mathcal{F}_{\boldsymbol{\omega}}$ is compactifiable
\item The period morphism $\mathrm{per}_\mathfrak{f}:\pi_1(M)\rightarrow \mathbb{R}$ factorizes through a group morphism $\pi_1(M)\rightarrow K$, where $K$ is a free group
\item $H_{\boldsymbol{\omega}}^\perp\subset \mathrm{ker} {\boldsymbol{\omega}}$
\item ${\rm rk} \mathcal{H}_{\boldsymbol{\omega}}=\rho({\boldsymbol{\omega}})$.
\end{enumerate}
The first criterion above is Proposition 2 in
\cite[Sec. 8.2]{FKL}. Since $\mathcal{H}_{\boldsymbol{\omega}}\subset H_{\boldsymbol{\omega}}$, we have
${\rm rk} \mathcal{H}_{\boldsymbol{\omega}}\leq {\rm rk} H_{\boldsymbol{\omega}}=c({\boldsymbol{\omega}})$ and the theorem shows
that compactifiability of $\mathcal{F}_{\boldsymbol{\omega}}$ requires:
\begin{equation}
\label{crank}
\rho({\boldsymbol{\omega}})\leq c({\boldsymbol{\omega}})~~.
\end{equation}
\paragraph{Remark.}
By its construction, the foliation graph discards topological
information about the restriction of the foliation to the minimal
components of the Novikov decomposition, which are represented in the
graph by exceptional vertices. As in the case
$\mathrm{Sing}({\boldsymbol{\omega}})=\emptyset$, the $C^\ast$ algebra of the foliation
should provide more refined information about the topology of ${\bar
\mathcal{F}}_{\boldsymbol{\omega}}$ than the foliation graph. To our knowledge, this
$C^\ast$ algebra has not been computed for foliations given by a Morse
1-form.
\paragraph{The oriented foliation graph.}
For each maximal component $C_j^\mathrm{max}$, the diffeomorphism \eqref{maxcyl} can be
chosen\footnote{The sign of $\int_{\gamma_j}{\boldsymbol{\omega}}$ does not depend on
the choice of $\gamma$ since ${\boldsymbol{\omega}}$ vanishes on the leaves of
$\mathcal{F}_{\boldsymbol{\omega}}$. If the sign is negative, then it can be made positive
by composing the diffeomorphism \eqref{maxcyl} with $\mathrm{id}_{L_j}\times
R$, where $R\in \mathrm{Diff}_-((0,1))$ is any orientation-reversing
diffeomorphism of the interval $(0,1)$.} such that the sign of the
integral $\int_{\gamma_j}{\boldsymbol{\omega}}$ is positive along any smooth curve
$\gamma_j:(0,1)\rightarrow C^\mathrm{max}_j$ which projects to the interval
$(0,1)$. Identifying the corresponding edge $e_j$ with this interval,
this gives a canonical orientation $\vec{e}_j$ of $e_j$ which
corresponds to ``moving along $e_j$ in the direction of increasing
value if $h_j$'', where $h_j$ is any locally-defined smooth function
on an open subset of $C^\mathrm{max}_j$ whose exterior derivative equals
${\boldsymbol{\omega}}$. It follows that the foliation graph $\Gamma_{\boldsymbol{\omega}}$ admits
a canonical orientation, which makes it into the {\em oriented
foliation graph} $\vec{\Gamma}_{\boldsymbol{\omega}}$.
\paragraph{Weights on the oriented foliation graph.}
Using the canonical orientation, the integrals:
\begin{equation}
\label{wdef}
w_j\stackrel{{\rm def.}}{=} \int_{\gamma_j}{\boldsymbol{\omega}}
\end{equation}
(whose value does not depend on the choice of $\gamma_j$ as above)
provide canonical positive weights on $\vec{\Gamma}_{\boldsymbol{\omega}}$
\cite{FKL,Honda}. These weights can be used \cite{Gelbukh3} to describe
the set of Morse 1-forms ${\boldsymbol{\omega}}$ which have the property that ${\bar
\mathcal{F}}_{\boldsymbol{\omega}}=\bar{\mathcal{F}}$ for a fixed singular foliation $\bar{\mathcal{F}}$.
\paragraph{Expression for the weights in terms of $\mathbf{b}$ and $\mathbf{f}$.}
In our application, the vector field $n={\hat V}^\sharp\in
\Gamma(T\mathcal{U})$ is orthogonal to the leaves of $\mathcal{F}$ and satisfies:
\begin{equation}
\label{nomega}
n\lrcorner {\boldsymbol{\omega}}=4\kappa e^{3\Delta}||V||=n\lrcorner \mathbf{f} -\partial_n \mathbf{b}\geq 0~~
\end{equation}
as a consequence of \eqref{meq}. Equality with zero in the right hand
side occurs only at the points of $\mathcal{W}=\mathrm{Sing}({\boldsymbol{\omega}})$. It follows
that the orientation of the edges of the foliation graph is in the
direction of $n$ and that we can take $\gamma_j$ to be any integral
curve $\ell_j$ of the vector field $n|_{C_j^\mathrm{max}}$. Relation
\eqref{nomega} gives:
\begin{equation*}
w_j=\mathbf{b}_j(\gamma_j(1))-\mathbf{b}_j(\gamma_j(0))+\int_{\gamma_j}\mathbf{f}~~.
\end{equation*}
When $\mathcal{F}_{\boldsymbol{\omega}}$ is compactifiable, this relation implies that the
sum of weights along all edges of a cycle of the {\em oriented}
foliation graph $\vec{\Gamma}_{\boldsymbol{\omega}}$ equals the period of $\mathfrak{f}$ along
the corresponding homology 1-cycle $\alpha\in H_1(M)$ of $M$:
\begin{equation*}
\sum_{\vec{e}_j~\mathrm{in~a~cycle~of~}\vec{\Gamma}_{\boldsymbol{\omega}}}w_j=\int_\alpha\mathfrak{f}~~.
\end{equation*}
\subsection{The fundamental group of the leaf space}
Even though the quotient topology of the leaf space $M/{\bar
\mathcal{F}}_{\boldsymbol{\omega}}$ can be very poor, one can use the classifying space
$\mathcal{G}$ of the holonomy pseudogroup of the regular foliation
$\mathcal{F}_{\boldsymbol{\omega}}$ \cite{Hclass} to define the fundamental group of the
leaf space through \cite{Levitt3}:
\begin{equation*}
\pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})\stackrel{{\rm def.}}{=} \pi_1(B\mathcal{G})~~.
\end{equation*}
Notice that $B\mathcal{G}$ is an Eilenberg-MacLane space of type $K(\pi,1)$ \cite{Hclass},
(i.e. all its homotopy groups vanish except for the fundamental group)
since $\mathcal{F}_{\boldsymbol{\omega}}$ is defined by a closed one-form and
hence the holonomy groups of its leaves are trivial. One finds
\cite{Levitt3}:
\begin{equation*}
\pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})=\pi_1(M)/\mathcal{L}_{\boldsymbol{\omega}}~~,
\end{equation*}
where $\mathcal{L}_{\boldsymbol{\omega}}$ is the smallest normal subgroup of $\pi_1(M)$
which contains the fundamental group of each leaf of $\mathcal{F}_{\boldsymbol{\omega}}$.
Notice that $M\setminus \mathrm{Sing}({\boldsymbol{\omega}})$ is connected (since $M$ is) and that the
inclusion induces an isomorphism $\pi_1(M\setminus
\mathrm{Sing}({\boldsymbol{\omega}}))\simeq \pi_1(M)$, since we assume $\dim M\geq 3$ and
hence $\mathrm{Sing}{\boldsymbol{\omega}}$ has codimension at least 3 in $M$. In particular,
the period map of ${\boldsymbol{\omega}}$ can be identified with that of
${\boldsymbol{\omega}}|_{M\setminus \mathrm{Sing}({\boldsymbol{\omega}})}$. Since ${\boldsymbol{\omega}}$ vanishes along
the leaves of $\mathcal{F}_{\boldsymbol{\omega}}$, this map factors through the projection
$\pi_1(M)\rightarrow \pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})$, inducing a map
$\mathrm{per}_0({\boldsymbol{\omega}}):\pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})\rightarrow \mathbb{R}$.
A minimal component $C^\mathrm{min}_a$ is called {\em weakly complete}
\cite{Levitt3} if any curve $\gamma\subset C^\mathrm{min}_a$ contained in
$C^\mathrm{min}_a$ and for which $\int_\gamma{\boldsymbol{\omega}}$ vanishes has its two
endpoints on the same leaf of $\mathcal{F}_{\boldsymbol{\omega}}$; various equivalent
characterizations of weakly complete minimal components can be found
in loc. cit. Let:
\begin{itemize}
\itemsep 0.0em
\item $N'_\mathrm{min}({\boldsymbol{\omega}})$ denote the number of minimal components which are
not weakly complete
\item $N''_\mathrm{min}({\boldsymbol{\omega}})$ denote the number of minimal components which are
weakly complete
\item $C^\mathrm{min}_{a_1},\ldots,C^\mathrm{min}_{a_k}$ (where $1\leq a_1<\ldots
<a_{N''_\mathrm{min}({\boldsymbol{\omega}})}\leq N_\mathrm{min}({\boldsymbol{\omega}})$) denote those minimal components
of the Novikov decomposition which are weakly complete
\item ${\boldsymbol{\omega}}_j\stackrel{{\rm def.}}{=} {\boldsymbol{\omega}}|_{C^m_{a_j}}$ denote the restriction of
${\boldsymbol{\omega}}$ to the weakly complete minimal component $C^\mathrm{min}_{a_j}$
\item $\Pi_j({\boldsymbol{\omega}}) \stackrel{{\rm def.}}{=} \Pi({\boldsymbol{\omega}}_j)$ denote the period group of
${\boldsymbol{\omega}}_j$. Then $\Pi_j({\boldsymbol{\omega}})$ is a free Abelian group of rank
${\rm rk}\Pi_j({\boldsymbol{\omega}})=\rho(\omega_j)\geq 2$ \cite{Levitt3}.
\end{itemize}
With these notations, it was shown in \cite{Levitt3} that
$\pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})$ is isomorphic with a free
product of free Abelian groups:
\begin{equation*}
\pi_1(M/\bar{\mathcal{F}}_{\boldsymbol{\omega}})\simeq F_{\boldsymbol{\omega}} \ast \Pi_1({\boldsymbol{\omega}})\ast\ldots \ast \Pi_{N''_\mathrm{min}({\boldsymbol{\omega}})}({\boldsymbol{\omega}})~~,
\end{equation*}
where $\ast$ denotes the free product of groups. Furthermore
\cite{Levitt3, Gelbukh2}, the free group $F_{\boldsymbol{\omega}}$ factors as:
\begin{equation*}
F_{\boldsymbol{\omega}}\simeq \pi_1(\Gamma_{\boldsymbol{\omega}})\ast \mathbb{Z}^{\ast K({\boldsymbol{\omega}})}~~,
\end{equation*}
where $\pi_1(\Gamma_{\boldsymbol{\omega}})\simeq \mathbb{Z}^{\ast c({\boldsymbol{\omega}})}$ is the
fundamental group of the foliation graph and $K({\boldsymbol{\omega}})$ is a
non-negative integer which satisfies $K({\boldsymbol{\omega}})\geq N'_\mathrm{min}({\boldsymbol{\omega}})$
and $K({\boldsymbol{\omega}})+c({\boldsymbol{\omega}})+N''_\mathrm{min}({\boldsymbol{\omega}})\leq b'_1(M)$. Here,
$b'_1(M)$ denotes the first noncommutative Betti number of $M$
\cite{Levitt1}, whose definition is recalled in Appendix
\ref{app:fol} (which also summarizes some further information
on the topology of $\bar{\mathcal{F}}$).
\subsection{On the relation to compactifications of M-theory on 7-manifolds}
One way in which one may attempt to think about our class of
compactifications is via a two-step reduction of eleven-dimensional
supergravity, as follows:
\begin{enumerate}
\item First, reduce eleven-dimensional supergravity along a leaf of
the foliation down to a supergravity theory in four dimensions; this
would of course be a {\em gauged} supergravity theory since the
restrictions of $F$ and $f$ to a leaf are generally non-trivial.
\item Further reduce the resulting four-dimensional theory down to
three dimensions, along the ``one-dimensional space'' orthogonal to
the leaf.
\end{enumerate}
\noindent This way of thinking, which corresponds to an attempt at
generalizing the well-known, but much simpler case of ``generalized
Scherk-Schwarz compactifications with a twist'' (see, for example,
\cite{Vandoren}), turns out to be rather naive, for the following
reasons:
\begin{itemize}
\item In the general case when $\mathcal{W}$ is nonempty and differs from $M$,
there is no such thing as a ``typical leaf'' of the regular
foliation $\mathcal{F}$ of $\mathcal{U}=M\setminus \mathcal{W}$, in the sense that the
leaves of this foliation are not all diffeomorphic with each
other. As explained above, what happens instead is that the leaves
of the restriction of $\mathcal{F}$ to each of the maximal or minimal
components of the Novikov decomposition of M are diffeomorphic with
each other, which means that for each component of the Novikov
decomposition one generally has a distinct diffeomorphism class of
leaves. As such, it is unclear which of these seven-manifolds one is
supposed to reduce on in step 1 above. Furthermore, the extended
foliation $\bar{\mathcal{F}}$ also contains singular leaves, and it is not
immediately clear (from a Physics perspective) how to correctly
reduce eleven-dimensional supergravity, in the presence of fluxes,
on such singular seven-manifolds. One should also note that the
leaves of the restriction of $\mathcal{F}$ to a minimal component of the
Novikov decomposition are non-compact, so the reduction along such
leaves cannot be understood as a Kaluza-Klein reduction in the
ordinary sense.
\item In general, there is no nice ``one-dimensional space''
transverse to the leaves. As explained above, the best candidate for
such a space is a non-commutative space whose ``commutative parts''
can be described by the foliation graph, but where some unknown
non-commutative pieces have to be pasted in at the exceptional
vertices. It is of course already unclear how to correctly reduce a
four-dimensional supergravity theory on a graph, let alone on a
non-commutative space.
\end{itemize}
As pointed out in \cite[Subsection 4.4.]{g2}, many of the issues
mentioned above already appear in the much simpler case when $\xi$ is everywhere
non-chiral. In that situation, the foliation graph is either a circle
(and the Novikov decomposition is reduced to a single maximal
component, all leaves being compact and mutually diffeomorphic, being the fibers of
a fibration over the circle) or a
non-commutative torus of dimension given by the projective
irrationality rank of ${\boldsymbol{\omega}}$ (in which case the Novikov
decomposition is reduced to a single minimal component, all leaves
being non-compact, mutually diffeomorphic and dense in $M$). Only the
first of these two cases has a chance at a meaningful interpretation
as a ``generalized Scherk-Schwarz compactification with a twist'',
where the twist is provided by the Ehresmann connection discussed in
\cite[Appendix E]{g2}, whose parallel transport generates the defining diffeomorphism $\phi_{a_\mathfrak{f}}$
which presents $M$ as a mapping torus in that case (see \cite[Subsection 4.2]{g2}). A proper analysis of that case (which is the
simplest of this class of compactifications) is already considerably
more subtle than might seem at first sight, for the following
reason. As shown in \cite[Subsection 2.6]{g2}, the restriction of
$\xi$ to a leaf $L$ of $\mathcal{F}$ induces the spinor $\eta_0$ of equation
\eqref{eta0} (see also \cite[eq. (2.21)]{g2}) which, as shown in
loc. cit., is the normalized Majorana spinor (in the seven-dimensional
sense) along the seven-manifold $L$ which induces its $G_2$ structure
and which should be used to perform the compactification of
eleven-dimensional supergravity on $L$ -- a reduction which would
constitute the first step outlined above. Notice,
however, that what one needs in our case is not the standard $\mathcal{N}=1$
compactification of eleven-dimensional supergravity on a 7-manifold with $G_2$ structure
which is usually considered in the literature following
\cite{Minasian}, since the latter is a compactification down to
four-dimensional Minkowski space --- while what would be needed in our
case would be a compactification down to a space
which is related to $\mathrm{AdS}_3\times S^1$. Also recall from
\cite[Subsection 2.6]{g2} that $\eta_0$ is a Majorana (a.k.a. real)
spinor on $L$ (in the seven-dimensional sense) with respect to a real
structure which is dependent of the precise leaf $L$ under consideration
and not only of its diffeomorphism class. In particular, the $G_2$
structure depends on the leaf $L$ (it varies from leaf to leaf) in the
complicated manner described by Theorems 1 and 2 of \cite{g2} and
it is not invariant under the parallel transport of the
Ehresmann connection mentioned above, so proper analysis of the second
step of the reduction is considerably more involved than what one might expect based on analogy with
previous work on Scherk-Schwarz-like constructions.
A conceptually better (and more uniform) way to think of the
``relation to seven-dimensional compactifications'' (beyond the
results of \cite{g2} and of this paper, which can be viewed as already
providing such a relation since they express very explicitly the
geometry of $M$ in terms of the seven-dimensional geometry of the
leaves of the foliation) is to consider the ``partial
decompactification limit'' in which the leaf space is ``large''. The
correct way to formulate this mathematically employs the theory of
adiabatic limits of foliations (see, for example, \cite{Kordyukov}),
which, in its most general form, concerns their behavior when the leaf
space (understood, in general, as a non-commutative space) is
``large'' in an appropriate spectral sense. This relates to extending
the ordinary adiabatic argument (which lies behind a proper
Kaluza-Klein formulation of the idea of ``two-step reduction'') to the
case of foliations. Though this subject is well-outside the scope of
the present paper, we mention that such a way of formulating the
problem leads to non-trivial mathematical questions given the fact
that the adiabatic limit of foliations is poorly understood for the
case of foliations which are not Riemannian, such as those which are
of interest in our case (see Remark 3 after Theorem 2 of reference
\cite{g2}). The adiabatic limit for the general situation when one has
to deal with a singular foliation $\bar{\mathcal{F}}$ does not seem to have been
investigated in the Mathematics literature.
\subsection{A non-commutative description of the leaf space ?}
Recall from \cite{g2} that the leaf space of $\mathcal{F}$ admits a very
explicit description as a non-commutative torus in the everywhere
non-chiral case (the case $\mathcal{U}=M$, when the foliation graph is reduced
either to a circle or to a single exceptional vertex). This leads to
the speculation \cite{Levitt3} that the topological information which
is lost when constructing the exceptional vertices of the foliation graph
in the general case could be encoded
by some sort of non-commutative geometry, as expected from the fact
that such vertices are constructed by collapsing at least one minimal
component of the Novikov decomposition to a single point; since the
minimal components are foliated by dense leaves, the $C^\ast$-algebra
of their leaf space must be non-commutative. Unfortunately, it is
non-trivial to make this expectation precise, because one also has to
take into account the effect of the singular leaves of $\bar{\mathcal{F}}$, so progress on this question would
first require giving a proper definition/construction of the
$C^\ast$-algebras of singular foliations in the sense of Haefliger, a
task which, to our knowledge, has not yet been carried out in the mathematics
literature. One may hope that some modification of the construction of
\cite{Androulidakis1, Androulidakis2} (which applies to singular
foliations in the sense of Stefan-Sussmann) would lead to a solution
of this problem for the case of Haefliger structures, a case which is
logically orthogonal to that considered in loc. cit.
\section{Conclusions and further directions}
We studied $\mathcal{N}=1$ compactifications of eleven-dimensional
supergravity down to $\mathrm{AdS}_3$ in the case when the internal part $\xi$
of the supersymmetry generator is not required to be everywhere
non-chiral, but under the assumption that $\xi$ is not chiral
everywhere. We showed that, in such cases, the Einstein equations
require that the locus $\mathcal{W}$ where $\xi$ becomes chiral must be a set
with empty interior and therefore of measure zero. The regular
foliation of \cite{g2} is replaced in such cases by a singular
foliation ${\bar \mathcal{F}}$ (equivalently, by a Haefliger structure on $M$)
which ``integrates'' a cosmooth singular distribution (generalized
bundle) $\mathcal{D}$ on $M$. The singular leaves of $\bar{\mathcal{F}}$ are precisely
those leaves which meet the chiral locus $\mathcal{W}$, thus acquiring
singularities on that locus.
We discussed the topology of such singular foliations in the generic
case when ${\boldsymbol{\omega}}$ is a Morse one-form, showing that it is governed
by the foliation graph of \cite{MelnikovaThesis,MelnikovaGraph,
FKL}. On the non-chiral locus, we compared the foliation approach of
\cite{g2} with the $\mathrm{Spin}(7)_\pm$ structure approach of
\cite{Tsimpis}, giving explicit formulas for translating between the
two methods and showing that they agree. It would be interesting to
study what supplementary constraints --- if any --- may be imposed on
the topology of $\bar{\mathcal{F}}$ (and on its foliation graph) by the
supersymmetry conditions; this would require, in particular, a
generalization of the work of \cite{FKL, Honda}.
The singular foliation $\bar{\mathcal{F}}$ is defined by a closed one-form
${\boldsymbol{\omega}}$ whose zero set coincides with the chiral locus. Along the
leaves of $\bar{\mathcal{F}}$ and outside the intersection of the latter with
$\mathcal{W}$, the torsion classes are determined by the fluxes \cite{g2}. For
the singular leaves in the Morse case, this leads to a more
complicated version of the problems which were studied in
\cite{Kcones1, Kcones2} for metrics with $G_2$ holonomy (the case of
torsion-free $G_2$ structures).
The backgrounds discussed in this paper display a rich interplay
between spin geometry, the theory of G-structures, the theory of
foliations and the topology of closed one-forms \cite{Farber}. This
suggests numerous problems that could be approached using the methods
and results of reference \cite{g2} and of this paper --- not least of
which concerns the generalization to the case of singular foliations
of the non-commutative geometric description of the leaf space. In
this regard, we note that a complete solution of this problem requires
extending the construction of the $C^\ast$ algebra of regular
foliations to the case of singular foliations in the sense of
Haefliger --- a generalization which would be different from
(and, in fact, ``orthogonal'' to) that performed in
\cite{Androulidakis1, Androulidakis2} for the case of singular
foliations in the sense of Stefan-Sussmann. This problem is unsolved
already for the case of singular foliations defined by a Morse
one-form (the difficulty being in how to deal with the singular
leaves). It would be interesting to study quantum corrections to this
class of backgrounds, with a view towards clarifying their effect on
the geometry of $\bar{\mathcal{F}}$. As mentioned in the introduction, the class of
backgrounds discussed here appears to be connected with the proposals
of \cite{Grana} and \cite{Bonetti}, connections which deserve to be
explored in detail.
One of the reasons why the class of backgrounds studied in this paper
may be of wider interest is because, as pointed out in \cite{Tsimpis},
the structure group of $M$ does {\em not} globally reduce to a a proper
subgroup of $\mathrm{SO}(8)$. This is the origin of the phenomena discussed in
this paper, which illustrate the limitations of the theory of
classical G-structures as well as of the theory of regular
foliations. In its classical form \cite{Chern}, the former does not
provide a sufficiently wide conceptual framework for a fully general
{\em global} description of all flux compactifications.
\acknowledgments{The work of C.I.L. is supported by the research grant
IBS-R003-G1 while E.M.B. acknowledges support from the strategic
grant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed
by the European Social Fund within the Sectorial Operational Program
Human Resources Development 2007--2013. This work was also financed
by the CNCS-UEFISCDI grants PN-II-ID-PCE 121/2011 and 50/2011 and
also by PN 09 37 01 02. The authors thank M.~Grana, D.~Tsimpis and
especially I.~Gelbukh for correspondence and suggestions.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,991
|
\section{Introduction}
A set of edges of a graph is an \emph{odd cycle (edge) transversal} if its removal results in a bipartite graph; the smallest size of an odd cycle transversal of $G$ is denoted by $\tau_{\odd}(G)$. Finding a minimum odd cycle transversal of a graph is equivalent to partitioning the vertex set into two parts, such that the number of edges between the two parts is maximum; this is known as the \emph{max-cut problem} in the literature.
Erd\H os~\cite{Erd65} observed that every graph has an odd cycle transversal containing at most half of its edges, and conjectured that every \emph{triangle-free} graph on $n$ vertices has an odd cycle transversal with at most $\frac1{25}n^2$ edges. Hopkins and Staton~\cite{HopSta82} proved that every triangle-free \emph{cubic} graph on $n$ vertices has an odd cycle transversal with at most $\frac3{10}n$ edges. For triangle-free cubic \emph{planar} graphs, the bound was improved to $\frac7{24}n+\frac76$ by Thomassen~\cite{Tho06}, and subsequently to $\frac9{32}n+\frac9{16}$ by Cui and Wang~\cite{CuWa09}.
A widely studied class of triangle-free cubic planar graphs is the class of \emph{fullerene graphs}: these are cubic bridgeless plane graphs with all faces of size $5$ or $6$. Do\v sli\'c and Vuki\v cevi\'c~\cite[Conjecture 13]{DoVu07} conjectured that every fullerene graph on $n$ vertices has an odd cycle transversal with at most $\sqrt{\frac{12}5n}$ edges, and showed that this bound is attained by fullerene graphs on $60k^2$ vertices, where $k \in \mathbb N$, with the icosahedral automorphism group. Dvo\v r\'ak, Lidick\'y and \v Skrekovski~\cite{DvLiSk12} have recently verified the conjecture asymptotically by proving that $\tau_{\odd}(G) = O(\sqrt n)$. The main result of this paper is a proof of the conjecture of Do\v sli\'c and Vuki\v cevi\'c.
\begin{theorem}\label{thm:main}
If $G$ is a fullerene graph on $n$ vertices, then $\tau_{\odd}(G) \leq \sqrt{\frac{12}5n}$. Equality holds if and only if $n=60k^2$, for some $k \in \mathbb N$, and $\Aut(G) \cong I_h$.
\end{theorem}
The rest of the paper is organised as follows. In Section~\ref{sec:terminology}, we cover the basic notation and terminology. In Section~\ref{sec:T-joins}, we recall the concepts of $T$-joins and $T$-cuts, and establish a bound on the minimum size of a $T$-join in a plane triangulation in terms of the maximum size of a packing of $T$-cuts in an auxiliary plane triangulation. In Section~\ref{sec:patches}, we introduce the notions of patches and moats, and prove bounds on the number of edges in moats. In Section~\ref{sec:proof}, we combine the bounds from the preceding two sections to complete the proof of Theorem~\ref{thm:main}. In Section~\ref{sec:independence}, we deduce a number of conjectures about the independence number of fullerene graphs. Finally, in Section~\ref{sec:eigenvalues}, we compute a new upper bound on the smallest eigenvalue of a fullerene graph.
\section{Notation and terminology}\label{sec:terminology}
Most terminology used in this paper is standard, and may be found in any graph theory textbook. All graphs considered are simple, that is, have no loops and multiple edges. The vertex and edge set of a graph $G$ is denoted by $V(G)$ and $E(G)$, respectively. If $X\subseteq V(G)$ or $X\subseteq E(G)$, we let $G-X$ be the graph obtained from $G$ by removing the elements in $X$, and $G[X]$ the subgraph of $G$ induced by $X$.
A graph is \emph{planar} if it can be drawn in the plane $\mathbb R^2$ so that its vertices are points in $\mathbb R^2$, and its edges are Jordan curves in $\mathbb R^2$ which intersect only at their end-vertices. A planar graph with a fixed embedding is called a \emph{plane graph}. If $G$ is a plane graph, the connected regions of $\mathbb R^2\setminus G$ are the \emph{faces} of $G$. A face of a plane graph $G$ bounded by three edges is a \emph{triangle} of $G$; if every face of $G$ is a triangle, then $G$ is a plane \emph{triangulation}. If $G$ is a plane graph, the \emph{dual graph} $G^*$ is the multigraph with precisely one vertex in each face of $G$, and if $e$ is an edge of $G$, then $G^*$ has an edge $e^*$ crossing $e$ and joining the two vertices of $G^*$ in the two faces of $G$ incident to $e$.
The \emph{distance} $\dist_G(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a shortest path in $G$ connecting $u$ and $v$. The \emph{open} and \emph{closed $k$-neighbourhood} of a subset $X \subseteq V(G)$ in $G$ are the sets $N_G^k(X)= \{v \in V(G) \mid \dist_G(v,X) = k\}$ and $N_G^k[X]= \{v \in V(G) \mid \dist_G(v,X) \leq k\}$, respectively. The usual open and closed neighbourhood is defined as $N_G(X)=N_G^1(X)$ and $N_G[X]=N_G^1[X]$, respectively. When $X=\{x\}$, we simply write $N_G^k[x]$ and $N_G^k(x)$. The size of the open neighbourhood $N_G(x)$ is the \emph{degree} $d_G(x)$. We let $\delta_G(X)$ be the set of edges of $G$ with exactly one end-vertex in $X$; if $H=G[X]$ we may also write $\delta_G(H)$ for $\delta_G(X)$. A set $C$ of edges is a \emph{cut} of $G$ if $C=\delta_G(X)$, for some $X \subseteq V(G)$. When there is no risk of ambiguity, we may omit the subscripts in the above notation.
An \emph{automorphism} of a graph $G$ is a permutation of the vertices such that adjacency is preserved. The set of all automorphisms of $G$ forms a group, known as the \emph{automorphism group $\Aut(G)$}. The \emph{full icosahedral group} $I_h \cong A_5 \times C_2$ is the group of all symmetries (including reflections) of the regular icosahedron.
\section{\texorpdfstring{$T$-joins and $T$-cuts}{T-joins and T-cuts}}\label{sec:T-joins}
To prove Theorem~\ref{thm:main}, we will consider the dual of a fullerene graph, that is, a plane triangulation $G$ with all vertices of degree $5$ and $6$. We denote by $T$ the set of $5$-vertices of $G$; it follows from Euler's formula that $|T|=12$. The problem is to find a minimal set $J$ of edges such that $G-J$ has no odd-degree vertices. Such a set of edges is known as a $T$-join.
More generally, let $G$ be any graph with a distinguished set $T$ of vertices such that $|T|$ is even. A \emph{$T$-join} of $G$ is a subset $J \subseteq E(G)$ such that $T$ is equal to the set of odd-degree vertices in $G[J]$. The minimum size of a $T$-join of $G$ is denoted by $\tau(G,T)$.
A \emph{$T$-cut} is an edge cut $\delta(X)$ such that $|T\cap X|$ is odd. A \emph{packing} of $T$-cuts is a disjoint collection $\delta(\mathcal F)=\{\delta(X) \mid X \in \mathcal F\}$ of $T$-cuts of $G$; the maximum size of a packing of $T$-cuts is denoted by $\nu(G,T)$. For more information on $T$-joins and $T$-cuts, the reader is referred to~\cite{CCPS98, LoPl86, Sch03}.
Since every $T$-join intersects every $T$-cut, $\nu(G,T) \leq \tau(G,T)$. If $G$ is bipartite, we in fact have equality.
\begin{theorem}[Seymour~\cite{Sey81}]\label{thm:seymour}
For every bipartite graph $G$ and every subset $T \subseteq V(G)$ such that $|T|$ is even, $\tau(G,T) = \nu(G,T)$. \end{theorem}
A family of sets $\mathcal F$ is said to be \emph{laminar} if, for every pair $X,Y \in \mathcal F$, either $X \subseteq Y$, $Y \subseteq X$, or $X \cap Y = \emptyset$. A packing of $T$-cuts $\delta(\mathcal F)$ is said to be laminar if $\mathcal F$ is laminar. A $T$-cut $\delta(X)$ is \emph{inclusion-wise minimal} if no $T$-cut is properly contained in $\delta(X)$. The following proposition can be found in~\cite{FHRV07}.
\begin{proposition}\label{prop:laminar}
For every bipartite graph $G$ and every subset $T \subseteq V(G)$ such that $|T|$ is even, there exists an optimal packing of $T$-cuts in $G$ which is laminar and consists only of inclusion-wise minimal $T$-cuts.
\end{proposition}
Let us remark that the problem of finding a minimum $T$-join is equivalent to the minimum weighted matching problem, which can be solved efficiently using Edmonds' weighted matching algorithm. The problem of finding a maximum packing of $T$-cuts may be considered as the dual problem in the sense of linear programming. Using Theorem~\ref{thm:seymour} and Proposition~\ref{prop:laminar}, it can be shown (see e.g.~\cite{CCPS98}) that there exists an optimal solution of the dual linear program which is half-integral and laminar. Intuitively, this would correspond to a packing of $T$-cuts where the $T$-cuts consist of `half-edges'. This idea was used, in conjunction with the Four Colour Theorem, by Kr\'al' and Voss~\cite{KrVo04} to show that if $G$ is a planar graph and $T\subseteq V(G)$ is the set of odd-degree vertices of $G$, then $\tau(G,T) \leq 2\nu(G,T)$.
Our approach is similar, but rather than dealing with half-edges, we consider a suitable transformation of the graph $G$. Namely, given a plane triangulation $G$, construct the graph $G'$ by subdividing the edges of $G$, that is, replacing the edges of $G$ by internally disjoint paths of length $2$; the graph $G'$ is clearly bipartite. Now construct the graph $G^{\vartriangle}$ from $G'$ by adding three new edges inside every face of $G'$, incident to the three vertices of degree $2$, as shown in Figure~\ref{fig:refinement}. We call $G^{\vartriangle}$ a \emph{refinement} of $G$. Observe that all the vertices in $V(G^{\vartriangle})-V(G)$ have degree $6$ in $G^{\vartriangle}$, so if $T$ is the set of odd-degree vertices of $G$, then $T$ is also the set of odd-degree vertices of $G^{\vartriangle}$.
\begin{figure}
\centering
\null\hfill
\subfloat[$G$]{\label{fig:triangulation}
\begin{tikzgraph}[scale=0.6,ultra thin]
\path (90:2) coordinate (a1);
\path (210:2) coordinate (a2);
\path (330:2) coordinate (a3);
\draw (a1)--(a2)--(a3)--cycle;
\foreach\i in {1,...,3}
{
\draw (a\i) node[vertex] {};
}
\foreach\i in {0,...,3}
{
\draw (a1)--+(60*\i:0.5);
}
\foreach\i in {2,...,5}
{
\draw (a2)--+(60*\i:0.5);
}
\foreach\i in {4,...,7}
{
\draw (a3)--+(60*\i:0.5);
}
\end{tikzgraph}
}
\hfill
\subfloat[$G'$]{\label{fig:subdivision}
\begin{tikzgraph}[scale=0.6,ultra thin]
\path (90:2) coordinate (a1);
\path (210:2) coordinate (a2);
\path (330:2) coordinate (a3);
\path (150:1) coordinate (s1);
\path (270:1) coordinate (s2);
\path (30:1) coordinate (s3);
\draw (a1)--(a2)--(a3)--cycle;
\foreach\i in {1,...,3}
{
\draw (a\i) node[vertex] {};
}
\foreach\i in {0,...,3}
{
\draw (a1)--+(60*\i:0.5);
}
\foreach\i in {2,...,5}
{
\draw (a2)--+(60*\i:0.5);
}
\foreach\i in {4,...,7}
{
\draw (a3)--+(60*\i:0.5);
}
\foreach\i in {1,...,3}
{
\draw (s\i) node[subdivision] {};
}
\end{tikzgraph}
}
\hfill
\subfloat[$G^{\vartriangle}$]{\label{fig:refinement2}
\begin{tikzgraph}[scale=0.6,ultra thin]
\path (90:2) coordinate (a1);
\path (210:2) coordinate (a2);
\path (330:2) coordinate (a3);
\path (150:1) coordinate (s1);
\path (270:1) coordinate (s2);
\path (30:1) coordinate (s3);
\draw (a1)--(a2)--(a3)--cycle;
\draw (s1)--(s2)--(s3)--cycle;
\foreach\i in {1,...,3}
{
\draw (a\i) node[vertex] {};
}
\foreach\i in {0,...,3}
{
\draw (a1)--+(60*\i:0.5);
}
\foreach\i in {2,...,5}
{
\draw (a2)--+(60*\i:0.5);
}
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\hfill\null
\caption{A face of a triangulation $G$, its subdivision $G'$, and its refinement $G^{\vartriangle}$.}
\label{fig:refinement}
\end{figure}
\begin{lemma}\label{lem:refinement}
For every planar triangulation $G$ and every subset $T \subseteq V(G)$ such that $|T|$ is even, $\tau(G,T) =\frac12\nu(G^{\vartriangle},T)$. Moreover, there exists an optimal laminar packing of inclusion-wise minimal $T$-cuts in $G^{\vartriangle}$.
\end{lemma}
\begin{proof}
Let $G'$ be the subgraph obtained from $G$ by subdividing every edge of $G$. For the first part, it suffices to prove the chain of inequalities
\begin{equation*}\label{eq:tau=nu/2}
\tau(G,T) \leq \tfrac12\tau(G',T) \leq \tfrac12\nu(G',T) \leq \tfrac12\nu(G^{\vartriangle},T) \leq \tau(G,T).
\end{equation*}
Clearly, any $T$-join $J'$ of $G'$ corresponds to a $T$-join $J$ of $G$ such that $|J|=\frac12|J'|$, so $\tau(G,T) \leq \frac12\tau(G',T)$. The second inequality $\tau(G',T)\leq \nu(G',T)$ holds by Theorem~\ref{thm:seymour}. To prove the final inequality $\frac12\nu(G^{\vartriangle},T) \leq \tau(G,T)$, observe that any $T$-join $J$ of $G$ corresponds to a $T$-join $J^{\vartriangle}$ of $G^{\vartriangle}$ such that $|J|=\frac12|J^{\vartriangle}|$. Hence, $\tau(G,T) \geq \frac12\tau(G^{\vartriangle},T) \geq \frac12\nu(G^{\vartriangle},T)$.
It remains to prove the third inequality, namely $\nu(G',T) \leq \nu(G^{\vartriangle},T)$. Let $\mathcal F$ be a laminar family on $V(G')$ minimising $\sum_{X \in \mathcal F}|\delta_{G'}(X)|$, such that $\delta_{G'}(\mathcal F)$ is an optimal packing of inclusion-wise minimal $T$-cuts in $G'$; such a family exists by Proposition~\ref{prop:laminar}. Suppose $\delta_{G^{\vartriangle}}(\mathcal F)$ is not a packing of $T$-cuts in $G^{\vartriangle}$. Then there exist $X_1, X_2 \in \mathcal F$ and an edge $e \in E(G^{\vartriangle}) - E(G')$ such that $e \in \delta_{G^{\vartriangle}}(X_1) \cap \delta_{G^{\vartriangle}}(X_2)$. Therefore $e=x_1x_2$, where $x_1$ and $x_2$ are vertices of $V(G')-V(G)$. By the laminarity of $\mathcal F$, $X_1 \cap X_2 = \emptyset$. Therefore, there exists $i \in \{1,2\}$ such that $x_i$ has a neighbour in $V(G') - X_i$. But then $\delta_{G'}(X_i-\{x_i\})$ is a $T$-cut in $G'$ which is disjoint from all other $T$-cuts of $\delta_{G'}(\mathcal F)$, and $|\delta_{G'}(X_i-\{x_i\})| < |\delta_{G'}(X_i)|$, contradicting the minimality of $\sum_{X \in \mathcal F}|\delta_{G'}(X)|$. Hence, $\delta_{G^{\vartriangle}}(\mathcal F)$ is a laminar packing of $T$-cuts in $G^{\vartriangle}$, so $\nu(G',T) \leq \nu(G^{\vartriangle},T)$.
For the `moreover' part, simply note that the packing $\delta_{G^{\vartriangle}}(\mathcal F)$ from the previous paragraph is an optimal laminar packing of inclusion-wise minimal $T$-cuts in $G^{\vartriangle}$.
\end{proof}
\section{Patches and moats}\label{sec:patches}
Throughout this section, $G$ is a plane triangulation with all vertices of degree $5$ and $6$, and $T$ is the set of $5$-vertices of $G$. A $2$-connected subgraph $H \subset G$ such that all faces of $H$, except the outer face, are triangles, is called a \emph{patch} of $G$. If $C$ is the outer cycle of $H$, and the number of vertices in $T\cap V(H-C)$ is $p$, then $H$ is a \emph{$p$-patch}. We define the \emph{area} $A(H)$ as the number of triangles in $H$. An example of a $3$-patch is shown in Figure~\ref{fig:patch-moat}.
Every $p$-patch with $1 \leq p \leq 5$ satisfies the following isoperimetric inequality, which is an immediate corollary of a more general theorem of Justus~\cite[Theorem~3.3.2]{Jus07}.
\begin{theorem}[Justus~\cite{Jus07}]\label{thm:justus}
Let $G$ be a plane triangulation with all vertices of degree $5$ and $6$, and let $T$ be the set of the $5$-vertices of $G$. If $H \subseteq G$ is a $p$-patch with outer cycle $C$, and $1 \leq p \leq 5$, then
\[
|V(C)| \geq \sqrt{(6-p)A(H)}.
\]
If equality holds, then $p=1$.
\end{theorem}
A \emph{moat} of width $k$ in $G$ surrounding $X \subseteq V(G)$ is a subset of $E(G)$ defined as
\[
\delta_G^k(X)=\bigcup_{i=0}^{k-1}\delta_G\left(N^i[X]\right).
\]
In particular, $\delta_G^1(X)=\delta_G(X)$. If $|T\cap X|=p$, then $\delta_G^k(X)$ is a \emph{$p$-moat} of width $k$. See Figure~\ref{fig:patch-moat} for an example of a $3$-moat of width $2$. If $u \in T$, the $1$-moat $\delta_G^k(\{u\})$ is simply denoted by $\delta_G^k(u)$, and is called a \emph{disk} of radius $k$ centred on $u$. To every moat $\delta_G^k(X)$ corresponds a set of $|\delta_G^k(X)|$ faces, namely the faces incident to at least one edge of $\delta_G^k(X)$. We say that these faces are \emph{spanned} by $\delta_G^k(X)$.
\begin{figure}
\centering
\begin{tikzgraph}[scale=0.5,ultra thin]
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(b1)--(c1)
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\caption{A $3$-patch (shaded in grey) surrounded by a $3$-moat of width $2$ (shown by the thick edges).}
\label{fig:patch-moat}
\end{figure}
The number of edges in a disk is easy to determine.
\begin{lemma}
\label{lem:disk}
Let $G$ be a plane triangulation with all vertices of degree $5$ and $6$, and $T$ the set of $5$-vertices of $G$. If $u \in T$, and no edge of $\delta^{k-1}(u)$ is incident to a vertex of $T-\{u\}$, then $\left|\delta_G^k(u)\right| = 5k^2$.
\end{lemma}
\begin{proof}
It is easy to see that $\left|\delta(N^k[u])\right|=5(2k+1)$, so $\left|\delta^k(u)\right|=\sum_{i=0}^{k-1}\left|\delta(N^i[u])\right|=5\sum_{i=0}^{k-1}(2i+1)=5k^2$.
\end{proof}
For more general moats, we can prove the following inequality.
\begin{lemma}
\label{lem:perimeter}
Let $G$ be a plane triangulation with all vertices of degree $5$ and $6$, $T$ the set of $5$-vertices of $G$, and $X \subset V(G)$. If $G[X]$ is a $p$-patch such that $0<p<6$, and no edge of $\delta^{k-1}(X)$ is incident to a vertex of $T$, then
\[
\left|\delta_G^k(X)\right| \geq (6-p)k^2+2k\sqrt{(6-p)A(G[X])}.
\]
If equality holds, then $p=1$.
\end{lemma}
\begin{proof}
Let $C$ be the outer cycle of $G[X]$, and denote by $n$, $m$ and $f$ the number of vertices, edges, and faces (including the outer face) of $G[X]$, respectively. Summing the vertex degrees of $G[X]$ gives $2m=\sum_{v \in V(C)} d_{G[X]}(v)+6(n-|V(C)|)-p$, so
\begin{equation}\label{eq:vertices}
\sum_{v \in V(C)} d_{G[X]}(v)=6|V(C)|+p-6n+2m.
\end{equation}
Summing the face degrees gives $2m=3(f-1)+|V(C)|$, so
\begin{equation}\label{eq:faces}
0=-2|V(C)|+4m-6f+6.
\end{equation}
Adding~\eqref{eq:vertices} and~\eqref{eq:faces},
\begin{equation}\label{eq:boundary}
\sum_{v \in V(C)} d_{G[X]}(v)= 4|V(C)|+p-6(n-m+f-1) =4|V(C)|+p-6,
\end{equation}
where the last equation follows from Euler's formula.
Applying~\eqref{eq:boundary} to the $p$-patch $G[X]$ and the $(12-p)$-patch $G-X$,
\begin{align*}
2|V(C)|+6-p &= \sum_{v \in V(C)}(6-d_{G[X]}(v)) \\
&= \sum_{v \in N(X)}(6-d_{G-X}(v)) \\
&= 2|N(X)|-6+p,
\end{align*}
whence $|N(X)|=|V(C)|+6-p$, so by induction,
\begin{equation}\label{eq:N^k(X)}
|N^k(X)|=|V(C)|+(6-p)k.
\end{equation}
By~\eqref{eq:boundary} and~\eqref{eq:N^k(X)}, the number of edges in $\delta(N^k[X])$ is
\begin{align*}
\left|\delta(N^k[X])\right| &= \sum_{v \in N^k(X)}(6-d_{G[X]}(v)) \\
&= 2|N^k(X)|+6-p \\
&= 2|V(C)|+(6-p)(2k+1),
\end{align*}
\enlargethispage{2\baselineskip}
so the number of edges in $\delta^k(X)$ is
\begin{align*}
\left|\delta^k(X)\right| &= \sum_{i=0}^{k-1}\left|\delta\left(N^i[X]\right)\right|\\
&= \sum_{i=0}^{k-1}\left(2|V(C)|+(6-p)(2i+1)\right)\\
&= 2k|V(C)|+(6-p)k^2.
\end{align*}
By Theorem~\ref{thm:justus}, $|V(C)| \geq \sqrt{(6-p)A(G[X])}$, with equality only if $p=1$.
\end{proof}
\section{Packing moats in plane triangulations}\label{sec:proof}
When $G$ is a plane triangulation, there exists, by Lemma~\ref{lem:refinement}, an optimal laminar packing $\delta_{G^{\vartriangle}}(\mathcal F)$ of inclusion-wise minimal $T$-cuts in the refinement $G^{\vartriangle}$. We may furthermore assume that the family which gives rise to this packing satisfies $|T\cap X|\leq 5$ for all $X \in \mathcal F$, and minimises $\sum_{X \in \mathcal F} |X|$. We call such a packing a \emph{moat packing}. Let us remark that Kr\'al', Sereni and Stacho~\cite{KrSeSt11+} considered moat packings in bipartite graphs (they used the name \emph{moat solution}). The reason for choosing this name is the following.
For every odd-cardinality subset $U \subset T$, the union of all $T$-cuts in $\delta_{G^{\vartriangle}}(\mathcal F)$ which separate $U$ from $T-U$ is of the form $\delta_{G^{\vartriangle}}^k(X)$, where $U \subseteq X \in \mathcal F$ and $k \in \mathbb N$, i.e., it is a moat of width $k$ surrounding $X$. By the minimality of $\sum_{X \in \mathcal F} |X|$, every $1$-moat in $\delta_{G^{\vartriangle}}(\mathcal F)$ is a disk centred on a vertex $u \in T$, and every vertex of $T$ is the centre of a disk of radius at least $1$. Also by the minimality of $\sum_{X \in \mathcal F} |X|$, if $X \in \mathcal F$ is such that $|X|>1$, then $G[X]$ is $2$-connected. Since every $T$-cut in $\delta_{G^{\vartriangle}}(\mathcal F)$ is inclusion-wise minimal, precisely one face of $G[X]$---the outer face---is not a triangle. Hence, $G[X]$ is a patch, for every $X \in \mathcal F$ such that $|X|>1$.
Therefore, a moat packing of $T$-cuts may be considered as a packing of disks, $3$-moats and $5$-moats. Figure~\ref{fig:tetrahedron} shows an example of such a packing.
\begin{figure}
\centering
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\draw (A3)--(D2);
\draw (A2)--(B1);
\draw (B3)--(C2);
\draw (B3)--(D1);
\end{tikzpicture}
\caption{A triangulation of the truncated tetrahedron, with a packing of twelve disks and four $3$-moats. The faces spanned by disks are shaded in dark grey, and those spanned by $3$-moats are shaded in light grey. The incidence vectors of this particular packing are $r=1$, $s=1$ and $t=0$.}
\label{fig:tetrahedron}
\end{figure}
We are at last ready to prove Theorem~\ref{thm:main}. To be exact, we first prove the following dual version.
\begin{theorem}\label{thm:T-join}
Let $G$ be a plane triangulation with $f$ faces and all vertices of degree $5$ and $6$. If $T$ is the set of $5$-vertices of $G$, then $\tau(G,T) \leq \sqrt{\frac{12}5f}$, with equality if and only if $f=60k^2$, for some $k \in \mathbb N$, and $\Aut(G) \cong I_h$.
\end{theorem}
\begin{proof}
Let $G^{\vartriangle}$ be the refinement of $G$; so $G^{\vartriangle}$ is a plane triangulation with $4f$ faces and all vertices of degree $5$ and $6$. By Lemma~\ref{lem:refinement}, there exists a moat packing $\delta_{G^{\vartriangle}}(\mathcal F)$. Let $m_1$, $m_3$ and $m_5$ be the number of edges in all disks, $3$-moats, and $5$-moats of $\delta_{G^{\vartriangle}}(\mathcal F)$, respectively. Define the incidence vectors $r, s, t \in \mathbb R^{12}$ as follows: for every $u \in T$, let $r_u$, $s_u$ and $t_u$ be the radius of the disk centred on $u$, the width of the $3$-moat surrounding $u$, and the width of the $5$-moat surrounding $u$, respectively. By the optimality of $\delta_{G^{\vartriangle}}(\mathcal F)$,
\begin{equation}\label{eq:moat_packing}
\tau(G,T) = \tfrac12\nu(G^{\vartriangle},T) = \tfrac12\left\langle r + \tfrac 13 s + \tfrac 15 t, 1\right\rangle,
\end{equation}
where $\langle \cdot,\cdot \rangle$ denotes the inner product.
So to prove the inequality in Theorem~\ref{thm:T-join}, it suffices to find an upper bound on $\left\langle r + \frac 13 s + \frac 15 t, 1\right\rangle$ in terms of $f$. To do so, we compute lower bounds on $m_1$, $m_3$ and $m_5$ in terms of the vectors $r$, $s$ and $t$, and then use the fact that the sum $m_1+m_3+m_5$ cannot exceed $4f$, the number of faces of $G^{\vartriangle}$.
First suppose that $\delta_{G^{\vartriangle}}^{r_u}(u)$ is a disk of $\delta_{G^{\vartriangle}}(\mathcal F)$, for some $u \in T$. Recall that by Lemma~\ref{lem:disk},
\begin{equation}\label{eq:onedisk}
\left|\delta_{G^{\vartriangle}}^{r_u}(u)\right| = 5r_u^2,
\end{equation}
so summing over all disks,
\begin{equation}\label{eq:disk}
m_1 = 5\sum_{u\in T}r_u^2 = 5\|r\|^2,
\end{equation}
where $\|\cdot\|$ denotes the norm.
Now, suppose $\delta_{G^{\vartriangle}}^{s_u}(X)$ is a non-empty $3$-moat of $\delta_{G^{\vartriangle}}(\mathcal F)$, where $u\in T \cap X$ and $|T \cap X| = 3$. The graph $G^{\vartriangle}[X]$ contains $|\delta_{G^{\vartriangle}}^{r_u}(u)|$ triangles spanned by $\delta_{G^{\vartriangle}}^{r_u}(u)$, for every $u \in T \cap X$. All the triangles are pairwise disjoint, so by~\eqref{eq:onedisk} and the Cauchy-Schwarz inequality,
\[
A(G^{\vartriangle}[X])\geq \sum_{u\in T \cap X}\left|\delta_{G^{\vartriangle}}^{r_u}(u)\right| = 5\sum_{u\in T \cap X}r_u^2 \geq \frac53\left(\sum_{u\in T \cap X}r_u\right)^2.
\]
Hence, by Lemma~\ref{lem:perimeter},
\begin{align}
\left|\delta_{G^{\vartriangle}}^{s_u}(X)\right| &\geq 3s_u^2+2s_u\sqrt{3A(G^{\vartriangle}[X])} \notag\\
&\geq 3s_u^2+2\sqrt{5}s_u\sum_{u\in T \cap X}r_u \notag\\
&= \sum_{u\in T \cap X}s_u^2+2\sqrt{5}\sum_{u \in T \cap X}r_us_u\label{eq:one3-moat}.
\end{align}
Summing over all $3$-moats,
\begin{equation}\label{eq:3-moat}
m_3 \geq \|s\|^2+2\sqrt 5\langle r,s \rangle.
\end{equation}
Finally, suppose $\delta_{G^{\vartriangle}}^{t_u}(Y)$ is a non-empty $5$-moat of $\delta_{G^{\vartriangle}}(\mathcal F)$, where $u \in T \cap Y$ and $|T \cap Y| = 5$. By the laminarity of $\delta_{G^{\vartriangle}}(\mathcal F)$, $G^{\vartriangle}[Y]$ contains at most one $3$-moat $\delta_{G^{\vartriangle}}^{s_u}(X)$ of $\delta_{G^{\vartriangle}}(\mathcal F)$, where $X \subset Y$ and $|T \cap X| = 3$. The graph $G^{\vartriangle}[Y]$ contains $|\delta_{G^{\vartriangle}}^{r_u}(u)|$ triangles spanned by $\delta_{G^{\vartriangle}}^{r_u}(u)$, for every $u \in T \cap Y$, as well as at least $\left|\delta_{G^{\vartriangle}}^{s_u}(X)\right|$ triangles spanned by $\delta_{G^{\vartriangle}}^{s_u}(X)$. All the triangles are pairwise disjoint, so by~\eqref{eq:onedisk}, \eqref{eq:one3-moat}, and the Cauchy-Schwarz inequality,
\begin{align*}
A(G^{\vartriangle}[Y]) &\geq \sum_{u\in T \cap Y}\left|\delta_{G^{\vartriangle}}^{r_u}(u)\right|+\left|\delta_{G^{\vartriangle}}^{s_u}(X)\right|\\
&\geq 5\sum_{u\in T \cap Y}r_u^2+2\sqrt{5}\sum_{u \in T \cap Y}r_us_u+\sum_{u \in T \cap Y}s_u^2\\
&= 5\sum_{u\in T \cap Y}\left(r_u^2+\tfrac1{\sqrt{5}}s_u^2\right)^2\\
&\geq \left(\sum_{u \in T \cap Y}r_u+\tfrac{1}{\sqrt 5}\sum_{u \in T \cap Y}s_u \right)^2.
\end{align*}
Hence, by Lemma~\ref{lem:perimeter},
\begin{align*}
\left|\delta_{G^{\vartriangle}}^{t_u}(Y)\right| &\geq t_u^2+2t_u\sqrt{A(G^{\vartriangle}[Y])} \\
&\geq t_u^2+2t_u\sum_{u \in T \cap Y}r_u+\tfrac{2}{\sqrt 5}t_u\sum_{u \in T \cap Y}s_u \\
&= \tfrac15\sum_{u \in T \cap Y}t_u^2+2\sum_{u \in T \cap Y}r_ut_u+\tfrac{2}{\sqrt 5}\sum_{u \in T \cap Y}s_ut_u.
\end{align*}
Summing over all $5$-moats,
\begin{equation}\label{eq:5-moat}
m_5 \geq \tfrac15\|t\|^2+2\langle r,t \rangle+\tfrac{2}{\sqrt 5}\langle s,t \rangle.
\end{equation}
The graph $G^{\vartriangle}$ has $4f$ triangles, and the disks, $3$-moats and $5$-moats span $m_1$, $m_3$ and $m_5$ triangles of $G^{\vartriangle}$, respectively. These triangles are mutually disjoint, so by~\eqref{eq:disk}, \eqref{eq:3-moat} and~\eqref{eq:5-moat},
\begin{align*}
4f &\geq m_1+m_3+m_5\\
&\geq 5\|r\|^2 + \|s\|^2 + 2\sqrt 5\langle r,s \rangle + \tfrac15\|t\|^2 + 2\langle r,t \rangle + \tfrac{2}{\sqrt 5}\langle s,t \rangle\\
&= \left\|\sqrt 5r + s + \tfrac 1{\sqrt 5}t\right\|^2.
\end{align*}
Hence, by the Cauchy-Schwarz inequality and~\eqref{eq:moat_packing},
\begin{align}
\sqrt{\frac{12f}5} & \geq \sqrt{3}\left\|r + \tfrac 1{\sqrt 5}s + \tfrac 15t\right\| \notag\\
& \geq \tfrac12\left\langle r + \tfrac 1{\sqrt 5} s + \tfrac 15 t,1\right\rangle \label{eq:cauchy-schwarz}\\
& \geq \tau(G,T). \notag
\end{align}
To prove the last part of Theorem~\ref{thm:T-join}, suppose that $\tau(G,T)=\sqrt{\frac{12}5f}$. Equality must hold in~\eqref{eq:3-moat} and~\eqref{eq:5-moat}, so by Lemma~\ref{lem:perimeter}, $s = t = 0$. Furthermore, equality must hold in~\eqref{eq:cauchy-schwarz}, so $r_u=r_v$ for every $u, v \in T$. Therefore $4f=5\cdot12 r_u^2$, so $f=15r_u^2$. Since $f$ is even, it follows that $r_u = 2k$, and therefore $f=60k^2$, for some $k \in \mathbb N$. To see that $\Aut(G) \cong I_h$, note that the graph $G$ may be constructed from the dodecahedron by inserting into each face a $1$-patch of the form $G[N^k[u]]$.
Conversely, if $G$ is a plane triangulation with $f=60k^2$ faces, all vertices of degree $5$ and $6$, and $\Aut(G) \cong I_h$, then $G$ may be constructed from the dodecahedron by inserting into each face a $1$-patch of the form $G[N^k[u]]$. Hence $\dist(u,v) \geq 2k$, for every pair of distinct vertices in $T$, so $\tau(G,T) \geq 12k=\sqrt{\frac{12}5f}$.
\end{proof}
By applying Theorem~\ref{thm:T-join} to the dual graph, we obtain a proof of Theorem~\ref{thm:main}.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $G$ be a fullerene graph on $n$ vertices. The dual graph $G^*$ is a plane triangulation with $n$ faces and all vertices of degree $5$ and $6$. Let $T$ be the set of vertices of degree $5$, $J^*$ a minimum $T$-join of $G^*$, and $J$ the set of edges of $G$ which correspond to $J^*$. Since $G^*-J^*$ has no odd-degree vertices, $G-J=(G^*-J^*)^*$ is bipartite, and by Theorem~\ref{thm:T-join}, $|J|=|J^*| \leq \sqrt{\frac{12}5n}$, with equality if and only if $n=60k^2$, for some $k \in \mathbb N$ and $\Aut(G) \cong I_h$.
\end{proof}
\section{Independent sets in fullerene graphs}\label{sec:independence}
Recall that a set $X \subseteq V(G)$ is \emph{independent} if the graph $G[X]$ has no edges; the maximum size of an independent set in $G$ is the \emph{independence number $\alpha(G)$}. By the Four Colour Theorem, every planar graph on $n$ vertices has an independent set with at least $\frac 14n$ vertices, and by Brooks' Theorem, every triangle-free, cubic graph on $n$ vertices has an independent set with at least $\frac 13n$ vertices. For triangle-free, cubic, planar graphs, the bound can be improved a little further.
\begin{theorem}[Heckman and Thomas~\cite{HeTh06}]\label{thm:heckman-thomas}
If $G$ is a triangle-free cubic planar graph on $n$ vertices, then $\alpha(G) \geq \frac38n$.
\end{theorem}
Daugherty~\cite[Conjecture~5.5.2]{Dau09} conjectured that every fullerene graph on $n$ vertices has an independent set with at least $\frac12n-\sqrt{\frac35n}$ vertices. He also conjectured~\cite[Conjecture~5.5.1]{Dau09} that every fullerene graph attaining this bound has the icosahedral automorphism group and $60k^2$ vertices, for some $k \in \mathbb N$. Andova et al.~\cite{ADKLS12+} recently proved that every fullerene graph on $n$ vertices has an independent set with at least $\frac12n-78.58\sqrt{n}$ vertices. Theorem~\ref{thm:main} immediately implies both conjectures of Daugherty.
\begin{corollary}
If $G$ is a fullerene graph on $n$ vertices, then $\alpha(G) \geq \frac 12n-\sqrt{\frac35n}$, with equality if and only if $n=60k^2$, for some $k \in \mathbb N$, and $\Aut(G) \cong I_h$.
\label{cor:daugherty}
\end{corollary}
\begin{proof}
Every graph $G$ contains an odd cycle vertex transversal $U$ such that $|U| \leq \tau_{\odd}(G)$, so $\alpha(G) \geq \alpha(G-U) \geq \frac12n-\frac12\tau_{\odd}(G)$. Therefore, by Theorem~\ref{thm:main}, $\alpha(G) \geq \frac12n-\sqrt{\frac35n}$, for every fullerene graph $G$. When $J^*$ is a minimum $T$-join of $G^*$, every face of $G^*$ is incident to at most one edge of $J^*$. This means that the set $J \subset E(G)$ corresponding to $J^*$ is a matching of $G$. Therefore, by Theorem~\ref{thm:main}, equality holds if and only if $n=60k^2$, for some $k \in \mathbb N$, and $\Aut(G) \cong I_h$.
\end{proof}
The \emph{diameter} of a graph $G$, denoted $\diam(G)$, is defined as the maximum distance over all pairs of vertices $u, v$ of $G$. The diameter of fullerene graphs satisfies the following upper bound.
\begin{theorem}[Andova et al.~\cite{ADKLS12+}]\label{thm:diameter}
If $G$ is a fullerene graph on $n$ vertices, then $\diam(G) \leq \frac15n+1$.
\end{theorem}
Corollary~\ref{cor:daugherty}, in conjunction with Theorems~\ref{thm:heckman-thomas} and~\ref{thm:diameter}, allows us to prove a conjecture of Graffiti~\cite[Conjecture 912]{FRFHC01}. Let us remark that the conjecture was proved for fullerene graphs on at least 617 502 vertices by Andova et al.~\cite{ADKLS12+}.
\begin{corollary}
If $G$ is a fullerene graph, then $\alpha(G) \geq 2(\diam(G)-1)$.
\end{corollary}
\begin{proof}
Let $G$ be a fullerene graph on $n$ vertices. It is easy to check that $\left\lceil \frac38n \right\rceil \geq \left\lfloor \frac25n \right\rfloor$ if $n < 40$, and $\left\lceil \frac12n-\sqrt{\frac35n}\right\rceil \geq \left\lfloor \frac25n \right\rfloor$ if $n \geq 36$. In the former case, we apply Theorems~\ref{thm:heckman-thomas} and~\ref{thm:diameter}, and in the latter case, we apply Corollary~\ref{cor:daugherty} and Theorem~\ref{thm:diameter}, to show that $\alpha(G) \geq 2(\diam(G)-1)$.
\end{proof}
Motivated by H\"uckel theory from chemistry, Daugherty, Myrvold and Fowler~\cite{DaMyFo07} (see also~\cite{Dau09}) defined the \emph{closed-shell independence number} $\alpha^-(G)$ of a fullerene graph $G$ as the maximum size of an independent set $A$ of $G$ with the property that exactly half of the eigenvalues of $G-A$ are positive. Recall that an \emph{eigenvalue} of a graph $G$ is an eigenvalue of its \emph{adjacency matrix}, the square $n \times n$ matrix $(a_{uv})$ where $a_{uv}=1$ if $uv \in E(G)$, and $a_{uv}=0$ otherwise.
\begin{theorem}[Daugherty, Myrvold and Fowler~\cite{DaMyFo07}]\label{thm:closed-shell}
If $G$ is a fullerene graph, then $\alpha^-(G)\leq \frac38n+\frac32$.
\end{theorem}
Daugherty, Myrvold and Fowler~\cite{DaMyFo07} (see also~\cite[Conjecture~7.7.1]{Dau09}) conjectured that the equality $\alpha^-(G)=\alpha(G)$ holds only when $G$ is isomorphic to one of the three fullerene graphs in Figure~\ref{fig:shell}, and verified the conjecture for all fullerene graphs on $n \leq 100$ vertices. Corollary~\ref{cor:daugherty} and Theorem~\ref{thm:closed-shell} imply the conjecture for all fullerene graphs on $n>60$ vertices, so the conjecture is now proved completely.
\begin{corollary}
\label{cor:shell}
A fullerene graph $G$ satisfies $\alpha^-(G)=\alpha(G)$ if and only if $G$ is one of the graphs in Figure~\ref{fig:shell}.
\end{corollary}
\begin{proof}
Let $G$ be a fullerene graph on $n$ vertices. The conjecture was verified for $n \leq 100$ in~\cite{Dau09}, so it suffices to consider the case $n > 100$. Since $\left\lfloor \frac38n+\frac32 \right\rfloor < \left\lceil 12n-\sqrt{\frac35n} \right\rceil$ for $n>60$, it follows by Corollary~\ref{cor:daugherty} and Theorem~\ref{thm:closed-shell} that $\alpha^-(G)<\alpha(G)$ for $n>60$.
\end{proof}
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\path (162:2.5) coordinate (d4);
\path (186-5:2-0.2) coordinate (d5);
\path (210+5:2-0.2) coordinate (d6);
\path (234:2.5) coordinate (d7);
\path (258-5:2-0.2) coordinate (d8);
\path (282+5:2-0.2) coordinate (d9);
\path (306:2.5) coordinate (d10);
\path (330-5:2-0.2) coordinate (d11);
\path (354+5:2-0.2) coordinate (d12);
\path (18:2.5) coordinate (d13);
\path (44-5:2-0.2) coordinate (d14);
\path (68+5:2-0.2) coordinate (d15);
\path (90:3) coordinate (e1);
\path (162:3) coordinate (e2);
\path (234:3) coordinate (e3);
\path (306:3) coordinate (e4);
\path (18:3) coordinate (e5);
\draw (a1)--(a2)--(a3)--(a4)--(a5)--cycle
(b1)--(b2)--(b3)--(b4)--(b5)--(b6)--(b7)--(b8)--(b9)--(b10)--(b11)--(b12)--(b13)--(b14)--(b15)--cycle
(a1)--(b2) (a2)--(b5) (a3)--(b8) (a4)--(b11) (a5)--(b14)
(c1)--(c2)--(c3)--(c4)--(c5)--(c6)--(c7)--(c8)--(c9)--(c10)--(c11)--(c12)--(c13)--(c14)--(c15)--(c16)--(c17)--(c18)--(c19)--(c20)--cycle
(b1)--(c2) (b3)--(c3) (b4)--(c6) (b6)--(c7) (b7)--(c10) (b9)--(c11) (b10)--(c14) (b12)--(c15) (b13)--(c18) (b15)--(c19)
(d1)--(d2)--(d3)--(d4)--(d5)--(d6)--(d7)--(d8)--(d9)--(d10)--(d11)--(d12)--(d13)--(d14)--(d15)--cycle
(c1)--(d2) (c4)--(d3) (c5)--(d5) (c8)--(d6) (c9)--(d8) (c12)--(d9) (c13)--(d11) (c16)--(d12) (c17)--(d14) (c20)--(d15)
(e1)--(e2)--(e3)--(e4)--(e5)--cycle
(d1)--(e1) (d4)--(e2) (d7)--(e3) (d10)--(e4) (d13)--(e5)
;
\foreach\i in {1,...,5}
{
\draw (a\i) node[vertex] {};
}
\foreach\i in {1,...,15}
{
\draw (b\i) node[vertex] {};
}
\foreach\i in {1,...,20}
{
\draw (c\i) node[vertex] {};
}
\foreach\i in {1,...,15}
{
\draw (d\i) node[vertex] {};
}
\foreach\i in {1,...,5}
{
\draw (e\i) node[vertex] {};
}
\end{tikzgraph}
}
\hfill\null
\caption{The three graphs in Corollary~\ref{cor:shell}, with the nomenclature of~\cite{FoMa95}. The graph $20$:$1$ is the dodecahedral graph, $40$:$40$ is the unique fullerene graph on $40$ vertices with the tetrahedral automorphism group $T_d$, and $60$:$1812$ is the buckminsterfullerene graph.}
\label{fig:shell}
\end{figure}
\section{Smallest eigenvalues of fullerene graphs}\label{sec:eigenvalues}
As the final application of Theorem~\ref{thm:main}, we compute an upper bound on the smallest eigenvalue of a fullerene graph $G$. Recall that the \emph{Laplacian} of a graph with adjacency matrix $(a_{uv})$ is the $n \times n$ matrix $(c_{uv})$, where $c_{uv}=d(u)$ if $u=v$, and $c_{uv}=-a_{uv}$ if $u \neq v$. A \emph{Laplacian eigenvalue} of a graph is an eigenvalue of its Laplacian. The smallest eigenvalue and the largest Laplacian eigenvalue of $G$ are denoted by $\lambda_n(G)$ and $\mu_n(G)$, respectively.
The maximum size of a cut in a graph can be bounded in terms of its largest Laplacian eigenvalue. The following is a corollary of a more general theorem of Mohar and Poljak~\cite{MohPol90}.
\begin{theorem}[Mohar and Poljak~\cite{MohPol90}]\label{thm:mohar-poljak}
If $G$ is a graph on $n$ vertices, then $|\delta(X)| \leq \frac 14n\mu_n(G)$, for every $X \subseteq V(G)$.
\end{theorem}
Andova et al.~\cite{ADKLS12+} have recently used Theorem~\ref{thm:mohar-poljak} to show that $\lambda_n(G) \leq -3+\frac{157.16}{\sqrt{n}}$ for every fullerene graph $G$. Their bound can be improved by applying Corollary~\ref{cor:daugherty}.
\begin{corollary}\label{cor:eigenvalue}
If $G$ is a fullerene graph on $n$ vertices, then $\lambda_n(G) \leq -3+8\sqrt{\frac3{5n}}$.
\end{corollary}
\begin{proof}
Since $G$ is $3$-regular, the smallest eigenvalue of $G$ is $\lambda_n(G)=3-\mu_n(G)$, and there exists a cut $\delta(X)$ such that $|\delta(X)| \geq \frac32n-\tau_{\odd}(G)$. Therefore, by Theorem~\ref{thm:mohar-poljak}, $\lambda_n(G) \leq -3+\frac 4n\tau_{\odd}(G)$, so by Theorem~\ref{thm:main}, $\lambda_n(G) \leq -3+8\sqrt{\frac3{5n}}$.
\end{proof}
Fowler, Hansen and Stevanovi\'c~\cite{FoHaSt03} showed that the smallest eigenvalue of the truncated icosahedron (see Figure~\ref{fig:C60}) is equal to $-\phi^2$, where $\phi$ is the golden ratio $\frac{1+\sqrt 5}2$, and conjectured that, among all fullerene graphs on at least $60$ vertices, the truncated icosahedron has the maximum smallest eigenvalue. By Corollary~\ref{cor:eigenvalue}, any fullerene graph on at least $264$ vertices satisfies the conjecture.
\section*{Acknowledgements}
The authors would like to thank Andr\'as Seb\H o for teaching them about $T$-joins and $T$-cuts, to Louis Esperet for reading an earlier draft of this paper, and to Dragan Stevanovi\'c for pointing out a gap in the proof of Theorem~\ref{thm:T-join}.
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,444
|
using System;
namespace OpenMetaverse.Imaging
{
public class ManagedImage
{
[Flags]
public enum ImageChannels
{
Gray = 1,
Color = 2,
Alpha = 4,
Bump = 8
};
public enum ImageResizeAlgorithm
{
NearestNeighbor
}
/// <summary>
/// Image width
/// </summary>
public int Width;
/// <summary>
/// Image height
/// </summary>
public int Height;
/// <summary>
/// Image channel flags
/// </summary>
public ImageChannels Channels;
/// <summary>
/// Red channel data
/// </summary>
public byte[] Red;
/// <summary>
/// Green channel data
/// </summary>
public byte[] Green;
/// <summary>
/// Blue channel data
/// </summary>
public byte[] Blue;
/// <summary>
/// Alpha channel data
/// </summary>
public byte[] Alpha;
/// <summary>
/// Bump channel data
/// </summary>
public byte[] Bump;
/// <summary>
/// Create a new blank image
/// </summary>
/// <param name="width">width</param>
/// <param name="height">height</param>
/// <param name="channels">channel flags</param>
public ManagedImage(int width, int height, ImageChannels channels)
{
Width = width;
Height = height;
Channels = channels;
int n = width * height;
if ((channels & ImageChannels.Gray) != 0)
{
Red = new byte[n];
}
else if ((channels & ImageChannels.Color) != 0)
{
Red = new byte[n];
Green = new byte[n];
Blue = new byte[n];
}
if ((channels & ImageChannels.Alpha) != 0)
Alpha = new byte[n];
if ((channels & ImageChannels.Bump) != 0)
Bump = new byte[n];
}
#if !NO_UNSAFE
/// <summary>
///
/// </summary>
/// <param name="bitmap"></param>
public ManagedImage(System.Drawing.Bitmap bitmap)
{
Width = bitmap.Width;
Height = bitmap.Height;
int pixelCount = Width * Height;
if (bitmap.PixelFormat == System.Drawing.Imaging.PixelFormat.Format32bppArgb)
{
Channels = ImageChannels.Alpha | ImageChannels.Color;
Red = new byte[pixelCount];
Green = new byte[pixelCount];
Blue = new byte[pixelCount];
Alpha = new byte[pixelCount];
System.Drawing.Imaging.BitmapData bd = bitmap.LockBits(new System.Drawing.Rectangle(0, 0, Width, Height),
System.Drawing.Imaging.ImageLockMode.ReadOnly, System.Drawing.Imaging.PixelFormat.Format32bppArgb);
unsafe
{
byte* pixel = (byte*)bd.Scan0;
for (int i = 0; i < pixelCount; i++)
{
// GDI+ gives us BGRA and we need to turn that in to RGBA
Blue[i] = *(pixel++);
Green[i] = *(pixel++);
Red[i] = *(pixel++);
Alpha[i] = *(pixel++);
}
}
bitmap.UnlockBits(bd);
}
else if (bitmap.PixelFormat == System.Drawing.Imaging.PixelFormat.Format16bppGrayScale)
{
Channels = ImageChannels.Gray;
Red = new byte[pixelCount];
throw new NotImplementedException("16bpp grayscale image support is incomplete");
}
else if (bitmap.PixelFormat == System.Drawing.Imaging.PixelFormat.Format24bppRgb)
{
Channels = ImageChannels.Color;
Red = new byte[pixelCount];
Green = new byte[pixelCount];
Blue = new byte[pixelCount];
System.Drawing.Imaging.BitmapData bd = bitmap.LockBits(new System.Drawing.Rectangle(0, 0, Width, Height),
System.Drawing.Imaging.ImageLockMode.ReadOnly, System.Drawing.Imaging.PixelFormat.Format24bppRgb);
unsafe
{
byte* pixel = (byte*)bd.Scan0;
for (int i = 0; i < pixelCount; i++)
{
// GDI+ gives us BGR and we need to turn that in to RGB
Blue[i] = *(pixel++);
Green[i] = *(pixel++);
Red[i] = *(pixel++);
}
}
bitmap.UnlockBits(bd);
}
else if (bitmap.PixelFormat == System.Drawing.Imaging.PixelFormat.Format32bppRgb)
{
Channels = ImageChannels.Color;
Red = new byte[pixelCount];
Green = new byte[pixelCount];
Blue = new byte[pixelCount];
System.Drawing.Imaging.BitmapData bd = bitmap.LockBits(new System.Drawing.Rectangle(0, 0, Width, Height),
System.Drawing.Imaging.ImageLockMode.ReadOnly, System.Drawing.Imaging.PixelFormat.Format32bppRgb);
unsafe
{
byte* pixel = (byte*)bd.Scan0;
for (int i = 0; i < pixelCount; i++)
{
// GDI+ gives us BGR and we need to turn that in to RGB
Blue[i] = *(pixel++);
Green[i] = *(pixel++);
Red[i] = *(pixel++);
pixel++; // Skip over the empty byte where the Alpha info would normally be
}
}
bitmap.UnlockBits(bd);
}
else
{
throw new NotSupportedException("Unrecognized pixel format: " + bitmap.PixelFormat.ToString());
}
}
#endif
/// <summary>
/// Convert the channels in the image. Channels are created or destroyed as required.
/// </summary>
/// <param name="channels">new channel flags</param>
public void ConvertChannels(ImageChannels channels)
{
if (Channels == channels)
return;
int n = Width * Height;
ImageChannels add = Channels ^ channels & channels;
ImageChannels del = Channels ^ channels & Channels;
if ((add & ImageChannels.Color) != 0)
{
Red = new byte[n];
Green = new byte[n];
Blue = new byte[n];
}
else if ((del & ImageChannels.Color) != 0)
{
Red = null;
Green = null;
Blue = null;
}
if ((add & ImageChannels.Alpha) != 0)
{
Alpha = new byte[n];
FillArray(Alpha, 255);
}
else if ((del & ImageChannels.Alpha) != 0)
Alpha = null;
if ((add & ImageChannels.Bump) != 0)
Bump = new byte[n];
else if ((del & ImageChannels.Bump) != 0)
Bump = null;
Channels = channels;
}
/// <summary>
/// Resize or stretch the image using nearest neighbor (ugly) resampling
/// </summary>
/// <param name="width">new width</param>
/// <param name="height">new height</param>
public void ResizeNearestNeighbor(int width, int height)
{
if (width == Width && height == Height)
return;
byte[]
red = null,
green = null,
blue = null,
alpha = null,
bump = null;
int n = width * height;
int di = 0, si;
if (Red != null) red = new byte[n];
if (Green != null) green = new byte[n];
if (Blue != null) blue = new byte[n];
if (Alpha != null) alpha = new byte[n];
if (Bump != null) bump = new byte[n];
for (int y = 0; y < height; y++)
{
for (int x = 0; x < width; x++)
{
si = (y * Height / height) * Width + (x * Width / width);
if (Red != null) red[di] = Red[si];
if (Green != null) green[di] = Green[si];
if (Blue != null) blue[di] = Blue[si];
if (Alpha != null) alpha[di] = Alpha[si];
if (Bump != null) bump[di] = Bump[si];
di++;
}
}
Width = width;
Height = height;
Red = red;
Green = green;
Blue = blue;
Alpha = alpha;
Bump = bump;
}
/// <summary>
/// Create a byte array containing 32-bit RGBA data with a bottom-left
/// origin, suitable for feeding directly into OpenGL
/// </summary>
/// <returns>A byte array containing raw texture data</returns>
public byte[] ExportRaw()
{
byte[] raw = new byte[Width * Height * 4];
if ((Channels & ImageChannels.Alpha) != 0)
{
if ((Channels & ImageChannels.Color) != 0)
{
// RGBA
for (int h = 0; h < Height; h++)
{
for (int w = 0; w < Width; w++)
{
int pos = (Height - 1 - h) * Width + w;
int srcPos = h * Width + w;
raw[pos * 4 + 0] = Red[srcPos];
raw[pos * 4 + 1] = Green[srcPos];
raw[pos * 4 + 2] = Blue[srcPos];
raw[pos * 4 + 3] = Alpha[srcPos];
}
}
}
else
{
// Alpha only
for (int h = 0; h < Height; h++)
{
for (int w = 0; w < Width; w++)
{
int pos = (Height - 1 - h) * Width + w;
int srcPos = h * Width + w;
raw[pos * 4 + 0] = Alpha[srcPos];
raw[pos * 4 + 1] = Alpha[srcPos];
raw[pos * 4 + 2] = Alpha[srcPos];
raw[pos * 4 + 3] = Byte.MaxValue;
}
}
}
}
else
{
// RGB
for (int h = 0; h < Height; h++)
{
for (int w = 0; w < Width; w++)
{
int pos = (Height - 1 - h) * Width + w;
int srcPos = h * Width + w;
raw[pos * 4 + 0] = Red[srcPos];
raw[pos * 4 + 1] = Green[srcPos];
raw[pos * 4 + 2] = Blue[srcPos];
raw[pos * 4 + 3] = Byte.MaxValue;
}
}
}
return raw;
}
/// <summary>
/// Create a byte array containing 32-bit RGBA data with a bottom-left
/// origin, suitable for feeding directly into OpenGL
/// </summary>
/// <returns>A byte array containing raw texture data</returns>
public System.Drawing.Bitmap ExportBitmap()
{
byte[] raw = new byte[Width * Height * 4];
if ((Channels & ImageChannels.Alpha) != 0)
{
if ((Channels & ImageChannels.Color) != 0)
{
// RGBA
for (int pos = 0; pos < Height * Width; pos++)
{
raw[pos * 4 + 0] = Blue[pos];
raw[pos * 4 + 1] = Green[pos];
raw[pos * 4 + 2] = Red[pos];
raw[pos * 4 + 3] = Alpha[pos];
}
}
else
{
// Alpha only
for (int pos = 0; pos < Height * Width; pos++)
{
raw[pos * 4 + 0] = Alpha[pos];
raw[pos * 4 + 1] = Alpha[pos];
raw[pos * 4 + 2] = Alpha[pos];
raw[pos * 4 + 3] = Byte.MaxValue;
}
}
}
else
{
// RGB
for (int pos = 0; pos < Height * Width; pos++)
{
raw[pos * 4 + 0] = Blue[pos];
raw[pos * 4 + 1] = Green[pos];
raw[pos * 4 + 2] = Red[pos];
raw[pos * 4 + 3] = Byte.MaxValue;
}
}
System.Drawing.Bitmap b = new System.Drawing.Bitmap(
Width,
Height,
System.Drawing.Imaging.PixelFormat.Format32bppArgb);
System.Drawing.Imaging.BitmapData bd = b.LockBits(new System.Drawing.Rectangle(0, 0, b.Width, b.Height),
System.Drawing.Imaging.ImageLockMode.WriteOnly,
System.Drawing.Imaging.PixelFormat.Format32bppArgb);
System.Runtime.InteropServices.Marshal.Copy(raw, 0, bd.Scan0, Width * Height * 4);
b.UnlockBits(bd);
return b;
}
public byte[] ExportTGA()
{
byte[] tga = new byte[Width * Height * ((Channels & ImageChannels.Alpha) == 0 ? 3 : 4) + 32];
int di = 0;
tga[di++] = 0; // idlength
tga[di++] = 0; // colormaptype = 0: no colormap
tga[di++] = 2; // image type = 2: uncompressed RGB
tga[di++] = 0; // color map spec is five zeroes for no color map
tga[di++] = 0; // color map spec is five zeroes for no color map
tga[di++] = 0; // color map spec is five zeroes for no color map
tga[di++] = 0; // color map spec is five zeroes for no color map
tga[di++] = 0; // color map spec is five zeroes for no color map
tga[di++] = 0; // x origin = two bytes
tga[di++] = 0; // x origin = two bytes
tga[di++] = 0; // y origin = two bytes
tga[di++] = 0; // y origin = two bytes
tga[di++] = (byte)(Width & 0xFF); // width - low byte
tga[di++] = (byte)(Width >> 8); // width - hi byte
tga[di++] = (byte)(Height & 0xFF); // height - low byte
tga[di++] = (byte)(Height >> 8); // height - hi byte
tga[di++] = (byte)((Channels & ImageChannels.Alpha) == 0 ? 24 : 32); // 24/32 bits per pixel
tga[di++] = (byte)((Channels & ImageChannels.Alpha) == 0 ? 32 : 40); // image descriptor byte
int n = Width * Height;
if ((Channels & ImageChannels.Alpha) != 0)
{
if ((Channels & ImageChannels.Color) != 0)
{
// RGBA
for (int i = 0; i < n; i++)
{
tga[di++] = Blue[i];
tga[di++] = Green[i];
tga[di++] = Red[i];
tga[di++] = Alpha[i];
}
}
else
{
// Alpha only
for (int i = 0; i < n; i++)
{
tga[di++] = Alpha[i];
tga[di++] = Alpha[i];
tga[di++] = Alpha[i];
tga[di++] = Byte.MaxValue;
}
}
}
else
{
// RGB
for (int i = 0; i < n; i++)
{
tga[di++] = Blue[i];
tga[di++] = Green[i];
tga[di++] = Red[i];
}
}
return tga;
}
private static void FillArray(byte[] array, byte value)
{
if (array != null)
{
for (int i = 0; i < array.Length; i++)
array[i] = value;
}
}
public void Clear()
{
FillArray(Red, 0);
FillArray(Green, 0);
FillArray(Blue, 0);
FillArray(Alpha, 0);
FillArray(Bump, 0);
}
public ManagedImage Clone()
{
ManagedImage image = new ManagedImage(Width, Height, Channels);
if (Red != null) image.Red = (byte[])Red.Clone();
if (Green != null) image.Green = (byte[])Green.Clone();
if (Blue != null) image.Blue = (byte[])Blue.Clone();
if (Alpha != null) image.Alpha = (byte[])Alpha.Clone();
if (Bump != null) image.Bump = (byte[])Bump.Clone();
return image;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,060
|
What's for dinner Today? Tomorrow? Wednesday? Thursday? Friday?
We deliver ingredients and recipes to your doorstep.
Are you done wondering what to cook for dinner every night? We are here to help! By telling us what you do and don't like, we learn about your family's preferences when it comes to dinner. This enables us to select new recipes which work for your family, which means you can stop worrying about what's for dinner! Gone is the frustation of wondering what to cook or purchasing expensive pre-planned meal kits.
✔ Be easy and fast.
✔ Go easy on the $$.
Curious? Why not try out our recipe wizard. Recipe Wizard takes your cravings and returns to you two kinds of recipes: Ones that most closely match your desired ingredients and others that may not match your initial desires but the Wizard thinks you might like.
All prices per month, recipes serve four people.
In 2018 three smart ladies met at the ATX Machine Learning hackathon. Our project won the competition, and that's where What's for Dinner was born. We are passionate about cooking, math, and machine learning!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,173
|
Select the print you would like from the drop-down list and click "add to cart". The cart will appear on the top right-hand side of the screen. You will receive a beautiful 8 x 10 inch Matte of your choice for $100. This price includes packaging. A $3.00 shipping charge will apply to all orders (regardless of how many photos are purchased).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,934
|
Q: Jersey MessageBodyWriter not found for media type=text/plain I'm trying to follow the Jersey docs to enable a non-200 response if an error occured (https://jersey.java.net/documentation/latest/representations.html#d0e3586)
My code looks like :
@POST
@Produces(MediaType.TEXT_PLAIN)
@Consumes(MediaType.APPLICATION_FORM_URLENCODED)
public ResponseBuilder getData(@FormParam("one") String one,@FormParam("two") String two,@FormParam("three") String three) {
if(one.isEmpty() || two.isEmpty() || three.isEmpty()) {
logger.error("Missing params for getData");
throw new WebApplicationException(501);
}
return Response.ok();
}
}
This unfortunately yields the following error :
[2015-02-01T16:13:02.157+0000] [glassfish 4.1] [SEVERE] [] [org.glassfish.jersey.message.internal.WriterInterceptorExecutor] [tid: _ThreadID=27 _ThreadName=http-listener-1(2)] [timeMillis: 1422807182157] [levelValue: 1000] [[
MessageBodyWriter not found for media type=text/plain, type=class org.glassfish.jersey.message.internal.OutboundJaxrsResponse$Builder, genericType=class javax.ws.rs.core.Response$ResponseBuilder.]]
A: The problem is the return type of your method. It has to be Response instead of ResponseBuilder.
Change your code to the following and it should work:
@POST
@Produces(MediaType.TEXT_PLAIN)
@Consumes(MediaType.APPLICATION_FORM_URLENCODED)
public Response getData(@FormParam("one") String one,@FormParam("two") String two,@FormParam("three") String three) {
if(one.isEmpty() || two.isEmpty() || three.isEmpty()) {
logger.error("Missing params for getData");
throw new WebApplicationException(501);
}
return Response.ok();
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,453
|
{"url":"http:\/\/simbad.u-strasbg.fr\/simbad\/sim-ref?querymethod=bib&simbo=on&submit=submit+bibcode&bibcode=2011A%26A...530A..10Q","text":"2011A&A...530A..10Q\n\n other querymodes : Identifierquery Coordinatequery Criteriaquery Referencequery Basicquery Scriptsubmission TAP Outputoptions Help\n\n Query : 2011A&A...530A..10Q\n\n2011A&A...530A..10Q - Astronomy and Astrophysics, volume 530A, 10-10 (2011\/6-1)\n\nCharacteristics of thick disks formed through minor mergers: stellar excesses and scale lengths.\n\nQU Y., DI MATTEO P., LEHNERT M.D. and VAN DRIEL W.\n\nAbstract (from CDS):\n\nBy means of a series of N-body\/SPH simulations we investigate the morphological properties of thick stellar disks formed through minor mergers with, e.g. a range of gas-to-stellar mass ratios. We show that the vertical surface density profile of the post-merger thick disk follows a sech function and has an excess in the regions furthest away from the disk mid-plane (z> 2 kpc). This stellar excess also follows a sech function with a larger scale height than the main thick disk component (except at large radii). It is usually dominated by stars from the primary galaxy, but this depends on the orbital configuration. Stars in the excess have a rotational velocity lower than that of stars in the thick disk, and they may thus be confused with stars in the inner galactic halo, which can have a similar lag. Confirming previous results, the thick disk scale height increases with radius and the rate of its increase is smaller for more gas rich primary galaxies. On the contrary, the scale height of the stellar excess is independent of both radius and gas fraction. We also find that the post-merger thick disk has a radial scale length which is 10-50% larger than that of the thin disk. Two consecutive mergers have basically the same effect on heating the stellar disk as a single merger of the same total mass, i.e., the disk heating effect of a few consecutive mergers does not saturate but is cumulative. To investigate how thick disks produced through secular processes may differ from those produced by minor mergers, we also simulated gravitationally unstable gas-rich disks (clumpy disks''). These clumpy disks do not produce either a stellar excess or a ratio of thick to thin disk scale lengths greater than one. Comparing our simulation results with observations of the Milky Way and nearby galaxies shows that our results for minor mergers are consistent with observations of the ratio of thick to thin disk scale lengths and with the Toomre diagram'' (the sum in quadrature of the vertical and radial versus the rotational kinetic energies) of the Milky Way. The simulations of clumpy disks do not show such agreement. We conclude that minor mergers are a viable mechanism for the creation of galactic thick disks and investigating stars at several kpc above the mid-plane of the Milky Way and other galaxies may provide a quantitative method for studying the (minor) merger history of galaxies.\n\nJournal keyword(s): galaxies: interactions - galaxies: formation - galaxies: evolution - galaxies: structure - galaxies: kinematics and dynamics\n\nFull paper\n\n Number of rows : 1\nN Identifier Otype ICRS (J2000)\nRA\nICRS (J2000)\nDEC\nMag U Mag B Mag V Mag R Mag I Sp type #ref\n1850 - 2023\n#notes\n1 NGC 4244 GiG 12 17 29.659 +37 48 25.60 \u00a0 10.71 \u00a0 9.99 \u00a0 ~ 541 1\n\n Query : 2011A&A...530A..10Q\n\nBasic data :\nNGC 4244 -- Galaxy towards a Group of Galaxies\nOrigin of the objects types :\n\n(Ref) Object type as listed in the reference \"Ref\"\n(acronym) Object type linked to the acronym according to the original reference\n() Anterior to 2007, before we can link the objet type to a reference, or given by the CDS team in some particular cases\n\nOther object types:\nG (2014AJ,AGC,...), GiG (2007ApJ,[CHM2007]), IR (IRAS,ISOSS), X (2E), HI ([KOV2009])\nSyntax of coordinates is : \"ra dec (wtype) [error ellipse] quality bibcode\" :\n\u2022 ra dec : right ascension and declination (unit and frame defined according to your Output Options)\nGrey values are increasing the original precision due to the computation of frame transformations\n\u2022 (wtype) : wavelength class for the origin of the coordinates (Rad, mm, IR, Optical, UV, Xray, Gam)\n\u2022 [error ellipse] : measurement uncertainty, on (ra,dec) if the positional angle is 90 degrees, on (majaxis,minaxis) otherwise (in mas at defined epoch in the original catalogue),\nposition angle (in degrees North celestial pole to East)\n\u2022 quality : flag of quality\n\u2022 E \u2265 10\"\n\u2022 D : 1-10\" (and some old data)\n\u2022 C : 0.1-1\"\n\u2022 B : 0.01-0.1\" + 2MASS, Tyc\n\u2022 A : VLBI, Hipparcos\n\u2022 bibcode : bibcode of the coordinates reference\nICRS coord. (ep=J2000) :\n12 17 29.659 +37 48 25.60 (Infrared) [ ] C\nSyntax of coordinates is : \"ra dec (wtype) [error ellipse] quality bibcode\" :\n\u2022 ra dec : right ascension and declination (unit and frame defined according to your Output Options)\nGrey values are increasing the original precision due to the computation of frame transformations\n\u2022 (wtype) : wavelength class for the origin of the coordinates (Rad, mm, IR, Optical, UV, Xray, Gam)\n\u2022 [error ellipse] : measurement uncertainty, on (ra,dec) if the positional angle is 90 degrees, on (majaxis,minaxis) otherwise (in mas at defined epoch in the original catalogue),\nposition angle (in degrees North celestial pole to East)\n\u2022 quality : flag of quality\n\u2022 E \u2265 10\"\n\u2022 D : 1-10\" (and some old data)\n\u2022 C : 0.1-1\"\n\u2022 B : 0.01-0.1\" + 2MASS, Tyc\n\u2022 A : VLBI, Hipparcos\n\u2022 bibcode : bibcode of the coordinates reference\nFK4 coord. (ep=B1950 eq=1950) :\n12 14 59.592 +38 05 05.11 [ ]\nSyntax of coordinates is : \"ra dec (wtype) [error ellipse] quality bibcode\" :\n\u2022 ra dec : right ascension and declination (unit and frame defined according to your Output Options)\nGrey values are increasing the original precision due to the computation of frame transformations\n\u2022 (wtype) : wavelength class for the origin of the coordinates (Rad, mm, IR, Optical, UV, Xray, Gam)\n\u2022 [error ellipse] : measurement uncertainty, on (ra,dec) if the positional angle is 90 degrees, on (majaxis,minaxis) otherwise (in mas at defined epoch in the original catalogue),\nposition angle (in degrees North celestial pole to East)\n\u2022 quality : flag of quality\n\u2022 E \u2265 10\"\n\u2022 D : 1-10\" (and some old data)\n\u2022 C : 0.1-1\"\n\u2022 B : 0.01-0.1\" + 2MASS, Tyc\n\u2022 A : VLBI, Hipparcos\n\u2022 bibcode : bibcode of the coordinates reference\nGal coord. (ep=J2000) :\n154.567616 +77.157096 [ ]\nSyntax of radial velocity (or\/and redshift) is : \"value [error] (wavelength) quality bibcode\"\n\u2022 value : radial velocity or\/and redshift (Heliocentric frame) according to your Output Options\n(redshift may be not displayed if the data value is <0 and the database inside value is a radial velocity)\n\u2022 [error] : error of the corresponding value displayed before\n\u2022 (wavelength) : wavelength range of the measurement : Rad, mm, IR, Opt, UV, Xray, Gam or\u00a0\u00a0'\u223c'(unknown)\n\u2022 quality : flag of quality ( A=best quality -> E=worst quality, \u001b{\u001f\ufffd \u001b} =unknown quality)\n\u2022 bibcode : bibcode of the value's origin\nRadial velocity \/ Redshift \/ cz :\nV(km\/s) 244 [8] \/ z(~) 0.000814 [0.000027] \/ cz 244.10 [8.00]\nD\nSyntax of morphological type is : mtype quality bibcode\n\u2022 mtype : Hubble morphological class (spirals, ellipticals, etc)\n\u2022 quality : flag of quality (A=best quality -> E=worst quality, \u001b{\u001f\ufffd \u001b} =unknown quality)\n\u2022 bibcode : bibcode of the morphological type reference\nMorphological type:\nS C\nSyntax of angular size is : \"maj-axis min-axis angle (wtype) quality bibcode\"\n\u2022 maj-axis : major axis size (arc minutes)\n\u2022 min-axis : minor axis size (arc minutes)\n\u2022 angle : orientation angle (in degrees)\n\u2022 (wtype) : wavelength class for the origin of the angular size (Rad, mm, IR, Opt, UV, Xray, Gam)\n\u2022 quality : flag of quality of the angular size values ( A=best quality -> E=worst quality, \u001b{\u001f\ufffd \u001b} =unknown quality)\n\u2022 bibcode : bibcode of the angular size reference\nAngular size (arcmin):\n15.85 6.31 41 (Opt) D\nSyntax of fluxes (or magnitudes) is : \"filter-name (System) flux-value [error] quality MultVarFlags bibcode\"\n\u2022 filter-name : U, B, V, R, I, G, J, H, K, u, g, r, i, z\n\u2022 (System) : may be AB (default is Vega)\n\u2022 flux-value : value of flux or magnitude\n\u2022 [error] : error value\n\u2022 quality : flag of quality of the flux value ( A=best quality -> E=worst quality, \u001b{\u001f\ufffd \u001b} =unknown quality)\n\u2022 MultVarFlags : Mult is zero or one char (J) for joined photometry ; Var can be zero or two chars (V[0-4])\n\u2022 bibcode : bibcode of the flux reference\nFluxes (10) :\n B (AB) 10.71 [0.03] C R (AB) 9.99 [0.03] C J 8.564 [0.027] C H 7.978 [0.033] C K 7.724 [0.046] C u (AB) 13.940 [0.006] C g (AB) 12.616 [0.002] C r (AB) 12.018 [0.002] C i (AB) 11.689 [0.002] C z (AB) 11.470 [0.003] C\n', {sourceSize:12, color:'#30a090'})); aladin.on('objectClicked', function(object) { var objName=object.data.MAIN_ID; aladin.showPopup(object.ra,object.dec,'',''+ objName+''); });\" title=\"Show Simbad objects\"> Overlay points in this preview\n All (CDSPortal) Send to within arcsec The VizieR photometry tool allows for easy visualization of photometry points extracted around the Simbad position from photometry-enabled catalogues in VizieR. The search radius has to be specified by the user. It is currently limited to a maximum of 30 arcsec. It depends mostly on the precision or quality of the coordinates (SIMBAD and VizieR catalogs), the resolution of the images from which the sources were extracted, source extent, and source crowding. Suggestions are: crowded field: 0.5 to 1.5 arcsec, 3 arcsec otherwise; uncertain coordinates (SIMBAD quality E or coordinates without reference): 5 to 30 arsec (risky!).\n Some important notes on this object about identifications and objects associations. notes: seem IRAS 12150+3804\n\nHierarchy : number of linked objects\nwhatever the membership probability is (see description here ) :\n\n : 6 The count displayed here is the number of children objects. The list obtained by clicking the button may be larger, as some children may be linked with different references or probability. %This number is the number of distinct objets linked, by using this button, you will obtain all links (may be more than one) from that object to his children : 3 : 155 Display criteria : All >=90% stars galaxies radio IR <2 arcsec #ref>50\n\n The link on the acronym of the identifiers give access to the information for this acronym in the dictionary of nomenclature. Identifiers (24) :\nAn access of full data is available using the icon Vizier near the identifier of the catalogue\n\n NGC 4244 ISOSS J12173+3747 SDSS J121729.43+374826.4 [CHM2007] HDC 706 J121729.65+3748255 AGC 29932 LEDA 39422 TC 777 [CHM2007] LDC 867 J121729.65+3748255 2E 2636 LJHY 32 UGC 7322 [KOV2009] 69 2E 1214.4+3800 2MASX J12172965+3748255 UZC J121729.0+374819 [M98c] 121459.9+380506 FGC 1402 MCG+06-27-045 Z 1215.0+3805 [SLK2004] 679 IRAS F12149+3805A RFGC 2245 Z 187-35 [VDD93] 152\n\n References (541 between 1850 and 2023) (Total 541) Simbad bibliographic survey began in 1850 for stars (at least bright stars) and in 1983 for all other objects (outside the solar system). Follow new references on this object\nReference summaries :\n\nfrom: to:\n\nor select by : (not exhaustive, explanation here)\n\nCollections of Measurements\n\nvelocities : 14 \u00a0\u00a0 distance : 12\n\nObserving logs\n\nherschel : 6 \u00a0\u00a0 ISO : 5 \u00a0\u00a0 IUE : 4\n\nExternal archives :\n\nLink by name to the catalogue in VizieR :\n\n NGC 4244 2E 2636 FGC 1402 IRAS F12149+3805A 2MASX J12172965+3748255 RFGC 2245 SDSS J121729.43+374826.4 UGC 7322\n\nSearch by coordinates in Vizier (radius: 30 arcsec)\n\nAnnotations :\nAnnotations allow a user to add a note or report an error concerning the astronomical object and its data. It requires registration to post a note. See description .\nPlease, have a look at Best practices. The list of all annotations to SIMBAD objects can be found here .\n\nCurrently no annotations available\n\nadd an annotation to this object\n\nreport an error concerning the data of this object\n\nTo bookmark this query, right click on this link: simbad:objects in 2011A&A...530A..10Q and select 'bookmark this link' or equivalent in the popup menu\n\n2023.02.04-13:07:30","date":"2023-02-04 12:07:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4984162449836731, \"perplexity\": 13188.353330048485}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500126.0\/warc\/CC-MAIN-20230204110651-20230204140651-00013.warc.gz\"}"}
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{"url":"https:\/\/delhisaraswatsangh.org\/sfx3b3\/deebe5-android-app-for-writing-math-equations","text":"Home android app for writing math equations\n\n# android app for writing math equations\n\nThis is an incredibly fun math fluency iOS app that uses storytelling to help young learners develop math skills over time. Also, logged in to the same Google account on your computer running Google Chrome. Until now, writing equations and math expressions on your computer has been slow and laborious. The Calculator + Scientific app is versatile, easy-to-use and beautifully designed. This app is designed for the high-school students who struggle with math. iPhone application development company| Infinite App. One of these tools, TeX equation editor, is integrated with Google Drive so you can easily paste TeX equations right into your Google Docs. Graphing Calculator + Math. Working in tandem with Read&Write, EquatIO lets students have their maths equations read outloud. Teen. Solves systems with up to 5 linear equations. Receives cards added by the group's manager app. It is best Handwriting to text App Android\/ios 2021. Equations is an app that helps you to solve equations, find common constants, and get common formulas. SUB and SUP are used to specify subscripts and superscripts. Some of the scientific methods supported by this app are matrices, equation solving, financial equations, graphing, complex numbers, calculus, and 64-bit programming mode. Sharpen your math skills with the best math apps for Android! Here i developed one Android Application to solve mathematical solutions automatically in simple manner. Math Tricks is a brilliant app that contains a lot of these tricks and its method of teaching is admirable. This is one of the hardest apps to develop that is mentioned in this how to make an educational app article. Praceta Jose Cardoso Pires N7 Android APK; Linux; News & reviews; Softpedia > Mac > Math\/Scientific > FX Equation. Online-Mathematik L\u00f6ser mit kostenlosen schrittweisen L\u00f6sungen f\u00fcr Algebra, Analysis und andere mathematische Probleme. Solving maths problem is easy if you use free maths problem solver app like Photomath. Share this post: Educatorstechnology Thursday, May 07, 2020 2020, google docs, math resources The popular Google Docs add-on, MathType, is now being offered for free for educational institutions. Math Equations allows you to take your typeset languages and convert them to images to use inside of your slideshow. Learning things via games is an enjoyable method of education and the information lasts longer in mind and it is exactly what this app is trying to do. Math Equations allows you to take your typeset languages and convert them to images to use inside of your slideshow. Kids Numbers and Math (Free) For all pre-school kids, it is difficult to teach them math in a way that \u2026 Such as using the latest MS Word's built-in MathML\/XML based equation editor, and then converting the pages to web. Until now, writing equations and math expressions on your computer has been slow and laborious. Equation solver with steps Are you in search of an equation calculator for free in order to solve math equations step-by-step? Math \u2013 o \u2013 mir is another free math equation editor software, which lets you write math equations easily. Other features include the ability to reload a equation image and make changes and re upload to your presentation. Challenge and improve your math skills! I recommend this to all who are interested in going from WYSIWYG (What you see is what you get) to LaTex. Not just that, but Chegg Math Solver also shows a step-by-step solution on how to solve the equation. Tip 1: Insert Math Equation. This app let you to handwrite text on your phone or tablet and that too in about 100 languages. Write beautiful math equations & symbols easily in MathMagic, use them widely in your word processors, Presentations, DTP software. What is EquatIO? MathMagic is a WYSIWYG math editor with Graphic user interface, with support for MathML, LaTeX, MS Equation Editor, and more. This is useful for displaying complex formulas on your web page. 402, Gowri Nilayam You can write maths documents and including geometric and physics circuit diagrams. The cool thing is that you don\u2019t have to write down the math problem, simply photo it and the app will do the rest. MyScript Calculator is one of the best android apps for providing mathematical solutions. Easy maths formula platform for Engineering and others. Screen shots. HMS deals with Factorisation, Differentiation, Integration & Linear Equation.. Write your math equations in linear format, for example like a2+b2=c2 and Word will convert it into professional display format a\u00b2+b\u00b2=c\u00b2. Mathematical operators (plus, minus, multiply, divide, modulus, dot, factorial, etc. This is a powerful math learner app for solving math homework problems. If you need a scientific calculator that can help you with all types of calculations and math problems, you will have to go for a third-party calculator app. It consists of three sections: solver, constants, and formulas. Write your math equations in linear format, for example like a2+b2=c2 and Word will convert it \u2026 Math is a difficult subject for a lot of people, but it's one of the most important. 1. Brainly. \u201cComplete Mathematics App covers basic and high school maths with many options to easily learn mathematics with a maths solver. For some students, simply writing an equation out isn't enough, and even the most instinctive of online equation editors can go over their heads. Wolfram Alpha. Recent changes added the math equations data into the alt text, which you can see by clicking using control + alt + y. Math Equations Practice Game Android latest 1.4 APK Download and Install. You can also generate an image of a mathematical formula using the TeX language (pronounced \"tek\" or \"tech\"). Price: Free. It is an engaging math app ideal for kids at the elementary level. More than just a calculator app, Mathway (Android, iOS) is a math learning aid that allows you to plug in equations and display step-by-step solutions to get to the final answer. If you don't want to deploy and maintain code in this article, you can use my server for free. To get started, install the Google lens app on your Android or Google app for iPhone. Photomath Apps On Google Play. Maths input panel of windows 8 is a handwriting recognition software; you need to have a touchpad and a stylus to be able to work with it properly. - Systems Solver: to solve systems with two unknowns, three unknowns, or four unknowns. This feature is available to Office Insiders only right now. This tutorial demonstrates how to write a math equation using Microsoft Word 2010. Home Features Solutions Enterprise Education Support Blog Get Nebo Take your notes faster with Nebo Pen and keyboard working in tandem. Pen Keyboard Pen Words are recognized while you write. INerd. Written document can be saved as PNG or PDF. This app is an equation editor for writing math equations, expressions, mathematical characters, and operations. Learn more Type or handwrite virtually any mathematical expression directly on your keyboard or touchscreen. PhotoMath is best maths equation solver app for Android, iOS, Windows But EquatIO is different. Equations, integrals, derivatives, limits and much more. A couple that can do the job (certainly there are more! In fact it is going to solve the equation \u2026 You can also cross out stuff you've already written to \u2026 This app will prove a way to enter emojis by drawing. All you need in math, geometry, equations. Considered to be one of the conventional math solving apps, it would have been a shame to exclude Khan Academy from our list of best math solver apps. Get math problems solution in real time with the MyScript Calculator. I mean, is there any keyboard or anything to write mathematical symbols? Go to Insert tab, find Symbols group and click Equation button. EquatIO makes math digital, helping teachers and students at all levels create math expressions quickly and easily. By purchasing this item, you are transacting with Google Payments and agreeing to the Google Payments. Also it\u2019s very bad at anything but mathematical equations, for example you can\u2019t draw shapes and lines and place them in an arbitrary part of the page. It offers learning modules for different skill levels and comes packed with text tools for writing equations. Because I'm unfamiliar with LaTex formatting, this equation editor has helped me immensely. The app includes a repository of over 40,000 questions and 10,000 videos that answer different types of maths. Download: iPhone. Calculator + Scientific. Even though this doesn\u2019t come with a free version, we found that it is worth it. This app also supports more than 1000 emojis so that you can express your feeling in any android app. Inequality Calculator for solving linear and quadratic inequalities. Word for Android and Word Mobile supports writing and editing math equations. Also solves 1 quadratic equation. This offer, mandated by the current health crisis, is valid till the first of August 2020. MathMagic for Android, the ultimate Equation Editor for Android tablets! Meera Bai ke Dohe, Pad with Vyakhya or shabdarth . It's pretty much impossible to write equations with the native keyboard and even if I somehow manage to type they don't get recognised. If the equations are created by some limited writers, there are many options to combine to get the best results. All tools to write a mathematical equation are accessible on the left side of the screen. Microsoft Math solver combines the power of multiple solvers such as math problem solver, equation solver, math answer scanner, percentage calculator, scientific calculator, word problem solver, math photo solver and math handwriting solver. Recent changes added the math equations data into the alt text, which you can see by clicking using control + alt + y. Add to Wishlist. Double tap to convert or use pen gestures to edit. 1,942. Solve equations and plot graphs in a few clicks with this free app! Solve4x. The app lets you write equations within the system. Use your phone as a contactless bank card. If learning maths is like a life-and-death situation for you, this is the shot you should take. EquatIO digital math can only be inserted to and extracted from Microsoft Word. Enable Your Users To Write Math Equations In Web And Desktop Apps Codeproject. Added In-App purchase to enable removing watermark from images. Input LaTeX, Tex, AMSmath or ASCIIMath notation (Click icon to switch to ASCIIMath mode) to make formula. Get notified of available surebets. 7 Best Math Problem Solver App for Android and iPhone Photomath. Math is a hard and complicated thing for most of us, but not for this awesome app! The Android app is expected next year, according to the developer's website. Hyderabad 500084, Cookies help us deliver our services. 2.Marble math: this app allows you to practice math while working on it through moving a marble through various mazes. Your private math tutor, solves any math problem with steps! Tip: How to show equations without any coding or even having your own server? This app supports printed and cursive writing with or without a stylus. Even though the default calculator app in most Android smartphones today is fairly feature packed and capable enough to handle a few complex equations. The app lets you write and solve math problems right onto the touch screen of an iPad using the custom keypad, and you can print, email, or save to cloud services like Dropbox. Type or handwrite virtually any mathematical expression directly on your keyboard or touchscreen. Use your phone as a contactless bank card, NFC mobile payment app Makes NFC payments. How To Display Math Formula And Equation In Android Inducesmile Tutorial Apps Studio Sdk Development. Math Equation Solver Android latest 4.5 APK Download and Install. TeX equation editor that creates graphical equations. On the other hand Math-o-mir can really speed up *typing* of mathematical equations. You can also use it on ANY blogs, Wikis or websites which do not support TeX and math equations. This is an equation editor for phone and tablets. To solve a math equation, users need to scan and take a photo of the question through the device camera. , After 6\" Mobiles and Portable Laptops, I still believe Tablets are the best fit. LaTeX math and equations Learn to typeset and align equations, matrices and fractions in LaTeX. New features: Multiplication, division and parentheses. TeX equation editor that creates graphical equations. You can also generate an image of a mathematical formula using the TeX language (pronounced \"tek\" or \"tech\"). A straightforward and efficient cross-platform application that makes it easy for you to write and create equations using your Mac. MathMagic is a WYSIWYG math editor with Graphic user interface and more. 1- Daum Equation Edior 'Edit the formula in this program is designed to quickly and easily. 4,145 downloads Updated: November 20, 2020 Trial . Once done, Chegg Math Solver automatically solves the mathematical equation. Silpa Park, Kondapur 2. Other features include the ability to reload a equation image and make changes and re upload to your presentation. You came to the right place! EquatIO makes math digital, helping teachers and students at all levels create math expressions quickly and easily. It offers solutions on the basics of every discipline along with bits of inverse trigonometry, trigonometry, logarithms, and constants. ): CalcuPad App (free and Pro- 1.99) \u2013 Provides a viewable writing area to display the math equation. With Nebo, handwriting is at the core of your digital workflow. EquatIO makes math digital, helping teachers and students at all levels create math expressions quickly and easily. Complete Mathematics is categorized into tutorials, formulas, equations, practical application with solver, questions and solutions, maths dictionary and quiz.\u201d 4- Mathematics Dictionary (update: no longer working) MyScript is an Android application that lets you write your notes as well as math formulas. Holen Sie sich Hilfe im Internet oder mit unserer Mathe-App. Fraction calculator with step-by-step operations and algebra. It lets people ask questions from \u2026 There\u2019s no need for any complicated code or programming languages. All Maths formulas for your work and study, TechCalc - the full-featured scientific and graphing calculator app for Android. If the built-in equations don\u2019t meet the actual needs, you could edit or modify them by equation tools or even write math equation by using handwriting board, especially those with complex symbols and structures. 1. SOLVER: - Equation solvers: to solve quadratic equation solvers, cubic equation solvers. Brainly is a social networking app for students. This is useful for displaying complex formulas on your web page. BaseColumns; CalendarContract.AttendeesColumns; CalendarContract.CalendarAlertsColumns; CalendarContract.CalendarCacheColumns; CalendarContract.CalendarColumns Consider the equation: This can be represented as: $H\\left(s\\right) = \\int$0 e-st h(t) dt The mathematical symbols are given with their standard ISO entity names. Photomath is a math solver for Android and iPhone app that simply works just by placing the smartphone over... MyScript Calculator. Until now, writing equations and math expressions on your computer has been slow and laborious. 10 Best Math Apps For Android Better Skills Authority. Write beautiful math equations & symbols easily in MathMagic, use them widely in your word processors, Presentations, Web. Card data to NFC How do I write MathJax on an Android device? Solves systems with up to 5 linear equations. Sail through Math. SPDroid - Android Apps And Tutorials 3,566 views Then it changes the equations to text and solves it. Download: Android. Solves systems with up to 5 linear equations. Contains Ads. 2660-530 Sao Juliao do Tojal I use this app to type equations, have them translated into LaTex, then I copy and paste the LaTex into my ibook I'm making for my Middle School Math class. Here\u2019s a iOS app that does this from images: Mathpix - Solve and graph math using pictures on the App Store Website is here: Mathpix Disclaimer: I\u2019m the founder of Mathpix. This app can instantly solve almost any math equation and calculate the \u201cx\u201d. FX Equation for Mac . Snip can convert images into LaTeX for inline equations, block mode equations, and numbered equations. Android Apps; Our Sources; Contact ; A Good Google Docs Add-on for Typing and Handwriting Math Equations. Tinkutara Education. Step by step solutions and formulas. Portugal, Cookies help us deliver our services. New features: Multiplication, division and parentheses. 9. It supports basic arithmetics, fractions, decimal numbers, linear equations, and several functions like logarithms, although London-based MicroBlink, the company behind the app, says that new math is constantly being added to the app with every new release. Snip also supports some text mode LaTeX, like the tabular environment. 10 Best Math Apps For Android Better Skills Authority. Well, Chegg Math Solver is one of the best math solver app that you can use on your Android smartphone. 4.3 \/ 5 4. Review Free Download specifications 100% CLEAN report malware. #1 Android Tutorial: Addition of Numbers (Make app to perform Arithmetic Operations) For Beginners - Duration: 17:48. Overview of basic math features, with live-rendering and sandbox in your browser. It also \u2026 Inline math; Equations; Fractions; Matrices; Scaling of Parentheses, Brackets etc. The math solver app \u2026 Find the best bet odds from your bookmakers. In this post I am going to show you how you can build an android app that can solve any kind of linear equation in a systematic way. Simply type, handwrite or dictate any expression, and EquatIO will convert it to accurate digital math which can be added into a Microsoft Word doc or G Suite apps with a click. 2. The first and only app that lets you mix both by using interactive ink. Learn math tricks on Android and have fun, playing math games; a cool way of learning. Math fun: A simple, fun and quick way to solve your math with in no time. Currently, MyScript supports mathematical symbols +, -, \u00d7, \/, \u221a, Pi, parentheses, and exponentiation. By using our services, you agree to our use of cookies, By purchasing this item, you are transacting with Google Payments and agreeing to the Google Payments, Calculator for solving linear, quadratic, cubic and quartic equations, Solves quadratic equations\/ formulas and gives you the step-by-step solution. Once done, open the Google lens app on your smartphone and capture the handwritten text, highlight it \u2026 This is an equation editor for phone and tablets. Word for Android and Word Mobile supports writing and editing math equations. #09 \u2013 Calculator ++ Calculator ++ is the most efficient calculator you are going to use on your Android device. What's new in FX Equation \u2026 EquatIO software allows you to create mathematical equations, formulas and more directly on your computer. It recognizes math objects and can export to LaTeX It recognizes math objects and can export to LaTeX Edit: this seems to have deprecated and it no longer available for download. This app has high rating as well so can try it with any hesitation. Here area a few solutions for creating math problems electronically \u2013 yes with apps for that! These apps are ideal for writing, editing and viewing math formula whether they are simple or complex. The Graphing Calculator + Math app does exactly what the name suggests: calculate long math equations with the push of a few buttons. Equation solving and graphing are as easy as shelling peas for this new math equation solver and helper for android! Also solves 1 quadratic equation. This app is an equation editor for writing math equations, expressions, mathematical characters, and operations.","date":"2021-03-01 03:18:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.1759873926639557, \"perplexity\": 2963.234878672909}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178361849.27\/warc\/CC-MAIN-20210301030155-20210301060155-00408.warc.gz\"}"}
| null | null |
Marsh's Library, situated in St. Patrick's Close, adjacent to St. Patrick's Cathedral, Dublin, Ireland is a well-preserved library of the late Renaissance and early Enlightenment. When it opened to the public in 1707 it was the first public library in Ireland. It was built to the order of Archbishop Narcissus Marsh and has a collection of over 25,000 books and 300 manuscripts.
History
Foundation
The library was built for the Most Rev. Narcissus Marsh, Church of Ireland Archbishop of Dublin, and formerly Provost of Trinity College, Dublin. It was long claimed that the Library opened in 1701, but this is untrue. The Cathedral agreed in 1701 to provide a plot of land for a library, but building work only commenced in 1703. The First Gallery and the Old Reading Room seem to have been completed by 1705. The library was formally established by an Act of the Irish Parliament in 1707 (6 Anne c. 19), and the Second Gallery was added in 1708 or 1709.
The design was by the then Surveyor General of Ireland, Sir William Robinson, also the architect of the Royal Hospital Kilmainham. Marsh donated his own library, which included the former library of Bishop Edward Stillingfleet, of over 10,000 volumes, regarded as one of the finest in England, which he had bought for 2,500 pounds.
Dr. Elias Bouhereau, a Huguenot refugee from La Rochelle who fled from France after the Revocation of the edict of Nantes, was the first librarian or Keeper, and also donated his personal library.
The Library was formally incorporated in 1707 by Parliament, which vested the house and books in a body known as the Governors and Guardians of the Library, comprising religious and state dignitaries and officials, and their successors still oversee it. Narcissus Marsh died in 1713, and is buried just beyond the library, in the grounds of the cathedral.
In 1745, John Stearne, Bishop of Clogher, bequeathed half of his collection of books to the library. The other half of Stearne's book collection (and all of his manuscripts) were given to Trinity College, Dublin.
Recent history
In 1989, Muriel McCarthy became the first female Keeper, holding the position until her retirement in 2011. The current Keeper (now re-titled as Director) is Dr Jason McElligott, who was educated at University College Dublin and St John's College, Cambridge.
When the Guinness family sold Farmleigh House to the State, the Benjamin Iveagh Library was donated to Marsh's Library, although its documents remain housed at Farmleigh.
The catalogue of the library is online at www.marshlibrary.ie/catalogue. Scholars and students can read the books and manuscripts by appointment. Tourists come from across the world to see the old library, and in 2013 there were a total of 16,000 tourist visitors. The figure for 2014 was 17,000 visitors. 2015 saw a rise in visitors to 23,000 people.
Holdings
The library contains over 25,000 books from the 16th, 17th and 18th centuries, in addition to around 300 manuscripts, and around 80 books (incunabula) from before 1501. Subjects covered include medicine, law, science, travel, navigation, mathematics, music, surveying and classical literature, and especially theology.
In addition to the eighty books dated before 1501, the Marsh collection contains four hundred and thirty books from Italy dated before 1600, twelve hundred English works produced before 1640, as well as another five thousand English books printed before 1700. Marsh's collection also includes Irish manuscripts purchased from Dudley Loftus in 1695. The Loftus acquisition included many works of Irish History, an Irish – Latin Dictionary from 1662, royal grants in Ireland from 1604 to 1631, and letters of Thady O'Doyne from 1159, 1590, and 1606. In 1941, Dean Webster donated a collection of Irish manuscripts and deeds and relating to County Cork as well as the Bishop William Reeves' collection of books and manuscripts.
The Marsh collection includes works in oriental languages, and in Hebrew, Arabic, Turkish and Russian, as well as an important collection of Latin Judaica. The Marsh collection of Jewish texts which were primarily his own personal books contain over two hundred and fifty works of Hebrew Bibles, writings from the Talmud, rabbinic works, and Yiddish books. Many of Marsh's personal collection of Hebrew works as well as oriental texts were purchased from the estate of the Edward Stillingfleet (1635-1699), a Protestant clergyman and lecturer. The Marsh Middle Eastern personal collection is robust in Near Eastern languages, biblical studies, philosophy, astronomy, and math.
The Bouhéreau collection relates especially to France, and French religious controversies, and also medicine. The collection of Elie Bouhereau, the first librarian of the Marsh Library, donated about 1703, is quite diverse. Covering multiple subjects many of his works were gifts from authors and religious leaders such as William Molyneux, Bishop Wettenhall, and Archbishop William King. The Bouhereau medical texts were the most updated works of the day and the French History works included the Massacre of St. Bartholomew's Day and the Siege of La Rochelle.
Among the manuscripts is a volume of the Lives of the Irish Saints in Latin from ca. 1400, as well as 16th century madrigals and other musical pieces, and manuscripts on theological, legal and medical matters.
Interior design
The library still features its original fittings, including seating and shelving. The bookcases are made of quarter-plained Baltic oak with carved and lettered gables. In some of the bookcases there are bullet holes from the Easter Rising when Jacob's Biscuit factory next door was occupied. There are three wire alcoves, known as 'cages', which came into use in the 1770s in response to thefts in the library.
Today
The library is one of the last 18th-century buildings in Ireland still used for its original purpose. It is open to visitors for a fee of €5, or €3 for students and senior citizens. Please see https://www.marshlibrary.ie/visit for current opening hours.
Researchers are admitted free of charge, but must apply in advance to reserve a place in the Reading Room. The reading room is expected to be closed until October 1, 2020.
The Library holds exhibitions, and occasional conferences, and has published a range of material, primarily related to exhibitions and the catalogue.
As a charitable institution the library accepts donations, which are recorded in a special ledger which dates back to 1707.
The Library has an active social media presence, and every day posts at least one image from the collections on Instagram, Facebook and Twitter. These images give a sense of the breadth and depth of the Library holdings.
See also
Bolton Library
References and notes
External links
Digitized manuscripts from Marsh's Library at ISOS (Irish Script on Screen)
Library buildings completed in the 18th century
Archives in the Republic of Ireland
Libraries in Dublin (city)
Buildings and structures completed in 1707
Cultural infrastructure completed in 1707
Museums in Dublin (city)
1700s establishments in Ireland
1707 establishments in the British Empire
Libraries established in 1707
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,000
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Cody Martin may refer to:
Zack and Cody Martin, fictional characters from the TV series The Suite Life of Zack & Cody
Cody Martin (baseball) (born 1989), American baseball pitcher
Cody Martin (basketball) (born 1995), American basketball player
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,832
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package skadi.beans
/**
* Use this class to define a single bean in the container. The container will
* create an instance of this bean, inject it with declared dependencies and in
* turn will inject it to other beans that depend on it. An example bean:
*
* <pre>
* new Bean named 'userService
* implementedWith classOf[com.sample.app.UserServiceImpl]
* constructorArgs 'userDao
* inject 100 -> 'maxPosts
* initializeWith('init, 5)
* </pre>
*
*/
class Bean {
/**
* Define the name for this bean using this method. The name must be unique
* for each bean as it used to resolve dependencies between beans. If a bean
* name is not defined it cannot be injected to other beans as a dependency.
*
* @param name
* unique name for this bean
* @return the instance of this bean
*/
def named(name: Symbol): Bean = {
this.name = name
beanNamed = true
this
}
/**
* The implementing class of this bean is declared using this method. This
* method MUST be invoked when defining a bean.
*
* @param clazz
* this bean's defining class
* @return the instance of this bean
*/
def implementedWith(clazz: Class[_]): Bean = {
this.clazz = clazz
this
}
// def implementedWith[T](implicit m: Manifest[T]): Bean = {
// this.clazz = m.erasure
// this
// }
/**
* Any parameters that are required by the constructor that will be used to
* create an instance of the bean should be defined here, in the same order
* as they are expected in the actual constructor. Note that the parameters
* can value literals, property placeholders or references to other beans.
* If this method is not invoked, the container will use the default
* (parameterless) constructor to create an instance of this bean.
*
* @param args
* arguments that are used when invoking the constructor of this
* bean
* @return the instance of this bean
*/
def constructorArgs(args: Any*): Bean = {
this.args = args.toList
this
}
/**
* Use this method to inject dependencies through setter methods. First Scala
* setters are taken into the account when scanning the class for viable methods.
* If none are found, Java setters are looked up.
*
* An example:
*
* <pre>
* class User {
*
* var username: String
* private var password: String
*
* def setPassword(password: String) {
* this.password = password
* }
* }
*
* new Bean named 'user
* implementedWith classOf[User]
* inject("admin" -> 'username,
* "adminadmin" -> 'password)
* </pre>
*
* Value literals, property placeholders and other dependencies are all viable
* injectables and will be resolved by the container when creating an
* instance of this bean.
*
* @param pairs
* pairs of setter dependency definitions, first item is the value
* that will be injected while the second item is the actual
* property of the class that will be injected through a setter
* method
* @return the instance of this bean
*/
def inject(pairs: (Any, Symbol)*): Bean = {
injectables = pairs.toList ::: injectables
this
}
/**
* Invoke this with a method that you wish this bean to be initalized
* with. Method with the given name will be invoked on the bean after an
* instance has been created and all necessary dependencies have been
* injected.
*
* @param initMethod
* name of the method this bean should be initialized with
* @param args
* optional arguments that should be supplied to the initialization
* method. They cannot be references to other beans, only literal
* values.
* @return the instance of this bean
*/
def initializeWith(initMethod: Symbol, args: Any*): Bean = {
this.initMethod = initMethod
initArgs = args.toList
this
}
/**
* Sets the scope of this bean as <tt>singleton</tt>, meaning that only one
* instance of this bean will be created by the container and will be shared
* among other beans that depend on it.
* Scope singleton is default scope and invoking this method is purely
* optional, use it only if you want to emphasize the scope of the declared
* bean to other readers of your code.
*
* @return the instance of this bean
*/
def scopedAsSingleton: Bean = {
this.scope = Scope.Singleton
this
}
/**
* Sets the scope of this bean as <tt>prototype</tt>, meaning that a new
* instance of this bean is created whenever this bean should injected as a
* dependency or an instance has been explicitly requested from the container.
* Also, <tt>prototype</tt> scope implies that this bean will not be eagerly
* instantiated.
*
* @return the instance of this bean
*/
def scopedAsPrototype: Bean = {
this.scope = Scope.Prototype
this
}
/**
* If this method is invoked on a bean, it will not be eagerly instantiated
* during the container's loading phase. It will only be created when it is
* required by another bean or it has been requested from the container.
* Use this only on the beans that are expensive to create resource-wise,
* otherwise eager instantiation allows you to catch any possible errors in
* initalization earlier.
* Note that this method has no effect if invoked on a bean that has bean
* scoped as prototype.
*
* @return the instance of this bean
*/
def loadLazily: Bean = {
this.lazyBean = true
this
}
override def toString(): String = {
val builder = new StringBuilder
builder.append("Bean [")
val beanName = if (beanNamed) "name = " + name.name
else "Unnamed bean"
builder.append(beanName)
builder.append(", ")
if (clazz != null) builder.append("class = " + clazz + ", ")
builder.append("scope = " + scope)
builder.append(", ")
builder.append("lazy = " + lazyBean)
builder.append("]")
builder.toString
}
/**
* Unique name of the bean, used to identify the bean within the application
* context. Initialized with an UUID, in case the bean is not named explicitly
* by the user.
*/
private[skadi] var name = Symbol(java.util.UUID.randomUUID.toString)
/**
* Actual class of this bean.
*/
private[skadi] var clazz: Class[_] = null
/**
* Constructor arguments that will be used to create an instance of this bean
* when necessary.
*/
private[skadi] var args: List[Any] = Nil
/**
* Values that will be injected after creation of the instance via setter
* methods.
*/
private[skadi] var injectables: List[(Any, Symbol)] = Nil
/**
* Initialization method that will be invoked on the instance of this bean
* after it has been constructed and injected with setter dependencies.
*/
private[skadi] var initMethod: Symbol = null
/**
* Optional arguments that go in the initialization method.
*/
private[skadi] var initArgs: List[Any] = Nil
/**
* Scope of this bean, defaults to <tt>Singleton</tt>.
*/
private[skadi] var scope = Scope.Singleton
/**
* Is the bean loaded lazily, defaults to false.
*/
private[skadi] var lazyBean = false
/**
* Determines if the bean was named explicitly by the user.
*/
private[skadi] var beanNamed = false
/**
* Constructor that is used to instantiate the target class.
*/
private[skadi] var constructor: java.lang.reflect.Constructor[_] = null
/**
* The instance of the target class, created using the given constructor
* arguments and injected with the setter arguments.
*/
private[skadi] var instance: Any = null
/**
* Determines if the target class is abstract or not.
*/
private[skadi] var abstractClass = false
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,505
|
Алгоритм Коменц-Вальтер () — запропонований Беатою Коменц-Вальтер алгоритм пошуку рядка. Подібно до алгоритму Ахо-Корасік може шукати водночас декілька підрядків у рядку.
Оснований на алгоритмі Бояра-Мура. Алгоритм Коменц-Вальтер важливий зокрема тим, що був реалізований у другій версії Юнікс-утиліти grep.
Оцінки практичної швидкодії алгоритму різняться: за одними оцінками, його швидкодія не перевищує швидкодію алгоритму Ахо-Корасік. За іншими оцінками, його швидкодія в багатьох випадках значно перевищує швидкодію алгоритму Ахо-Корасік.
Якщо замість алгоритму Бояра-Мура взяти за основу алгоритм Бояра-Мура-Горспула, то алгоритм Коменц-Вальтер можна дещо спростити, однак його швидкодія для рядка довжиною n та найдовшого ключа L, в найгіршому випадку залишатиметься на рівні .
Примітки
Література
.
Посилання
Beate Commentz-Walter A String Matching Algorithm Fast on Average. Extended Abstract — оригінальна публікація алгоритму.
Рядкові алгоритми
Алгоритми пошуку
Алгоритми пошуку рядків
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,187
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Come August this year, Nationstar Mortgage Holdings will become "Mr. Cooper," according to a news release.
The change comes after a company-wide focus, starting in early 2016 on fostering cultural change and making the homeownership experience better for its customers.
"Mr. Cooper" was chosen as the new brand name for its more personal feel and to enable its customers to feel more at home with their loan providers.
The company's almost three million customers will be notified of the name change later this month and they can visit the official site for more information.
Nationstar has faced bad press recently, with a high volume of consumer complaints – it has regularly been among the most-complained-about companies in the Consumer Financial Protection Bureau's consumer complaints reports – and regulatory penalties for alleged compliance failures.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,704
|
\section{Introduction}
This report describes and analyses sojourn time distributions which can then be applied to parametrise the semi-Markov model (SMM). Soujourn time distributions are the distributions that govern how long a semi-Markov model remains in each state. Unlike usual Markov-models where the sojourn time distribution is geometric, semi-Markov models allow any distribution to be used.
Such sojourn time distributions have been modelled before by using known distributions such as the gamma, Poisson, multinomial, more general exponential family members and Coxian phase-type distributions particularly in the context of hidden semi-Markov models (HSMMs) \cite{Mitchell1993,Yu2010,Sansom2001,Bulla2006,Guedon2003,Duong2008,BullaThesis}. An R-package is available for modelling some of these distributions \cite{Bulla2013}. Non-parametric versions have also been studied in the context of expectation maximisation for HSMMs \cite{Bulla2006,Yu2010}, however the underlying properties of these distributions have not been studied in the literature and we seek to do this \Cref{sec:genGeometric}. Further, the two subfamilies of the distributions we study have not previously appeared in the literature.
In general, the sojourn time distributions we study can be considered discrete generalisations of the geometric distribution in finite or infinite time. There are many approaches for generalisations of the geometric distribution \cite{Rattihalli2021,Tripathi2016,Gomez2010}, although they usually only consider infinite time. For example, considering the exponential as the continuous analog of the geometric distribution and the gamma distribution as generalising the exponential distribution, then discretisations of the gamma distribution can be considered generalisations of the geometric distribution as in \cite{Chakraborty2012}.
Unlike other geometric generalisations, ours will involve the parameters $\rho$. These parameters can be considered as the marginal probabilities that a semi-Markov chain stays in the same state for the next time step, as in \Cref{sec: SemiMarkovRelationship}. For a Markov chain, the geometric sojourn time distribution corresponds to constant $\rho$ for each state, and here we extend this to consider linear and simple polynomial factor models for $\rho$. We are not aware of any other research on generalisations of the geometric distribution that has such an approach.
Markov models are often applied to model behaviours where only small amounts of data may be available, so we require parsimonious models which nevertheless have sufficient generality to capture different kinds of qualitative sojourn time behaviours. We also require models for which statistically efficient estimation methods can be defined. At this stage, we make the obvious point that whilst maximum likelihood estimators (MLEs) are statistically unbiased and efficient for large sample sizes \cite{DeGroot}; this may not be the case for small sample sizes such in our experimental data. So the study of the performance of MLEs forms an important part of this study which we undertake in \Cref{sec:MLEsandCRLBs}.
Statistical inference methods for hidden SMMs (HSMMs) are typically more computationally demanding than the equivalents for HMMs, so it is also of interest to determine whether an observed process has Markovian state transitions or the more general semi-Markov ones. We show in \Cref{sec:experidata} how we approached this for our experimental data, which we can use as a methodology for testing this kind of hypothesis in general. We also see a strong fit between this experimental data and the simple polynomial factor model, suggesting the theory performs well in experimental settings.
Overall, the contents of this report gives a novel approach to modelling sojourn time distributions that will be of use for (hidden) semi-Markov models, and already holds appeal for modelling experimental data. The results concerning the sojourn time distributions may be of independent interest for the discrete distribution literature.
\subsection{Layout of the report}
The contents of this report is as follows. In \Cref{subsec:defnOfGeneralisation} we define the family of distributions we aim to study, which generalises the geometric distribution. We give general properties including the mean, variance, moment and probability generating functions as well as results that determine when the moments exist in general. These results are new. In \Cref{subsec:linearex} and \Cref{subsec:simplePolyexample} we define the subfamilies of interest, the linear factor model and simple polynomial factor models, and describe their distributions, which is also new research. In \Cref{sec: SemiMarkovRelationship} we show the relationship to semi-Markov models and discuss further results in this area.
In \Cref{sec:MLEsandCRLBs} we study how to determine the MLEs of the subfamilies of interest, studying the linear factor model in \Cref{subsec:linearMLEs} and the polynomial factor models in \Cref{subsec:polyMLE}. In each case, we determine numerically the MLEs from various samples and show how the variance of the MLEs closely follows the inverse Fisher information which is related to the Cram\'{e}r-Rao lower bound \cite{DeGroot} suggesting our numerical methods are behaving efficiently. We study how this changes with different sample sizes. We also discuss methods to determine the size of the support of the distribution, i.e. the maximum sojourn time.
In \Cref{sec:experidata} we analyse our experimental data. We show how the data has sojourn times which do not appear to follow a geometric distribution, and why a simple polynomial factor model is a better fit. We then find the MLEs for such a model for each of the three tasks, including determining the maximum sojourn time, and discuss how well these models fit. We are satisfied with the model for tasks 1 and 3, and suggest some improvements for task 2. In \Cref{sec:conclusion} we conclude and discuss future work.
In \Cref{appendix} the second author expands on the work for the linear factor model. He confirms the results in \Cref{subsec:linearMLEs} with separate simulations, as well as shows how the expected log-likelihood explains some of the behaviour of the MLEs and how to further understand estimation when the maximum sojourn time is unknown. In \Cref{appn:ExperimentalData} we include the experimental data analysed in \Cref{sec:experidata}.
\section{Generalising the geometric distribution}\label{sec:genGeometric}
Here we define a family of discrete distributions that generalise the geometric distribution in both finite and infinite state cases. We give properties of this family, including formulae for the moments, and prove results relating to their existence in the infinite support case. We note that this family is particularly general as it can be used to characterise all discrete distributions that are supported on $1,2,3,\ldots, T$ with $T$ a positive integer and possibly infinite.
We then focus on two specific subfamilies, the linear factor model in \Cref{subsec:linearex} and some simple polynomial factor models in \Cref{subsec:simplePolyexample}. Finally we detail the relationship to semi-Markov models in \Cref{sec: SemiMarkovRelationship} and the relationship to matrix analytic methods.
As far as we are aware, all results in sections \Cref{subsec:defnOfGeneralisation}, \Cref{subsec:linearex} and \Cref{subsec:simplePolyexample} are new contributions to the literature on discrete distributions, which we expect will also lead to new contributions in Markov processes research.
\subsection{Definition and properties} \label{subsec:defnOfGeneralisation}
We define the family of discrete distributions that we intend to study then study properties of the distributions.
\begin{dfn}\label{defn:discreteRho}
Take any $\rho(1),\rho(2),\ldots, \rho(T-1)\in (0,1]$ with $\rho(1)>0$, and set $\rho(T)=0$, where $T$ is a positive integer (which we may allow to go to $\infty$). Then we can define a discrete distribution with support contained in $\{1,2,\ldots T\}$ having probability mass function (PMF)
\begin{align}
\label{eqn:PMF}
f(k)=(1-\rho(k))\rho(k-1)\rho(k-2)\ldots \rho(1) \\\nonumber
= (1-\rho(k))\prod_{t=1}^{k-1}\rho(t) \quad \text{for }k=1,2,\ldots, T,
\end{align}
where we use the convention that product $\prod_{t=1}^{0}\rho(t)$ is equal to 1.
\end{dfn}
Note that we could allow some $\rho(k)=0$ for $1 \le k<T$, but then $f(k) =1$ and then $f(k+i) = 0$ for all $i=1,2,\ldots,T$, reducing the support of $f$. Also, whenever $\rho(k)=1$ we have $f(k) = 0$, so as we see below, this is sufficiently general to capture all discrete distributions with support contained in $\{1,2,\ldots \infty\}$. We can also shift the distribution of $f$ to distribution $f_t$ with support contained within $\{1+t,2+t,\ldots, T+t\}$ for any integer $t$ by setting $f_t(t+k) = f(k)$. We will consider this in \Cref{sec:experidata}.
It is clear that each $f(k)\in [0,1]$ as products of numbers in $[0,1]$ remains in $[0,1]$. We have
\begin{align*}
\sum_{k=1}^Tf(k) &= \sum_{k=1}^T(1-\rho(k))\rho(k-1)\rho(k-2)\ldots \rho(1)\\
&= 1 + \sum_{k=2}^T\prod_{t=1}^{k-1} \rho(t) - \sum_{k=1}^T\prod_{t=1}^{k} \rho(k)\\
&= 1 + \sum_{k=1}^{T-1}\prod_{t=1}^{k} \rho(t) - \sum_{k=1}^T\prod_{t=1}^{k} \rho(k)\\
&= 1-\prod_{t=1}^{T} \rho(k) = 1,
\end{align*}
as $\rho(T)=0$. So $f$ gives a valid PMF.
If we let $\rho(k)=p$ for some fixed $p\in (0,1)$ and take $T\to \infty$ then this is gives the geometric distribution with PMF for $k=1,2,\ldots$ \begin{equation} f(k) = p^{k-1}(1-p).\label{eqn:PMFgeometric} \end{equation}
In general, we can interpret the $\rho(k)$ as the probability of an event occurring at time $k$ given it also occurred at time $k-1,k-2,\ldots,1$. Then $f(k)$ is the probability of the event not occurring at time $k$ given it occurred at time $1,2,\ldots,k-1$. In the geometric case, the time steps are independent and $\rho(k)$ does not change, however the general $f$ described above allows for dependence between each time. We expand on this in \Cref{sec: SemiMarkovRelationship} where we show how this relates to a semi-Markov model.
\begin{lem}\label{lem:relationship}
We have the relationship
\begin{equation} \label{eqn:relationship}
\sum_{t=k}^{T}f(t)=1-\sum_{t=1}^{k-1}f(t) = \frac{f(k)}{1- \rho(k)} = \prod_{t=1}^{k-1}\rho(t).
\end{equation}
\end{lem}
\begin{proof}
By induction, we can show
\begin{equation} \label{eqn:rhoInTermsOff}
\rho(k) = \frac{1- f(1) - f(2) - \ldots - f(k)}{1- f(1) - f(2) - \ldots - f(k-1)}=1-\frac{f(k)}{\sum_{t=k}^{T}f(t)}=1-\frac{f(k)}{1-\sum_{t=1}^{k-1}f(t)},\end{equation}
and the result follows.
\end{proof}
Note that \Cref{eqn:rhoInTermsOff} shows that any discrete PMF supported $k=1,2,\ldots,T$ (with $T$ possibly infinite) can be defined in terms of $\rho(k)$, so in fact \Cref{defn:discreteRho} is an alternative way to describe all such distributions. If $F$ is the corresponding cumulative distribution function to $f$, then
\[ \rho(k) =\frac{1 - F(k)}{1-F(k-1)}\] and also
\begin{align} F(k) &=\sum_{t=1}^kf(k) = \sum_{t=1}^k(1-\rho(t))\prod_{s=1}^{t-1} \rho(s) \nonumber\\
&= 1+ \sum_{t=1}^{k-1}\prod_{s=1}^{t-1}\rho(s) - \sum_{t=1}^k\prod_{s=1}^{t-1}\rho(s)\nonumber\\
&= 1-\prod_{t=1}^k\rho(t).
\label{eqn:FinTermsOfrho}\end{align}
Note that if $F(k) = 1$ then $\rho(k) = 0$ and this means the distribution has reached the end of its support. We use this in \Cref{sec:experidata}. The next proposition characterises all moments of the distribution in terms of the $\rho(k)$.
\begin{prop} \label{prop:MGFPGF}
For a random variable $X$ with PMF $f$ as in \Cref{eqn:PMF} we have moment generating function (MGF)
\[M_X(t) = \sum_{k=1}^{T-1} (e^{t(k+1)}-e^{tk})\prod_{t=1}^k \rho(k)+e^t.\]
The $n$-th derivatives are
\[M_X^{(n)}(t) = \sum_{k=1}^{T-1} ((k+1)^{n}e^{t(k+1)}-k^{n}e^{tk})\prod_{t=1}^k\rho(t) +e^t.\]
So we have
\[\E(X^n) =M_X^{(n)}(0) = \sum_{k=1}^{T-1} ((k+1)^{n}-k ^{n})\prod_{t=1}^k\rho(t) +1,\] and
then
\begin{align*} \E(X) &= \sum_{k=1}^{T-1} \prod_{t=1}^k\rho(t) +1,\\
\Var(X) &= \E(X) +2 \sum_{k=1}^{T-1}k \prod_{t=1}^k\rho(t)- \E(X)^2.
\end{align*}
The probability generating function is
\[P_X(z) = \sum_{k=1}^{T-1} (z^{k+1}-z^k)\prod_{t=1}^k\rho(t) + z\]
with $n$-th derivatives
\[P_X^{(n)}(z)= \sum_{k=n}^{T-1} \left(\frac{(k+1)!z^{k+1-n}}{(k+1-n)!}-\frac{k!z^{k-n}}{(k-n)!}\right)\prod_{t=1}^k\rho(t) + \frac{dz}{dz^n}.\]
The factorial moments are
\begin{align*}\E(X(X-1)\cdots (X-n+1))& = P_X^{(n)}(1)= \sum_{k=n}^{T-1} \frac{k!n}{(k+1-n)!}\prod_{t=1}^k\rho(t) + \frac{dz}{dz^n}.\end{align*}
\end{prop}
\begin{proof}
The moment generating function is
\begin{align*}
M_X(t) &= \sum_{k=1}^{T}e^{kt}f(k)\\
&= \sum_{k=1}^{T}e^{kt}(1-\rho(k))\prod_{t=1}^{k-1}\rho(t)\\
&=\sum_{k=1}^{T}e^{kt}\prod_{t=1}^{k-1}\rho(t)-\sum_{k=1}^{T}e^{kt}\prod_{t=1}^{k}\rho(t)\\
&= e^t+ \sum_{k=2}^{T}e^{kt}\prod_{t=1}^{k-1}\rho(t)-\sum_{k=1}^{T}e^{kt}\prod_{t=1}^{k}\rho(t)\\
&= e^t+ \sum_{k=1}^{T-1}e^{(k+1)t}\prod_{t=1}^{k}\rho(t)-\sum_{k=1}^{T-1}e^{kt}\prod_{t=1}^{k}\rho(t)\\ &=\sum_{k=1}^{T-1} (e^{t(k+1)}-e^{tk})\prod_{t=1}^k \rho(k)+e^t
\end{align*}
The formulas for the derivatives $M_X^{(n)}(t)$ and $E(X)$ follow directly. We then have the variance
\begin{align*}\Var(X) &= \E(X^2)-\E(X)^2 = \sum_{k=1}^{T-1} (2k +1)\prod_{t=1}^k\rho(k) +1 - \E(X)^2 \\
&= \E(X) +2 \sum_{k=1}^{T-1}k \prod_{t=1}^k\rho(k)- \E(X)^2.\end{align*}
The probability generating function is
\[P_X(t) = \sum_{k=1}^{T}z^kf(k) = M_X(\log(z))= \sum_{k=1}^{T-1} (z^{k+1}-z^{k})\prod_{t=1}^k \rho(k)+z.\]
The $n$-th derivatives $P_X^{(n)}(z)$ are straightforward to calculate, so the factorial moments are
\begin{align*}\E(X(X-1)\cdots (X-n+1))& = P_X^{(n)}(1)= \sum_{k=n}^{T-1} \left(\frac{(k+1)!}{(k+1-n)!}-\frac{k!}{(k-n)!}\right)\prod_{t=1}^k\rho(t) + \frac{dz}{dz^n} \\
&= \sum_{k=n}^{T-1} \frac{k!n}{(k+1-n)!}\prod_{t=1}^k\rho(t) + \frac{dz}{dz^n}.\end{align*}
\end{proof}
If $T$ is finite, it is clear that all moments are finite. Below we consider the case when $T$ is infinite.
\begin{prop} \label{prop:existenceOfMoments}
Let $X$ be a random variable with PMF $f$ as in \Cref{eqn:PMF} with $T= \infty$, and say \[
\lim_{k\to \infty} \rho(k) = r\] where we must have $r\in [0,1]$ as $\rho(k)\in[0,1]$. Then all moments exist whenever $r<1$. If $r = 1$ then some moments may exist.
\end{prop}
\begin{proof}
The ratio test says that a series $\sum_{i=1}^\infty a_i$ for $a_i\in \mathbb{R}$ converges whenever
\[ \lim_{k\to \infty} \left | \frac{a_{k+1}}{a_k}\right | <1.\] It diverges if the limit is $>1$ and may or may not converge if this limit is $1$. Say we have $\lim_{k\to \infty} \rho(k) =r$, then we can consider the series
\[\sum_{k=1}^\infty k^n \prod_{t=1}^k\rho(t)\] for some $n=1,2,\ldots, \infty$, which appears in the formula for the moments $E(X^n)$ in \Cref{prop:MGFPGF}. Then
\begin{align*}
\lim_{k\to \infty} \left | \frac{a_{k+1}}{a_k}\right |&= \lim_{k\to \infty} \left | \frac{(k+1)^n \prod_{t=1}^{k+1}\rho(t)}{k^n \prod_{t=1}^k\rho(t)}\right |\\
&= \lim_{k\to \infty} (1+\frac{1}{k^n}) \rho(k+1) \\
& \to 1\cdot r
\end{align*}
therefore all moments centred at $0$ converge when $r<1$ by the ratio test. As all moments of order $n$ can be written in terms of moments centred at $0$ of up to order $n$, then all moments converge for $X$ when $r<1$. Similarly, they diverge if $r>1$ (which is not possible as we have $\rho(k)\in[0,1]$ for all $k$) and may or may not converge if $r=1$.
\end{proof}
Note that we do not necessarily have that $\lim_{k\to \infty} \rho(k)$ converges, so a more general statement would be than any subsequence of the $\rho(k)$ that converges must converge to something less than $1$ to guarantee existence of the moments. This is equivalent to the following stronger corollary of \Cref{prop:existenceOfMoments}.
\begin{cor}
Let $X$ be a random variable with PMF $f$ as in \Cref{eqn:PMF} with $T= \infty$, and say \[
\limsup_{k\to \infty} \rho(k) = r\] where we must have $r\in [0,1]$ as $\rho(k)\in[0,1]$. Then all moments exist whenever $r<1$. If $r = 1$ then some moments may exist.
\end{cor}
If $\limsup_{k\to \infty} \rho(k) = 1$, that is, there are (sub)sequences of the $\rho(k)$ which converge to $1$, more complicated arguments must be considered as in the following example.
\begin{ex}
If $\rho(k) = \frac{k}{k+1}$ for $k =1,2,\ldots, \infty$ then $\lim_{k\to \infty} \rho(k) =1$ and we have \[\sum_{k=1}^\infty\prod_{t=1}^{k}\rho(t) = \sum_{k=1}^\infty \frac{1}{k+1},\] which diverges and no moments exist. These $\rho$ correspond to PMF \[f(k)=\frac{1}{k}-\frac{1}{1-k} = \frac{1}{k(k+1)}.\]
More generally, if $\rho(k) = \frac{k^m}{(k+1)^m}$ for some positive integer $m$ then still $\lim_{k\to \infty} \rho(k) =1$, however we have
\[\sum_{k=1}^\infty k^n\prod_{t=1}^{k}\rho(t) = \sum_{k=1}^\infty \frac{k^n}{(k+1)^m}.\] When $n+1<m$, we have
\begin{align*}
\sum_{k=1}^\infty \frac{k^n}{(k+1)^m} &= \sum_{k=1}^\infty \frac{1}{(1+\frac{1}{k})^n}\frac{1}{(1+k)^2}\frac{1}{(1+k)^{m-2}}\\
&\le \sum_{k=1}^\infty \frac{1}{(1+k)^2} = \frac{\pi^2}{6} -1
\end{align*}
so this sum converges and this means moments up to order $m-2$ exist. When $n = m-1$ we have
\begin{align*}
\sum_{k=1}^\infty \frac{k^n}{(k+1)^m} &= \sum_{k=1}^\infty \frac{1}{(1+\frac{1}{k})^{m-1}}\frac{1}{1+k}\\
&\ge \frac{1}{2^{m-1}} \sum_{k=1}^\infty \frac{1}{1+k} \\
& \to \infty
\end{align*}
so this diverges, and all moments of order $m-1$ or higher do not exist. These $\rho$ correspond to PMF \[f(k) = \frac{1}{k^m} - \frac{1}{(1+k)^m}.\]
\end{ex}
\begin{ex}
For a random variable $X$ with the geometric distribution $\rho(k)=p$, $T\rightarrow \infty$ as in \Cref{eqn:PMFgeometric} then using \Cref{prop:MGFPGF} we have mean
\[ \E(X) = 1+\sum_{j=1}^{\infty}p^j = 1+\frac{1}{1-p}-1=\frac{1}{1-p},\]
and variance
\begin{align*}
\Var(X) & = \E(X) +2\lim_{T\to \infty}\sum_{k=1}^{T-1} k\prod_{t=1}^{k}\rho(t)- \E(X)^2\\
&=\frac{1}{1-p} + 2\lim_{T\to \infty}\sum_{k=1}^{T-1}kp^k - \frac{1}{(1-p)^2}\\
& = \frac{1}{1-p} + 2\lim_{T\to \infty}p\frac{d}{dp}\sum_{k=1}^{T-1}p^k - \frac{1}{(1-p)^2}\\
& = \frac{1}{1-p} + 2\lim_{T\to \infty}p\frac{d}{dp}\left(\frac{1-p^T}{1-p}-1\right) - \frac{1}{(1-p)^2}\\
& = \frac{1}{1-p} + 2p\frac{d}{dp}\left(\frac{1}{1-p}-1\right) - \frac{1}{(1-p)^2}\\
& = \frac{1}{1-p} + \frac{2p}{(1-p)^2} - \frac{1}{(1-p)^2}\\
& = \frac{1-p+2p-1}{(1-p)^2}\\
&=\frac{p}{(1-p)^2}.
\end{align*}
Both of these results are as expected from the geometric distribution.
\end{ex}
\subsection{Linear factor model} \label{subsec:linearex}
In this section we consider the case where we have a linear $\rho$, that is
\[ \rho(k)=ak+b \quad \text{for all } k=1,2, \ldots, T.\] For this to be valid, we require that $ak+b \in [0,1]$ for all $k$. This then means if we send $T\to \infty$ we must make $a=0$, resulting in the geometric distribution. If we consider $T$ finite then we require the endpoints $k=1,T-1$ are in $[0,1]$, which means we need $a+ b\in [0,1]$, so $a\in [-b,1-b]$ and $a(T-1)+b \in [0,1]$, so $a\in [-\frac{b}{T-1},\frac{1-b}{T-1}]$. Combining these we have
\[ \max\{-b,\frac{-b}{T-1}\} \le a \le \min\{1-b,\frac{1-b}{T-1}\}.\] To make sure the end points are non-overlapping, we then also require that $b\in [-\frac{1}{T},\frac{T-1}{T-2}]$ for $T\ge 3$.
A natural choice in this case would be to set $a,b$ such that $a(T)+b = 0$, which gives $a=-b/T$ and $b\in (0,\frac{T}{T-1}]$. Then \[ \rho(k) = b\left(1-\frac{k}{T}\right)\] and the corresponding $f$ is as follows
\begin{align*}
f(k)&=(1-\rho(k))\rho(k-1)\rho(k-2)\ldots \rho(1) \\
&= (1+b(\frac{k}{T}-1))(b(1-\frac{k-1}{T}))(b(1-\frac{k-2}{T}))\cdots(b(1-\frac{1}{T}))\\
&=(1+b(\frac{k}{T}-1))\left(\frac{b}{T}\right)^{k-1}(T-k+1)(T-k+2)\cdots (T-1)\\
&= (1+b(\frac{k}{T}-1))\left(\frac{b}{T}\right)^{k-1}\frac{(T-1)!}{(T -k)!}.
\end{align*}
This is a natural extension of the geometric distribution over a finite domain.
Note that we can similarly use $a$ instead of $b$, where we have \[ \rho(k) = a\left(k-T\right),\] with $b=-aT$, $a\in [\frac{1}{1-T},0)$.
We can graph $f$ for certain values of $b$ (and corresponding $a$) and $T=10$, as in \Cref{fig:LinearGeomExtension}.
\begin{figure}[!hpt]
\centering
\includegraphics[width=13cm]{PlotOfFRhoLinear.jpg}
\caption{Plot of $f$ with $\rho(k)=b(1-\frac{k}{T})=a(k-T)$ with $T=10$ and various $b$.}
\label{fig:LinearGeomExtension}
\end{figure}
We see quite different behaviours depending on the value of $b$. The larger $b$ exhibit a non-monotonic, uni-modal distribution with maximum near the centre of the support, while the smaller $b$ exhibit more geometric-like qualities, with maximum at $k=1$ and decreasing for larger $k$. Varying $T$ gives similar results.
\subsection{Simple polynomial factor models} \label{subsec:simplePolyexample}
Here we concentrate on models of the form \[\rho(k) = a(k-c)^n+b\] with $\rho(T)=0$. We require $n$ to be a fixed positive integer.
We again set $\rho(T)=0$ which gives $a(T-c)^n+b=0$, so we must have $a=\frac{-b}{(T-c)^n}$ provided we assume $b\ne 0$. Note that if $b=0$ and $\rho(T) = 0$ then either $c=T$ and we are considering linear cases $\rho(T) = a(k-T)$ from the previous section, or $a =0$ and we have $\rho = 0$.
Given this we assume $c\ne T$ then writing in terms of $b$ we have
\begin{align*}
\rho(k)& = b\left(1-\frac{(k-c)^n}{(T-c)^n}\right),\\
f(k) &= \left(1+b\left(\frac{(k-c)^n}{(T-c)^n}-1\right)\right)\left(\frac{b}{(T-c)^n}\right)^{k-1}\prod_{t=1}^{k-1}((T-c)^n-(t-c)^n).
\end{align*}
Similarly, writing in terms of $a$ we have
\begin{align*}
\rho(k) &= a\left((k-c)^n - (T-c)^n\right),\\
f(k) &= \left(1+a\left((T-c)^n-(k-c)^n\right)\right)a^{k-1}\prod_{t=1}^{k-1}((t-c)^n-(T-c)^n).
\end{align*}
We require $\rho(k)\in (0,1]$ for all $k=1,\ldots, T-1$. Whether $n$ is even or odd changes the requirements on $b$ and $c$ to ensure this. We get the conditions that $c\in \mathbb{R}\setminus\{T\}$ and that
\begin{align*}
n \text{ even}, \quad c \le \frac{T+1}{2}, \quad& 0\le b \le \frac{(T-c)^n}{(T-c)^n -(k-c)^n}, \quad& \frac{1}{(k-c)^n-(T-c)^n}\le a \le 0\\
n \text{ even}, \quad c > T, \quad & \frac{(T-c)^n}{(T-c)^n-(k-c)^n}\le b \le 0, \quad & 0\le a \le \frac{1}{(k-c)^n-(T-c)^n}\\
n \text{ odd}, \quad c < T, \quad & 0\le b \le \frac{(T-c)^n}{(T-c)^n -(k-c)^n}, \quad & \frac{1}{(k-c)^n-(T-c)^n}\le a \le 0\\
n \text{ odd}, \quad c > T, \quad & \frac{(T-c)^n}{(T-c)^n -(k-c)^n} \le b \le 0, \quad & 0 \le a \le \frac{1}{(k-c)^n-(T-c)^n}.
\end{align*}
Note that $n$ even and $c\in [\frac{T+1}{T},T)$ implies $b=0$, which we will again exclude here. With certain applications in mind as in \Cref{sec:experidata}, we are most interested in the odd $n$ cases particularly $n=3$ with $c<T$, however we will consider all cases in generality when possible.
In \Cref{fig:PolyGeomExtension} we graph $f$ for certain values of $b$ (and corresponding $a$) with $n=3$, $c=4$ and $T=15$.
\begin{figure}[!hpt]
\centering
\includegraphics[width=13cm]{PlotOfFRhoSimplePoly.jpg}
\caption{Plot of $f$ with $\rho(k)=b\left(1-\frac{(k-c)^n}{(T-c)^n}\right)=a\left((k-c)^n - (T-c)^n\right)$ with $T=15$, $n=3$, $c=4$ and various $b$.}
\label{fig:PolyGeomExtension}
\end{figure}
Note that here we see quite different behaviours depending on the value of $b$. The larger $b$ exhibit a non-monotonic, uni-modal or even bi-modal distributions with maxima towards $T$, while the smaller $b$ exhibit more geometric-like qualities, with maximum at $k=1$ and decreasing for larger $k$. Varying $T$ gives similar results.
We also graph $f$ while varying $c$ and $b$ with fixed $n=3$, and $T=15$, as in \Cref{fig:PolyGeomExtensionC}
\begin{figure}[!hpt]
\centering
\includegraphics[width=13cm]{PlotOfFRhoSimplePolyVaryC.jpg}
\caption{Plot of $f$ with $\rho(k)=b\left(1-\frac{(k-c)^n}{(T-c)^n}\right)=a\left((k-c)^n - (T-c)^n\right)$ with $T=15$, $n=3$, $c=-5,-4,\ldots,14$ and various $b$. Note that $\max(b) = \min_{k} \frac{(T-c)^n}{(T-c)^n -(k-c)^n}$.}
\label{fig:PolyGeomExtensionC}
\end{figure}
Note that here we see quite different behaviours depending on the value of $b$. The larger $b$ exhibit a non-monotonic, uni-modal or even bi-modal distributions with maxima towards $T$, while the smaller $b$ exhibit more geometric-like qualities, with maximum at $k=1$ and decreasing for larger $k$. That such a varied behaviour is achieved through only two parameters will be helpful for fitting to real data as in \Cref{sec:experidata}. Varying $T$ gives similar results.
In \Cref{fig:PolyGeomExtensionC2} we further separate out the different behaviours for fixed $b$ and changes in $c$.
\begin{figure}[!hpt]
\centering
\includegraphics[width=15cm]{PlotOfFRhoSimplePolyVaryCSubplots.jpg}
\caption{Plot of $f$ with $\rho(k)=b\left(1-\frac{(k-c)^n}{(T-c)^n}\right)=a\left((k-c)^n - (T-c)^n\right)$ with $T=15$, $n=3$, $c=-5,-4,\ldots,14$ and various $b$ that are fixed in each plot. Note that $\max(b) = \min_{k} \frac{(T-c)^n}{(T-c)^n -(k-c)^n}$.}
\label{fig:PolyGeomExtensionC2}
\end{figure}
\subsection{Relationship to SMM} \label{sec: SemiMarkovRelationship}
In this section we describe the relationship between the family of distributions in \Cref{defn:discreteRho} and semi-Markov models. This is one of the motivations for this paper. This section also contains details on how semi-Markov models can be considered using Matrix analytic methods. This will be important in future applications of the sojourn time distributions studied so far.
\subsubsection{Sojourn time distributions arising from SMMs}
In a semi-Markov model, we have a random process $(X_n,\tau_n)$ $n=1,2,\ldots$. Here $X_n$ is a random variable for each $n$ with state space $S=\{1,2,3,\ldots,s\}$ and $\tau_n$ is a random variable for each $n$. Either $\tau_n$ has a continuous state space in $(0,T_{\max})$ for some $T_{\max}>0$ possibly infinite or a discrete state space in $1,2,\ldots,T_{\max}$ for some integer $T_{\max}>0$ possibly infinite. Here we will only consider discrete state spaces.
Note that we differ from some traditional presentations of semi-Markov models as in \cite[\S1.9]{Medhi2003} where often $X_n$ is only the transitions of the process and $\tau_n$ is the times these transitions occur, assumed to be continuous times, giving a \emph{Markov renewal process} $(X_n,\tau_n)$. Instead, here our $X_n$ will be a discrete random process that may remain in the same state as $n$ increments, and the $\tau_n$ will record the number of times the process has remained in the same state continuously.
We require the independence condition that
\begin{align*} P((X_{n+1},\tau_{n+1})=(j,t)| (X_{n},\tau_{n}),(X_{n-1},\tau_{n-1}),\ldots, (X_{1},\tau_{1})) \\
= P((X_{n+1},\tau_{n+1})=(j,t)|(X_{n},\tau_{n})).
\end{align*}
and the time-homogenous condition that
\[ P((X_{n+m+1},\tau_{n+m+1})|(X_{n+m},\tau_{n+m})) = P((X_{n+1},\tau_{n+1})|(X_{n},\tau_{n}))\]
for all $m=1,2,\ldots$.
This means that $Z_n = (X_n,\tau_n)$ is a Markov chain. We require that $\tau_n$ count the number of consecutive time steps $X_n=j$ has spent in state $j$, so that if we observe that both $X_n$ and $X_{n+1}$ are equal to $j$, then
\[P((X_{n+1},\tau_{n+1})=(j,t)|(X_{n} =j,\tau_{n})) = 0 \qquad \text{if } \tau_{n}\ne t-1, t\ge 2,\] and otherwise if $X_n=i\ne j$ then
\[P((X_{n+1},\tau_{n+1})=(j,t)|(X_{n} =i,\tau_{n})) = 0 \qquad \text{unless } t=1.\]
Then $(X_n,\tau_n)$ is specified by the following probabilities
\begin{align*}
P(X_{n+1}=j,\tau_{n+1}=t_2|X_{n+1}=i,\tau_{n+1}=t_1)=\begin{cases} A_{i,j}(t_1) & \text{if } t_2 =1, j\ne i,\\
\rho_i(t_1) & \text{if } i=j, t_2 = t_1+1, \\
0 & \text{otherwise.} \end{cases}
\end{align*}
Here the $\rho_i(k)$ can be considered as the margin probability that the chain stays in the same state for the next time step. With this set up, we call $X_n$ a \emph{semi-Markov process}. \Cref{fig:Z_trans2} shows the allowable state transitions for the process $Z_n$.
\begin{figure}[!htp]
\centering
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(4.,0.5) rectangle (12.5,7.5);
\draw [rotate around={0.:(7.83,6.38)},line width=2.pt] (7.83,6.38) ellipse (1.0453503525729777cm and 0.901031275608368cm);
\draw [rotate around={0.:(10.98,6.36)},line width=2.pt] (10.98,6.36) ellipse (1.0480183729244286cm and 0.9041252733925524cm);
\draw [rotate around={0.:(5.58,4.58)},line width=2.pt] (5.58,4.58) ellipse (1.0295649597816658cm and 0.8826686843942214cm);
\draw [rotate around={0.:(5.6,1.68)},line width=2.pt] (5.6,1.68) ellipse (1.0103414909727655cm and 0.8601685464960177cm);
\draw [->,line width=2.pt] (6.914880793461978,5.9444726038992215) -- (6.21075932820925,5.277623444471598);
\draw [->,line width=2.pt] (8.88,6.44) -- (9.937153222134956,6.4497087863121845);
\draw [->,line width=2.pt] (7.64678916372626,5.492915139373649) -- (6.14,2.42);
\draw (7.2195015639526835,6.773992246644394) node[anchor=north west] {$(j_1,t)$};
\draw (10.01583752600971,6.773992246644394) node[anchor=north west] {$(j_1,t+1)$};
\draw (5.000907793200267,5.000105511465399) node[anchor=north west] {$(j_2,1)$};
\draw (5.009896074623204,2.045910172333133) node[anchor=north west] {$(j_{s},1)$};
\draw (5.301685140236067,3.526996123322203) node[anchor=north west] {$\vdots$};
\draw (8.902493766325249,7.3816172521783715) node[anchor=north west] {$\rho_i(t)$};
\draw (5.208767170275509,6.413214899608595) node[anchor=north west] {$A_{j_1j_2}(t)$};
\draw (6.8058062469563265,3.754855500397444) node[anchor=north west] {$A_{j_1j_s}(t)$};
\end{tikzpicture}
\caption{Shows allowable transitions out of a state for the augmented chain $Z_n$. \label{fig:Z_trans2}}
\end{figure}
\begin{ex}\label{ex:MarkovSMM}
Note semi-Markov processes are an extension of Markov processes. Given a time homogenous Markov process $X_n$ specified by transition probabilities $P(X_{n+1}=j|X_n=i) = A_{i,j}$, then we can define $\tau_n$ as the consecutive time $X_{n}$ has spent in the current observed state since it entered this state. We then get that
\[P(X_{n+1}=i,\tau_n=t+1|X_{n}=i,\tau_n=t)=P(X_{n+1}=i|X_n = i)=A_{i,i} = \rho_i(t)\] and for $j\ne i$ we get that
\[P(X_{n+1}=j,\tau_n=1|X_{n}=i,\tau_n=t)=P(X_{n+1}=j|X_n = i)=A_{i,j}.\] Note that neither of these probabilities depend on the time $t$.
\end{ex}
We have suggestively used $\rho_i$ in the notation above. This is due to the $\rho_i$ satisfying the conditions in \Cref{defn:discreteRho} for each $i$. We have that
\begin{align*}
P(&\text{$X_n$ enters state $i$ then stays at state }i\text{ for exactly }k\text{ consecutive steps})\\
&=\sum_{j_1\ne i, j_2\ne i}P(X_{n+k}=j_1\ne i,X_{n}=X_{n+1}=\ldots =X_{n+k-1}=i, X_{n-1}= j_2)\\
&=\sum_{j_1\ne i, j_2\ne i}P(X_{n+k}=j|X_{n}=\ldots =X_{n+k-1}=i,X_{n-1}= j_2)\cdot\\
& \quad P(X_{n}=\ldots =X_{n+k-1}=i, X_{n-1}= j_2)\\
&=(1-\rho_i(k))\sum_{j_2\ne i}P(X_{n+k-1}=i|X_{n}=\ldots =X_{n+k-2}=i, X_{n-1}= j_2)\cdot\\&\quad P(X_{n}=\ldots =X_{n+k-2}=i, X_{n-1}= j_2)\\
&=(1-\rho_i(k))\rho_i(k-1)\sum_{j_2\ne i}P(X_{n}=\ldots =X_{n+k-2}=i, X_{n-1}= j_2)\\
&\quad \vdots\\
&=(1-\rho_i(k))\rho_i(k-1)\rho_i(k-2)\ldots \rho_i(1)\sum_{j_2\ne i}P(X_n=i,X_{n-1}=j_2),
\end{align*}
Then we have that
\begin{align*} P(&X_{n+k+1}\ne i, X_{n+k}=X_{n+k-1}=\ldots=X_{n+1}=i|X_n=i,X_{n-1}\ne i) \\
\quad & = (1-\rho_i(k))\rho_i(k-1)\rho_i(k-2)\ldots \rho_i(1).\end{align*}
This is a PMF and the corresponding $f_i$ to the $\rho_i$ from \Cref{defn:discreteRho}. We call this the \emph{sojourn time} distribution of a state $i$, with support $t=1,2,\ldots, T_i$. An aim of our research is to be able to characterise such distributions. Note that if the semi-Markov process is a Markov-process as in \Cref{ex:MarkovSMM} then the sojourn time distributions are geometric with parameters $A_{ii}$.
See \cite[\S1.9]{Medhi2003} and \cite[\S8]{Sahner1996} for further details on semi-Markov models, although they mainly focus on continuous models. Note that in that case a Markov-process would have an exponential sojourn time distribution.
\subsubsection{Matrix analytic form} \label{subsec:MAM}
Matrix analytic methods (MAMs) \cite{LR99} are a convenient way of manipulating probabilities associated with Markov chains having a two-dimensional state space as is the case for the augmented chain $Z_n$. MAMs are generally associated with {\it quasi birth-death} processes but we can apply the ideas here. Here we associate the {\it levels} with the sojourn time variable, and the {\it phases} with the state variable.
Here we assume $T_i=T$ are equal for all sojourn time distributions, although we can extend this when they are not equal. Then using the MAM approach, we let $z_n =$ Vec$(Z_n)$ so the transition matrix $\mathcal{A}$ for the $z_n$ process is specified by
\begin{align*}
\mathcal{A}^\prime & = \left[ \begin{array}{ccccc} A(1)^\prime & A(2)^\prime & \cdots & \cdots & A(T)^\prime \\
D(1) & 0&&& 0\\
0 & D(2) & 0& &0\\
\vdots&&\ddots&&\vdots\\
0&&& D(T-1) & 0\end{array} \right] \ .
\end{align*}
Each block in $\mathcal{A}$ is of size $s \times s$ and given by
\begin{align*}
\left[ A(t) \right]_{i,j} & = \left\{ \begin{array}{ll} A_{i,j}(t) & i \neq j \\ 0 & i =j \end{array} \right. \\
D(t) & = \text{diag} \left( A_{i,i}(t) \right) = \text{diag} \left( \rho_i(t)\right) \ .
\end{align*}
The process can thus either (i) go up one level and stay in the same state (phase) or (ii) drop back to level 1 and go to some other state.
From any state $(i,T)$, the process must jump to some $(j,1), j \neq i$.
Note that $\mathcal{A}$ is row stochastic and all blocks of $\mathcal{A}$ are row sub-stochastic apart from $A(T)$ which is row stochastic.
To extend this definition when the $T_i$ are not equal, then set $T=\max_iT_i$. When $t\ge T_i$ then set $A(t)=A(T_i)$, otherwise each $A$ has the definition above. When $t\ge T_i$ then $[D(t)]_{i,i}=0$ and otherwise each $D$ has the definition above.
\subsubsection*{Stationary Distribution}
The MAM formulation admits a conceptually simple process for finding the stationary distribution $\Pi$ for $z_n$ via the usual approach of solving $\mathcal{A}^\prime \, \Pi = \Pi$. Since $\mathcal{A}$ is row stochastic, $\Pi$ exists and is unique. The block form of $\mathcal{A}$ leads to :
\begin{align*}
\Pi(t) & = D(t-1) \, \Pi(t-1), \ t = 2, \ldots, T \\
\Pi(1) & = \underbrace{\left( \sum_{t=1}^{T} \ Q(t)^\prime \right)}_{\mathcal{B}^\prime} \ \Pi(1) \,
\end{align*}
where
\begin{align*}
Q(t) = D(1) \, D(2) \, \cdots \, D(t-1) \, A(t) \ .
\end{align*}
The quantities $\left[ Q(t)\right]_{i,j}$ are the conditional probabilities of the process $X_n$ staying in a state $i$ for exactly $t$ steps then going to $j \neq i$ given it entered $i$, i.e.
\begin{align*}
\left[ Q(t)\right]_{i,j} & = \Pr \left\{ X_{n+1} = X_{n+2} = \cdots X_{n+t-1} = i, X_{n + t} = j | X_n = i, X_{n-1}\ne i \right\}\\
&= A_{i,j}(t)\rho_i(t-1)\rho_i(t-2)\ldots \rho_i(1)
\end{align*}
and note that
\begin{align*}
\sum_{j\ne i} \left[ Q(t)\right]_{i,j} & = \sum_{j\ne i} \Pr \left\{ X_{n+1} = X_{n+2} = \cdots X_{n+t-1} = i, X_{n + t} = j | X_n = i, X_{n-1}\ne i \right\}\\
&= \sum_{j\ne i} A_{i,j}(t)\rho_i(t-1)\rho_i(t-2)\ldots \rho_i(1)\\
&= (1-\rho_i(t))\rho_i(t-1)\rho_i(t-2)\ldots \rho_i(1)
\end{align*}
is our sojourn time distribution as in \Cref{sec:experidata}.
Thus the matrix $\mathcal{B}$ contains the probabilities over all possible sojourn times in state $i$, of $X_n$ transitioning from state $i$ to some state $j \neq i$ at a later time. So $\mathcal{B}$ is the transition probabilities for the {\it embedded Markov Chain}.
Then $\mathcal{B}$ is row stochastic, and we can find a non-negative solution $\Pi(1)$ with $\Pi(1) = \mathcal{B}^\prime \, \Pi(1)$. The remaining levels of $\Pi$ are then easily computed. The process is computationally efficient and of order $O(s^2 \, T) + O(s^3)$.
\subsubsection*{Dynamics and Forgetting}
The forgetting properties for the augmented process are determined by the second largest magnitude eigenvalue of $\mathcal{A}^\prime$. Consider the equation $\mathcal{A}^\prime \, \Phi = \lambda \, \Phi$, then from the block form we have
\begin{align*}
\Phi(t) & = \lambda^{-1} \, D(t-1) \, \Phi(t-1), \ t = 2, \ldots, T \\
\Phi(1) & = \lambda^{-1} \, \left( \sum_{t=1}^{T} \ \lambda^{1-t} \, Q(t)^\prime \right) \ \Phi(1) \,
\end{align*}
Consider the matrix polynomial
\begin{align*}
\mathcal{P}(\lambda) & = \lambda^{T} \, I - \sum_{t=1}^{T} \ \lambda^{T-t} \, Q(t)^\prime
\end{align*}
then the eigenvalues are given by det $\mathcal{P}(\lambda) = 0$. Whilst we have not as yet studied forgetting properties of SMCs, these remain an important issue and the above formulation permits their study with reduced computational complexity. We will return to this issue in future work.
\section{Maximum likelihood estimation and Cram\'{e}r-Rao bounds} \label{sec:MLEsandCRLBs}
In this section we study maximum likelihood estimators (MLEs) of the parameters for the linear factor model and the simple polynomial factor models in \Cref{subsec:linearex} and \Cref{subsec:simplePolyexample}. To do this, we first write formulae for the likelihoods, then determine their properties in an effort to understand how to solve for the MLEs.
We then numerically simulate drawing from such distributions with known parameters and determining the MLEs numerically. We compare our results for different sample sizes and initial parameters, including discussing the Cram\'{e}r-Rao lower bound for the variance of estimates for the parameters and how closely this aligns with the variance of our MLEs.
\subsection{Linear factor model} \label{subsec:linearMLEs}
Recall the linear factor model from \Cref{subsec:linearex} with $\rho(k)=ak+b$. We will write this in terms of the parameter $b$ so that $\rho(k) = b(1-\frac{k}{T})$. Given samples $x_1,\ldots, x_n$ we wish to know the MLE for the parameters for $f$. We first consider $T$ fixed. We then have log-likelihood
\[ l(b;\mathbf{x}) = \sum_{i=1}^n \left(\log(1+b(\frac{x_i}{T}-1)) + (x_i-1)\log(\frac{b}{T}) + \log\frac{(T-1)!}{(T -x_i)!}\right) .\]
Differentiating with respect to $b$ we have
\begin{align}
\frac{d l}{db} & = \sum_{i=1}^n \left(\frac{\frac{x_i}{T}-1}{1+b(\frac{x_i}{T}-1)} + \frac{x_i-1}{b}\right)\nonumber\\
& = \sum_{i=1}^n \frac{\frac{x_i}{T}-1}{1+b(\frac{x_i}{T}-1)} + \frac{n
\bar{x}-n}{b}, \label{eqn: derivativeLinearb}
\end{align}
where $\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i$ denotes the sample mean.
When we send $b$ towards zero, we have that the right most term goes to infty and the left most term is finite, so this means we have a positive slope around $b=0$, while the sign at $b=\frac{T}{T-1}$ can be positive or negative.
We can check the second derivative of $l$ w.r.t. $b$ is
\begin{align*}
\frac{d^2 l}{db^2} & = \sum_{i=1}^n \frac{-(\frac{x_i}{T}-1)^2}{(1+b(\frac{x_i}{T}-1))^2} + -\frac{n\bar{x}-n}{b^2}.
\end{align*}
which is negative for all values of $b$. This means that the likelihood is concave, so that the maximum occurs either at the end point with $b=\frac{T}{T-1}$ or at the (only) point (if it exists) where the first derivative is equal to zero for $b\in (0,\frac{T}{T-1}]$.
We can use numerical methods to solve for this $b$, which is the MLE, using the formula for the first derivative. We can also reduce further to a polynomial form before solving.
To do this, set the first derivative equal to zero and then we have
\begin{align*}
\frac{n-n\bar{x}}{b}& = \sum_{i=1}^n \frac{\frac{x_i}{T}-1}{1+b(\frac{x_i}{T}-1)}\\
& = \frac{\sum_{i=1}^n(\frac{x_i}{T}-1)\prod_{j\ne i}(1+b(\frac{x_j}{T}-1))}{\prod_{i=1}^n(1+b(\frac{x_i}{T}-1))}
\end{align*}
so that we require to find $b$ such that it solves the polynomial equation
\begin{align*}
0&= b\sum_{i=1}^n(\frac{x_i}{T}-1)\prod_{j\ne i}(1+b(\frac{x_j}{T}-1))+ (n\bar{x}-n)\prod_{i=1}^n(1+b(\frac{x_i}{T}-1)) \\
&= n\bar{x}-n + (n\bar{x}-n+1) (\sum_{i=1}^nb^i\sum_{j=1}^i\sum_{k_1\ne\ldots\ne k_j=1}^n c_{k_1}c_{k_2}\cdots c_{k_j})
\end{align*}
with $b\in (0,\frac{T}{T-1}]$ and $c_{i}= \frac{x_i}{T}-1 $. As this is a $n$-order polynomial we can use numerical methods to solve.
Solving for a root of the polynomial within $(0,\frac{T}{T-1}]$ works well in practice, for small sample sizes. Note that from our previous discussions there is at most 1 root in this domain, and if there is no root the MLE is $\frac{T}{T-1}$. However when the sample size is large, as $|c_i|<1$, then the coefficients of $b$ become exceptionally small for all low order terms, and then function may then return zero up to precision level of the programming language, causing an issue. Solving \Cref{eqn: derivativeLinearb} numerically instead does not have the same issues, although we note that the log-likelihood is often extremely flat around the MLE requiring the tolerance for numerical optimisation to be reduced.
We can similarly do this with respect to the variable $a$, where $\rho(k) = a(k-T)$. Note that there are minimal changes as $b=-aT$ and $a$ is negative. With $T$ again fixed and $x_1,\ldots, x_n$ a sample, we have the log-likelihood
\[ l(a;\mathbf{x}) = \sum_{i=1}^n \left(\log(1+a(T-x_i)) + (x_i-1)\log(-a) + \log\frac{(T-1)!}{(T -x_i)!}\right) .\]
Differentiating with respect to $a$ we have
\begin{align}
\frac{d l}{d a} & = \sum_{i=1}^n \left(\frac{T-x_i}{1+a(T-x_i)} + \frac{x_i-1}{a}\right)\nonumber\\
& = \sum_{i=1}^n \frac{T-x_i}{1+a(T-x_i)} + \frac{n\bar{x}-n}{a}. \label{eqn: derivativeLineara}
\end{align}
If we consider sending $a$ to zero, then the first term tends to a positive number while the second term tends to negative infinity, so the slope is negative as we send $a$ to zero. The second derivative is
\begin{align}
\frac{d^2 l}{d a^2} & = \sum_{i=1}^n \frac{-(T-x_i)^2}{(1+a(T-x_i))^2} - \frac{n\bar{x}-n}{a^2}. \label{eqn: 2ndderivativeLineara}\\
&\le 0
\end{align}
As this second derivative is negative, then the function must be concave everywhere. Given the slope is positive as $a$ tends to zero, then there is a unique MLE which occurs either where the derivative is equal to zero or at the lower boundary where $a=\frac{1}{1-T}$.
We can again solve for the MLE numerically. We can also apply the same technique as above to write this as a polynomial equation to find the roots, however the coefficients will now be of the form $(T-x_i)$ which for large sample sizes will create very large numbers that may not be computable by the operating system.
\subsubsection{Numerical analysis}
While we can solve the above equations numerically to determine the MLE for $a$ and/or $b$ for a given sample and given $T$, we do not know if the MLE is a good estimate or not, given the finite sample sizes. We cannot analytically determine the bias nor the variance of the MLE from the formulae above. However, we have simulated the bias and variance of the MLE as below using Matlab. For the definitions of Cram\'{e}r-Rao Lower Bound and Fisher information used in this section see any standard statistics textbook such as \cite[pg.~517--527]{DeGroot}.
Note that the Cram\'{e}r-Rao Lower Bound $(\mathrm{CRLB})$ is a lower bound for the variance of that estimator under certain regularity conditions. If $T(X)$ is an estimator for a parameter $a$ then we have
\[ \Var(T) \ge \frac{\frac{d}{da}E_X(T(X))}{\mathcal{I}(a)}= \mathrm{CRLB}(a)\] where $\mathcal{I}$ is the Fisher information.
This gives a measure of how consistently the estimator estimates the parameter (up to the bias). If the estimator is unbiased, the expected value of the estimator is equal to the parameter and the derivative is equal to one. In this case, the lower bound is exactly the inverse Fisher information. So, for unbiased estimators, the CRLB provides a performance bound which is only dependent on the model, and not the form of estimator itself.
In our case, we cannot determine the expected value of the MLE nor the derivative of this analytically, so we have chosen not to estimate this. We can determine the Fisher information as follows, where it is the negative of the expected value of the second derivative of the log-likelihood.
With respect to $a$ this is
\begin{align*} \mathcal{I}(a)=\E_X\left(\frac{d}{d a}\log(f(X;a))\right)^2 & = \E_X\left(\sum_{i=1}^n \left (\frac{(T-X_i)}{1+a(T-X_i)} + \frac{X_i-1}{a^2}\right)\right)^2
\end{align*}
which is an analytic formula given we know the true $a$ and $T$. Similarly for $b$.
For the numerical simulations, we set the parameters $T$ and $a$ (or $b$). We assume $T$ is known. Using Matlab, we sample $n$ points from the probability distribution $f$ and then numerically determine the MLE for $a$ using {\tt fmincon}. We iterate this 10,000 times. We determine the sample mean of the MLEs of the iterates and the sample variance of the MLEs of the iterates numerically.
In \Cref{fig:linearExtensiona} we plot the sample bias and see that it is close to zero for all sample sizes, and decreases to zero as the sample size increases. We also plot the sample variance along with the inverse Fisher information and mean squared error. Note that the Fisher information is not a lower bound in all cases, which we understand is due to the bias of the MLE and also due to regularity conditions that may break for $a$ very small, as the likelihood is undefined for $a=0$.
\begin{figure}[]
\vspace{-14pt}
\centering
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14.5cm]{PlotBiasVarFisherInfo_a0.01T10_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14.5cm]{PlotBiasVarFisherInfo_a0.0556T10_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14.5cm]{PlotBiasVarFisherInfo_a0.1T10_2.jpg}
\end{subfigure}
\caption{Mean of the sample bias of the MLE estimates for $a$ over all iterations, and separately the sample variance and mean squared error of the estimates with the Fisher information. Sample sizes ranged from $2$ to $50,000$, and samples were taken from the distribution $f$ with $\rho(k) = a(k-T)$ and true $a=-0.01,-0.0556,-0.1$, $T=10$. $10,000$ iterations for each sample size. We assumed $T$ was known. Note the log scales where used.}
\label{fig:linearExtensiona}
\end{figure}
We see that small sample sizes for $a=-0.01$ result in the variance being lower than the inverse Fisher information, while for larger samples the Fisher information and the variance are almost the same. For the largest sample size of $50,000$ we have the inverse Fisher information slightly lower than the sample variance. Plotting for other $a$ as per \Cref{fig:linearExtensiona} we see that the larger $a$ is the more the Fisher information is below the sample variance, acting as a lower bound.
\subsubsection{\texorpdfstring{Unknown $T$}{Unknown T}} \label{subsec:linearunknownT}
In the previous section we assumed $T$ was known. In this section, we consider how to estimate $T$ when $T$ is unknown, as would be likely to occur in practice. Note that given a sample $x_1,\ldots, x_m$ we need at least $T\ge\max_i{x_i}$, as the support of the PMF must allow for the sample to be obtained. We numerically estimate the MLE for $T$ using a grid based method.
We sample from the linear model with $T=10$, and consider the true $a$ equal to $-0.1,-0.01,-0.0556$. We take a sample $x_1,\ldots, x_m$ of size $m$ and then for each of
\[T =\max_i{x_i}, \max_i{x_i}+1,\max_i{x_i}+2,\ldots, T_{\max}\]
we estimate the MLEs using Matlab's {\tt fmincon}. Here we took $T_{\max} = \max_i{x_i}+200$.
We pick the value of $T$ and corresponding {\tt fmincon} output for $a$ which have the maximum likelihood for the sample as our MLEs. We iterate this 100 times and record the bias of the means of the MLEs of the iterates and the variance of the MLEs of the iterates. We do this for various different sample sizes $m$ including $m=2, 3, 4, 5, 6, 8, 10, 15, 20, 50, 100, 500,$ $1000, 5000, 10000, 50000$. We plot in \Cref{fig:linearExtensionMLEUnknownTfora} and \Cref{linearExtensionMLEUnknownTforT} the bias of the MLEs against $m$, and we plot the variance, the inverse Fisher information and the mean squared error against $m$. We see that variance is larger when $m$ is small, but otherwise it decreases and begins to align with the inverse fisher information for $a$.
Note that as we are undertaking a grid search for $T$, then $T$ is an integer and at large sample sizes for some $a$ we observe the MLE achieving the correct result of $T=10$ for every iteration, giving a variance of zero. This is unable to be plotted on the axes due to the log scale. Such is the case when $a=0.1$. In this case, we have also included the single parameter inverse Fisher information. Here, it is clear that the MLE estimates for $a$ tend to follow the single parameter inverse Fisher information when the sample size is large enough, which is when $T$ is correctly predicted at every iteration. Note that such integer parameters such as $T$ are often analysed by considering how frequently they achieve the true value, and converting this into a probability --- in this case they are achieving the true value every time, so the probability would be 1 with some confidence level. We have not used this approach here.
The inverse Fisher information is derived by assuming $T$ to be continuous and is only included as a guide. We use that the derivative of $l(a,T;x)$ for a sample $x_1,\ldots, x_m$, with respect to $T$ is
\[ \frac{dl}{dT} = \sum_{i=1}^m \left(\frac{a}{1+a(T-x_i)} + \sum_{k=1}^{x_i-1}\frac{1}{T-k} \right) .\]
\begin{figure}[]
\vspace{-8pt}
\centering
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoa_a0.1T10.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoa_a0.0556T10.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoa_a0.01T10.jpg}
\end{subfigure}
\caption{Mean of the sample bias of the MLE estimates for $a$ over all iterations, and separately the sample variance and mean squared error of the estimates against the Fisher information. Sample sizes ranged from $2$ to $50,000$, and samples were taken from the distribution $f$ with $\rho(k) = a(k-T)$ and true $a=-0.1,-0.0556,-0.01$, $T=10$ and $100$ iterations for each sample size. We assume $T$ is unknown. Note the log scales where used.}
\label{fig:linearExtensionMLEUnknownTfora}
\end{figure}
\begin{figure}[]
\vspace{-8pt}
\centering
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoT_a0.1T10.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoT_a0.0556T10.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotTBiasVarFisherInfoT_a0.01T10.jpg}
\end{subfigure}
\caption{Plot of the mean of the sample bias of the MLE estimates for $T$ over all iterations, and separately the sample variance and mean squared error of the estimates against the Fisher information. Sample sizes ranged from $2$ to $50,000$, and samples were taken from the distribution $f$ with $\rho(k) = a(k-T)$ and true $a=-0.1,-0.0556,-0.01$, $T=10$ and $100$ iterations for each sample size. Note the log scales where used.}
\label{linearExtensionMLEUnknownTforT}
\end{figure}
Note that for $a$ close to zero, then for small sample sizes the sample that contains all ones is frequently selected. In this case, the MLE is $T=1$ and $a=0$. Given the frequency of this result, the variance for T is reduced for these small sample sizes (although the bias is large).
\subsection{Simple polynomial factor model} \label{subsec:polyMLE}
We now repeat the process in \Cref{subsec:linearMLEs} for the simple polynomial factor model in \Cref{subsec:simplePolyexample}. With respect to $b$ the log-likelihood for an i.i.d. sample $x_1,\ldots, x_m$ is
\begin{align*} l(b,c, T) =& \sum_{i=1}^m\Bigg(\log\left(1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)\right)+(x_i-1)\log(b) -n(x_i-1)\log(T-c) \\
&+ \sum_{t=1}^{x_i-1}\log((T-c)^n-(t-c)^n)\Bigg).
\end{align*}
The derivative with respect to $b$ is as follows
\[ \frac{\partial l}{\partial b} = \sum_{i=1}^m\left( \frac{\frac{(x_i-c)^n}{(T-c)^n}-1}{1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)} + \frac{x_i-1}{b}\right).\]
The second derivative with respect to $b$ is as follows
\[ \frac{\partial^2 l}{\partial b^2} = \sum_{i=1}^m\left( \frac{-(\frac{(x_i-c)^n}{(T-c)^n}-1)^2}{(1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right))^2} - \frac{x_i-1}{b^2}\right).\]
The derivative with respect to $c$ is as follows
\[ \frac{\partial l}{\partial c} = \sum_{i=1}^m \left(\frac{\frac{-nb(x_i-c)^{n-1}}{(T-c)^n} +\frac{nb(x_i-c)^n}{(T-c)^{n+1}} }{1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)} + \frac{n(x_i-1)}{T-c} + \sum_{t=1}^{x_i-1}\frac{-n(T-c)^{n-1}+n(t-c)^{n-1}}{(T-c)^n-(t-c)^n}\right).\]
The second derivative with respect to $c$ is as follows
\begin{align*} \frac{\partial^2 l}{\partial c^2} &= \sum_{i=1}^m \left(\frac{nb\left(\frac{-(n-1)(x_i-c)^{n-2}}{(T-c)^n} -\frac{2n(x_i-c)^{n-1}}{(T-c)^{n+1}} +\frac{(n+1)(x_i-c)^n}{(T-c)^{n+2}}\right) }{1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)} - \frac{\left(\frac{-nb(x_i-c)^{n-1}}{(T-c)^n} +\frac{nb(x_i-c)^n}{(T-c)^{n+1}} \right)^2}{\left(1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)\right)^2} \right.\\
&+\frac{n(x_i-1)}{(T-c)^2}\\
& \left.+ \sum_{t=1}^{x_i-1}\left(\frac{n(n-1)((T-c)^{n-2}-(t-c)^{n-2})}{(T-c)^n-(t-c)^n}- \frac{(-n(T-c)^{n-1}+n(t-c)^{n-1})^2}{((T-c)^n-(t-c)^n)^2}\right)\right).\end{align*}
The mixed partial derivative with respect to $b$ and $c$ is as follows
\[ \frac{\partial^2 l}{\partial b \partial c} = \sum_{i=1}^m \left(\frac{\frac{-n(x_i-c)^{n-1}}{(T-c)^n} +\frac{n(x_i-c)^n}{(T-c)^{n+1}} }{1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)}-\frac{\frac{-nb(x_i-c)^{n-1}}{(T-c)^n} +\frac{nb(x_i-c)^n}{(T-c)^{n+1}} }{\left(1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)\right)^2}\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)\right).\]
The Hessian of the likelihood is
\[ H(b,c) = \begin{bmatrix} \frac{\partial^2 l}{\partial b^2} &\frac{\partial^2 l}{\partial b \partial c} \\\frac{\partial^2 l}{\partial b \partial c} & \frac{\partial^2 l}{\partial c^2} \end{bmatrix}. \] Noting that $\frac{\partial^2 l}{\partial b^2}<0$ for all values of $b,c$ then at any point where the first partials are equal to zero, this point is either a maximum or a saddle point. Then the MLE must be found either at such a point that is a maximum or along the boundary. Numerical solvers can be used to find the maximum, solving for the maximum of the likelihood function in terms of $b$ and $c$ to determine the MLEs.
Similarly, in terms of $a$, then $b = -a(T-c)^n$ and $\rho(k) = a((k-c)^n - (T-c)^n)$.
We have
\[ f(k) = \left(1+a\left((T-c)^n-(k-c)^n\right)\right)(-a)^{k-1}\prod_{t=1}^{k-1}((T-c)^n-(t-c)^n).\]
With respect to $a$ the log-likelihood for a sample $x_1,\ldots, x_m$ is
\begin{align}\label{eqn:loglikepolyac} l(a,c, T) =&\\\nonumber \sum_{i=1}^m\Bigg(\log&\left(1+a\left((T-c)^n-(x_i-c)^n\right)\right)+(x_i-1)\log(-a)
+ \sum_{t=1}^{x_i-1}\log((T-c)^n-(t-c)^n)\Bigg).
\end{align}
The derivative with respect to $a$ is as follows
\[ \frac{\partial l}{\partial a} = \sum_{i=1}^m\left( \frac{(T-c)^n-(x_i-c)^n}{1+a\left((T-c)^n-(x_i-c)^n\right)} + \frac{x_i-1}{a}\right).\]
The derivative with respect to $c$ is as follows
\[ \frac{\partial l}{\partial c} = \sum_{i=1}^m \left(\frac{-na(T-c)^{n-1} + na(x_i-c)^{n-1} }{1+a\left((T-c)^n-(x_i-c)^n\right)} + \sum_{t=1}^{x_i-1}\frac{-n(T-c)^{n-1}+n(t-c)^{n-1}}{(T-c)^n-(t-c)^n}\right).\]
These derivatives are required to write the Fisher information which we use in \Cref{subsec:PolyNumAnaly}
\subsubsection{Numerical analysis}\label{subsec:PolyNumAnaly}
Similar to the linear factor model case, here we take various samples from the simple polynomial factor model in terms of a true $a$ and $c$. We consider the case $n=3$ assumed to be known and $T=10$ also assumed to be known, and $c=5$ assumed to be unknown while $c<T$ is known. We then consider the different cases of $a=0.1\min(a), 0.5\min(a),0.9\min(a)$. For a set sample size $m$ and a fixed true $a$, we take samples from the true distribution and numerically determine the MLEs of $a$ and $c$ using the Matlab command {\tt fmincon}. We iterate this 10,000 times and then take the mean of the bias and variance of these MLEs over the iterations. We consider $m=2, 3, 4, 5, 6, 8, 10, 15, 20, 50, 100, 500, 1000, 5000, 10000, 50000$.
In \Cref{fig:PolyGeomExtensionMLEa} and \Cref{fig:PolyGeomExtensionMLEc} the mean of the bias against the sample sizes for $a$ and $c$ respectively. We see that the bias is large when $a$ is small, however it approaches zero as $a$ gets larger.
In \Cref{fig:PolyGeomExtensionMLEa} and \Cref{fig:PolyGeomExtensionMLEc} we also plot the variance, mean squared error (MSE) and the Fisher information against the sample sizes, noting that in general where there is no bias the Fisher information is a lower bound for the variance of an estimator. In our case, it is clear that there is a bias of the estimator, however the Fisher information remains informative, with the variance and MSE tracking closely to the Fisher information as the sample size $m$ increases. We also plot a corrected variance that considers the variance with the outliers removed.
In general, we see that when the true $a$ is close to zero, the sample variance for $a$ is lower than the inverse Fisher information for small sample sizes. As $a$ becomes larger in absolute value, the sample variance starts to become larger than the inverse Fisher information for small sample sizes. Similar results hold for the MLEs of $c$.
All results in this section used the following options for {\tt fmincon}: {\tt options =} \\{\tt \allowbreak optimoptions("fmincon","Algorithm","interior-point","EnableFeasibilityMode",} \\{\tt true,"SubproblemAlgorithm","cg","SpecifyObjectiveGradient",true,}\\ {\tt "StepTolerance",1e12,"ConstraintTolerance",1e15).} We supplied the gradient of the likelihood using the derivatives of the likelihood given at the start of \Cref{subsec:polyMLE}.
\begin{figure}[]
\vspace{-20pt}
\centering
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolya_a0.1maxT10c5_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolya_a0.5maxT10c5_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolya_a0.9maxT10c5_22FixEvenForceComplex.jpg}
\end{subfigure}
\caption{Mean of the sample bias, sample variance and Fisher information for the MLE estimator for $a$ for various sample sizes from distribution $f$ with $\rho(k) = a((k-c)^n-(T-c)^n)$ with $n=3$, $a=0.1\min(a),0.5\min(a),0.9\min(a)$, $T=10$ and $10,000$ iterations taken for each sample size. Note the log scales where used. Assumed $T$ known.}
\label{fig:PolyGeomExtensionMLEa}
\end{figure}
\begin{figure}[]
\vspace{-20pt}
\centering
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolyc_a0.1maxT10c5_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolyc_a0.5maxT10c5_2.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=14cm]{PlotBiasVarFisherInfoPolyc_a0.9maxT10c5_22FixEvenForceComplex.jpg}
\end{subfigure}
\caption{Mean of the sample bias, sample variance and Fisher information for the MLE estimator for $c$ for various sample sizes from distribution $f$ with $\rho(k) = a((k-c)^n-(T-c)^n)$, $n=3$, $a=0.1\min(a),0.5\min(a),0.9\min(a)$, $T=10$ and $10,000$ iterations taken for each sample size. Note the log scales where used. Assumed $T$ known.}
\label{fig:PolyGeomExtensionMLEc}
\end{figure}
\subsubsection{\texorpdfstring{Unknown $T$}{Unknown T}}
Similar to \Cref{subsec:linearunknownT} we can apply a grid search to determine the MLE for $T$ from the data, along with $b$ and $c$. We do not do this with simulations here, however we do apply this method to experimental data as in \Cref{sec:experidata}. Note that as usual we require that $T\ge \max_i\{x_i\}$ for a sample $x_1,\ldots, x_m$. Helpful formulae for the derivative of the log-likelihoods with respect to $T$ are given below, noting that $T$ is assumed to be integer so these are only intended as a guide to be used for the inverse Fisher information.
We leave further simulations for future work.
In terms of $b$, the derivative with respect to $T$ is as follows
\[ \frac{\partial l}{\partial T} = \sum_{i=1}^m \left(\frac{\frac{-(x_i-c)^n}{(T-c)^{n+1}} }{1+b\left(\frac{(x_i-c)^n}{(T-c)^n}-1\right)} - \frac{n(x_i-1)}{T-c} + \sum_{t=1}^{x_i-1}\frac{n(T-c)^{n-1}}{(T-c)^n-(t-c)^n}\right).\]
In terms of $a$, the derivative with respect to $T$ is as follows
\[ \frac{\partial l}{\partial T} = \sum_{i=1}^m \left(\frac{na(T-c)^{n-1} -na(x_i-c)^{n-1}}{1+a\left((T-c)^n-(x_i-c)^n\right)} + \sum_{t=1}^{x_i-1}\frac{n(T-c)^{n-1}}{(T-c)^n-(t-c)^n}\right).\]
\subsection{Discussion}
In this section we further discuss the results of \Cref{subsec:linearMLEs} and \Cref{subsec:polyMLE}. We note that for the linear factor model, overall we observe the variances of MLEs for the parameters when $T$ is known closely follow the inverse Fisher information in all cases, and we only observe it deviating below the inverse Fisher information when the $a$ is particularly small and the sample size is small.
However, when estimating both $a$ and $T$ using the grid search method, we do see deviations of the variance away from the inverse Fisher information for small samples sizes, and in the case where $T$ becomes correctly identified in every sample. In this latter case, the variance for the MLE for $T$ becomes zero, while the variance for $a$ starts to track the one-parameter inverse Fisher information instead, which shows the limitations for using the inverse Fisher information (or CRLB) for integer variables. Further investigation of the PMFs of the distributions produced from these parameters shows little visual difference. In \Cref{appendix:unknownT} the second author discusses this further.
In the simple polynomial factor model with $n=3$ we see similar behaviour although slightly more extreme than the 1-parameter case, when estimating $a$. We observe the variance for $a$ tracking closer to the inverse Fisher information for large sample size, however we see some deviation away from the inverse Fisher information for small sample sizes. The behaviour for the MLE estimates of $c$ is also similar, although deviates further from the inverse Fisher information in small sample sizes. We suspect that some of this may result from the programming behind the Matlab function {\tt fmincon}, and we discuss this further in \Cref{appn: NumericalPolyIssues}. Additionally, the true CRLB requires knowing the square of the derivative of the bias of the variable, which we have not estimated --- we note that the bias is larger in these cases and this may contribute to the differences observed. Further examination we leave for future work.
\section{Applying to experimental data} \label{sec:experidata}
In this section we apply the linear and polynomial factor models to experimental data. This experimental data consists of the consecutive times a number of participants took to complete three screen based tasks. We consider these times to be the sojourn times and each screen based task to be a separate state. A key objective of our analysis is to charaterise the distributions of the times taken. The data is available in \Cref{appn:ExperimentalData}.
After collecting the data and plotting histograms of the data, we do not see distributions that appear to be geometric. Instead, the distributions more closely match the linear and simple polynomial factor models studied in the previous sections. We can determine the empirical PMF and cummulative density functions (CDFs) through bin counting, and then determine the empirical $\rho(k)$ using the formulae in \Cref{lem:relationship}. Plotting this in \Cref{fig:RhoPlotsExperi} we see immediately that the plot is not constant, so not indicating a geometric distribution, and is non-linear, which indicates a simple polynomial factor model may be appropriate.
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1\textwidth]{RhoPlotsTask1.jpg}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1\textwidth]{RhoPlotsTask2.jpg}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1\textwidth]{RhoPlotsTask3.jpg}
\end{subfigure}
\caption{Empirical $\rho$ of the data from tasks 1, 2 and 3. Only values not equal to 1 were plotted. A smoothed version was also plotted, where the PMF was smoothed so that each bin was the average of it and the two bins either side of it. As each task displays non-constant and non-linear for each task $\rho$ this suggests that a simple polynomial factor model may be a good fit.}
\label{fig:RhoPlotsExperi}
\end{figure}
Below, we plot histograms of the consecutive time spent on task for each task. We apply the linear factor model as in \Cref{fig:PDFExperi} and the simple polynomial factor model with $n=3,c<T$ as in \Cref{fig:PDFexperipoly} to each task. We note that the number of data points for each task were 178, 187, 187 respectively.
A first observation of the data is that the data do not have minimum of 1 for each task, instead the minimums are 4,3 and 12 respectively. This makes sense as the tasks had a certain level of complexity that would ensure more than one time unit would need to be counted for any attempt at the task.
Given this, we extended our model to consider distributions of the linear and polynomial factor models that were also shifted to start at an integer $t_1>1$ with $t_1$ less than or equal to the minimum of the sample. We compared the different possible shifts by comparing the value of the likelihood for the MLEs. Whichever one had the largest likelihood we considered to be the estimated MLE. A table of the results is included for each analysis, see \Cref{tab:linearExperi200} and \Cref{tab:polyExperi200}. In this case we used a grid search to determine $T$ by searching up to 200 integers higher than the minimum possible value for $T$, i.e. up to 200 higher than the max of the data.
\begin{table}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline
Task &Maximum &Minimum & Shift & MLE $a$ & MLE $T$ \\ \hline
1 &164& 4 & 3 & $-0.002786$& $360$\\
2 &201& 3 & 2 & $-0.002519$ & $398$\\
3 &221& 12 & 11 & $-0.002451$& $409 $ \\
\hline
\end{tabular}
\caption{This table details the output from our optimisation process for each task from applying the simple linear factor model. We have also included the maximum and the minimum of the data for that task, as well as the shift of the distribution that gave the largest likelihood. Here $T$ was allowed to range up to 200 integers higher than the maximum data value after the shift.}
\label{tab:linearExperi200}
\end{center}
\end{table}
We use PMF, CDF and QQ-plots to compare the empirical distribution with the MLE estimates, as in \Cref{fig:CDFexperi} and \Cref{fig:CDFexperipoly}. Here the QQ-plots plot the quantiles of the empirical distribution against the distribution using the MLEs estimates. To indicate a good fit we expect that this plot should be very close to the straight line drawn on the graph, which is the quantiles of the MLE against itself.
We note that the simple linear factor model was not a good fit when comparing to the plots of the PMFs as well as to the QQ-plots and CDF plots. We observed this as well in the optimisation procedure that the $T$ value was at the far end of the range we were searching. We increased the grid search to include $T$ values up to 1000 higher than the maximum data point, which greatly increased the visual fit of the data, however the optimisation procedure still picked the largest value of $T$ possible. This suggests that this particular linear factor model may not be the right fit for this data. A table of results for this is include in \Cref{tab:linearExperi1000}.
\begin{table}[]
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline
Task &Maximum &Minimum & Shift & MLE $a$ & MLE $T$ \\ \hline
1 &164& 4 & 3 & $-8.6204\times 10^{-4}$& $1160$\\
2 &201& 3 & 2 & $-8.3542\times 10^{-4}$ & $1198$\\
3 &221& 12 & 11 & $-9.1503\times 10^{-8}$& $409 $ \\
\hline
\end{tabular}
\caption{This table details the output from our optimisation process for each task from applying the simple linear factor model. We have also included the maximum and the minimum of the data for that task, as well as the shift of the distribution that gave the largest likelihood. Here $T$ was allowed to range up to 1000 integers higher than the maximum data value after the shift.}
\label{tab:linearExperi1000}
\end{center}
\end{table}
\begin{table}[]
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c|c| }
\hline
Task &Maximum &Minimum & Shift & MLE $a$ & MLE $c$ & MLE $T$ \\ \hline
1 &164& 4 & 3 & $-9.1503\times 10^{-8}$& $74.8998$& $294$\\
2 &201& 3 & 3 & $-4.2160\times 10^{-8}$& $92.9539$ & $377$\\
3 &221& 12 & 9 & $-3.6970\times 10^{-8}$& $89.4028$ & $387 $ \\
\hline
\end{tabular}
\caption{This table details the output from our optimisation process for each task from applying the simple polynomial factor model with $n=3$. We have also included the maximum and the minimum of the data for that task, as well as the shift of the distribution that gave the largest likelihood. Here $T$ was allowed to range up to 200 integers higher than the maximum data value after the shift. Note however that the MLE for T was less than the maximum minus the shift.}
\label{tab:polyExperi200}
\end{center}
\end{table}
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task1_PDFslinear1000.jpg}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task2_PDFslinear1000.jpg}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task3_PDFslinear1000.jpg}
\end{subfigure}
\caption{Histogram of the data from tasks 1, 2 and 3, along with the MLE estimated PMFs using the linear factor model. Here $T$ was allowed to range up to 1000 integers higher than the maximum data value after the shift.}
\label{fig:PDFExperi}
\end{figure}
\begin{figure}[]
\centering
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task1_CDFs_QQPlot_linear1000.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task2_CDFs_QQPlot_linear1000.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task3_CDFs_QQPlot_linear1000.jpg}
\end{subfigure}
\caption{QQ plots of the quantiles of the data from task 1,2, and 3 against the quantiles of the MLE estimated distribution, along with plots of the CDF of the sample against the CDF of the MLE estimated distribution. This used the linear factor model. Here $T$ was allowed to range up to 1000 integers higher than the maximum data value after the shift.}
\label{fig:CDFexperi}
\end{figure}
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task1_PDFs.jpg}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task2_PDFs.jpg}
\hfill
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}\centering
\includegraphics[width=1.1\textwidth]{Task3_PDFs.jpg}
\end{subfigure}
\caption{Histogram of the data from tasks 1, 2 and 3, along with the MLE estimated PMFs using the simple polynomial factor model with $n=3$. Here $T$ was allowed to range up to 200 integers higher than the maximum data value after the shift.}
\label{fig:PDFexperipoly}
\end{figure}
\begin{figure}[]
\centering
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task1_CDFs_QQPlot.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task2_CDFs_QQPlot.jpg}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}\centering
\includegraphics[width=14cm]{Task3_CDFs_QQPlot.jpg}
\end{subfigure}
\caption{QQplots of the quantiles of the data from task 1,2, and 3 against the quantiles of the MLE estimated distribution, along with plots of the CDF of the sample against the CDF of the MLE estimated distribution. Here MLE parameters were estimated for the simple polynomial factor model with $n=3$.}
\label{fig:CDFexperipoly}
\end{figure}
For the simple polynomial factor model with $n=3$ visual observation of the PMF plot seems to show close alignment between the recorded data and the MLE estimated PMF. In the QQ-plots we see mostly good alignment between the recorded data quantiles and the MLE estimated quantiles except perhaps task 2, where the data appears slightly more peaked than the distribution.
Note that applying to the simple polynomial factor model with $n=5$ did not markedly change the results of the plots, particularly for task 2. This may suggested trying a more complicated model if needed, with additional parameters. The other suggestion is that there is a relatively small data set (under 200 points) that this is applied to, so collecting more data could help improve the models. Overall we are happy with the fit for the simple polynomial factor model with $n=3$ given this small data set.
\section{Conclusion and future work} \label{sec:conclusion}
In this report we have described a new way to extend the geometric distribution and studied specific parametrised subfamilies including the linear factor model and the simple polynomial factor models in \Cref{sec:genGeometric}. We have studied maximum likelihood estimation of the parameters for these subfamilies, and also considered how to estimate the maximum sojourn time $T$ in the linear factor model case as in \Cref{sec:MLEsandCRLBs}. Finally in \Cref{sec:experidata} we applied our techniques to experimental data, where we conclude that the polynomial factor model of order three appears to offer very good performance in estimating sojourn time distributions for these experiments.
While already a contribution to research in this area, with the aim to be applicable to semi-Markov models as in \Cref{sec: SemiMarkovRelationship}, there are many avenues of future research. These include analysis on estimating unknown $T$ in the simple polynomial factor models and examining the bias, variance and mean squared errors of the MLE estimators in greater detail.
Another avenue of research is to consider other generalisations, perhaps with additional parameters for the $\rho$. Examples include more general polynomial factor models for $\rho$, having $\rho$ of the form $e^{-g(k)}$ where $g:\mathbb{N}\to\mathbb{N}$, or a trignometic $\rho$ such as $\rho(k) = \frac{1}{2}\cos(k)+\frac{1}{2}$, provided $\rho\in [0,1]$ is guaranteed. This may improve the fit for task 2 of the experimental data, as discussed in \Cref{sec:experidata}.
We would also like to understand more about these sojourn time distributions and how the MLEs behave in general. For example, in the simple polynomial factor models we found that there were larger variances observed for small sample sizes, however our observations of the different distributions generated by these MLEs showed little visual differences. Future work can include making this precise, for example by considering the KS-Statistics generated by the MLE distributions, as in \cite{DAgostino}, which give an measure of the distance between the CDFs (in this case using a $\sup$ norm).
The second author has further explored this further in the appendix, where he replicates some of the results for the linear factor model then explores further modelling behaviour. In \Cref{appendix:unknownT} where he considers the $\ell_1$ distance between the PDFs for the linear factor model, showing that while the variance may be elevated the true distributions are converging as desired. This leaves open further questions around the connection of this approach to KS-statistics as well as how this applies in the simple polynomial factor model. In \Cref{appendix:exploglik} he also considers how the expected log-likelihood can also give indications of how the MLE procedure will results in different outcomes for different parameters of $a$. This we aim to explore this further in future work.
Finally, we intend to broaden this research by considering how to determine the parameters in online settings, including when estimating parameters for hidden semi-Markov models. This will likely mean adapting the expectation-maximisation algorithm usually used by these models to account for these sojourn time distribution parameters. We also seek to expand on the matrix analytic methods of \Cref{subsec:MAM}, including studying forgetting times and the general dynamics of the models. Part of this is to continue with the work in \Cref{sec:experidata} that seeks to determine when a sojourn time distribution is not geometric (i.e. non-Markovian) implying that we should use semi-Markov methods.
\section{Acknowledgements}
This research is supported by the Commonwealth of Australia as represented by the Defence Science and Technology Group of the Department of Defence. We also thank Prof. Anna Ma-Wyatt and Dr Jessica O'Rielly for their involvement in collecting the data used in \Cref{sec:experidata}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}\label{Intro}
A physical system interacting with the environment is referred to as {\it open} \cite{BreuerBook}.
In reality, such an interaction is unavoidable, so every physical system is affected by the presence of its surroundings, leading to dissipation, thermalization, and, in the quantum case, decoherence \cite{GardinerBook, SchlosshauerBook}.
During the recent few years, interest in open quantum systems increased because of several reasons.
On the one hand, both decoherence and dissipation generally constitute the main obstacle to the realization of quantum computers and other quantum devices.
On the other hand, recently there has been a series of very interesting proposals to exploit the interaction with the environment to dissipatively engineer challenging states of matter \cite{Griessner2006, Kraus2008, Diehl2010, Diehl2010b},
and of works where the engineering of environments paved the way to the creation of entanglement and superpositions of quantum states \cite{Poyatos1996, Turchette2000, Plenio2002}.
Moreover, open quantum system techniques are often adopted to investigate problems lying at the core of the foundations of quantum mechanics.
Here, one of the unsolved issues regards the problem of the quantum-to-classical transition, i.e. the question how do classical features we experience in the macroscopic world arise from the underlying quantum phenomena \cite{SchlosshauerBook, Schlosshauer2005, Zurek2003, Haroche}.
Most of the theories addressing the emergence of the classical world deem it a consequence of the coupling of quantum systems with the environment
\cite{Maniscalco2006, Blume2008, Zurek2009, Korbicz2012, Maniscalco2014, Korbicz2014, Tuziemski2015}.
In this work we focus on an ubiquitous model of open quantum system, the quantum Brownian motion (QBM), which describes the dynamics of a particle (playing the role of the open system) coupled with a thermal bath made up by a large number of bosonic oscillators (the environment) \cite{Caldeira1983a, Caldeira1983b, Haake1985, Grabert1988, Hu1992, Hu1993}.
QBM is in many situations the default choice for evaluating decoherence and dissipation processes, and in general it provides a way to treat quantitatively the effects experienced by an open system due to the interaction with the environment \cite{Marshall2003, Bose1997, Bose1999, Groblacher2015}.
Hereafter, we will refer to the central particle studied in the model as \textit{the Brownian particle}.
The main tool for the investigation of the dynamics of the system is the master equation (ME), which describes the evolution of the reduced density matrix of the Brownian particle, obtained by taking the trace over the degrees of freedom of the bath.
The ME allows to compute various physical quantities, such as the time scales of decoherence and dissipation processes, as well as the average values of variables like position and momentum.
Widely used in the literature is the so-called Born-Markov master equation (BMME) \cite{SchlosshauerBook,BreuerBook}. The latter, however, does not always preserve positivity of the density matrix, leading to violations of the Heisenberg uncertainty principle (HUP), i.e., $\sigma_X \sigma_P \geq \hbar/2$, especially at very low temperatures \cite{FlemPRE2011, Massignan2015}. Here $\sigma_X$ and $\sigma_P$ are the standard deviation of the position and momentum, respectively. The MEs in the so-called Lindblad form preserve the positivity of the density operator at all times \cite{Lindblad1976, SchlosshauerBook,BreuerBook}, and this in turn guarantees that the HUP is always satisfied. A brief, self-contained demonstration of the latter is given in the Appendix.
Various ways of addressing this difficulty have been put forward \cite{Lindblad1976393, Diosi1993, Isar1994, Sandulescu1987, Gao1997, Wiseman1998, Gao1998, Ford1999, GaoReply, Vacchini2000}. In this paper we add a term to the BMME, that vanishes in the classical limit, bringing the equation to the Lindblad form and, in particular, ensuring that the HUP is always satisfied \cite{Lindblad1976}. We study the dynamics of the obtained equation, in particular its stationary state.
An important purpose of the current paper is to investigate models of QBM more general than those usually studied in the literature.
Usually, the particle-bath coupling is assumed to be linear in both the coordinates of the particle and the bath oscillators, and for definiteness in the following we will refer to this specific case with the name {\it linear QBM}.
We are here also interested in a more general case, where the coupling is still a linear function of the positions of the oscillators of the bath, but depends nonlinearly on the position of the Brownian particle.
This situation arises when dealing with inhomogeneous environments, in which damping and diffusion vary in space.
An immediate application of this generalization concerns the physical behavior of an impurity embedded in an ultra cold gas. In this case spatial inhomogeneities are due to the presence of trapping potentials and, possibly, stray fields \cite{Shashi2014, Tempere2009}.
Here, we will study in detail the case when the coupling depends quadratically on the position of the test particle, and we will refer to this case with the name {\it quadratic QBM}.
The paper is organized as follows.
In Sec.\ \ref{LinCase} we consider QBM with a linear coupling.
We first introduce the BMME, and briefly discuss the lack of positivity preservation mentioned above. We then add a term to obtain a LME according to the procedure proposed in \cite{Gao1997}, and rewrite it in the Wigner function representation. The Wigner function defines a quasi-probability distribution on the phase space \cite{GardinerBook}. We derive the time-dependent equations for the moments of this distribution, show that they have an exact Gaussian solution, and study in detail its long-time behavior. In particular, we analyze the correlations induced by the environment, which cause a rotation and distortion of the distribution, as well as squeezing effects expressed by the widths and the area of the distribution's effective support.
In Sec.\ \ref{QuadraticCase} we study a non-linear QBM, corresponding to an inhomogeneous environment.
In particular we consider an interaction which is a quadratic function of the position of the Brownian particle.
This model has been studied in \cite{Massignan2015}, by means of a BMME.
We again modify the BMME to obtain an equation in the Lindblad form and we study its stationary solutions in the phase space (Wigner) representation.
For the quadratic QBM, the exact stationary state is no longer Gaussian but a Gaussian approximation can be used in certain regimes. However, when the damping is strong, the Gaussian ansatz does not converge for large times, showing that it is not a good approximation to a stationary state.
\section{Linear QBM}\label{LinCase}
\subsection{The Model}
The QBM model describes the physical behavior of a particle interacting with a thermal bosonic bath of harmonic oscillators. In general the potential of the Brownian particle can be arbitrary,
but we will study the harmonic case only.
The model is described by the Hamiltonian:
\begin{equation}\label{Ham}
\hat{H}=\hat{H}_{S}+\hat{H}_{E}+\hat{H}_{I},
\end{equation}
in which:
\begin{align}
&\hat{H}_S=\frac{\hat{P}^2}{2m}+\frac{m\Omega^2\hat{X}^2}{2},\\\nonumber
&\hat{H}_E=\sum_k\frac{\hat{p}_k^2}{2m_k}+\frac{m_k\omega^2_k\hat{x}_k^2}{2},\\\nonumber
&\hat{H}_I=\sum_k g_k \hat{x}_k \hat{X}
\end{align}
where $m$ is the mass of the Brownian particle, $\Omega$ is the frequency of the harmonic potential trapping it, $m_k$ and $\omega_k$ are the mass and the frequency of the $k^{\rm th}$ oscillator of the environment, and $g_k$ are the bath-particle coupling constants.
In this Section, the interaction term $\hat{H}_I$ depends linearly on the positions of both the Brownian particle and the oscillators of the environment. We refer to this model as a {\it linear QBM}.
The Hamiltonian is the starting point to derive the ME.
We are interested in a BMME, obtained by making two approximations \cite{SchlosshauerBook}.
In the first one, the \textit{Born approximation}, we assume that the influence of the particle on the bath is negligible, so that the two systems remain uncorrelated (i.e. their joint state is a tensor product) at all times (including the initial one).
In the second, the \textit{Markov approximation}, we neglect memory effects, namely we require that the self-correlations created within the environment due to the interaction with the Brownian particle decay over a time scale much shorter than the relaxation time scale of the particle.
Under these hypotheses, one derives from the Hamiltonian Eq.\ (\ref{Ham}) the following ME \cite{BreuerBook, SchlosshauerBook, Massignan2015},
\begin{align}\label{MEQBMLin}
\frac{\partial\hat{\rho}(t)}{\partial t}=&-\frac{i}{\hbar}\left[\hat{H}_S+C_x \hat{X}^2,\hat{\rho}(t)\right]\\
&-\frac{D_x}{\hbar}[\hat{X},[\hat{X},\hat{\rho}(t)]]
-\frac{D_p}{\hbar m\Omega}[\hat{X},[\hat{P},\hat{\rho}(t)]]\nonumber\\
&-\frac{iC_p}{\hbar m\Omega} [\hat{X},\{\hat{P},\hat{\rho}(t)\}].\nonumber
\end{align}
To avoid ambiguities, we wish to stress that, throughout the whole paper, we will be working in the Schr\"odinger picture, where the time-dependence is carried by the state of the system (rather than by the operators), and average values of operators are calculated as usual as $\ave{A}_t\equiv{\rm Tr}[\rho(t) A]$.
The effects of the bath on the motion of the Brownian particle are encoded in the spectral density of the bath. In the following, we will focus on the commonly used \textit{Lorentz-Drude} spectral density:
\begin{equation}\label{LDSD}
J(\omega)=\frac{m\gamma}{\pi}\frac{\omega}{1+\omega^2/\Lambda^2}
\end{equation}
which is linear at low frequencies, and decays as $1/\omega$ beyond the cut-off $\Lambda$, introduced to regularize the theory. With this choice of the spectral density the coefficients of the BMME at a bath's temperature $T$ read:
\begin{align}
\label{CxCoeff}
C_p&=\frac{m\gamma\Omega}{2}\frac{\Lambda^2}{\Omega^2+\Lambda^2}\\
C_x&=-\frac{\Lambda}{\Omega}C_p\\
D_x&=C_p\coth\left(\frac{\hbar\Omega}{2k_B T}\right)\\
D_p&
=\frac{2C_p}{\pi}\left[\frac{\pi k_B T}{\hbar \Lambda}+
z(T,\Lambda,\Omega)\right],
\end{align}
with:
\begin{equation}
z(T,\Lambda,\Omega)=\psi\left(\frac{\hbar\Lambda}{2\pi k_B T}\right)-{\rm Re}\left[\psi\left(\frac{i\hbar\Omega}{2\pi k_BT}\right)\right]
\end{equation}
where
$\psi(x)$ is the DiGamma function \cite{Massignan2015}.
In the following, we will refer to $T$, $\Lambda$, and $\gamma$ as the model's parameters.
The term proportional to $C_x$ in Eq.\ (\ref{MEQBMLin}) leads to a renormalization of the harmonic trapping frequency. Following the usual approach \cite{BreuerBook}, the latter may be cancelled by including in the Hamiltonian the counter-term $\hat{V}_c=-C_x\hat{X}^2$.
The evolution defined by Eq.\ (\ref{MEQBMLin}) does not preserve positivity of the density matrix.
As discussed in detail in, e.g., Ref.\ \cite{Massignan2015, FlemPRE2011}, the lack of positivity leads to violations of the Heisenberg uncertainty principle away from the Caldeira-Leggett limit discussed below. In particular, this prevents the study of the dynamics in the regime of very low temperatures.
In fact, these violations in Eq.\ (\ref{MEQBMLin}) are driven by the logarithmic divergence at low temperatures of $D_{p}$ (which is itself proportional to $\gamma$, i.e. to $g^2_{k}$).
Overcoming this problem is a fundamental step towards a correct description of the dynamics of a Brownian particle. In this section we propose a modified ME for the linear QBM which has the Lindblad form and, consequently, preserves positivity of the density matrix.
It is well known that the ME\ (\ref{MEQBMLin}) cannot be expressed in the Lindblad form \cite{BreuerBook,SchlosshauerBook}. Our equation differs from it by two terms, one of which can be naturally absorbed into the system's Hamiltonian.
Adopting a LME is not the only possible manner to deal with the violations of the Heisenberg uncertainty principle.
From a formal point of view, the ME\ (\ref{MEQBMLin}) is the result of a perturbative expansion to the second order in the strength of the bath-particle coupling (actually, expanding to second order requires weaker assumptions than the Born and Markov ones; the resulting equation may still take into account some non-Markovian effects which vanish in the limit of large times \cite{BreuerBook}).
In \cite{FlemPRE2011} it has been shown that Heisenberg principle violations in the stationary state disappear if one performs a perturbative expansion beyond the second order in the coupling constant.
Obviously, if the exact ME is used, violation of Heisenberg principle cannot occur in any parameter regime.
\subsection{Lindblad Master Equation}
A LME has the form:
\begin{align}\label{LindEq}
\der{\hat{\rho}}{t}=&-\frac{i}{\hbar}\comm{\hat{H}_{S}}{\hat{\rho}}
+\sum_{i,j}\kappa_{ij}
\left[\hat{A}_{i}\hat{\rho}\hat{A}^{\dagger}_{j}-\frac{1}{2}\{\hat{A}^{\dagger}_i\hat{A}_j,\hat{\rho}\}\right],
\end{align}
where $\hat{A}_i$ are called Lindblad operators and $(\kappa_{ij})$ is a positive-definite matrix.
Following the approach proposed in \cite{Gao1997} we will replace the BMME (\ref{MEQBMLin}), which cannot be brought to a Lindblad form, by an equation of the form Eq.\ (\ref{LindEq}) with a single Lindblad operator of the form
\begin{equation}\label{LindbladOP}
\hat{A}_{1}=\alpha\hat{X}+\beta\hat{P},\qquad \textrm{ with } \kappa_{11}=1.
\end{equation}
Substituting this operator into Eq.\ (\ref{LindEq}) we obtain:
\begin{align}
\der{\hat{\rho}}{t}=&-\frac{i}{\hbar}\comm{\hat{H}'_{S}}{\hat{\rho}}-i\frac{\Gamma}{\hbar}\comm{\hat{X}}{\{\hat{P},\rho\}}\label{MELindNew}\\
&-\frac{D_{XP}}{\hbar^2}\comm{\hat{X}}{\comm{\hat{P}}{\hat{\rho}}}-\frac{D_{PP}}{2\hbar^2}\comm{\hat{P}}{\comm{\hat{P}}{\hat{\rho}}}\nonumber\\
&-\frac{D_{XX}}{2\hbar^2}\comm{\hat{X}}{\comm{\hat{X}}{\hat{\rho}}},\nonumber
\end{align}
with:
\begin{equation}\label{ShiftedHam}
\hat{H}'_{S}=\hat{H}_S-\frac{\Gamma}{2}\{\hat{X},\hat{P}\}\equiv\hat{H}_S+\Delta\hat{H}
\end{equation}
and:
\begin{align}
D_{XX}=&\hbar^2|\alpha|^2, &&D_{XP}=\hbar^2 \rm{Re}\left(\alpha^{*}\beta\right),\label{CoefficientsGenerator}\\\nonumber
D_{PP}=&\hbar^2|\beta|^2, &&\Gamma=\hbar \rm{Im}\left(\alpha^{*}\beta\right).
\end{align}
One could obtain the same result employing two Lindblad operators, proportional to $\hat{X}$ and $\hat{P}$ respectively.
Without loss of generality, we may take $\alpha$ to be a positive real number
since multiplying $\hat{A}_1$ by a phase factor does not change Eq.\ (\ref{LindEq}), and we will restrict ourselves to ${\rm Im}\beta > 0$, because, as seen from Eq.\ (\ref{CoefficientsGenerator}), $\alpha{\rm Im}(\beta)$ is the damping coefficient $\Gamma$, which must be positive.
Eq.\ (\ref{MELindNew}) differs from Eq.\ (\ref{MEQBMLin}) just by two extra terms,
involving $D_{PP}$ and $\Delta\hat{H}$. Equating the coefficients of the remaining terms with those of the analogous terms appearing in Eq.\ (\ref{MEQBMLin}),
one finds:
\begin{align}
&D_{XX}=2\hbar D_x, &&D_{XP}=\frac{\hbar D_p}{m\Omega},\\\nonumber
&\Gamma=\frac{C_p}{m\Omega}, &&D_{PP}=\frac{(\hbar\Gamma)^2+D^2_{XP}}{D_{XX}}.
\end{align}
In the Caldeira-Leggett (CL) limit $k_BT\gg\hbar\Lambda
\gg\hbar\Omega$, these reduce to:
\begin{align}
&\Gamma\approx\gamma/2,\\
&D_{XX}\approx2 m\gamma k_B T,\nonumber\\
&D_{XP}\approx-\gamma \frac{k_B T}
{\Lambda},\nonumber\\
&D_{PP}\approx\frac{\gamma k_BT}{2m \Lambda^2}\nonumber
\end{align}
Following \cite{SchlosshauerBook}, since the quantities represented by $P$ and $m\Omega X$ have generally the same order of magnitude, one can argue, as in Eq. (5.56) of \cite{SchlosshauerBook}, that the terms proportional to $D_{XP}$ and $D_{PP}$ are negligible in the CL limit, recovering the structure of the usual CL ME.
The operator $\Delta\hat{H}$ can be absorbed into the unitary part of the dynamics defined by Eq.\ (\ref{MELindNew}), so it can be eliminated by introducing a counter term into the system's Hamiltonian. More generally, we will add to $\hat{H}_S$ a counter term
\begin{equation}
\hat{H}_{C}=(r-1)\Delta\hat{H},
\end{equation}
which depends on a parameter $r\in\mathbb{R}$, leading to the modified Hamiltonian:
\begin{align}
\hat{H}'_{S}=&\hat{H}_{S}-(r\Gamma/2)\{\hat{X},\hat{P}\}\\\nonumber
=&\frac{(\hat{P}-mr \Gamma \hat{X})^2}{2m}+\frac{m(\Omega^2-r^2\Gamma^2)\hat{X}^2}{2}.
\end{align}
The effect of $r$ is twofold: it introduces a gauge transformation which shifts the canonical momentum $\hat{P}$, and it renormalizes the frequency of the harmonic potential.
In the rest of the section we shall study the dynamics defined by equation Eq.\ (\ref{MELindNew}), first for general values of $r$ and then, for the discussion of the stationary state, focusing on the case $r=0$.
We stress that the introduction of a counter term in the Hamiltonian does not affect the Lindblad character of the LME in Eq.\ (\ref{MELindNew}), since it just enters in its unitary part.
\subsection{Solution of the LME}
We are interested in studying the long-time dynamics of the Brownian particle.
In particular, we consider its representation in the phase space, employing the Wigner function representation \cite{GardinerBook}.
In terms of the Wigner function, Eq.\ (\ref{MELindNew}) becomes $\dot{W}=\mathcal{L} W$, with
\begin{align}\label{LMEWF}
\mathcal{L}W=&-\frac{P}{m}\der{W}{X}+m\Omega^2X\der{W}{P}\\
&+\Gamma\left[r\der{}{X}(XW)+(2-r)\der{}{P}(PW)\right]\nonumber\\
&+\frac{1}{2}\left[D_{XX}\dsec{W}{P}+D_{PP}\dsec{W}{X}\right]-D_{XP}\frac{\partial^2 W}{\partial X\partial P}\nonumber.
\end{align}
Equivalently, one can look at the equations for its moments:
\begin{align}
\label{EqMotMomW}\der{\ave{X}_t}{t}&=\frac{\ave{P}_t}{m}-r\Gamma{\ave{X}_t}\\
\der{\ave{P}_t}{t}&=-m\Omega^2 \ave{X}_t-(2-r)\Gamma\ave{P}_t\nonumber\\
\der{\ave{X^2}_t}{t}&=-2r\Gamma\ave{X^2}_t +\frac{2\ave{XP}_t}{m}+D_{PP}\nonumber\\
\der{\ave{XP}_t}{t}&=-m\Omega^2 \ave{X^2}_t -2\Gamma \ave{XP}_t + \frac{\ave{P^2}_t}{m} - D_{XP}\nonumber\\
\der{\ave{P^2}_t}{t}&=-2m\Omega^2 \ave{XP}_t -(4-2r)\Gamma \langle P^2 \rangle_t + D_{XX},\nonumber
\end{align}
where the moments of the Wigner function are calculated as
\begin{equation}
\ave{f(X,P)}_t=\int^{\infty}_{-\infty}dX\int^{\infty}_{-\infty}dP \; f(X,P)W(X,P,t).
\end{equation}
These moments correspond to symmetric ordering of the quantum mechanical operators $\hat X$ and $\hat P$ \cite{SchBook}. In particular, note that the time-dependence is solely contained in the Wigner function, in agreement with the fact that we work in the Schr\"odinger picture.
The solutions for the first moments are:
\begin{align}
\ave{X}_t= & e^{-\Gamma t }\left[X_{0}\cos(\beta_{r} t)+x^0_r\sin(\beta_{r} t)\right],\\ \nonumber
\ave{P}_t= & e^{-\Gamma t }\left[P_{0}\cos(\beta_{r} t)-p^0_r \sin(\beta_{r} t)\right],
\end{align}
where:
\begin{align}
&x^0_r=\frac{m\Gamma X_{0}(1-r)+P_{0}}{m\beta_r}\\
&p^0_r=\frac{\Gamma P_{0}(1-r)+m X_{0}\Omega^2}{\beta_r}\nonumber
\end{align}
with:
\begin{equation}
X_0\equiv\ave{X}_0,\quad P_0\equiv\ave{P}_0,
\end{equation}
and:
\begin{equation}
\beta_r\equiv\sqrt{\Omega^2-\Gamma^2(r-1)^2}.
\end{equation}
Similar solutions have been presented in \cite{Kumar2009, Sandulescu1987, Isar1994}.
Eqs.\ (\ref{EqMotMomW}) may alternatively be written in terms of the kinetic momentum $\ave{\tilde{P}}_t=\ave{P}_t-m r \Gamma\ave{X}_t$:
\begin{align}
&\der{\ave{X}_t}{t}=\frac{\ave{\tilde P}_t}{m},\label{EqXPtilde}\\
&\der{\ave{\tilde P}_t}{t} = -m\left[\Omega^2 -r(r-2)\Gamma^2\right]\ave{X}_t -2\Gamma\ave{\tilde P}_t,\nonumber
\end{align}
or equivalently gathered in the compact form
\begin{equation}
\dsec{\ave{X}_t}{t}+2\Gamma\der{\ave{X}_t}{t}+\left[\Omega^2 -r(r-2)\Gamma^2\right]\ave{X}_t=0.
\end{equation}
which, of course, can be derived directly from the equations Eq.\ (\ref{EqMotMomW}).
For both $r=0$ and $r=2$ one obtains a damped oscillator with the original frequency of the harmonic trap, $\Omega$.
For other values of $r$ the frequency is renormalized, with the maximal renormalization corresponding to $r=1$.
In Eqs.\ (\ref{EqMotMomW}) we see that $r$ introduces apparent damping in the position, as already noted in \cite{Wiseman1998}.
Because of this, in the following we will set $r=0$.
The extra term proportional to $D_{PP}$, not present in the starting BMME, appears only in the equation for $\dot{\ave{X^2}}$, without affecting the other equations, and in particular those for the first moments, so that it may be interpreted as a \textit{position diffusion coefficient}.
We wish now to focus on the stationary solution of Eq.\ \eqref{LMEWF}. The latter may be found by means of the following Gaussian ansatz:
\begin{equation}\label{RGWF}
W_{ST}=\zeta\exp\left[\frac{1}{2(\rho^2-1)}\left(\frac{X^2}{\sigma^2_X}+\frac{P^2}{\sigma^2_P}+\frac{2\rho XP}{\sigma_X\sigma_P}\right)\right],
\end{equation}
which is normalized to one taking:
\begin{equation}
\zeta\equiv\frac{1}{2\pi\sigma_X\sigma_P\sqrt{1-\rho^2}},\quad |\rho|\leq1,
\end{equation}
with:
\begin{equation}
\sigma_X=\sqrt{\ave{X^2}},\quad\sigma_P=\sqrt{\ave{P^2}},\quad\rho=-\frac{\ave{XP}}{\sigma_X\sigma_P},
\end{equation}
and, in the remainder of this Section, the variances are computed using the time-independent Gaussian Ansatz in Eq.\ \eqref{RGWF} \cite{Weedbrook2012}.
Inserting the Gaussian ansatz in Eq.\ (\ref{RGWF}) into Eq.\ (\ref{LMEWF}) we find:
\begin{align}
&\sigma^2_{X}=\frac{D_{XX}-4m\Gamma D_{XP}+m^2(4\Gamma^2+\Omega^2)D_{PP}}{4m^2\Gamma\Omega^2}\label{PosUnc}\\\nonumber
&\sigma^2_{P}=\frac{D_{XX}+m^2\Omega^2D_{PP}}{4\Gamma}\\\nonumbe
&\sigma_{P}\sigma_{X}\rho=mD_{PP}/2
\end{align}
We introduce the adimensional variables:
\begin{equation}
\delta_x=\sqrt{\frac{2m\Omega \sigma^2_X}{\hbar}},\quad\delta_p=\sqrt{\frac{2\sigma^2_P}{m\Omega\hbar}}
\end{equation}
With this parametrization, the Heisenberg inequality $\sigma_X \sigma_P \geq \hbar/2$ reads $\delta_x\delta_p\geq1$.
The Lindbladian character of Eq.\ (\ref{LMEWF}) guarantees that the second moments will satisfy the Heisenberg relation at all times. We furthermore note that the term with coefficient $D_{PP}$, i.e. the extra term induced by the Lindblad form of the ME, leads to a correlation between the two canonical variables.
Geometrically, this correlation can be interpreted as a rotation of the stationary solution in the phase space, see the black sketches in Fig.\ \ref{densityPlotAngle}.
In the CL limit, the term with the coefficient $D_{PP}$ is negligible, and the solution is an ellipse with its axes parallel to the canonical ones, reproducing the well-known results.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_1.pdf}
\caption{\label{densityPlotAngle} Plot of the angle $\theta/\pi$ at $\gamma/\Omega=0.8$.
This angle is represented in the ellipse at the bottom of the picture. Here, the orange-solid (green-dashed) line represents the minor (major) axis of the Wigner function, i.e., that related to $\delta_l$ ($\delta_L$). The axes $X$ and $P$ are those of the phase space.
}
\end{center}
\end{figure}
To analyze the properties of the stationary state in the phase space, we consider the variances of the major and minor axes of the Wigner function. These axes are defined as the eigenvectors of the covariance matrix:
\begin{equation}
\text{cov}(X,P)=\left(\begin{array}{cc}
\delta^2_x&-\rho\delta_x\delta_p\\
-\rho\delta_x\delta_p&\delta^2_p\\
\end{array}\right)
\end{equation}
The smaller and larger eigenvalues of this matrix, $\delta_l$ and $\delta_L$, are given respectively by:
\begin{equation}
\delta^2_{l,L}=\frac{1}{2}\left(\delta^2_x+\delta^2_p\mp\sqrt{\left(\delta^2_x-\delta^2_p\right)^2+4\delta^2_x\delta^2_p\rho^2}\right)
\end{equation}
We now aim to quantify such a rotation, calculating the angle $\theta$ between the major axis of the Wigner function (i.e. the eigenvector corresponding to $\delta_L$), and the $X$-axis of the phase space.
In Fig.\ \ref{densityPlotAngle} we present the behavior of $\theta$ as function of $T$ and $\Lambda$, at fixed $\gamma$.
At high $\Lambda$ the major axis aligns approximately with the $P$-axis of the phase space ($\theta = \pi/2$),
while at low $\Lambda$, it is close to the $X$-axis ($\theta=\pi$),
in agreement with the behavior of the BMME discussed in \cite{Massignan2015}, where $\ave{XP}$ was identically zero.
On the other hand, at low temperatures the Wigner function associated to the stationary solution of the LME may be significantly rotated with respect to the axes of the phase space.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig_2.pdf}
\caption{Eccentricity of the Wigner function introduced in Eq.\ (\ref{RGWF}), at $\gamma/\Omega = 0.8$.
The red dashed line represents the values of $T$ and $\Lambda$ yielding $\delta^2_l=1$, and we have genuine squeezing below it.
}
\label{sqWF}
\end{figure}
In \cite{Massignan2015} it has been shown that, going to low temperature, the position of the Brownian particle governed by the BMME experiences \textit{genuine squeezing} along $x$ in the Wigner function representation, i.e. $\delta_x<1$. Similar squeezing effects are pointed out in \cite{Maniscalco2014}, by studying the numerical solution of the exact ME. In the case of the LME, it was checked numerically that $\delta_x$ introduced in Eq.\ (\ref{PosUnc}) is always bigger than one. However, the minor axis of the ellipse describing the Wigner function can display genuine squeezing.
To quantify the degree of squeezing of the Wigner function,
Fig.\ \ref{sqWF} shows the values of eccentricity defined as
\begin{equation}
\eta=\sqrt{1-(\delta_l/\delta_L)^2},
\end{equation}
computed for different values of temperature $T$ and UV-cutoff $\Lambda$.
The eccentricity is largest at low temperatures. In particular, below the red dashed line, we find an area where $\delta_l<1$, corresponding to genuine squeezing along the minor axis of the Wigner Function, while in the CL limit the eccentricity $\eta$ approaches zero, and we obtain a Wigner function with circular symmetry.
In Fig.\ \ref{minimalSqueezing} we present the minimal value of $\delta^2_l$ obtained by choosing the appropriate (low) temperature.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig_3.pdf}
\caption{Minimum value of $\delta^2_l$ over all temperatures, as a function of the cut-off frequency, at several values of the damping constant.}
\label{minimalSqueezing}
\end{figure}
This picture highlights the range of values of $\Lambda$ and $\gamma$ where genuine squeezing occurs.
We find that the eccentricity is an increasing function of the damping constant, i.e. squeezing becomes more pronounced as $\gamma$ grows. In particular, at least $\gamma/\Omega>0.5$ is needed to obtain $\delta_l<1$.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_4.pdf}
\caption{\label{coolPlot} Cooling parameter $\chi$ introduced in Eq.\ (\ref{heatingCondition}), plotted for $\gamma/\Omega=0.8$.
The system exhibits cooling to the right of the solid line, and heating to its left.
For comparison, the dashed line represents the cooling/heating boundary obtained with the BMME (\ref{MEQBMLin}), which is independent of $\gamma$.
}
\end{center}
\end{figure}
We may say that the Brownian particle experiences an effective heating if the effective phase space area
is larger than the one occupied by a quantum Gibbs-Boltzmann distribution at the same temperature.
We thus define the system to be cooled if\footnote{For the Gibbs-Boltzmann distribution we have $\ave{X^2}_{GB}\ave{P^2}_{GB}\sim\coth^2{(\hbar\Omega/2k_BT)}$. So the denominator of Eq.\ (\ref{heatingCondition}) provides an information regarding the area of the Gibbs-Boltzmann distribution.}:
\begin{equation}
\chi=\frac{\delta_l\delta_L}{\coth\left(\frac{\hbar\Omega}{2k_BT}\right)}<1,
\label{heatingCondition}
\end{equation}
and heated otherwise.
The degree of heating/cooling $\chi$ is shown in Fig.\ \ref{coolPlot}.
In Fig.\ \ref{minimalCooling} we present the minimal value achieved by $\chi$ as the temperature is varied.
We note that to obtain small values of $\chi$ one needs to choose large values of both $\Lambda$ and $\gamma$.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_5.pdf}
\caption{\label{minimalCooling}
Minimum value of the cooling parameter $\chi$ over all temperatures, as a function of the cut-off frequency, at several values of the damping constant.
}
\end{center}
\end{figure}
There is a difference between the configuration of the cooling areas arising in the Lindblad dynamics studied here, and the ones produced by the BMME (\ref{MEQBMLin}) studied in \cite{Massignan2015}.
In the latter, the cooling/heating boundary coincides with the line defined by $\delta_x=\delta_p$, and this condition does not depend on $\gamma$, while
in the present Lindblad model, the location of the boundary varies with $\gamma$.
However, the boundary calculated within the LME converges to the BMME one in the $\gamma\rightarrow0$ limit. Moreover, the LME discussed here displays heating at very low temperatures.
In Figs.\ \ref{minimalSqueezing} and \ref{minimalCooling} we have not extended the range of values of the damping constant beyond $\gamma=1$.
In fact, the expressions for the coefficients of the equation Eq.\ (\ref{MELindNew}) have been obtained by comparing it with the equation Eq.\ (\ref{MEQBMLin}). The latter is perturbative to second order in the strength of the coupling between the Brownian particle and the environment.
The square of the coupling constant is proportional to the damping coefficient, so the validity of the perturbative expansion fails for $\gamma$ large.
In particular, in the case of QBM this perturbative expansion holds for $\gamma\lesssim\Omega$ \cite{BreuerBook,Haake1985}.
\subsubsection*{Low Temperature Regime}
We consider here in detail the stationary state in the low temperature regime $k_BT<\hbar\Omega$.
Such a study was impossible in \cite{Massignan2015} because solutions violated the Heisenberg principle there.
Here, the Lindblad form of the ME in Eq.\ (\ref{MELindNew}) ensures the positivity of the density matrix at all times, so no violations of the Heisenberg principle occur.
In the discussion above, we noticed that the time-dependent equations of motion of the LME admit as an exact solution a Gaussian with non-zero correlations between the two canonical variables $X$ and $P$. In the stationary state, in particular, one finds $\ave{XP}=-mD_{PP}/2\neq0$.
This is a novelty in comparison with the stationary solution of the BMME in Eq.\ (\ref{MEQBMLin}), which shows no correlations between $X$ and $P$.
In the range of $\Lambda$ explored in Fig.\ \ref{densityPlotAngle}, the correlation between $X$ and $P$ becomes noticeable for $k_BT\lesssim0.5\hbar\Omega$. So, an important feature of the stationary solution of our LME at low temperature is that its major axis is rotated with respect to those of the phase space.
In Fig.\ \ref{sqWF} we analyze the eccentricity of the stationary state.
We point out that as the temperature decreases, the distribution becomes increasingly more squeezed.
In particular, at low temperature we find a region displaying genuine squeezing of the probability distribution in the direction of $l$.
In Fig.\ \ref{coolPlot} we also note the presence of a cooling area in the low temperature regime.
Nevertheless, in the zero-temperature limit the stationary state shows again heating.
The zero-temperature limit of the Lindblad model deserves special attention, as the two limits $T\rightarrow 0$ and $\gamma\rightarrow 0$ do not commute.
Taking first the zero-coupling and then the zero-temperature limit, one simply finds $\delta_x=\delta_p$ (in agreement with the general result for a free harmonic oscillator), but no further information on their specific value.
If instead one takes first $T\rightarrow 0$ and then $\gamma\rightarrow0$, one finds $\delta_x=\delta_p$ and the additional condition:
\begin{equation}\label{HeisenbergThreshold}
\delta_x\delta_p=\delta_l\delta_L=\frac{5}{4}+\frac{\left[\log(\Lambda/\Omega)\right]^2}{\pi^2}>1,
\end{equation}
indicating that for the Lindblad model the Heisenberg inequality is not saturated in the limit when the particle becomes free.
This is in contrast with the behavior of the non-Lindblad BMME (\ref{MEQBMLin}), for which, in this limit, we have $\delta_x\delta_p=1$.
Summarizing, the effect of $D_{PP}$ is to introduce extra heating at low temperatures and couplings, manifested by a small constant, and a weak logarithmic dependence on the UV cut-off $\Lambda$.
\section{Quadratic QBM}\label{QuadraticCase}
\subsection{The Hamiltonian and the Lindblad ME}
In this section we consider the quadratic QBM, whose coupling is still linear in the positions of the oscillators of the bath, but is quadratic in the position of the Brownian particle:
\begin{equation}\label{QuadraticCoupling}
\hat{H}_I=\sum_k \frac{g_k}{R} \hat{x}_k\hat{X}^2.
\end{equation}
Here $R$ is a characteristic length related to the motion of the Brownian particle and we set it to be
$ R = \sqrt{\hbar/m \Omega}.$ The interaction term in Eq.\ (\ref{QuadraticCoupling}) describes an interaction of the particle with an inhomogeneous environment, giving rise to position-dependent damping and diffusion.
A concrete example where we may encounter this kind of nonlinearity is the model of an impurity in a Bose-Einstein condensate.
In \cite{Shashi2014, Tempere2009} it has been shown that the dynamics of such a system can be described by the \textit{Fr\"ohlich Hamiltonian}.
In an inhomogeneous gas, i.e. a gas with a spatially dependent density profile, this Hamiltonian differs from the QBM one due to the nonlinear dependence of the interaction term on the position of the impurity.
When we consider, for instance, a Thomas-Fermi density profile, i.e. a density profile varying quadratically with the position, the interaction Hamiltonian is an even function of the position.
Here, the coupling in Eq.\ (\ref{QuadraticCoupling}) provides the first-order correction to the zero-order term in the expansion of the interaction between the impurity and the bath.
In short, QBM with a quadratic coupling is not just a mathematical exercise, but opens modeling possibilities in new contexts. In Appendix F of Ref.\ \cite{Massignan2015} it has been shown in detail that the Hamiltonian of an impurity in a BEC can be expressed in the form of that of QBM with a generic coupling.
The dynamics induced by the interaction term in Eq.\ (\ref{QuadraticCoupling}) has already been discussed in detail in \cite{Massignan2015}. There, the ME for the Brownian particle has been derived, in the BM approximations, for a Lorentz-Drude spectral density. Nevertheless, this ME is not in a Lindblad form, nor is exact.
Accordingly, the stationary solution is not defined for some values of the model's parameters because of violations of the Heisenberg uncertainty principle at low temperatures.
In this Section, we aim to find a LME as similar as possible to that derived in \cite{Massignan2015}.
Just like in the case of linear QBM, we expect it to differ from the BMME by some extra terms. To achieve this goal we consider a single Lindblad operator:
\begin{equation}\label{LindOpQuadQBM}
\hat{A}_1=\mu\hat{X}^2+\nu\{\hat{X},\hat{P}\}+\epsilon\hat{P}^2
\end{equation}
where $\mu$, $\nu$ and $\epsilon$ are nonzero complex numbers.
Substituting it into Eq.\ (\ref{LindEq}) we obtain:
\begin{widetext}
\begin{align}\label{LMEwithquadraticcoupling}
\der{\hat{\rho}}{t}=&-\frac{i}{\hbar}\comm{\hat{H}_{S}+\Delta\hat{H}_{2}}{\hat{\rho}}-\frac{D_{\mu}}{2\hbar^2}\comm{\hat{X}^2}{\comm{\hat{X}^2}{\hat{\rho}}}
-\frac{D_{\nu}}{2\hbar^2}\comm{\{\hat{X},\hat{P}\}}{\comm{\{\hat{X},\hat{P}\}}{\hat{\rho}}}
-\frac{D_{\epsilon}}{2\hbar^2}\comm{\hat{P}^2}{\comm{\hat{P}^2}{\hat{\rho}}}\\
&-\frac{D_{\mu\nu}}{\hbar^2}\comm{\hat{X}^2}{\comm{\{\hat{X},\hat{P}\}}{\hat{\rho}}}
-\frac{D_{\mu\epsilon}}{\hbar^2}\comm{\hat{X}^2}{\comm{\hat{P}^2}{\hat{\rho}}}
-\frac{D_{\epsilon\nu}}{\hbar^2}\comm{\hat{P}^2}{\comm{\{\hat{X},\hat{P}\}}{\hat{\rho}}}\nonumber\\
&-i\frac{C_{\mu\nu}}{\hbar}\comm{\hat{X}^2}{\{\{\hat{X},\hat{P}\},\hat{\rho}\}}
-i\frac{C_{\mu\epsilon}}{\hbar}\comm{\hat{X}^2}{\{\hat{P}^2,\hat{\rho}\}}
-i\frac{C_{\epsilon\nu}}{\hbar}\comm{\hat{P}^2}{\{\{\hat{X},\hat{P}\},\hat{\rho}\}},\nonumber
\end{align}
\end{widetext}
where:
\begin{equation}
\frac{D_{\mu}}{\hbar^2}\equiv|\mu^2|,\quad\frac{D_{\mu\nu}}{\hbar^2}\equiv {\rm Re}(\mu^*\nu),\quad\frac{C_{\mu\nu}}{\hbar}\equiv {\rm Im}(\mu^*\nu),
\end{equation}
and similarly for the other combinations of indices.
We could have obtained the same result by means of three Lindblad operators (rather than a single one), each proportional to one of the terms appearing on the right-hand side of Eq.\ (\ref{LindOpQuadQBM}).
Similarly to Sec.\ \ref{LinCase}, there is a term which appears in the unitary part of the ME:
\begin{align}
\Delta\hat{H}_2&=2D_{\mu\nu}\hat{X}^2-2D_{\epsilon\nu}\hat{P}^2+2D_{\mu\epsilon}\{\hat{X},\hat{P}\}\\
&-\frac{1}{2}C_{\mu\nu}\{\{\hat{X},\hat{P}\},\hat{X}^2\}-\frac{1}{2}C_{\mu\epsilon}\{\hat{P}^2,\hat{X}^2\}\nonumber\\
&+\frac{1}{2}C_{\epsilon\nu}\{\{\hat{X},\hat{P}\},\hat{P}^2\}.\nonumber
\end{align}
We eliminate it by introducing appropriate counter terms in the Hamiltonian.
The ME in Eq.\ (\ref{LMEwithquadraticcoupling}) is in a Lindblad form.
Proceeding as in Sec.\ \ref{LinCase}, equating the coefficients on the right hand side of Eq.\ (\ref{LMEwithquadraticcoupling})
to the corresponding ones in the BMME for quadratic QBM derived in \cite{Massignan2015}, we obtain:
\begin{align}
&D_{\mu\epsilon}=\frac{ D_{pp}}{m\Omega},\quad D_{\mu\nu}=D_{xp},\\
&C_{\mu\epsilon}=\frac{C_{pp}}{\hbar m\Omega},\quad C_{\mu\nu}=\frac{C_{xp}}{\hbar},\nonumber
\end{align}
and $D_{\mu}=2 m \Omega D_{xx}$.
The remaining coefficients are then uniquely determined as:
\begin{align}
&D_{\epsilon\nu}=\frac{1}{D_{\mu}}\left[D_{\mu\nu}D_{\mu\epsilon}+\hbar^2C_{\mu\nu}C_{\mu\epsilon}\right],\\\nonumber
&C_{\epsilon\nu}=\frac{1}{D_{\mu}}\left[C_{\mu\nu}D_{\mu\epsilon}-D_{\mu\nu}C_{\mu\epsilon}\right],\\\nonumber
&D_{\epsilon}=\frac{1}{D_{\mu}}\left[D^2_{\mu\epsilon}+\left(\hbar C_{\mu\epsilon}\right)^2\right],\\\nonumber
&D_{\nu}=\frac{1}{D_{\mu}}\left[D^2_{\mu\nu}+\left(\hbar C_{\mu\nu}\right)^2\right].
\end{align}
It is easy to check that in the CL limit $k_BT\gg\hbar\Lambda \gg\hbar\Omega$, the coefficients of all extra terms vanish,
and Eq.\ (\ref{LMEwithquadraticcoupling}) recovers the structure of the BMME introduced in Ref.\ \cite{Massignan2015}. \\
\subsection{Stationary State of the Quadratic QBM}\label{sec:StationaryStateQuadratic}
We turn now to the study of the stationary state of the Brownian particle in the case of quadratic coupling.
To this end we express the LME in Eq.\ (\ref{LMEwithquadraticcoupling}) in terms of the Wigner function $W$,
and obtain an equation of the form $\dot{W}=\mathcal{L} W$, with:
\begin{widetext}
\begin{align}\label{Wigner-quadratic}
\mathcal{L} =&-\frac{\partial_X
P}{m}+m\Omega^2\partial_P X
+2D_{\mu} \partial_P^2 X^2+2D_{\nu}\left(\partial_P P-\partial_X X\right)^2+2D_{\epsilon} \partial_X^2 P^2\\\nonumber
&+4D_{\mu\nu}(\partial_P^2 XP-\partial_P\partial_X X^2 +\partial_P X)-4 D_{\mu\epsilon} (\partial_X X -1) \partial_P P {-4 D_{\epsilon\nu}P\partial_X\left(\partial_P P-\partial_X X\right)}\\\nonumber
&+8C_{\mu\nu}\left[\partial_P P X^{2} +\frac{\hbar^2}{4}\partial_P^{2}(\partial_X X-1)\right]+C_{\mu\epsilon}\Big[4\partial_P XP^{2}-\hbar^2\partial_P\partial_X^{2}X+2\hbar^2\partial_P\partial_X\Big]\nonumber\\
&-2C_{\epsilon\nu}P\partial_X\left(4XP+\hbar^2\partial_P\partial_X\right). \nonumber
\end{align}
\end{widetext}
We now find the stationary solution of the above equation. In this case the Gaussian ansatz in Eq.\ (\ref{RGWF}) may at best provide an approximate solution, in contrast with the case of the linear QBM, since the system of equations for the second moments is not closed. We approximate higher-order moments by their Wick expressions in terms of second moments (which would be exact in a Gaussian case), obtaining the following closed, nonlinear system of equations in the variables $\delta_x$, $\delta_p$ and $\rho$:
\begin{widetext}
\begin{align}
\label{eqdelta1}\frac{1}{2}\der{\delta^2_{x}}{t}&=4m\hbar\Omega C_{\epsilon\nu}[1+\delta^2_x\delta^2_p(1+2\rho^2)]+2m^2\Omega^2D_{\epsilon}\delta^2_p+4D_{\nu}\delta^2_x
-\Omega\delta_x\delta_p\rho\\
\label{eqdelta2}\frac{1}{2}\der{\delta^2_{p}}{t}&=\frac{2D_{\mu}}{m^2\Omega^2}\delta^2_x-\frac{4\hbar}{m\Omega}C_{\mu\nu}+6\hbar C_{\mu\epsilon}\delta_x\delta^3_p\rho+\Omega\delta_x\delta_p\rho
+4\delta^2_p\left[D_\nu-D_{\mu\epsilon}-\frac{\hbar C_{\mu\nu}}{m\Omega}\left(1+2\rho^2\right)\delta^2_x\right],
\end{align}
and:
\begin{multline}
\label{eqdelta3}-\frac{1}{2}\der{(\delta_x\delta_p\rho)}{t
=4\hbar C_{\mu\epsilon}+\Omega\delta^2_p-8m\Omega D_{\epsilon\nu}\delta^2_p+\frac{12\hbar}{m\Omega}C_{\mu\nu}\delta_p\delta^3_x\rho\\
+\left(8D_{\mu\epsilon}-12m\hbar\Omega C_{\epsilon\nu}\delta^2_p\right)\delta_x\delta_p\rho
-\left[\Omega+8\frac{D_{\mu\nu}}{m\Omega}+2\hbar\left(1+2\rho^2\right)C_{\mu\epsilon}\delta^2_p\right]\delta^2_x.
\end{multline}
\end{widetext}
This system of equations could admit more than one stationary solution,
so we have to study the proper one.
We choose the solution that coincides with that obtained with the non-Lindblad dynamics in the CL limit, since in this limit the coefficients of the extra terms of the LME in Eq.\ (\ref{LMEwithquadraticcoupling}) vanish.
In \cite{Massignan2015} the stationary state in the case of the non-Lindblad dynamics has been studied in detail, and the variances have been calculated analytically.
Similarly to the linear QBM studied in the previous section, we characterize the stationary state in terms of the variances of the Wigner function, and define the eccentricity, the cooling parameter, and the angle between the major axis and the X axis of the phase space as before.
These quantities are shown in Figs.\ \ref{Eccentricity01}, \ref{Cooling01}, and \ref{Angle01}, as functions of $\Lambda$ and $T$, when $\gamma/\Omega=0.1$.
In Fig.\ \ref{Eccentricity01} we point out that the eccentricity tends to zero in the CL limit, while it increases away from it. This behavior is similar to that found for the linear QBM.
We found that for $\gamma/\Omega\leq0.1$ the Brownian particle experiences neither cooling nor genuine squeezing.
In contrast to the linear case, we do not find a noticeable rotation at low temperature in the quadratic one. We would expect to observe this at larger values of $\gamma$, as in the case of linear coupling.
However, for larger values of the damping constant the many stationary solutions of the system of Eqs.\ (\ref{eqdelta1}-\ref{eqdelta3}) cross, and therefore it is not straightforward to determine the stationary solution of
\eqref{Wigner-quadratic} that coincides with the one obtained in the CL limit.
Moreover, for larger values of $\gamma$ the Gaussian ansatz given in Eq.\ (\ref{RGWF}) may fail to approximate any stationary states.
To show this point, in Fig.\ \ref{nostazstate} we plotted the time dependence of $\delta^2_x$ for several values of $\gamma$, at fixed values of $T$ and $\Lambda$.
Above a certain value of $\gamma$, the position variance does not converge to a stationary value. This suggests that in these cases the Gaussian solution of Eq.\ \eqref{Wigner-quadratic} is not stationary.
Fig.\ \ref{nostazstate} is plotted for the initial conditions $\delta^2_x=\delta^2_p=1$, corresponding to the case when the harmonic oscillator is in its ground state.
The choice of the initial conditions is not crucial, as we observe a very similar behavior with quite different initial conditions.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_6.pdf}
\caption{\label{Eccentricity01} Eccentricity $\eta$ of the Wigner function at $\gamma/\Omega=0.1$, for quadratic coupling.
}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_7.pdf}
\caption{\label{Cooling01} Cooling parameter $\chi$ for quadratic coupling, at $\gamma/\Omega=0.1$.
}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_8.pdf}
\caption{\label{Angle01} Angle $\theta/\pi$ between the major axis of the Wigner function, and the X axis of the phase space at $\gamma/\Omega=0.1$, for quadratic coupling.
}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_9.pdf}
\caption{\label{nostazstate} Time dependence of $\delta^2_x$ for several values of $\gamma$, at $\Lambda/\Omega=16$ and $k_B T/\hbar\Omega=4$. The thin solid lines represent the stationary value of $\delta^2_x$ in the state, namely the stationary solution of Eqs.\ (\ref{eqdelta1}-\ref{eqdelta3}) for such a quantity.
}
\end{center}
\end{figure}
We conclude this Section pointing out that, although in Eqs.\ (\ref{eqdelta1}-\ref{eqdelta3}) we performed the Gaussian approximation at the level of the equations for the moments, it is possible to obtain exactly the same result applying the approximation directly on the original LME in Eq.\ (\ref{LMEwithquadraticcoupling}), or on that LME expressed in terms of the Wigner function,
Eq.\ (\ref{Wigner-quadratic}).
In Appendix \ref{appendixGauss} we show, by a very general analytical demonstration, that the Gaussian approximation applied to the original LME yields again a ME of the Lindblad form, guaranteeing therefore that the approximated solutions will preserve the HUP at all times.
We provide further numerical evidence of this fact in Fig.\ \ref{Heisenberg01}, where we plot the product of the two uncertainties $\delta_x$ and $\delta_p$ resulting by Eqs.\ (\ref{eqdelta1}-\ref{eqdelta3}), on which the Gaussian approximation has been carried out. As may be noticed in the figure, the approximated equations do not produce any violation of the HUP.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig_10.pdf}
\caption{\label{Heisenberg01}
Plot of the product $\delta_x\delta_p$ at $\gamma/\Omega=0.1$, for quadratic coupling. This quantity is always larger than 1, in accord with the HUP.
}
\end{center}
\end{figure}
\section{Conclusions and Outlook}
We studied a modification of the QBM model, focusing on the description of the stationary state of the Brownian particle in the phase space, using the Wigner function representation.
To perform this analysis we considered a ME of the Lindblad form, which ensures the positivity of the density matrix at all times.
In this way we got rid of the Heisenberg principle violations discussed in \cite{Massignan2015}, which prohibited the study of the dynamics in the low temperature regime.
In Sec.\ \ref{LinCase} we dealt with QBM with a linear coupling. In this case, the stationary state can be represented exactly by a Gaussian Wigner function.
We put particular emphasis on the analysis of its properties in the low temperature regime, where its properties are most interesting.
At low temperature we found that the Brownian particle exhibits genuine squeezing of the probability distribution.
An important feature of the stationary state in this regime is its rotation in the phase space, a direct consequence of the extra terms introduced to obtain a Lindblad form for the equation.
Another important effect experienced by the stationary state can be quantified by the degree of cooling, expressing the ratio of the area of the effective support of the Wigner function to that of the Gibbs-Boltzmann distribution at the same temperature.
A concrete physical system where cooling and squeezing can be encountered is suggested in \cite{Maniscalco2004}.
In Sec.\ \ref{QuadraticCase} we performed the same analysis for QBM with a coupling which is quadratic in the coordinates of the test particle. Importantly, we found that there exists a critical value of the damping constant over which the Gaussian ansatz fails to approximate any stationary solutions.
Our procedure of adding extra terms to the BMME derived in \cite{Massignan2015}, so that the resulting equation is in a Lindblad form, is just one of the ways to obtain a Markovian dissipative LME. Other approaches have been presented, e.g., in Refs.\ \cite{Taj2008, Pepe2012}.
Moreover, for Gaussian dynamics an exact (non-Markovian) closed master equation with time dependent coefficients can be derived \cite{Ferialdi2014,Ferialdi2016March,Carlesso16}.
In a forthcoming work we plan to derive a LME describing QBM with a general class of couplings, and study its various limiting behaviors, in particular the small mass limit of the Brownian particle.
The method which we used to treat the LME in this manuscript is not the only suitable one. Another possible manner to solve this kind of equations, and in particular to characterize the stationary solution has been presented in Ref.\ \cite{Englert1994}. The core of this procedure is turning LMEs into partial first-order differential equations for a phase-space distribution (PSD) which generalizes well-known ones such as the Wigner function. The main point lies in removing the evolution generated by the free Hamiltonian by including it in the interaction representation. Accordingly, the time dependence of the PSD originates solely from the interaction term.
Although the interaction picture adopted in \cite{Englert1994} could be used in the context we are treating, its usefulness is not necessarily guaranteed. In fact, the interaction picture represents a suitable tool when the free part of the Hamiltonian describes a dynamics much faster than that induced by the interaction term. In general this is not the case for the Brownian motion of a trapped particle, where the time scales related to both processes can approach the same order of magnitude.
On the other hand, employing this method looks like a very interesting task, which maybe can allow us to go beyond the Gaussian approximation underlying Sec. (\ref{QuadraticCase}). This task, however, lies outside of the focus of the current paper. We thus reserve it for future works.
There are other methods to correct the Heisenberg principle violations
highlighted in \cite{Massignan2015}. The BMME for the quadratic QBM derived in \cite{Massignan2015} is based on a second order perturbative ME in the bath-particle coupling constant. Going to higher orders permits one, in principle, to get rid of the violations of the Heisenberg principle. This task can be pursued by means of the time-convolutionless method presented in \cite{BreuerBook}. An advantage of this approach is that the resulting ME incorporates non-Markovian effects.
Nevertheless, since it arises from a perturbative expansion, it does not allow to investigate the strong coupling regime $\gamma>\Omega$, where cooling and squeezing effects are expected to be stronger.
The ideas presented here can be used to investigate the physical behavior of an impurity in a BEC by open quantum systems techniques.
In this framework the impurity plays the role of the Brownian particle, while the set of the BEC Bogoliubov modes represents the environment.
The linear QBM provides a useful tool to study the dynamics of the impurity in a uniform medium, while the QBM with a generic coupling may be used to investigate impurities immersed in an inhomogeneous background, such as the one provided by an harmonic trap.
In conclusion, in Appendix \ref{appendixGauss} we proved that the Gaussian approximation preserves the Lindblad form of a ME, and so it does not yield any HUP violation, regardless of whether it is performed on the equations for the moments or directly on the LME.
We developed this demonstration starting from a LME related to a Lindblad operator which is just quadratic in the creation and annihilation operators, because it is enough to cover the situation analysed in Sec.\ \ref{QuadraticCase}.
In general, one could extend the proof to LMEs associated to Lindblad operators containing $n^{\rm th}$ powers of creation and annihilation operators.
This, as far as we know, has never been shown and constitutes an interesting motivation for future projects. Also, a generalization of this proof to LMEs for fermionic systems \cite{Kraus2009} is apparently possible and interesting.
\acknowledgments This work has been funded by a scholarship from the Programa M\'{a}sters d'Excel-l\'{e}ncia of the Fundaci\'{o} Catalunya-La Pedrera, ERC Advanced Grant OSYRIS, EU IP SIQS, EU PRO QUIC,
EU STREP EQuaM (FP7/2007-2013, No. 323714), Fundaci\'o Cellex, the Spanish MINECO (SEVERO OCHOA GRANT SEV-2015-0522 and FOQUS FIS2013-46768), the Generalitat de Catalunya (SGR 874).
P.M. is funded by a ``Ram\'on y Cajal" fellowship. S.H.L. and J.W. were partially supported by the NSF grant MS 131271. They are grateful to L. Torner and ICFO for hospitality in the Summer of 2015.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How do I loop through usersettings in user.config C# First of all I am an idiot who cant format questions appearantly so im gonna have to post this all in one codeblock.
I have settings in my program that are programmatically added.
These are added like this:
SettingsProperty SP = new SettingsProperty("LibImage" + AmountOfImages);
SP.PropertyType = typeof(string);
SP.DefaultValue = "goat";
SP.Provider = Settings.Default.Providers["LocalFileSettingsProvider"];
SP.Attributes.Add(typeof(UserScopedSettingAttribute), new UserScopedSettingAttribute());
Settings.Default.Properties.Add(SP);
Settings.Default.Reload();
Settings.Default.Save();
Settings.Default["LibImage" + AmountOfImages] = OFD.FileName;
MessageBox.Show(Settings.Default["LibImage" + AmountOfImages].ToString());
These get added to user.config and show up like this:
<setting name="LibImage1" serializeAs="String">
<value>C:\Users\User\Background\Biggie.jpg</value>
</setting>
<setting name="LibImage2" serializeAs="String">
<value>C:\Users\User\Background\BUSTA-RHYMES.jpg</value>
</setting>
When I restart the program I want to add all these images added to a panel like this:
int i = 0;
Settings.Default.Reload();
foreach (SettingsProperty P in Settings.Default.Properties)
{
MessageBox.Show(P.Name);
//part below not relevant for question
if (P.Name.StartsWith("LibImage"))
{
i++;
IMG = Image.FromFile(P.DefaultValue.ToString());
PanelImgAr[AmountOfImages] = new SelectablePanel()
{
Size = new Size(150, 84),
Location = new Point(0, -84 + (94 * i)),
BackgroundImage = IMG,
BackgroundImageLayout = ImageLayout.Stretch
};
PanelImgAr[AmountOfImages].Click += new EventHandler(SelectablePanel_Click);
PanelImages.Controls.Add(PanelImgAr[AmountOfImages]);
}
}
But the MessageBox gives me no names.
This is probably because Settings.Default.Properties loops through App.config.
Can anyone tell me how I loop through user.config?
Or how I add the settings in user.config to App.config?
A: parse the full path of your user.config file to an XDoc.
from here on you can read it as a char array.
string path = ConfigurationManager.OpenExeConfiguration(ConfigurationUserLevel.PerUserRoamingAndLocal).FilePath;
if (File.Exists(path))
{
XDocument XDoc = XDocument.Load(path);
foreach (var node in XDoc.Nodes())
{
if (!string.IsNullOrEmpty(node.ToString()))
{
string S = node.ToString();
string Word = "";
for (int i = 0; i < S.Length; i++)
{
if (IsAcceptedChar(S[i])) Word += S[i];
else
{
if (Word == "setting name")
{
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,384
|
The $3 trillion US healthcare industry is at a huge risk. It's an industry that hasn't thought about cybersecurity seriously, mostly because they use old systems with timeworn security mechanisms.
Forget your credit card records, the hackers are after your medical records. The medical records are 10 times more valuable than the stolen financial information. FBI issued warnings to healthcare service providers after Chinese cybercriminals hacked into the computer network of Community Health Systems Inc. As a result, 4.5 million patients lost their personal information. The hospitals' network is vulnerable and it's easy for hackers to lay their hands on personal data to perform a medical fraud.
The HIPAA Journal reported 33 data breaches in June 2018. It put 356,000 patients' healthcare records at stake. Majority of the attacks were owing to IT infrastructure failure, unauthorized access and hacking.
With the patient record, the hackers get information related to names, date of birth, insurance policy numbers, diagnosis codes and billing information. Hoaxers use this information to create fake ids so that they can buy medical equipment and medicines that can be resold. The other fraud cases that have come into light are hackers trying to claim made-up claims with insurers as they combine patient numbers and bogus provider number. Unfortunately, victims with stolen information are not able to uncover the data theft for a long time unlike stolen financial information (that are instantly discovered and reported to banks for them to take immediate action). Digitizing medical information has increased the security risks much higher.
What are top reasons that put the healthcare industry at a risk?
Healthcare leader still see cyber-attacks an issue of the IT department rather than an issue for the entire organization. Not everyone working in the healthcare industry understands the importance of data security. As a result, there are often weak passwords that can be easily hacked and non-existent authentication practices. Majority of the security flaws in the system are introduced by the people working in the healthcare facility or hospital.
Solution: There must be a lot of focus on training the staff and organizational awareness on authentication policies and adoption of stricter authorization measures.
The healthcare staff use emails extensively and usually don't have a foolproof mechanism to protect their emails. Since the volume is large and awareness low, healthcare professionals will open a phishing email (most likely).
Solution: As mentioned above, the staff needs to be made aware of the risks involved in opening unsolicited emails. Training sessions can help them identify the phishing emails.
There isn't enough staff to manage access controls and, therefore, everyone can access everything. It's internal staff members who are primarily responsible for data breaches (whether they do it intentionally or unintentionally).
Solution: Sensitive patient data must be protected always. There must be detailed and proper authorization documentation available to enable authorized access and appropriate action should be taken once the employee leaves the organization.
It's a known fact that the healthcare industry lags in the adoption of latest software systems. The operating systems are never updated, the backups are not taken, the security policies are not updated, and the software versions are old. If the systems are obsolete, the risk and vulnerability of bugs and cyber-attacks increases multi-fold.
Solution: It is critical to update the software, operating system and security policies periodically. Cyber attackers often think of new ways of cracking the security systems, therefore, it's important to be one step ahead of them.
There's been a sharp rise in the use of mobiles, laptops and tablets at workplaces and healthcare industry is no exception. People bring their own devices to work and such devices might be exposed to data security risk.
Solution: There must be strict BYOD or Bring Your Own Device policy for staff members who wish to carry their own devices or access data over their mobiles. Data encryption is required for all devices, which will take care of the mobile application security.
There are several other reasons for the healthcare industry to be at a high risk of data theft. If, as someone from the healthcare industry, you wish to get more insight into cybersecurity and are serious about protecting your patient records, you can connect with us.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,502
|
{"url":"https:\/\/math.stackexchange.com\/questions\/1369511\/if-i-randomly-generate-a-string-of-length-n-from-an-alphabet-a-b-c-whats-t","text":"# If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that exactly k characters will be the same?\n\n\u2022 I have an alphabet: {A, B, C}.\n\u2022 I'm randomly generating strings of length N from that alphabet.\n\nExamples: Examples: N=5, AACBC, AAAAA, BBCAA\n\n\u2022 What is the likelihood that exactly k characters of that string are the same? (k <= N)\n(k corresponds to the maximum number of similar characters...\nExample: With string AABCAAA: N=7, k=5 because there are 5 A's.\nString AABBCC: N=6, k=2 because there are equally-sized groups of A's, B's, and C's.)\n\nInitially, my solution looked like this:\nP(k characters are the same) = $(\\frac{1}{3})^k * (\\frac{2}{3})^{n-k}$\nUntil I realized that this solution wasn't robust enough-- it doesn't matter WHICH characters are the same, only that k characters are the same.\n\n\u2022 Not only that, you will have to consider permutations of each string. Hint: Since there are three categories instead of two, you can't use the binomial coefficient. \u2013\u00a0true blue anil Jul 22 '15 at 3:49\n\u2022 Did you mean $N=6$ for AABBCC? \u2013\u00a0user940 Jul 22 '15 at 3:51\n\u2022 @ByronSchmuland yes I meant N = 6! Thanks for pointing that out. \u2013\u00a0jdmcpeek Jul 22 '15 at 7:10\n\u2022 Do you mean \"at least $k$ characters of the string will be the same\"? \u2013\u00a0DanielV Jul 22 '15 at 8:04\n\u2022 @DanielV I mean exactly k characters. \u2013\u00a0jdmcpeek Jul 22 '15 at 8:13\n\nLet k, l, m be the # of occurrences for different letters in decreasing order.\n\nUsing the multinomial distribution formula, to illustrate,\n\nfor string AAABBC, Pr = $\\frac{6!}{3!2!1!}\\cdot\\left(\\frac{1}{3}\\right)^6$\n\nand for string AABBCC, Pr = $\\frac{6!}{2!2!2!}\\cdot\\left(\\frac{1}{3}\\right)^6$\n\nIn general, Pr = $\\frac{N!}{k!l!m!}\\cdot\\left(\\frac{1}{3}\\right)^N$, k+l+m = N\n\ncontinued...\n\nI have taken that you want probabilities for particular values of k. For k = 3 and N = 6, for example, you will need to sum up probabilities for permutations of 3-3-0 (3#s) & 3-2-1 (6#s)\n\nedited by the questioner...\nThe final solution comes down to this. Imagine that we have three bins, each representing the number of times each character appears in a string. For AAABC, the bins would be {A:3, B:1, C:1} For:\n\n\u2022 N = the length of the string,\n\u2022 k = the maximum bin value (there can be ties),\n\u2022 l = the next bin value,\n\u2022 m = the last bin value,\n\u2022 d = the number of bin values that are different from k's bin value (max 2)\n\u2022 Examples:\n1. ABC: bins = {A: 1, B: 1, C: 1}. d = 0\n2. AAA: bins = {A: 3, B: 0, C: 0}. d = 1\n3. AAC: bins = {A: 2, B: 0, C: 1}. d = 2\n\u2022 C = the number of letters in our alphabet (always 3),\n\nPr = $\\frac{N!}{k!l!m!}\\cdot\\left(\\frac{1}{3}\\right)^N \\cdot\\frac{C!}{(C - d)!}$, k+l+m = N\n\n\u2022 I'm not sure if this works for a few cases I've been using to test on. For example, if I generate a three-letter string (N=3), the probability that two of them are the same (k = 2) is $\\frac{6}{9}$. I know this because I counted it out by hand. When I apply the multinomial distribution formula: $\\frac{3!}{2!*1!*0!} * (\\frac{1}{3})^3 = 1\/9$ \u2013\u00a0jdmcpeek Jul 22 '15 at 7:24\n\u2022 Read my \"continued\" portion. Since 2-1-0 has 6 permutations, you will need to multiply by 6, and you will get the correct answer. \u2013\u00a0true blue anil Jul 22 '15 at 8:00\n\u2022 Do you mean that I multiply by the number of permutations of k? Like: Pr = **$\\frac{N!}{k!l!m!}\\cdot\\left(\\frac{1}{3}\\right)^N * k!$ \u2013\u00a0jdmcpeek Jul 22 '15 at 8:08\n\u2022 Also, what about in the case of 2 cards that are both different? $\\frac{2!}{1!*1!*0!} * (\\frac{1}{3})^2 * 2! = 4\/9$, which isn't the correct probability. \u2013\u00a0jdmcpeek Jul 22 '15 at 8:10\n\u2022 Not multiply by k!, there can only be 3 patterns for N = 3: all identical (1 permutation), 2 identical (3 permutations) and all different (6 permutations). Thus for the 1-1-0 case, you need to multiply by 3,(which arises as 3!\/2!) not 2! \u2013\u00a0true blue anil Jul 22 '15 at 8:39","date":"2019-11-20 14:08:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7579469084739685, \"perplexity\": 964.7018687575104}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670559.66\/warc\/CC-MAIN-20191120134617-20191120162617-00257.warc.gz\"}"}
| null | null |
Tool specialist KS Tools now includes a line of fastening clips in its product range and at the same time has developed a tool for locating them. The Clipfinder at www.clipfinder.com assists workshops in quickly and effortlessly identifying the right clip for a vehicle. As well as a whole host of clip varieties sorted by vehicle makes, cross-brand universal varieties are also offered. So the professional always has the correct clip to hand. The main attraction of the Clipfinder is the option of being able to hold an existing clip up to the screen to compare it with the 1:1 illustration of the part identified in the internet. This works like a charm on the monitor screen.
"Sometimes the devil is in the detail", declares the KS Tools project management. "Clips often look similar and also appear at first glance to suit the corresponding application. Later it transpires that the wrong part has indeed been resorted to." Then in the extreme case the panelling comes loose or a loose fitting clip can cause irritating rattling noises which are also difficult to localise later on. With the Clipfinder, identifying the clips is quite simple. The most important step when using for the first time is to calibrate the monitor screen. Using a ruler, the Clipfinder is set so that the photos lodged in the program can be compared directly 1:1 with the original clip. A huge help in selecting clips. Alternatively, simply enter the screen diagonal measurement.
Besides this, the mechanic can identify the clips by entering the OE number. You only have to enter the correct number in the appropriate field and the corresponding KS Tools clip is displayed immediately. This works not only on a PC plus monitor but also on a laptop, tablet or Smartphone.
Clips mostly have to be renewed after a repair as they are invariably damaged on removal. It is especially at low temperatures that the material is brittle so that the clip breaks at the latest when being refitted. If that isn't noticed, the clip has no function and rattles or falls from its holder at the first opportunity.
KS Tools now offers a whole array of clips which cover the most important applications of these small plastic parts. A total of 21 sets form the right inventory to be armed for all eventualities.
The clips are available singly or as a set in a practical carrying case. These contain the most common models so that the correct clip is always at one's fingertips.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,968
|
\section{Introduction}
The study of elastically strained semi-conductor thin films displays a lot of challenges both from a theoretical and applied point of view \cite{Pimpinelli1998,Politi2000,Ayers2007}. The observations of the self-organized strained islands, which arise on these semi-conductor films, has attracted a lot of interest due to their opto-electronic properties for light emitting diode and quantum dots laser \cite{Shchukin1999,Stangl2004,Brault2016}. In addition, the development of a model that explains the shape and the dynamics of strained islands (quantum dots) remains a fascinating challenge since it involves the dynamical interplay of elastic, capillary, wetting and alloying effects\cite{Spencer1991,Muller2004,Chiu2006,Aqua2013,Aqua2013-PR,Spencer2013,WeiSpencer2016,Rovaris2016,WeiSpencer2017}.
During the deposition or the annealing of the semiconductor film in heteroepitaxy on a substrate, the atomic lattice difference between the film and the substrate induces an elastic stress which can lead to a morphological instability \cite{Srolovitz1989,Mo1990,Eaglesham1990,Floro1990,Sutter2000,Tromp2000,Floro2000,Berbezier2009,Misbah2010,Aqua2013-PR}.
This instability, known as the Asaro-Tiller-Grinfeld (ATG)\cite{Asaro1972,Grinfeld1986} instability, leads to the formation of parabolic-shaped islands (prepyramid) \cite{Tersoff2002}, which have been observed in the nucleation less regime \cite{Sutter2000,Tromp2000}. The prepyramids later evolve in pyramids as more volume is deposited \cite{Tersoff2002}.
Later on, the self-organized strained islands displays a coarsening dynamic for which it has been shown theoretically that the surface energy anisotropy slows down the coarsening \cite{Zhang2000,Aqua2010}.
The cause of the slowing down of the coarsening can be attributed to several effects is still under investigation and is addressed in this article \cite{Zhang2000,Aqua2010,Korzec2010465,Aqua2013}.
In this present work, we raise the following question: what is the effect of the amplitude of the surface energy anisotropy on the dynamics of coarsening. We show, using a one-dimensional continuum model and a set of numerical simulations, that coarsening is slowed down as the amplitude of the anisotropy of surface energy increases. Furthermore, we develop a simple model to quantify the effect of the anisotropy of surface energy on the coarsening time.
We propose that the main cause of the slowing down of the coarsening is due to the effect of the anisotropy of surface energy. The main effect of the anisotropy of surface energy is to favorize a specific orientation of the surface. This leads to an influence on the shape of the island and thus this affect the distribution of the elastic field in the island. As we shall show, this modification of the shape of the island leads to a change in the dependency of the chemical potential with respect to the island height $h_0$, and as a consequence this slows down the coarsening dynamics.
We first present the one dimensional dynamical model, which takes into account the anisotropy of surface energy and is based on the resolution of the equation of continuum elasticity. Secondly, we describe analytically the equilibrium shape of one dimensional pyramidal island.
From our model, we estimate the dependency of the chemical potential as a function of the island height. Thirdly, we use the relations obtained in the previous part to propose a simple dynamical model in order to explain how the coarsening time of two anisotropic strained islands increases as a function of the anisotropy strength.
We conclude our article by illustrating our results with the numerical simulations of the coarsening of an array of islands in the presence of anisotropy.
\section{Continuum model}
Semiconductors film dynamics can be modelled by a mass conservation equation which takes into account the surface diffusion. This surface diffusion current is proportional to gradients of the surface chemical potential $\mu$. In the absence of evaporation the $1D$ equation for the top surface of the film $h(x,t)$ reads:
\begin{equation}
\frac{\partial h}{\partial t} = \mathcal{D} \sqrt{1+ h_x^2} \frac{\partial^2 \mu}{\partial s ^2}\ ,
\label{eqgeneral}
\end{equation}
where $\mathcal{D}$ is the diffusion coefficient, $h_x$ is the slope of the surface height $\partial_x h(x,t)$ and $\partial/\partial s$ the surface gradient \citep{Levine2007,Schifani2016}.
The chemical potential $\mu$ at the surface is defined by:
\begin{equation}
\mu=\delta \mathcal{F}/\delta h \, .
\label{eq:mu1}
\end{equation}
Here $\mathcal{F}$ is the free energy of the system which encompasses the surface and the elastic contribution $\mathcal{F}=\mathcal{F}_s+\mathcal{F}_{el}$ and $\mu=\mu_s+\mu_{el}$. The surface energy reads
\begin{equation}
\mathcal{F}_s=\int \gamma(h,h_x) \sqrt{1+|h_x|^2}dx .
\label{eq:Fs}
\end{equation}
It includes both wetting effects and surface energy anisotropy.
The elastic energy is given by the integration over both the film and the substrate of the elastic energy density, it reads:
\begin{equation}
\mathcal{F}_{el}=\int_{z<h(x)} \mathcal{E}_{el}(x,z)dxdz.
\end{equation}
The elastic energy density $\mathcal{E}_{el}$ can be computed using the values of the stress tensor $\sigma_{ij}$ and of the strain tensors $e_{ij}$. It reads,
\begin{equation}
\mathcal{E}_{el} = \frac{1}{2}\sigma_{ij}e_{ij}\,.
\end{equation}
As a first approximation, we examine a decomposition of the surface energy $\gamma(h,h_x)$ where the wetting and anisotropic effect are independent:
\begin{equation}
\gamma(h,h_x)=\gamma_f\left[\gamma_h(h)+\gamma_a(h_x)\right].
\label{eq:gamma-s}
\end{equation}
The wetting effects are linked to the film thickness $h$ through $
\gamma_h (h)=c_w \exp (-h/\delta_w)
$, where $c_w$ and $\delta_w$ are respectively the amplitude and the range of the wetting potential \cite{Muller1996}.
We choose the anisotropy term in the surface energy to have a single minimum at a value $\tan(\theta)=\tan(\theta_e)$, as shown in Fig. \ref{fig1}:
\begin{equation}
\gamma_a (h_x)=1-\alpha h_x^2\left( 1- \frac{h_x^2}{2\tan^2(\theta_e)}\right)
\, .
\label{surf-aniso}
\end{equation}
Here $\alpha$ is the anisotropy strength and $h_x=\tan(\theta)$ is the surface slope.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure1.pdf}
\caption{\label{fig1} Surface energy described by the anisotropy function given by Eq. (\ref{surf-aniso}) as a function of $h_x$, for $\alpha=0.15$ and $\theta_e=\pi/6$. The minimum is given at $h_x=\tan (\theta _e)$ and represents the characteristic island slope. The vertical axis is in unit of $\gamma_f$.}
\end{center}
\end{figure}
\newpage
Using Eq. (\ref{eq:mu1}) and Eq. (\ref{eq:gamma-s}), the surface chemical potential $\mu_s$ is found to be:
\begin{equation}
\begin{aligned}
\mu_{s}=-\left[1-2 \alpha+A(\alpha,\theta_e)h_x^2+B(\alpha,\theta_e)h_x^4 \right] h_{xx}-\frac{c_w}{\delta_w}e^{-h/\delta_w}.
\end{aligned}
\label{eq:mus}
\end{equation}
Here the parameters $A(\alpha,\theta_e)$ and $B(\alpha,\theta_e)$ are defined as
\begin{eqnarray}
A(\alpha,\theta_e)=\frac{3}{2}\left[-1-4\alpha+4\alpha\cot^2(\theta_e) \right]\,,\\
B(\alpha,\theta_e)=\frac{15}{8} \left[1+2\alpha+4 \alpha \cot ^2(\theta_e) \right]\, .
\end{eqnarray}
The elastic chemical potential can be written as
\begin{equation}
\mu_{el}=-\mathcal{H}(h_x)\, .
\label{eq:muel}
\end{equation}
where $\mathcal{H}(h_x)$ is the Hilbert transform of the spatial derivative of $h(x,t)$, defined as $\mathcal{F}^{-1}(|k|\mathcal{F}(h))$, where $\mathcal{F}$ is the Fourier transform \cite{Aqua2007}. Thus the total chemical potential $\mu$ reads,
\begin{equation}
\mu=\mu_s+\mu_{el}\,.
\end{equation}
The evolution equation for the surface $h(x,t)$ will merely follow from Eq. (\ref{eqgeneral})
and from the expression of the surface chemical potential Eq. (\ref{eq:mus}) and of the elastic chemical potential Eq. (\ref{eq:muel}).
We first consider the space scale
\begin{equation}
l_0=\gamma_f / [2(1 + \nu)\mathcal{E}_0]\,,
\label{eq:l0}
\end{equation}
resulting from the balance between the typical surface energy $\gamma_f$ and the elastic energy $\mathcal{E}_0$ density.
Here $\mathcal{E}_0 = E \, \eta^2/(1-\nu)$ where $\eta = (a_f - a_s)/a_s$ is the misfit parameter where $a_f$ (resp. $a_s$) is the film (resp.
substrate) lattice spacing, $E$ is the Young's modulus of the film and the substrate, and $\nu$ the Poisson's coefficient.
Secondly we consider the time scale
\begin{equation}
t_0 =l^4_0/(\mathcal{D} \gamma_f)\,,
\label{eq:t0}
\end{equation}
where $\mathcal{D}$ is the surface diffusion coefficient. In units of $l_0$ and $t_0$, the evolution equation reads
\begin{equation}
\frac{\partial h}{\partial t}=\frac{\partial^2 \mu}{\partial x^2}=\frac{\partial^2 (\mu_{s}+\mu_{el})}{\partial x^2}\,,
\label{eq:equil}
\end{equation}
where $\mu_{s}$ and $\mu_{el}$ are given by Eqs. (\ref{eq:mus}) and (\ref{eq:muel}) respectively.
The numerical integration of Eq. (\ref{eq:equil}) is performed using a pseudo-spectral method as used in \cite{Aqua2007,Aqua2010} on a periodic domain of size $L_T$. For example, for a ${\rm Si}_{0.75}{\rm Ge}_{0.25}$ film on ${\rm Si}$, we find $l_0=27\, {\rm nm}$ and $t_0=23\,{\rm s}$ at $700^{\circ}\, {\rm C}$ (see \cite{Chason1999} for an estimate of surface diffusion coefficients). Eq. (\ref{eq:equil}) is parametrised by the wetting constant $c_w$ and $\delta_w$ and the anisotropy constants $\alpha$ and $\theta_e$. It is a non-linear equation and its evolution is dominated by a coarsening phenomenon in which small islands disappear at the benefit of larger islands. The quantity $S$ defined as the surface of the system
\begin{equation}
S=\int_{-L_T}^{L_T} h(x)dx\,,
\label{eq:surfacem}
\end{equation}
is conserved during the dynamic as a simple consequence of the form of Eq. (\ref{eq:equil}).
\section{Analytical model and numerical simulations}
In this section, we first study the equilibrium shape of one island. Using a simple \textit{ansatz}, we analytically determine the characteristics parameters of the island such as its size, height and energy. Our predictions are in good agreement with our numerical computation.
We show that there is a smooth transition from parabolic-like to pyramid-like shapes as S increases.
Secondly, we derive a dynamical model which shows that the influence of the anisotropy strength $\alpha$ is to increase the coarsening time. Finally, we illustrate our article with the numerical simulations
of an array of islands displaying coarsening.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure2.pdf}
\caption{\label{fig2} Island like solution given by the numerical simulation of Eq. (\ref{eq:equil}) represented by dots, compared to the \textit{ansatz} proposed in Eq. (\ref{eq:anzats}) represented by the continuous curve.
$L$ is the half-width of the island and $x_1$ is the top island parabola width.
We use as parameters $c_w=0.045$ and $\delta=0.005$. The initial condition is given by a small random perturbation around a constant value of $h=0.4$. The value of the surface is $S=6.1$. The anisotropic parameters are $\alpha=0.1$ and $\theta_e=\pi/6$. The system size is $L_T=16$.}
\end{center}
\end{figure}
\subsection{Equilibrium}
The equilibrium shape of one island is given by the time independent stationary solution of Eq. (\ref{eq:equil}) as shown in Fig. \ref{fig2}. This solution is characterised by the constant value of the chemical potential $\mu$ along the $x$ axis. The island-like shape can be described using the following \textit{ansatz} for the function $h(x)$:
\begin{eqnarray}
h(x)=
\begin{cases}
h_w+\tan(\theta_e) (x+L) & -L<x<-x_1 \, , \\
h_0-\frac{\tan(\theta_e) }{2 x_1} x^2 & -x_1\le x\le x_1 \, , \\
h_w-\tan(\theta_e) (x-L) & x_1<x<L \, , \\
h_w & |x| \geq L \, .
\end{cases}.
\label{eq:anzats}
\end{eqnarray}
This pyramidal-like shape describes an island of maximum height $h_0$ which sits on a wetting layer of height $h_w$.
The island sides are described by straight lines of slope $ \pm \tan(\theta_e)$ for $-L<x<-x_1$ and $x_1<x<L$.
The island top is described by a parabola for $-x_1<x< x_1$, which satisfies the continuity of the first derivative of $h(x)$ at $x=\pm x_1$.
At the foot of the island $|x|=L$ the function $h(x)$ is continuous and has a value $h_w$. Here $x_1$ is the half-width of the parabola.
This {\it ansatz} is characterised by four unknown parameters $(h_0, h_w, x_1, L)$ which can be deduced from four relations.
The first relation is the continuity of $h(x)$ at $x=\pm x_1$, it imposes:
\begin{equation}
h_0=h_w+\tan (\theta_e)(L-x_1/2)\,.
\label{eq:condh}
\end{equation}
After the substitution of the \textit{ansatz} (\ref{eq:anzats}), we expand Eq. (\ref{eq:equil}) around $x=0$ in a polynomial series up to second order in $x$.
At order $0$ in $x$, we obtain the value of the half-width island $L$
\begin{equation}
L= \exp{\left(\frac{\pi(1 -2 \alpha) -\pi \mu x_1 \cot (\theta_e )+2 x_1( \log (x_1)-1)}{2 x_1}\right)}\,.
\label{eq:condL}
\end{equation}
Here the chemical potential $\mu$ reads:
\begin{equation}
\mu=-\frac{c_w}{\delta_w}e^{-h_w/\delta_w} \, .
\label{eq:mu-hw}
\end{equation}
This previous relation is due to the fact that far from the island the film is flat, so that $h_{x}$ and $h_{xx}$ vanish, and only the wetting potential term remains dominant in Eq. (\ref{eq:mus}) and (\ref{eq:muel}). The chemical potential $\mu$ being fixed, the wetting layer value $h_w$ reads:
\begin{equation}
h_w=-\delta_w \log\left(\mu\frac{\delta_w}{c_w}\right)\,.
\label{eq:mu-hw1}
\end{equation}
From the expansion at second order in $x$ of Eq.(\ref{eq:equil}), we obtain a transcendental equation for $x_1$, it reads:
\begin{equation}
\frac{\tan (\theta_e ) \left(\frac{1}{L^2}+\frac{6 \pi \alpha +x_1}{x_1^3}\right)}{\pi }-\frac{3 (4 \alpha +1) \tan ^3(\theta_e )}{2 x_1^3}=0\,.
\label{eq:condx1}
\end{equation}
Combining Eq. (\ref{eq:condL}) and Eq. (\ref{eq:condx1}), we obtain the following transcendental equation for the parameter $x_1$
\begin{equation}
\begin{aligned}
&\left(\frac{1}{\pi}\exp \left(-\frac{\pi(1 -2 \alpha) -\pi \mu x_1 \cot (\theta_e )+2 x_1(\log (x_1)-1)}
{x_1}\right)+\frac{6 \alpha }{x_1^3}+\frac{1}{\pi x_1^2}\right)\\
&-\frac{3 (4 \alpha +1) \tan ^2(\theta_e )}{2 x_1^3}=0\,. \end{aligned}
\label{eq:setx1}
\end{equation}
After substitution of Eq. (\ref{eq:mu-hw}) in Eq. (\ref{eq:setx1}), we can solve Eq. (\ref{eq:setx1}) numerically using a simple root finding algorithm to obtain the parameter $x_1$ for different values of $h_w$.
The island half-width $L$ can then be deduced from Eq. (\ref{eq:condL}). Furthermore the value of the island height $h_0$ can be deduced from Eq. (\ref{eq:condh}).
For each value of the wetting layer height $h_w$, we can compute the value of the surface $S$ (mass) using Eq. (\ref{eq:surfacem}) and the \textit{ansatz} Eq. (\ref{eq:anzats}).
Finally, from the knowledge of the values ($h_0$, $L$, $x_1$, $\mu$), we can compute the value of the surface $S$, it reads:
\begin{equation}
\begin{aligned}
S = { } 2 [h_0 x_1+h_w (L_T-x_1)]+\left(3 L^2-6 L x_1+2 x_1^2\right)\frac{\tan (\theta_e )}{3}\,.
\end{aligned}
\label{eq:surface}
\end{equation}
We present in Fig. \ref{fig2} the island shape numerically integrated from Eq. (\ref{eq:equil})
and compare it with the \textit{ansatz} (\ref{eq:anzats}).
The agreement
is quite satisfactory and there are no free parameters.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figure3.pdf}
\caption{\label{fig3} Numerical island profile computed from Eq. (\ref{eq:equil}) for different initial values of the surface $S$. For small surface $S$, we obtain a parabolic-shaped island, with constant island width. For island surfaces larger than $S_m=3.7$, the islands present a pyramidal-like shape. Its characteristic half-width $L_m=2.26$, given by Eq. (\ref{eq:lm}), is represented in the figure.
From bottom to top: Red-Curve bottom ($S=1.80$), Orange ($S=2.66$), Green ($S=3.48$), Blue ($S=5.84$) , Purple ($S=7.58$), Black ($S=9.74$), color on-line.
The dashed-dotted curve represent the characteristic slope given by $\tan(\theta_e)$, in this case $\theta_e=\arctan(\sqrt{3/4})$. }
\end{center}
\end{figure}
When the horizontal size of the parabola $2 x_1$ becomes smaller than the island size $L$, the island morphology changes from pyramid-like shape to a parabolic-like shape. Using Eq. (\ref{eq:condL}) and Eq. (\ref{eq:surface}) we obtain
\begin{equation}
L_m = \frac{3}{5} \pi \sec ^2(\theta_e ) (1-(8 \alpha +1) \cos (2 \theta_e ))\,,
\label{eq:lm}
\end{equation}
\begin{equation}
S_m = \frac{2}{3} L_m \left[3(h_0+ h_w)+L_m \tan (\theta ) \right]\,.
\label{eq:sm}
\end{equation}
Therefore, the pyramidal shape can only exist for $S>S_m$ and $L>L_m$. Below this value of $S_m$, the islands are parabolic-like shaped and the anisotropy can be neglected.
We plot in Fig. \ref{fig3} various island profiles obtained by numerical simulation for different initial values of the surface $S$. Our numerical simulation shows that for a small island surface $S$, the island shape is parabolic-like and its widths $L$ is rather constant. As the surface of the system increases the islands become pyramid-like and their widths increase smoothly with respect to their height. The transition from parabolic-like shape to pyramid is smooth as the control parameter $S$ is varied.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure4}
\caption{\label{fig4} Chemical potential $\mu$ as a function of the island height $h_0$ at equilibrium with $\theta_e=\pi/6$. The $\bullet$, $\blacktriangle$ and $\blacksquare$
symbols are computed by numerical integration of Eq. (\ref{eq:equil})
for $\alpha=0$, $\alpha=0.2$ and $\alpha=0.4$ respectively. The continuous, dashed and dotted curves represent the solution given by Eq. (\ref{eq:muhapp}) for $\alpha=0$, $\alpha=0.2$ and $\alpha=0.4$ respectively. The approximation given by Eq. (\ref{eq:muhapp}) fits the results of the numerical solution.}
\end{center}
\end{figure}
Finally, we compute the value of the chemical potential as a function of the island height for various value of the anisotropy strength $\alpha$.
As shown in Fig. \ref{fig4}, the chemical potential decays quasi-linearly as a function of the island height as shown below in Eq. (\ref{eq:muhapp}).
Using Eq. (\ref{eq:setx1}) the chemical potential $\mu$ can be expressed easily as a function of the parameter $x_1$. In the same way using Eq. (\ref{eq:condh}) the island height can be expressed as a function of $x_1$. Finally using the relation $\frac{\partial \mu}{\partial h}=\frac{\partial\mu}{\partial x_1}\frac{\partial x_1}{\partial h}$ the slope of the chemical potential versus the height of the island for small values of $\alpha$ is found to be:
\begin{equation}
\partial \mu/\partial h \simeq -\frac{1}{1+3\alpha}\,.
\label{eq:muhapp}
\end{equation}
\newpage
\subsection{Dynamics}
\subsubsection{Numerical simulation of the coarsening of two islands and dynamical model}
In this subsection, we characterise the dynamic of coarsening of two islands. In Fig. \ref{fig5}, we show the time evolution of two pyramidal-shaped islands obtained by the numerical simulation of Eq. (\ref{eq:equil}). During the coarsening, the larger island increases at expense of the smaller island until it disappears.
Ultimately at a time $t_c$ defined as the coarsening time only one island remains in the system.
The initial conditions are prepared following \cite{Schifani2016}, by replicating a pyramid with a slight difference in amplitude.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure5.pdf}
\caption{\label{fig5} Spatio temporal evolution of two islands, deduced by numerical simulation of Eq. (\ref{eq:equil}).
The initial condition are two islands of height $h_1(0)=0.89$ and $h_2(0)=0.8$ separated a distance $d=L_T/2$, where $L_T=32$ represents the system size. The anisotropic parameters are $\alpha=0.1$ and $\theta=\pi/6$. The continuous curve (black) represents the island profile at $t=1$, dashed curve (orange) $t=30$, dotted curve (blue) $t=80$ and dotted-dashed curve (red) $t=100$. After a time $t_c=88$, there remains only one island in the system.}
\end{center}
\end{figure}
In order to analyse this phenomenon, we propose a simple model for the coarsening of two islands, inspired by the work presented in \cite{Schifani2016}
\begin{equation}
\Resize{5.5cm}{\begin{array}{ll}
h_1\partial_t h_1=\tan(\theta_e)\frac{\mu(h_2)-\mu(h_1)}{d},\\
h_2\partial_t h_2=\tan(\theta_e)\frac{\mu(h_1)-\mu(h_2)}{d}.
\end{array}}
\label{eq:dyn-mod}
\end{equation}
Here $h_1(t)$ is the height of the large island, $h_2(t)$ is the height of the small one and $d$ is the distance separating them. If we consider the island width as $h_i/\tan(\theta_e)$, we recover the model proposed in \cite{Schifani2016}. The advantage of this model is that its resolution requires only the resolution of a differential equation instead of the resolution of a partial differential equation.
In Fig. \ref{fig6}, we represent the time evolution of each islands heights $h_1$ and $h_2$, corresponding to the result displayed in Fig. \ref{fig5}. We also compare in Fig. \ref{fig6} the results obtained from the resolution of the model Eq. (\ref{eq:dyn-mod}) with results of the numerical simulation of Eq. (\ref{eq:equil}).
The model predictions is in good agreement with the numerical simulation of Eq. (\ref{eq:equil}).
The resolution of the model can be done in two ways: a simple numerical integration of Eq. (\ref{eq:dyn-mod}) or an analytical resolution of Eq. (\ref{eq:dyn-mod}) as explain in section $III.B.2$.
The slight discrepancy between the numerical result (numerical simulation of Eq. (\ref{eq:equil})) and theoretical result (resolution of the puntual model Eq. (\ref{eq:dyn-mod})) for the final height $h_1$
is mostly due to the fact that our model is based on a simple pyramidal-like shape {\it ansatz} during all the coarsening dynamic.
This small discrepancy in the height does not affect the coarsening time $t_c$.
\subsubsection{Effect of the anisotropy on the coarsening time}
In Fig. \ref{fig7}, we present the coarsening time $t_c$ of two strained islands as a function of the anisotropy strength $\alpha$.
We compare the analytical coarsening time $t_c$ obtained by the resolution of the model Eq. (\ref{eq:tc-analitical}) and the results obtained by numerical simulation of Eq. (\ref{eq:equil}). There is a good agreement between both results. As shown in Fig. \ref{fig7} we find that the coarsening time increase linearly as a function of the anisotropy strength $\alpha$. We can explain this effect in the following way
Using Eq. (\ref{eq:dyn-mod}) it can be easily shown that $\partial_t(h_1^2+h_2^2)=0$. We thus propose the following change of variables in order to solve analytically Eq. (\ref{eq:dyn-mod}):
\begin{equation}
\begin{array}{ll}
h_1(t)=h_0\sin(\phi(t)),\\
h_2(t)=h_0\cos(\phi(t)).
\end{array}
\,
\label{eq:chanvar}
\end{equation}
Here $h_0$ is related to the initial islands heights as $h_0=\sqrt{h_1(0)^2+h_2(0)^2}$. Substituting Eq. (\ref{eq:chanvar}) into Eq. (\ref{eq:dyn-mod}) yields:
\begin{equation}
\partial_t\phi=\frac{\tan(\theta_e)}{h_0d(1+3\alpha)}\left(\frac{1}{\cos(\phi)}-\frac{1}{\sin(\phi)}\right)\, ,
\label{eq:phi}
\end{equation}
submited to the initial condition $\phi(0)=\phi_0=\arctan\left(\frac{h_1(0)}{h_2(0)}\right)$.
Eq. (\ref{eq:phi}) can be integrated analytically, its solution is
\begin{eqnarray}
t(\phi)=(1+3 \alpha)d\frac{h_0}{\tan(\theta_e)} \mathcal{T(\phi)}.
\label{eq:tvsphi}
\end{eqnarray}
The analitical form of $\mathcal{T}(\phi)$ is given in \footnote{$\mathcal{T}(\phi)=(\sin (\phi )-\sin (\phi_0)+\cos (\phi )-\cos (\phi_0)
-\sqrt{2} \tanh ^{-1}\left[\frac{1}{\sqrt{2}}\left(\tan \left(\frac{\phi
}{2}\right)+1\right)\right]+\sqrt{2} \tanh ^{-1}\left[\frac{1}{\sqrt{2}}\left(\tan \left(\frac{\phi_0}{2}\right)+1\right)\right])$ where $h_0=\sqrt{h_1(0)^2+h_2(0)^2}$ and $\phi_0=\phi(0)=\arctan\left(\frac{h_1(0)}{h_2(0)}\right)$}.
The coarsening time $t_c$ is defined by the following criteria: when the height of the small island reaches the wetting layer height $h_w$. This implies the following implicit relation for $t_c$:
\begin{equation}
\phi(t_c)=\arcsin\left(\frac{h_w}{h_0}\right).
\end{equation}
This previous relation derives from Eq. (\ref{eq:chanvar}) easily. Using Eq. (\ref{eq:tvsphi}) we obtain the coarsening time $t_c$. It reads:
\begin{equation}
t_c=(1+3 \alpha)d\frac{h_0}{\tan(\theta_e)} \mathcal{T}\left[\phi(t_c)\right],
\label{eq:tc-analitical}
\end{equation}
this result agrees with the numerical simulation presented in Fig. \ref{fig7}. In the limit of $(h_w/h_0)\ll1$, the expression $\mathcal{T}(\phi(t_c))$ can be simplified as given in \footnote{$\mathcal{T}(\phi(t_c)) \simeq -\frac{\text{h1}}{\text{h2} \sqrt{\frac{\text{h1}^2}{\text{h2}^2}+1}}-\frac{1}{\sqrt{\frac{\text{h1}^2}{\text{h2}^2}+1}}+\sqrt{2} \tanh ^{-1}\left(\frac{\tan
\left(\frac{1}{2} \tan ^{-1}\left(\frac{\text{h1}}{\text{h2}}\right)\right)+1}{\sqrt{2}}\right)+1-\sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right)$. This result is obtained assuming that the wetting layer height is smaller than the initial islands heights $(h_w/h_0)\ll1$. With this approximation $\phi(t_c)\simeq 0$. }.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure6.pdf}
\caption{\label{fig6} Time evolution of the height $h_1$ and $h_2$, corresponding to the result shown in Fig. \ref{fig5}. The dashed curves represent the islands heights given by the numerical simulation of Eq. (\ref{eq:equil}), and the continuous curves represent the result obtain by the numerical resolution of the dynamical model presented in Eq. (\ref{eq:dyn-mod}). The same results for the continuous curve can be obtained analytical as explained in section $III.B.2$. The coarsening time is $t_c= 88$ for both solutions. The vertical axes is in units of $l_0$ and the time scale is $t_0$.}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure7.pdf}
\caption{\label{fig7} Coarsening time $t_c$ as a function of the anisotropy strength $\alpha$. The system under study is the same as shown in Fig. \ref{fig5}. The dots represents the numerical simulation of Eq. (\ref{eq:equil}) and the curve is the solution for $t_c$ given by Eq. (\ref{eq:tc-analitical}). The coarsening time $t_c$ depends linearly on $\alpha$.}
\end{center}
\end{figure}
\newpage
\subsubsection{Numerical simulation of an array of islands with anisotropy}
For illustration, we present the numerical simulation of the coarsening of an array of islands. The numerical simulation of Eq. (\ref{eq:equil}) reveals mostly two phenomena. A first instability regime which arises for an initial film height higher than the critical layer.
A second regime in which coarsening takes place and is not interrupted. As shown in Fig. \ref{fig8} after the initial instability the smaller islands vanish by surface diffusion through the wetting layer at the benefit of the bigger islands until the system reach the equilibrium. The equilibrium state is characterised by a large island whose characteristic size can be deduced from the parameters of the Eq.(\ref{eq:anzats}). This phenomenon is observed numerically with or without the presence of the surface energy anisotropy.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure8.pdf}
\caption{\label{fig8} Evolution of an anisotropic strained islands according to Eq. (\ref{eq:equil}), where the initial condition is a flat film of height $0.11$ with a small random perturbation. First the ATG instability evolves, and the coarsening start after the islands have formed. Finally, there is only one pyramidal-shape island with the characteristic slope shown in Fig. \ref{fig1}.}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figure9.pdf}
\caption{\label{fig9} Ensemble average for the maximum height $h$ as a function of time for three different values of $\alpha$. $\alpha=0$ (continuous line), $\alpha=0.01$ (dashed line) represented in Fig. \ref{fig8} and $\alpha=0.05$ (dashed-dotted line) obtain by the numerical simulation of Eq. (\ref{eq:equil}). The initial condition for the systems is a flat film of height $0.1$ with a small random perturbation. The maximum value of $h$ is calculated using an ensemble average of twenty simulations for each different $\alpha$.}
\end{center}
\end{figure}
We show in Fig. \ref{fig9}
an ensemble average for the maximum height of $h$ as a function of time computed by numerically integrating Eq. (\ref{eq:equil}). The results are presented for three different values of the anisotropy strength ($\alpha=0$, $\alpha=0.01$ and $\alpha=0.05$). We have performed twenty numerical simulation for each value of $\alpha$. We observe that the rate coarsening of an anisotropic system ($\alpha=0.01$ and $\alpha=0.05$) is slower than the system without anisotropy ($\alpha=0$). This effect is due to the increase of the coarsening time $t_c$ as described previously in Fig. \ref{fig7}. Finally, we note that the determination of the coarsening exponent reported in \citep{Aqua2007} is still under investigation in presence or absence of surface anisotropy. As matter of fact the understanding of the dynamics between the islands could serve to elaborate an analytical model to describe the coarsening dynamic of a many islands system.
\newpage
\clearpage
\section{Conclusion}
This article presents a numerical and analytical study of the shape and of the dynamics coarsening of strained anisotropic islands.
We have characterised analytically strained islands using a simple \textit{ansatz}. We have introduced a dynamical model to investigate the dynamics of coarsening of two islands This models compares favorably with our numerical simulation.
We have shown that the coarsening dynamics of strained island in hetero-epitaxy is slowed down by the presence
of the surface energy anisotropy.
Our results are in good agreement with our numerical simulations.
For future work the comparison to experiments will be investigated in three dimensions.
\begin{acknowledgements}
We would like to thank Franck Celestini, Jean-No\"el Aqua, Pierre M{\"u}ller and Julien Brault for useful discussions. We thank the ANR NanoGaNUV for financial support.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,142
|
package org.araqne.logstorage.file;
import java.io.IOException;
import java.nio.ByteBuffer;
import java.util.Arrays;
import net.jpountz.lz4.LZ4Compressor;
import net.jpountz.lz4.LZ4Factory;
import net.jpountz.lz4.LZ4FastDecompressor;
import net.jpountz.util.Native;
public class Lz4Compression implements Compression {
private static final LZ4Factory factory;
static {
org.slf4j.Logger slog = org.slf4j.LoggerFactory.getLogger(Lz4Compression.class);
try {
Native.load();
slog.info("araqne logstorage: loaded native lz4 library successfully, result [{}]", Native.isLoaded());
} catch (Throwable t) {
slog.warn("araqne logstorage: cannot load native lz4 library, cause []", t.toString());
} finally {
factory = LZ4Factory.fastestInstance();
}
}
@Override
public ByteBuffer compress(byte[] b, int offset, int limit) throws IOException {
LZ4Compressor compressor = factory.fastCompressor();
int maxCompressedLength = compressor.maxCompressedLength(limit);
byte[] compressed = new byte[maxCompressedLength];
int compressedLength = compressor.compress(b, offset, limit, compressed, 0, maxCompressedLength);
return ByteBuffer.wrap(Arrays.copyOf(compressed, compressedLength));
}
@Override
public void uncompress(byte[] output, byte[] b, int offset, int limit) throws IOException {
LZ4FastDecompressor decompressor = factory.fastDecompressor();
decompressor.decompress(b, offset, output, 0, output.length);
}
@Override
public void close() {
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,942
|
DEMOCRACY AND THE FOREIGNER
BONNIE HONIG
DEMOCRACY AND THE FOREIGNER
Princeton University Press
Princeton and Oxford
Copyright © 2001 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
3 Market Place, Woodstock, Oxfordshire OX20 1SY
All Rights Reserved.
Library of Congress Cataloging-in-Publication Data
Honig, Bonnie.
Democracy and the foreigner / Bonnie Honig.
p.cm.
Includes bibliographical references and index.
eISBN : 978-1-40082-481-6
1. Democracy. 2. Immigrants. 3. Nationalism.
4. Internationalism. I. Title.
JC423 .H748 2001
325.1—dc212001016373
This book has been composed in ITC Garamond Light
Printed on acid-free paper. ∞
www.pup.princeton.edu
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
_For my parents,_
_David and Schewa Honig,_
_whose foreignness shaped their lives_
_and mine_
Politics exists because those who have no right to be
counted as speaking beings make themselves of some
account, setting up a community by the fact of placing
in common a wrong that is nothing more than this
very confrontation, the contradiction of two worlds
in a single world: the world where they are and
the world where they are not, the world where there
is something "between" them and those who
do not acknowledge them as speaking beings
who count and the world where there is nothing.
–Jacques Rancière
Contents
Acknowledgments
1 NATIVES AND FOREIGNERS: Switching the Question
2 THE FOREIGNER AS FOUNDER
Dorothy and the Wizard
Rousseau's Lawgiver
Freud's Moses
Girard's Scapegoat
Democracy and Foreignness
3 THE FOREIGNER AS IMMIGRANT
The Book of Ruth as a Foreign-Founder Text
Ruth
Immigration and Founding
Ozick's Ruth: Convert or Migrant?
Kristeva's Ruth: The Ideal Immigrant
Gender and the Foreign-Founder
Kristeva's Orpahs:Cosmopolitanism without Foreignness
Mourning, Membership, Agency, and Loss:Ruth's Lessons for Politics
4 THE FOREIGNER AS CITIZEN
The Myth of an Immigrant America
Class Mobility as American Citizenship
Ethnic Bases of Social Democracy:Michael Walzer's Immigrant America
Foreign Brides, Family Ties, and New World Masculinity
Dramatizing Consent: The Universal Charms of American Democracy
Taking Liberties: Intimations of a Democratic Cosmopolitanism
5 THE GENRES OF DEMOCRACY
Does Democracy Have a Genre?
Democracy's Romance: A Tale of Gothic Love
Notes
Bibliography
Acknowledgments
"There's no place like home." The magical phrase, repeated three times while she clicks together heels gripped by hard-won ruby slippers, returns Dorothy home from her nightmare-dream in the land of Oz. On its face, the phrase expresses a heartfelt home-yearning, as in "There's _no_ place like home, [sigh]." Here the phrase suggests that home is so unique, wonderful, and irreplaceable a place (a "place where people know me, where I can just be, " in Minnie Bruce Pratt's phrasing), that no other place ever lives up to it. However, the selfsame phrase unmasks this yearning for home as a fantasy. Switch the emphasis to "There's no place _like_ home, " and the phrase now seems to suggest that there is no place as wonderful as home is mythologized to be, and that includes home itself. The uncanny, punlike character of the phrase combines an unrelenting yearning for home together with an awareness that the home so yearned for is a fantasy.
Such ambivalence—its political-cultural organization into the sepa-rate, distinct desires we call home-yearning and escapism—is one of this book's points of departure. The desires for "a place where people know me, where I can just be" and for a place where I can never "just be" because my sense of my self is always challenged and expanded often drive the politics of foreignness traced here. Or better, perhaps these desires are themselves the effects of the politics of foreignness, symptoms of larger issues in democratic theory having to do with the freedom, agency, community, and solidarity that are the daily work of life in a more or less democratic polity.
Little did I know when I wrote the above scene-setting sentences several years ago that by the time this book was truly finished I would have written it in thirteen different studies in four different states. Such nomadism has enabled me to present my work in progress to a wide range of audiences who generously shared their perspectives with me at talks and conferences. Many others read all or parts of the manuscript along the way and provoked me to think further about issues I thought I had already settled. First and foremost, for giving me the gift of their time, and for reading drafts of chapters with an intellectual generosity to which I have already grown happily accustomed, I am grateful to some of the members of my new Chicago circle of theory colleagues, Eric Santner, Patchen Markell, Peter Fenves, and Sara Monoson. Our newest member, Miguel Vatter, arrived in time to make me attend once more to Machiavelli. Also in that circle, Linda Zerilli has been my partner in thinking about this book since its inception, and to prove it I have a stack of emails from her that is taller than I am. I thank her for her many, many, many contributions to the project.
Amy Gutmann and Sanford Levinson read the manuscript for Prince-ton University Press and provided helpful guidance and encourage-ment for the last stages of revision. Others who heard or read parts of the manuscript and commented on them to my profit include Seyla Benhabib, Lauren Berlant, Josh Cohen, Bill Connolly, Mo Fiorina, Richard Flathman, Jill Frank, Marcie Frank, Marjorie Garber, Ruth Grant, Barbara Johnson, Michael Jones-Correa, George Kateb, Steve Macedo, Harvey Mansfield, Pratap Mehta, Martha Minow, Bruce Robbins, Michael Rogin, Michael Sandel, George Shulman, Rogers Smith, Michael Warner, Patrick Weil, Stephen White, and Iris Marion Young. Late in the day, Joan Cocks graciously made time to read almost the whole manuscript and helped give it its final contours.
I have had great good fortune in research assistance over the years:thanks to Patchen Markell, Michaele Ferguson, Arash Abizadeh, Michael Bosia, Torrey Shanks, and Ella Myers.
Drafts of all the chapters were presented at various colloquia, panels, and conferences: at the Ford Foundation, MIT, several APSAs, Harvard, Rutgers, and Duke Universities, the University of Iowa, ICA in London, the American Studies Association in 1998, among others. Some of the chapters appeared in much abbreviated form as articles, and I am grateful to Sage Publications, _Strategies_ (http://www.tandf.co.uk), and _Social_ _Text_ for permission to reuse material here that was first published there.
I am especially grateful to the students at Northwestern University. The first to join the new graduate program in political theory, in particular, Torrey Shanks and Ella Myers, have inaugurated here a new and exciting theory conversation of which I am fortunate to be part.
My recent association with the American Bar Foundation made me appreciate the forceful role of the iconic lawgiver in the symbolic politics of foreignness. Chapter 2 exists because of the ABF. The ABF also provided me with research funding, teaching relief, and intellectual company. I am grateful to John Comaroff, Bryant Garth, Carol Heimer, Chris Tomlins, Joanne Martin, Beth Mertz, Annelise Riles, Vicki Woeste, and, briefly, Janet Gilboy, as well as such visitors as Felicia Kornbluh, Michelle Landis, Ronen Shamir, Saskia Sassen, and Kunal Parker. Also at ABF, Renee Brown prepared the manuscript for publication with courtesy and professionalism for which I thank her.
At Princeton University Press, Ann Wald first acquired the manuscript, and Ian Malcolm adopted it as if it were his own. I am very grateful to both—to Ann for her early interest in this project and to Ian for the great care he took of it.
Michael Whinston designed and oversaw the construction of the last three studies in which this book was written, thus doing literally in those moments what he always also does figuratively: supporting my writing, work, and personal life with characteristic warmth and good planning. He also helped hammer into shape the final drafts of chapters 1 and 5. My son, Noah Whinston, was born at the start of this project, and my daughter, Naomi Honig, arrived five years later at its close. It seems to me appropriate that they should sandwich the project in this way because they teach me daily one of this book's most vital lessons: how to live with alienness. Noah also provided several key insights into the _Wizard of Oz_ and the biblical story of Moses, for which I thank him most sincerely.
Finally, I am enormously indebted to Tereza Almeida, Audra Jestes, Jane Merriam, and, most recently, Carol Paine for caring for my children while I worked and for teaching me some of the skills of child care upon which I now rely. I cannot thank them enough.
Evanston, Illinois, and Warren, Vermont
December 2000
DEMOCRACY AND THE FOREIGNER
Would it still make sense to speak of democracy when it would no longer be a question . . . of country, nation, even of state or citizen?
–Jacques Derrida
What is a foreigner? A [wo]man who makes you think you are at home.
–Edmond Jabes
1 NATIVES AND FOREIGNERS: Switching the Question
_"How should we solve the problem of foreignness?"_ The question under-lies contemporary discussions of democracy and citizenship. Proposed solutions vary. Political theorists deliberate about whether or to what extent social unity is necessary to sustain social democracy. Courts rule on the extent of government's obligations to its noncitizen residents. Economists debate the costs and benefits of immigration. Sociologists argue about the (in)effectiveness of multilingual education. But, notwithstanding their differences, participants in contemporary debates about foreignness all reinscribe foreignness as a "problem" that needs to be solved by way of new knowledge, facts, or politics. In so doing, they reiterate the question that has dominated political theory for centuries.
In classical political thought, foreignness is generally taken to signify a threat of corruption that must be kept out or contained for the sake of the stability and identity of the regime.This somewhat xenophobic way of thinking about foreignness endures in the contemporary world, though other options—from assimilation to the many varieties of multiculturalism—are now also considered viable. All of these options persist in treating foreignness as a problem in need of solution, however. Even many of the most multiculturally minded contributors to diversity debates treat foreignness as a necessary evil and assume that we would be better off if only there were enough land for every group to have its own nation-state.
There are some who take a more positive view of foreignness. In _Nations Unbound_ , Basch, Glick Schiller, and Szanton Blanc endorse a new transmigrant politics that is, in their view, bringing to an end the nation-state's privileged position as the central organizing institution of modern cultural, political, juridical, and administrative life. In the same spirit, James Holston and Arjun Appadurai claim that in many places "the project of a national society of citizens . . . appears increasingly exhausted and discredited."4 Analogously, for Iain Chambers, increased encounters with transnational others have the following, desirable con-sequence: "The earlier European intertwining of national language, literature and identity is unpicked, and the epic of modern nationalism is forced open to meet the exigencies that emerge from more complex patterns."
For celebrants of postnational politics, foreignness does not seem to be a problem in need of solution. It is a welcome agent of welcome changes. But these thinkers, wittingly or unwittingly, rearticulate the classical position on foreignness noted above. That is, on the postnationalist account, too, foreignness is a threat to the stability and identity of established regimes. Postnationalists differ from their predecessors only in their valuation of that threat. They celebrate it and valorize the very fragmentation that earlier political theorists took to be a problem.
Motivated by these ongoing debates, I take foreignness as a topic, a question, rather than a problem. What does it mean? What sort of work does it do in cultural politics? In the chapters that follow, I read texts of democratic theory looking at the roles (often heretofore unnoticed) played in those texts by strangers or foreigners, and I read popular and high cultural stories about strangers or foreigners, looking for the lessons they might have for democratic theory. Again and again, I find foreignness used in familiar ways, as a device that gives shape to or threatens existing political communities by marking negatively what "we" are not.6 But I also find foreignness operating in a less convention-ally familiar way, with a seldom-noted positive content and effect.
Sometimes, the figure of the foreigner serves as a device that allows regimes to import from outside (and then, often, to export back to out-side) some specific and much-needed but also potentially dangerous virtue, talent, perspective, practice, gift, or quality that they cannot provide for themselves (or that they cannot admit they have). This supple-ment of foreignness gives receiving regimes something different from the novelty, cultural breadth, and depth identified by theorists of immigration and multiculturalism such as Bhikhu Parekh.7 Indeed, it is often their foreignness itself—not, as Parekh suggests, the culturally inflected talents, skills, or perspective that individual foreigners happen to have—that makes outsiders necessary even if also dangerous to the regimes that receive them. Indeed, sometimes foreignness operates as an agent of (re)founding.
In the classic texts of Western political culture (both high and low), the curious figure of the foreign-founder recurs with some frequency:established regimes, peoples, or towns that fall prey to corruption are restored or refounded (not corrupted or transcended) by the agency of a foreigner or a stranger. Moses appears as an Egyptian prince to lead the Israelites out of Egypt and bring to them the law from the mountain. The biblical Ruth's migration from Moab to Bethlehem reanimates the alienated Israelites' affective identification with their god while also beginning the line that will lead to King David. Oedipus arrives from else-where to solve the riddle of the Sphinx and save Thebes (temporarily) with his wise leadership. In _The Statesman_ , it is the Eleatic Stranger who teaches us how to know the true statesman. In _The Republic_ , the founding dialogue of political theory's interminable debate about justice takes place in the house of a foreign merchant, Cephalus, who is originally from Syracuse. Why is this the setting? Does Plato mean to imply that justice, or perhaps philosophical dialogue itself, is occasioned by engagement with foreignness?8 Later in _The Republic_ , Plato has Socrates say casually that the myth of the metals, the Republic's founding myth, is a "Phoenician thing," not unfamiliar and yet of foreign origin. In the _Social Contract_ , Jean-Jacques Rousseau's lawgiver comes from elsewhere to found an ideal democracy. And in the contemporary United States, a variety of American institutions and values, from capitalism to community to family to the consenting liberal individual, are seen to be periodically reinvigorated by that country's newest comers, its idealized citizens: naturalizing immigrants. Again and again, the cure for corruption, withdrawal, and alienation is . . . aliens.
This finding invites us to switch the question that has long dominated our thinking about foreignness. Rather than " _How should we solve the_ _problem of foreignness?_ " and "What should 'we' do about 'them'?" (questions that never put the "we" into question and this, surely, is part of their point and attraction), the question that animates this book is: _What problems does foreignness solve for us?_ Why do nations or democracies rely on the agency of foreignness at their vulnerable moments of (re)founding, at what cost, and for what purpose?
As we will see in the chapters that follow, foreign-founder scripts use foreignness in a dazzling variety of ways: the foreign-founder may be a way of marking the novelty that is necessarily a feature of any (re)-founding. The same mythic figure may be a way of illustrating a psycho-logical insight that stale or corrupt patterns cannot be broken without the injection of something new. The novelties of foreignness, the mysteries of strangeness, the perspective of an outsider may represent the departure or disruption that is necessary for change.9 The foreignness of the founder might also be a way of marking and solving a perennial problem of democratic founding in which the people must be equal under the law and cannot therefore receive it from any one of their own number. Some theorists, such as Julia Kristeva, speculate that stories of foreign-founders are a culture's way of marking its inextricable relation to otherness, its strangeness to itself. Finally, the foreignness of the foreign- founder might be a way of modeling the impartiality, breadth of vision, objectivity, and insight that a founder must have. Who but an outsider could be trusted to see beyond the established lines of conflict and division that make shared governance difficult?
Some might argue that none of these hypotheses is really needed because the reason we tell stories of foreign-founding is quite simply that they are true. For example, in answer to the question: "Isn't it curious that some stories of founding feature a foreign-founder?" they might say: "Not really. Or at least not necessarily. After all, some origin stories feature foreign-founders because some peoples really were founded by a foreigner. Take Russia, for example."11 They might then go on to detail the events that led up to the A.D. 862 invitation by quarreling Slavic tribes to the Varangian Rus': "Our whole land is great and rich, but there is no order in it. Come to rule and reign over us!"
But can the facts of a foreign-founding story decide the question of its meaning and power? The facts do not explain why the story is retold and recirculated, nor to what effect. The truth and meaning of Russia's origin story have, for example, been heatedly debated for centuries. Historiographers date the inauguration of the debate to September 6, 1749, the day Gerhard Friedrich Müller gave a lecture on the origins of Russia to members of the Imperial Academy. Omeljan Pritsak captures what must have been a dramatic scene: "Müller never finished his lecture. As he spoke, tumult arose among the Russian members and 'adjuncts' of the Imperial Academy in protest against the 'infamous' words they were hearing." Müller was charged with dishonoring the nation, and a special committee was appointed "to investigate whether Müller's writings were harmful to the interests and glory of the Russian Empire."The result? "Müller was forbidden to continue his research in Old Rus' history, and his publications were confiscated and destroyed. The intimidated scholar eventually redirected his scholarly work to a less incendiary subject: the history of Siberia."
This academic exile to Siberia did not put the matter to rest, however. Following these events, eighteenth and nineteenth-century anti-Normanists denied the foreignness of the Rus', claiming they were really Slavs, not Swedes. One Slavophile even rewrote the founding story: in Khomiakov's long poem, _Vadim_ , the foreign Rus' leader, Riurik, is driven out of Novgorod by a good Slav, Vadim. In _The Chronicles_ , Vadim is mentioned only in passing, as having led a _failed_ resistance to Riurik. But Khomiakov was "one of the leading spokesmen for the Slavophile view of history." For him, Andrew Wachtel explains, "the undesirability of a foreign-born ruler and the need for a native Slavic element to triumph over him becomes clear."
Unsurprisingly, a century earlier, Catherine the Great, herself a foreign ruler of Russia, gave the story a rather different spin, in which the foreign Riurik is an enlightened ruler and in which (in Wachtel's words) "Russian patriotism and national pride are not incompatible with borrowing from the outside."15 Others insisted on the incompatibility that Catherine tried to minimize, but their agenda was neither nationalist nor xenophobic. Nineteenth-century Normanists such as Vladimir Solovyev affirmed and valorized Russia's foreign-founding because, they argued, it meant that Russia was particularly well positioned for a cosmopolitan or universalist mission.
Pritsak suggests that these debates are ongoing because each side of the argument has its weaknesses. The problem, he says, is a problem of knowledge: "H]istorians have too often substituted political (or patriotic) issues for improved techniques of historical methodology in their discussions; they have had a limited knowledge of world history; and they have been biased in their use of source materials."[17 It is not clear, however, whether the objective and thorough history that Pritsak calls for could put a quick end to the centuries of to-ing and fro-ing on the question of the Rus'. Do historical facts have that kind of power? Even if the empirical question regarding Riurik's foreignness could be put to rest, the question of his expressive significance as a foreign-founder would still be an open question whose debate would arouse passions and land some unfortunate souls in a Siberia of one kind or another.
Confronted with the fact of Riurik's foreignness, nationalists would undoubtedly engage in a symbolic politics of foreignness: How foreign was he? What does his foreignness mean for Russia? In all likelihood, the fact of Ruirik's foreignness would drive some to argue that Riurik, though really a Scandinavian, was possessed of Slavic features and temperament. Alternatively, scholars might redate what they take to be the beginning of a Slavic people, so that Riurik would be removed from his position of central importance in the Slavs' origin story. In short, the questions raised by the foreignness of the founder are not, _pace_ Pritsak, empirically soluble; they are symbolic questions. Similarly, in contemporary debates about immigration, the facts can inform but they cannot resolve the question of whether immigrants are good or bad for the nation because the question is not, at bottom, an empirical question. The question of why the founder (or the refounder) is a foreigner points not to the origins of the origin story in question, but rather to the daily workings of that story in the life of a regime.
Similarly, when I ask why a (re)founder is a foreigner, I am asking not "Where does he or she come from and why?" but rather "What symbolic work is the story of foreign-(re)founding doing for the regime in question?"19 Of what practices and programs of renationalization and legitimation are the symbolic politics of foreignness a part? And, in the cases of _democracy's_ foreign-(re)founders in particular: Is foreignness a site at which certain anxieties of _democratic_ self-rule are managed? At bottom, these questions are not about foreignness per se, but about the work that foreignness does, the many ways in which it operates as a way to frame other issues of democratic theory and citizenship. 20 The answers to these questions vary in relation to different texts and contexts.
In Chapter Two, where I read Rousseau's _Social Contract_ together with Freud's _Moses and Monotheism_ and Girard's _Violence and the Sacred_ , I find that the figure of the foreign-founder may be a way of managing some paradoxes of democratic founding, such as the alienness of the law, an especially charged problem in democratic regimes. In Chapter Three, the foreignness of Ruth is what enables her to supply the Israelites with a refurbishment they periodically need: she chooses them in a way that only a foreigner can (and the more foreign the better) and thereby re-marks them as the Chosen People. She also domesticates by way of her apparently freely felt love for the law the alien and violently imposed law that Moses brought to the Israelites from the mountain. In Chapter Four, I trace how foreignness works, in the American exceptionalist literature, contradictorily and simultaneously to reinstill popular but always shaky beliefs in a meritocratic economy, heartfelt community, patriarchal family structures, and a consent-producing liberal individualism, all of which undergird the sense of choice-worthiness that immigrants are positioned and required to enact for the United States. Immigrants' new membership in the United States is not only celebrated, it is also endorsed as iconic of good citizenship, with problematic consequences for the native born and for all would-be democratic citizens.
All of these uses of foreignness are double edged, however. Foreignness operates in each instance as both support of and threat to the regime in question. Moses' foreignness and that of Rousseau's lawgiver, the biblical Ruth, and America's immigrants do not only solve certain problems. In each case, their foreignness is itself a problem for the regimes that seek to benefit from its supplement. What I find, therefore, and what I call attention to in each of the chapters below, is what we just saw at work in the case of Russia's origin story: a _politics_ of foreignness in which different parties to the debate try to mobilize a founder's foreignness on behalf of their ideal, while also striving somehow to solve or manage the problem of the founder's foreignness.
The cultural organization of foreignness as threat and/or supplement is not exhausted by the types of foreignness examined here. To the foreigner as founder, immigrant, and citizen, one could add other categories—the foreigner as refugee, boundary crosser, terrorist, outlaw, repository of irrationality, erotic excess, madness, anarchy, and so on. But my goal is not to offer a complete catalog of the symbolic figurations of foreignness. Instead, my goal is to study in depth some of the uses to which foreignness is daily put on behalf of democracy. Since democracy is still thought of in predominantly national terms, this means we must look not only at texts of democratic theory in which foreignness figures, but also (as in Chapter Three on the Book of Ruth) at texts in which foreignness is put to work on behalf of national or subnational communities. Since much of the contemporary democratic theory literature theorizes democracy as a form of liberalism, we must look also at texts in liberal theory that use foreignness to shore up specifically liberal institutions and values such as consent (as in Chapter Four, when I discuss Peter Schuck's and Rogers Smith's part in American immigration debates).
It is worth mentioning here, however, that one counterimage of foreignness does keep surfacing in each of the chapters: that of the _taking_ foreigner. This taking is not the criminal activity of an outsider (though it is not immune to such depictions) but an honorific democratic practice—that of demanding or, better yet, simply enacting the redistribuion of those powers, rights, and privileges that define a community and order it hierarchically. Here the iconic taking foreigner puts foreignness to work on behalf of democracy by modeling forms of agency that are transgressive, but (or therefore) possessed of potentially inaugural powers. Carried by the agencies of foreignness, this revalued "taking" stretches the boundaries of citizenship and seems to imply or call for a rethinking of democracy as also a cosmopolitan and not just a nation-centered set of solidarities, practices, and institutions. One might object that such a move to locate democracy also on cosmopolitan registers itself amounts to a "use" of foreignness on behalf of a political aspiration. The point cannot be denied. But the hope is that this particular use might better serve the needs of democracy than the mostly nation-centered alternatives whose promises and insufficiencies I track in the chapters that follow.
In Chapter Five, I raise the issue of genre, which provides one way of understanding why political theorists have not heretofore attended to the politics of foreignness as I cast them here. Most readers of democratic theory tend to bring certain romantic genre expectations to texts, often treating the narrative as a series of arguments intended to bring about a reconciliation, a happy (or at least resigned) marriage between a people and their law or institutions. In Chapter Five, I ask: What if we read democratic theory gothically instead of romantically? Gothic novels depend on the reader's uncertainty as to whether the heroine's would-be lover is really a hero or a villain. Similarly, a gothic approach to democratic theory presses us to attend to the people's perpetual uncertainty about the law and their relation to it: Is it really part of us or an alien thing, an expression of our intimate will or a violent imposition?That gothics tend to represent and deepen our uncertainty about the hero by making him a foreigner, and the setting a foreign (often a Catholic) place only adds to the appositeness of this genre to democratic theory, in which anxieties about empirical foreigners and (in more abstract terms) the alienness of the law are always at work, even if seldom in a way that is noted by many scholars.
Another genre choice also shapes this book. What unites texts as disparate as Rousseau's _Social Contract_ , Freud's _Moses and Monotheism_ , the biblical Book of Ruth, Michael Walzer's _What It Means to Be_ _an American_ , and Schuck and Smith's _Citizenship without Consent:_ _Illegal Aliens in the Polity_ with one another and with the various films I discuss as well, including _The Wizard of Oz_ , _Strictly Ballroom_ , and _Shane_ , is that all are—whatever else they may be— _myths of foreign-_ _(re)founding_.21 Reading them as such makes certain salient but hereto-fore relatively unnoticed characteristics of the texts rise to the surface, while others, though perhaps more often noted, recede into the background.
For example, the Book of Ruth is usually read as a conversion story, so Ruth is often compared to Abraham, the first convert to Judaism. Reading Ruth as a myth of foreign-founding invites a comparison instead to Moses, the possibly foreign lawgiver who formed the tribes of Israel into a people of the law. That new comparison calls attention to the fact that Moses died in the land where Ruth was born: Moab. Suddenly, it seems possible that one of the many effects of this great short story is its implication that the law may be reborn as a woman, that one of Ruth's many functions may be to rescript our affective relation to the law from a relation of violent imposition or awe to one of loving devotion, from the sublime to the beautiful.22 The justification for reading Ruth as a myth of foreign-founding is that she is, indeed, the agent of a (re)founding. Her virtuous example returns the Israelites from a period of corruption to devotion to the one true God. Through her son, Obed, she inaugurates the monarchic line of David.
My analysis of foreign-founder scripts is motivated by several goals. It is not my intention to make a general claim about the necessity of stories of foreign-founding to successful refoundings, nor, indeed, to recommend the telling of such stories. Not all regimes tell such stories, and those that do retell them with varying frequencies and intensities. I aim merely to ask what sort of work is done by such stories where they do exist. The genre is a curious one and seems to beg for some sort of explanation. Why do regimes tell stories of themselves in which they are depicted as dependent upon the kindness of strangers? What effect might such stories have on the democratic aspirations of a regime? Aren't democracies particularly threatened by such accounts, given the still widely held belief that democracy presupposes and requires social unity?
Second, entering into the interpretative fray over the significance of myths of foreign-founding is a way to vie for the political-cultural capital that such stories offer the interpreters that claim them. One of the most interesting things to come out of this study is the fact that foreignness in and of itself is neither a cosmopolitan nor a nationalist resource. A foreign-founder is not, as such, an obstacle to a national project. Nationalists find in the figure a vehicle of renationalization. Cosmopolitans find in it a resource for denationalization. Since the symbolic powers of foreignness are capacious enough to be mobilized by both sides, those who would like to expand the reach of democracy beyond the nation's borders must enter the interpretive fray and not just count on the facts of foreignness to do the world-building work of politics.
Third, this genre, in which foreignness does positive work (even if not _only_ positive work) for a regime, might be a useful resource for those who would like to address tendencies to xenophobia that are part and parcel of modern democratic life. But how? Will attending to the iconic foreign founder open up democracies to the foreignness we now encounter, not for the first time, as part of the processes of globalization and migration? Maybe, but as I argue in Chapter Three contra Julia Kristeva, such an awareness all by itself is not a sufficient condition of generating a more open and magnanimous democratic politics. In truth, it may not even be a necessary condition. There is no logic that requires that relatively homogeneous societies are less tolerant than relatively heterogeneous ones, and there is no empirical evidence to support such a claim, either. If the foreign-founder helps us to combat xenophobia, it is by inviting us to see how a fraught relationship to foreignness may be generated or fed by certain needs and demands of democracy itself (in different ways in its various settings, theorizations, and practices) rather than, say, stemming exclusively from deep psychological needs or from separate, independent tendencies to nativism or xenophobia.
With this last phrase, I take issue with Rogers Smith's argument in _Civic Ideals_. In that book, Smith advances what he calls a "multiple traditions thesis" according to which the United States is not—contra Alexis de Tocqueville, Louis Hartz, and other American exceptionalists—a purely or essentially liberal democratic regime. Instead (as evidenced by its citizenship laws, whose history Smith traces in detail), the United States is a regime constituted by many competing, incompatible, sometimes cooperating ideologies such as liberalism, republicanism, racism, patriarchalism, and nativism. Liberalism and republicanism are, according to Smith, egalitarian, while the other traditions are ascriptive. This pluralization of America's ideological base allows Smith to argue against critics of American liberalism, and also against critics of liberal theory more generally, that the shortcomings of liberal theory and practice do not stem from liberalism per se. The admirable moments of progress, liberation, and justice that punctuate American history are attributable to its liberal or republican commitments, on Smith's account, and America's less fine moments can be traced to its ascriptive traditions.
What is striking about Smith's reading of American liberalism is that its structure replicates the very mode of thinking that the author seeks to criticize. Out to discredit "ascriptive mythologies that can easily become demonologies, " Smith produces an argument that is itself demonological in structure. The many violent crimes and injustices that mark American national history are not essential to its character as a partly liberal democratic regime. Those violences come from elsewhere, from other parts of the American polity. Ascriptive ideologies distinct from liberal-ism are responsible for the nativist, sexist, and racist citizenship laws and arguments catalogued by Smith. Thus, liberalism is insulated from implication in the unsavory elements of American political history. The real culprits, those other "traditions, " are set up as Girardian scapegoats. Made into the bearers of all that liberalism seeks to disavow, they can now be cast out of the polity, which is then (re-)unified around this purging of its pollutants.25 That is to say, they are rendered foreign to the would-be, still-hoped-for, liberal democratic body politic.
My point is not that, contra Smith, liberalism is in fact the real culprit, after all. What evidence could decide _that_? On the contrary, my point is that setting the problem up as a search for the single, causal culprit is misguided.27 Rather than join this argument, pro or contra, I want to point out how Smith's multiple-traditions thesis works to direct our critical scrutiny away from the object being defended (in this case, liberal values and institutions), while encouraging a demonizing attitude to-ward the objects of critique (in this case, more explicitly ascriptive forms of life).28 The foreign-founder invites us to set the problem up differently. Because the figure puts foreignness at the center of some democracies' daily (re)foundings, the foreign-founder presses us to look beyond xenophobic beliefs to explain xenophobic politics. What if such politics are _also_ driven by pressures that come from within democracy itself, as it is variously practiced and theorized?
By inviting us to switch the question—from "How should we solve the problem of foreignness?" to "What problems does foreignness solve for us?"—the foreign-founder gives us a more promising way to proceed in our efforts to study the diverse, intimate relations between liberal democracy and its would-be others. Such an alternative analysis shows how certain anxieties endemic to liberal democracy—the paradox of democratic power (given up just as it might have been gained), the alienness of the law, the lack of a sense of choiceworthiness or the periodic need to have that sense refurbished, the distance or inaccessibility of consent— themselves generate or feed an ambivalence that is then projected onto the screen of foreignness. This ambivalence is testified to by Rousseau's curiously foreign lawgiver who is both loved and feared by the people he founds.
The iconic foreign-founder also presses us to ask whether democracy, in its origins and daily refoundings, may presuppose not only the reconstruction of the national (as theorists such as Smith, Beiner, Miller, and others assume) but also the violation of the national.29 To counter-act the still deep-going assumption that democracy is necessarily a national form, I refer occasionally in the pages that follow to an alternative conception of democracy: democratic cosmopolitanism. Nationalists often resort to the specter of an international government in order to discount cosmopolitanism and reprivilege the state as the center of any future democratic politics.30 But democracy is not just a set of governing institutions.31 It is also a commitment to generate actions in concert that exceed the institutional conditions that both enable and limit popular agencies.
At their most successful (as with some international human rights, labor, and environmental organizations), such actions in concert open up and even institutionalize spaces of public power, action, and dis-course that did not exist before. In short, democratic cosmopolitanism is a name for forms of internationalism that seek not to govern, per se, but rather to widen the resources, energies, and accountability of an emerging international civil society that contests or supports state actions in matters of transnational and local interest such as environmental, economic, military, cultural, and social policies.32 This is a democratic cosmopolitanism because democracy, in the sense of a commitment to local and popular empowerment, effective representation, accountability, and the generation of actions in concert across lines of difference, is its goal. In that sense it is also a rooted cosmopolitanism, rooted not (contra a range of cosmopolitans from Julia Kristeva to David Hollinger) in a national ideal but rather in a democratic ideal, one that seeks out friends and partners even (or especially) among strangers and foreigners.
Jean-Jacques Rousseau, himself the great theorist of democracy as a national form, makes a surprising gesture in this direction when he refers in passing to the apparent doubles of his domestic foreign-founder, the "few great cosmopolitan souls" who are the last remaining persons in the modern world to be moved by the pity and the natural goodness characteristic of the Rousseauvian state of nature:
The law of nature no longer operated except between the various societies, where under the name of the law of nations, it was tempered by some tacit conventions in order to make intercourse possible and to take the place of natural commiseration which, losing between one society and another nearly all the force it had between one man and another, no longer dwells in any but _a few great cosmopolitan souls_ , who surmount the imaginary barriers that separate peoples and who, following the example of the sovereign Being who created them, include the whole human race in their benevolence.
The analysis developed here of the intricate relations between democracy and foreignness just might open up more room for the admirable impulses personified by Rousseau's few great cosmopolitan souls and today enacted by such admirable groups as Médecins du Monde. But I am motivated most centrally by a more humble and academic desire. How shall we understand the following puzzles? The texts that I read, from the _Wizard of Oz_ to _Shane_ , from Rousseau's _Social Con-tract_ to the Hebrew Bible and American liberal and democratic theory all suggest in one way or another, that democratic citizens (not the Bible's original audience, but definitely Moses' and Ruth's contemporary readers), are often threatened and supported by dreams of a foreigner who might one day come to save us and enable us finally to abdicate or perhaps reassume the abundant responsibilities of democracy. Why? Why do these fears and hopes take shape through the figure of a foreigner? And what can that foreigner, the iconic foreign-founder, teach us about the insufficiencies, challenges, dramas, and dreams of democracy?
What good is a legend to a people that makes their hero into an alien?
–Sigmund Freud
2 THE FOREIGNER AS FOUNDER
Dorothyand the Wizard
In a book about _The Wizard of Oz_ , Salman Rushdie offers a fresh read-ing of that canonical American film as a tale not of home yearning—the conventional reading—but of adventurism. Rushdie's Dorothy is an eager émigrée, anxious to be off and away from the stultifying gray of her Kansas home.1 Boredom and colorlessness (but also a certain felt neglect at the hands of her Aunt Em and Uncle Henry ["Not now, Doro-thy!"]) prepare the way for Dorothy's eventual home leaving. But injustice and felt powerlessness provide the final push. Dorothy finally leaves because her aunt and uncle bend helplessly before the force of a law that serves the wealthy, powerful Miss Gulch, allowing her to take away Dorothy's dog, Toto. Home is not only an uninteresting place, Rushdie rightly stresses, it is also unsafe, unfair, and unjust.
But Dorothy is not only an adventurous émigrée.3 She is also (both wittingly and unwittingly) the vehicle of a welcome regime change. The injustice and unfairness that marked the Kansas home from which Dorothy fled in frustration are no less present in Oz. And the well-intentioned are for the most part as ineffective in Oz as they were in Kansas. The peoples of Oz and its environs are far removed from democratic self-governance. Dwarfed by arbitrary and unaccountable power, the Munchkins suffer deeply under the tyranny of the Wicked Witches of the East and West, against which the curtained, bureaucratic rule of the Wizard is no match. They are left dependent for occasional unpredictable salvation upon good witches with limited powers. Dorothy, the émigrée, the stranger, the foreigner who arrives inexplicably from elsewhere, becomes the vehicle of Oz's peoples' reempowerment. She kills two ruthless witches and unmasks the well-meaning but distantly inept and fraudulent Wizard, whose departure she also occasions. Her final gift to the peoples of Oz is her own departure.
The conventional view of the _Wizard of Oz_ sees Dorothy's departure as a testimony to the value of home which, complicated as it is, never stops tugging at Dorothy's heartstrings. This is the reading Salman Rush-die is out to offset. But Dorothy's departure acts out another quite different script at the same time, that of the foreign-founder, a figure that pops up repeatedly in Western culture, high and low. It is by virtue of her power as a stranger and a naive that Dorothy can do what no native of Oz would dare to. Unsocialized by the reign of terror that has molded the locals into servile abjection, Dorothy topples the forces of corruption and alienation. Then, to top that off, she leaves the people to sort out the terms of their own self-governance. Rather than stay to rule this grateful population, she leaves them under the loose charge of the Scarecrow, Tin Man, and Lion, three characters who represent the virtues necessary for successful self-rule: brains, heart, and courage.
Dorothy does not turn the local population's debt to her into an opportunity for domination.5 The Wizard in his day did just that, and the consequence was the alienating governance whose weaknesses and blowhard corruptions made Dorothy's adventure so necessary for Oz. Between them, then, the Wizard and Dorothy may be seen to represent two sorts of foreign-founder and the uncertainty of the gifts they bear for the regimes they (re)found.
But what about the peoples whose founding is at stake? We can learn more about them by contrasting Dorothy's fantasy and its fulfillment in Oz with the peoples of Oz's fantasy and its fulfillment by Dorothy. In Oz, Dorothy is big rather than little, she has power (represented by the ruby slippers), and she is looked up to and worshiped rather than ignored. Hers is a child's fantasy of being as important and powerful as a grown-up, and the mostly diminutive peoples of Oz (even the Wizard is short) satisfy her longing.6 For their part, the various peoples of Oz and its environs (from the Munchkins to the enslaved soldiers of the Wicked Witch of the West) are likely moved by a rescue fantasy. They are infantilized and, although they dream of being liberated, their dream includes the fantasy of being relieved of the adult responsibilities they have thus far failed to live up to. Dorothy, this magical foreigner who fell like manna from the sky, does their dirty work for them.
The two fantasies meet on common ground: in both, the local peoples are infantilized. Like good Tocquevillians, they belong to many leagues and civic associations, and they have a lot of offices (as well as some officiousness: recall the comic scene in which the Munchkins insist that the death of the Wicked Witch of the East must be legally, officially, verifiably certified—"Is she really most sincerely dead?"). But the Munchkins take no democratic responsibility for freeing themselves from the forces that dominate their lives. What does this mean for their political future? How likely are they to take up the democratic self-governance that could follow in the wake of Dorothy's rescue of them and her timely departure?
The people's paradoxical dependence upon a founder at what could be the very moment of their assumption of democratic power and responsibility is a deep and knotty problem in the history of political thought and culture. By what magic are dependent, not yet fully formed followers supposed to become the responsible, active citizens that democracy requires? The problem is crystallized at the moment of founding, but it recurs daily in a regime as new members enter fully into citizenship (whether by reaching adulthood or crossing the border to resettle) and as established citizens are renormalized into accepted expectations of belonging. Might the figure of the _foreign_ -founder, who appears not just in such popular-culture texts as _The Wizard of Oz_ but also in such high-culture texts as the Hebrew Bible, Rousseau's _Social Contract_ , and Freud's _Moses and Monotheism_ , be a way of work-ing through this and other problems that plague democratic political culture?
Rousseau's Lawgiver
"What good is a legend to a people that makes their hero into an alien?" asks Sigmund Freud in _Moses and Monotheism_ , the book in which he unmasks Moses as a foreigner (make that two foreigners).8 A legend that makes a people's hero into an alien is no good at all—this is the implied answer to Freud's rhetorical question. But what if we suppose otherwise? Might the foreign-founder's foreignness be, perhaps, a necessary condition of his performance of his founding functions?
The lawgiver of Rousseau's _Social Contract_ is a foreign-founder. This foreign-founder's seldom-noted figuration of foreignness at the heart of Rousseau's ideal democracy invites us to ask whether democracy itself—at its origins and in its daily refoundings—might require not just the (re)construction of the national (in order, say, to broaden the sympathies of the masses for one another, as Benedict Anderson might argue), but also the violation of the national. In contrast to neo-Rousseauvians like David Miller, who argue that democracy depends upon socionational unity, Rousseau's foreign-founder invites us to ask whether democracy might not also presuppose and require some deep relation to foreignness.9 Is there some deep connection between democracy and foreignness? If so, the figure of the foreign-founder might serve as a political-cultural resource for those seeking to open up the reach of democracy in late modernity, to multiply affinities with others both here, in the temporal space of the nation-state, and elsewhere.
When Rousseau introduces a foreign-founder to the _Social Contract_ , he reiterates a classic script familiar to readers of founding myths from Greece to Rome to the Hebrew Bible. But Rousseau's reliance on a foreign-founder is particularly curious given the xenophobia for which he is otherwise quite famous, and given his belief that a legitimate regime, a regime in which the people are sovereign, presupposes and requires social unity. So what is a theorist of socionational unity, like Rousseau, doing relying on a _foreign_ -founder?
Moreover, why would a radical democrat like Rousseau turn to a lawgiver at all? Why not allow a people to constitute itself, as such?What problems of founding are managed by the introduction of a founder? What problems of founding are managed by way of the founder's foreignness? Are there any problems of democracy, in particular, that are being worked through, symbolically, by way of the device of foreignness? And why doesn't Rousseau, for whom social unity was a most important goal, worry that the introduction of a foreigner into his ideal polity might undermine rather than animate his experiment in radical democracy?
Rousseau comes to perceive the need for a lawgiver when he considers for the first time, in Book II, Chapter 3, of the _Social Contract_ , the possibility that the General Will can err. Although he insists that the General Will as such cannot err (if it erred, it would simply not be the General Will), Rousseau's consideration of the possibility opens up a troublesome gap between the people doing the willing, and the General Will that is supposed to be the effect (but also the cause) of their willing.
"[T]he general will is always right and always tends toward the public utility. However, it does not follow that the deliberations of the people always have the same rectitude" (ibid., p. 155). Here Rousseau evidences an anxiety that plagues most radical democrats who agitate to give the people power. Popular sovereignty is supposed to _solve_ the problems of (il)legitimacy and arbitrariness. But once the people have power, that "solution" suddenly looks like a problem, for the people, too, can be a source of arbitrariness. The rhetorical dynamic goes some-thing like this: "We need to give the people power! The people must have power! Omigod, the people have power! What are they _going to_ _do_ with it?"
Rousseau tries to maintain his faith in the people even under the pressure of his anxiety—"the populace is never corrupted, but it is often tricked"—but he cannot finally fend off the concern that the people, upon whose will the legitimacy of the regime is supposed to rest, will not necessarily exercise their will in the _right_ way.10 At the same time, Rousseau seems both to trust in the ability of ideal circumstances to secure a mutual transparency for the populace (necessary for the emergence of a General Will that can be known as such), and to despair of ever overcoming the stubborn opacity and mutual strangeness that mark human relations even under ideal circumstances. When under the sway of the former assumption, Rousseau characterizes the process of general willing as a process of public deliberation and debate. But under the sway of the latter assumption, responding to the conditions of mutual opacity and mistrust, Rousseau insists that discussion will only fan the flames of factionalism, and concludes that the General Will will be produced, if at all, only by way of silent introspection (Book IV, Chapter 2).
Either way, Rousseau does not believe that the people can be the solution to arbitrary, illegitimate, and coercive rule without also becoming a problem that rules or laws must solve.11 You cannot have good laws without good people, and you cannot have good people without good laws, he says as he presents the problem as a problem of founding (though in fact it recurs daily as subjects are [re]socialized into citizenship): "For an emerging people to be capable of appreciating the sound maxims of politics and to follow the fundamental rules of statecraft, the effect would have to become the cause. The social spirit which ought to be the work of that institution, would have to preside over the institution itself. And men would be, prior to the advent of laws, what they ought to become by means of laws" (Book II, Chapter 7, p. 164). A _deus_ _ex machina_ is necessary to solve the problem, and this _deus ex machina_ arrives in the figure of the founder, a good person prior to good law, a miraculous lawgiver.
From what Rousseau says about the founder, others have concluded that he must be a god, a divinity. Indeed, Rousseau himself seems to say as much:
The discovery of the best rules of society would require a superior intelligence, who saw all of men's passions yet experienced none of them; who had no relationship at all to our nature yet knew it thoroughly: whose happiness was independent of us, yet who was nevertheless willing to attend to ours; finally one who, preparing for himself a future glory with the passage of time, could work in one century and enjoy the reward in another. Gods would be needed to give men laws.(Ibid.)
But Rousseau does not himself depend upon gods. He goes on to give a great deal of advice to the "one who dares to undertake the founding of a people" and it is here that he mentions, as did Aristotle before him, that "it was the custom of most Greek towns to entrust the establishment of their laws to foreigners."13 Rousseau adds to Aristotle's list, noting that "the Republics of modern Italy in many cases followed this example; Geneva did the same [here Rousseau is thinking of Calvin as a founder] and profited by it."
Rousseau's mention of famous foreign-founders at this crucial point in his text suggests that he sees foreignness as a way to manage some of the challenges that face a founder: Who besides a god or a godlike man would be able to discover the best rules for a society, see all of men's passions yet experience none of them; have no relationship at all to our nature yet know it thoroughly and, perhaps most important of all, have a happiness that is independent of us? These characteristics might be found only in a man of perfect virtue. But they—or something enough like them—might just as well attach themselves to a foreigner.15 Someone who comes from somewhere else is familiar with human nature, intrigue, and ambition but is not himself captivated by the particular intrigues at work here, in this new place, in which he has no invest-ment or history. A foreign-founder's foreignness secures for him the distance and impartiality needed to animate and guarantee a General Will that can neither animate nor guarantee itself.16 Moreover, because he is not one of the people, his lawgiving does not disturb the equality of the people before the law. And finally, his foreignness may well add to his charms and enhance his leadership. No known genealogy demystifies his charismatic authority.
But this solution comes at a price, particularly for a democratic founding. The lawgiver leads the citizens to a legitimate set of arrangements, but he also positions citizens in a relation of heteronomy that is deeply at odds with Rousseauvian legitimacy. As Alan Keenan puts it, "Rousseau's recourse to the legislator" is "surprising, even farfetched, " because it appears "in a text devoted to laying the foundations of popular autonomy."
The problem is not unique to Rousseau's democratic ideal. The dependence of weak, dependent, democratic citizens upon strong, heroic, independent strangers is played out repeatedly in American popular culture as well, most noticeably and ritually in movie Westerns. As in one classic of the genre, _Shane_ , a heroic stranger arrives from else-where to save a weak town from powerful and corrupt bandits.19 (In _Shane_ , these bandits are led by a man named Riker.) The hero saves the town through the sheer power of his exemplary if flawed personality, innate sense of justice, and his mighty prowess with firearms. By the film's end, the sources of corruption have been excised (Shane kills Riker), and the townspeople are restored to virtue and reempowered for self-governance. Although the people usually beg him to stay (sometimes with no small degree of trepidation), the hero moves on when his (re)founding work is done.20 In _Shane_ , the hero explains that he cannot stay because murder never goes away.
Similarly, in Rousseau's _Social Contract_ , the founder leaves after founding the polity. There is no provision for the office of legislator in Rousseau's _Social Contract_. "This function, which constitutes the re-public, does not enter into its constitution." The relief experienced by the locals when the foreign-founder, whom they may well have begged to stay, decides rather to leave recalls the "terror" which, according to Jacques Derrida, attends the eternal waiting for the Messiah: "Who has ever been sure that the expectation of the Messiah is not, from the start, by destination and invincibly, a fear, an unbearable terror—hence the hatred of what is thus awaited?"
If the fortuitous arrival of the foreign-founder seems too good to be true, his timely departure seems almost beyond belief.22 This timely departure is one of the conveniences of the classic foreign-founder script, and it is necessary for several reasons. The energies of the foreign-founder, so valuable and much-needed in a time of (re)founding, are threats to established regimes. In the case of Shane, the very mysteriousness, individuality, audacity, charisma, passion, violence, and power that enable him to save the town he passes through also make him dangerous to the community on a daily basis. The violence he used to excise the town's oppressors was indispensable to the fragile community, but that same community, still grateful, is now endangered by the continued presence of that violence in its midst.
Shane's threat to the restoration of the ordinary is figured as a threat to the nuclear family at the film's center. The wife-mother is attracted to Shane, and the son clearly prefers the stranger to his own father, who is less manly, more human-all-too-human, than the visiting hero. As long as Shane stays in town, life cannot be restored to its routine heteronormative safeties.
A distinctively republican perspective on the matter is articulated by Abraham Lincoln, who notes that the same men who are great founders in extraordinary times are fated to be criminals in ordinary times. From a republican perspective, the danger posed by would-be founders stems not so much from their individuality, per se, as from their greatness and ambition, which can be boundless. "M]en of ambition and talents . . . naturally seek the gratification of their ruling passion."[23 However, this particular republican concern is not evident in _Shane_ nor in the other foreign-founder texts examined here, and this raises a question about what exactly this founder "ambition" is supposed to signify. Is the problem always with the founder, as the republican account seems to suggest? Or is it possible that what gets characterized as a founder's ambition is in fact (also) a projection of the people's own (illicit and therefore denied) desire to submit in a very nonrepublican way to the will of the inspiring and charismatic founder rather than to the law he founds?
Either way, whether it is he himself who desires to rule or the people themselves who desire to be ruled over, the lawgiver must leave. Rousseau's lawgiver must leave not so much because he represents a danger to the community (as in the case of the stranger of the classic Westerns), but because the people must at some point be left to their own devices. This is supposed to be a democratic people. Law and order may be put (back) into place by way of the founder's agency, but the regime now depends upon his departure (and, indeed, upon his disappearance, lest he develop a posthumous cult following. This is the significance of the fact that, as Harold Bloom points out, "No one knows Moses'] burial place to this day").[24 The founder must leave (or, as in the worthy example of Lycurgus, he must abdicate),25 and he must not return. In Rousseau's _Social Contract_ , the lawgiver's foreignness may even function to reassure the people that he really will leave when his time comes. Why should he stay? He has no ties to this place.
But there is surely something too neat about this script. Can the foreignness of the founder only serve the regime well and not also unsettleor disturb it at the same time? The foreignness of Rousseau's lawgiver may stand for his ability to—it may even even enable him to—\found wisely, but wouldn't that same foreignness also make him (and perhaps also his law) a threat to the social unity that is a necessary condition of Rousseauvian democracy?
Rousseau's assumption of a timely departure is problematic at the level of theory because it seeks to prevent the foreignness of the founder from ever becoming a problem for the regime that profits from it. A greater sensitivity to the dilemmas of the foreign-founder's position and the limited efficacy of timely departures is exhibited by Yasushi Akashi who, as U.N. Special Representative to Cambodia, was himself a kind of foreign-founder. Akashi's job was to enter into the "wreckage of a country] trying to start over after civil wars" and provide ways out of political impasses that warring parties could not resolve for themselves. Responding to criticisms that he ought to have been more zealous in "razing Cambodia's Communist-dominated government departments to clear space for more democratic institutions," Akashi explained that during his tenure in Cambodia a certain foreign-founder from America was always at the back of his mind: Douglas MacArthur. The MacArthur-led American postwar occupation of Japan was "enlightened and generous and liberal, " Akashi said. "But some of the democratic policies were changed, some even abolished by subsequent, more conservative, Japanese governments—not because they were opposed to the policies, but because they were given to us by foreigners."[ 26 For Akashi, the foreign-founder is a radically undecidable figure. His foreignness may benefit the regime he (re)founds, but it is also a threat to the regime at the same time.
The biblical Moses, one of three lawgivers most admired by Rousseau, was also a foreign-founder, and he, too, was marked by this undecidability. 28 Raised in Pharaoh's household, Moses was thought initially to be an Egyptian. True, Exodus explains that Moses was really an Israelite, abandoned by his mother in the hope that he might escape the death sentence decreed for male Israelite infants. But even if Moses' origins can be so neatly purified by genetic fiat, it remains the case that he was brought up as an Egyptian, not an Israelite. He must have spoken differently from the people he ultimately led. (Freud suggests that Moses' famous stammer, his "heavy tongue," might not be a simple"historical truth" but rather a way of recalling, in "slightly distorted"form, "the fact that Moses spoke another language.")29 Moses must have carried himself in a foreign way. This may have enabled him better to represent the Israelites to Pharaoh, who had until recently been a member of Moses' family, after all.30 And Moses' foreignness may have contributed to the Israelites' willingness to be led by him. After generations of slavery, how could the Israelites not think that an Egyptian accent, bearing, and affect bespoke authority and perhaps even superiority?Surely, following the lead of an Egyptian was established habit by now.
But that same foreignness, so enabling to Moses' leadership, must also have attenuated Moses' relation to the Israelites. Indeed, it may well be that it was only Moses' timely death, before the Israelites entered the Promised Land, that prevented these awkward questions about membership and identity from coming up for a real interrogation."Is Moses really one of us, or not?" the Israelites might well have asked once settled in the Promised Land. Either way, Moses would have been in trouble, of course. As a mere member of the people, his power to found them and give them the law might have been undone. As an alien, he and perhaps even his law might have had to have been immediately ejected from the Israelites' midst.
Rousseau never confronts the problems that might be posed by the lawgiver's foreignness and its undecidability. His assumption of a timely departure allows him to avoid the issue, and he never asks about the effects the founder's foreignness might have on a regime even long after the founder's departure. Nor does Rousseau inquire into the particular circumstances of the lawgiver's departure. Does he jump, or is he pushed? For a consideration of these issues, we have to turn to Freud, who sees the foreignness of Moses as the object of one of the greatest cover-ups of all time.
Freud's Moses
Notwithstanding Rousseau's idealized version of the foreign- founder script, the founder's foreignness does not only reassure. If the foreign-founder must leave after the work of founding is done, that need stems from the very foreignness that so enables foreign-founders, and not just from their necessary but denied violence, not just from their structurally awkward role as lawgivers in an otherwise egalitarian democracy, and not just from the founders' off-putting personal quirks, ambitions, or vices. Sigmund Freud, in _Moses and Monotheism_ , offers a reading of the Moses story in which the foreignness of Moses runs much deeper than in the speculative account I offered above; and, sure enough, in Freud's version, Moses' leadership ends not by natural, if ordained death (as in the official version), but by his murder at the hands of the people he founded.
According to Freud, Moses is "an Egyptian—probably of noble origin—whom the myth undertakes to transform into a Jew" (ibid., p. 13).31 Beginning by noting that Moses' name is not Hebrew but Egyptian (pp. 4ff), Freud goes on to draw on the structural similarities between his own account of the Family Romance (in which the child, disappointed in his parents but especially in his father, fantasizes that he is adopted and that his real family is one superior to that from which he is presently trying to escape) and Rank's account of exposure myths, in which the hero, such as Cyrus or Romulus, is thought to be of low birth but turns out to be of high birth.32 In both the Family Romance and the exposure myths, Freud points out, it is the supposedly adoptive family that is the real one; the supposedly real (lost and hoped-for) family is a figment of the individual or cultural imagination. Moses' story reverses the chronology of humble and noble, but it is otherwise structurally similar to these other fantasies and myths.33 So, Freud concludes, here too the supposedly adoptive family must be the real one; Moses is really a member of Pharaoh's household. Moses is foreign to the people he founds.
Rousseau's foreign lawgiver is guided by his removed judgment when he decides which mores are best suited to a people given their histories, cultures, territory, climate, population size, and other similar considerations. Rousseau assumes that the biblical Moses was similarly guided, attentive to considerations of fit and proportionality when he gave the law to the ancient Israelites.34 But Freud's Moses is guided by a quite different agenda.
Freud's Egyptian Moses is not primarily interested in looking for the mores that will best fit a group of individuals and help mold them, lastingly, into a people. Instead, Freud's Moses is looking for a people to vivify a moral code bereft of followers, the suppressed religion of Ikhnaton to which he, Moses, remained loyal. An Egyptian follower of Ikhnaton's monotheism, Moses was deeply affected by the Egyptian rebellion against that religion after Ikhnaton's death: "If Moses] was not to recant the convictions so dear to him, then Egypt had no more to give him; he had lost his native country." Homeless at home, Moses stumbled in his "hour of need" upon "an unusual solution." He "conceived the plan of founding a new empire, of finding a new people, to whom he could give the religion that Egypt disdained" (ibid., pp. 31–32). Moses molds the Israelites into Ikhnaton's religion, passing on to them mores that are utterly foreign to them; he imposes upon them alien customs (such as circumcision, a practice that was associated with the ancient Egyptians, Freud argues);[35 and he commands them to obey a god that is not theirs.
_In sum, Rousseau tells the story of a people in search of religion, while_ _Freud tells the story of a religion in search of a people:_ it's a great image—a homeless religion wanders around Egypt and its environs looking for followers. While the foreignness of Rousseau's founder is a sign of his ability judiciously to find the right fit between a people and the mores that will bind them together, the foreignness of Freud's founder is a sign of the harsh ill-fittedness of the religion whereby he founds his people.36 While there is violence in both foundings—Rousseau's lawgiver forces people to be free and soon introduces a death penalty; Freud's Moses suppresses resistance with "savage chastisement"—Rousseau treats that founding violence as both justified and incidental, while Freud sees it as savage, and significantly so.37 The violence of Rousseau's founding is not a sign of the law's illfittedness to the people; rather, it is a sign of some people's recalcitrance even when faced with good law. Freud, however, sees the violence of the Mosaic founding as a sign of his law's fundamentally impositional, harsh, and foreign character. Monotheism is too abstract to satisfy the instinctual longings of this people, and it is also a foreign import. Or, better, abstract monotheism, because it is so strict and renunciatory, is experienced and represented as an alien (and, more concretely, a foreign) thing.
We might capture the difference between Rousseau and Freud in psychoanalytic terms: Rousseau's foreign-founder is like the good father whose primary concern is his children and their needs, which he is always trying to serve.38 Freud's foreign-founder is like the bad father who uses his children to fulfill needs that are his own.
As it happens, these good and bad fathers correspond to the rather different natural fathers imagined by Rousseau and Freud, respectively, in _On the Social Contract_ and _Totem and Taboo_. In Rousseau's state of nature, "children remain bound to their father only so long as they need him to take care of them. As soon as the need ceases, the natural bond is dissolved." But who decides when the "need ceases, " and how? And what if the parties disagree about this?39 (One cannot help but recall here the plaintive cries of the young boy in _Shane_ as he watches his hero ride away: "Shane, come back!")40 Rousseau never raises these questions. They probably never occurred to him since he would have assumed that these things are decided uncontroversially in a state of nature. But the pattern of timely withdrawal is repeated outside of the state of nature. The idealized father, who withdraws at the indisputably correct time (and who always knows what the indisputably correct time is) so his children can live as independent beings, free of paternal authority, is Rousseau's model for the lawgiver who makes his appearance thirteen chapters later in the _Social Contract_.
By contrast, Freud's natural father, who makes his appearance in _Totem and Taboo_ , has excessive desires, and the needs of his children are the last thing on his mind. He is in no way self-limiting, and he rules over his sons with authoritarian force. He does not leave when his sons reach maturity, and, finally, when they can no longer tolerate his prohibitions, particularly those against sharing the women of the clan, they murder him. Even that does not rid them of his overbearing presence, however. They internalize the very prohibitions against which they chafed in his lifetime—they eat him—and they are haunted by his will and their violence ever after.41 Freud's Moses reperforms this script: "The great deed and misdeed of primeval times, the murder of the father, was brought home to the Jews, for fate decreed that they should repeat it on the person of Moses, an eminent father substitute" (ibid., p. 113).
There can be no adjudicating the "truth" of these two imaginings. Indeed, the issue here is not about "fathers" per se but about how these different imaginings of the natural father shape our thinking about the problems of authority, freedom, and founding that foreign-founder scripts are intended to manage. In both Rousseau and Freud, the pater–nal figure—the natural father—stands for the law, authority, or powers by which we are governed but which we have not ourselves authored and to which we have never consented. For Rousseau, theorist of the general will, paternal power represents the very problem he is trying to solve, the problem of the law's arbitrariness and illegitimacy. ("Man is born free and everywhere he is in chains" Book I, Chapter 1].) But rather than solve that problem, Rousseau merely avoids it by imagining that fathers whose will is impositional and never justified will both recognize the moment of departure when it comes and then obligingly disappear. For Freud, that obliging disappearance is a fantasy, at best a screen memory behind which the murder of the foreign-founder, the man Moses, is concealed.[42 Rousseau's account of the natural father who leaves when his sons are ready invites us to imagine that arbitrary and impositional powers simply withdraw when children—or citizens—reach the age of majority. Freud's account promises us that paternal authority never withdraws voluntarily; we always have to take it rather than wait for it to be handed over, and even if we think we have rid ourselves of its power, it will return to haunt us.
That haunting occurs in Rousseau as well, however. Indeed, we might see Rousseau's foreign-founder as a kind of return of the re-pressed father. On this interpretation, the felt but unacknowledged alienness of the law of the natural father by whom we were governed, however benevolently, without our consent finds expression in the curious foreignness of the founder who also governs us without our con-sent until, in his judgment, we reach the age of political maturity.
In other words, Rousseau teaches us that even when the terms of our governance are good for us, we cannot but experience (paternal) authority as alien (it does not come from us). Perhaps the concept of _foreignness_ is a way of symbolically marking and making sense of that alien experience, a way of giving it cultural organization while also displacing it outward and disowning it; it comes from somewhere else. If the founder—who shapes our identity as a people, who ushers us into maturity and enables us to be autonomously self-governing—is foreign, that is because the law by which we were founded is always lingeringly alien to us since we did not (indeed, we could not) will it for ourselves.43 (That is why we needed a founder in the first place!) And we never will—at least not in the full, unvarnished sense required by would-be radical democrats like Rousseau.
Such a reading of Rousseau offers one possible explanation for the more general frequency and circulation of foreign-founder scripts in political culture. On this account, the foreign-founder script both domesticates and preserves in its entire uncanniness the alienness of the law. That Rousseau, a xenophobic theorist of self-identity, should him-self rely on the figure of the foreign-founder shows just how inescapable and thoroughgoing is this sense of the law's alien character, even in a particularly well-ordered regime, which is what Rousseau thought he was theorizing in the _Social Contract_.
Indeed, Rousseau knew this: noting that the task of the legislator is to "change human nature, " Rousseau says that in order to do this the legislator must "deny man his own forces in order to give him forces that are _alien_ [in another translation, the term—"étrangères" in the original—is rendered " _foreign_ "] to him and that he cannot make use of without the help of others" (Book II, Chapter 7; emphasis added). Might the foreignness of the foreign-founder, his alienness, be a sign—and a vehicle—of the (self-)alienation that is the necessary basis of life among others under law? It may be that one function of the mythic foreign-founder (in this context) is to make some sort of sense of that felt alienness of the law, marking that alienness but domesticating it at the same time by way of a story of a nice (if somewhat short-tempered) foreign-founder, which is what Moses was before Freud got his hands on him.
_In short_ , contra _Freud, it is not because the tablets were handed to_ _the Israelites by an Egyptian Moses that the Israelites experienced the_ _law as alien and impositional (which is why Moses had to use force to_ _secure its implementation). Rather, it is because of the law's alien and_ _impositional character that the story of the law's origins had to be one_ _in which it was an Egyptian—a foreign-founder—who handed the tablets_ _to the Israelites._
(Freud misses this point because he is so intent in _Moses and Monotheism_ on trying to prove that Moses was _really_ an Egyptian, as a matter of empirical fact, that he never poses the symbolic question: _Why_ was Moses an Egyptian?44 What cultural, symbolic or political work was accomplished by way of his "foreignness"? This omission is astonishing coming from the author of _The Interpretation of Dreams_ , though it is understandable given the exigencies that must have driven the Jewish Freud's project of writing about Moses and monotheism in the 1930s, first in Austria and then, as an exile, in England.)
But the law is not just alien; it is both strange and familiar at the same time. And that is why the foreignness of Moses is not simply concealed, as Freud assumes—it is _poorly_ concealed, poorly concealed by way of a genetic fiat that had to fail: Moses' foreignness is concealed in order to signal and secure the people's identification with the law; but it is poorly concealed in order to preserve a marker of the law's alienness to the people who live by it.
There is no way to avoid this sense of the law's alienness. But its character as a problem is severely aggravated when we are dealing with _democratic_ law (which is supposed to be coming from the people, after all), and when democracy is conceived of in Rousseauvian republican terms as a politics of radical self-authorship and self-identity.46 However, there are good, radical democratic reasons for wanting to preserve rather than heal the sense of alienation that Rousseau—and certain contemporary theorists of legitimation and deliberation47 —try so hard to overcome.
Legitimation theorists worry that alienation can be a source of civic cynicism and withdrawal. It can. But it can also be a source of civic activism, unrest, and protest. The positive side of "alienation" is that it marks a gap in legitimation, a space that is held open for future refoundings, augmentation, and amendment. That gap is closed by those who read alienness out of Rousseau's text (whether by ignoring or domesticating the foreignness of the lawgiver), are blind to the haunting opacity of the people to one another in Rousseau's polity, and are inalert to the ambiguity of the law that both is and is not the product of the General Will, produced both by public deliberation and by silence, generated by the people but also imposed by the lawgiver.
In sum, Rousseau and Freud together present a fuller picture of the symbolic functions of foreignness than does either theorist alone. From Rousseau, we learn that foreignness models founding virtues such as objectivity, disinterestedness, and impartiality, while also symbolizing a denaturing self-alienation that life under law enacts and requires. From Freud we learn that foreignness signals and maybe also aggravates the trauma of norm transmission or the imposition of law. These perspectives are not necessarily in conflict. What is (allegedly) impartial and objective invariably appears alien and violent from the vantage point of the particular. But these perspectives are not necessarily compatible, either. What is allegedly impartial and objective may also appear alien and violent from a vantage point that understands itself to be impartial and objective, too.
Similarly, Rousseau's and Freud's respective theorizations of the figure of the foreign-founder together produce a more complete picture than does either alone. Rousseau imagines that the supplement of foreignness is entirely successful and not also unsettling to the order founded by its agency. Freud, by contrast, sees the founder's foreignness as entirely unsettling to the nation and as an aggravator of the already traumatic process of norm transmission. Each thinker provides an important corrective to the other. Taken together, they teach us that the supplement of foreignness is undecidable: it both shores up (Rousseau) _and_ unsettles (Freud) the people or the law being founded.
As I argue in the chapters that follow, the foreign-founder's undecidability is inescapable and, indeed, necessary (even if also threatening) to his founding mission. In the case of the biblical Ruth, for example, it is simply not possible for her to shore up the nation without also unsettling it at the same time. This undecidability sets in motion a politics of (re)founding, which involves the plural efforts by postfounding generations to (re)define their collective identity by retelling their origin stories or by inventing new ones. For those whose origin stories feature a foreign-founder, the politics of refounding often involve a contest to erase that figure from memory or to position him as _either_ foreign to _or_ founder of the nation. Rarely is the foreign-founder celebrated as such. The political-cultural struggles to mark or resolve the undecidability of the foreign-founder figure are a major focus of the chapters that follow. For now let us complete this stage of our analysis of the foreign-founder figure by putting it into proximity with another liminal figure who is also a vehicle of social or political reconstitution: the scapegoat.
Girard's Scapegoat
The setting for the foreign-founder's appearance may be described, in René Girard's terms, as a sacrificial crisis, a crisis in which "the whole cultural foundation of the society is put in jeopardy" ( _Violence and the_ _Sacred_ , p. 49). What Rousseau saw as the threatening divisiveness of factionalism or self-interest, and what _Exodus_ describes as a lapse into idolatry, Girard describes as a loss of common distinctions, a loss of communal unanimity. The loss of unanimity returns the community to an unending cycle of violence. Rousseau and Girard both think the solution is renewed social unity or unanimity and both see an outsider as a necessary vehicle of that solution. But for Rousseau, social unity is achieved in response to the leadership and direction provided by an outsider, a foreign-founder, while for Girard, unanimity is achieved by way of opposition to an outsider—a sacrificial victim or a scapegoat:"The scapegoat is the innocent party who polarizes a universal hatred"( _Job, the Victim of His People_ , p. 5). What counts is not the hatred but—as in Rousseau at the founding of the social contract—"the communal gesture of unanimity" ( _Violence and the Sacred_ , p. 101). In Girard, that unanimity is achieved only in relation to an outsider because "otherwise the community might find it difficult to unite against it" (ibid., p. 102).
Sacrificial victims are chosen from among those "who are either out-side or on the fringes of society," such as "prisoners of war, slaves, small children, unmarried adolescents, and the handicapped." The list, which "ranges from the very dregs of society, such as the Greek _pharmakos_ , to the king himself," covers a diverse group, but its members have something in common: all are "exterior or marginal individuals, incapable of establishing or sharing the social bonds that link the rest of the inhabitants."49 The scapegoat's outsider status ensures that the violence performed against him will not be avenged: "Between these victims and the community a crucial social link is missing, so they can be exposed to violence without fear of reprisal. Their death does not automatically entail an act of revenge"(ibid., p. 13).
Girard repeatedly emphasizes the beauty and completeness of the scapegoat solution, noting that violence is symbolically expunged or absorbed by a ritual that never exceeds the economy to which it is assigned. But this account, no less than Rousseau's vision of a lawgiver who leaves at the perfectly right time, is surely too neat.50 Moreover, Girard never inquires into the outsider status of those taken to be safe objects of violence. It is as if their empirical marginality is what causes their scapegoating. And yet Job, perhaps Girard's lengthiest example of a scapegoat, was not an outsider. He was a member of the community that came to despise him, probably because his run of bad luck made his neighbors nervous about their own fate.51 Then they cast him out. With this turn of events, he _becomes an outsider._ As he himself says, "The serving maids look on me as a foreigner, a stranger, never seen before" (quoted but not commented upon by Girard, _Job_ , p. 4).
Now, this suggests that the scapegoat need not be an actually existing foreigner but rather anyone whom the community can successfully and unanimously cast as one. And, indeed, in his book on Job, Girard tracks the social processes that produce Job's marginalization. In that book, if not in _Violence and the Sacred_ , Girard seems to understand that the individual is not a scapegoat because he is a foreigner; instead (as Job found out), he is (or becomes) a foreigner because he is a scapegoat. Or, better, these two of Girard's books in tandem rightly teach us that the practice of scapegoating sometimes chooses its objects from an al-ready existing, available pool of outsiders and at other times produces its objects from among the members of the community in crisis. The important point here is that scapegoating is not caused by scapegoats—an already existing pool of outsiders. Scapegoating is a social practice that finds or produces the objects it needs.
A scapegoat is a figure made to represent some taint borne by the community as a whole, in particular, the loss of distinctions that defines the sacrifical crisis from which the community is trying to recover. The attribution of that taint to a scapegoat allows the community unanimously to disavow it, and the ritual murder of the scapegoat cleanses the community and reestablishes the lines of proper order that had be-come so dangerously attenuated. Girard's scapegoat theory is valuable to an analysis of the politics of foreignness because it presses us to attend to the _politics_ of foreignness—the cultural symbolic organization of a social crisis into a resolution-producing confrontation between an "us" and a "them." Moreover, Girard offers us a new angle from which to reread our foreign-founder texts.
In a Girardian retelling of the Moses story, Moses would be a scape-goat (especially if we take seriously Freud's suggestion that Moses was murdered by his people), and Moses' foreignness would be a projection (not, contra Rousseau, Freud, and Girard in _Violence and the Sacred_ , an empirical fact, or at least not necessarily so). Such a retelling might go like this. In the desert of Sinai, the Israelites long for the "fleshpots of Egypt" and worship idols while awaiting Moses' return from the mountaintop. After Moses returns and reinterpellates them into the law of monotheism, the Israelites seek to disavow and expunge their forbid-den lapse into identification with Egypt. They project that forbidden identification onto Moses, calling _him_ an Egyptian, later inscribing him into their cultural narratives as a (n adopted) son of Pharoah. Then they kill him in order to cleanse themselves of their sin, and they say he died of natural causes.52 Poor Moses is a good candidate for scapegoating not because he is a foreigner (on this reading, he isn't . . . yet) but perhaps because, as a leader, he is, as Girard points out, a liminal figure, a quasi insider who is already in violation of the social links that scapegoat rituals seek to shore up. Alternatively, in more Freudian terms, Moses may be chosen out of the group because it was he who instituted the law that occasioned the Israelites' collective transgression and violence.
Girard's account suggests new readings of our other texts as well. Earlier, we thought that the Munchkins need Dorothy to do their dirty work for them because they are not bold enough to take on the witches that terrorize them, and we thought of Shane as a hero who comes in to save weak, would-be democratic citizens from bandits or bullies like Riker. We took these accounts at their face value, and we worried that citizens in need of rescue at their origins ( _or, more pointedly, citizens_ _who tell themselves such stories about themselves_ ) could hardly be prepared for the daily challenges of democracy in which, after all, bandits or bullies surface all the time while powerful rescuers do not. But what if neither the Munchkins of Oz nor the local citizens of _Shane are_ weak? What if the stories they tell about themselves suggest instead that they are actually quite powerful, powerful enough to do the violence that their (re)founding requires (let us say they do take on the wicked witches or face off with Riker's men), and powerful enough to blame a scapegoat for it, even generating a cover story that almost conceals their implication in the violence and seeks to relieve them of responsibility for it?
The people's unwillingness to take responsibility for their (re)found-ing violence might be psychological, but it might also be political. An-other way of putting Girard's point about the need to perform sacrificial violence in order to end cyclical violence without inaugurating a new cycle of violence is to say, as many democratic theorists do, that ordinary democratic life demands a measure of stability and routine that might be impossible to secure for citizens known to be capable of great passion and violence.53 It was in the spirit of this belief that Abraham Lincoln said of the first American founders: "Passion has helped us; but can do so no more. It will in future be our enemy."54 What if, believing this, the locals invent a scapegoat, a Shane or a Moses or a Dorothy, who is then said to have been the agent of the violence that was really committed by the would-be citizens but which they now need to dis-avow? Or perhaps the Shane or the Moses or the Dorothy figure is in-deed one of their number and a party to their collective violence but is then said to have been the leader and even (in an effort to externalize the problem) the outside agitator of it?55 Whether wholly invented or chosen out of the collectivity, or a little bit of both, the figures that personify the violence take it upon themselves (or suffer its projection onto themselves) and thereby, Christlike, absolve the collectivity of implication in it.
Now what if we apply this approach to Rousseau's _Social Contract_? We might see the arrival of the figure of the lawgiver not (contra William Connolly, Alan Keenan, Geoffrey Bennington, and others, including myself above)57 as a sign that the project of General Willing has failed, but rather as a sign that it has succeeded. Think of what the lawgiver is said to bring to the people—not just perspicacity but also a death penalty, not just heroic beneficence but also stern discipline. What if these are in fact willed by the General Will, generated by the people them-selves, imposed upon themselves, and exacted from among their own number? What if the too-good-to-be-true arrival of the lawgiver is just a story we tell ourselves so we can disavow, rather than take responsibility for, those violences, those ritual and nonritual sacrifices on which the founding and daily maintenance of our democratic polity depends?
This interpretation of the _Social Contract_ may seem a bit far-fetched, but there are several things that can be said in its favor. First, it allows us to treat as continuous the aspiration of _The Government of Poland_ and the _Social Contract_. In _The Government of Poland_ , Rousseau ad-vises us to build national unity by telling and performing stories of he-roes we admire. What if the _Social Contract_ is just such a story, the story of our valiant but failed effort to legitimate our chains by willing the General Will and our happy rescue by a fortuitous foreign-founder?
Another virtue of such a reading is that it solves a problem that has puzzled Rousseau's readers for ages: Why does this radical democrat for whom popular sovereignty is the only solution to the problem of legitimacy turn, at the last minute, to a founder, and to a _foreign_ -founder at that? Not (contra Connolly, Keenan, Bennington, and myself above) because he needs an external source to animate a General Will that cannot animate itself. Rousseau himself certainly never admits _that_. True, he puzzles over whether the General Will can err. But were the General Will really unable to set itself in motion, that would mean the whole project of the _Social Contract_ had failed, and that would be quite an admission to make only halfway through the argument in its favor. Instead, what if the lawgiver is brought in because of the General Will's _success_? What if Rousseau calls on the lawgiver because he seeks to externalize the General Will's violence, the willed violence of (re)-founding?
That externalization may be best achieved by way of a story, and a figure, rather than by way of an analytic argument. Perhaps, rather than being a philosophical justification of political violence (most famously answering the question: Under what circumstances can a person be "forced to be free"?), Rousseau's text is performative in character. What if, rather than _argue_ for the legitimation Rousseau seeks, Rousseau's text tries to _bring that legitimacy into being with an origin story_? ("Rousseau had always considered texts about politics to be political acts, " says Neil Saccamano in a reading of Rousseau that has great affinities with the one developed here.)58 Taken as a public myth of origins, Rousseau's _Social Contract_ performs the exorcism of a local violence on which citizens depend, but which they must disavow lest it consume them.59 The price of that exorcism is high: the introduction of the law-giver-scapegoat and the erasure of the people's miraculously successful willing of the General Will.
Perhaps the greatest virtue of this reading is that the foreignness of Rousseau's lawgiver suddenly makes sense. No longer is that foreignness a puzzling departure from Rousseau's famous xenophobia. Instead, in keeping with that xenophobia, the founder's foreignness can now be seen to enhance his effectiveness as a lawgiver-scapegoat. What quality, other than foreignness (viewed from a xenophobic perspective like Rousseau's) could better secure the lawgiver's role as scapegoat?
As is now clear, this new take on the _Social Contract_ switches our sense of the lawgiver. No longer is he a repetition of the idealized, independence-granting father of the state of nature (Book I, Chapter 2). Instead, the lawgiver, on this account, is the mirror opposite of that first father. Where the first, natural father's self-limiting nature absolved us of implication in any violence against him by making such violence (unbelievably) unnecessary, the second father figure absolves us of implication in violence by appearing to do our dirty work for us. Moreover, his timely departure or murder helps to sustain the fiction that whatever violence we are involved in is confined to the generation of founding. Thus, we tell ourselves, the violence that touches our regime is contained and final rather than boundless and cyclical. Our clean hands come at a price, however. The price is our own democratic power, the power to act in concert as a sovereign people. The story we tell ourselves about ourselves is an infantilizing origin story in which we abdicate democratic responsibility for our common life together. Also on this reading, then, Rousseau's introduction of a paternal figure continues to puzzle, even if its foreignness does not.
Democracy and Foreignness
As we shall see in the chapters that follow, a thorough analysis of the symbolic politics of foreignness draws upon the multiple, sometimes conflicting insights of all three of the thinkers addressed here: Rousseau, Freud, and Girard. But what if we treat these authors' texts not just as analyses of a problem but also as alternative origin stories? Which of these texts would serve most effectively as a generator of democratic agency? Curiously, given that Rousseau was theorizing democracy and given that Girard is trying to understand sacrificial violence for the sake of our collective life together, it may be Freud, the one most removed from democratic theory, who provides the best resources out of which to generate a model of agency for would-be democrats.
If we treat Freud's _Moses and Monotheism_ as itself an origin story (that is to say, if we treat that text as a countermyth and not just as a study of an origin story), and if we set aside for a moment Freud's severe demand that the subject renounce instinctual life for the sake of a rational life under abstract law, then certain salient elements of a potentially democratic agency emerge from Freud's account: here we have subjects who exist in agonistic relation to a founder whose alienness is a poorly kept secret; subjects who do not expect power to be granted to them by nice authorities with their best interests at heart; subjects who know that if they want power they must take it and that such taking is always illegitimate from the perspective of the order in place at the time;60 subjects who know that their efforts to carve out a just and legitimate polity will always be haunted by the violences of their founding; subjects who experience the law as a horizon of promise but also as an alien and impositional thing.
Such subjects are, I would argue, better prepared for the ongoing, always changing demands of democracy than those Rousseauvian sons of kinder, gentler fathers who expect the reigns of power to be handed over to them when the time is somehow objectively, impossibly right. Such modified Freudian subjects seize power themselves and convert their experience of alienation into a source of transgressive (because unauthorized) democratic energy. They are prone to repeat and relive their originary traumas. They do not expect their father figures or their own violences and transgressions to confine themselves to preassigned locations or economies.
Democracy is always about living with strangers under a law that is therefore alien (because it is the mongrel product of political action—often gone awry—taken with and among strangers). Even at its very best, or especially so, democracy is about being mobilized into action periodically with and on behalf of people who are surely opaque to us and often unknown to us. We can see this, as Rousseau himself usually did, as a problem that needs to be solved. For example, we can, in the spirit of Rousseau, focus, as Benedict Anderson does, on how our mu-tual strangeness is overcome by technologies of simultaneity (like mass, national newspapers). Or, like RenéGirard, we can tell ourselves that those scapegoats—often cast as foreign—that used to check the rupturous tendencies of our social violence have been effectively replaced by modern legal institutions.61 Or we can focus on our lingering alienness to one another even in the face of such technological and institutional remedies and even in the face of our actions in concert.
If we take the last position, then the strangers with whom and on behalf of whom we struggle, and the felt strangeness of the institutions that aim to define the terms of democratic contest, might stand not simply as obstacles to a democratic project. They might also stand as markers of the fact that democracy's energies and origins always point beyond the (national) borders and commonalities that have heretofore presented themselves as democracy's necessary conditions. Insofar as he compels us to this insight, the figure of the foreign-founder (perhaps like one of Rousseau's "few great cosmopolitan souls"?) might be an agent of one more (re)founding, inaugurating and animating a democratic politics that seeks to broaden the distributions of goods, freedoms, powers, accountability, and justice within and across borders without presupposing a unified demos stabilized by a metaphorics of national kinship.
In the next two chapters, I look at two other multiply and contestably retold origin stories, the biblical Book of Ruth and the myth of an immigrant America. Both of these origin stories reiterate the foreign-founder script analyzed here. I map out the field of contesting interpretations and iterations and then offer my own. My interventions are motivated by a desire to redirect and harness the energies of foreign-founder scripts on behalf of a democratic politics that does not renationalize the state. This is a democratic politics that seeks, instead, to multiply the sites of affect, coordination, and organization that move people into (and sometimes out of) politics on their own behalves and on behalf of others.
And we Americans are the Israelites of our time. –Herman Melville
3 THE FOREIGNER AS IMMIGRANT
The Bookof Ruth as a Foreign-Founder Text
The Book of Ruth is not usually thought of as a foreign-founder text, but all the basic elements are there. The Israelites are in a period of corruption. A foreigner arrives and her presence among them works to effect two significant changes. Ruth, the Moabite, is the vehicle of a regime change from rule by judges to rule by kings. In that sense, she is a kind of founder, even if not exactly a lawgiver. But Ruth is also a (re)founder in Rousseau's other sense: she (re)founds a "people." Traditional Jewish readers see in Ruth a shining example of virtuous devotion to the one true God or to her Israelite mother-in-law, or both, and Ruth's example is said to be so powerful as to return the Israelites from corruption and set them again on their own true spiritual path. That happy outcome is represented in this text by a return to plenitude: the end of a famine and also the end of barrenness. The Book of Ruth ends with the birth of a boy, Obed, whose line will lead to David, the first line of kings to rule Israel. (That same line will later produce Jesus.)
Ruth is different from Rousseau's foreign-founder in that she is not a lawgiver per se, and her foreignness is not a way of modeling distant impartiality, objectivity, or neutrality. Her function is not to lead a people nor to address directly the narrowness of a people caught up in corrupt factionalism and self-interest. Ruth is different from Girard's scapegoat, too. If she (re)generates communal unanimity, it is not by serving as a magnet for a unifying hatred that takes her as its object. Rather, through indirection—through her example—Ruth inspires and reenchants a jaundiced nation. Foreignness as distance does play a role in Ruth's ability to do that. Her ability to inspire is in direct proportion to her distance (conceptual, epistemological) from the people she joins. The more foreign she is, the more apparent is the universality of the divinity and the people to which she is drawn.
Ruth does introduce two new wrinkles into the foreign-founder script. She is a woman, not a man, and she does not leave when her work of refounding is done. She stays and so becomes an immigrant. Is there any connection between these two departures from the more standard foreign-founder texts? Why does Ruth stay, and does her staying support or subvert the (re)founding she enables? Ruth's status as not just a foreigner but also an immigrant means that she poses for us some slightly different questions than those posed by our other foreign-founders: Is there some sort of connection that we might grasp through Ruth, not just between foreignness and founding, but between _immigration_ and founding?
Ruth
The Book of Ruth begins with a flashback. A few years earlier, a man named Elimelech, his wife, Naomi, and their two sons left Bethlehem to escape famine. They moved to Moab, having heard that Moab was flourishing while Bethlehem suffered. The move to Moab is controversial. Elimelech has abandoned his community in a time of need, and worse yet, he has gone to live in Moab, the home of the historical enemies of the Israelites. This terribly forbidden move, and the famine that occasions it, suggests that the Israelites have fallen away from their fundamental moral principles. Elimelech's emigration, in particular, signals a loss of social unity among the Israelites, and his emigration to Moab, of all places, suggests a diminution of respect for proper boundaries. Both are signs of what RenéGirard calls a sacrificial crisis.
The Moabites are corrupt as well, but in their case the condition is not temporary. They refused water to the Israelites as they wandered in the desert from Egypt to the Promised Land. And when the Israelites camped at Beth Peor, some Moabite women are said to have tried to seduce the Israelite men into illicit relations and idol worship. For this, the prohibition in Deuteronomy against marrying Moabites is uncompromising: "None of the Moabites' descendants, even in the tenth generation, shall ever be admitted into the congregation of the Lord."
Elimelech dies soon after settling in Moab. His sons, Mahlon and Chilion, marry two Moabite women, violating the biblical prohibition against such marriages. These men also die within ten years, leaving behind three childless widows, Naomi and her Moabite daughters-in-law, Ruth and Orpah. Naomi hears that the famine in Bethlehem is over, and she decides to return home. Her daughters-in-law accompany her initially, but she soon tells them to "turn back, each of you to her mother's house" in Moab.3 They refuse, Naomi insists, and finally Orpah, weeping, agrees to return to Moab; but Ruth remains. And when Naomi tells her again to leave ("See, your sister-in-law has returned to her people and her gods; return after your sister-in-law"), 4 Ruth responds poignantly:
Whither thou goest, I will go Whither thou lodgest, I will lodge Thy people shall be my people Thy god shall be my god Whither thou diest, I will die, and there I will be buried.
Naomi says nothing in response, but she stops protesting and Ruth ac-companies her on her journey.
In Bethlehem, Naomi is welcomed back by the women of the community. She announces her losses to them and declares her name changed from Naomi (which means "pleasant") to Mara (which means "bitter").6 Naomi and Ruth establish a joint household. Ruth supports them by harvesting the remnants left in the field of a man named Boaz, who, as it turns out, is a relative of Naomi. Having heard of Ruth's remarkable loyalty to Naomi, Boaz welcomes Ruth to his field and sends her home with extra grain.
But Naomi and Ruth conspire together to achieve a more certain protection than that. Ruth seeks out (and perhaps seduces) Boaz one night on the threshing-room floor and calls on him to extend his protection to her through marriage, while also redeeming a piece of land that was left to Naomi by Elimelech. Boaz notes that there is another male relative who has prior right or obligation to redeem the land, but he promises to do what he can for Ruth. He goes the next morning to find the next of kin and convenes a meeting of the town elders to resolve the question of Elimelech's land. The next of kin's interest in redeeming the land dwindles when he hears that Boaz intends to marry Ruth. Knowing that if they have a son, the child could claim the redeemed land as his own inheritance without recompense, the next of kin offers his option/obligation to Boaz.
Boaz and Ruth marry and have a son who is given to Naomi to nurse. The women's community celebrates, proclaims the child Naomi's son and protector in old age, pays Ruth the highest compliment, declaring her to be of more value to Naomi than seven sons, and names the child Obed. Ruth never speaks again, and she is, of course, absent from the Book of Ruth's closing patrilineal genealogy which ends with David, later to be the king of Israel. Ruth's precarious position in the Israelite order is stabilized by a marriage and birth that provide the founding energy for a new monarchic regime. In turn, Ruth's migration seems to be the vehicle of this welcome regime change. The Book of Ruth opens "In the days when the judges ruled, " a time of famine, barrenness, and corruption, and closes amidst plentiful harvest and a newly born son with a genealogy anticipating the coming monarchy. This is a community in dire need of the inspirational example of Ruth's virtue and perspicacity. The emigration of Elimelech and the intermarriages of Mahlon and Chilion suggest that the Israelite character has been corrupted under the rule of the judges. Ruth helps to solve the problem, not only with her inspiring example but also by founding the line of David, a single monarchy that claims to be superior to the plural jurocracy it replaces.
But this regime (re)founding leaves us nonetheless uncertain about Ruth's status as an immigrant. How should we read Ruth's closing silence? Has she been successfully assimilated, or has she been left stranded? More generally, what connections between foreignness and founding and between immigration and founding are presupposed and consolidated by this great short story? What is a Moabite woman—a forbidden foreigner—doing at the start of the line of David?
Immigration and Founding
According to two representative readers of the Book of Ruth—Cynthia Ozick and Julia Kristeva—Ruth is a model immigrant.8 Ozick reads the Book of Ruth as a tale of reinvigoration by way of conversion or assimilation. (This is in line with the dominant, traditional reading of the story.) Ruth's conversion to Judaic monotheism from Moabite idolatry testifies to the worthiness of the Jewish God. Ruth's devotion to Naomi exemplifies Ruth's virtue, which is an example for everyone and a ground for the rule of David. Ruth, the model immigrant and convert, supplements the Israelite order and saves it from its wayward rule by judges by beginning the line that will lead to a new sovereign monarchy. Ozick's Ruth exmplifies the universal appeal of monotheism and its progress in time, while also marking the compatibility of a now both promisingly and dangerously unlocatable monotheism with a particular, located, statist, and royal lineage.
For Kristeva, by contrast, Ruth unsettles the order she joins. A new monarchy is inaugurated by Ruth, but it is also riven by her, by the moment of otherness she personifies as a Moabite. While Ozick's Ruth completes the Israelite order, Kristeva's Ruth makes it impossible for the order ever to attain completeness. And this, Kristeva argues, is Ruth's great service to the Israelites: she disabuses them of their fantasies of wholeness and makes them more open to difference and otherness, preparing the way for a welcome cosmopolitan identity. But Kristeva's Ruth does not only disrupt the order she joins. She also adopts its customs and rituals and tries to get along. From Kristeva'sperspective, that makes Ruth a valuable model for those contemporary Moslem immigrants who are seen as resistant to absorption into their receiving regimes.
Ozick's and Kristeva's redeployments of the Book of Ruth exhibit two of the dominant and enduring responses we have to immigrants. Either immigrants are valued for what "they" bring to "us"—diversity, energy, talents, industry, innovative cuisines, and new recipes, plus a renewed appreciation of our own regime whose virtues are so great that they draw immigrants to join us—or they are feared for what they will do to us: consume our welfare benefits, dilute our common heritage, fragment our politics, undermine our democratic culture. Both responses judge the immigrant in terms of what she will do for—or to—us as a nation.
The first (welcoming) response models immigration as an occasion for citizens (who are perhaps jaded) to reexperience the fabulous wonder of founding, the moment in which the truth or power of their regime was revealed or enacted for all the world to see. Notably, Moab is (as President Bill Clinton put it in a speech in the Middle East in the fall of 1994) "the land where Moses died and Ruth was born."10 Ruth is a vehicle through which the law comes alive again generations after the death of the lawgiver, Moses. She repeats the foreign-founder script first acted out for the Israelites by Moses. She returns this people to their origins but without the violence that Machiavelli thought was a virtually necessary feature of such a return. In so doing, she occludes the violences of this people's origins. Her immigration reperforms the social contract of Sinai and allows the Israelites to reexperience the official version of their own beginnings: not the "savage chastisement" of Freud's Moses, but rather the wondrous experience of awe before the law. With Ruth, the law is not violently imposed, it is instead lovingly chosen. And Ruth's choice of the Israelites re-marks them as the Chosen People, a people worthy of being chosen.11 Here, the immigrant's choice of "us" makes us feel good about who we are.
The second (wary) response to immigrants also suggests a reexperience of the founding. Highlighted here, though, is the impulse to secure a new regime's identity by including some people, values, and ways of life and excluding others.12 By moving into Bethlehem, Ruth reverses the trajectory of the Girardian scapegoat. She brings pollutants in rather than carrying them out. But she serves the same purpose as the Girardian scapegoat, nonetheless. She manages or conceals the violence of founding and reinstalls the unanimity that grounds the social order. By way of her conversion, the Israelites are brought back from corruption not, as at Sinai, by a stern Moses, but by a kind and virtuous woman who, without violence, makes visible to everyone the universally magnetic power of the one true god. Through Ruth, we might say, the sub-lime law is made beautiful. But this immigrant foreign-founder is nonetheless also deeply threatening to the people she (re)founds. Her choice of the Israelites and her presence among them _endangers_ their sense of who they are. If a Moabite—the most foreign of all foreigners, a member of an idolatrous and murderous people—can move to Bethlehem, does that mean that Israel is now a borderless community open to all comers?
The Israelites turn to a scapegoat to help solve the problem. The text's contrast between Ruth and Orpah highlights the extraordinariness of Ruth's border crossing, as Ozick points out.14 But the contrast also has another effect: it makes clear that Israel is not open to all comers. It is open only to the Moabite who is exceptionally virtuous, to the good Ruth but not to the threatening Orpah. (Is Orpah not threatening?Traditional interpreters give expression to their fears when they claim that Goliath is her descendant.) The Book of Ruth invites readers to project Ruth's frightening foreignness onto Orpah, the one who leaves to return to her gods, the one who did not choose the Israelites. Orpah is the vehicle whereby the Israelites expel outward to Moab a foreignness that is inside their social order (Moses might have been an Egyptian, Jahve might be an alien god, Ruth is surely a Moabite).
Together, Ruth and Orpah personify the coupling of wonder _and_ fear, opportunity _and_ threat, the sense of supplementation _and_ fragmentation that marked the foreign-founders of Chapter Two and that immigrants often excite in the orders that absorb or exclude them. Personified by the two distinct characters of Ruth and Orpah, these impulses may seem to be attached to different objects—the good immigrant versus the bad, for example, or the welcome newcomer versus the frightening stranger. But what if we read Orpah as part of Ruth, a personification of the part of Ruth that cannot help but remain a Moabite even in Bethlehem?15 There are grounds to do so. _It is, after all,_ _Ruth's very foreignness—her likeness to Orpah—that enables her to_ _choose the Israelites in a meaningful way._ Indeed, the more radical Ruth's foreignness, the more meaningful the sense of chosenness that results from her choosing.16 The more deep the enmity between Moab and Israel, the more profound the friendship that is declared in its midst. The more radically particular the convert, the more obviously universal the divinity that compels her to join up. The Israelites' own insistence that _their_ god is uniquely universal is what puts them in need of periodic new testimony to his charms. Even as they eschew converts, they rely on them in this deep way. The most powerful testimony to Judaic monotheism's attractions is the testimony provided by the _most_ unlikely person, the one coming from the most radically particular and hostile culture. It is because Ruth is a Moabite that her conversion—if a conversion it was—is fabulous. Indeed, were the scapegoating of Orpah really to work, were it somehow possible to cleanse Ruth of her foreign Moabite identity, the price of such a cleansing would be the very gift Ruth has to offer. There is no way around it: with all of its good resonances and bad, a _Moabite_ has come to live in Bethlehem.
Ozick's Ruth: Convert or Migrant?
For Cynthia Ozick, the Book of Ruth illustrates the choiceworthiness of the (now, once again) Chosen People, the universal pull of their monotheism and the force of its assimilative power. Ruth is a model émigrée because she leaves behind the idolatry and barbarism of her native Moab. When she says to Naomi "thy god shall be my god, " Ruth announces her fidelity to a god more advanced than those of the Moabites, one that cannot be seen and may not be physically imagined or represented.
Ozick's reading of the Book of Ruth is indebted to the rabbinical interpretations but departs from them significantly. "I mean for the rest of my sojourn in the text to go on more or less without the rabbis], " she says at one point.[18 Where earlier readers interpreted Orpah in terms of her unfavorable comparison with Ruth, Ozick pauses to look at Orpah in her own right. "Let us check the tale, fashion a hiatus, and allow normality to flow in: let young stricken Orpah not be over-looked."Orpah is noteworthy not just for her failure, by contrast with Ruth, to emigrate to Bethlehem for the sake of Naomi and monotheism. Orpah stands out for her own admirable action: she married an Israelite in Moab (not a popular thing to have done, certainly) and came to love Naomi. Orpah may not have been up to the tests of monotheism and emigration, but she was an "open-hearted" woman,20 beyond the confines of "narrow-minded, " conventional prejudice.
Ozick's Orpah is special, but ultimately, in the crucible of the decision to emigrate and convert or not, the true principle of her character is revealed. She represents "normality," not "singularity." Her wants are mundane; her imagination does not soar. In returning to her mother's house, she returns also to her idols. Orpah "is never, never to be blamed for" her choice, Ozick says, but she suggests nonetheless that history has, indeed, judged Orpah ("Her mark is erased from history; there is no Book of Orpah"). Ozick resists the judgment of history by pausing to reflect on Orpah. But Ozick also consolidates history's judgment by depicting Moab's (and Orpah's) disappearance from the world stage as deserved rather than contingent, and by figuring Orpah's decision, as ordinary and immature by contrast with Ruth's decision, which is "visionary." "Ruth leaves Moab because she intends to leave childish ideas behind."
The contrast between Ruth and Orpah, though softened by Ozick's appreciative hiatus, instantiates Ozick's distinction between the normal and the singular. But it also does something else. Ozick's contrast be-tween Ruth and Orpah effectively works to undo the undecidability of the immigrant who both supports and threatens to undermine the order that both depends upon and is threatened by her. Ozick positions Ruth, the immigrant, to reinvigorate the Israelite order without at the same time threatening to corrupt it. The threat of corruption, along with the specter of unconvertible foreignness, is projected onto Orpah, whose failure to emigrate symbolizes a failure to convert (and vice versa). If by staying home Orpah stayed with her gods, then by leaving home Ruth left her gods behind. The contrast leaves no doubt about Ruth's conversion. There is no danger in her presence in Bethlehem. She is surely one of "us."
The unthreatening character of Ruth's reinvigorative immigration is further consolidated by another moment in Ozick's essay. In a lovely insight into Naomi, Ozick sees her instruction to Ruth to follow Orpah and return to "her people and to her gods" as evidence that Naomi "is a kind of pluralist, " _avant la lettre_.27 Naomi is not a zealot, Ozick says. Orpah has her gods, Naomi has hers, and Naomi knows and accepts that. But Ozick stops short of noting the significance of the fact that Naomi's acceptance of Moabite idolatry is tied to the fact that Moabite idol worship occurs in Moab. Her pluralism is territorial. When Naomi says that Orpah has returned to her people and to her gods, Naomi implies (and Ruth surely picks up on this) that it is not possible to go to _her_ people in Bethlehem with Moabite gods. In Naomi's pluralism, people and their gods are tied together and positioned in their proper territorial places. Ozick is right that this is a valuable pluralism by contrast with the forms of imperialism and zealotry that tolerate difference nowhere on earth. Its limits are more evident, however, by contrast with forms of pluralism that demand a more difficult toleration, that of differences that live among us, in our neighborhoods, right next door, in our own homes.
Ozick's positioning of Ruth and Orpah as personifications of singularity and normality, combined with her territorialization of cultural difference, establishes a safe and secure distance between Ruth and Orpah. This distance (intentionally or not) works to enable Ruth to serve as a vehicle of the reinvigoration Ozick seeks without also jeopardizing the identity of the Israelites. Ozick's Ruth is able to supplement the Israelite order without at the same time diluting or corrupting it because the undecidable figure of the (Moabite) immigrant, both necessary for renewal and dangerous to the community, has been split in two: Orpah—the practical, material, Moabite who stayed at home with her idols in her "mother's house"—figures the other whose absence keeps the community's boundaries and identity secure; while Ruth—loyal, devoted to Naomi, possessed of the mature, abstract imagination needed to be faithful to the one invisible god—refurbishes the order's boundaries through her conversion to it. This splitting protects the Israelite order from the corruptions of foreignness while allowing the regime to profit nonetheless from the supplement of Ruth's migration.
But Ruth's incorporation into the Israelite order is less complete and more ambivalent than Ozick suggests. Where Ozick sees virtue, conversion, and assimilation, the text of the Book of Ruth suggests complication, recalcitrant particularism, and prejudice. The following four examples illustrate how this radically undecidable immigrant resists Ozick's decisive narration.
First, the Book of Ruth repeatedly refers to Ruth as "Ruth, the Moabitess," 29 suggesting that she in some sense _stays_ a Moabite, forbidden, surely noticed, and perhaps despised by her adopted culture even while also celebrated by it.
Second, the Book of Ruth makes a point of the fact that Naomi takes Obed from Ruth to nurse. Why? The taking is reminiscent of the story reported by Herodotus of the "Pelasgian inhabitants of Lemnos, who carried off Athenian women from Brauron and had children by them. When their mothers brought them up in the Athenian way, the fathers became afraid and killed both mothers and their children." The Israelites' appreciation of Ruth's reenchantment of their way of life finds expression in the women's community's celebration of her. But when Naomi takes Obed from Ruth, that signals the community's continuing concern about Ruth's foreignness. Ruth, the Moabite, cannot be trusted to raise her son properly, in the Israelite way.
Third, another pivotal scene, this one misread rather than ignored by Ozick: What happened that night on the threshing-room floor? Most commentators, including many of the rabbis, treat the scene as a seduction. Ozick, however, says that the scene depicts "a fatherly tenderness, not an erotic one—though such a scene might, in some other tale, burst with the erotic."33 Indeed. Another commentator, Jack Sasson, does better. Focusing on Boaz's initial fright upon awakening, Sasson speculates that Boaz mistook Ruth for a "Lillith." A Lillith is a demonic woman/ spirit thought to be responsible for nocturnal emissions and male impotence. "[U]pon awakening, Boaz discerns the figure of a woman. Fear-ing that it might be that of a Lillith, he shudders in fear. The storyteller's joke is that Ruth turns out to be equally as aggressive in her demands to be accepted as a mate. In this case, we shall be shortly reassured (if we do not know it already) that matters will turn out well for all concerned."
The "joke" of the scene depends upon Boaz's "mis"identification of Ruth as a Lillith. But the joke of the scene is not on Boaz. It's on this commentator. Because of course Ruth _is_ a Lillith. What Sasson does not note is that Boaz's "error" is overdetermined not simply by Ruth's sex-gender but also by her Moabite identity. Moabite women were seen by the Israelites as fearsome temptresses and seductresses. This scene is much more (or less) than a joke, then. In it, Boaz is allowed to experience his worst fears about Ruth: that, her conversion/immigration not-withstanding, she is truly a Moabite after all, a bearer of desire that will not respect the proper boundaries of male, Israelite subjectivity.
The key to the scene is Boaz's question upon awakening: "Mee at?"— "Who are you?" as in "Who goes there?" It is a border guard's question. Boaz may ask it because he really does not know who this figure is. It _is_ dark. But we know he can see _some_ thing, because he says "Mee at?" which addresses the question to a female. He would otherwise have asked "Mee atah?" or "Mee zeh?" using the universal masculine.
Still, Boaz may ask because he really doesn't know. Or, he may ask because in this nighttime encounter it occurs to him for the first time as a really pressing concern that he really does not know _who Ruth is_! Is she a new Moses, risen from the dead in Moab, come to save and inspire and regenerate the Israelites? Or is she a Moabite? Is she friend or enemy? Founder or foreigner? Who _is_ she? The answer comes: "I am Ruth, your handmaid." Not just "I am Ruth" but also "your handmaid." She tries to reassure. Nothing to fear here, she seems to say. But what does she know? She can hardly reassure in this matter. Besides, if the rabbis are right, Boaz will soon die, on the night of his wedding to Ruth. It seems there was something to fear, after all.
Finally, let us turn to the most famous scene of this short story, the scene in which Ruth declares herself to Naomi. _What_ is Ruth saying when she says: "Whither thou goest, I will go, Whither thou lodgest, I will lodge, Thy people shall be my people, Thy god shall be my god, Whither thou diest, I will die, and there I will be buried"?
Some commentators, such as Julia Kristeva, treat this speech as a declaration of woman-to-woman love and friendship. Ruth will stick with Naomi, no matter what, 'til death does them part. (Indeed, in our own time, this speech serves as a wedding or commitment vow for many lesbian couples.) For Ozick, however, it is noteworthy that Ruth is saying not only that she loves Naomi _but also that she_ feels the pull of the one true god. Why would Ruth say "Thy god shall be my god" if she were not moved by faith? Why would she even move to Bethlehem? "Everything socially rational is on the side of Ruth's remaining in her own country."
Ozick's reading is not implausible, but there is nothing _in the text_ to rule out other rival readings: the social rationalities of the situation are unclear, after all. It cannot have been easy to return to Moab as the childless widow of an Israelite.36 Desperate to get out of there, Ruth may have spoken to Naomi neither out of love, nor faith, but rather out of immigrant practicality: please take me with you, she pleads, knowing that Naomi does not want to. Naomi has just said to her, "See, your sister-in-law has gone back to _her people_ and to _her gods_ ; return, too, as your sister-in-law has done." Ruth may detect in this instruction a concern that she, Ruth, a Moabite, with her own people and her own gods, will be unacceptable and unassimilable in Bethlehem. Do not worry, Ruth responds. I may not know all the customs but I will go where you go, live where you live, _your people shall be my people_ and _your god shall be my god_. As far as the text is concerned, Ruth may simply be reassuring Naomi—as so many immigrants have reassured their hosts and sponsors before and since—that she will be no trouble.
These three readings of Ruth's speech—Kristeva's, Ozick's, and my own—suggest that it trafficks in all three of the kinds of friendship distinguished by Aristotle in the _Nicomachean Ethics_ : virtue (Ozick's version, in which Ruth's declaration is a conversion), pleasure (Kristeva's version, in which Ruth's speech declares a deep love for Naomi), and use (my own reading, in which Ruth's speech is an expression of immigrant practicality). Aristotle claims that only friendship as usefulness is _political_ friendship, but in _Politics of Friendship_ Jacques Derrida suggests instead that what marks friendship as a political relation are the perpetual confusions among its three registers.37 Enmities arise "be-tween friends who, as it were, have been misled, and have misled each other because they have first mistaken friendships, confusing in one case friendship based on virtue with friendship based on usefulness, in another, legal and ethical friendship, etc." (p. 206).
Reading Ruth under the sign of virtue and pleasure, respectively, both Ozick and Kristeva seek to position Ruth's relation to Naomi on a single register of friendship. They want Ruth to be only a friend, not an enemy at the same time. Or better, they want her to be only one kind of friend, not another at the same time. Ozick maps Ruth's relation to Naomi as one of strictly virtue friendship for the sake of a pure Israelite monotheism, which is reliable and universal, in Ozick's view, only to the extent that it is untainted by eros (hence her rejection of any erotic quality to the nighttime meeting with Boaz) and untouched by any instrumental calculations of usefulness. Kristeva privileges the woman-to-woman eros reading for the sake of a cosmopolitanism that needs the animation of an erotic motivation but seeks to avoid an overly universalizing virtue, on the one hand, and an inadequately passionate—merely instrumental—regard for others, on the other hand. In both of these readings, Ruth's Moabite identity is transcended, whether by the pull of virtue or love. Without the continuing taint of her foreignness, however, Ruth's capacity to (re)found the people, or as in Kristeva's case, Ruth's capacity to model any meaningful kind of cosmopolitanism, is severely diminished.
Reading Ruth's speech to Naomi as an expression of friendship as usefulness—as I do—may serve as an antidote to these other readings insofar as it resists the impulse to transcend Ruth's foreignness and seeks not to overstate her membership and acceptance in the Israelite community. But this reading is also too univocal. For the _undecidability_ _of Ruth's speech_ —the fact that the speech is precariously perched simultaneously on all three registers of friendship (virtue, pleasure, and use)—is what accounts for the deep uncertainty that surrounds her. Convert, devoted and loving daughter-in-law, practical and reliable immigrant—which kind of friend is Ruth to the Israelites? Ruth is always, undecidably, both friend and enemy (in Derrida's sense of the wrong kind of friend) at the same time. That is what positions her possibly to inaugurate a politics, not just a monarchic line but a politics, a set of struggles about meanings and powers and futures, a reconfiguration of "the space [or spaces] where parties, parts, or lack of parts have been defined."
The nuances of Ruth's speech are absent from Ozick's reading be-cause she positions Ruth on only one register of friendship (virtue-con-version) and because she splits the undecidable figure of the immigrant into two distinct figures: the one who shores up the order (Ruth), and the one who might corrupt it (Orpah). Ozick sees things this way be-cause she counts on Ruth, the immigrant, to perform a function not unlike that of the foreign-founder in Rousseau's _Social Contract_ : he, too, comes from elsewhere to return a wayward order to its forgotten first principles. He, too, as Neil Saccamano puts it, enacts the "scene"of lawgiving and "ravishes] the assembled public with a passion for law that gives them all the virtues they do not have."[39 As we saw in Chapter Two, Rousseau tries to solve the problem of the foreign-found-er's unsettling foreignness by having him leave when his restorative work is done. Ruth is less accommodating. She cannot leave without undoing the very gift she has to offer, that of refurbishing the Israelites' sense of chosenness by choosing to live among them. So she stays. And her foreignness—so necessary to her refounding function—remains a problem.40 Ozick tries to solve the problem as many multicultural Western democracies have done: by having the helpful (part of the) foreigner (Ruth) assimilate and by ensuring that the dangerous (part of the) foreigner (Orpah) leaves or stays behind.
Kristeva's Ruth: The Ideal Immigrant
The true opposite of Ozick's reading of the Book of Ruth is developed by Andre Lacocque in _The Feminine Unconventional: Four Subversive_ _Figures in Israel's Tradition_. Playing Freud to Ozick's Rousseau, Lacocque presents a foreign Ruth that is all bad, providing an effective counter to Ozick's presentation of the foreign Ruth as all good. As his book's title suggests, Lacocque sees only the _un_ settling and none of the supplementary effects of Ruth's inclusion in the Davidic line. Ac-cording to Lacocque, Ruth could not have been intended to help found the line of David, nor could that text conceivably have been used to support his reign, for "[o]n the basis of Ruth, the great king could . . .be considered as an alien, a mongrel, a parvenu, the outcome of unspeakable mating affairs." Lacocque concludes, therefore, that the Book of Ruth must have been written much later, not as an ad hoc legitimation text for David but rather as a "postexilic parable," a "lesson to the Temple-based ideologists in Jerusalem" who opposed intermarriage or marriage to converts.
Julia Kristeva tries to move beyond the simple poles represented by Ozick and Lacocque in an appreciative reading of Ruth that sees her as both a supporter and a disrupter of the regime she (re)founds. In _Strangers to Ourselves_ , Kristeva reads the Book of Ruth as a potentially alternative and disruptive model of a founding myth. (It is disruptive, but it still founds.) Noting Ruth's love for Naomi, Kristeva calls attention to the woman-to-woman passion at the base of the Davidic line, a passion that flies in the face of structuralist assumptions about the order-constituting function of the male homosocial exchange of women.42 And she points out, further, that Ruth's disturbing foreignness has a positive and generative effect on the regime she joins.
Kristeva does not look to foreignness as a way to model objectivity or impartiality. Instead, echoing another aspect of Rousseau as well as Freud in her appreciation of the fact that life under law requires a degree of self-alienation, Kristeva argues that the sense of strangeness and self-difference excited in the self by an encounter with an other is an important experience for would-be democratic and cosmopolitan citizens. Ruth personifies an otherness that is said to make impossible the identitarian nationalism to which Kristeva's cosmopolitanism is opposed. Ruth, "the outsider, the foreigner, the excluded, " founds a monarchic line that is riven by difference from the beginning. The rift is generative: "If David is also Ruth, if the sovereign is also a Moabite, peace of mind will never be his lot, but a constant quest for welcoming and going beyond the other in himself."
There is, however, little trace of Kristeva's idealized ("welcoming") relation to the other in David's lament, cited by Kristeva, that "the people often speak to him wrathfully, saying 'Is he not of unworthy lineage?Is he not a descendant of Ruth, the Moabite?'" nor in David's wish, also cited by Kristeva, to be rid of his Moabite ancestry so that the people might properly revere him.44 David's impure origins are unsettling to him and to his people. David was more zealous than Kristeva suggests in dealing with others. He certainly outdid Saul in his willingness to destroy his enemies. And later rabbinic interpreters imagine David complaining about being identified with Ruth because he thinks (certainly the later interpreters think) the foundation of his regime will be more stable and more secure without her. Like Sophocles' Creon, who thought the stability of his new regime depended upon building it on one figure and not two—Eteocles without Polynices—David thinks that if he can build on one figure, Boaz without Ruth (as in the Bookof Ruth's closing genealogy), then the foundation of his regime will be secure.
Contra Lacocque, David does need Ruth, however. But not to "worry"his sovereignty, as Kristeva puts it.45 Instead, he looks to Ruth to supplement his own well-known deficiencies of character with her exceptional virtue. Moreover, David's genealogical connection to a Moabite likely suited him in another respect, too, given his and later Solomon's efforts to expand Israel's sphere of influence to include Moabite territory. In short, Ruth's position in relation to the Israelite order is neither unambivalently supportive (contra Ozick) nor unambivalently subversive (contra Lacocque).
This thoroughgoing ambivalence suggests that the Israelites are far less comfortable with their undecidable foreign (re)founder than Kristeva suggests. Kristeva argues that Ruth's gift to the regime _is_ her foreignness and its worrying of Israelite sovereignty. But this misses the fact that the Israelites, and Ozick, think that Ruth's virtue is in spite of her foreignness or apart from it. Her gift to the regime is her exemplary character, faith, and conversion-immigration. Ruth's foreignness is what makes her choice of the Israelites so powerful, but her foreignness per se is no gift.
I note these textual and historical details not just to correct Kristeva on matters of fact, but to raise the political-theoretical question of whether the simple fact of a divided sovereignty or subjectivity, riven by differences (that are perhaps personified by others), is a sufficient condition (as opposed to being merely a necessary one, _if that_ ) to se-cure a properly open relation to strangeness in ourselves and others. Kristeva's use of the Book of Ruth is reminiscent of the strategy used by the Russian westernizer, Vladimir Sergeyevich Solovyev, who looked to Russian history to ground an outward and universal orientation for Russia. As we saw in Chapter One, Solovyev claimed to have found that ground in the ninth-century request for a foreign-founder made to the Norse by several feuding Slavic communities. But what is the significance of this supposed Scandinavian origin for Russia, an origin that is much disputed to this day? The mere awareness of our own internal divisions may make us more tolerant of others (who may personify those divisions for us). But it may just as well engender and feed a determination to extinguish or contain that strangeness, to scapegoatit, in order to (re)establish the unity and, as in Girard, the unanimity we crave. Something _else_ is needed to propel the move from divided subjectivity to an acceptance of strangeness in others.
Like Solovyev, Kristeva seems to count on the ethics-generating power of stories about strangers to move us out of our inward-looking national or ethnic identities. But, in the end, Kristeva's own acceptance of strangeness turns out to depend upon the stranger's willingness to affirm the existence and the worth of the order she supplements and disturbs. As we shall see below, when she discusses Ruth again, this time in the context of contemporary French immigration politics, Kristeva makes it quite clear that for her, no less than for Ozick, Ruth is a model immigrant because of her willingness to leave behind Orpah and all she represents.
Gender and the Foreign-Founder
The Book of Ruth does not only celebrate Ruth, as we have already seen, it also expresses the Israelites' fear of her. But the fact that Ruth—a frighteningly foreign Moabite—is an Israelite heroine would not have surprised Freud. Freud was aware, after all, that the foreign Moses who governed the Israelites so savagely is also celebrated by them. Freud might have seen the celebration of Ruth as a sign that her dangerous foreignness, her character as a much-feared Lillith, had been massively repressed, just as was the Egyptianness of Moses for so long. But Freud also has the resources within his account of Moses both to see that the Israelites need to have their sense of choiceworthiness periodically shored up by a foreigner, and to know that a frightening Moabite would be best positioned to provide the Israelites with what they need.
In _Moses and Monotheism_ , Freud argues that the Israelites covenanted under Moses with an "alien" god, Jahve, whose alienness was then hidden by Jahve's retrodictive claim that "he had been the God of those patriarchs, " Abraham, Isaac, and Jacob (p. 53). Freud finds "astonishing" the idea of a "god suddenly 'choosing' a people, making it 'his' people and himself its own god." He says: "I believe it is the only case in the history of human religions. In other cases the people and their gods belong inseparably together; they are one from the beginning" (pp. 54–55). (Also astonishing is the fact that Freud never asks whether these other peoples' original unitariness is any truer than that of the Israelites. He only questions the latter's.) "Sometimes, it is true, we hear of a people adopting another god, but never of a god choosing a new people." Freud interprets this unheard of divine choosing as a way of remembering the repressed choice made by the Egyptian Moses, who "had stooped to the Jews, [and] had made them his people" (p.55). Whatever its real source (whether Jehovah, the original alien god of Abraham or the foreign god of Ikhnaton, or Moses, the foreign law-giver), Ruth replays the script: rising from the shadow of Moses' un-known grave in Moab, this frighteningly foreign Moabite chooses the Israelites as her people and thereby reperforms the original choice that made them "chosen." This is a repetition with a difference: Ruth is seen to model a relation to the law that is one of loving choice, not violent submission. Either way, however, she re-marks for the Israelites, as Moses did generations before, the alienness of the law under which they live (although, as we saw in Chapter Two, that felt alienness will be most salient, most problematic, and most in need of cultural organization in a democratic regime).
But Ruth reiterates Moses' script with another difference as well. She does not die a timely death, nor is she killed by the people she (re)-founds. Instead, she seems to be absorbed by them. As we saw earlier, she cannot leave without undoing the gift she brings: she cannot remark the Israelites as choiceworthy unless she sticks to her choice to live among them. And thus her foreignness—always with them and forever the condition of the meaningfulness of her choice—seems to be even more difficult to repress or manage than was that of the foreign-founders of Chapter Two. There, the problem of the foreign-founder's foreignness (and of the regime's founding violence, which the founder's foreignness masked) was solved (if it was solved) by the founder's timely departure or murder. In Chapter Three, thus far, we have seen how the problem of Ruth's foreignness is solved by the contrast between the bad foreigner (Orpah) and the good (Ruth), which works to reassure members of the receiving regime that the latter's foreignness (and maybe also the foreignness of her predecessor, Moses) is not really threatening.46 But in the case of Ruth and her readers, another device of domestication is also at work.
Ozick and Kristeva both use Ruth's identity as a woman to soften the impact of her foreignness. Each treats Ruth as a good convert or a good immigrant, but the goodness of this convert or immigrant is a gendered goodness. For both Ozick and Kristeva, Ruth is an agent of care, a giving, maternal or daughterly woman. Both readers emphasize Ruth's love for Naomi, her devotion to her, her care for her.47 These feminized traits subtly position Ruth as a giver and not a taker in relation to the Israelites, a support and not a threat to the regime. In short, Ozick and Kristeva domesticate Ruth's foreignness by way of her supposed femininity. If Ruth can stay but the foreign Moses could not, that is because Ruth is a woman and is more available, therefore, to be absorbed into Israelite life in Bethlehem.
But Ruth is not the woman Ozick and Kristeva take her to be. She is not merely an accommodating, caring giver. She is a taker, too. Ruth's flair for taking becomes apparent once we unfasten her text from con-temporary conventional expectations of feminine virtue.49 Then we notice that all of Boaz's famous gifts to Ruth and Naomi are actually initiated by Ruth. She turns up in his field; he responds benevolently. She thanks him for his kindness, pointing out her foreignness. He offers her even more. But he does not offer her the full protection that is his to give. So she shows up in his bed and calls upon him to act on behalf of two women who cannot represent themselves in a land deal. Again, he responds positively. One can only assume that marriage was first proposed by Ruth as well.
Of all of Ruth's readers, only the feminist biblical scholar Phyllis Trible sees fully Ruth's character as a taker. Accenting Ruth's agency and initiative, Trible revalorizes the biblical figure of the grasping woman. The Bible's female takers are usually depicted as conniving, scheming, and manipulative. But, says Trible, in a patriarchal society women have to be inventive; all the more so Ruth, who is not just a woman but a foreign widow. So we must admire what just a few other commentators have noticed: Ruth's tendency repeatedly to exceed and even violate the instructions given to her by Naomi. Told by Naomi to go glean with the women, Ruth goes to Boaz's field and gleans with the men, whereupon she is welcomed by Boaz but is told again to stick with the women. Naomi tells Ruth to follow Boaz's lead in their threshing-room floor encounter. He will know what to do. But Ruth herself takes the initiative. Moreover, upon returning to Naomi, Ruth tells her that the gifts of grain with which she has once again returned from Boaz were sent to Naomi with Boaz's best wishes. But the text does not support Ruth's claim. Where most commentators see devotion and obedience, the text of the Book of Ruth repeatedly suggests invention and transgression.50 For Trible, these departures from the conventional feminine virtues of devotion and obedience are what make Ruth an admirable heroine and an exemplary woman.
Trible's rejection of the conventional assumption that women are and ought to be the giving agents of care positions her to develop a power-ful and insightful reading of Ruth. But Trible's approach also highlights a problem. Trible's positively charged image of woman as taker gets its energy from the dominant identification of women as caregivers. Good female takers, no less than bad, Lillith-like takers, are judged against and thereby reinforce the same essential expectation: that women are normally nurturant and caring.52 Her departure from that conventional expectation is what makes Trible's Ruth exceptional and heroic. But that exceptionalism also reinscribes the very standard that Trible, through Ruth, would like to upset.
In any case, it will not do to force Ruth into being either a giver or a taker, a good woman or an admirably (or not so admirably) bad one. For multiple and probably contradictory reasons, Ruth moves to Bethlehem and does what she can to claim for herself and Naomi the land, marriage, and maternity that will provide them with security. As a Moabite in Bethlehem, Ruth has no right to these things, but she claims them anyway. She acts in advance of the categories that might legitimate her actions, and so models a kind of political agency that is appropriate for those who seek to make claims in the absence of proper legal standing. It was for people such as these that Hannah Arendt developed the idea of "the right to have rights," a most basic right which she attributed to the stateless who are, as she knew, always one step (or more) away from the still largely state-secured rights that so many call simply "human."53 What is important about the right to have rights in this context is that it invites us to distinguish the status of the immigrant as an object of charity or hospitality (Ozick rightly makes much of this, commending Boaz for his charity to Ruth, a foreigner) from an alternative status, one that does better from the vantage point of democratic theory, that positions the immigrant as a full agent empowered to make (always contestable) claims or take rights on her own behalf. Ruth never exhibits such a full agency. At best, her efforts locate her some-where on the spectrum between full agency as a taker and the more passive object of her hosts' sometimes ambivalent magnanimity.
Kristeva's Orpahs: Cosmopolitanism without Foreignness
In _Nations without Nationalism_ , Kristeva returns to Ruth, whom she sees as a daring but also accommodating border-crossing convert, to model a cosmopolitanism that Kristeva directs at French nationalists and at recent immigrants to France such as the Maghrebi denizens and citizens who "wear the Muslim scarf to school."54 These immigrants resemble Ruth in their willingness to emigrate from their original homes, but they also resemble Orpah insofar as they remain attached to their particular home cultures.55 They lack a Naomi to help them make what Kristeva figures as a transition from particularism to a more abstract table of values. They migrate to France but do not endorse its more universal Enlightenment ideals. And so Kristeva demands that immigrants be asked, "What motivated them (beyond economic opportunities and approximate knowledge of the language propagated by colonialism) to choose the French community with its historical memory and traditions as the welcoming lands?" With this question, Kristeva means to elicit an appreciation that is otherwise not apparent or forthcoming.
Is there nothing French that is choiceworthy and to which immigrants might feel allegiance?56 Kristeva looks among them for evidence of any willingness to do what Ruth did: to swear allegiance to Naomi, her host, and to her god. Indeed, Kristeva's cosmopolitanism depends upon similar pledges of allegiance from French citizens and immigrants alike.
The enduring attachment of many Algerian immigrants to their culture and homeland and their option since 1963 of citizenship in an independent Algeria have led many of them either to reject French citizenship or to relate to it in purely instrumental terms. In response, those on the French Right have in the last fifteen years been calling for tighter controls on immigration and demanding that citizenship be awarded only to those who relate to France affectively. Those on the French Left resist efforts to control immigration and reject attempts to inscribe citizenship as an affective practice.
Charging that the first response is too "nationalist" and the second too "world-oriented" (the Left is too ready to "sell off French national values"), 59 Kristeva carves out a middle ground between them and offers up a cosmopolitanism that is distinctively French in which the nation is still an important but not all-encompassing site of identity, centered not on _Volk_ but on compact.60 Kristeva resignifies the nation from a final site of affiliation to, in psychoanalytic terms, a _transitional object_. (The object is a device, such as a favorite blanket or stuffed animal, that empowers the child to separate from the mother[land] and eventually, in theory, anyway, move on to an independent—blanketless/postnationalist—existence.) Brilliantly cutting across the French Right-Left divide, Kristeva's cosmopolitanism is rooted and affective but attached finally to a transnational, not a national, object.
Kristeva's cosmopolitanism secures and is secured by affective relations to a series of "sets"—specifically: self, family, homeland, Europe, and mankind—in which each set operates as a transitional object for the next.61 By locating the sets in a progressive, sequential, trajectory of transition, Kristeva avoids the issue of possible conflicts among them. She also avoids the question of a specifically French affiliation by using the abstract term "homeland" for _that_ set. But her call for an identification with _Europe_ positions French and Maghrebi subjects asymmetrically in relation to her cosmopolitanism.62 And because her cosmopolitanism, as she says repeatedly, "makes] its way through France, "[63 specifically by way of Montesquieu, it works to shore up a uniquely French identity, even while claiming to overcome or transcend it."T]here is no way for an identity to go beyond itself without first asserting itself in satisfactory fashion, " she says.[64 But this generous recognition of the need to affirm identity before overcoming it is not ex-tended to France's immigrant communities.
There is surely no way out of this paradox, in which cosmopolitanism must be striven for through the particular, albeit heterogeneous, (national) cultures that shape us. (Indeed, this is one way of describing the project of Chapter Four, below, which seeks to recover the nationalist myth of an immigrant America on behalf of a democratic cosmopolitanism.) But Kristeva does not explore the paradox, and she tends to leave the heterogeneity of France behind in her embrace of one particular strand of French Enlightenment thought. She is right to say we must "pursue a critique of the national tradition without selling off its assets." But her account of French cosmopolitanism ultimately protects what she sees as the nation's assets from critique and from critical engagement with others: "Let us ask, for instance, where else one might find a theory and a policy more concerned with respect for the _other_ , more watchful of citizens' rights (women and foreigners included, _in spite of_ _blunders and crimes_ ), more concerned with individual strangeness, in the midst of national mobility?"
The limits of Kristeva's cosmopolitanism emerge again when, echoing Ozick's preference for Ruth over Orpah, Kristeva suggests that the " 'abstract' advantages of a French universalism may prove to be superior to the 'concrete' benefits of a muslim scarf, " implying that the scarf, unlike the nation, is essentially a fetish and is therefore unable, as such, to serve as a healthy transitional object.66 She seems to have those who wear the scarf in mind when she says there "are mothers (as well as 'motherlands' and 'fatherlands') who prevent the creation of a transitional object; there are children who are unable to use it."67 Kristeva sees these veiled women much as Ozick sees Orpah: tethered to their idols, their mothers and motherlands, capable of some bold mobility but ultimately incapable of proper and mature transition, they mark (what Kristeva calls) the "melancholy" of nationalism.
Kristeva quite rightly sees a generative possibility in a differently conceived French _nation_.69 Why not accord the same possibility to the Moslem scarf? In _Women and Gender in Islam_ , Leila Ahmed highlights the progressive properties of veiling. Arguing that for Moslem women in contemporary Egypt the veil, worn increasingly by professional and university women, operates as a kind of transitional object, Ahmed shows how it enables upwardly mobile women to move from the familiar settings of their rural homes "to emerge socially into a sexually integrated world" that is "still an alien, uncomfortable social reality for both women and men."70 Thus, rather than stand for an unhealthy attachment to one's culture—which is how Kristeva and other critics of veiling figure the practice—veiling, on Ahmed's account, actually enables transition and separation; it provides the distance and insulation that enable women securely to enter the public realm.
Kristeva's and Ahmed's assessments of veiling mirror each other and serve as a synecdoche for broader debates about whether foreignness is good or bad for the nation. In veiling, Kristeva sees a threat of immigrant dilution of national identity, but Ahmed sees the possibility of a supportive and animating diversity; Kristeva sees backward particular-ism and female confinement, but Ahmed sees progress, in the form of women's entry into the public sphere; Kristeva sees fetish, but Ahmed posits a transitional object. Similarly, in contemporary France, the practice of veiling is figured simultaneously and without any sense of contradiction as both a sign of the powerlessness of Moslem women (who are controlled by their domineering fathers) and as a sign of those same women's great power (to resist the French colonial enterprise, first in Algeria and now in France).
Neither one nor the other as such, practices of veiling, precariously and variously positioned somewhere between patriarchal confinement and female empowerment, harbor both the possibilities laid out by Kristeva and Ahmed (and others still).73 But the practice's centrality to debates about immigration and foreignness raises another question, beyond that of whether veiling is good or bad for women, good or bad for the receiving regime in question: _What are we doing when we_ _express our concerns about immigration and foreignness through the_ _bodies of women?_ This is not a new question—Fanon asked it in "The Unveiling of Algeria"—and when another commentator on veiling answers it, her response echoes Fanon's. Winifred Woodhull explains:"In the eyes of many French people, girls of Magrebian descent are generally diligent students and compliant people—in short the most assimilable element of the immigrant population; if they begin to de-fend their right to 'difference,' the whole project of integration seems to be jeopardized."
Once again, as with Ozick's and Kristeva's figurations of Ruth, femininity (scripted as compliance) is assumed to soften foreignness. The consequences of that assumption are unmistakable. Set up as good-girl markers of the French Enlightenment project's success, Moslem women are easily cast as the foreign causes of its failure.75 The Enlightenment project's success depends upon its ability to convert others to its values, so its failure is easily externalized, available to be blamed on unconvertible others who are now scapegoated, cast as unusually recalcitrant, Lillith-like creatures, or immature and even "autistic" girls. This prevents anyone from asking, self-critically, whether the failure of the project of integration in this instance has anything to do with that project's particular historical articulation in this setting, or with its values or ambition or scope. And it plays into the hands of patriarchal powers by casting women as passive and weak victims of paternal powers, whether religious or familial.
Ironically, however, if Ahmed is at all right and veiling _can_ function as a healthy transitional object, then Kristeva's figuring of the veil as a concreteness that must be relinquished in order to accede to the welcome abstraction of cosmopolitanism puts her in the very position of those mothers whom she criticizes, those "mothers (as well as 'mother-lands' and 'fatherlands') who prevent the creation of a transitional object." The pleasing irony of this insight should not, however, blind us to the fact that the problem with Kristeva is not simply her failure to explore the transitional properties of veiling while managing nonetheless to see the transitional possibilities of the nation. Were that the case, she could simply change her position on veiling and the problem would be solved.76 Instead, the problem with Kristeva is her failure to engage others in her deliberations about the project, goals, and instruments of a cosmopolitanism she values too much to risk by including it in the conversation as a question rather than as the answer.77 Kristeva ends up in this awkward position because she neglects what Judith Butler calls the "difficult labor of translation," an ongoing project of political work that always also involves a critical self-interrogation and courts the risk of transformation.78 Without a commitment to such a labor, Kristeva's cosmopolitanism already knows what it is—and what it isn't, and so it _risks_ becoming another form of domination, particularly when it confronts an other that resists assimilation to it, an other that is unwilling to reperform for "us" the wonder of our conversion to world or French citizenship. This is the other that most worries Kristeva, the migrant other "whose autistic withdrawal into their originary values is not easy to deal with."
When Kristeva does invite an exchange with "foreigners, which] we all are (within ourselves and in relation to others), " she imagines it will "amplify and enrich the French idea of the nation."[80 But this imagined exchange, in which others join to complete the French idea, calls attention to the need for a different cosmopolitanism in which cosmopolitans risk their cosmopolitan (and nationalist) principles by engaging others in their particularities, while _at the same time_ defending, (re)dis-covering and (re)articulating located universalisms such as human rights and the equal dignity of persons. There is not enough evidence of such a risk in the questions put to immigrants by Kristeva: "What does each immigrant community contribute to the lay concept of _national_ _spirit as esprit général_ reached by the French Enlightenment? Do these communities recognize that _esprit général_ or not?"
Mourning, Membership, Agency, and Loss:Ruth's Lessons for Politics
I return to _Ruth_ by way of a psychoanalytic account of transitional objects. Transitional objects play a role both in Kristeva's account of immigration and cosmopolitanism and in Ozick's reading of the Book of Ruth, in which Naomi is in effect the transitional object that enables Ruth to make the progressive move from Moab to Israel.82 However, a transitional object account of Ruth can also generate conclusions that are quite different from those reached by Ozick and Kristeva.
Modeling issues of separation and autonomy in terms of the child's developing independence from the mother, the object relations school of psychoanalysis emphasizes the role of transitional objects in the process of individuation.83 Drawing on the work of D. W. Winnicott, who emphasizes the loss that attends and occasions individuation and separation, Eric Santner argues that transitional objects enable successful separation only if certain necessary conditions are met. First, the separation must not be traumatic; it must be temporary. Second, there must be a healthy environment conducive to transitional object play. And third, that play must have an intersubjective dimension; that is to say, it must be witnessed periodically by the figure whose temporary absences are being borne. If these conditions are met, the space of object play can serve as a site of healthy mourning for the loss entailed by transition. At play with the transitional object, the subject acts out her bereavement and is thereby empowered for separation and individuation (as in the "fort-da" game—a kind of peek-a-boo—described by Freud). There is empowerment here, not just mourning: the play provides the subject not simply with a substitute (for the loss being mourned), but with a lesson in what Peter Sacks calls "the very means and _practice_ of substitution." At best, the subject learns _agency_ in the face of loss (perhaps even as a result of it, if the conditions are right for such a learning).
If these conditions are not met, neither mourning nor empowerment will ensue. Instead, the subject will first make a fetish of the object, engaging it in a furious and hyperbolic play that signals her denial of her loss. Second, the object will ultimately lose all meaning for the subject, and she will abandon the object entirely, leaving it stranded. The evacuation of the object's meaning can resulht in "signification trauma," which leaves the subject stranded, silent, and speechless, outside the world of language, play, and mourning. Emphasizing all three dimensions of transitional object play—mourning, empowerment, and inter-subjectivity—Santner summarizes Winnicott's view with the aphorism, "Mourning without solidarity i.e., transitional object play in the absence of inter-subjective witnessing] is the beginning of madness."[85 (Her own debts to psychoanalysis [albeit not to Winnicott] explain why Kristeva describes what she sees as immigrants' failed transition in terms of speechless autism and melancholy.)
How might this account apply to the Book of Ruth? If successful transitions are determined not by the nature of the transitional object itself but by the context in which it operates, then we must attend to the role of institutions, culture, community, and politics in projects of transition, something Kristeva does not do in her critique of immigrant particular-ism. Moreover, Santner's focus on mourning, empowerment, and inter-subjectivity calls attention to the fact that none of these three components of successful transition is available to Ruth. Ruth's separation from Orpah (who, on my account, personifies Moab) is traumatic, not temporary. There is no healthy space for transitional object play, no intersubjective witnessing, and no possibility of proper mourning because Ruth is not given cultural, juridical, or psychological permission to mourn Orpah-Moab. Nor are we. Ruth made the right choice. Ozick and Kristeva agree on that. What could there be to mourn?
Ozick and Kristeva both seem to assume that their affirmation of the rightness of Ruth's choice (and their marginalization of Orpah) is what secures Ruth's transition from Moab. But, if Santner and Winnicott are correct, the opposite is true: Naomi's power as a transitional object for Ruth _depends upon_ the proper mourning of Orpah, even upon a kind of continued (perhaps hyphenated?) relation with her (Ozick makes a move in this direction when she says we should "pause" over Orpah, but she then hurries right on past her), even upon recognizing that Orpah (Moab) is part of Ruth. We might even say that Ruth's insight into a universality (as Ozick and Kristeva would put it) is touched by a particularity with which it may be in tension but by which Ruth and her insight are nonetheless also nourished.
Indeed, contra Ozick and Kristeva, the Book of Ruth can be read as a tale of incomplete mourning and failed transition. Seen through the lens provided by Santner, Ruth's famous loyalty to Naomi now can no longer signal simply the selfless devotion of a virtuous or passionate woman, nor is it only a mark of Ruth's immigrant practicality (a possibility I myself raised earlier, along with Fewell and Gunn). What if this clinging is a symptom of Ruth's denial of her loss of Orpah-Moab, a sign of Santner's first stage in which the subject's denial of her loss leads to a frenzied attachment in which the transitional object is fetishized?
And Ruth's closing silence can no longer be taken to signal merely successful absorption.86 Instead, what if that silence is a mark of Santner's second stage, in which the subject suffers from a "signification trauma"? In Ruth's case, the trauma is produced by the separation from Orpah-Moab and the loss of any meaningful relationship to Naomi, Ruth's transitional object. That second loss is symbolized by Naomi's adoption of Obed (in Ruth's place?) but it is foreshadowed by, among other things, Naomi's failure to introduce or even mention Ruth to the women who welcome Naomi back to Bethlehem.
These two moments in Santner's theory and in Ruth's story mark two familiar moments of immigration dynamics. One, a furious assimilationism in which all connections to the motherland are disavowed. And two, a refusal of transition and a retreat into an enclave that leaves the immigrant stranded in relation to the receiving country _and_ in relation to the lost homeland. The two moments are figured developmentally by Santner and Winnicott, but immigrants and their receiving regimes may experience them simultaneously.
The binary of absorption versus enclavism is not driven by foreignness nor by individual foreigners. It is animated by our own efforts to recuperate foreignness for national projects. As we saw earlier, femininity gets similarly driven into categories of goodness or badness by the demand that it support national projects. It is the not very well hidden nationalism of Kristeva's cosmopolitanism that leads her to see new-comers to France in terms of this stern binary. She does not explicitly invoke the sense of kinship that David Miller thinks is a necessary condition of social democracy, but neither does she see foreignness itself as an occasion (and, indeed, a product) of democratic refashioning. The history of interpretative engagements with the Book of Ruth illustrates the consequences of this approach. Democracy is unexpanded and untested by the insistence that others become "us" or go back whence they came. That often punitive insistence itself plays a (never acknowledged) role in producing the very tendencies it excoriates—withdrawalism, recalcitrant particularism, separatism.
We learn a great deal by treating the Book of Ruth not only as a myth of foreign-founding and immigration but also as a parable of mourning and membership. Ruth's role as mourner is signaled early in the text when Naomi, attempting to take leave of Ruth and Orpah, says: "Go, return each to your mother's house: the Lord deal kindly with you, as ye have dealt with the dead, and with me." With the dead, and with me. In that order. It is Antigone's order. First, the dead Polynices (and Jocasta) and then the living Haemon, Creon, and Ismene. It is an order-ing that Creon learns from Antigone, at great cost to her and to Haemon. When Creon finally sees the light and rushes to undo his mistakes, he goes first to bury the dead Polynices and only second to save the by then already dead Antigone. In that order: "With the dead, and with me." It was a widows' household that Ruth, Naomi, and Orpah shared, a house built on death.
"The homeopathic constitution and (reconstitution) of the self takes place not in a vacuum, " Santner says, "but always in a particular social context."89 Similarly, the Book of Ruth suggests that there are institutional and cultural conditions for the proper work of mourning, and it teaches the importance to a meaningful and empowered agency of intersubjective spaces, actions in concert, multiple solidarities, civic powers, and (always contested) connections to the past. Because such spaces, actions, powers, and connections are available to Naomi in Bethlehem (Boaz is her relative, the women of Bethlehem are her friends, and her connection to Moab is preserved by Ruth), Naomi is restored to plenitude and agency (symbolized by her adoptive nursing of Obed). Ruth's fate is different because Bethlehem positions her and Naomi asymmetrically in relation to their losses. Naomi's dead sons and husband can be mourned in Bethlehem, but one of Ruth's losses cannot.90 In the margins of the women's community in Bethlehem, does Ruth mourn Orpah, the sister-in-law who stands for all Ruth left behind in Moab? Ruth's resources and context are limited because the loss of Orpah-Moab is not seen as such, and her transnational connections to Orpah-Moab (a potentially alternative site of support and power) are severed.
Like Antigone's mourning of Polynices, Ruth's mourning of Orpah is forbidden for the sake of a regime's stability and identity. Thus, Ruth's mourning—like Antigone's—is endless, melancholic. Her losses get in the way of the closure this community seeks to attain through her _and_ in spite of her. Indeed, the fact that Naomi's restoration to the community is finally marked by her occupation of Ruth's position as mother to Obed suggests that the reinvigoration of this community and the stabilization of David's monarchy depend not only upon the supple-ment of Ruth's inspiring example but also, and at the same time, upon her marginalization. The Israelites need Ruth's foreignness to shore up their identity as a Chosen People but that identity is also deeply threat-ened by her foreignness (which must then be hidden or managed under the umbrella of her supposed conversion and assimilation), and there seems to be no way out of this dynamic.
The problem is not Ruth's foreignness, per se. As a foreigner she could be many things: exotic, desirable, mysterious, wise, insightful, dangerous, objective, treasured, and so on. Foreignness will signify different things depending on what work it is being made to do, depending on what goal the community is trying to achieve through the foreigner. The need to have Ruth as a supplement and simultaneously to banish her to the margins is driven by the fact that the community's goal here is not a challenged and contested democracy but rather a kinship-style national identity that needs to have its sense of Chosenness periodically shored up.
In Chapter Two, I suggested that it is important to rethink democracy in non-kinship terms, as a politics among strangers. Here, I make the converse suggestion: what if we redeployed the affective energies of kinship on behalf of a democratic politics that is more cosmopolitan than nationalist in its aspirations? Ruth's severed sororal relation to Orpah calls to mind in particular one example of such an effort: sister-cities use the model and the rhetoric of sororal relation to establish affective sites of transnational connection that bypass state apparatuses in order to pursue shared goals and establish relations of long standing. Usually founded by local civic energies and initiatives, sister-cities are not limited "to carrying out a single project," and this makes them an important complement to more temporary, issue-oriented forms of local and international solidarity that are coalitional.91 Most important, sister-cities interrupt projects of (re)nationalization by generating practices of affective citizenship and solidarity that exceed state boundaries and sometimes even violate state foreign policy.92 They are one site of enacted cosmopolitanism, sites of leverage in national politics, sites at which alternative (non-state-centered) forms of membership and affiliation develop.93 Sister-cities commemorate Ruth and Orpah by enacting a sometimes forbidden sorority rather than inheriting the permitted "genetic" kind. In so doing, they contest the tendency to model state citizenship in terms of kin relations. Instead, they use "kin" relations to model extraterritorial solidarities. They disperse the sites of democratic politics beyond and within the states that would like to be democracy's privileged and exclusive centers. And they thereby reperform a kind of taking for which Ruth herself is rightly famous and should be more so. Sister-cities invite and sometimes enable people to "cross over, " as did Moses—not the Moses of Freud, who was unmoved by principles of solidarity, but rather the Moses of Zora Neale Hurston, who, in her own 1930s rewrite of the Moses story, cast Moses as an Egyptian prince whose heart finally went out to the enslaved subjects of his kingdom.
Nothing is more annoying in the ordinary intercourse of life than this irritable patriotism of the Americans.A foreigner will gladly agree to praise much in their country, but he would like to be allowed to criticize something, and that he is absolutely refused. –Alexis de Tocqueville
4 THE FOREIGNER AS C I T IZEN
The Myth of an Immigrant America
"A hero is missing from the revolutionary literature of America, " says Louis Hartz in _The Liberal Tradition in America_. "He is the legislator, the classical giant who almost invariably turns up at revolutionary moments to be given authority to lay the foundations of the free society"(p. 46). Hartz may be right that the lawgiver is absent from the scene of American founding, but the figure of the foreign-founder is not. Again and again, the American democratic theory literature turns to foreignness to found the regime or return it to itself. True, the vehicle of this foreign supplement is not a lawgiver, per se. Instead, American exceptionalists, from Tocqueville to Hartz to Walzer, treat immigrants as the agents of founding and renewal for a regime in which member-ship is supposed to be uniquely consent based, individualist, rational, and voluntarist rather than inherited and organic.1 For these and many other thinkers, the future of American democracy depends not on the native born but on the recent arrival, not on someone with a past to build on but rather on someone who left his past behind. In short, exceptionalist accounts of American democracy are inextricably inter-twined with the myth of an immigrant America.
The myth of an immigrant America depicts the foreigner as a supplement to the nation, an agent of national reenchantment that might res-cue the regime from corruption and return it to its first principles. Those first principles may be capitalist, communal, familial, or liberal.2 In the capitalist version of the myth, the immigrant functions to reassure workers of the possibility of upward mobility in an economy that rarely delivers on that promise, while also disciplining native-born poor, domestic minorities, and unsuccessful foreign laborers into believing that the economy fairly rewards dedication and hard work.
The communitarian immigrant responds to the dissolution of family and community ties, or the prevention of community formation that results in large part from a capitalist economy's unresisted need for a mobile labor force. Periodic infusions of community by way of immigration are said to soften the alienating effects of capitalism's mobilities and of the American liberal individualism that eases their way.
Still others position immigrants as the saviors of traditional patriarchal family arrangements that have been variously attenuated by capitalist mobility and materialism, liberal individualism and feminism. The patriarchal immigrant models proper gender roles and relations for a nation that has lost its sexual bearings. New World American families depend upon Old World masculinities and femininities—family values—that need to be imported periodically from elsewhere.
Finally, liberal consent theorists look to immigrants to solve the problem that Rousseau addressed in the _Social Contract_. Recall that for Rousseau, a legitimate regime is one in which the law, which is always alien, can be made our own by our willing it. Rousseau understood that merely periodic practices such as voting do not position citizens to experience the law as their own. Hence, he argued, the law must be willed frequently, and this was possible under certain, elusive circumstances (by relatively homogeneous citizenries in small polities, and so on). Those circumstances do not exist in the mass, heterogeneous democracy of the United States. While some liberals have argued that American democracy legitimates itself through tacit consent, there is also another mechanism of legitimation at work here, one that operates through the agency of foreignness. The regime's legitimacy is shored up by way of the explicit consent of those celebrated foreigners—immigrants—who, almost daily, are sworn into citizenship in the nation's naturalization ceremonies. More Ruth than Moses (as Hartz's observation about America's lack of a heroic lawgiver would lead us to expect), the liberal consenting immigrant addresses the need of a disaffected citizenry to experience its regime as choiceworthy, to see it through the eyes of still-enchanted newcomers whose choice to come here also just happens to reenact liberalism's own cleaned-up Sinai scene: its fictive foundation in individual acts of uncoerced consent. Simultaneously, the immigrant's decision to come here is seen as living proof of the would be universality of America's liberal democratic principles.
In all four versions, the myth of an immigrant America recuperates foreignness, en masse, for a national project. It does so by drawing on and shoring up the popular exceptionalist belief that America is a distinctively consent-based regime, based on choice, not on inheritance, on civic not ethnic ties. The exceptionalists' America is anchored by rational, voluntarist faith in a creed, not ascriptive bloodlines, individualism, not organicism, mobility, not landedness.4 The people who live here are people who once chose to come here, and, in this, America is supposedly unique. In short, the exceptionalist account normatively privileges one particular trajectory to citizenship: from immigrant (to ethnic, as in Walzer but not in Tocqueville) to citizen.
The exceptionalist account captures something about American democracy while also missing a great deal. American democracy is founded not only on immigration but also on conquest (Native Americans) and slavery (the forced importation of African slave labor) and, in the postfounding era, on expansion (Hawaii, Alaska, Puerto Rico, etc.), annexation (French settlements in Illinois, St. Louis, and New Orleans as well as a significant Spanish-speaking population in the South-west as a result of the war with Mexico), and more slavery. These other foundings are often obscured by the hegemonic myth of an immigrant America. In Charleston, South Carolina, for example, a tourist pamphlet announces that Sullivan Island, off the coast of the city, "might well be viewed as the Ellis Island of black Americans."
In its favor, we might say that at least the myth generates an open and inclusive tolerance of diverse immigrants. But things are not so simple. In fact, the myth generates a sense of quite _anxious_ dependence upon the kindness of strangers. The foreigners whose immigrations to the United States daily reinstall the regime's most beloved self-images are also looked on as threats to the regime. And this is no accident.
American political culture is marked by a play of xenophobia and xenophilia that is not simply caused by periodic power changes from nativists to inclusionists, as Michael Walzer and Rogers Smith both suggest. 6 Nor is it merely a sign of changing economic "realities, " from expanding to shrinking labor needs.7 These may be parts of the story, but there is a deeper logic at work here.
In the various versions of the myth of an immigrant America, it is—as was the case with Ruth—the immigrant's _foreignness_ that positions him to reinvigorate the national democracy, and that foreignness is undecidable: our faith in a just economy, our sense of community or family, our consent-based sense of legitimacy, or our voluntarist vigor are so moribund that only a foreigner could reinvigorate them. But the dream of a national home, helped along by the symbolic foreigner, in turn animates a suspicion of immigrant foreignness at the same time. "Their" admirable hard work and boundless acquisition puts "us" out of jobs. "Their" good, reinvigorative communities also look like fragmentary ethnic enclaves. "Their" traditional family values threaten to overturn our still new and fragile gains in gender equality. "Their" voluntarist embrace of America, effective only to the extent that they come from elsewhere, works to reaffirm but also endangers "our" way of life. The foreigner who shores up and reinvigorates the regime also unsettles it at the same time. Since the presumed test of both a good and a bad foreigner is the measure of her contribution to the restoration of the nation rather than, say, to the nation's transformation or attenuation, nationalist xenophilia tends to feed and (re)produce nationalist xenophobia as its partner.
Ali Behdad is the only critic who comes close to seeing this undecidability of foreignness. He describes the "nation's mode of identification" as always "ambivalent: on the one hand, we are a nation of immigrants; on the other hand, we identify ourselves against our immigrants as we try to control them." That ambivalence is worth attending to, Behdad astutely argues, because it is a productive site for the state's develop-ment of myriad "strategies of discipline, normalization, and regulation." 8 In other words, rather than strive to undo that ambivalence—by attributing it to different parties (as Walzer does) or to different traditions (as Smith does) or to different time periods (as many historians do)—Behdad asks about the performative effects of that ambivalence. What productive energies are unleashed at its site? How does that site serve as "a space of contestation where concepts of nationality as citizenship and state as sovereignty can be re-articulated and re-affirmed"?
Behdad's account and mine work these issues through different texts, and we use different analytic lenses, but we form an obvious and, I hope, productive alliance that calls for the reexamination of some staple assumptions in the study of American nationalism and democracy. However, we differ on one crucial point: when Behdad traces out American ambivalences about immigrants, he misses the pole that is, to my mind, most important. Pointing out that both pro- and anti-immigration movements in the United States are marked by an "us" and "them" mentality, he argues that even those who favor immigration tend to cast immigrants in "symbolically violent" terms.10 He illustrates the American ambivalence regarding foreigners with the following list of "contradictory stereotypes: on the one hand, the immigrant is weak and wretched [and therefore possessed of a claim on our 'humanitarian' sentiments], and, on the other, powerful and dangerous [and therefore a threat to our nation]; on the one hand an opportunist who steals our jobs, and, on the other, a lazy parasite who abuses our social welfare funds." In sum, Behdad concludes, "these stereotypes point to the ambivalence of the nation toward its immigrants, an ambivalence marked by both knowledge and disavowal, control and defense, exclusion and amnesty, acceptance and rejection."
But one stereotype is missing here: the supercitizen immigrant.12 Neither needy nor threatening, as such, but always mirrored by and partnered with those others, the supercitizen immigrant is the object of neither American hostility nor charity but of outright adoration. The stereotypically weak immigrant and the stereotypically powerful one both elicit disavowal. But the supercitizen immigrant is an object of identification. He is the screen onto which we project our idealized selves. He works harder than we do, he values his family and community more actively than we do, and he also fulfills our liberal fantasy of membership by way of consent. Somehow, this iconic good immigrant manages to have it all—work, family, community, and a consensual relation to a largely nonconsensual democracy—even though these very goods are experienced by the rest of us as contradictory or elusive: work in late modern capitalist economies often demands hours and mobilities that are in tension with family and community commitments; meaningful consent eludes the native born for reasons I discuss below.
The immigrant as supercitizen is a staple of the exceptionalist literature and is worth attending to now because he is still very much alive as a political-cultural resource today. In recent years, with the rise of xenophobic initiatives in the United States, Americans on both the Right and the Left have sought to recover the iconic good immigrant who once helped build this nation and whose heirs might contribute to the national future. Both political theorists and activists have responded to renewed anti-immigrant sentiment by stressing the gifts that foreigners have to offer receiving regimes. But what if their xenophilia is intimately connected with the xenophobia they deplore and seek to combat? De-ploying the supercitizen immigrant on behalf of a national ideal, do these xenophiles feed the fire they mean to fight?
Another perhaps useful way of exploring the potentially intimate connections between xenophobia and xenophilia might be to recast what I am calling the undecidability of foreignness in terms of the politics of friendship elucidated by Jacques Derrida. Recall Aristotle's distinction among three kinds of friendship—virtue, pleasure, and use—and Derrida's claim, contra Aristotle, that politics is not confined to the register of use but arises instead when mistakes are made among the different kinds of friendship. In Chapter Three we saw how Aristotle's three kinds of friendship correspond to three readings of Ruth's famous speech to Naomi. The same three kinds of friendship also correspond to the varieties of immigrant supplement traced here. Friendship as use is represented by the capitalist immigrant who comes here to make money. Friendship as virtue is represented by the communitarian and familial immigrants who model proper community and family devotion. And friendship as pleasure is represented by the consenting immigrant who exhibits an exemplary love for the law. The lines of demarcation are not perfectly clear; there are traces of all three kinds of friendship in each of the four supplements of foreignness mapped out here. For example, there is virtue in loving the law and pleasure in capitalist success. Nonetheless, the trichotomy works well enough to enable us to map out the patterns of misunderstanding and disappoint-ment that generate a politics among would-be friends (and not, _pace_ Behdad, between an us and a them, per se). Again and again, as we shall see throughout this chapter, from community-oriented foreigners who live in enclaves to those women who come to the United States as foreign brides, the play of xenophilia and xenophobia is accelerated or renewed when the one friend's (or nation's) expectations of a particular sort of relationship are disappointed and met instead by another. Often, this disappointment is expressed by way of the charge that the other is a taker who is just using us rather than a giver who really wants to be one of us. As we saw with Ruth, however, the "taking" foreigner actually has something very important to give to democratic theorists and citizens.
Because the myth of an immigrant America is very powerful and its effects are quite real, it is important to ask whether democracy is well or ill-served by it.13 And are there any alternative, also normative but less nationalist and more cosmopolitan uses to which the myth of an immigrant America might be put in a counterpolitics of foreignness?These questions are particularly pressing because the success of the myth of an immigrant America in setting the thoroughly nationalist terms of the contemporary immigration debate in the United States suggests that those who look to the mere fact of heightened migration as a bellwether of a new, _post_ national order are falsely confident. If left unchallenged, national imaginations (and the U.S. national imagination in particular) are creative enough and well funded enough to recuperate symbolic immigrant energies for national projects, while also often mistreating actual immigrants.
Mere facts—the mere fact of heightened migration—cannot be counted upon to do the world-building work of politics. People cross borders all the time. As we saw with Russia's Riurik and with the Israelites' Ruth, it is not the fact but the significance of those crossings, the meanings and causes on behalf of which those crossings can be pressed into service, that is the stuff of politics. Riurik was mobilized on behalf of nationalist and cosmopolitan causes alike. Similarly, Ruth was avail-able for capture on behalf of diverse causes ranging from Ozick's ethnonationalist Judaism to Kristeva's still nationalist cosmopolitanism. American immigrants are just as variously available for capture on behalf of diverse causes, and this is what the symbolic politics of immigration is all about: the struggle and counterstruggle to define the terms of foreignness in relation to the always shifting terrain and values of national or democratic polities.
Class Mobility as American Citizenship
It is by now commonplace to hear the capitalist success of (a small minority of) immigrant and ethnic groups explained in terms of their immigrant drive (often said to be lacking in domestic minorities) and in terms of their large extended families and communities who provide cheap labor and pool their resources. What is valued here are the resources available to be sacrificed for financial success, not the affective family or community relations themselves, nor their potential to serve as sites of associational political power.
The capitalist immigrant helps keep the American Dream alive, upholding popular beliefs in a meritocratic economy in good times and bad. If he can do it, starting with nothing and not knowing the language, surely anyone can. At the same time, however, the use of foreignness to supplement the national economy and discipline the domestic poor engenders resentment of foreigners for competing with the native born for scarce resources. Because the capitalist foreigner is depicted as someone who is interested only in material things, he quickly turns from someone who has something to offer us into someone who only wants to take things from us.15 His virtuosic acquisitiveness slides easily into a less admirable, crass, and self-serving materialism. The nationalist, xenophilic deployment of the foreigner to model the American Dream does not just offset these xenophobic reactions, it itself helps to generate them.
The effects of the capitalist version of the myth of an immigrant America on American democracy are particularly unwelcome. The resources of democratic citizenship are diminished, not enhanced, by a supplement of foreignness that is made to stand for privatization, the accumulation of extreme wealth, and a complete disinterest in civic and political life. The myth undermines potential interethnic and transnational coalitions of labor, and it celebrates radical inequalities that are in deep tension with democratic citizenship. The new model minorities do not just "make it"; they become outlandishly wealthy. This version of the myth identifies citizenship with materialism, capitalist production, and consumption. The foreigners depicted here are not politically engaged. They are too busy living the American Dream.
Hence the tone of surprise governing a typical _New York Times_ article reporting on the politicization of Asian Americans: "Marty Shih is the kind of person who has earned Asian Americans the widespread characterization as the model minority, " writes Steven A. Holmes, perversely assigning to Mr. Shih the responsibility for the media's label. In just eighteen years, Mr. Shih, "through grit and hard work" turned the $500 with which he arrived in America into a $40 million business."But Mr. Shih's rags-to-riches story _took an unusual path_ last month when he established the Asian American Association to, among other things, campaign against legislation that would drastically reduce the levels of legal immigration, an issue that has galvanized Asian Americans like no other in recent times."
The "usual" trajectory of Asian American incorporation is commercial, not political.16 Immigrants, especially America's model minorities, stay away from politics. But do they? Completely absent from this now conventional picture are noncitizen or new citizen political actors as diverse as the Haymarket activists (imprisoned or deported), Sacco and Vanzetti (executed), Harry Bridges (leader of the 1934San Francisco general strike who fought deportation efforts in the courts and won), Emma Goldman (expelled), Harry Wu, and a whole slew of others involved in contemporary labor, local and school politics, from undocumented workers in Southern California active in unionization politics to Cambodians agitating for decent public schooling for their children in Lowell, Massachusetts, to Chinese locals involved in "educational struggle" in San Francisco, to aliens stumping for local candidates in New York.
Contemporary depictions of immigrants as concerned only with material acquisition and not with empowered democratic agency are not only misleading. Worse yet, they are often _enforced_ in response to immigrants who become politicized enough to trouble this dominant normative image of quiescence. Take California's Proposition 187, for example. Given the local economy's dependence upon foreign labor, it makes little sense to think that the intended effect of that proposition was simply and merely to deter immigrants from crossing the border. Deterrence may have been part of the intended effect. But another effect is surely also counted upon: the recriminalization of the alien population and new, heightened costs of alien visibility. The result is not just to reduce illegal immigration but to quash the potential power of the undocumented as political actors, labor organizers, and community activists.
Ethnic Bases of Social Democracy:Michael Walzer's Immigrant America
Michael Walzer's communitarian version of the myth of an immigrant America is tailored to respond to the private realm withdrawalism wrongly valorized by the capitalist version of the myth. Given the success of the capitalist economy and America's liberal ideology in individuating, uprooting, and alienating most of the regime's members, Walzer argues, only newcomers can be counted upon to have and to foster the social, civic, and familial ties that social democracy presupposes. For Walzer, then, the model immigrant is not the capitalist overachiever but the family member who cares for his own and builds community institutions. The communitarian immigrant imports a form of citizenship that liberal capitalist America is always in danger of losing or consuming. Walzer's iconic immigrant reinvigorates civil society and the mediating institutions upon which social democracy depends.
In _What It Means to Be an American_ , Walzer observes that "citizens are not effective one by one but only when they are bound together in states or freely associated in parties, interest groups, or social movements. And culture is not sustained by private men and women but by families, nations, and communities of faith."20 The health and vigor of social democratic pluralism depends upon new waves of immigration because the newest hyphenates are the most zealous in their community- sustaining activities.21 But activists get battle fatigue. Community members get distracted by private concerns and withdraw their energies from one another and from public concerns over time. The black feminist activist, Bernice Johnson Reagon, responds to these inevitabilities with the instruction to keep our eyes on the oldest activists whose commitments have somehow endured.22 Walzer's counsel is to focus on the newest comers: "Continued large-scale immigration . . . creat[es] new groups of hyphenate Americans and encourag[es] revivalism among activists and believers in the old groups."
Walzer's immigrants import the family and community ties that life in capitalist America destroys. They tend to their own and—with federal government help in the way of funding and support for continued immigration—they are empowered to build and run much-needed institutions. 24 Walzer's America is dotted by Jewish hospitals, Moslem schools, and Swedish old-age homes. If ethnic communities are allowed to deteriorate, or if they are prevented from forming (by way of enforced assimilation, lack of funds, or the elimination of immigration), then, Walzer worries, the basic institutions of American social democracy will vanish as well. For Walzer, America's immigrants and ethnics moderate the excesses of American individualism (the form of corruption that attaches to liberalism) while also refusing the fragmentation of subnationalism and separatism (the forms of corruption that attach to communitarianism).
Walzer's image of the immigrant as, effectively, a refounder of American civil society is powerful, and its worthy aim is to generate a tolerance and magnanimity toward newcomers that is all too often absent from the American political landscape. But, positioned as the bearers of a "communitarian corrective" to American liberal capitalism, Walzer's immigrant communities attract not only gratitude but also, inevitably, suspicion.25 These much-lauded organic communities of virtue, positioned as so contributive to the national democracy, are also seen as threatening enclaves that reject American values even while living in our midst.26 The communitarian xenophilic deployment of foreignness _on behalf of a national project_ itself plays into the hands of and, indeed, helps to feed this xenophobic response.
That xenophobic response may in fact be amplified by the other gift borne by Walzer's immigrants. For Walzer, the supplement of immigrant foreignness perpetually resecures the character of American liberal democracy as thinly patriotic rather than zealously so. American national affect consists in little more than "the flag and the pledge" _because_ it is a nation of immigrants, Walzer says. "However grateful they are for this new place, immigrants] still remember the old places."[27 But what is the significance of their memory? Does Walzer mean to say that it stands in the way of immigrants' becoming nationalized to the point of zealotry? A powerful if literal illustration to the contrary is the organization by myriad ethnic groups of their members into volunteer units to fight in the Civil War: the "German 18th Regiment, the Polish Legion, the Cameron Rifle Highlanders, the Guard de La Fayette, the Netherlanders' Legion, and the [more multicultural] Garibaldi Guard, which was made up of Hungarians, French, Spaniards, and Croats, as well as Italians."
Is there a singular "experience" to which Walzer can be referring when he says: "This is not Europe; we are a society of immigrants, and the experience of leaving a homeland and coming to this new place is an _almost_ universal 'American' experience. It should be celebrated"?29 Perhaps Walzer is not trying to refer to an antecedent experience so much as he is trying to generate a new one: a thinly national sense of commonality around a not yet shared but perhaps now soon to be shared sense of immigrant journey. What could be wrong with that?
One problem is that the celebration of America's "almost universal" immigrant experience does not simply limit American nationalism; it is also a vehicle of it. The myth of an immigrant America is a nationalist narrative of choiceworthiness. In the American context, the pleasure and reinvigoration of having been chosen is illustrated and produced by the _New York Times'_ s periodic publication of a photograph of new citizens taking the oath. That pleasure is further protected by the failure of the United States to keep any continuous official statistics on remigration or emigration.30 And as Walzer's self-conscious "almost"indicates, the universalization of America's immigrant "experience" has effects on those minorities whose membership in the regime does not map on to the immigrant trajectory to citizenship normatively privileged by Walzer.
In particular, when landed and racial minorities "still remember the old places," the political import of their memory is quite different from the nostalgic yearnings of Walzer's immigrants. Unlike America's traditional ethnic groups, some blacks, Native Americans, and Hispanics have legitimate land-based claims. Unlike America's traditional ethnics, these groups have sometimes sought more than mere recognition. Contra Walzer, who says this never happens, these groups have at times sought secession, or even self-government.31 It is no accident that these forms of political activism are obscured by Walzer's redeployment of the myth of an immigrant America.32 Their demands might divide or fragment the nation-state rather than reanimate it from below. For Walzer, as for many on the Left, the nation-state must be protected from such divisive claims because it is the most likely organizing force of any social-democratic politics.
Landed and racial minorities are not the only ones whose claims are marginalized by Walzer's account, however. Also obscured from view are the many nonethnic institutions for health, education, and welfare in the United States. Especially noteworthy in the last decade or two have been such groups as Planned Parenthood, ACT UP, and the Gay Men's Health Crisis. Why aren't the rather substantial democratic energies of such groups also granted a privileged place in Walzer's immigrant-invigorated civil society? If "citizens are not effective one by one but only when they are bound together in states or freely associated in parties, interest groups, or social movements," why not include as many groups as possible, as long as those groups contribute to the furtherance of social democratic projects?
Walzer's broad commitment to a vigorous civil society suggests he does support such groups. If he does not mention them explicitly, that may be because gay, lesbian, and feminist movements high-light the formation of secondary associations not just out of new migrations (Walzer's preferred source in the U.S. case) but also (as in feminisms or gay rights movements, for example) out of _injuries_ wrought by established, traditional groups.33 Feminists, gays, and lesbians establish alternative institutional resources because their needs are not met and their ways of life are often not tolerated by the ethnic and civic communities with whom they might otherwise identify. In short, the autonomy of these extraethnic groups is itself a _symptom_ of the sometime injustices of the various immigrant groups whose energies animate Walzer's civil society.
Others, more socially conservative than Walzer, share his concern about the rootlessness and mobility of late modern life, but they associate these explicitly with the loss of the very traditional family and community structures against which many feminist, gay, and lesbian groups often define themselves. For many pro-immigration conservatives, immigrants import the roles and expectations that maintain traditional, patriarchal structures. Here, new immigrants are mobilized symbolically to renormalize the native born into traditional heterosexual gender roles while "we" supposedly normalize "them" into a new national citizenship. This dynamic is powerfully illustrated in a popular fable of immigration and national renewal: the Australian film, _Strictly_ _Ballroom_.
Foreign Brides, Family Ties, and New World Masculinity
_Strictly Ballroom_ , a campy comic Australian fable of immigration and national renewal, tells the story of an atrophied community of ballroom dancers saved from corruption by a Spanish immigrant, Fran, who brings new life and virtue to their practices and new energy to their flamenco.34 Initially, Fran seeks assimilation. She assiduously studies the forced steps that are the unquestioned ground upon which the community's dancers are judged. But her quest for inclusion is bound to fail. She has no connections in this corrupt community in which connections are necessary for success, and she has little to recommend her. Dancing "their" steps, she is awkward. She is also unattractive, weighed down by the thick glasses that film heroines have forever removed to reveal a stunning but somehow hitherto unsuspected beauty.
There is an opening for her, however. The powdery white, desiccated community is not only corrupt, it is also riven. One of its members, their star dancer, is a renegade who dares to depart from the community's fetishized steps. When Scott does his own thing, Fran is thrilled and impressed, but the community is aghast. From their perspective, he is too undisciplined, wild, all over the place. His dance seems to have no structure. The choice seems to be between the structure and discipline of a corrupt and unjust but orderly and established community, and a radical individualism that is irresponsible, chaotic, and nihilistic. 35 (In short, the film replays the most caricatured versions of con-temporary political theory's liberal-communitarian debate.)
Scott's free dance style represents a self-seeking individualism that is symptomatic of the community's larger corruptions. Scott's mother, a disciplinary agent who consistently tries to renormalize Scott into the extant ballroom community, herself acts as a self-seeking individualist, too: she is cheating on Scott's father, having an affair with an oily man of superior standing in the dance community.
These corruptions are healed by the foreigner, Fran, and her family. Scott's individualism is tamed and structured by Fran's father, an Old World patriarch. This dark Spanish immigrant gives the couple lessons in authentic flamenco dancing.36 (Fran's father used to dance with her mother[land], but his partner passed away.) At the same time, Fran's father teaches the youths two other lessons: his daughter learns to affirm her roots rather than deny them, and Scott learns that his dance and life choices are not exhausted by the options of the "strictly ballroom"community versus a renegade individualism. In the authentic flamenco of this immigrant community, Scott finds a Walzerian resource that provides his innovative dance (and his life) not only with the energy he craves but also with a shaping structure that distinguishes that newly energized dance from the chaotic individualism Scott's home community fears.
At the final dance contest, Scott and Fran dance an energized and innovative flamenco that is not undisciplined and is capable, therefore, of finally felling the corrupt leaders of the strictly ballroom community whose lies and deceits are exposed. Scott's (Australian) individualism, now moderated and anchored by Fran's émigr ée authenticity and familial bonds, refounds the dance community, rescuing it from its pallid fetishisms and restoring to it its original energy and its founding principles of elegance, honesty, creativity, and fairness.
Fran functions as the communitarian/ethnic corrective of Scott's love-less individualism. But the film features a second supplementary relation as well: Fran's father, an empowered father-figure and a representative of the old patriarchal order, takes the place of Scott's father, a hopelessly henpecked, feeble, and feminized man who is utterly powerless to help his troubled son. Indeed, it often seems that Scott is more drawn to Fran's father than to Fran, that Scott values Fran because she is a way for Scott to get closer to a real father. This immigrant patriarch's foster fathering does not only benefit Scott; it also frees Scott's father from his dominating, castrating wife. The energies unleashed by these foreigners and, in particular, the example of Fran's Old World father benefit Scott's father: they make a man out of him.
In short, the supplement of foreignness works on at least two registers in _Strictly Ballroom_ : through the agency of foreignness, proper virtue is restored to the social world of the ballroom while proper order is restored to the patriarchal family at Scott's house. With the proper containment of the feminine (in the form of Scott's outrageously ambitious mother), Scott's father can be a father again, and the world is made safe for the (re)emergence of an Australian masculinity from within the confines of the feminized, suburban household. The agents of all this are the foreigners who import proper masculinity and proper femininity to a place that has lost its gender bearings. That is to say, _Strictly Ballroom_ replays the classical republican identifications of corruption with female ambition and male emasculation and of refounding with a return to proper gender identities and roles.
But the importation of a real masculinity from elsewhere does not only save Australian masculinity. It also stands as a perpetual reminder of the inadequacy of Australian masculinity. By comparison with Fran's father, who personifies an authentic, Old World masculinity, won't Australian masculinity always be a mere copy? And yet, without Fran's fa-ther, Australian masculinity will continue to be consumed by the feminized household of suburbia. There is no way out of this quandary.
Perhaps the point is that Australian masculinity needs not just the supplement of Fran's father but also that of Fran herself, who is enough of an Old World woman to provide Scott in marriage and in dance with the sort of adoring feminine prop that proper masculinity requires. The young couple's relationship, the film implies, will be different from Scott's parents' marriage because Scott's immigrant girlfriend comes from a family that values family more than the instrumental goods and status that led Scott's mother astray. By modeling immigration politics in terms of this new relationship, the film suggests that it may be possible for immigrants and members of the receiving regime to relate to each other without politics, as two Aristotelian friends somehow positioned on a single, unambiguous (and safely heterosexual) register of friendship.
The desire for an Old World wife to prop up New World masculinity and restore the patriarchal family is evident not only in film. These days the demand is met (and fed) by companies such as Scanna International Worldwide Introductions, which "introduce" American men to foreign women. As one of their clients, David Davidson of Fairlawn, Ohio, explains: " 'There's an exodus of men leaving this country to find wives, ' Davidson said. 'They're looking for women with traditional values like we had 40 years ago.40 They're finding Russian women have those values. Family comes first for them—not work or the Mercedes or the bank account,' said Davidson who has been married and divorced four times." Of his own Russian fianc ée, Davidson said, "She is the most feminine young woman that I've been in the company of. She knows how to be a lady."
Davidson's confident opposition between family values and rank materialism is called into question by another American man interviewed for the same article: "In one form or another [American men] are sick and tired of the princess attitude of American women. . . . Russian women are old-fashioned. . . . Their husband and family come first."But he added that "Russian women see marriage to U.S. men as a way to improve their impoverished lives."
The existence of a foreign bride _trade_ already suggests that—the protests of American men notwithstanding—these marriages are not simply romantic. Indeed, the trade highlights the nature of the institution of marriage in general as not only an institutionalized form of heterosexual intimacy but also always a site at which all sorts of goods and services are exchanged, including citizenship, legal residence status, money, companionship, and sex.42 Moreover, the fact that diverse American, Japanese, Taiwanese, and Arabic men locate a real femininity in places as diverse as Russia, Thailand, and the Philippines suggests that none of these places is a wellspring of true femininity. What if, instead, the foreignness of the imported brides functions to produce a set of relations and inequalities that are available to be (mis)read as femininity?43 This would account for how it is that, somehow, the purchase of a foreign bride—for $7,500 and a residency permit—is said to put the romance back into an institution that is losing its charm.
A foreign bride's perceived family priorities may be less a matter of feminine affect than a matter of necessity. Isolated from others and de-pendent upon her husband, the foreign bride is ignorant of local customs and languages. Her subject position mimes that of the traditional, feminine wife, but foreignness abets or trumps femininity as the real and reliable cause of a foreign bride's dependence and acceptance, her so-called family values. What is labeled "feminine" and eroticized is the foreign bride's would-be powerlessness, her confined agency and her limited alternatives. That perceived powerlessness is why the husbands, who believe that their foreign wives are feminine and unmaterialistic, are undisturbed by the knowledge that these women—who are seeking to escape poverty and limited opportunities, after all—are actually quite interested in the very thing to which they are supposed to be indifferent: their husbands' proverbial "bank account" and the size of it. What is most important is not finally whether the woman is interested in money but whether she has the power to pursue that interest by way of employment for herself or ambition on her husband's behalf.
The xenophilic embrace of foreignness to reenchant traditional fam-ily structures generates two xenophobic responses. Increasingly, the popular press has been publishing stories of foreign brides who turn out to have been using the husbands who sponsored their entry into the United States. Instead of self-sacrificing caregivers, these women are said to be untrustworthy takers. Acting as Lilliths rather than Ruths, they cheat their husbands, rob them, and leave them. More fundamentally, they wrong not only their husbands; they cheapen the institution of marriage by treating it instrumentally. An Aristotelian reading of the situation would say that it is because these wives relate to their husbands on the register of use rather than on the register of pleasure or virtue that the institution of marriage is politicized. Derrida would prob-ably suggest, however, that it is the inevitable confusion of pleasure, virtue, and use (clearest here but attached to the institution of marriage as such) that is responsible for the politicization of marriage.
Such loveless marriages are seen as doubly dangerous (certainly more dangerous than all the other loveless marriages in the nation) because they disenchant two of the nation's most beloved institutions:the institution of marriage, which foreign brides are supposed to help prop up, as well as the institution of citizenship, which is supposedly damaged when immigrants acquire it improperly.44 The affective health of both institutions depends upon immigrants' being attracted to them not for the sake of money or other worldly goods but rather for the sake of a love, devotion, or virtue that is seen as prior to the institutions in question and not as one of their ideological effects. To the critical question—Do these passions give legitimacy to the state (or marriage), or does the state (or marriage) itself generate and legitimate these passions?—this first xenophobic response has an emphatic if conventional answer: "First comes love, then comes. . . ."
The second xenophobic response generated by this particular xenophilia is audible in my own text. Here patriarchal immigrants are seen as threats to the rough (very rough) gender equalities that are American liberal democracy's ambiguous achievement.45 The xenophilic deployment of foreignness on behalf of traditional family structures is particularly troublesome for social democrats because the foreign bride trade promises to resecure and revalorize female powerlessness and male power. The xenophilic deployment of foreignness to solve the problems of gender politics generates these xenophobic responses. This is what happens when foreigners are pressed into service on behalf of institutions—capitalism, community, family—that seem incapable of sustaining themselves. The deployment of foreignness as a restorative supplement itself positions foreigners also as the original cause of the very institutional illness they are supposed to be curing. Where foreign women are figured as exemplary wives who can save the institution of romantic marriage, they inevitably fail, and then they are also set up as betrayers of that and other ideals: the self-interested corrupters of increasingly devalued institutions whose downfall can now be safely attributed to the institutions' abuse at the hands of untrustworthy outsiders who never really loved us but were only out to use us all along.
Dramatizing Consent: The Universal Charms of American Democracy
The demand that foreign women bring feminine romance to American marriages is paralleled by the demand that immigrants romance America and help to reenchant another institution that many feel is in danger of losing its affective charms: the institution of citizenship. The fourth and final redeployment of the myth of an immigrant America, the liberal version, looks to immigrants to reperform the official social contract by naturalizing to citizenship. In the case of the United States, this means (re)enacting for established citizens the otherwise too abstract universalism of America's democratic constitutionalism.
Immigrants not only testify to the universality of American constitutional principles, they are also the only Americans who actually _consent_ explicitly to the regime. Since liberal democracies draw their legitimacy from their consent base, the failure of the native born to consent explicitly seems to pose a deep problem for liberal democracies. Some liberals solve the problem by way of tacit consent. Others, like Peter Schuck and Rogers Smith, have sought instead to provide heightened opportunities for the native born to consent explicitly.
In _Citizenship without Consent: Illegal Aliens in the Polity_ , Schuck and Smith argue that native-born citizens should be offered the opportunity to self-expatriate at the age of majority.47 Although a right of expatriation now exists, few know about it and it is not easy to exercise. Schuck and Smith favor routinizing the choice (by way of automatic mailings to native-born citizens at the age of majority) and lowering the costs (citizens might choose permanent resident status, not necessarily emigration).48 Why make these changes? "In a polity in which actual consent is expressed symbolically only through periodic elections, these proposals can impart a new social meaning and integrity to the tacit consent that must suffice during the intervening periods."
It is possible that these changes may heighten consent for the native born, and they may help relegitimate the liberal state, as Schuck and Smith say.50 But such changes may have other effects as well. For the sake of a heightened affect and legitimacy, Schuck and Smith are willing to risk the creation of a rather substantial class of resident aliens, which is what will become of those who eschew the new invitation to consent. How will Schuck and Smith's revalorized citizenship benefit from the development of a potentially large class of persons willing to live here and consume goods and services without partaking of the rights (voting) and obligations (military service) of citizenship?51 What will be-come of state citizenship when it is transformed from a supposedly universal category into a property of a self-chosen few? (This is not to say that citizenship should be preserved without change but rather to ask, genuinely, what would happen to its meaning and practice if the changes called for by Schuck and Smith were actually instituted.)
More to the point, what sort of power can we expect to come out of a mailed-in consent form? Consent by mail, an action, typically liberal, taken in private, is not likely ever to have the same affective symbolic-cultural effect as the public scene it is intended to mime: that of new citizens taking the oath of citizenship. As Sanford Levinson suggests, immigrant naturalization ceremonies function as a kind of "national liturgy." 52 With a hope and a prayer and an oath, the gap of consent is filled. Immigrant naturalization ceremonies—frequently publicized on the front pages of the nation's newspapers—testify to the fundamental consentworthiness of the regime by symbolically representing the consent that is effectively unattainable for native-born citizens of a liberal regime.
Does this mean that new citizen oath takers act as consenters by proxy, giving voice to the (supposed) silent, tacit consent of the native born? There is something odd about thinking that immigrants can fill the gap of consent when immigrants are so often infantilized (they can't speak English, they need help) and seen as desperate.53 How could such (symbolic) persons be positioned to enact the mature, balanced, and reasonable reflection of rational consent? If the immigrant is desperate, infantile, or "too foreign," his speech act will misfire (it may look like parody to the native born). Indeed, liberals who want immigrants to help solve liberal democracies' legitimation problem are pressed by their own demands to distinguish impossibly between sincere and fraudulent speech acts, admirable immigrant idealism and rough practicality, and among virtue, pleasure, and use—is it true love or are they just using us?54 As of April 1, 1997, elderly or mentally ill immigrants who cannot utter the words of the citizenship oath can no longer be-come citizens. Is it the inaccessibility of immigrant intentions that drives the last decade's obsession with the quite literal performance of the speech act of citizenship? Or is that obsession, perched on the "paradox of intention and capacity," itself a symptom of the modern liberal effort to (in Elizabeth Wingrove's words) "theorize an individualism consistent with new standards of political legitimacy: consent"?
The intractability of these problems (is the naturalizing immigrant sincere or is he just out for himself?) suggests that if immigrants and their swearing-in ceremonies are doing some symbolic-cultural-political work, that work must be something other than the simple provision of consent by proxy. What else might it be?
First, these ceremonies give the abstract value of consent a material and embodied form, thus addressing a problem that Elizabeth Win-grove identifies by way of Rousseau: "that consent makes sense only in its material enactments and that it remains unintelligible when divorced from worldly—institutional, bodily—conditions."56 As we saw with Ruth, so too here the abstract universal (the invisible god of Judaic monotheism, the universal and formal rights and powers of American constitutionalism) requires its abstractness in order to be what it is, but it also requires, paradoxically, concretization by way of particular, empirical manifestations of its power; hence Ruth's endlessly retold "conversion" story, hence the dissemination of the now iconic photo of new immigrants taking the pledge of citizenship. The American need for periodic testimony to the true universality of its principles and the choiceworthiness of its democracy is met by new immigrant foreigners. Indeed, as we saw with Ruth, the more foreign the new consenter, the more powerful the impact of her consent as testimony to the universal's universal attractiveness.
At a deeper level, the rite of naturalization does not just reenact or embody consent. It reperforms the origin of the regime _as_ an act of consent. The oft-disseminated spectacle of new citizens taking the oath of citizenship—a scene in which the new citizen and the state embraceeach other in an act of speech—recenters the regime on its fictive foundation of voluntarist consent. Two effects are achieved thereby:
1. First, an emphatic answer is given to the question of who comes first, the law or the subject, by depicting a subject who exists as such prior to the law and is able therefore to consent to it without apparently being always already formed by it. In this regard, the iconic scene of new immigrants taking the pledge of citizenship has an ideological effect. It privileges a choosing subject as a natural subject prior to the law, and it grounds the law in a choice that is its foundation and its raison d'étre.
2. Second, rites of renaturalization reenact the regime's ideologically approved origins, obscuring the nonconsensual and ascriptive bases and present-day practices of American democracy. The broadcasting (on television, in the nation's newspapers) of this verbal, visible path to citizenship remarginalizes the varied, often violent, sources of the republic (slavery, conquest, appropriations, and constitutional conventions), and it recenters the regime on a voluntarism that most citizens and residents never experience directly. The scene may even excite in some citizens a sympathetic denaturalization that enhances their sympathetic renaturalization (just as many married couples effectively renew their vows when they go to other people's weddings, reexperiencing the pleasure of the gaze of the state and the community upon marital union).
But this (symbolic) "solution" to the problem of consent generates problems of its own. It places the legitimacy of the regime (and its claimed universality as a republic or a democracy) in the hands of foreigners who may or may not close the gap of consent for "us." This is a problem because many newcomers do not satisfy the national need to be chosen—many do not seek citizenship. And those who do naturalize do not simply solve the legitimacy problem; they also inadvetently highlight it by simultaneously calling attention to the fact that most American citizens never consent to the regime. (We saw the same dynamic at work in _Strictly Ballroom_ , where Australian masculinity was both refurbished and also perpetually undone by the importation of masculinity from the Old World.)
In any case, even (or especially) when immigrants do prop up the national fantasy of consentworthiness, the regime's fundamental (un-acknowledged) dependence upon foreigners produces an anxiety that finds expression in a displaced anxiety about foreigners' dependence upon us (an anxiety that, of course, erases the regime's dependence upon foreignness). Thus, it comes as no surprise that in Schuck and Smith's book (and in American political culture, more generally: the book is deeply symptomatic), the good, consenting immigrant, the model of proper, consensual American citizenship, is shadowed by the bad immigrant, the illegal alien who undermines consent in two ways: He never consents to American laws, and "we" never consent to his presence on "our" territory. Schuck and Smith's illegal takes things from us and has nothing to offer in return. He takes up residence without permission; he is interested in social welfare state membership (the proverbial bank account), not citizenship (except for instrumental purposes having to do with securing access to social welfare goods); she takes services without payment (the example repeatedly invoked is that of illegals' unpaid maternity bills at Los Angeles hospitals). 58 In short, the "illegal" in Schuck and Smith's text slides from being a person defined by a juridical status that positions him as always already in violation of the (immigration) law into being a daily and wilfull lawbreaker.
The illegal's threat to consent is crystallized most vividly in _Citizenship_ _without Consent_ (a book widely touted at the 1996 Republican convention) by the American-born children of illegal aliens (hence, perhaps, the authors' displaced] obsession with unpaid maternity bills). Schuck and Smith argue, against a century of Supreme Court decisions, that American-born children of illegals have no constitutional right to citizenship. The Fourteenth Amendment applies to people born in the "jurisdiction" of the United States. Illegal aliens, in the United States without the approval of the state or the consent of its citizens, are on American territory but not in its jurisdiction.[59 Schuck and Smith do not argue that this means that these children should _not_ receive birthright citizenship. It simply means that this right is not constitutionally entrenched and that the decision about whether or not to grant birthright citizenship to the children of illegal aliens is available for democratic (popular and legislative) debate and consent.
The rhetorical weight of the rest of the book, however, is on the side of excluding children of the undocumented from birthright citizenship.For Schuck and Smith, the goal is to revalue American citizenship, to (re)gain control over its distribution.62 Schuck and Smith frame the issue in terms of consent and depict the state as the nonconsenting victim of wayward migrants, but it is not at all clear that the state does _not_ consent to the presence on its territory of large numbers of illegal immigrants. Illegal migration is not only combatted by the state; it is also simultaneously enabled, covertly courted, often managed, and certainly tolerated by it.63 Established citizens profit from the subsidies that cheap migrant labor provides to their child-care costs and food prices.
More to the point, the liberal xenophilic deployment of the foreigner as the truest citizen because the only truly consenting one actually feeds the xenophobic backlash against the nonconsenting immigrant—the illegal alien—to whom we supposedly do not consent and who does not consent to us.65 If this analysis is correct, then the iconic good immigrant—the supercitizen—who upholds American liberal democracy is not accidentally or coincidentally partnered with the iconic bad immigrant who threatens to tear it down. Popular ambivalences about foreignness are not, as Rogers Smith has argued elsewhere, the product of distinct, nativist ideologies that are unconnected in any deep or significant way to American liberal democracy.66 The co-presence in American political culture of xenophilia and xenophobia comes right out of America's fundamental liberal commitments, which map a normatively and materially privileged national citizenship onto an idealized immigrant trajectory to membership. This means that the undecidability of foreignness—the depiction of foreigners as good and bad for the nation—is partly driven by the logic of liberal, national consent, which, in the case of the United States, both produces and denies a fundamental dependence upon foreigners who are positioned symbolically so that they must and yet finally cannot fill the gaps of consent and legitimacy for us.67 That is, nativist ideologies may shape, direct, and accelerate the xenophobia in question. But, contra Rogers Smith, it is misleading to see them as the external corrupters of an otherwise fundamentally egalitarian and tolerant liberal tradition whose only weakness is its failure to inspire in communitarian terms.68 Indeed, as we saw in Chapter One, Smith's characterization of the problem as one of liberalism's corruption at the hands of an outside agitator itself replays the xenophobic script that Smith is out to criticize.
But xenophobia is not the only problem here. The iconic bad immigrant is also problematic because he distracts attention from democracy's real problems.69 Schuck and Smith's deployment of the figure of the illegal exceeds their apparent intent and highlights a different, more tenacious corruption than that of "illegal aliens in the polity"—that of the withdrawal of most American citizens and residents from political life.70 The illegal imagined by Schuck and Smith turns out to stand for the much rehearsed corruption of American citizenship from an active liberal voluntarism to a nonconsenting, passive social welfare consumerism in which good citizens—"givers"—have been replaced by self-interested maximizers and free-riders, "takers." No more than a minority of American citizens votes in American elections; fewer still involve themselves directly in politics. Schuck and Smith externalize these corruptions of American democratic citizenship and, in good Girardian style, project them onto a foreigner who can be made to leave. These Girardian scapegoats represent our best virtues and our worst vices. They become the occasion of a new social unity that Schuck and Smith hope, somehow, to achieve by way of some small policy changes, periodic mailings, constitutional reinterpretations, and better border policing. In short, Schuck and Smith's iconic foreigners, both good and bad, are problematic because they invite unfair treatment of foreigners but also because they mislead us into believing that the solution to liberal democracy's problems and the right response to heightened migrations are a politics of national retrenchment.
Taking Liberties:Intimations of a Democratic Cosmopolitanism
To change a story signals a dissent from social norms as well as narrative forms. –Rachel Duplessis
Tracking the varied workings of the hegemonic myth of an immigrant America helps identify sites at which it may be possible to evaluate, interrupt, and reinhabit dominant figurations of foreignness. The next step is to ask: "How might the myth of an immigrant America be redeployed as part of a counterpolitics of foreignness?"
Fundamentally, the various versions of the myth of an immigrant America all seek to renationalize the state and to position it at the center of any future democratic politics. By pressing the foreign immigrant into service on behalf of the nation and its iconic economy, community, family, and liberal individual citizen, the myth positions the immigrant as either a _giver_ to or a _taker_ from the nation. Indeed, the xenophilic insistence that immigrants are givers to the nation itself feeds the xenophobic anxiety that they might really be takers from it. We saw this dynamic at work in Chapter Three, as well, where it threatened to be viciously circular. I suggested that we might break the vicious circle by thinking about immigrants in relation to democracy, rather than the nation, and by thinking of "taking" as the very thing that immigrants have to give us. I reconsider those possibilities here in a bit more detail and in the somewhat different context of American democratic theory.
Does the ostensibly pejorative symbolic depiction of immigrants as "takers" have a positive dimension that democratic theorists could mobilize? In 1792, Madison said: "In Europe charters of liberty are granted by power [while] in America . . . charters of power are granted by liberty." Madison's insight is that democracy is a form of politics in which power is not received by grateful subjects but rather is taken, redistributed, reenacted, and recirculated by way of liberty, that is, by way of popular political action. Might the negative depiction of immigrants as those who take things from the nation (possibly a projection of a returning, repressed guilt for the original takings on which the regime is founded) be available for recuperation on the part of those who, like Madison, think democracy _always_ involves some sort of taking?
Not all takings are performed by immigrants or foreigners, but they are all performed by subjects who are not fully included in the system of rights and privileges in which they live. The practice of taking rights and privileges rather than waiting for them to be granted by a sovereign power is, I would argue, a quintessentially democratic practice. Indeed, it is one of the practices whereby the American experiment in democracy itself began. As Alexis de Tocqueville points out in _Democracy in_ _America_ , American "settlers" began "exercising rights of sovereignty"without the prior knowledge or authorization of the "motherland." Says Tocqueville, "The new settlers, without denying the supremacy of the homeland, did not derive from thence the source of their powers, and it was only thirty or forty years afterward, under Charles II, that a royal charter legalized their existence."
Jacques Rancie`re, in _Dis-agreement_ , offers several other examples of the same sort of practice in which new rights and standing are taken and then recognized only later (if at all). Working with Pierre-Simon Ballanche's nineteenth-century retelling of Livy's tale of the Roman plebeians' secession on Aventine Hill, Rancie`re notes that, contra Livy, this is a battle not about poverty and anger but about who has the status of a speaking being and about how those who are denied such a status can nonetheless make their claims or make room for themselves. Rancie` re puts it beautifully: "Between the language of those who have a name and the lowing of nameless beings, no situation of linguistic exchange can possibly be set up, no rules or codes of discussion." Is armed battle, then, the only recourse for the nameless class? The plebeians found another way: "They do not set up a fortified camp in the manner of the Scythian slaves. They do what would have been unthinkable for the latter: they establish another order, another partition of the perceptible, by constituting themselves not as warriors equal to other warriors but as speaking beings sharing the same properties as those who deny them these." They mime the speech acts of their would-be superiors and "through transgression, they find that they too, just like speaking beings, are endowed with speech that does not simply express want, suffering or rage, but intelligence." All of this, Rancie`re refers to as the "staging of a nonexistent right."72 Ranci ére gives another example of the practice when he cites the case of Jeanne Déroin who, in 1849, "presents herself as a candidate for a legislative election for which she cannot run," thus staging the contradiction at the heart of the French republic which is a regime founded on both an "equality that does not recognize any difference between the sexes" and on "complementarity in laws and morals, " where the latter is the proper sphere of women.73 Yet another example, this one not in Rancière, is that of Victoria Woodhull, a nineteenth-century American feminist who, instead of campaigning to have women's right to vote added to the constitution, asserted that the right to vote was already implicit and (along with other women) simply began voting and was arrested for it.
These examples of nonimmigrant democratic takings invite a reassessment of the much-reviled figure of the bad immigrant taker. A positive valuation of the taking immigrant as a _democratic_ taker anchors a fifth way of looking at the myth of an immigrant America, this one on behalf of a democratic cosmopolitanism. Here the myth is a narrative of democratic activism whose heroes are not nationals of the regime but insist, nonetheless, on exercising national citizen rights while they are here. Historically, such immigrants have banded together to take or redistribute power. Their demands were resisted, denied, misunderstood, sometimes grudgingly granted or yielded, often greeted with violence, once in a while ceded without fanfare. The people who made the demands were sometimes deported, imprisoned or executed. Others sometimes stayed, sometimes left to go elsewhere, sometimes returned to their points of origin, sometimes died. The nation was not their telos. But they were all engaged in what Rancie`re calls, honorifically, "political activity," a form of activity that "shifts a body from the place assigned to it or changes a place's destination. It makes visible what had no business being seen, and makes heard a discourse where once there was only place for noise; it makes heard as discourse what was once only heard as noise."
The democratic aspect of this version of the myth of an immigrant America lies not in its aspiration to tell a story of ever broadening _national_ inclusion, nor in its effort to expose the "lie regarding the universal" enshrined in the nation's constitution. The peculiarly democratic character of the reinhabited myth inheres in its character as a history and a continuing present of empowerment, frame shifting, and world building. We have here a story of illegitimate demands made by people with no standing to make them, a story of people so far outside the circle of who "counts" that they cannot make claims within the existing frames of claim making. They make room for themselves by staging nonexistent rights, and by way of such stagings, sometimes, new rights, powers, and visions come into being.
Because the myth of an immigrant America is a narrative of demands made by outsiders, it is not just a nationalist story; it is also, potentially, a myth of denationalization. Reinhabited as a democratic rather than a nationalist narrative, the recovered myth of an immigrant America might push late modern democratic actors to pursue two conflictingaims simultaneously: (1) to insist on the inclusion of immigrants and migrants in democracy's national future, while also (2) pressing for the (symbolic and institutional) denationalization of democracy at the same time.
One way to include immigrants in democracy's national future while resisting the recuperation of immigrant energies for the state's renationalization is to expand alien suffrage. Contrary to popular belief, the history of suffrage is not one of ceaseless expansion. The United States has a long history of alien suffrage ("Finally undone by the xenophobic nationalism attending World War I"), in which democratic participation is linked not to the juridical status of citizenship but to the fact of residence.76 At present, several cities allow noncitizen residents to vote in local, school board (Chicago and New York), or municipal (several Maryland localities, such as Takoma Park) elections.
Promoting social and worker movements might help win for presently unrepresented populations a voice in institutional self-governance as well as greater autonomy in daily life. One excellent example of such an effort is the Workplace Project, an organization that provides legal representation and advice to the undocumented while also training them to advocate and organize on their own behalves, representing themselves to bosses, landlords, school administrators, and state officials. The Workplace Project extends citizenship practices to noncitizens. It includes aliens in democracy's national future while also transforming citizenship from a state-granted juridical status to a civic practice.78 Here is an education in democratic citizenship far worthier of the name than the citizenship classes offered by the state in preparation for naturalization. As Michael Walzer says, citizenship cannot be learned "just by watching."
At the same time, the denationalization of democracy must be furthered by enacting transnational ties to empower local minorities. Groups like Women Living Under Islamic Law, Amnesty International, or Greenpeace press states and hold them accountable for their treatments of persons and public goods. In the name of fair and equal treatment of all persons, such groups provide state residents with alternaive, not state-originated sites of support and power.
The point of this democratic cosmopolitanism is not to replace the state with an international government. The state remains an important potential and actual organizer of social welfare and justice as well as a potentially powerful ally to citizens and groups struggling to hold accountable certain powerful local and international institutions. But the state also remains the institutional source of a great deal of injustice, inequity, and violence in the lives of its citizens and residents. Thus, it is important for social democrats to find ways to offset the still too singular power of the state by multiplying the memberships and affiliations of state residents. The goal is to empower people who are among the weakest and least empowered residents of the regime because they are weak politically and vulnerable to exploitation. For example, given the U.S. government's role in creating, administering, and obscuring the problem, the situation of undocumented workers and their availability as exploitable labor can only be effectively addressed by mobilizing social and political energies to counteract the effects of the state's criminalization and denationalization of this class of persons. Of migrant laborers, Labor Secretary Ray Marshall said in 1978, "These people work scared and hard." They still do, but twenty years later farmworkers work for about 20 percent less money than when Marshall spoke out on their behalf.81 Michael Walzer is right to say that political effectiveness depends upon people joining together in groups to act.82 People who are scared, denationalized, and criminalized are less likely to take the risk of visibility that joining together entails.
The goal of a democratic cosmopolitanism is to offset the risks and vouchsafe the benefits of state (non)membership by widening the resources and energies of an emerging international civil society to contest or support state actions in matters of transnational and local interest such as environmental, economic, military, cultural, and immigration policies. This is a _democratic_ cosmopolitanism because democracy—in the sense of a commitment to local, popular empowerment, effective representation, and the generation of actions in concert across lines of difference—is its goal. Such actions are generated out of a sense of solidarity that may be located on any of a number of registers—local, national, or international. Forms of national unity may sometimes support a cosmopolitan commitment to democracy, sometimes not.
Movements need myths. Activists can make up new myths, or they can take those already in existence and recycle them. The latter strategy is preferable because it takes advantage of existing cultural resourcesand simultaneously deprives opposing forces of the powerful narratives that would otherwise continue, uncontested, to support them in their nationalist objectives. The myth of an immigrant America can be turned from its nationalist functions to serve a democratic cosmopolitanism in which citizenship is not just a juridical status distributed (or not) by states, but a _practice_ in which denizens, migrants, residents, and their allies hold states accountable for their definitions and distributions of goods, powers, rights, freedoms, privileges, and justice.
As we saw in Chapter One, political theorists have given little serious attention to cosmopolitanism because they identify it too quickly (and wrongly) with world government. But there is also another reason for their reluctance to explore democracy's cosmopolitan impulses. Political theorists tend to figure the question of democracy's past, present, and future in terms of a debate between nationalism versus cosmopolitanism that is deeply caricatured and so abstract that it erases the long history of internationalism in politics.85 Mirroring its predecessor debate between liberalism and communitarianism, this new set of engagements figures cosmopolitanism ( _qua_ individual ethic or governance) as rootless, abstract, shallow, lacking depth, meaning, and purpose, fundamentally untrustworthy. Nationalism or patriotism is figured as rooted, bounded, structured, full of trust and meaning but capable of erring on the side of zealotry.86 The same sort of debate circulates with reference to citizenship: Is it instrumental or affective? Should it operate on the register of use or virtue? The caricature of cold cosmopolitanism versus warm nationalism (often revalued as cool, level-headed cosmopolitanism versus hot, irrational nationalism) is seductive enough that many liberals have recently given up their seemingly natural coalition with cosmopolitanism in order to forge a partnership with the nation. Liberal nationalism, the resulting hybrid, claims to be able to offer both individual liberty and group solidarity, freedom of movement, and meaningful bonds of civic and social trust. The claim has been con-tested, of course, by unreconstructed liberals and nationalists alike.
Rather than adjudicate these debates as they now exist, I turn to practices of democratic cosmopolitanism as a way to negotiate the divide between the binaries of cosmopolitanism versus patriotism, instrumental versus affective citizenship. A democratically activist cosmopolitanism scrambles the linkages on either side of this opposition. Such a scrambling renders visible already existing sites of sub- and international activisms and memberships that are affective, but not nationalist, rooted but not simply in culture, deep but not particularist, transnational but not simply disloyal. Some of these have already been mentioned: new social movements such as migrant workers' rights groups that operate in domestic and international arenas; professional internationalisms such as Médecins du Monde that act on, in, and among sovereign states; extranational, citizen-forged institutional linkages such as sister-cities and NGOs; gay and lesbian movements for civic equality, health care, international human rights, or queer world-buildings; human rights watch groups that in the last generation have helped to change the structures of accountability governing the behavior of would-be sovereign states; and various feminisms, many of whose members or affiliates develop affective ties to one another precisely by exposing the merely instrumental character of existing would-be affective national citizenships.
In other words, rather than renationalize the state, democratic cosmopolitans seek to denationalize the state, not because they do not value affective ties and memberships but precisely because they do. They denationalize the state in order to make room for the generation of alternative sites of affect and identity against which states often guard. Just one typical example of such guardedness may suffice to make the more general point: "Canada has explicitly challenged attempts by Indians to use UN forums for indigenous peoples on the grounds that Indians are Canadian citizens."88 Because aboriginals are citizens of Canada, the claim is that they do not need alternative or competing extranational affiliations, identities, or forums. They are already represented as citizens. However, it is precisely because they do belong to states in some way (whether as citizens, as in the case of Canada, or not) that aboriginals (and other citizens and residents as well) seek other sites of affiliation and affect. They seek to offset the power that states have over them. Hence, the resistance of states like Canada to aboriginal efforts to go into transnational coalitions. The resistance of states to their citizens' or residents' efforts to organize along extranational axes of identity suggests that the denationalization of the state from an affective to an instrumental set of institutions may be a necessary step on the road toward a more vibrant and empowered democratic politics.
This insight seems to me to be at the heart of Gayatri Spivak's ideal of a feminist internationalism that does not simply broaden the circle of already existing nationalisms. Spivak says about groups such as Women Living Under Islamic Law:
These feminist internationalists must keep up their precarious position within a divided loyalty: being a woman and being in the nation without allowing the West to save them. Their project, menaced yet alive, takes me back to my beginning. It is in their example that I look at myself as a woman, at my history of womaning. Women can be ventriloquists but they have an immense historical potential of _not_ being (allowed to remain) nationalists; of knowing, in their gendering, that nation and identity are commodities in the strictest sense: something made for exchange. And that they are the medium of that exchange.
Spivak's hortatory call puts women's national citizenship on the register of use, marks it as specifically instrumental, and on that basis calls women to an internationalism that need not be itself instrumental; it is, after all, perched on a "loyalty," even if a divided one. Spivak's call is effective in part to the extent we are aware of the traffic in women, the long histories of state productions and uses of sexual difference for their own purposes: in other words, to the extent that we are aware of the state's own instrumental relation to its citizens.
But it is not enough to challenge the binary of affective nationalism versus instrumental cosmopolitanism by showing that cosmopolitanism can be affectively charged and that the nation can be related to instrumentally. This leaves in place the governing, limiting binary of affective versus instrumental citizenship and merely switches the objects of our (non)attachments—we now have an instrumental relation to the nation- state and an affective relation to the cosmopolitan. Is there a way out of this problem? What if we refigured our understanding of civic attachment as a kind of passionate ambivalence? It is partly in quest of such a refiguration that I turn in Chapter Five to consider the literary genre that gives play to such ambivalence: the genre of gothic romance.
I feel as if I'm gonna keel over at any minute and die. That is often what it feels like if you're really doing coalition work.Most of the time, you feel threatened to the core and if you don't, you're not really doing no coalescing. . . . You don't go into coalition because you just like it. The only reason you would consider trying to team up with somebody who could possibly kill you, is because that's the only way you
can figure you can stay alive.
– Bernice Johnson Reagon
5 THE GENRES OF DEMOCRACY
Reading foreign-founder texts like Rousseau's _Social Contract_ , Freud's _Moses and Monotheism_ , the Book of Ruth, and the myth of an immigrant America together suggests that sometimes the (re)construction of the national may require or depend upon the violation of the national. Democratic law, which is said to be fully willed by the people but never truly is, may call on an iconic foreign-founder to make sense of the felt alienness of that law and of the ongoing mutual opacity of a people that is supposed to develop (but rarely does) a sense of kinship and commonality in the joint enterprise of self-government. Alternatively, or in addition, democratic regimes implicated in violences that could delegitimate them may risk their democratic power in order to restore or preserve its innocence, by telling themselves stories about them-selves in which a scapegoatlike foreign-founder takes the people's violences upon himself. National culture may look to have its sense of distinctive choiceworthiness refurbished periodically, and the agency of an extranational may be the best—if also most threatening—vehicle of that reassurance. Democracies, especially the American exceptionalist variety with its emphasis on individualism, liberty, and voluntarism, may need to have not just their sense of choiceworthiness but also their mediating institutions (like the family) and their ground of consent periodically recemented. Newcomers, as opposed to the native born, seem best positioned to serve—even while they also undermine—those ideological and cultural needs.
The questions are: Why, then, in all the contemporary political theoretical debates about citizenship, and even about immigration, has no one paid attention to the complex, constitutive, role of foreignness as an undecidable supplement in national democratic imaginations? Why has the foreignness of Rousseau's lawgiver gone largely unremarked? Why has Ruth been seen as only a support but not also a threat to the people she rejuvenates? Why haven't democratic theorists paid attention to the ambivalent role of the myth of an immigrant America in sustaining liberal democratic values and institutions? What is it that makes (in)visible the role of foreignness in shaping democratic imaginations? Or better, in Girardian terms, what is it that makes (in)visible the organization of pervasive, indeterminate social crises into particular, concrete, positive and negative engagements between an "us" and a "them"? One possible if partial answer to these questions has to do with the genre in which we read the texts of democratic theory.
Does Democracy Have a Genre?
Drawing on Northrup Frye's taxonomy in _Anatomy of Criticism_ , Hayden White has argued that historians, contrary to their self-representation as authors of objective prose, write in modes.1 Historians narrate events like the French Revolution as romance, comedy, tragedy, or satire, casting history itself or historical actors as heroes, villains, or buffoons. Does democratic culture or democratic theory have a mode, too?
Unlike Hayden White, whose aim is to persuade historians to become more self-conscious about their writing practices, I ask this question in order to call attention to the _reading_ practices of political theorists. Most democratic theorists approach the texts they interpret through the mode of romance. But they bring a particular set of genre expectations to their texts. They read democratic theory according to the genre conventions of a popular or modern romance, as a happy-ending love story.(Indeed, some explicitly invoke marriage—happy marriage—as a key metaphor for social contract or social unity.)3 From their perspective, the problem of democratic theory is how to find the right match between a people and its law, a state and its institutions. Obstacles are met and overcome, eventually the right match is made and the newly wed couple is sent on its way to try to live happily ever after.
But what genre best fits a work of political thought in which a people with a great deal in common decide to share the burdens and pleasures of a life together only to find that they have cast in their lot with a bunch of untrustworthy strangers? What genre dictates that, when their joint project founders, a mysterious foreigner will appear on the horizon to rescue these wayward people from their misfortune? What genre then trades on the reader's uncertainty as to whether that apparently rescuing foreigner is really a hero or villain? In what genre would the reader be unsurprised to find that soon the foreigner is using violence and even the penalty of death to make the local people accept his gifts to them? In what genre would the reader be right to expect this mysterious foreigner suddenly to disappear one day, never to return? These are the plot pieces of Jean-Jacques Rousseau's _On the Social Contract_ (on our first reading of it) and the genre that seems to fit them best is not modern romance but _gothic_ romance (also called modern gothic or female gothic).
"Gothic" might also be the most parsimonious description of the reading of the Book of Ruth developed here, in contrast to other readings that are more romantic. In the gothic reading, troubling questions about Ruth's motivations and about the nature of her relations to Naomi and Boaz are allowed to disturb the reader's romantic assumptions and expectations. The Book of Ruth is thoroughly hospitable to a female gothic reading given how it plays with Ruth's foreignness as an undecidable sign of support and threat to the Israelites who welcome and fear her. But if there is one single scene that gives strongest support to a gothic treatment of the text, it is that in which a frightened Boaz awakens unable to identify the shadowy figure who has disturbed his sleep. Boaz's anxiety that the virtuous Ruth might after all really be a Moabite— even a Lillith—never goes away.6 So, too, the myth of an immigrant America, with its perpetual play of xenophobic and xenophilic impulses, positions the native born as the anxious spouse of the mysterious, undecidably safe and/or treacherous foreigner.
In _Loving with a Vengeance_ , her 1984 book on gothic and Harlequin romance, Tania Modleski explains that the (anti) hero in " 'pure Gothics'" is always undecidable, " 'a handsome magnetic suitor or husband who may or may not be a lunatic and/or murderer.'"7 He is also often a foreigner or someone who lives in a foreign (often a Catholic) place. "In the typical [female] Gothic plot, the heroine comes to a mysterious house, perhaps as a bride, perhaps in another capacity, and either starts to mistrust her husband or else finds herself in love with a mysterious man who appears to be some kind of criminal." While Harlequin romances trace the transformation of the young heroine's feelings for the hero from fear into love, in gothics the heroine is older and "the transformation is from love into fear." In short, Harlequins are preoccupied with "getting a man" but "in Gothics the concern is with understanding the relationship and the feelings involved once the union has been formed."
If we allow marriage to stand as a metaphor for the social contract, we can see how Rousseau's _Social Contract_ (again, on our first reading of it) explores the community's feelings about the union it has formed. Then Rousseau's text goes on to follow this female gothic path of dangerous disenchantment. Rousseau begins with a full faith in the people and their power to will the general will. But once the union is in place, he becomes suspicious of them, as indeed they become suspicious of each other as well. Playing the coltish new wife to the people's Maxim de Winter, Rousseau begins to suspect his beloved republicans of engaging in plot and intrigue. The people's devotion to the polity may not be sincere or total, after all. Rampant mutual mistrust puts the dream of an intimate democracy at risk. The narrative shifts from a mood of comfortable identity to one of unease, suspicion, and mistrust. And then the author, like many classic gothic heroines, casts frantically about for a rescuer. At the very moment when the project of legitimation through the General Will seems least likely to succeed, Rousseau—otherwise intent upon crafting a socially unified regime—introduces a foreignerinto the text. A foreign-founder (who just might be a charlatan) arrives to animate a General Will that either cannot animate itself (our first reading) or can found its polity only by way of a delegitimating violence, which it must then disavow and project onto an outsider who can be made to leave (our second reading).
Tania Modleski relies on Norman Cameron's psychosocial account of paranoia to explain the rise of female gothic romance novels. The move from love to mistrust, intimacy to fear, symbolized in gothics as an encounter with the foreign, the strange, or the ghostly, is produced by extreme isolation, itself often the product of marrying and moving away from one's family of origin (effects, therefore, likely to be found among foreign brides, which, in the gothic formula, is what all newly married women are, in some sense).These are the effects of living a life out of the reach of any public sphere, "without the corrective effects that the talk and action of others normally provide."
For Cameron, the social may serve as an inoculation against the development of such anxieties into full-fledged paranoia; the social is a source of security, sanity, and reassurance. Others have written about the social's power to generate and feed paranoia, not just assuage it, a point Modleski does not consider.10 But Modleski's use of Cameron is still suggestive insofar as it points to historical institutional changes as an occasion of genre innovation. Did gothic romances arise because women lost access to the "social"? Did changes in family structure from extended to nuclear family forms isolate individuals, especially women, in ways that were conducive to the development of paranoid conditions to which female gothics, in turn, gave vent?
A variation on this theme might take seriously the role of female gothics in working out certain ambivalences that are social and political rather than strictly familial.11 Modleski turns to psychoanalysis to show how gothics work through the reader's ambivalence toward her parents and her anxiety about separating from them by telling a story in which the heroine finds a _real_ enemy on whom to hang her discomfort—not her remote father, but her really strange or foreign husband. Echoing Freud's reading of "The Sandman, " Modleski says, "Often the attempt to find an enemy and the attempt to exonerate the father are part of the same project."12 But we might think of the anxiety about the father to which Modleski alludes as itself symptomatic, the product of an anxiety about the regime to which the subject belongs. In other words, what if the issues being worked through by way of female gothics have less to do with separation anxiety from familial parents than with an anxious need to separate oneself from democratic or national communities that implicate their members in practices and violences that the subject can not abide and seeks to disavow?
Perhaps we ought to interpret the rise of the female gothic genre not simply or primarily as a response to changes in family structures, but rather as a response to changes in political arrangements ongoing from around the time of the publication of Rousseau's _On the Social_ _Contract_ up to the early twentieth century. Especially given the prominence of marital metaphors for citizenship, should we be surprised to find that anxieties about the identities and agendas of one's (often recently arrived) compatriots in increasingly mass democracies might find expression by way of novels that are set in the uncanny domestic terrain of the (notably, often foreign) household? Similarly, why wouldn't the democratic citizen's (especially the eighteenth and nineteenth centuries' nonvoting woman citizen's) anxieties—about the always erased and increasingly distant violences of the nation-state's birth and the remoteness of the law that is repeatedly declared to be his/her own—find expression in a genre that focuses on remote fathers and husbands and seeks to call them to account for or absolve them of possible past and hidden crimes?13 Such historical speculations might perhaps be supported by the coincidence of timing between the development of a mass (reading) public in the nineteenth century and the nation-state building projects with which that century's democracies busied themselves.
But even if such an historical wager should fail to pay off, it could still be the case, as I would indeed argue, that contemporary democratic agendas of openness, equality, inclusion, national and extranational solidarity, and accountability are well served by the project of (re)reading democracy's romantic narratives through a female gothic lens. In other words, my aim is less to assign the correct genre identity (is it _really_ gothic?) to Rousseau's _Social Contract_ than it is to start an argument about what genre it, and other texts of democratic theory, should be read in (that is, regardless of what genre they were written in).14 The issue here is the normative import of our reading practices as political theorists and as citizens and residents of democratic regimes.
By rereading national and democratic myths of lawgiving, (re)founding, and national democratic renewal through a female gothic lens rather than through the lens of modern romance, we have trained our eyes on the power of foreignness as a symbolic force. We have seen how foreignness is used to figure and perhaps manage enduring problems in democratic theory. We have been made alert to the problem of the alienness of the law, and we have seen how various efforts to domesticate or conceal that alienness work to open some and close other lines of solidarity among citizens and residents, natives and foreigners.
But our gothic lenses do not only enable us to generate fuller and more textured readings of democratic theory's received texts and democracy's popular ones. We are also, as gothic readers and as fans of gothic heroines, better positioned for the responsibilities and challenges of democratic citizenship. In du Maurier's _Rebecca_ , the new Mrs. De Winter desires to be "older" or simply "old" (pp.22, 196, 299).She succeeds. In the end, her early naivét éis replaced by a mature self-assurance, testified to by a clear, calm sense of her new class identity in relation to her inferiors ("I am Mrs. De Winter now, you know") and by a newly maternal relation to her husband ("I held my arms out to him and he came to me like a child") (pp.290, 352, 375).Smug social mobility and maternalism are not democratic virtues, it is true. But might the maturation plot typical of this genre offset the infantilizing effects of democracy's scapegoat narratives? Those scapegoat narratives grow out of the desire of some democratic peoples to preserve their innocence and abjure their power (as well as their violence) even at great cost to their own democratic agency. The new Mrs. De Winter, by contrast, seeks to lose her innocence in order to acquire power and agency.
_Jane Eyre_ has some of the same _bildungsroman_ qualities of _Rebecca_ , but Jane has some valuable counters of her own to offer democratic readers. Jane's story is also one of entry into mature, independent adulthood, but some critics have argued that Jane's power depends in the end upon the enfeeblement of Rochester (oedipally handicapped, rendered lame and blind by fire).It is, however, not the case that Jane's power and independence are established by way of her marriage and in relation to Rochester. They come as a result of Jane's good fortune and hardy effort in establishing for herself some significant, unusualattachments. She adopts and is adopted by the Rivers family (uncannily strange and, indeed, as it turns out, related to her) and acquires two sisters and a brother as her own. Notably, these relations are discovered and made possible by the death of an uncle, Jane's sharing of her inheritance from him, and the departure and eventually the death of the brother. This leaves Jane, formerly an orphan, with a nuclear family that is distinctly sororal (and probably a bit too perfect: siblings without rivalry?).It is from the stage of this achievement of social place by way of something like—yet unlike—kinship that Jane marries Rochester.
This reading of _Jane Eyre_ harkens back to our reading of the Book of Ruth. Ruth is also put into closer relation to Naomi by way of a fortuitous inheritance (the unredeemed land) and by way of the deaths of several male family members in Moab and in Bethlehem. But Ruth's story inverts Jane's. Ruth fails to achieve a social place, and her failure seems to be related to her inability to establish sororal relations with Orpah (Moab).And, of course, in both these stories, the newly included woman's (quasi or would-be) membership seems to depend in some deep way upon the exclusion of an other who is her double. In the Book of Ruth it is Orpah, and in _Jane Eyre_ it is, as Gayatri Spivak first pointed out, Bertha Mason.16 Similarly, in _Rebecca_ , the new Mrs. De Winter takes up her new social place only after learning that her double, Maxim's first wife, was not in fact cherished by him but was instead a cruel, Lillith-like creature—sexually appetitive, disloyal, and perhaps, therefore, deserving of her husband's (unintended?) violence against her.
In short, from _Rebecca_ , we could learn to counter the impulse to self-infantilize, and from _Jane Eyre_ , we could learn the importance to independence of relations of solidarity and enacted sororal kinship. From these two novels in tandem, we learn again the lesson first taught by Spivak by way of Bertha Mason: that circles of solidarity and kinship are usually drawn in ways that not only include but also always exclude as one of their enabling conditions. Gothic romances do not tell us how to solve the problem of exclusion, but they do make it visible.
Gothic romances have a great deal to offer democratic readers. As we first saw in Chapter Two, and then with some variations in Chapters Three and Four, the subjects best prepared for the demands of democracy are those who exist in agonistic relation to a founder (or a father or law) whose alienness is a poorly kept secret; subjects who do notexpect power to be granted to them by nice authorities (fathers or husbands) with their best interests at heart (or, if they do harbor such an expectation, they are the sort that is able to rally after an initial disappointment); subjects who know that if they want power they must take it (isn't this often the lesson of female gothics many of which are, at bottom, basically _bildungsromane_?); subjects who know that such takings are always illegitimate from the perspective of the order in place at the time (the new wife's real or paranoid fears often drive her to violate conventional expectations); subjects who know that their efforts to carve out a just and legitimate polity will always be haunted by the violences of their founding (can the marriage of the gothic heroine flourish after the intrigue, suspicion, and perhaps real violence the couple has suffered?); subjects who experience the law (the father, the husband) as a horizon of promise but also as an alien and impositional thing. In Chapters Two, Three, and Four, we did not have a name for them yet, but such subjects might best be termed _gothic subjects_.
Democracy's Romance: A Tale of Gothic Love
A democratic theorist who advocates a gothic perspective can hardly ignore Richard Rorty's recent diatribe against gothics. In _Achieving Our_ _Country_ , his most sustained argument in favor of the renationalization of the democratic state, Richard Rorty rails against the "temptation to gothicize" as a "stumbling block to effective political organization."What does he mean?
Rorty is referring to the genre of horror gothics, not female gothics, and he has in mind the trademark tendency of horror gothics to generate and explore feelings of paranoia and paralysis in the face of pervasive powers that determine human fate from places beyond the reach of human agency."Gothic" is Rorty's metaphor for a worldview in which human agency is nugatory, hope is naive, it is useless to struggle, and the future holds no promise of change. Rorty's diatribe against gothics is intended to open up room for renewed leftist political activity in America. The irony is that his diatribe is itself written in the most conventional horror-gothic terms. Rorty casts many on the Left as foreign and dangerous to his project, missing an opportunity to make coalition with a wide range of national and international democratic actors.
In the last third of the twentieth century, Rorty says, there has been no real leftist politics at work in America. The problem is that old Left elites have been replaced by the so-called cultural Left, academics seduced by popular culture, theory, and identity politics. The new cultural elite squanders opportunities for a real politics that addresses issues of class and social justice, preferring instead to disunite America with its focus on past wrongs and difference (Vietnam, racism, sexism, anti–Semitism). Theirs is "a Gothic world in which democratic politics has become a farce" because in it human destiny is governed by a "preternatural force."17 Too haunted by past wrongs to be able to love their patria, too haunted by amorphous powers to be able to act, the new cultural Left is unable to offer a positive vision of an American future that might motivate others to join them in a fight for justice. Unable or unwilling to dream, the cultural Left can bring about no change, for in order to "urge national political renewal . . . [y]ou have to be loyal to a dream country rather than to the one to which you wake up every morning."
There are no proper names attached to these charges.(The lack is significant in someone like Rorty, who loves to throw around proper names in sometimes long, dizzying signifying chains.) That is because Rorty's cultural Left is itself a gothic monster, an amorphous, dreamless creature so caught up in a past that it has no future, a creature haunted by powers beyond its control and yet itself so powerful that it is almost single-handedly responsible for the success of the Right in the last twenty-five years, a creature so pervasive and monstrous that it takes possession of the souls of young girls, as in this scene early in _Achieving_ _Our Country_ :
A contemporary American student may well emerge from college less convinced that her country has a future than when she entered. She may also be less inclined to think that political initiatives can create such a future. The spirit of detached spectatorship, and the inability to think of American citizenship as an opportunity for action, [all hallmarks of the "cultural Left"] may already have _entered such a student's_ _soul_.
Here Rorty casts the university as the haunted castle to which unsuspecting parents send their innocent young daughter. In the castle of the academy, the girl's soul is lost. Doomed to a state of nightmarish dreamlessness, she is—like the victims in the 1950s horror movie _Invasion_ _of the Body Snatchers_ —fated to follow but never to lead or initiate action for the rest of her life.
Rorty's nightmarish scenario takes place in the United States. In a more self-conscious and classic gothic narration, the corrupted university would be in a foreign place. But true to the horror-gothic form he decries, Rorty soon puts the people responsible for the university's corruption outside the circle of Americanness and renders them foreign to the body politic. The cultural Left is "unpatriotic, " not part of mainstream America.20 "Outside the academy, Americans still want to feel patriotic. They still want to feel part of a nation which can take control of its destiny and make itself a better place."21 Inside the academy, behind the closed doors of the English Department, something very different is going on. If only, in place of the academy's dark gothic musings, we could have the "daylit cheerfulness" of Whitman, who, along with Dewey, gives "us all the romance, and spiritual uplift, we Americans need to go about our public business."
Predictably, Rorty's horror-gothic call to romanticize rather than gothicize the nation sets in motion the very dynamics of demonization I have been tracing throughout this book.23 As we saw in Chapters Two, Three, and Four, the insistence on total identification with an idealized object—the nation, the demos, the General Will, a Ruth, the lawgiver, the good citizen—tends to drive the subject to split the beloved object into two (the good object and the bad) and to defend the former against the now externalized threat of the latter: the nation against the foreigner; the demos against the outsider; the good, old, nation-loving Left versus the bad, new, cultural and unpatriotic Left; the General Will against the particular will; Ruth against Orpah; the kind lawgiver against the brutal tyrant; the good citizen against the bad immigrant; the giver against the taker.
Again and again, we have seen how the politics of foreignness are driven by failed efforts to insist on the unity of the nation or the demos and to insure that the supplement of foreignness only supports regimes that are, however, always also unsettled by it. One result of this insistence is the loss of opportunities to pass one of democracy's strictest tests, the challenge to work and live and share not just with people with whom we have a great deal in common but also with those with whom we just happen to be bound up; that is, in Bernice Johnson Reagon's phrasing, the challenge to work and live and share "with somebody who could possibly kill you . . . because that's the only way you can figure you can stay alive."
Rather than return home to the nation and love it (or its idealization) fully and completely, democratic actors could move their fellows into democratic action along multiple registers: subnational, antinational, transnational, and national. At the same time, they would do well to nurture some ambivalence regarding their principles, their leaders, and their neighbors and to put that ambivalence to good political use, relying on the gothic lenses whose aid to vision I have been advocating in this chapter. These lenses come from female gothics, not horror gothics. What they provide us with is not a sense of paralyzing paranoia in the face of monstrous forces beyond our control, nor a clear distinction between the forces of good and evil, but a healthy caution to be wary of authorities and powers that seek to govern us, claiming to know what is in our best interests. From female gothics, we get a valuable exhortation to take matters into our own hands.
The best female gothic heroines are takers. They take it upon themselves to leave their suitors or to discover the truth about their husbands, who may or may not be out to get them. Whatever they discover—it doesn't matter what, really—the exercise of detection teaches them agency, and they become less vulnerable to their husbands (good or bad) because they have learned their powers. Gothic heroines never take total control of their lives. They are _gothic_ heroines, after all, and so they usually remain vulnerable: they end up with a man who still might be a murderer, or they uncover only some but not all the secrets that haunt them. Female gothics teach us not only the powers but also the limits of self-conscious agency. Democratic actors should find both lessons valuable.
Similarly, we, as readers of female gothics, can make our own, sometimes limited, discoveries.26 We can discover that the devices used by gothics to heighten our sense of uncanniness—devices such as foreignness—are devices that work on us every day and that we, as democratic agents, do not just have to yield to them. Instead, we have some power to explore, unmask, detect, and even expose them. Often in gothics, itturns out that it is not the apparently scary foreigner but the nice man next door, meek and mild, who is the real murderer. In real life, gothic readers know, we don't get such neat resolutions. The nice guy and the scary one are often the same person. The president who introduces vast new social welfare programs is the same one who escalates the war in Vietnam. Or better, since there need not be a tight or necessary link between those two of Lyndon Johnson's legacies, the iconic supercitizen foreigner who reenchants the nation can only do that for us because he is foreign in the same way as that other foreigner, the one who might be a Lillith, the one who is here illegally and seeks to take our wealth, pleasure, and politics away from us. These two can be foreign in the same way because their foreignness does not come from them, it comes from or through us.
Nationalists like Richard Rorty and Rogers Smith find it difficult to find a public place for the ambivalence that female gothics deepen and explore. Philosophical ruminations on such things may, Rorty says, "be useful to some of us in our quest for private perfection, " but they can only do damage to a worthwhile politics.27 Similarly, Smith, whose goal is also to reawaken contemporary citizens to politics (albeit to a liberal rather than a social-democratic politics such as that envisioned by Rorty), himself offers clear-cut distinctions between good and evil, not gothic ambiguity or undecidability. Proposing a national history for committed, nationalist citizens, Smith explains that such a history should be "understood to include serious struggles among people, movements, principles, and causes with different aims and interests—struggles in which the actors a particular citizen decides to regard as the 'good guys' may not always, perhaps even not often, win."
But why tether the passion that politics requires to the nation? As we have seen throughout, passion, involvement, and identification are daily called into action on behalf of many extra- and subnational affiliations and memberships and causes. Moreover, even if we do tether such passions to the nation, why then split the nation's history into a battle between bad guys and good (distinct and separable as the "traditions" they represent), as if these are not usually found combined, inextricably, in a single person, ideal, or movement? The need for passion, involvement, and identification in politics does not require us to move in these directions. Even if, along with Smith and Rorty, we believe that we must romance the nation, the question still remains: Should ours be a Harlequin or a gothic romance?
Smith and Rorty are right that passion, involvement, and identification are necessary to a vibrant (and redistributive) democratic politics. They provide an important corrective to my own reactive tendency toward mere instrumentalism as an antidote to the nationalisms I reject. Recall that in my first reading of Ruth, I cast Ruth as a merely practical émigr ée who related to the Israelites solely on the register of friendship as use. Then, motivated by Derrida's interpretation of Aristotle, I moved instead to think of Ruth as undecidably perched on all three registers of friendship—virtue, pleasure, and use. In Chapter Four, however, I suggested that "the denationalization of the state from an affective to an instrumental set of institutions may be a necessary step on the road toward a more vibrant and democratic politics." As I noted there, that apparent embrace of instrumentalism requires modification, too, for it risks merely relocating the binary of instrumental versus passionate attachment (our relation to the nation-state is now instrumental, and the international is now the real location of passionate attachment) rather than breaking out of that binary. Can we move beyond that statement, beyond Spivak (whose hortatory call, quoted at the end of Chapter Four above, put women's national citizenship on the register of use) and beyond Reagon, who imbues coalition politics with urgent passion ("do it everyday you get up and find yourself alive") but also seems to slide toward instrumentalism when she says that we work together only because we have to ("because that's the only way you can figure you can stay alive" [368])?
I have been trying throughout this book to pluralize our sense of the _objects_ of our attachments, but as we have just seen, this can leave the binary of instrumentalism versus passionate attachment still in place. What if we pluralized passion itself? Gothic readers knows all about passion, but they understand it differently from their romantic counter-parts. Gothic readers know that we may passionately support certain heroes (or principles or institutions) in political life while also knowing that we ought not take our eyes off them for fear of what they might do to us if we did. They know that one can be passionately attached to something—a nation, a people, a principle—and be deeply and justifiably (and even therefore!) afraid of it at the same time. As we have seen, again and again, it is the refusal of this ambivalence regarding democratic law (the foreigner as founder), the nation (the foreigner as immigrant), and national or liberal democracy (the foreigner as citizen) that drives the troubling politics of foreignness traced here.30 What if, instead, democratic subjects related ambivalently, gothically, and, yes, passionately, to their leaders, their nations, their state institutions, and all their sites of belonging?
An example of such a relation is provided by James Baldwin, who, in _Notes of a Native Son_ , resists the temptations of mere instrumentalisms like Reagon's for the sake of a more fully gothic romantic (neither merely instrumental nor conventionally patriotic) account of blackwhite relations in America.
In every aspect of his being, [the American Negro] betrays the memory of the auction block and the impact of the happy ending. In White Americans he finds reflected—repeated, as it were, in a higher key—his tensions, his terrors, his tenderness.... Now he is bone of their bone, flesh of their flesh; they have loved and hated and obsessed and feared each other and his blood is in their soil. Therefore he cannot deny them, nor can they ever be divorced (pp.122–123).
Like the more conventionally romantic interpreters of American democracy, Baldwin uses marital and familial metaphors to describe relations among the polity's members but, in his hands, these are not in the service of a happy-ending romance nor on behalf of a simple sense of national belonging. What he provides instead is a sense of the terror of belonging, the hope and betrayal that come with the inextricable intertwining of people in one another's lives across lines of difference and power. What he captures and distills and animates is the impossibility of being dispassionate and merely instrumental in relation to institutions and structures that govern and shape us as thoroughly as our national and state institutions do.
Exhorting citizens to return to the nation and relate to it or to its good guys in unambivalent terms is not the way to (re)inaugurate a vital and magnanimous democratic politics, though it may serve the rather different cause of nationalism or patriotism. Instead, we need a politics that acknowledges our passionate ambivalences and engages them by pluralizing our attachments so that the nation-state is just one of several sites of always ambivalent attachment rather than the sole and central site of simple romantic love.32 The democratic cosmopolitanism that results from such efforts may not escape the paradoxes and conundra of which the symbolic politics of foreignness are symptomatic, but it might relieve some of the pressures that intensify those paradoxes. Per-haps it might even stop us from rescripting those paradoxes into political problematics that usually end up pitting "us" against "them."
Notes
Chapter One
Natives and Foreigners
1. Plato, _Laws_ , 705a; Aristotle, _Politics_ 1272b15.
2. Arthur Schlesinger sees assimilation as necessary to the maintenance of American liberal institutions ( _The Disuniting of America_ ). David Miller worries that foreignness—insofar as it attenuates the viability of national membership—puts social democracy at risk: social democracy requires a unity of community and purpose ( _On Nationality_ ). See also Lind, _The Next American Nation_ , and Brimelow, _Alien Nation_ , for arguments in favor of limiting immigration in order to increase social democracy or preserve existing identities and ways of life. On multiculturalism, see Tully, _Strange Multiplicity_ ; Kymlicka, _Multicultural_ _Citizenship_ ; and Carens, _Culture, Citizenship, and Community_.
3. This is the case both in Kymlicka's _Multicultural Citizenship_ and in Tully's _Strange Multiplicity_ , two books that are otherwise quite different. Kymlicka's liberal principles are expanded but not really tested by their encounters with otherness. Tully's book is more interested in exploring the transformative con-sequences of engagements with otherness.
4. Holston and Appadurai, "Cities and Citizenship, " p. 188.
5. Chambers, _Migrancy, Culture, Identity_ , p. 30.
6. As is suggested by Laclau and Mouffe's theorization of the "constitutive outside, " in _Hegemony and Socialist Strategy_. In its role as "constitutive outside," the other gives definition to an identity by marking what it is not. Interestingly, given my view that the "constitutive outside" lacks a sufficiently positive content, Laclau has since gone on to theorize another site of (supposed) contentlessness in the form of the (supposedly) "empty signifier." See his _Emancipations_ and Linda Zerilli's review, "This Universalism Which Is Not One."
Tina Loo and Carolyn Strange provide an unusually dense and layered example of communal and national (re)constitution by way of contrast with foreignness in "The Travelling Show Menace."
7. Although these other contributions, novelty, cultural breadth, and depth, are also part of the picture, Parekh is right to say, in the spirit of John Stuart Mill's experimentalism, that "new ways of life also bring with them new talents, skills, sensitivities, ways of looking at things, different kinds of imagination, new psychological and moral resources, new sources of spiritual energy and give their receiving society a cultural breadth and depth" (Parekh, "Three Theories of Immigration," p. 108). Portes and Rumbaut also emphasize the reenergizing effect of immigrants on America in _Immigrant America_. This gift of foreigners is not restricted to states like the U.S. that see themselves as immigrant regimes, as Parekh's Millian argument about immigrants' effects on Britain suggests. Similarly, although we hear a great deal in the popular press about the failure of immigrants and their children to acquire or master English, we also get the following, lovely insight from Hugh Kenner (himself a Canadian immigrant to the U.S.), writing about Joseph Conrad: "Though his spoken English is reported to have been often impenetrable, his fiction abounds in local brilliances no native speaker would have thought of, like 'He was densely distressed'" (Kenner, "Between Two Worlds," p. 14).
8. The answer to this rhetorical question may well be "yes." As Thomas Pangle points out with reference to Plato's _Laws_ (a dialogue, set in Crete, in which the main character, an Athenian, is a foreigner): "A political philosopher who wishes to bring about fundamental changes is more likely to succeed if he appears to be not merely an old conservative, but a 'foreigner' in some sense, whose circumstances compel him to defend the ways of 'his people' against the implicit and explicit criticisms of his 'hosts' (the persons he wishes to change or to be the agents of change). Only thus can he openly introduce and defendalien ways and yet appear to be neither a traitor nor a man without loyalty"(Pangle, "Interpretive Essay," p. 396).
9. This use of foreignness appears in Mill's assessment of Chinese culture:remarking on the great wisdom of the Chinese, their excellence in educating community members, and their success at putting the best educated into power, Mill worries that the Chinese "have become stationary, have remained so for thousands of years." How to improve and get themselves back on the road of progress which they themselves initially charted so well? "If they are ever to be improved," Mill says, "it must be by foreigners" (Mill, "On Individuality, " p. 67).
10. All of these explanations are considered by Jean Carbonnier, who also notes with curiosity the foreign-founder phenomenon, in his "A beau mentir qui vient de loin ou le mythe du legislateur étranger."
11. I amindebted to Henry Bienen for making this objection and for pressing me, therefore, to think through the ideas in the next few paragraphs.
12. Zenkovsky, _Medieval Russia's Epics, Chronicles and Tales_ , pp. 49–50.See also Riasanovsky, _A History of Russia_ , p. 24.
13. According to Wachtel, drawing on Anatole Mazour, " 'The founding of [the] Norman school dates back to the first half of the 18th century, traced largely to a group of German scholars' that included 'Bayer, Müller, Schlozer, and others.' [As Pritsak points out, Müller's lecture drew on an article of Bayer that had been published thirteen years earlier, in 1736 (p. 3).] Historians of Russian descent, including Tatishchev and Lomonosovin the 18th century, rejected this theory, primarily because of 'its tendency to consider the Slavs not only a back-ward people, but as incapable of governing themselves.'" According to Pritsak, it was Lomonosov's testimony before the committee investigating Müller that was most devastating to his cause (p. 4). See Wachtel, _An Obsession with History_ , p. 26, and Pritsak, _The Origin of Rus'_ , pp. 3–5.
14. Wachtel, _An Obsession with History_ , pp. 42–43 and passim. I amindebted to Andrew Wachtel for his help in figuring out the Slavic sources.
15. See Wachtel's very interesting discussion of Catherine's identification with Riurik as a foreign-born ruler but more importantly as a literary object (Wachtel, _An Obsession with History_ , pp. 26ff).
16. Said Solovyev: "Try to think of Russian history from the exclusive national point of view. Even if the Scandinavian origin of our state could somehow be explained away, it cannot be denied that the introduction of Christianity into Russia by the Greeks at once brought our nation into the sphere of the supernational life of the world" (Solovyev, _The Justification of the Good_ , p. 428).
17. Pritsak, _The Origin of Rus'_ , pp. 6–7.
18. This argument recirculates points made by Wittgenstein in his critique of James Frazer's treatment of the Fire-festivals in _The Golden Bough_. Frazer argued that Fire-festivals, in which effigies are burned or in which there is a pretense of throwing a victim into a fire, "may naturally be interpreted as traces of an older custom of actually burning human beings on these occasions." Wittgenstein countered that Frazer's search for an origin to explain the power of a contemporary practice missed the point. "But why should it not really be (partly, anyway) just the _thought_ (of the festival's sacrificial origin) that makes the impression on me? Aren't ideas frightening? . . . Hasn't the thought something terrible?" The meaning of our stories and practices is related not to a privileged referent that causes them in some way but to all kinds of evidence, "including such evidence as does not seem directly connected with them from the thought of man and his past, from the strangeness of what I see and what I have seen and heard in myself and others."
Of course, Wittgenstein being Wittgenstein, the argument does not stop here. Not only do those who ask after the meaning of an event have to take account of infinite possible sources; they also have to direct their attention away from sources as such and toward the conduct of the event itself in its daily practice. Meaning, as Wittgenstein said famously in the _Philosophical Investigations_ , is in use. In Frank Cioffi's summary: "Preoccupation with the origins of the Fire festivals is mistaken, not because it is impertinent to 'the inner character of the ritual' but because it is impertinent to the impressions [the ritual] engenders." And later: "That men should pretend to burn men gets its 'depth' from our prior knowledge that men have burned men, not from our conviction that in this particular ritual men were once burned" (Cioffi, _Wittgenstein on Freudand_ _Frazer_ , pp. 90–91).
Ren éGirard is also vulnerable to the criticism Wittgenstein levels at Frazer. Rene In _Violence and the Sacred_ , Girard treats all ritual violence as a way of working through (in modified form) an original violence: "The community is both attracted and repelled by its own origins. It feels the constant need to reexperience them, albeit in veiled and transfigured form . . . [T]he ritualistic imagination . . . allows violence a certain amount of free play _as in the original instance_ , but not too much" (p. 99; emphasis in original). Wittgenstein's point is that we need not think of the "original instance" as an actually existing event or referent that explains or causes the later ritual. Instead, the "original instance" may be thought of as itself the _product_ of the ritual that itself projects an original as its (sometimes unconscious) cause. This is one reason for my own insistence, throughout this book, that the politics of founding is always the politics of (re)-founding.
19. Regarding my use of the term "regime" above, I should note that throughout this book, I use the term in the Straussian and Foucaultian senses, which connote not just government institutions but the widest array of political, cultural, and ethical practices, ways of life, powers, and knowledges that make up the world of citizenship.
20. Consider another example: in and of itself, unlodged in any symbolic context, the fact that many Americans came from elsewhere could mean many different things. In Canada, as in the United States, the vast majority of the population is descended from immigrants, and yet Canadians have not historically taken an "immigration experience" to be essential to their identity. Canada has no Statue of Liberty. Immigrants are officially welcomed these days but they have not historically been idealized. In the Canadian context, the _facts_ of immigration simply do not bear the same symbolic meaning as in the United States. Why not?
An explanation is suggested by Gerard Noiriel's analysis of the United States and France. Noiriel argues that, contrary to conventional understandings, France is actually an immigrant country: "Approximately 20 percent of people born in France have at least one parent or grandparent of immigrant origin. If we take great-grandparents into account and include the foreign population born outside French territory, we reach a total of nearly one-third of the total population." And yet, no one thinks of France as a nation of immigrants, and the figure of the immigrant in France does not possess the same idealized symbolic import that it has in the United States. (This is not to say that it lacks symbolic import, however. On immigration politics in France, see my discussion in Chapter Three.) According to Noiriel, the difference between France and the U.S. is not simply that one has few immigrants and the other has a great many. The difference is the way in which each regime mobilizes and shapes the facts of immigration at the symbolic level. In the United States, that mobilization is part of a _myth of origins_ ; in France it is not: "In all countries, the nation-state's constitution is accompanied by a certain number of 'myths of origin' designed to reinforce the cohesion of a population which has divided itself into antagonistic groups. In countries where immigration played a decisive role in the initial populating, the theme of 'immigrants' often occupies an important place in the constitution of the 'myth of origins.'" In France, "Mass immigration only began in the 19th century," well after the republican founding, "at a time when the structures of the French State had already been in place for quite some time"(Noiriel, "Immigration: Amnesia and Memory," pp. 368–371). Thus, where the United States turned Ellis Island into a national icon, France razed the point of disembarkation used by immigrants in the nineteenth century.
But did immigration in fact play "a decisive role in the initial populating" of the United States? Is that the fact that explains the later symbolization? Although it is true that the original U.S. inhabitants came from elsewhere, one could well argue that they thought of themselves not as immigrants but as Puritans, and that therein lies an alternative myth of origins for the United States in which it is not the original travels and travails of the newcomers that made for what ultimately became their supposedly quintessentially American character but rather their common mission, religion, and purpose. (Indeed, American immigration policy reflected this self-understanding on and off—mostly on—until 1965, a fact largely approved by Peter Brimelow in his _Alien Nation_.) In short, notwithstanding the facts of its origins, even the American self-understanding as a nation of immigrants is at least underdetermined and is as in need of explanation as the nonimmigrant-centered self-understanding of countries with substantial immigrant populations like France and Canada.
21. As this list shows, both high and popular culture feature stories of foreign-founding. Some of these stories are true, as may be the case with the story of Russia's foreign-founder, and others are mythic. Still others, like _Shane_ or Clint Eastwood's spaghetti Westerns, are merely stories, but they recirculate a foreign-founder script that precedes them and whose power exceeds (but is also bolstered by) their particular iteration of the script. Indeed, that is surely part of what motivates the retelling. Repeated stories, from _The Wizardof Oz_ to the _Social Contract_ , interest me because they frame (and are symptomatic of) our expectations and assumptions about our own powers. The power of such stories is most visible when we see them operating as cultural shorthands. For example, a management company posted the following advertisement on a billboard en route to Chicago's O'Hare airport: picturing a pair of ruby slippers and some emerald green towers, the copy read, "If M-B [the management company], managed Emerald City, she would have stayed." The ad can allude successfully to Dorothy without naming or depicting her because the iconography of the ruby slippers operates as a fairly universal symbol in our culture.
22. Kant famously argued that the Jewish people were a people of the sublime, incapable of relating to the beautiful. Only Europeans could apprehend both, and only the German people could find the right balance between the two (Kant, _Observations on the Feeling of the Beautiful and Sublime_ ). My reading of Ruth as a beautiful reinhabitation of the Mosaic sublime may be a way of developing an alternative to that reading of the Jews, though I do not pursue that thought in this book. (In his forthcoming book, _On the Psychotheology of Everyday_ _Life_ , Eric Santner argues that Franz Rosenzweig was also trying to find a way to liberate the Jewish people from this prejudice.)
23. As I suggested above: those who rely on mere population movements, increasing in proportion and tempo in late modernity, to produce a postnational politics wrongly rely on a mere fact to do the work of politics. If new and increased population movements are going to bring about a postnational politics, that will be because proponents of a postnational politics win a(n) (ongoing) contest over the symbolic significance of such population movements. In the absence of popular participation in that contest, nation-states will exercise their uncontested right of persuasive definition, and these "foreigners" will be assimilated into the traditional, nation-centered and nation-centering paradigm of immigration. The United States, with its (re)founding myth of immigration, is particularly well positioned to do this.
24. Thus, against the feminist critic of social contract theory, Carole Pateman, Rogers Smith argues that although she "provides undeniable evidence that liberal writers endorsed conventional beliefs in natural sexual inequality," she does not succeed in showing that the problem lay with their liberalism per se. According to Smith, Pateman's evidence suggests not that liberalism is itself apatriarchal ideology, but "that theorists like Locke" were themselves wedded to multiple, sometimes conflicting traditions and simply "did not really reconcile their inherited patriarchal beliefs with the implications of their more novel, distinctively liberal arguments." In line with his own multiple-traditions thesis, Smith contends that liberalism and patriarchy are "two intertwined but relatively autonomous systems of ideas and practices that . . . many Americans [like their great influence, Locke] have often inconsistently endorsed" (Smith, _Civic Ideals_ , p. 29).
Smith tries to make the point again in response to Uday Mehta's claim that Lockean liberalism, in spite of its universal and " 'inclusionary character' has 'spawned practices' involving 'the political marginalization of various people.'"But here Smith makes an important concession: Mehta is right, he says, to connect these marginalizations to liberalism itself. There is no disputing Mehta's observation that the universality of Lockean liberalism refers to the universal _potential_ of all humans to be rational. Equal political rights were to be awarded only to those who realized their potential, and this usually meant that women, children, workers, foreigners, and others were excluded. Says Smith: "Such liberal arguments have played a role in America's history of hierarchical citizenship laws." But ascriptive arguments have played a larger role, he insists, and in failing to address these, Mehta "reinforces conventional views that the liberal sources matter far more" (Smith, _Civic Ideals_ , p. 517n.48, citing Mehta's "Liberal Strategies of Exclusion" and _The Anxiety of Freedom_ ). Here at least (by contrast with Smith's critique of Pateman), the issue is whether liberal sources of exclusion matter "more" or less, by contrast with elsewhere in _Civic Ideals_ , where the question seems to be whether they matter at all. If they matter at all, which is all I am in a position to claim in this book (the question of whether they matter more or less than other ideologies is not my question, nor indeed was it Mehta's), then they and their shortcomings are surely worthy of attention.
25. Smith, _Civic Ideals_ , p. 505. Smith notes that readers of articles published prior to his book's appearance had accused him of "wrongly exonerating liberal values and institutions from any share in promoting unjust inequalities." In response, he points out that he himself argues that "liberalizing and democratizing changes have often created the conditions for the resurgence of inegalitarian ideologies and institutions" (Smith, _Civic Ideals_ , p. 5). That is, the main problem is still those contending, multiple traditions. Liberalism's fault is its creation of an opening for them by producing changes and dislocations in the name of overly universal ideals, such that those displaced by the changes can not identify with the end goals. Those citizens become bitter and ripe for (re)-capture by ascriptive traditions. (See also Smith, _Civic Ideals_ , pp. 474–500).
26. Or, more precisely, the democratic body politic, with its zealous passions and ascriptive traditions, is rendered foreign to the liberal egalitarian project, which is forced to secure its own relevance by way of an elitist disavowal of the actual people it claims to champion.
27. No less than was Frazer's search for the real referent of the Fire-festivals.
28. Smith's argument is, in any case, tautological. He defines liberal values as egalitarian (which is the question, not the answer), and then everywhere he finds egalitarianism he claims to have found liberalism and only liberalism, and he never finds liberalism anywhere else. This means that in spite of its apparently historical character, Smith's argument is effectively a logical (or illogical) one, notwithstanding its social science trappings. (He presents his reading as a hypothesis that he is testing in the lab of history in which the civic ideologies offered by "the Tocquevillian and multiple traditions accounts [are] independent variables and the citizenship laws" of each era are "dependent variables" [Smith, _Civic Ideals_ , p. 8]). The point is worth noting since in a 1995 article Smith casts Michael Rogin, an earlier and much different sort of critic of American "political demonology, " as making unhistorical claims about the "logic of liberalism" when Rogin's arguments are, in fact, more relentlessly historical (but not, it is true, social scientific) and for the most part eschew ideal types of the sort upon which Smith relies (Smith, response to "Beyond Tocqueville, Please!" by Jacqueline Stevens).
29. Or, as Jean Carbonnier puts it in regard to law: "La xenophilie juridique serait-For the more nationalist assumptions elle donc, quoique moins consciente, aussi naturelle que son contraire, la nationalisme du droit?" ("Could one say that the xenophilia of the law, albeit in a less conscious manner, is as natural as its contrary, the nationalism of the law?") (Carbonnier, "A beau mentir qui vient de loin ou le mythe du législateur étranger (1)," pp. 228–229). For the more nationalist assumptions regarding democracy, see Beiner, _Theorizing Citizenship_ ; Smith, _Civic Ideals_ ; and Miller, _On Nationality_.
30. See, for example, Beiner, _Theorizing Citizenship_ , and Smith, _Civic Ideals_.
31. This is another respect in which I differ from Smith, whose analysis is in the service of the problem of governance, not politics qua civic action. It is from the perspective of governance that he issues a call, at the beginning and end of his book, for a new national myth: political leadership requires "a population to lead that imagines itself to be a 'people'" (Smith, _Civic Ideals_ , p. 6). And again: "Political elites must find ways to persuade the people they aspire to govern that they are a 'people' if effective governance is to be achieved" (p. 9). And again: "Because the imperative to constitute a people that feels itself to be a people is politically necessary, it is also a weighty though certainly not absolute moral imperative" (p. 474).
I move away from this focus on the problem of governance to an alternative emphasis on the problem of political action, from a focus on the state and its needs to a focus on democracy and its needs (which may be quite different from and may even conflict with those of the state). Therefore, the assumed "necessity" of national identity and affect to democratic politics is very much in question in this book.
32. Thus, contra Beiner, Smith ( _Civic Ideals_ , p. 9), and others, who treat cosmopolitanism as aiming to _replace_ states with an international government, states remain important potential and actual organizers of social justice as well as potentially useful allies against powerful international institutions that can be difficult to hold accountable. But cosmopolitan politics offers another register at which to hold states and other institutions accountable for their actions in the world.
33. Rousseau, _Discourse on the Origin of Inequality_ , p. 160; emphasis added.
Chapter Two
The Foreigner as Founder
1. Salman Rushdie points out the repeated grayness (Rushdie, _The Wizard_ _of Oz_ , p. 16) and also notes that the home-yearning phrase—"There's no place like home" (in Rushdie's estimation, "the least convincing idea of the film")—that now functions as a synecdoche for the film was not itself part of the book upon which the film is based (pp. 14–16). "There's no place like home" was a Hollywood addition, intended (futilely, given the phrase's fundamental undecidability [see p. xiii above]) to bring comfort and closure to what is otherwise, in Rushdie's estimation, fundamentally an open-ended fable of emigration and adventurism: "At the heart of _The Wizardof Oz_ is a great tension between . . .the human dream of _leaving_ [and] its countervailing dream of roots, " but, when Dorothy sings "Over the Rainbow, " "can anyone doubt which message is the stronger?" That song, says Rushdie, "is, or ought to be, the anthem of all the world's migrants, all those who go in search of a place where 'the dreams that you dare to dream really do come true.' It is a celebration of Escape, a grand paen to the Uprooted Self [strikingly, initials that stand as well for the United States, land of the Uprooted Self], a hymn— _the_ hymn—to Elsewhere" (p. 23).
2. Salman Rushdie captures, with an émigr é's eye, the film's depictions of the many rites and rituals whereby people try to secure the safety of their homes against the vagaries of nature, contingency, injustice, inequality, and power politics. In the opening scenes of the film, for example, Dorothy's Auntie Em and Uncle Henry are too busy to pay attention to their distressed niece. They are, as they say impatiently, "trying to count." Counting eggs they have collected, they are literally—given the coming tornado—counting their chickens before they've hatched. They rely on their counting to impart a certain safety and reliability to their windswept and politically tumultuous world. (Home, as Marina Warner points out, is always the fragile product of "men's and women's labour together" [Warner, _Managing Monsters_ , p. 112]).
This ritual of addition is supported by the film's geometry. Rushdie notes that the "world of Kansas . . . is shaped into 'home' by the film's use of simple uncomplicated shapes (triangles, circles, parallel lines); none of your citified complexity here. Throughout _The Wizardof Oz_ , home and safety are represented by such geometrical simplicity, whereas danger and evil are invariably twisty, irregular, misshapen." Think of the hunchbacked wicked witch of the West by contrast with the perfect-postured good witch, Glinda. And recall that the aunt's and uncle's mundane defenses are undone by a _twister_ (Rushdie, _The_ _Wizardof Oz_ , pp. 21–23).
3. Adventurous, yes, but also, as Rushdie points out, rather "meek, " especially by contrast with the Wizard, who is also an immigrant: "These two immigrants have adopted opposite strategies of survival in a new and strange land. Dorothy has been unfailingly polite, careful, courteously 'small and meek,' whereas the Wizard has been fire and smoke, bravado and bombast, and has hustled his way to the top, floated there, so to speak, on a cloud of his own hot air" (Rushdie, _The Wizardof Oz_ , p. 54). These two immigrant strategies—points on a spectrum that ranges from submission to the host culture over to the immigrant demand that the host culture meet him on his own terms—are often gendered feminine and masculine, respectively. We will revisit them in Chapter Three by way of a different immigrant tale, the biblical Book of Ruth.
4. In Baum's book, the story ends differently: Dorothy's three companions are assigned different sections of Oz to rule over. Nonetheless, Henry M. Littlefield reads even Baum's _Oz_ as a populist parable in which the cowardly lion represents William Jennings Bryan. (Littlefield, "The Wizard of Oz: Parable on Populism").
5. Dorothy does not leave because she _wants_ to avoid dominating the peoples of Oz. She leaves simply because she wants to go home. In so doing, however, she unwittingly acts out a foreign-founder script in which foreigners reenchant the regimes they visit without overstaying their welcome.
6. Actually, as Rushdie points out, Dorothy's fantasy is to be _more_ powerful than the grown-ups she knows. Her Auntie Em and Uncle Henry are weak, unable to protect Dorothy and Toto from Miss Gulch, and this leads to Dorothy's departure. Later, when she is "confronted by the weakness of the Wizard of Oz, she doesn't run away, but goes into battle, first against the Witch, and then against the Wizard himself. The Wizard's ineffectuality is one of the film's many symmetries, rhyming with the feebleness of Dorothy's folk; _but Dorothy's_ _difference of reaction is the point_ " (Rushdie, _The Wizardof Oz_ , pp. 10–11).
7. Is it odd to treat such disparate texts as the _Wizardof Oz_ and the Hebrew Bible as if they had some sort of relation to each other? Both high and low culture contribute to contemporary democracies' cultural unconscious. As I my-self write about Dorothy en route to writing about Moses, I am reminded of my own deep association of the _Wizardof Oz_ and Moses. When I was a child, _The_ _Wizardof Oz_ was broadcast on television every spring, right around the same time as Passover, the holiday that memorializes Moses' founding of the Israelites as a people of the law. As I write this, Moses is ascending (or descending?) toan even larger cultural iconicity, however, by way of the first animated film to be produced by Dreamworks: _Prince of Egypt_.
8. Freud argues that there were in fact two leaders named Moses: the first, an Egyptian prince, was murdered by his people; the second introduced Jehovah to the Israelites in the desert.
9. Miller, _On Nationality_ , pp. 94–95 and passim. Others whose work proceeds from such neo-Rousseauvian assumptions are Michael Walzer ( _What It_ _Means to Be an American_ ) and Michael Sandel ( _Democracy's Discontents_ ).
10. That is, there is a tension in Rousseau between a formal conception of the General Will, where the General Will is whatever the people will under ideal circumstances (small territory and population, regular assembly meetings, relative material equality, etc.), and a more substantive conception of the General Will, where the General Will is whatever is really and truly in the public interest, regardless of whether the people will it or not. On this latter view of the General Will, one can never fully entrust power to the people except, paradoxically, at the risk of the General Will's corruption.
11. Rousseau does not just rely on rules or laws to solve the problem. As Steven Johnston points out, the General Will is supported by a host of practices, beliefs, festivals, and rituals that operate, deniably, below the official registers of the General Will ( _Encountering Tragedy: Rousseau and the Project of Democratic_ _Order_ ). On Rousseau's reliance on "woman" and (hetero)sexuality to contain the social order, see Zerilli, _Signifying Woman_ ; and Wingrove, _Rousseau's_ _Republican Romance_.
12. I discuss the _aporia_ of founding in detail in "Declarations of Independence: Arendt and Derrida on the Problem of Founding a Republic."
13. Rousseau, _On the Social Contract_ , Book II, Chapter 7. See also Aristotle, _Politics_ , Book II, 1274a23–1274b25.
14. Thomas Pangle argues that Rousseau's references to Geneva are an instance of using foreignness as an agent of change, given that Rousseau, himself Swiss, was addressing French society in this text. In this, Pangle notes, Rousseau was preceded by Plato, who, as we saw in Chapter One, also exhibits some appreciation of the usefulness of foreignness as an agent of change ("Interpretive Essay," p. 396).
15. Beyond Rousseau's signal reference to prior foreign-founders, the possibility that Rousseau's lawgiver is a foreigner is also supported by Rousseau's insistence that the legislator, "[h]e who frames the laws, " cannot be a member of the sovereign body (Book II, Chapter 7). Since the sovereign body is the entire people in Rousseau's _Social Contract_ , this means that the legislator can not be a member of the people for whom he proposes legislation. He must come from elsewhere.
Further evidence for this reading of the legislator as a foreigner is provided by the fact that the issue of translation comes up in this context—his language "cannot be understood by the masses." True, Rousseau himself attributes the need for translation not to the founder's foreignness, per se, but to his "overly general perspective," his focus on " overly distant objects" and the abstractness of his ideas, which "are impossible to translate into the language of the populace" (Book II, Chapter 7). Nonetheless, the language of translation suggests that a linguistic difference is also operating here or at least that the difference between the abstract and the particular is being metaphorized in terms of the foreign and the familiar. Similarly in Freud, as Patchen Markell reminds me, the harshness of the Mosaic founding of monotheism is attributed not solely to Moses' foreignness, but also and perhaps even primarily to the abstractness of Mosaic monotheism, which demands a painful instinctual renunciation. But the point is surely not to have to choose between abstractness or foreignness as the real cause of the Rousseauvian lawgiver's opacity or of the Israelite resistance to Freud's Moses. One could focus on the abstractness of the law as the real issue and treat foreignness or alienness as a mere metaphor for that abstractness. Or one could, as I do here, _take that metaphorization seriously_ and ask after the goals and consequences of figuring the law (abstract or otherwise) or the lawgiver as foreign. Why is foreignness enlisted as a vehicle or device of the abstract law's domestication? And what politics of foreignness follows from that?
16. Hence the reliance of Italian city-states on foreigners to serve in the office of _podesta_ , a magistracy. "Justice was the province of the official known as the Podesta, who was invariably the citizen of another place, since Italian cities were so torn with feuds that they could not rely on a local citizen to give impartial justice" (Mark Girouard, _Cities and People: A Social and Architectural_ _History_ , p. 53). Weber, too, makes note of the podesta and then goes on to discuss the use of foreigners as both adjudicators and legislators: the former is "summoned from outside the group, not for the purpose of creating a new social order, but to provide a detached, impartial arbitrator, especially for cases in which the adversaries are of the same social status. On the other hand, the legislators were generally, though not always, called to their office when social tensions [ _between_ classes] were in evidence" (Weber, _Sociology of Religion_ , p. 49).
Hannah Arendt also makes note of the practice, not in Italian city-states, but in ancient Greek ones, of relying on foreigners to be lawgivers. She offers an entirely different explanation: "The Greeks, in distinction from all later developments, did not count legislating as among the political activities. In their opinion, the lawmaker was like the builder of the city wall, someone who had to do and finish his work before political activity could begin. He therefore was treated like any other craftsman [like a migrant worker?] and could be called from abroad and commissioned without having to be a citizen, whereas the right to _politeusthai_ , to engage in the numerous activities which eventually went on in the _polis_ , was entirely restricted to citizens. To them, the laws, like the wall, around the city, were not results of action but products of making. Before men began to act, a definite space had to be secured and a structure built whereall subsequent actions could take place, the space being the public realm of the _polis_ and its structure the law; legislator and architect belonged in the same category. But these tangible entities themselves were not the content of politics (not Athens, but the Athenians were the _polis_ ) and they did not command the same loyalty we know from the Roman type of patriotism" (Arendt, _The Human_ _Condition_ , p. 195).
Jean Carbonnier would probably call Arendt's a rationalist explanation of the foreign-founder phenomenon. He is more interested in making symbolic sense of it and so draws, in his own analysis, on features such as those I listed above (Carbonnier, "A beau mentir qui vient de loin ou le mythe du legislateur étranger (1)"). Were Arendt interested in giving a symbolic reading, she might see how a foreign-founder is positioned to solve what she sees as the insoluble problem of (secular) founding: that the "we" has no authority to do what it sets out to achieve. (On this problem in Arendt, see my "Declarations of Independence: Arendt and Derrida on the Problem of Founding a Republic.")
17. Hence Machiavelli's observation that the best founders are foundlings, people whose origins are mysterious. Machiavelli was not so sure that foreigners made good founders, however. Of the two origin stories of Rome, he preferred the foundling version, represented by Remus and Romulus, to the foreign-founder version, represented by Aeneid (Machiavelli, _Discourses_ I.1). In Rome, both myths of founding coexisted. But it was the Aeneid version that was influential in America at the time of the founding. (Thanks to Sacvan Bercovitch on this last point.)
18. Keenan, _The Democratic Question_ , p. 17.
19. For my purposes here, it is not important to distinguish strangers from foreigners. The point is simply to note the recurring theme of (re)founders who come from elsewhere to save hapless citizens for democracy. In connection with this, see Lauren Berlant's brilliant analysis of the "infantilization of citizenship" in the United States in _The Queen of America Goes to Washington City_.
20. "Within the parable of the western, the hero is a man with sure moral bearings who plays the role of enforcer only in the absence of law enforcement," says Kiku Adatto in _Picture Perfect_ , p. 129.
21. Derrida, _Politics of Friendship_ , p. 173.
22. Of course, his reliance on a foreign-founder could be Rousseau's way of highlighting the virtual impossibility of real democracy in the modern world. Such a reading of Rousseau would be buttressed by attending to the list of other necessary conditions for democracy given by Rousseau, including small size of population, territory, and relative isolation, all of which, Rousseau knew, were extremely unlikely in the modern world and have only become more unlikely since the time of his writing.
23. Lincoln, "The Perpetuation of Our Political Institutions," p. 18.
24. Bloom, "Interpretive Essay," p. 4.
25. George Washington exhibited the discipline necessary to secure such a timely departure. Or better, the ideal of the citizen-farmer that influenced Washington's decision not to seek a third term in office may have counterbalanced the ambition that such an office tends to generate (Wills, _Cincinnatus_ ).
26. Crossette, "And You Thought the Age of Viceroys Was Over," p. 3.
27. Geoffrey Bennington, one of the few to note the foreignness of Rousseau's founder, also notes the founder's "radical undecidability," but Bennington makes no connection between the founder's undecidability and his foreignness. Instead, Bennington says the lawgiver is undecidable because it is impossible to know for sure whether the lawgiver is authentic or a charlatan. Neither signs nor durability of the institutions founded (Rousseau's two criteria) can help a people faced "with a possible legislator, " says Bennington, because signs are ambiguous and the people cannot assess durability in advance. "Legislator and charlatan thus remain radically undecidable" (Bennington, _Legislations_ , p. 222). Rousseau concedes the unreliability of signs, but, contra Bennington, Rousseau himself treats durability as a reliable, corrective criterion: " 'Vain wonders [miracles, signs] form a transient bond, but only wisdom makes it durable'" (quoted in Bennington, _Legislations_ , p. 220). While Bennington is right to say that durability cannot be assessed ahead of time, it is also the case that it cannot happen that a people should be in the position of deciding ahead of time about the authenticity of the legislator prior to the legislator's formation of them into a people. The issue of the lawgiver's authenticity is post hoc, an issue for the judgment of posterity (as Hannah Arendt might put it), not for deliberations of a group deciding by whom it should be founded. That never happens. That such a judgment, when made post hoc, can be wrongly made—"a durable state may always be a mere simulacrum of a good one" (p. 222)—is part of the structure of judgment itself and not a sign of a _radical_ undecidability that has anything in particular to do with Rousseau, lawgiving, or democracy (as Bennington shows he knows when he later says "any event of thought . . . involves this undecidability" [p. 222]). Samuel Weber also makes note of the foreignness of Rousseau's lawgiver, but he does not pause to analyze it at length (Weber, "In the Name of the Law").
28. Rousseau also admired Lycurgus and Numa ( _Government of Poland_ , Chapter II, "The Spirit of Ancient Institutions"). I learn from Jan Assmann that this treatment of Moses as a lawgiver or founder is decidedly un-Jewish: "It is not a particularly Jewish project to make Moses the creator of the Jewish nation, " and, notably, it is this Moses, "the lawgiver and political creator, who needs his Egyptian education" (Assmann, _Moses, the Egyptian_ , p. 165).
29. Freud, _Moses and Monotheism_ , p. 38. Rousseau echoes Moses' protest to God that he cannot lead the Israelites because he has a heavy tongue and will not be understood: Rousseau's legislator has a problem because his language "cannot be understood by the masses." It seems that one thing these foreign-founders have in common is that they are precisely not "great communicators."
30. That Moses' foreignness fitted him for his task of liberating the Israelites is suggested in _Acts_ 7:22: "And Moses was learned in all the wisdom of the Egyptians, and was mighty in words and deeds" (quoted in Assmann, _Moses,_ _the Eyptian_ , p. 149).
31. Most responses to Freud's text focus on his claim that Moses was an Egyptian, often using it as a way to explore Freud's relation to Judaism, rather than treating Freud, as I do, as a contributor to an independent genre of foreign-founder scripts. For example, Jonathan Boyarin suggests that Freud, in this text about a father-killing, actually enacts a parricide of his own, killing off the inconveniently Jewish Moses in order to replace him with a more acceptably assimilative (in the context of 1930s Vienna) Egyptian version. If the non-Jewishness of one's name signals a non-Jewish identity, as Freud argued regarding Moses, then, Boyarin points out, not only Moses' Jewishness, but also Sigmund's, is conveniently undone.
Barbara Johnson also sees Freud's Egyptianization of Moses as a self-erasure but, contra Boyarin, she insists the erasure is paradoxically affirmative: "For Freud, the nature of his Jewishness was to be _sous rature_ , under erasure. And yet that erasure was somehow _itself_ the very erasure of Jewishness" (Johnson, "Moses and Intertextuality, " p. 21). This paradox probably provides too neat a 'solution' to Freud's difficult relation to Judaism. (If the very act of denying his Jewishness is Freud's way of affirming it, what would one have to do really to deny it? Or is "the essence of Jewishness" precisely the impossibility of [non] identification? In that case, however, what could one possibly do to affirm it? And [why] is this paradox the essence of _Jewish_ identity and not of _every_ form of [non]identification?) But Johnson considers another possibility as well. Freud's rewriting of Moses, she argues, marks Freud's refusal "to confine the notion of difference within a logic of identity." Describing "participation in a people as an experience of _self-difference_ " (p. 23), Freud's text (which was published in German in 1935 and in English in 1937) is, in effect, Johnson suggests, a secret letter to Hitler, one that challenges the identitarian premises of Nazi anti-Semitism.
Contra Johnson, Jan Assmann argues that Freud aims to identify the real reasons for—rather than the unreal objects of—anti-Semitism. Freud's Egyptian Moses does not challenge anti-Semitism simply by showing that the supposedly unitary object of anti-Semitism, the "Jew, " does not in fact exist because identity is riven all the way down. Instead, on Assmann's account, Freud uses the foreign Moses to locate the source of anti-Semitism's hatred in (in Assmann's words) the " 'hostility' inherent in monotheism as a religion of the father. Not the Jew but monotheism had created this undying hatred. By making Moses an Egyptian, [Freud] deemed himself able to shift the sources of negativity and intolerance out of Judaism and back to Egypt and to show that the defining fundamentals of Jewish monotheism and mentality came from outside of it"(Assmann, _Moses the Egyptian_ , p. 167). In short, the "struggle against the Mosaicdistinction could" be anti-Semitic, but it could "also assume the character of a fight against anti-Semitism." That it could be the latter is evidenced, on Assmann's account, by the fact that "the most outspoken destroyer of the Mosaic distinction was a Jew: Sigmund Freud" ( _Moses the Egyptian_ , p. 5). I think Assmann is right about the undecidability of the struggle against Mosaic distinction, but he is clearly quite wrong to assume that if a Jew propounded the argument, that is—all by itself—evidence that the argument is not anti-Jewish.
32. The exposure myths help to manage not only class difference but also the problem of foreignness: "Thus Cyrus is for the Medes an alien conqueror; by way of the exposure myth he becomes the grandson of their king" (Freud, _Moses and Monotheism_ , p. 11).
33. Why does the Moses story reverse the usual chronology? "Whereas in all other cases the hero rises above his humble beginnings as his life progresses, the heroic life of the man Moses began by descending from his eminence to the level of the children of Israel" (Freud, _Moses and Monotheism_ , p. 13).
34. This is an _assumption_ on Rousseau's part. He does not study Moses. He simply infers Moses' authenticity as a judicious lawgiver from his success in founding a people capable of such sheer durability. That is, as I argued above contra Bennington, the judgment of the lawgiver's authenticity is post hoc.
35. Rousseau admires Moses for using "countless prohibitions" and practices to make the Jews unable to be absorbed by other peoples, to preserve their distinctiveness, to make them "outsiders forever." So it is funny to note that circumcision, the very practice that was touted as marking the Israelite difference, might actually have made the Israelites more like their foreign masters than unlike them, and would certainly have provoked in the Israelites not just a sense of distinctness but also an uncanny memory of their former foreign masters.
36. Indeed, there is an oddly (for Freud) identitarian premise doing unacknowledged work here. Freud repeatedly suggests some sort of connection between the violence with which Moses founded the Israelites and the impositional quality of Mosaic law as _foreign_ law. The implication, that if a code of norms is your own its transmission is less impositional, is contradicted by Freud in _Civilization and Its Discontents_ , where the transmission of norms (there the transmission is cross-generational rather than cross-cultural) is traumatic as such.
37. The foreign-founders of Rousseau and Freud _both_ find it necessary to use violence to form their followers into a people, but the violence of Freud's founder is more explicit, more extreme, more colorfully described, and more centrally related to the process of founding. Quoting from the Bible, Freud says Moses "directed the Exodus 'by strength of hand'" (Freud, _Moses and Monotheism_ , p. 32). Freud goes on to report (with Machiavelli hovering in the background) that "Moses, trained in Ikhnaton's school, employed the same methods as the king; he gave commands and forced his religion on the people" (p. 57). In those times, Freud notes in a footnote, "any other form of influence would scarcely have been possible." Exodus reports "a series of grave revolts" against Moses while the people were "wandering in the wilderness." All of these "were suppressed with savage chastisement" (p. 58). In his reading of Rousseau, Johnston emphasizes the violence of Rousseauvian democracy's (re)founding ( _En-countering_ _Tragedy_ ).
38. Joan Copjec distinguishes between the slain primal father who is the principle of _jouissance_ and the ideal father, represented by the son who, by way of his "eviction of excess pleasure [the slain father], " is formed "as an ideal father, 'mild and provident' in the words of de Tocqueville, kinder and gentler; in the words of George Bush's speechwriter, Peggy Noonan. He is the place to which all our questions are addressed, the place of knowledge; he is therefore often imagined under the traits of the educator (take, for example, Noonan's ideal: America's new 'educational President'). The ideal father installs a badly needed certainty in the place of the devastating uncertainty, the crisis of legitimation, that follows in the wake of the primal father's murder" (Copjec, "The Unvermögender Other, " p. 36). Rousseau's lawgiver, who educates the people into the law, is like the ideal father. As we shall see soon, however, he can also be read as the primal (soon to be slain) father.
39. In other words, don't we here have a version of the paradox of founding?As with the father, so with the lawgiver: if it is up to the lawgiver in his wisdom to decide that the time is right for him to leave, then doesn't that mean that the people are not yet ready to be left? Were they ready to be self-governing, they would know the time was right and they would take it upon themselves to send the lawgiver packing (or, as in Freud's version, they would kill him). But Rousseau does not envision the lawgiver's departure that way. The paradox exemplifies Rousseau's tendency both to trust and mistrust the demos at the same time, a tendency that makes itself felt throughout the _Social Contract_.
40. And, as Bill Connolly reminds me, the boy also calls out: "Mother wants you. . . ."
41. Freud, _Totem and Taboo_ cite, summarized in _Moses and Monotheism_ , pp. 102–103.
42. Freud mentions the idea of screen memories in passing in _Moses and_ _Monotheism_ , albeit not with reference to the departure of the lawgiver (p. 93).It is interesting to note that where Rousseau's lawgiver's fantastically timely departure is what makes his law stick (and this accords with the traditional interpretation of the biblical Moses' timely death before entering the Promised Land), in Freud it is precisely the absence of such a timely gift that leads to the murder of the father, and it is that murder, and its haunting aftereffects, that gives his law its lasting quality.
43. Again, we are revisited by the paradox of founding.
44. But he does ask about the consequences of Moses' foreignness for later Christians and Jews. On this point, see Assmann, _Moses the Egyptian_. Seealso Connolly, "Freud, Moses, and Secularism, " and Connolly, _Brains, Techniques,_ _and Time: The Ethics of Nonlinear Politics_.
45. Freud explains why Moses' foreignness was concealed, but he never asks why it was _poorly_ concealed: the concealment was motivated by a desire "to glorify the new God and deny his foreignness" (Freud, _Moses and Monotheism_ , p. 85). But why did the Israelites not conceal their secret better? The question recalls Yerushalmi's engagement with Freud: Yerushalmi says, contra Freud, that if the Israelites had murdered Moses, they would not have concealed it, since they are presented throughout the Hebrew Bible as extremely recalcitrant, and the murder would fit with that picture. If the Israelites were really interested in concealing the foreignness of Moses, and their supposed murder of him, they had the power to do so, and they would have also then concealed what they chose instead explicitly to reveal: the fractious nature of the relationship between this founder and his people, who constantly challenged his authority and violated his rules (Yerushalmi, _Freud's Moses_ ). But, as Jacques Derrida argues, with and against Yerushalmi, why not assume that the Israelites could have both concealed their murder of Moses and confessed it at the same time? After all, what event vanishes with _out_ a trace? I suppose I am saying the analogous thing with reference to Moses' foreignness, which is only more obviously both concealed and disclosed at the same time (Derrida, _Archive Fever_ ).
46. Indeed, as Linda Zerilli points out, Rousseau's insistence on the need for self-identity ultimately drives his intense hostility to all forms of difference (Zerilli, _Signifying Woman_.) For a reading of Rousseau that focuses on his emphasis on self-identity, see Derrida, _Grammatology_.
47. I have in mind here Rawls and Habermas and their various devotees.
48. Rousseau, too, uses xenophobia as a way to generate social unanimity, not in the _Social Contract_ , but in _The Government of Poland_. In the _Social_ _Contract_ , Rousseau does note that a certain unitariness becomes characteristic of the body politic "in regard to the foreigner"; that is, when the polity operates in relation to other polities, "it becomes a simple being, an individual" (Book I, Chapter 7). Notably, in Girard, the social unanimity sought includes the agreement of the scapegoat himself: "What is required is their [the scapegoats'] enthusiastic agreement with the decision to destroy them" (Girard, _Job_ , p. 116).
49. Girard, _Violence and the Sacred_ , p. 12. In _Job_ , Girard also mentions or phans (p. 78). When Girard explains that kings are sometimes scapegoats be-cause they are liminal, he assimilates kings to a larger category that includes foreigners, children, mad people, and so on (Girard, _Violence and the Sacred_ , p. 12). (See also his reading of Job as a king figure.) In so doing, he offers some insight into the role of that which is figured as liminal in propping up the ordinary, but he also misses an opportunity to ask whether there is any particular, specific connection between _kings_ and scapegoating, such as that explored here by way of the idea of the alienness of the law. This is the flip side of the error made by Geoffrey Bennington who, because he always already knows that the law is alien, treats the alienness of the lawgiver solely as a symptom of that other alienness and never asks what other connections may tie together foreignness and founding, or foreignness and democracy in particular (Bennington, _Legislations_ ). Moreover, since the law may be alien in _any_ regime, Bennington's analysis begs the question we are trying to answer here: Is there a specific connection between _democracy_ and foreignness?
50. The neatness of Girard's theory leaves him unable to give an adequate account of what _causes_ sacrificial crises: Where do they come from? If they work so well, why do they need to be periodically repeated? Why should we assume that sacrificial rituals never exceed the economy to which Girard has assigned them? It seems that Girard overestimates the power, effectiveness, and ubiquity of scapegoating or ritual killing. (By p. 251, even foreign war is "merely another form of sacrificial violence" [ _Violence and the Sacred_ ]). For an implicit critique of Girard, especially on the count of ubiquity, see Giorgio Agamben's analysis of the Roman law category of bare life or _homo sacer_ , a form of life that is identified by law as available to be killed but not ritually or sacrificially (Agamben, _Homo Sacer_ ).
51. On the need of Job's friends to turn his bad luck into something deserved and meaningful rather than random or contingent, see Connolly, _The Augustinian_ _Imperative_. Girard discusses this in some detail too, generating a nice reading of _Antigone_ in relation to _Job_ (Girard, _Job_ , Chapter Sixteen).
52. Of course, the identification of Moses as foreign and his subsequent murder could be a way of cleansing other taints too, such as those explored above:the alienness of the law that the people nonetheless proclaim as their own, and/or the mutual opacity of a people who like to style themselves as kin. No matter what we identify as the troublesome taint, the solution could still be the scapegoating of Moses as foreign, and any one of these could still be a Girardian reading insofar as each treats Moses as a scapegoat whose "foreignness" (contra Rousseau, Freud, and Girard) is a symbolic device or _fiction_ that works—as part of a (re)founding event—to _solve_ a social crisis.
53. By attributing this view to many theorists of democracy, I mean to distance myself from it. As I argue elsewhere, the effort to distinguish qualitatively between the extraordinary and the ordinary, or between the politics of founding and the politics of maintenance, suggests misleadingly that these are opposite and separate phenomena, when instead they are deeply supportive of each other (Honig, _Political Theory and the Displacement of Politics_ , Chapter Four).It is in order to keep in mind the relationship between the two and to mark their similarity that I talk, throughout this book, about the politics of (re)founding and not just the politics of founding. The latter refers to an origin story; the former marks the role of that origin story (retold in myriad ways) in the daily reconstitution of citizenship.
54. Lincoln, "The Perpetuation of Our Political Institutions, " p. 21.
55. Yet another reading of _Shane_ suggests itself if, recalling that the story is told from the perspective of the young boy, we read it together with another story of expelled violence told from a boy's perspective: E.T.A. Hoffman's "The Sandman, " interpreted by Freud in "The Uncanny." Freud argues that in "The Sandman, " the boy addresses his fear of his father's horrifying power (in Freud's terms, his "castration anxiety") by splitting his father into two distinct figures, one (in Eric Santner's gloss) "nurturing and caring, the other demonic and castrating" (Santner, _My Own Private Germany_ , pp. 68–69). Analogously, we might say that the boy's father in _Shane_ (rather than the community as a whole) does the violence that their (re)founding calls for; but the boy, frightened by his father's power, then invents a departing figure onto whom he can safely project his father's violence, leaving himself a domesticated father figure, one who is nurturant and safe, if also a bit weak.
As Eric Santner points out, Freud argues that the splitting of the father-imago is "largely the product of the son's delusional elaboration of an inevitable and universal ambivalence _vis-a`-vis_ the father." But the American psychoanalyst William Niederland argues that there is (in the Schreber case, anyway) a " ' _nucleus_ _of truth_ ' in the son's paranoid productions" and it is possible to trace many of those delusions back "to the father's _actual handling_ of his son during childhood." Either way, whether the occasion of the splitting is produced or found, splitting is a strategy that allows the subject to save the object and preserve the ability to identify with it. See Santner, _My Own Private Germany_ , Chapter Two, esp. pp. 64–70. See also Freud, "The Uncanny, " p. 232n.1.
56. Girard offers a moving analysis of the need for the scapegoat to cooperate in his own scapegoating (Girard, _Job_ , Chapter Sixteen). The analysis here could be usefully read in tandem with Berlin's condemnation of positive liberty in "Two Concepts of Liberty." Both Berlin and Girard condemn the insistence that the victim participate in his own punishment and, in effect, will it; but Girard, in my view, offers a better analysis (but not approval) of the social needs served by that insistence.
57. In _Political Theory and Modernity_ , Connolly argues that the General Will cannot generate itself, and he recurs to Rousseau's _Geneva Manuscript_ for firm evidence of this. Geoffrey Bennington takes the same dim view of the General Will's chances of success ( _Legislations_ , _Dudding_ ) as, indeed, did I earlier in this chapter. I mean now not to replace the earlier reading with the later one, but to add this later interpretation to the earlier one as a second possibility.
58. Saccamano, "Rhetoric, Consensus, and the Law in Rousseau's _Contrat_ _Social_ , " p. 731.
59. In short, read this way, Rousseau's public myth of origins seeks to achieve the effects sought by Rogers Smith in his: the displacement of domestic violence onto a separable, external, and disavowable other.
60. This illegitimacy is symbolized by the murder of the founder in Freud.
61. "In the final analysis, then, the judicial system and the institution of sacrifice share the same function [i.e., to stem the cycle of vengeance], but the judicial system is infinitely more effective" (Girard, _Violence and the Sacred_ , p. 23).Cf. "we owe our good fortune to one of our social institutions above all: our judicial system which serves to deflect the menace of vengeance" (Girard, _Violence_ _and the Sacred_ , p. 15).
62. See Jacqueline Stevens's _Reproducing the State_ for an effective account of how the family and kinship are not models for the state but its products, or, better, both.
Chapter Three
The Foreigner as Immigrant
1. See the genealogy of Jesus in Matt. 1:1–17.
2. Deut. 23:4.
3. Ruth 1:8.
4. Ruth 1:15.
5. Ruth 1:16–17.
6. Ruth 1:20.
7. There is some debate about the details of this scene: Is the next of kin being asked to redeem the land through purchase or to redeem Ruth through marriage? For a summary of the debate and the single best reading of the scene, see Fewell and Gunn, _Compromising Redemption_.
8. In recent years, there has been a veritable explosion of commentary on the Book of Ruth (Kates and Reimer, eds., _Reading Ruth_ ; Brenner, ed., _A Feminist_ _Companion to Ruth_ ; and Brenner, ed., _Ruth and Esther_ ). Ozick's and Kristeva's readings, in which Ruth's migration to Bethlehem is motivated either by her conversion to Judaic monotheism or by her love for Naomi, largely typify the main approaches (Ozick, "Ruth"; Kristeva, _Strangers to Ourselves_ ). Most recently, commentators have begun to write about Ruth from a Moabite or subaltern sort of perspective. See especially Donaldson, "The Sign of Orpah"; Dube, "The Unpublished Letters of Orpah to Ruth"; McKinlay, "A Son Is Born to Naomi"; and Brenner, "Ruth as a Foreign Worker."
9. Some doubt that the Book of Ruth can be a resource for an analysis of immigration politics because the text tells the story of a single migrant, while the contemporary issue is concerned with hordes of people. My own view is that the text's success at dramatizing enduring issues of immigration politics is due partly to its use of the device of personification. Moreover, the story of Ruth has established connections to immigration politics that precede my analysis and Kristeva's. Marjorie Garber recalls playing Ruth in the late 1940s in the U.S. in a series of fund-raisers sponsored by Hadassah to help Jewish refugees make their way to Palestine after the war. Interestingly, given Kristeva's use of the head scarf to mark the recalcitrance of Moslem immigrants, Garber, as Ruth, wore a head scarf to mark her character's European, refugee identity (Garber, conversation with author, Cambridge, Mass., 1996).
10. Deuteronomy 34:5. Buber makes special note of Moses' burial place: "Yahweh buried him, Moses, in the valley of the land of Moab, near Beth Pe'or" (Buber, "Exodus 19–27, " p. 45).
11. As we saw in Chapter One, these elements of the Book of Ruth become apparent when we read Ruth as a foreign-founder (i.e., in relation to Moses) and not as a convert, per se. Reading Ruth as a convert is the traditional reading, which invites comparison with Abraham, a comparison made by both Ozick and Kristeva, as we see below (n. 48).
12. Toni Morrison calls particularly sharp attention to the exclusionary dimension of the (re)founding effect of American immigration in relation to American blacks (Morrison, "On the Backs of Blacks").
13. Some rabbinical commentators suggest that the Moabites practiced human sacrifice. Much is made of this in the 1950s Hollywood film version of the Book of Ruth in which Ruth is a priestess of the cult that delivers young girls to be consumed by the fires of the idol, Chemosh. The charge is not supported by historical evidence, however. A. H. Van Zyl argues that this was not common practice in Moab. In general, animals were sacrificed. King Mesha did sacrifice his eldest son to the Moabite god, Chemosh, but that was under extraordinary circumstances and not part of an ordinary practice (Van Zyl, _The_ _Moabites_ , p. 201). The emphasis in Judaic texts on Moabite human sacrifice is symptomatic of the figuring of the Moabites, in opposition to the Israelites, as a people lacking respect for proper boundaries and distinctions.
14. Ozick, "Ruth, " p. 221.
15. Jack M. Sasson notes this device of personification elsewhere in the Book of Ruth: "A didactic device frequently resorted to by Biblical writers is to limit the spectrum of choice to two alternatives, only one of which will prove to be correct. An obvious method of putting such a concept in effect is the creation of two brothers, only one of whom will ultimately fare well. Mahlon marries Ruth—he will live on" (through the posterity of Obed). Other biblical examples noted by Sasson are Cain and Abel, Jacob and Esau, Ishmael and Isaac—all male (Sasson, _Ruth: A New Translation_ , pp. 16–17). Why does Sasson not include Ruth and Orpah in his list? Perhaps because of his Proppian assumption that Orpah is a merely marginal character, not central to the tale and not worthy, therefore, of further interpretive attention.
16. Hence the _necessity_ of the link between inclusion and exclusion, contra Michael Walzer, who says he is "inclined to reject the _metaphysical_ belief that all inclusion necessarily entails exclusion." I agree with Walzer in hestitating to endorse the general "metaphysical" claim. One merit of this reading of the Book of Ruth, however, is that it illustrates the claim in a particular but also more generalizable context: that of immigration. See Walzer, _What It Means to Be an_ _American_ , pp. 44–45n.30. Another merit of this reading of the Book of Ruth is that it provides an example of an other who is both necessary to and forbidden by the community she supplements and disturbs, in which there is a deep and necessary relationship between that other's forbiddenness and necessity insofar as both stem from the _same_ feature—in this instance, Ruth's foreignness. This makes Ruth a superior example to Oedipus, on whom Girard relies in theorizing the doubleness of the other (Girard, _Violence and the Sacred_ ). That doubleness is weak in Oedipus's case. His necessity to Thebes stems from his wisdom as a ruler, while his forbiddenness stems from his incest (and maybe from his hubris). The relation between these two is contingent, not necessary: in principle, Oedipus could have been a wise king (necessary to the order) without also being a product of incest and a committer of parricide (forbidden by the order). The contingent coincidence of these two characteristics indicates that Sophocles and his audience were exploring the connections between necessity and forbiddenness through Oedipus, but that they had not conceived of these connections in the most truly tragic terms.
Antigone does better at figuring the combination of necessity and forbiddenness to the order that requires and bans her (though Girard takes little notice of her). Here the connections are tighter than in the case of Oedipus: Antigone's necessity to the order (her exemplary fidelity to the rites of burial and the gods of the underworld) is connected to her forbiddenness as a child of incest (which marks her as death identified insofar as she ought, as it were, never to have been born, hence her name). But even here, the coincidence of necessity and forbiddenness is not quite as tight as in the case of Ruth, whose story is not, however, usually treated as tragic. In principle, one could have fidelity to the gods of the underworld (necessary) without being an incest (forbidden). But for Ruth, it is her foreignness and conversion/immigration that make her necessary to the order (as loyalty-swearing foreigner) and dangerous to it (as foreigner). And you cannot have one without the other.
17. Ozick, "Ruth, " pp. 227–228.
18. Ibid., pp. 219–220.
19. Ibid., p. 221.
20. Ibid., p. 224.
21. Ibid., p. 222.
22. Ibid., p. 220.
23. Ibid., p. 221.
24. Ibid., p. 224.
25. Ibid., p. 227.
26. See Katrina Larkin, ( _Ruth and Esther_ ) who says that Ruth's declaration to Naomi "blunt[s] the issue of Ruth's foreignness" (quoted by McKinlay, who expressly disagrees with Larkin on this point in McKinlay, "A Son Is Born to Naomi, " p. 152).
27. Ozick, "Ruth, " p. 223.
28. In psychoanalytic terms, Orpah's (over)attachment to her mother( land)—represented by the phrase her "mother's house" (an unusual locution for the Bible)—prevents her, as it did Antigone (who clung to Polynices, the displaced site of her longing for her mother, Jocasta), from entering the (paternal or monotheistic) Law, the realm of the Symbolic (as Luce Irigaray argues in _Speculum of the Other Woman_ ). Irigaray, moved by a sensibility more tragic than Ozick's, finds a subterranean location for Antigone, who eternally unsettles the dominant order. Ozick pauses to reflect on Orpah, but she does not look to Orpah as a source of eternal dissonance or (in Irigaray's appropriation of Hegel's term) irony.
29. Ruth 1:22, 2:2, 2:6, 2:21, 4:5, 4:9.
30. One commentator argues that this is because childbirth was never Ruth's desire but, rather, Naomi's all along (Reimer, "Her Mother's House, " p. 105).
31. Holst-Warhaft, _Dangerous Voices_ , p. 211n.54, citing Herodotus, _Histories_ , 6.6.138.
32. Another possible reading, however, this one suggested to me by Harry Fleischman from San Francisco, might position Naomi as more corrupt than Ruth. Naomi, like Elimelech, left Bethlehem in a time of famine and returned only when the famine was over. This self-interested behavior suggests that she is morally compromised and that, perhaps, Ruth's foreignness effectively works to distract us from that. Naomi's morally compromised character might even foreshadow that of the line of kings that will follow, always flirting with material concerns and distracted from the one true god.
33. Ozick, "Ruth, " pp. 229–230. And yet the Hebrew term used here for "feet" is a pun for genitals. Ozick's claim echoes Hegel's that the brother-sister relation, of which he takes Polynices and Antigone to be exemplars, is unerotic. As Jacques Derrida points out, the claim is astonishing given the incestuous origins of this pair: "Antigone's parents are not some parents among others" (Derrida, _Glas_ , p. 165).
34. Sasson, _Ruth: A New Translation_ , p. 78.
35. Ozick, "Ruth, " p. 225.
36. Contra Ozick, Orpah's course was courageous, too. The difficulties of such a return are occluded by Ozick, who comments on the unusualness of Orpah's exogamy but then assumes that Orpah's life in Moab will be unproblematic: "Soon she will marry a Moabite husband and have a Moabite child" (Ozick, "Ruth, " p. 224). Fewell and Gunn have a better grasp of the situation:"What are Ruth's opportunities in Moab? Who would want to marry a barren widow, much less one that had been living with a foreigner? And would she be known as the 'Israelite-lover, ' the one too good for her own people? . . . In the end, we might ask, what takes more courage, the staying or the leaving?" (Fewell and Gunn, _Compromising Redemption_ , pp. 97–98). Cf. Kaplan, "The Noah Syndrome, " p. 167.
37. As Derrida points out, Aristotle knows that the borders among these kinds of friendship are porous and cannot be fully policed (p. 205). "Aristotle never gives up analysing the ruses that enable one friendship to be smuggled into another, the law of the useful into that of pleasure, one or the other into virtue's mask" (Derrida, _Politics of Friendship_ , p. 105).
38. Ranci ére, _Dis-agreement_ , p. 30.
39. Saccamano has in mind in particular the lawgiver's impersonation of a prophet in order to achieve this effect. (Saccamano, "Rhetoric, Consensus and the Law in Rousseau's _Contrat social_ , " p. 745).
40. Of course, as we saw in Chapter Two, the founder's foreignness remains a problem even if he leaves, since the law or institutions or norms whereby he (re)founded the people are associated with him and his foreignness.
41. Lacocque, _The Feminine Unconventional_ , pp. 86, 91, 107–108, passim. The historical evidence on the dating of the Book of Ruth is not decisive. Lacocque's conclusion depends finally on his assumption that a Moabite ancestor could only damage David, an assumption which, as we shall see momentarily, is questionable.
42. One might well add to this the observation that the order-constituting exchange in this text is that of a male—Obed—who is passed from one woman, Ruth, to another, Naomi. On the other hand, one could just as well say that Ruth is passed from Mahlon to Boaz by way of Naomi.
43. Kristeva, _Strangers_ , pp. 75–76.
44. Ibid., p. 74. Indeed, contra Kristeva, we may see David's heroic triumph over Goliath as a way of ridding himself of the taint of Moabite foreignness. Goliath, according to rabbinical commentators, is Orpah's grandson. By killing him, David may have done more than simply kill a giant in order to prove his manhood (a traditional folktale device). This test of David may have done double duty. By killing his Moabite cousin, David not only defended his people from attack and entered into manhood, he also proved his loyalty to the Israelites in opposition to Moab. Thus, he gives us Ruth without Orpah.
45. Ibid., p. 75.
46. Ruth's foreignness may also do something else: as Moses' double, Ruth may be a figure whereby Moses' lingering foreignness is dealt with. I.e., how could his repressed foreignness be a problem for the Israelites when they can take on board someone as foreign as this Moabite?
47. Kristeva, _Strangers_ , p. 71; and Ozick, "Ruth, " p. 226. By contrast, Gail Twersky Reimer argues that "Ruth had no desire for children and no interest in maternity." However, Reimer rightly notes, this has not prevented most commentators from reading Ruth in terms of a "single model of woman's relation-ship to motherhood" (Reimer, "Her Mother's House, " p. 105).
48. But Ruth's gender—more to the point, Ozick's gendering of Ruth—also positions Ruth, on Ozick's account, as second to Judaism's founding father, Abraham. "Abraham—the first Hebrew to catch insight—caught it as geniusdoes, autonomously, out of the blue, without any inculcating traditions." Ruth, by contrast, is brought to that vision by living among Hebrews, learning their ways, and loving them. Could she have been a second Abraham? Ozick asks. We can never know for sure, for the "story as it is given is perforce inflexible, not amenable to experiment. We cannot have Ruth without Naomi; nor would we welcome the loss of such loving-kindness." But given what we do know, Ozick says, "Ruth may not count as a second Abraham because her tale is en-folded in a way Abraham's is not: she has had her saturation in Abraham's seed." His is the active agency; she is the passive recipient, "inculcation cannot be expunged: there it is" (Ozick, "Ruth, " p. 378). (This difficulty of assessing the significance of a choice made under the influence of socialization comes up again in Chapter Four in relation to the issue of consent on the part of citizens born into the regime to which they are then asked to consent.)
Ruth's birth into monotheism is second to Abraham's, in Ozick's estimation, because his was an autogenetic birth while hers was enabled by a mother. This fantasy of a birth uncontaminated by a mother permeates Ozick's text. It is, in psychoanalytic terms, a masculine fantasy of invulnerability, self-sufficiency, and autonomy which, in its elimination of the mother from the moment of birth, also elides the primal scene and thereby tries to avoid entirely the question of sexual difference and of sexuality itself. It is worth recalling here Ozick's insistence that there is nothing erotic about the threshing-room floor scene.
Notably, Kristeva makes the opposite assessment of the same pairing, arguing that Ruth is superior to Abraham because she, the foreigner, "did it on her own initiative, " by contrast with Abraham, who "left his father's house in answer to a call from god." To his credit, Abraham responded positively, but Ruth sought out the Israelites and their god all on her own. That is why, ac-cording to Kristeva, Boaz says Ruth deserves "perfect recompense" for her virtue. He means to suggest that she is more meritorious than even Abraham (Kristeva, _Strangers_ , p. 73).
49. It is worth noting that Ruth's takings are all from Boaz or mediated by him. Ruth does not take up space in Bethlehem's public sphere. When the time comes for the legal wrangling over land ownership and inheritance issues, Boaz represents Ruth. She does not represent herself. She seems to accept that the law is not her sphere or that its workings are less available to be interrupted by her takings. This may be another reason why this foreign-founder can stay. She may be a taker who skirts established customs and violates gender norms, but she yields before the law.
50. The point is noted in the most detail by Fewell and Gunn, _Compromising_ _Redemption_ , but also by Bal, _Anti-Covenant_ ; Pardes, _Countertraditions in the_ _Bible_ ; and Newsom and Ringe, eds., the _Women's Bible Commentary_. Notably, this last source finds _fault_ with Ruth for her active seduction of Boaz and for her other daring innovations.
51. Trible, _Godand the Rhetoric of Sexuality_. In the end, however, Ruth gives birth to Obed and is folded into the Israelite order, according to Trible. Trible mourns this outcome and Ozick celebrates it. Regardless of how we value it, however, the outcome presses us to ask: Was it Ruth's gender that enabled her absorption?
52. The general category of demons that included Lilliths was comprised of figures who "were thought to share one thing in common: an inability to achieve destinies commonly attributed to members of their respective sex" (Sasson, _Ruth: A New Translation_ , p. 76, relies here on S. Lakenbacher, "Note sur l'ard–atlili, " esp. pp. 148–158). For example, the _lamastu_ , "constantly frustrated in her ability to produce children, " was a woman who could not be domesticated as a mother. "Enraged, this creature roamed far and wide, ready to attack and harm unsuspecting children and women in labor." In addition, the _lilu_ , _lilit/_ _ardat_ , _lili_ represented "individuals who were never able to consummate their marriages. Crazed with unquenched desires, these creatures sought to mate with humans of their opposite sex . . . to ruin marriages, and lure prospective mates into their own madness." A version of these boundary- and role-defying women was present in the ancient Semitic world as Lillith, whose name may link up etymologically with _layla_ , meaning night.
53. Arendt, _Origins of Totalitarianism_. One could think of the "right to have rights" as the political-theoretical ground for the model of political agency as taking that I develop throughout this book. That is, the right to have rights could be seen as an authorizing ground for the claims made by those without proper standing to make them. (I think that's what Arendt was thinking of when she developed this idea.) But I don't pursue this point further because in general I think such authorizing grounds tend to follow, post hoc, from the making of new claims rather than grounding them in advance.
54. Kristeva, _Nations without Nationalism_ , p. 36.
55. Kristeva, _Strangers_ , p. 194.
56. Kristeva, _Nations_ , p. 60.
57. Ibid., p. 63.
58. See Brubaker, _Citizenship and Nationhood in France and Germany_ , pp. 138–164; and Hollifield, _Immigrants, Markets, and States_ , Chapters Six and Seven.
59. Kristeva, _Nations_ , p. 37.
60. Ibid., p. 40.
61. Ibid., p. 41. But why not reverse the chronology? Affective relations to the state may well be what undergirds and secures the ties of family (rather than the other way around). Jacqueline Stevens poses this question pointedly in _Reproducing the State_. Similarly, Micheline Ishay troubles another point on the chronology when she suggests that cosmopolitanism or internationalism is historically and conceptually prior to nationalism (Ishay, _Internationalism and_ _Its Betrayal_ ).
62. See Moruzzi, "A Problem with Headscarves, " p. 665.
63. Kristeva, _Nations_ , p. 38.
64. Ibid., p. 59.
65. Ibid., pp. 46–47; my emphasis.
66. Ibid., p. 47. Kristeva does note the tenuousness of the distinction be-tween fetish and transitional object, though, when she concedes that the transitional object is "any child's indispensable fetish" (ibid., p. 41).
67. Ibid., pp. 41–42.
68. Ibid., p. 43.
69. Ibid., p. 47.
70. Ahmed, _Women and Gender in Islam_ , pp. 223–224. Ahmed studies veiling in Egypt, not France, but her argument was echoed by France's Federation of Councils of Parents of Pupils in Public Schools (FCPE), which opposed the expulsion of over seventy girls who wore head scarves to their schools in Lille and the Paris region: these expulsions carry with them "the immense inconvenience of confining these young girls to within their family circle and of limiting any possibility of emancipation" ( _Migration News Sheet_ , p. 2).
71. Ahmed explains: "[T]he fact that wearing it signals the wearer's adherence to an Islamic moral and sexual code has the paradoxical effect, as some women have attested, of allowing them to strike up friendships with men . . . without the fear that they will be dubbed immoral or their reputations damaged" (Ahmed, _Women and Gender in Islam_ , p. 224).
72. The contradiction is not unique to French treatments of veiling. As Marnia Lazreg points out, scholars everywhere treat Moslem women "either as embodiments of Islam, or as helpless victims forced to live by its tenets" (Lazreg, _The_ _Eloquence of Silence_ , p. 14).
73. The complexity of the situation is implied, if not explicitly theorized by Winifred Woodhull, who speculates about what would be the effects of a French ban on the _hijeb_ : "Will their parents simply take them out of school, with the result that they may marry sooner and have fewer professional opportunities than they might have had otherwise? Will the parents keep their daughters in school but be more steadfast in their refusal to allow them to participate in physical education and sex education classes, to go on class trips, or go to the movies with friends? Or will the ban have the opposite effect, reinforcing the legitimacy of the French school system so that the girls may continue their studies, participate fully in the curriculum, and so on?" (Woodhull, _Transfigurations_ _of the Maghreb_ , pp. 48–49).
74. See Fanon, _A Dying Colonialism_ ; and Woodhull, _Transfigurations_ , p. 48.
75. Indeed, Kristeva subjects women to special scrutiny because she relies on the figure of the good mother to figure her cosmopolitanism. There are those whose maternalism preserves cultural difference in a nationalist way, she says, and theirs is "a certain conformist 'maternalism'" which "lies dormant in everyone of us and can turn women into the accomplices of fundamentalisms and mystical nationalisms as they were of the Nazi mirage" (Kristeva, _Nations_ , p.34). And there are those whose maternalism ushers into existence new cosmopolitan "living spaces" that are neither too nationalist nor too world-oriented. Why frame these two options in terms of maternalism? Because women have the luck and the responsibility of being boundary-subjects . . . more dramatically so than men are, " and this positions women to be the mothers of a not yet born "polyvalent community" (Kristeva, _Nations_ , p. 35). Gayatri Spivak also positions women (but not qua mothers) in exceptional relation to cosmopolitanism. See p. 106 above.
76. Thanks to Pratap Mehta on this point.
77. Here there is an important difference between Kristeva and Ahmed. While Ahmed ultimately champions just one dimension of veiling and assesses it in terms of its potential contribution to a feminist metropolitanism that she herself values, she first stops to explore the heterogeneous meanings of the scarf for some of the Moslem women who wear it. Moreover, her feminist metropolitanism seems to be shaped by what she finds. The engagement works two ways. Kristeva, by contrast, already knows the significance of veiling, has no questions to ask of those who practice it, and has, apparently, nothing to learn from them. As I point out above, this suggests that the veil is less of a fetish for its Moslem wearers than for Kristeva herself. Indeed, on one account, a fetish precisely "stands for an absent articulating context" (Rogoff, "From Ruins to Debris, " p. 241).
78. Butler, "Kantians in Every Culture?" p. 18.
79. Kristeva, _Nations_ , p. 11.
80. Ibid., p. 47.
81. Ibid., p. 60; emphasis in original.
82. Ozick, "Ruth, " pp. 227–228. Ruth's famous speech to Naomi is interpreted by many as, effectively, a list of the transitional objects by way of which Ruth makes her passage: going where Naomi goes, living where she lives, Ruth will come to know her people and then finally her god. Kristeva approves of this transitional passage, while Ozick (as we saw earlier) sees it as inferior to Abraham's less-mediated route to monotheism.
The cultural-symbolic connections among nationalism, immigration, psychoanalysis, and transitional objects were on display when the _New York Times_ _Book Review_ used a flag-stuffed baby bottle to illustrate its review of Michael Lind's _The Next American Nation_.
83. I borrow from one version of this account, but I distance myself from psychoanalysis's reliance on the model of an original maternal relation. Separation and transition are issues not just for children or immigrants but for all of us throughout our lifetimes. I also seek to avoid the progressive trajectory of developmental accounts. That trajectory infantilizes the immigrants whose transitions are part of what is at issue here, and it works to affirm Western receivingregimes' perceptions of sending regimes as a "past that the West has already lived out" and can be left behind without loss (Visvanathan, "From the Annals of the Laboratory State, " p. 41). Kristeva's and Ozick's progressive accounts tend to feed these prejudices, too.
84. Santner, _Stranded Objects_ , pp. 19–26; and Peter Sacks, _The English Elegy_ , p. 8. Santner is working with Winnicott, _Playing and Reality_ ; and Freud, "Beyond the Pleasure Principle."
85. Santner, _Stranded Objects_ , pp. 26–27.
86. Ozick and Trible both see Ruth's silence as a sign that this extraordinary woman has been successfully absorbed into the ordinary structure of the regime. Ozick approves of this; Trible, with her emphasis on the patriarchal dimensions of the order in question, sees this as a tragic ending (Ozick, "Ruth"; Trible, _God and the Rhetoric of Sexuality_ ).
87. Moreover, the identification of such tendencies with outsiders absolves contemporary democracies from having to face the fact that enclavism is characteristic these days less of immigrants and ethnics (most of whom still become absorbed by the third generation, according to Portes and Rumbaut [ _Immigrant_ _America_ ]), than of the wealthy, who show a marked propensity to withdraw from public goods and services.
88. Patricia Karlin-Neumann notes the character of the Book of Ruth as a mourning narrative, but she identifies Naomi as the mourner and Ruth as the servant who brings her back to life. Karlin-Neumann never attends to Ruth's character as a mourner who is left bereft. As in Ozick and Kristeva, Ruth is treated as an other who has a service to grant the order (Karlin-Neumann, "The Journey toward Life").
89. Santner, _Stranded Objects_ , p. 24.
90. True, the rabbis say that Naomi's losses are in some sense not regrettable: the deaths of Elimelech and his sons are deserved because they abandoned the community in a time of need. But these men are, nonetheless, members, indeed once-respected members, of this group.
91. Zelinsky, "The Twinning of the World, " p. 1. This is not to suggest that the relations established under the umbrella of sorority are in any way not politicized. The following is one example of the impact sister cityhood can have and the kind of politics that can be generated by such ties. In late 1988, the Lion's Club International of Taipei donated 10, 000 Chinese language books to the Monterey Park, California, public library intending the gift to "reinforce the closeness they felt with their sister-city which many [had] begun to call 'Little Taipei'" Mayor Barry Hatch saw in this gift an assault on American values and fought to refuse it; however, he ultimately lost out to a coalition of local civic groups and Chinese-American community leaders (Crawford, _Language Loyalties_ , pp. 1–3).
92. Chilsen and Rampton, _Friends in Deed_.
93. Of course, as with any cultural resource, this one, too, is subject to inflation, as I was reminded recently when I spotted in Chicago's O'Hare Airport a string of banners declaring Chicago the sister-city of scores of other cities in every region of the world.
Chapter Four
The Foreigner as Citizen
1. I do not discuss Tocqueville in detail here, but see Volume One, Chapter Two, and later pp. 280–281ff of _Democracy in America_ for his views on how immigration and especially "the double movement of immigration" (p. 281) are fundamental to the shape, character, and success of the unique phenomenon of American democracy.
2. Sowell, _Migrations and Cultures_ ; Walzer, _What It Means to Be an American_ ; and Schuck and Smith, _Citizenship without Consent_.
3. The American Dream performs similar functions, as Jennifer Hochschild points out in _Facing Up to the American Dream_.My argument here is analogous in some ways to Hochschild's. She and I are both trying to find progressive possibilities in apparently conservative myths, rather than reject those myths outright.
4. Sacvan Bercovitch redeploys the exceptionalist interpretation of American identity even while subjecting it to greater critical scrutiny than is customary among exceptionalists: "Of all symbols of identity, only _America_ has united nationality and universality, civic and spiritual selfhood, secular and redemptive history, the country's past and paradise to be, in a single synthetic ideal" (BerShostakovitch, _The American Jeremiad_ , p. 176). My aim in this chapter is not to assay the historical success of this nationalist project, but to attend to some of its political and cultural _costs_.
5. The phrase is taken from Peter H. Wood, _Black Majority: Negroes in Colonial_ _South Carolina_ , p. xiv, itself cited in Rosen, _A Short History of Charleston_ , p. 63. Thanks to Paul Pierson for calling the South Carolina quote to my attention and to Michaele Ferguson for tracking it down. See also Daniels, _Coming_ _to America_ , p. 54: "The slave trade was one of the major means of bringing _immigrants_ to the New World in general and to the United States in particular"(emphasis added). Thanks to Kunal M. Parker for calling my attention to Daniels's work. For a thoughtful analysis of how the politics of race were mapped, historically, as an immigration politics by towns seeking to refuse financial responsibility for destitute former slaves, see Parker, "Making Blacks Foreigners"(paper on file with author).
6. See Walzer, _What It Means to Be an American_ , and Rogers Smith, "Beyond Tocqueville, Myrdal and Hartz: The Multiple Traditions Thesis in America."
7. The economistic explanation is also judged to be limited by Ali Behdad, whose work on immigration politics I discuss below. "The conventional liberal wisdom about the public reaction to immigration is, 'When things are going well and there's a shortage of labor, people either look the other way or are actively supportive of bringing cheaper labor into the United States. But when jobs are tight, and the cost of supporting people goes up, then we suddenly redo the calculus'" (political scientist Bruce Cain, quoted in Brownstein and Simon, "Hospitality Turns into Hostility, " p. A6). Behdad argues that "such an economic view of anti-immigration consensus . . . fails to address the role of immigration as both a necessary mechanism of social control in the formation of the state apparatus and an essential cultural contribution to the formation of national identity" (Behdad, "Nationalism and Immigration to the United States, " p. 155).
8. Behdad, "Nationalism and Immigration to the United States, " p. 175.
9. Ibid., pp. 165–166.
10. Ibid., p. 166. Although Behdad thinks I miss the most important pole, too: "The different functions of the immigrant, I would add, are the effect of an ambivalent mode of national identity in the United States, which simultaneously acknowledges the nation's immigrant formation and disavows it. When I say an ambivalent mode of national identity, I have in mind not only the general split between hospitality and hostility, xenophilia and xenophobia, that Honig convincingly discusses in her article, but also the particular ways in which the competing myths of American identity themselves are ambivalently articulated. As I will show in the cases of Crevecoeur's valorization of immigrant America (xenophilia) and the Know-Nothings in the mid-nineteenth century (xenophobia), every discourse of immigration espouses opposite notions of what constitutes an American identity. Forgetting in each instance allows for an ideologically divided response to the question of "Who is an American?" The idyllic and heterogeneous America presented in Crevecoeur, for example, is also revealed as a racially segregated community that excludes both Native Americans and enslaved Africans. Similarly, the reactionary attitude of Know-Nothings toward immigrants is also a progressive response to the industrialists' exploitative uses of immigrants" (Behdad, _Forgetful Nation_ ).
11. Behdad, _Forgetful Nation_.
12. Actually, more than one stereotype is missing here. Also absent is the leftist internationalist foreigner by way of whom public passions were inflamed during the trial of Sacco and Vanzetti as well as during the McCarthy era.
13. On the reality of the myth's effects, see Waters, _Ethnic Options_.
14. See, for example, Winnick, "America's 'Model Minority.' " Schlesinger, too, makes stereotypical note of the strong family relations of Jews and Asians, remarking the power of those relations as a resource for individuals ( _The Dis_ _uniting of America_ ). A recent, less stereotypical and more sustainedly empirical effort in this direction is Sowell, _Migrations and Cultures_.
Celebrants of model minorities highlight the ways in which extended families (and their cheap labor) are necessary for capitalist success, but they say nothing about how capitalist economies also attenuate such ties. Symptomatic was a front-page _New York Times_ story (Dobrzynski, "For More and More Job Seekers, an Aging Parent Is a Big Factor.") on the increasing reluctance of middle-income labor to move for employment, given their desire to remain close to aging parents. In the second paragraph, the language of the story switches. The phenomenon is now called a "problem, " and the perspective adopted for the rest of the report is that of the companies who have to deal with this resistance. The same story could, of course, have been written (also problematically) in a celebratory way with a headline such as: "The return of family ties."
15. For a psychoanalytic account of the foreigner as someone who only wants to take "our thing, " see Zizak, _Tarrying with the Negative_ , pp. 201ff.
16. Holmes, "Anti-Immigrant Mood Moves Asians to Organize, " p. A1; emphasis added. An example of the more usual story about immigrants is "Hospitality Is Their Business, " an account of Indian-American involvement in the hotel industry (popularized in the film _Mississippi Masala_ ). The role of these immigrants as supplements to the American Dream is made quite clear by Joel Kotkin, quoted in the _New York Times_ story as follows: "These Indians are modern Horatio Algers. They're willing to start in marginal and sometimes risky areas that native-born Americans are not interested in going into, and working [ _sic_ ] incredibly hard hours" ("Hospitality Is Their Business, " pp. D1 and D9).Success is here measured by the move in one to two generations from hands-on labor to office management and serious wealth.
The story does not note a small irony: these immigrants are in the _hospitality_ business at a time when the country is particularly inhospitable toward immigrants. Nor does it make much of one complication of the Horatio Alger comparison: some of these immigrants seem to have arrived with rather substantial reserves of capital. Mr. Patel, who "attributes the Indians' success to 'the way we were brought up'"—(whole families put their shoulder to the wheel and community members lend each other money without interest or collateral)—immigrated after "a 20-year career with Barclays Bank in Kenya" (D9).
17. The U.S. Supreme Court opinion claimed to have seldom seen "such a concentrated and relentless campaign to deport an individual" (Bernstein, "Harry Bridges: Marxist Founder of West's Longshoremen Union").
18. On undocumented worker involvement in unionization activities, see the cases of construction workers in Rothstein, "Immigration Dilemmas, " and mattress manufacturing workers in Delgado, _New Immigrants, Old Unions_. On workplace activism earlier in the century, see Greene, _The Slavic Community_ _on Strike_ , and Laslett, "Labor Party, Labor Lobbying, or Direct Action?" On school politics in Lowell, Massachusetts, see Perez-Bustillo, "What Happens When English-Only Comes to Town?" On Chinese political involvements, see Victor Low, _The Unimpressible Race_. On current alien political activism, see "Foreign Legions: Lots of Noncitizens Feel Right at Home in U.S. Political Races, " _Wall Street Journal_.
19. On this point, and others related to the arguments developed here, see Shapiro, _Cinematic Political Thought_.
20. Walzer, _What It Means to Be an American_ , p. 11.
21. Walzer provides no empirical evidence for this. But the claim fits well with Irving Howe's account of Jewish immigrants of an earlier generation in New York ( _The Worldof Our Fathers_ ) as well as with Ronald Takaki's account of Chinese and Japanese immigrants on the American West Coast ( _In a Different_ _Mirror_ ).
22. Reagon, "Coalition Politics: Turning the Century."
23. Walzer, _What It Means to Be an American_ , p. 48. See Holmes, "Anti-Immigrant Mood Moves Asians to Organize": "They want relatives to join them from overseas. They want their culture replenished with new arrivals" (p. A11).
24. Walzer, _What It Means to Be an American_ , p. 66. Cf. p. 18.
25. Walzer develops the idea of a "communitarian corrective" in "The Communitarian Critique of Liberalism."
26. Arthur Schlesinger gives voice to the fragmentation concern ( _The Disuniting_ _of America_ ). Others, like Randolph Bourne, see the fragmentary potential of immigrants and ethnics but differ from Schlesinger in their evaluation of that potential. Rather than decry fragmentation as a threat to citizenship, Bourne celebrates it as a healthy check on American nationalism ("Trans-National America"). In so doing, Bourne effectively shares in Lacocque's assessment of Ruth.
Although I am relying on two different figures to give voice to it, I nonetheless continue to insist—as I did with Ruth—that we see this play of xenophilia and xenophobia as a national ambivalence, rather than as a difference of opinion between two discrete parties. This will continue to be the case in the sections that follow, as well, in which we shall see again and again how two supposedly opposing, xenophilic and xenophobic, assessments not only mirror each other but also both feed the nationalism that is a necessary condition of their respective opponent's position.
27. Walzer, _What It Means to Be an American_ p. 24.
28. Binder and Reimers, _All the Nations under Heaven_ , p. 52.
29. Walzer, _What It Means to Be an American_ , p. 17; emphasis added. On the supposed contrast to Europe, see Gerard Noiriel on France's true character as an immigrant nation ("Immigration: Amnesia and Memory"). In a later book, Walzer acknowledges that France is "Europe's leading immigrant society, " but, he points out, it is different from the U.S. in that it is not friendly to immigrants as such and demands their rapid assimilation (Walzer, _On Toleration_ , pp. 37ff).
30. Estimates are that 195, 000 U.S. residents emigrate annually. See Labovitz, "Immigration—Just the Facts." Regarding the first decades of this century: "Intelligent estimates of how many foreigners returned to their native countries range from a high of nearly 90 percent for the Balkan peoples to a low of 5 percent for the Jews. We do know that in the period between 1908 and 1914, immigration officials recorded 6, 703, 357 arrivals and 2, 063, 767 departures. During these years, more than half the Hungarians, Italians, Croatians, and Slovenes returned to Europe. For the most part returnees included a high percentage of single men" who migrated back and forth, seasonally, until the 1920s quota system was put in place (Dinnerstein and Reimers, _Ethnic Americans_ , pp. 46–47).
31. On the black independence movement in Oklahoma, see Littlefield, _The_ _Chickasaw Freedmen_.
32. Walzer does invite these other groups to become part of his immigrant America. One need not have entered the United States as an immigrant in order to imagine one's citizenship along an immigrant trajectory. Walzer asks whether his citizenship model "can successfully be extended to the racial minorities now asserting their own group claims." Noting recent adaptive moves by (some) black Americans to be called African Americans, Walzer approves of the move. But he is not sure they will succeed. He worries that racism may get in the way and drive some groups to seek out the "anti-pluralist alternatives of corporate division and state-sponsored unification" ( _What It Means to Be an American_ , p. 76). Walzer never asks whether his normative privileging of the immigrant-ethnic- citizen trajectory to membership, and the invitation to adapt to it, may itself obscure particular claims, injustices, and bases of organization for specific groups.
33. For an analysis of new group formations out of injuries wrought by the old, see Connolly, _The Ethos of Pluralization_.
34. _Strictly Ballroom_ , directed by Baz Luhrmann (1992), is an Australian film, but it was very popular with U.S. audiences. Its story of heteronormative national renewal is not unique to Australia. As Peter Weir's _Green Card_ illustrates, the coupling of romance and immigration themes on behalf of the nation travels well from Australia to the United States. There are several American immigration movies that would illustrate the same basic themes: _Big Night_ is among the best. And since this chapter is focussed on American immigration politics, it would have been less risky to offer a reading of one of them. I chose _Strictly Ballroom_ , nonetheless, precisely because it is not usually seen as an immigration movie. Thus, in addition to deepening our understanding of the relation between the politics of foreignness and the politics of gender, reading this film in terms of its politics of foreignness is defamiliarizing and helps to show how the politics of foreignness is often at work in places where we least expect it. I am indebted to Samuel Fleishacker for first suggesting to me that _Strictly Ballroom_ might berelevant to my argument (though I think he thought the movie would serve my purposes less well than it does).
35. Interestingly, Scott, the individualistic renegade, is also a bearer of the community's standards. When he takes Fran as his partner, he begins by teaching her the basic steps upon which the community insists. Later, for the sake of a dance competition which Fran, a "nobody, " obviously cannot win, he allows himself to be partnered with a pale blond insider who knows how to dance properly. In the end, however, he returns to Fran.
36. These immigrants are subtly depicted as good immigrants by contrast with the stereotypical Spaniard pictured in the background taking perpetual siestas with a bottle of alcohol nearby.
37. Walzer himself uses dance to illustrate a similarly acceptable hybridity. In Gene Kelly's _An American in Paris_ , Walzer finds a delightful fusion of Irish and American (African-American, to be precise) that is a synecdoche for the other admirable social, civic, and cultural fusions he admires.
38. In effect, the film illustrates Louis Hartz's thesis about fragmented societies in the new world. The Australian dance community is like a Hartzian fragment. Separated from its organic origins and frozen in time, it is incapable of either innovation or restoration. The Old World, by contrast, is capable of innovation because it has dynamism, conflict, and multiplicity within it. See Hartz, _The Founding of New Societies_.
39. Although, given Machiavelli's account of (male) _virtu_ ` as the ability to be like (the female) _fortuna_ , there is always some essential gender confusion at the base of republican politics. On _virago_ , see my _Political Theory_ , chap. 1.
40. On the American fantasy of the traditional family, see Coontz, _The Way_ _We Never Were_.
41. "More U.S. Men Look for Love Overseas, " _Columbus Dispatch_ , p. 2C. See also Villapando, "The Business of Selling Mail-Order Brides."
42. It is no accident that the term "foreign bride" has already floated over to the financial pages, where it operates as a metaphor for international merger: "A merger flurry in the Swedish banking sector continued on Tuesday and analysts forecast more reshuffling at home before banks cast their eyes oversees for foreign brides" ("Swedish Bank Merger Flurry Seen Continuing, " _Reuter European_ _Business Report_ ).
43. The restoration of proper masculinity by way of the importation of truly feminine foreign brides is not exclusively practiced by American men. In Japan, Thai brides are a "sought-after commodity" for reasons that echo those given by the American men quoted here ("Here Come the Brides, " _Newsday_ , p. B04). And the same trend has been noted in Taiwan, where the government has recently set quotas "designed to slow the influx of foreign brides and boost the marriage prospects of Taiwanese women" ("Crackdown on Importing Foreign Brides, " _Chicago Tribune_ , p. 2). Business is flourishing as well in Saudi Arabia and elsewhere.
44. This intersection of the institutions of marriage and citizenship is significant. Concerns about both came together in a March 17, 1997, Letter to the Editor in the _New York Times_. The author responded to the recent spate of marriages between immigrants and American citizens (reported by the paper as part of an effort by foreigners to acquire residency) by calling attention to the "irony" of the fact that Americans allow this abuse of marriage for instrumental purposes while continuing to deny marriage to those who really value it, gay couples in love. Two critics who examine this intersection are Michael Warner, _The Trouble with Normal_ , and Lauren Berlant, "Face of America".
45. Susan Okin makes this argument without apparent ambivalence in _Is_ _Multiculturalism Bad for Feminism?_ See also my response to Okin, published in the same volume, "My Culture Made Me Do It."
46. As Kant was well aware, the universal cannot survive in the absence of particular enactments of its law. Hence Kant's repeated, transgressive use in the _Groundwork of the Metaphysics of Morals_ of particular examples to represent the moral law on whose unrepresentability he was otherwise insistent.
47. Schuck and Smith, _Citizenship without Consent_ , p. 130.
48. Ibid., pp. 123–124.
49. Ibid., pp. 131–132.
50. Although it is beyond me why we should seek further to legitimate an institution whose legitimacy ought properly always to be in question.
51. These are the sort of people, "passport holders, " that Benedict Anderson worries about in "Exodus."
52. Levinson, _Constitutional Faith_ , p. 99.
53. Schuck and Smith, _Citizenship without Consent_ , p. 109.
54. That is, new immigrants need to be taking on citizenship for the _right_ reasons. In short, what we have here is an uneasy dependence of the performative (consent) on the constative (the right reasons). On the final unsustainability of Austin's distinction between these two, see Derrida's critical appreciation of Austin in "Signature, Event, Context." For my own extended reading of Derrida and Austin on this topic, see Honig, _Political Theory and the Displacement of_ _Politics_ , Chapter Four.
55. Wingrove, _Rousseau's Republican Romance_ , p. 23. As Wingrove rightly points out, the "perceived tension between structure and act—between determining conditions and undetermined choice—is not an epistemological crisis of social theory that arises in the wake of Marx and Freud." Instead, as in her quoted passage above, it is a result of liberal efforts to theorize an individualism capable of performing the consent that liberal legitimation requires. Wingrove's book came out just as I was putting the finishing touches on mine. I find her theorization of "consensual nonconsensuality" in Rousseau to be quite valuable for thinking about consent more generally, and wish I had had access to it earlier, so as to have been able to engage this work in more detail. As my own reading of Rousseau would suggest, however, the idea that the desires forfreedom and domination are twinned (and not just connected, contra Wingrove, who in some moods, suggests that what consensual nonconsensuality comes down to is simply the fact that "the freedom [democratic politics] makes possible requires domination" [p. 23]) is worth pursuing not just through Rousseau's substantial writings on heterosexual romance (as Wingrove does). It is also worth pursuing through his figuration of the law as the both loved and feared paternal and alien figure of the lawgiver in _The Social Contract_ , the text to which Wingrove pays the least sustained attention in her own reading of Rousseau. I will pursue further that twinning of love and fear in the next chapter by way of an analysis of the gothic genre. It is perhaps no surprise that Wingrove also reads Rousseau with some sensitivity to genre—not to gothic, however, but to the related genre of romance.
56. Wingrove, _Rousseau's Republican Romance_ , p. 22.
57. French republicanism was founded with a similar turn to foreignness to testify to the power of France's would-be universal principles. The national legislative assembly approved granting the title of French citizen to Joseph Priestley, Thomas Paine, Jeremy Bentham, William Wilberforce, Thomas Clarkson, James MacKintosh, David Williams, George Washington, Alexander Hamilton, James Madison, and Kosciuszko, among others on August 26, 1792. (For the discussion, see _Archives parlementaires_ , August 24 and 26, 1792.) This was in recognition of their writing or actions on behalf of "la liberté, de la humanité, et des bonnes moeurs." An interesting historical analysis of the xenophilic and xenophobic moments of the Revolution is provided by Virginie Giraudon, who says: "For the first time in French history, an integration status was granted in the name of the universality of ideas to those who had done the most for humanity." But later, "the need for proselytism was replaced after 1792 by the necessity to survive against foreign threats." Giraudon reports the shift from xenophilia to xenophobia but does not ask after the possible logic of their inter-relation (Giraudon, "Cosmopolitanism and National Priority, " p. 593).
58. Here the undocumented immigrant is clearly gendered feminine _in connection_ to her character as a "taker." This figure is reminiscent of the conventional, conniving woman—often depicted in the Hebrew Bible—who takes what is not hers. As we saw in Chapter Three, it was against this devaluation of feminine taking that Phyllis Trible developed the figure of the admirably good and heroic female taker (Trible, _God and the Rhetoric of Sexuality_ ).
59. Schuck and Smith, _Citizenship without Consent_ , p. 122.
60. By stressing the right of the existing community of citizens to "consent" to newcomers, Schuck and Smith perversely turn Lockean consent from a device designed to limit state power into a device for its enhancement. (A similar move is made by Levinson and by Walzer, though at least their way of making the point does not press into service the device of consent. Levinson: "A 'double choosing' is involved: An immigrant's choice to 'adopt' an American identity is coupled with that immigrant's need to be chosen by the United States itself as a suitable member of the political community" (Levinson, _Constitutional Faith_ , p. 97). Cf. Walzer, _Spheres of Justice_ , pp. 31, 39.
61. It should be noted, though, that Schuck and Smith also say that "children (and perhaps their parents as well) may have legitimate moral or humanitarian claims upon American society" ( _Citizenship without Consent_ , pp. 98, 100, and passim) apart from whether they have a claim to citizenship. "It is enough for present purposes to affirm that the Constitution need not and should not be woodenly interpreted either to guarantee their children citizenship or to cast them into outer darkness" (p. 100).
62. Schuck and Smith, _Citizenship without Consent_ , p. 107. One measure of the devaluation of citizenship is Supreme Court decisions such as _Graham vs._ _Richardson_ , which insists that social welfare benefits cannot be restricted to legal residents. In a later article, Schuck is more resigned to the "devaluation of citizenship." He rightly situates this development in the context of increased international integration and migration, and he thinks, four years after _Citizenship_ _without Consent_ , that recent changes in national citizenship are "probably irreversible." But he is unwilling to let citizenship go: "It provides a focus of political allegiance and emotional energy on a scale capable of satisfying deep human longings for solidarity, symbolic identification and community. Such a focus may be especially important in a liberal ethos whose centrifugal, cosmopolitan aspirations for global principles and universal human rights must some-how be balanced against the more parochial imperatives of organizing societies dominated by more limited commitments to family, locality, region, and nation"(Schuck, "Membership in the Liberal Polity, " pp. 64–65).
Schuck is right that the nation-state sometimes balances the drives toward globalization and localization. But the contrary is also true. The nation state is often a _vehicle_ of both globalization and localization as well, as was clear in the United States's move to found NAFTA and in ongoing efforts to localize the administration of social services. Moreover, it is also the case that global and local affiliations are not necessarily disempowering or undemocratic. They can provide helpful, democratizing checks against the coercive powers of the nation- state. It is therefore important to think about the ways in which the emotional "human" satisfactions of citizenship can be appropriated for nonnational entities. Thus, I agree with the last line of Schuck's 1989 essay but take it as one of my _starting_ points: "Today's conception of citizenship may not be adequate to meet tomorrow's needs" (p. 65).
63. Calavita, _Inside the State_ , p. 167 and passim.
64. This is a practice of which Michael Walzer was rightly critical in "Membership, " Chapter Two of _Spheres of Justice_.
65. It should be noted, however, that consent and voluntarism are not obviously nor necessarily enhanced by moving away from _jus soli_. Such a move makes citizenship (contrary to the authors' stated intentions) more ascriptive, not less so; it becomes a status that is more obviously inherited (or not) fromone's parents. Moreover, the practice of _jus soli_ is no less consensual than other mechanisms (tolerated by the authors) that accord children citizenship and nationality at birth.
66. See Rogers Smith, "Beyond Tocqueville, Myrdal and Hartz." Here and in his recent book, _Civic Ideals_ , Smith positions himself as a critic of American exceptionalism so it might seem strange that I put him in the company of figures like Tocqueville, Myrdal, and Hartz when he wrote in opposition to them. As we saw in Chapter One, however, Smith's liberalism is an exceptionalist's liberalism, unhaunted by doubts, otherness, or violences that touch it at its core. Smith departs from the more usual exceptionalists in his insistence that such a pure liberalism has not found itself fully at home in the United States—not yet.
67. That is to say, I am hazarding a strong, logical claim, by contrast with Michael Rogin, who is wrongly charged by Rogers Smith with making claims about the logic of liberalism. Rogin's rather substantial arguments about America's history of exclusion and genocide are historical, not logical. See Smith's response to Jacqueline Stevens, "Beyond Tocqueville, Please!" and Rogin, _Ronald Reagan,_ _the Movie_.
68. Rogers Smith, "Beyond Tocqueville, Myrdal and Hartz" and _Civic Ideals_.
69. No less than all the attention paid to John Huang and other "foreign" or hyphenated lobbyists and contributors distracted attention from the real problems of money in American politics. The Center for Responsive Politics did a study of foreign money in 1996. Using some very generous definitions (e.g., U.S. firms that are subsidiaries of foreign companies), the Center found that these organizations contributed (hard and soft) a total of $12.6 million to federal candidates. A conservative estimate of the total cost of federal elections is $1.2 billion ($200 million in soft, $200 million in public, $800 million in hard money [Congressional races]). So, foreign money is no more and probably less than 1 percent—if that (Daly, "Global Connections"). I am indebted to Steve Ansolabehere for these figures and estimates.
70. The authors unwittingly call attention to this deeper problem when they say that they are seeking to complement the "actual consent [that] is expressed symbolically only through periodic elections" in America. Concerned only about the periodicity of election-based consent, they do not mention the fact that no more than a minority of American citizens vote in American elections.
This unself-conscious projection of the corruptions of American citizenship onto illegal aliens is paralleled by Michael Walzer's more self-conscious metaphorization of withdrawn American citizens as "psychological resident aliens." But Walzer's metaphor also misleads. Just as the metaphor of illegality slides from status to behavior in Schuck and Smith, so in Walzer, a juridical status assigned by the state (resident alien) slides into a political attitude imputed to the person (political withdrawalism). But there is no evidence to support the identification of resident alien status with political uninvolvement, at least not with any level of uninvolvement worthy of remark. Nor is there any evidencefor the converse: that naturalizing immigrants are prone to political involvement (Walzer, "Political Alienation and Military Service, " pp. 99–100, 112–113. Cited by Levinson, _Constitutional Faith_ ).
71. Tocqueville, _Democracy in America_ , pp. 40–41.
72. Rancie`re, _Dis-agreement_ , pp. 24–25.
73. Ibid., p. 41.
74. Ibid., p. 30.
75. The word "counts" and the phrase quoted above are both from Rancie`re. Rancie`re recasts class struggle in terms of those who—from Plato forward—"count" and the uncounted.
76. Raskin, "Legal Aliens, Local Citizens, " p. 1397.
77. It should be noted, however, that residency can be a restrictive rather than a permissive requirement. Long Island uses stringent proof of residency requirements to keep immigrants out of public schools (Carvajal, "Immigrants Fight Residency Rules Blocking Children in L.I. Schools"). In a way, the move to residency harkens back to the legal practice in the eastern United States of treating settlement, not citizenship, as the decisive category of inclusion. However, settlement was hardly a benign category, no more than residence is now. On the legal use of the category of settlement to make blacks into "foreigners, "see Parker, "Making Blacks Foreigners."
78. Another example of an organization devoted to immigrant worker empowerment is Choices, a domestic worker cooperative in the San Francisco Bay Area (Salzinger, "A Maid by Any Other Name"). Other examples include Asian Immigrant Women's Advocates and UNITE. The L.A. Committee for the Protection of the Foreign Born may be seen as an ancestor of these and other groups seeking alien empowerment and rights protection.
79. Walzer, _What It Means to Be an American_ , p. 33.
80. Such groups are not usually state originated but they sometimes are, and perhaps they ought to be state supported. See Lester Salomon's discussion of NGOs and other third-sector associations: "Finally, perhaps the most decisive determinant of third sector growth will be the relationship that nonprofit organizations can forge with government. The task for third sector organizations is to find a _modus vivendi_ with government that provides sufficient legal and financial support while preserving a meaningful degree of independence and autonomy" (Salomon, "The Rise of the Nonprofit Sector, " p. 122). I would add only that in the event of such cohabitations, the third sector would do well to relate to its new partner—government—gothically rather than romantically, that is to say, with healthy measures of caution, skepticism, and ambivalence. In Chapter Five I outline in a bit more detail what would be entailed by such a Gothic perspective.
81. Calavita, _Inside the State_ , p. 167, citing Ray Marshall, "Economic Factors Influencing the International Migration of Workers, " p. 169. See also "U.S. Surveys Find Farm Worker Pay Down for 20 Years, " _New York Times_.
82. Empowering aliens to act as citizens, even when they lack that juridical status (which is the goal of the democratic cosmopolitanism advocated here) attenuates the lines between aliens and citizens, and this is something Schuck and Smith are out to resist. They disapprove of Supreme Court decisions like _Plyler_ , which award social benefits and rights to noncitizens. Oddly, such developments are seen by Schuck and Smith as symptoms of a more communitarian judiciary (one would have thought "cosmopolitan" to be a better adjective). That "communitarian judiciary, " they argue, "increasingly compels government to consent by imposing obligations toward aliens that it has not voluntarily undertaken [but isn't this what courts _do_? Compel governments to provide services or respect rights in ways that they do not voluntarily undertake?]; sometimes, as in _Graham_ , courts override the legislature's explicit refusal to consent" (Schuck and Smith, _Citizenship without Consent_ , p. 109).
83. Of course, people often find themselves involved in political activity that is risky and even dangerous. This is because political involvement often does not happen as a matter of rational calculation based on incentives. As in my gloss on Hannah Arendt, calculation just produces a cycle of willing and nilling that would be endless were it not for the fact that "political action comes to us, it involves us in ways that are not deliberate, willful or intended." Political action "happens to the as yet unready and not quite willing (because still also nilling) subject in the private realm" and thrusts the subject into the risky visibility of the public sphere. See my "Toward an Agonistic Feminism," pp. 223–224.
84. As Randolph Bourne put it in his 1916 essay, "The Jew and Trans-National America": "We want no national unity that is not based on democratic and socialized and international goals" (p. 126). As opposed to the reverse order, which is the more commonly accepted view now: that democracy needs to be based on national unity.
Like Kristeva, Bourne sees the foreign immigrant as having a fragmenting effect, but he welcomes that effect because it staves off the further development of an American nationalism that is not conducive to the social-democratic and international goals on behalf of which Bourne was advocating. Bourne makes his position quite clear in another essay, "Trans-National America."
85. On internationalism and cosmopolitanism in politics, see Bruce Robbins's introduction to _Cosmopolitics_ and also his _Feeling Global_. My own small forays onto the terrain of cosmopolitanism are informed and motivated by these texts and by conversations with their author.
86. The caricature appears in Beiner, _Theorizing Citizenship_.
87. We could call all of these "rooted cosmopolitanisms"—the name given to another hybrid meant to overcome the opposition between nationalism and cosmopolitanism. Something by that name is championed by thinkers as different from each other as Julia Kristeva and David Hollinger. Kristeva's and Hollinger's ideals are not identical but, their differences notwithstanding, these two share a commitment to a cosmopolitanism that depends upon the renationalization of the state.
88. Carens, _Culture, Citizenship, and Community_ , p. 187. Carens rightly notes that "this is an important issue for aboriginal people, " but he seems to think that access to international forums is mostly a matter of gaining "recognition and respect on the world stage as distinct cultural communities and political actors" rather than (also and perhaps most important) being a matter of generating a social movement politics that is a source of leverage for native peoples involved in negotiations on myriad issues with would-be sovereign states. In the end, Carens, too, calls for the renationalization of the state, though unlike some nationalists, he sees no conflict between such a renationalization and forms of citizenship that are differentiated rather than unitary. It is, he acknowledges, paradoxical (but true) that the best way to heighten affect for the nation among minorities and immigrants is often to permit them greater freedom in self-governance and special rights and exemptions. In this, Carens differs from more unitary nationalists such as Richard Rorty and Rogers Smith, whom I discuss in Chapter Five.
89. Spivak, "Acting Bits/Identity Talk, " p. 803.
Chapter Five
The Genres of Democracy
1. Frye, _Anatomy of Criticism_ ; White, _Metahistory_. I follow White in acknowledging the problematic but still useful nature of Frye's taxonomy (p.8n.6). Stephen Greenblatt makes a similar argument about the genred nature of legal writing with regard to the Ken Starr report ("A Story Told with Evil Intent").
2. The question is particularly worth asking at this point in time, when (in the aftermath of Alasdair MacIntyre's _After Virtue_ ) recourse to narrative as a "solution" to political theoretical problems has become a virtual mainstay of contemporary theorizing. Recourse to narrative, however, begs the question of what _kind_ of narrative, what mode, and what genre of narration? Without answers to these questions, it is not possible to assess what sort of work is being done (or should be done) by narrative as such.
3. Sanford Levinson has argued in favor of analogizing marriage and social contract in _Constitutional Faith_. The analogy is also drawn in the Hebrew Bible (Levinson draws on this example), where God is said to have taken the Israelites as his bride on Mount Sinai, the scene of the social contract.
4. Some democratic theorists look instead to the tragic mode to organize and inform their reflections on democracy. William Connolly, for example, uses a tragic perspective on political thought precisely to dethrone the dominant, romantic approaches (though he would not put it in these literary terms). See his tragedy-sensitive readings of Nietzsche, Rousseau, and Hegel in _Political Theory_ _and Modernity_. See also Steven Johnston's treatment of Rousseau as a tragic thinker in _Encountering Tragedy_. In a different effort to break the spell of politi-cal theory's romantic assumptions, John Seery turns to irony in _Political Re-turns_. Irony is recommended by Richard Rorty as appropriate for private individuals in _Irony, Contingency, Solidarity_. For citizens, however, Rorty recommends romance (in _Achieving Our Country_ and _Philosophy and Social_ _Hope_ ), as we shall see below.
5. "Modern gothic" is Joanna Russ's name for the genre, "female gothic" is the term used by Tania Modleski. Russ explains: these modern gothics "bear no resemblance to the literary definition of 'Gothic.' They are not related to the works of Monk Lewis or Mrs. Radcliffe, whose real descendants are known today as Horror Stories. The modern Gothics resemble, instead, a crossbreed of _Jane Eyre_ and Daphne Du Maurier's _Rebecca_ and most of them advertise themselves as 'in the Du Maurier tradition, ' 'in the Gothic tradition of _Rebecca_ , ' and so on" (Russ, "Somebody Is Trying to Kill Me, " p. 666). Tania Modleski ( _Loving with a Vengeance_ ) is less extreme on this score. She sees female gothics as in the tradition of Lewis and Radcliffe.
It should be noted that when I say that gothic romance is the genre that fits Rousseau's _Social Contract_ "best, " I mean both to remark on the fit and on its imperfection, as in—best, but still not perfectly. In Russ's account of the genre, for example, the heroine is always passive. Not so, however, if we take Du Maurier's _Rebecca_ as exemplary (which Russ does not, in spite of her awareness of its importance in advertising Modern Gothic novels). In place of passivity, Rebecca has a _bildungsroman_ quality—the heroine overcomes obstacles, faces her fears, and enters maturity—which Modleski, contra Russ, identifies as central to female Harlequins and gothics (Modleski, _Loving with a Vengeance_ , pp. 20, 52). In democratic theory's received texts, the figure who occupies the place of the gothic heroine is often "the people, " whose maturity is their reward for grappling with the alienness of the law, though sometimes it is (also) the author who feels betrayed by "the people, " who are easily corrupted. Either way, the figure is not passive.
6. That same scene is, of course, available to be read as a simple love story (Ozick) or as comedy (as in Sasson's reading).
7. Modleski, _Loving with a Vengeance_ , p. 61, quoting from Russ, "Somebody Is Trying to Kill Me, " p. 667, herself quoting an ex-editor of Ace Books.
8. Modleski, _Loving with a Vengeance_ , pp. 59–61. This distinction does not apply to all female gothics, of course. In _Jane Eyre_ , Charlotte Bront é's fe-male gothic, which antedates the gothics Modleski is looking at, the heroine goes through the transformation that Modleski identifies as Harlequin: from fear to love.
9. Cameron, quoted in Modleski, _Loving with a Vengeance_ , p. 62. The same sorts of explanations about overisolation are often given nowadays to account for interest in the paranormal and belief in extraterrestrials, those other aliens.
Modleski's deployment of Cameron also fits well with the observation made by an ex-editor of Gothic romance novels: "The basic appeal is to women who marry guys and then begin to discover their husbands are strangers . . . so there's a simultaneous attraction/repulsion, love/fear going on." The editor is quoted by Russ ("Somebody Is Trying to Kill Me, " p. 667) and then quoted again by Modleski ( _Loving with a Vengeance_ , p. 39). A dynamic of attraction and repulsion is also how Kant describes the practice of respect for persons. See my discussion of Kant in _Political Theory and the Displacement of Politics_ , Chapter Two.
10. Hannah Arendt's analysis in _Rahel Varnhagen_ of a Jewish parvenu's maddeningly impossible efforts to achieve social status and belonging are particularly instructive here. For discussions of this theme in Arendt, see Morris Kaplan, "Refiguring the Jewish Question, " and Pitkin, _Attack of the Blob_. On the social as a cause of paranoia, see also Eric Santner, _My Own Private Germany:_ _Daniel Paul Schreber's Secret History of Modernity_.
11. Modleski also toys with sociopolitical explanations, but, unlike me, she tends to assume that the thing being explained or worked through is always in some way a women's issue. For example, what she calls the "gaslight" genre (developed in the 1940s) "may be seen to reflect women's fears about losing their unprecedented freedoms, " achieved in the absence of so many men during the war years ( _Loving with a Vengeance_ , p. 21).
12. Modleski, _Loving with a Vengeance_ , p. 75. In the end, the turn to psycho-analysis is necessary, Modleski insists, because, contra Joanna Russ, "romantic disillusionment and feelings of social isolation in the newly married woman [are] not sufficient to explain the particular kinds of fantasies encountered in female Gothics" (p. 65).
13. These speculations require that we suspend the common assumption that female gothics are, as such, only by, about, and for women. They also require us to ignore Russ's dating of the genre. She locates its origins in the 1950s, but she is talking not just about a conventional genre but about a particular paperback publishing phenomenon, represented in her text by Ace Books. The genre precedes the phenomenon, as Russ herself makes clear, when she alludes to female gothics as "in the tradition of" _Jane Eyre_ and _Rebecca_ ("Some-body Is Trying to Kill Me, " p. 666).
14. Thus, for example, even if Robert Burstein were right that American democracy developed along with a popular self-stylization of Americans as a uniquely sentimental people, the fact that America is (if it is), in Burstein's title, a "sentimental democracy" would not mean that citizens and residents of the United States cannot or ought not relate to that regime's institutions and practices gothically rather than sentimentally or romantically _(Sentimental Democracy:_ _The Evolution of America's Romantic Self-Image_ ).
15. Connections between the gothic form and democratic political culture have been noted by others, but they tend to focus on horror gothics, not on female gothics. Horror gothics were the first or the most popular— even mass—genre in the nineteenth century. They are in that sense said to be "democratic" and are for that reason often looked down upon by highbrow literary critics: "Associated with the sensational, the formulaic, and the popular, the gothic is seen to lack seriousness of purpose and connection to actual experience" (Goddu, _Gothic America_ , p. 187, n. 15). Cf. Botting, _Gothic_ , and Martin and Savoy, eds., _American Gothic_ ). But see also Punter on how the supposed massness of gothic's circulation is much overestimated (Punter, _The Literature of_ _Terror_ , p. 22).
Toni Morrison posits a different connection between horror Gothics and _American_ democracy in particular. This genre of haunting is America's national literary genre, she says, because America has always been haunted by the unjust and unacknowledged racist origins it has struggled, unsuccessfully, to bury (Morrison, _Playing in the Dark_ ). Teresa Goddu takes up this suggestion and explores it in detail in _Gothic America_. Like me, Goddu rereads as gothic (though, again, not as _female_ gothic) certain texts that are usually read as romantic or, in her case, sentimental. However, in her reading of Cre`vecoeur's _Letters of an American Farmer_ , for example, the gothic moments identified by Goddu are all connected to heretofore unnoted or suppressed passages in the text that have to do with figures who are black. Indeed, throughout, Goddu's gothic reading of America is identified with the scenes of America's racial horrors (mostly slavery but also, in Chapter Three, Indian removal). The uses of the horror gothic form to accent the seamy underside of American democracy are welcome insofar as they help to expose injustices that many would rather forget than confront. But such uses manage to leave democracy as such un-touched by the very monsters and ghosts that gothics seek to awaken. If the gothic is safely connected to the horrors of slavery, it leaves room for the main narrations of American democracy to continue, relatively undisturbed, in the genre of simple romance. The echo to Rogers Smith's _Civic Ideals_ (which I discussed in Chapter One) is unmistakable: indeed, we might say that Goddu implicitly and unwittingly replays Smith's "multiple traditions thesis" as a "multiple genres thesis."
Sure enough, Goddu makes a point of connecting American gothics and marginality: not only are _American_ gothics often cast as mere copies in relation to the real British genre (a move for which Goddu criticizes Eugenia DeLamotte [p. 162, n.3]). American _gothics_ are also marginalized in relation to "real" or serious literature. For Goddu, this double marginality of America's gothics may signal the genre's fittedness for its role, which is to give expression to the horrors at the margins of American democracy.
Contra Goddu and even Morrison, however, the real issue, in my view, is the need to locate the sources of democracy's hauntings in democracy itself, and not just in its attendant, possibly contingent injustices and repressions. That is, the point is to ask about the operations of the gothic in daily life rather than simply reinscribe sentimental romance as citizenship's dominant daily genre while acknowledging that this romance is occasionally interrupted by perversions that are gothic in character. It should be noted, however, that one could maintain the identification of gothics with slavery, but still achieve this demarginalization by arguing for the centrality of slavery to democracy (Morrison and Goddu don't make this case. Orlando Patterson argues for the centrality of slavery to democracy in _Freedom_ ).
Ronald Paulson hazards a different connection between gothics and democracy, but the effect is also a remarginalization of the genre and the insights it might harbor. Paulson traces the historical evolution of horror gothics and argues that they arise in response to the horrors of France's democratic revolution (Paulson, _Representations of Revolution_ ). Thus, the connection established is not between gothics and democracy in its quotidian character— Paulson is not looking at the role of the revolution in democracy's daily life—but rather, as in Burke, at the relation between exceptional revolutionary horror and the literary horror (Burke's "sublime") in which gothics trade. By contrast, I am asking after possible conceptual connections between quotidian gothics—so-called female gothics—and democracy (and democratic theory) in its everydayness.
From my perspective, Julia Stern does better than critics such as Goddu and Paulson in her gothic reading of Wilson's _Our Nig_. Arguing that those who have until now read the novel as "exclusively sentimental" ("Excavating Genre in _Our Nig_ , " p. 447) miss its point, Stern shows how Wilson uses horror gothic devices to show the inadequacy of the private sphere (Stowe's sentimental kitchen) as a resource for empowerment. According to Stern, Wilson shows, contra Stowe, how the kitchen can be not just hearth but also hell, and, in so doing, Stern argues, Wilson effectively insists on the importance of a public sphere for political action. Stern's critique seems to be right on the mark, but it may desentimentalize the iconic scene of kitchen domesticity while leaving uninterrogated (or does Wilson leave it uninterrogated?) a certain sentimentality that clings to the public sphere.
Finally, Joan Copjec also hazards a connection between gothics and democracy in a promissory footnote on which, to my knowledge, she has not yet delivered: regarding her essay, "The _Unvermögender_ Other, " she says: "This paper is an introduction to a much longer study of the contributions of detective and Gothic fiction to" modern democracy, which Copjec understands in Lefortian terms, "not simply as a form of government but more radically as a 'mutation of the symbolic order'" (Copjec, "The _Unverm_ _ö_ _gender_ Other, " p. 41, n. 6).
16. Spivak, _A Critique of Postcolonial Reason: Towarda History of the Vanishing_ _Present_. I should note that the two female gothics on which I amdrawinghere— _Rebecca_ and _Jane Eyre_ —do feature undecidable male (anti)heroes, but these are not figured as foreign, per se. I trust the points about the genre's contributions to democratic theory (points that are not, after all, specific to the politics of foreignness) can be made, nonetheless, by way of these two female gothic novels.
17. Rorty, _Achieving Our Country_ , pp. 94–95.
18. Ibid., p. 101 and passim.
19. Ibid., pp. 10–11; emphasis added.
20. The charge is completely unsubtle in Rorty's op. ed. in the _New York_ _Times_ headlined, "The Unpatriotic Academy." The piece is republished in _Philosophy_ _and Social Hope_.
21. Rorty, _Philosophy and Social Hope_ , p. 99. There is hope for the cultural Left, however, if they kick their theory "habit" and make an effort (as if some have not done so already) to make coalition with labor in America. Rorty's depiction of the cultural Left as both a power beyond its own control and as capable of self-help in the face of a bad habit reiterates what Mark Edmundson describes as two sides of the same coin: contemporary victims of gothic terror turn out to be terrorized by others who are themselves victims, whether of prior abuse or addiction (Edmundson, _Nightmare on Main Street_ , p. 57).
22. Rorty, _Philosophy and Social Hope_ , pp. 95, 97.
23. Edmundson would be unsurprised by this mimesis: with reference to contemporary social critics, he says: "In each case, an analytic method that might have as its object a critique of gothic culture, with all of its facile pessimism, un-self-consciously reproduces gothic assumptions" (Edmundson, _Nightmare on Main Street_ , pp. 42–43). Does Rorty's fall into gothicism in the midst of trying to critique it prove Edmundson's general thesis that ours is a thoroughly gothicized culture? It might. But it might also simply show how the insistent will to romance always awakens a gothic response.
24. Johnson, "Coalition Politics: Turning the Century, " pp. 356–357. For a detailed reading of this essay, see my "Difference, Dilemmas, and the Politics of Home."
25. "A deep ambivalence about authority lies near the heart of our culture of the Gothic, " notes Edmundson ( _Nightmare on Main Street_ , p. 21). That am-bivalence was an admirable feature of early gothic writers, like Monk Lewis, who "were, in the main, progressives, " says Edmundson (p. 63). But contemporary gothics lack the "enriching ambivalence of Monk Lewis' mode" (p. 21). Edmundson may or may not be right about the decline of the gothic genre from a complex to an overly simple form. Even if he is right about the decline, however, that would not make moot the question posed here, which is whether _some_ sort of gothic lens (female, not horror) would be useful to would-be democratic citizens.
26. Modleski is less optimistic than I am regarding the capacities of gothic readers to move beyond the confines of gothic narrative: female gothics playwith paranoid feelings, and they may "not employ, as elaborately as 'high' art the psychological and formal devices for distancing and transforming the anxieties and wishes of the readers" (Modleski, _Loving with a Vengeance_ , p. 31).Mark Edmundson faults horror gothics for the same reason; hence, his harsh judgment of the genre and his closing wish for new creative artists who might "take Gothic pessimism as a starting point and come up with visions that, while affirmative, never forget the authentic darkness that Gothic art discloses" ( _Nightmare_ _on Main Street_ , p. 179). It is notable, however, that Edmundson, unlike Rorty, does not jettison the genre altogether in favor of its opposite: romance. Instead, Edmundson hopes for better future practitioners of the genre.
27. Rorty, _Achieving Our Country_ , p. 96.
28. Rogers Smith, _Civic Ideals_ , p. 499.
29. Rorty takes advantage of the ambiguity of the term "Romance, " sometimes using it in Frye's sense, suggesting a mode in which Romantic heroic individuals transcend death, and at other times using it in the sense of female or Harlequin romance, suggesting a happy-ending love story, in particular with one's country. The former is dominant in "Religious Faith, Intellectual Responsibility and Romance" in _Philosophy and Social Hope_ , the latter in _Achieving Our_ _Country_. In _Achieving Our Country_ , Rorty sometimes refers to Emerson in ways that seem to connote Romantic individualism, but Rorty quickly puts the term to use on behalf of national romance or patriotism ( _Achieving Our Country_ , p.97 and passim).
30. Rousseau knew this. He saw the germ of this idea in Machiavelli, whose _The Prince_ Rousseau read not as a manual to guide princes but rather as a warning to republican peoples: "He professed to teach kings; but it was the people he really taught." On Rousseau's account, Machiavelli made it clear that a people that allowed itself to love their king had best be on guard, because although "the power which comes from a people's love is no doubt the greatest . . . it is precarious and conditional and princes will never rest content with it."Beloved kings will turn into tyrants at a moment's notice (the romance will turn gothic) because kings prefer the certainty of subjection to the vicissitudes of love. And the people who love them had best know this ( _Social Contract_ , Book III, Chapter 6).
31. The ambivalence recommended seems to share something with Rogers Smith's closing caution in _Civic Ideals_. In the last two pages of that long book, Smith briefly chastens the patriotism he had until then been stoking, perhaps so that it will not be mistaken for blind (illiberal) nationalism: the challenge for citizens now is that "their patriotism must be at once profound and qualified, recognized as something both necessary and dangerous and thus as an allegiance that is deepest when it harbours searching doubts. . . . Americans should in fact accept that a time may come when the United States itself, like preceding human political creations, is less rather than more useful as a way of constituting a political community that can engage people's loyalties and serve their finestaspirations. But they should give support and guidance to their country so long as it seems the best hope available to them for leading free and meaningful lives, and for allowing others to do so as well" (pp. 505–506). It is noteworthy that the object of potential ambivalence here in Smith is the nation alone and not the liberal tradition or principles that Smith is out to cleanse and defend and that we, in Chapter Four, saw as themselves part of the impetus for the paired xenophobia and xenophilia in American political culture. But, in any case, how much weight should be given to such a sentiment, arrived at so late in the day, when it is preceded by numerous references (in opposition to abstract "liberal democratic precepts" [p. 10]) to the moral imperative of leaders to shape the governed into a "people, " (pp. 6, 9, 500, 502, and 474: "Because the imperative to constitute a people that feels itself to be a people is politically necessary, it is also a weighty though certainly not an absolute _moral_ imperative" [emphasis added])? That the need for nationhood comes along with potential dangers is noted but then more quickly overridden earlier in the book, when Smith compares the nation to political parties: both are "ineradicably human creations, crafted to govern and assist some people more than others. . . . In light of the good they do, we may rightly value them highly and feel great loyalty toward them; but in light of their dangerous tendencies, we should understand them to be imperfect human instruments and not take them as the proper objects of our full trust or ultimate allegiance. _Despite these essential qualifications_ , liberal democracies that conceive of political communities in the ways I pro-pose can, I believe, legitimately capture some of the engaging features of ascriptive Americanism and other myths glorifying allegedly transcendent national identities" (p. 11; emphasis added). The need to be a people is so strongly felt and assumed in this text that it is difficult to imagine under what (humanist?) circumstances someone like Smith would suddenly give it up, as opposed, say, to being willing to criticize the nation, at any particular moment in time, in the name of liberal principles.
32. Hence my sympathy with George Shulman's claim in "Race and the Romance of American Nationalism in Martin Luther King, Norman Mailer, and James Baldwin." Regarding Mailer and Baldwin, Shulman says: "Their examples suggest the value in practices of citizenship that defeat idealization but not aspiration." What is the difference between idealization (as in Rorty's idealized nation) and aspiration? "[I]dealization flees actuality while aspiration finds in it gifts to exploit." Thus, idealization produces rage and horror when it is disappointed, which it inevitably is because actuality tends to reassert itself from time to time. Aspiration's hopes are less vulnerable to being dashed.
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How To dismantle Your Legacy As A Physician
Brett Favre was one of the greatest quarterbacks in NFL history. As physicians, we can learn a lot about the legacy we leave and about how we will be remembered from Brett Favre. After a rookie year with the Atlanta Falcons, Favre joined the Green Bay Packers where he spent the next 16 seasons amassing one football record after another. In Wisconsin, he was a hero: parents named their new-born sons Brett, Green Bay jerseys with #4 quickly sold out, and he was awarded the NFL's most valuable player three years in a row. On March 4, 2008, he announced his retirement but then a few months later, he changed his mind about retirement and asked to be traded so he went to the New York Jets for a year and then announced his retirement (again). A few months later, he signed with the Minnesota Vikings, the arch rivals of the Green Bay Packers. And Brett Favre went from being the most beloved man in Wisconsin to being the most hated man in Wisconsin.
So, what does this have to do with physicians? We do not have the fame of a Brett Favre, but we do build up a reputation in our hospitals, our communities, and our medical schools. If you look around at medical centers and colleges of medicine, buildings are named after those locally famous doctors who stayed at their institution for years or decades and then retired from that institution. They don't name buildings after doctors that practice at a hospital for 25 year and then leave to go practice somewhere else.
It is because our brains are wired to dislike someone more if we initially thought that they liked us but then later disliked us. As an example, think about the last ugly divorce that a neighbor, co-worker, or family member went through and how the former spouses now see each other. It also works the other way: we like someone more who we initially thought disliked us but then later liked us. As an example, think about every military sergeant and every high school coach that ever existed.
When a senior colleague, a mentor, a department chairman, or a division director leaves for a similar job elsewhere, they become a persona non grata. We perceive that the physician is leaving because he or she no longer likes us. Consequently, we no longer like him or her. Institutional history is always written by those who remain and not by those that leave and so those physicians who leave are remembered by institutional history not for all of the good that they did while they were here, but rather remembered just for leaving.
There are exceptions. For example, for a bona fide promotion, such as a division director who leaves to become a department chairman elsewhere else. Or for family reasons, such as a physician who moves to a different city because his/her spouse's job got transferred. Or for internal transfers, such as a physician in a large multi-hospital medical system who is asked by the corporate leadership to transfer to fill a clinical void at one of the other hospitals.
But it is the physicians who depart for seemingly lateral moves who we perceive as rejecting us and thus we in turn reject. And the longer a physician has been at one hospital before leaving for another, the more strongly we reject him/her. We tend to erase and forget all of their accomplishments. We find other physicians to elevate to the level of celebrity to replace those who left. The students, residents, fellows, and junior physicians who came to the institution because of them feel as if they were lied to. The physician's patients feel betrayed. To all, the departing physician becomes a pariah.
You can measure the qualifications of a physician by how he/she is recruited for a job. You can measure the integrity of a physician by how he/she leaves a job. It is better to leave an institution after only a few years than to leave after a few decades when you have become the face of that institution.
I want to be remembered not like Brett Favre but like Cal Ripken. He was born in Maryland and played every one of his 21 seasons with the Baltimore Orioles. His player number was retired by the Orioles in 2001 and the in the state where he was beloved as a player, he is still beloved.
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{"url":"http:\/\/marc-b-reynolds.github.io\/math\/2019\/08\/10\/Avalanche.html","text":"In a recent blog post \u201cA fast 16-bit random number generator?\u201d Daniel Lemire gives a proof-of-concept PRNG with 16-bits of state. The empirical search strategy used was estimating the \u201cAvalanche Effect\u201d of the cadidate parameter and choosing the best. In the comments someone questioned why using a hash measure is reasonable for PRNG. One purpose of the post is to answer that question. Additionally I\u2019ll eventually get around to using some of this stuff.\n\nLet\u2019s forget all about hashing and say we have some bit mixing function $f$ which transforms a register sized integer value to a same sized integer value (not a requirement but KISS) and let\u2019s just say 32-bit integers (for bit mixing bigger really is better\u2026but that\u2019s another story). Avalanche is vaguely defined as \u201csmall changes in input produce large changes in output\u201d. Since that\u2019s totally useless we define a couple of desirable properties (stated slightly differently from the common):\n\nstrict avalanche criterion (SAC): If we apply $f$ to an input $x$ and to $x$ with the $i^{\\text{th}}$ bit flipped then output bits flip with a probability of 1\/2.\n\nSo the code for computing the changes for the $i^{\\text{th}}$ bit might look like this: bits = f(x) ^ f(x^(1<<i)).\n\nbit independence criterion (BIC): If we apply $f$ to an input $x$ and to $x$ with the $i^{\\text{th}}$ bit flipped then all other pairs of bit positions should flip independently in the output.\n\nBIC is a stricter and harder measure to pass but after gathering counts (wait for it) proceeds like a SAC based test, so I\u2019m going to ignore that. Also there are variants of SAC tests (such as multiple input bits flipping or nearby inputs) which I\u2019m also going to (mostly) ignore. Changing a single bit has the most value since we\u2019re looking how well our mixing function propagates a single bit change in input to all output bits.\n\nASIDE: I\u2019m going to state that it\u2019s highly desirable for our function $f$ to be a bijection or \u201cevery input maps to a unique output\u201d. (I\u2019m gonna keep ignoring different sized input\/ouput). If $f$ not a bijection then for a given output there are two or more inputs which produce it this introduces a bias (both for SAC measures and pigeonhole principle flavored as well.\n\nNow Lemire\u2019s 16-bit generator does not use a bijection for the mix and the downside of this is the generator is biased. The upside is it performs amazingly well on a number of PRNG statistics (Yo, this is really hard with 16-bits of state). So the point of this aside is to remember Goodhart\u2019s law and that desirable properties can be blown off (have weak measures) if we meet our overall goal(s).\n\nBack on topic. SAC: output bits flip with probablity of 1\/2. For PRNG we expect the output bits to be set with a probability of 1\/2. So feeding a sequence of numbers into $f$ should produce a random number (and yeah there are a bunch of other tests we want to run to see how well it\u2019s working). In fact many state-of-the-art random number generators do exactly this. PCGs generate a LCG and run through a mix and there are quite of few that use Weyl sequences (additive recurrence) followed by mix. A quick reasoning why this is popular is a PRNG state update needs to be a bijection that is full period (this is a hard requirement) and the period of the mixing function doesn\u2019t matter at all. So we can use a crappy (in terms of randomness measures) state update and mix it with a very good mixing function.\n\nThe remainder of this is spitballing how to make tests out of the desired property. Let jump in a look at a stripped down version that Lemire used:\n\nSo he just walk all legal input, flips the bits, finds the changed bits, then:\n\n\u2022 computes the population count (aka Hamming weight)\n\u2022 subtracts 8. Why? Each bit should flip with probability of 1\/2 so (on average) half the bits should flip at each step\n\u2022 takes the abs of the previous result\n\u2022 add this value to the running sum\n\nNice, short and simple estimation of SAC. Small values are good and big are bad.\n\nLet\u2019s run with a skeleton for 32-bit to 32-bit which is not so short:\n\nThe outer structure is pretty much the same. Bullet-point time:\n\n1. We\u2019re not brute forcing all input and are instead sampling $n$ values. There are various sampling strategies that can be used, but we\u2019re going to ignore that in this post.\n2. The other change is that last loop. We\u2019re instead walking individually through the bits in our \u201cwhich ones flipped\u201d variable bits and updating a \u201cmatrix\u201d of counts in we which track how output bits flip for each changed input bit position. Additionally this skeleton computes the population count and tracks how many of each we see.\n\nOnce this process is done we need to compute a measure of how well we\u2019re meeting the expected property. Let\u2019s say we walk through each element of counts and divide by the number of samples $\\left(n\\right)$ we used. We\u2019d have a number on $\\left[0,1\\right]$. Zero if the given input bit never flipped the given output bit and one if it always flipped. Since our expectation result is $\\frac{1}{2}$ let\u2019s subtract that and multiply by two to remap the range to $\\left[-1,1\\right]$ and call that value the bias. If we were to walk through all the bins of counts and track the maximum magnitude bias we could return that as our measure and we\u2019ll call that the max bias (this is the $\\ell^{\\infty}$ or uniform\/supremum\/infinity\/Chebyshev norm of the bias). This gives us the magnitude of worst case bias (just to state the obvious).\n\nSomething completely different we could do with our values in counts is to perform a goodness-of-fit test. The SAC property says that all the values should be uniformly distributed if the function is a good bit mixer.\n\nAnother thing we could do with the bias is directly visualize its output as a heat map. As a baseline let\u2019s look at how an identity function would look:\n\nThe legend is off to the right with positive values in red, negative in blue and approaching zero is black. The identity function doesn\u2019t mix anything so the only bits that flip are those from performing the test. Reading the figure we have from left-to-right least to most significant output bit and bottom-to-top least to most significant bit flipped on the input sample.\n\nNow let\u2019s look at the MurmurHash3 32-bit finalizer at each step. There are really very quite many hash functions which have an identical structure simply with different choices of constants.\n\nThe figures are in the same order as the code: top-left is first xorshift, top-right the multiply, second row the next xorshift\/multiply and last row the final xorshift. The samples are high-entropy (a PRNG source) and the number of samples ($n=1048575$) a middle ground choice (enough that the bias is clear but no so many that the final figure becomes all black).\n\nAn older hashing function (Wang hash) is similar to new-jack MurmurHash3 sytle finalizers and is sometime still recommended. Let\u2019s look at that:\n\nLet\u2019s very quickly run through was we could do with the population count data. SAC says bits should flip with probability of 1\/2. If we think of each bit as a coin tossed in the air (tails=0 and heads=1) then the probability of getting a specific population count is described by the binomial distribution:\n\n$\\left(\\begin{array}{l}{n} \\\\ {k}\\end{array}\\right) p^{k}(1-p)^{n-k}$\n\nwhere $p$ is the probability (which is 1\/2), $n$ is the number of trials (the number of bits) and $k$ the number of successes (the population count). Renaming $n$ (we\u2019re already using that) to $b$ and plugging in our fixed $p$ gives:\n\n$2^{-b} \\left(\\begin{array}{l}{b} \\\\ {k}\\end{array}\\right)$\n\nAt this point we have enough information to perform a goodness-of-fit test.\n\n## Comments\n\n\u00a9 2022 Marc B. Reynolds.\nall original content is public domain under UNLICENSE","date":"2022-07-02 23:27:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6790320873260498, \"perplexity\": 1017.9044963457918}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104205534.63\/warc\/CC-MAIN-20220702222819-20220703012819-00199.warc.gz\"}"}
| null | null |
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<time class="listdate" datetime="2017-09-13T12:17:09Z" title="2017-09-13 12:17">2017-09-13 12:17</time><a href="../../posts/sending-json-array-using-request-library/" class="listtitle">Sending BULK DATA using REQUEST LIBRARY</a><a></a>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,261
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One of a kind semi-sheer hand printed tunic dress crafted from vintage fabrics. The Daisy Dress is a part of an ongoing collaborative series between Maryam Nassir Zadeh, Sophie Andesgascon and Claire McKinney. Each piece is one of a kind and may have slight variations in print and fabric.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,720
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Fruitless Pursuits: Pre-order Hot Toys Iron Man Mark XLII DIECAST 1/6th Scale Figure!
Pre-order Hot Toys Iron Man Mark XLII DIECAST 1/6th Scale Figure!
But fear not! This figure actually has interchangeable parts, so if you're not so keen on displaying an Iron Man that has just had the crap kicked out of it by the terrorist antics of Fu Manchu-sporting villain, then you can also display him clean and factory new.
Hot Toys Die Cast Marvel Series Iron Man Mark XLII Limited Edition Sixth Scale Figure - MMS Diecast Series (Hot Toys) Or to see more pictures, join me... after the jump!
I think I prefer this one myself, although there's been nothing shown to indicate that this faceplate opens up to reveal a non-battle damaged Robert Downey Jr. I'm pretty sure that if you want to open it up then you're stuck with a bloody Tony who has just been boxed. I guess you could pretend he just cut himself shaving.
I don't know though... all these exposed wires, mixed with gold, is giving me a bit of a C3PO vibe. Just for the record, if Tony Stark and C3PO ever decide to get married then I fully support it.
There's a few hidden bells and whistles too. Like these hinged panels that open to reveal emergency cheese graters.
Probably not the first flaps that Tony Stark has fully deployed.
And if you look really closely here you will see all the additional panels that can be switched out. As I said - there doesn't appear to be an extra face sculpt.
I think that personally I'll probably just stick to my upcoming Avengers Mark VII, but this is nice if you're a big Iron Man fan. And don't forget that if the metal versions are too costly, you can also grab this same armor far cheaper as part of the new Power Pose line. So many options! Let's hope the movie's good.
Ha! "Probably not the first flaps that Tony Stark has fully deployed" nice one dude.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,344
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Tegelsmora församling var en församling i Örbyhus kontrakt i Uppsala stift i Svenska kyrkan. Församlingen låg i Tierps kommun i Uppsala län och ingick i Örbyhus pastorat. Församlingen uppgick 2014 i Vendel-Tegelsmora församling.
Administrativ historik
Församlingen har medeltida ursprung och utgjorde till 1991 ett eget pastorat, för att därefter till 2014 ingå i ett pastorat gemensamt med Vendels församling, från 2001 benämnt Örbyhus pastorat.
Församlingen uppgick 2014 i Vendel-Tegelsmora församling.
Kyrkor
Tegelsmora kyrka
Se även
Tegelsmora socken
Källor
Historiska församlingar i Uppsala stift
Indelningar i Tierps kommun
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,700
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{"url":"http:\/\/stackoverflow.com\/questions\/14380861\/writing-validation-function-for-listbox","text":"# writing validation function for listbox\n\nI have a asp control listbox. And i have to validate this. This is described below:\n\n<div style=\"float:top; width:300px\">\n<span>Anrede<\/span>\n<asp:DropDownList id=\"dropdownListAnrede\" runat=\"server\" BorderStyle=\"Solid\"\nTabIndex=\"1\" Width=\"250px\" BackColor=\"White\" BorderColor=\"Silver\" BorderWidth=\"1px\" Height=\"22px\">\n<asp:ListItem >Bitte ausw\u00e4hlen<\/asp:ListItem>\n<asp:ListItem Value=\"Herr\">Herr<\/asp:ListItem>\n<asp:ListItem Value=\"Frau\">Frau<\/asp:ListItem>\n<\/asp:DropDownList>\n<asp:CustomValidator ID=\"CustomValidatorAnrede\"\n**ClientValidationFunction=\"\"** runat=\"server\"\nControlToValidate=\"dropdownListAnrede\" ValidateEmptyText=\"true\" SetFocusOnError=\"true\"\nForeColor=\"Red\" onservervalidate=\"CustomValidatorName_ServerValidate\"> W\u00e4hlen Sie bitte eine Anrede aus!<\/asp:CustomValidator>\n<\/div>\n\n\nI have to validate as if it dont have a value(Herr\/frau) the sumission will not take place. and the error message will show the message is written in the text. I have to write a ClientValidationFunction in javascript. but I wonder how?\n\n-\nDid you read the documentation? That said, isn't a simple RequiredFieldValidator working in your case (as the first has no value)? \u2013\u00a0 Steve B Jan 17 '13 at 14:11\n\nAdd a dummy value as the first item that's essentially no selection:\n\n<asp:DropDownList id=\"dropdownListAnrede\" runat=\"server\" ...>\n<asp:ListItem >-Select One-<\/asp:ListItem>\n\n\nGet the drop down list, and check if it has a selected value greater than zero (omit the first). This would be the clientvalidationfunction:\n\nfunction val(sender, e) {\nvar ddl = document.getElementById(\"<%= dropdownListAnrede.ClientID %>\");\ne.IsValid = ddl.selectedIndex > 0;\n}\n\n-\nas i know the ID (\"dropdownListAnrede\") can i directly place it in the JS function? How do I call the function? Do i have to sende peremeters(sender,e)? \u2013\u00a0 Abdur Rahim Jan 17 '13 at 14:17\nCopy my val function directly into javascript, and take the name of the function \"val\" and add a property to the Custom Validator: ClientValidationFunction=\"val\". \u2013\u00a0 Brian Mains Jan 17 '13 at 14:36\nIf i select an item which has index greater than 0, will the error message disappear? \u2013\u00a0 Abdur Rahim Jan 17 '13 at 14:40\nThanks. Your code is working fine. except the first time. when i expand the dropdown but dont change my selection, there are no error message showing. I want to execute this action right from the first dropdownlist close event. what should i change? \u2013\u00a0 Abdur Rahim Jan 17 '13 at 14:48\nIt will still prevent postback of the page; when you go to save the form, the validation will appear, even though the * doesn't appear next to it. You may have to manually invoke the validator using JavaScript if you want that to happen, as MS handles all of that for you and doesn't allow you to easily change the default functionality. To invoke the validator manually, you can use the client-side ValidatorValidate or Page_ClientValidate. To do that, check out this: msdn.microsoft.com\/en-us\/library\/aa479045.aspx \u2013\u00a0 Brian Mains Jan 17 '13 at 15:55","date":"2015-01-27 14:54:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.36219292879104614, \"perplexity\": 4218.919863598126}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-06\/segments\/1422121981339.16\/warc\/CC-MAIN-20150124175301-00045-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
| null | null |
PKG_NAME = util-linux-schedutils
PKG_VERS = 2.26
PKG_EXT = tar.xz
PKG_DIST_NAME = util-linux-$(PKG_VERS).$(PKG_EXT)
PKG_DIST_SITE = https://www.kernel.org/pub/linux/utils/util-linux/v$(PKG_VERS)
PKG_DIR = util-linux-$(PKG_VERS)
DEPENDS =
HOMEPAGE = https://github.com/karelzak/util-linux
COMMENT = Random collection of Linux utilities
LICENSE = GPL
GNU_CONFIGURE = 1
CONFIGURE_ARGS = --without-ncurses --without-python
CONFIGURE_ARGS += --disable-all-programs --enable-schedutils
include ../../mk/spksrc.cross-cc.mk
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,704
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\section*{ACKNOWLEDGMENT}
\bibliographystyle{IEEEtran}
\section{Introduction}\label{sec:introduction}
Stress is a term that describes bodily reactions to perceived physical or psychological threats \cite{jongyoon2012development}. Since the start of stress level recording among the population, these values have been on the rise, and the pandemic had a significant impact on them. There is a consistent increase of stress-related mental symptoms (anxiety, depression, general psychological distress) in the general population during the pandemic compared to before \cite{eu2021mentalhealth}.
While these facts are dire, stress in its inception is a good evolutionary response to dangerous situations, allowing our bodies to be better prepared to perform in the face of a ``fight-or-flight'' situation. An example of such a situation could be an encounter with a tiger. Nowadays, it is unusual to find tigers in a person's day-to-day life, and so, it is more prevalent in the case of deadlines or responsibilities, and its purpose is to help humans to be better prepared to deal with such events, using biological changes to face a recognized threat. It still can be beneficial, keeping us alert in dangerous situations and focused to meet challenges \cite{jongyoon2012development}.
On the other side, if such situations keep adding up and stress does not subside, it stops being classified as an acute stress response, and it starts entering the chronic stress realm. At this stage, our bodies are producing hormones to keep the stress response up, but the outcome starts being more negative than positive. This chronic stress can lead to the atrophy of the brain mass and decrease of its weight. These structural changes bring about differences in the response to stress, cognition and memory\cite{habib2017stressimpact}
Mental health problems exist along a continuum, from mild, time limited stress, to severe mental health conditions, and while mental illnesses and stress are not the same, they are closely related. Stress and anxiety affect most people at some point in their lives, but the regularity at which that happens is one of the key points of classifying such events as a disease. Focusing on the anxiety and anxiety disorders, they are the most common type of mental illness in the world, affecting 264 million people worldwide as of 2017, with an increase of $14.9\%$ per decade \cite{elgendi2019assessinganxiety}. While, the rise of both stress and anxiety is related, so are their symptoms.
Anxiety is one of the most pervasive and ubiquitous human emotions, in all cultures \cite{sarason1990testanxiety}. It is considered a basic negative emotion, such as sadness, anger, worry and fear. Anxiety, fear and stress all share similarities and might even overlap to some extent, but they are different states: Stress has a clear cause, which is called a stress-causing factor or a stressor, such as the tiger mentioned before. Fear also shares some similarities to stress, but it is classified as an emotion and might trigger a stress response, it is associated with danger and/or insecurity, and it is also focused on immediate present danger. Anxiety, by contrast, corresponds to a state of uncertainty, and it is more closely
related to a future-oriented mood state associated with preparation for possible, up-coming negative event.
Measuring anxiety and stress has a big overlap, due to a propensity of one to cause the other, common risk factors, as well as the bodily reactions being similar. Choosing which biological data to capture and analyze to target each situation becomes paramount for detection. Nevertheless, the reason for the association between these psychological syndromes is yet to be established \cite{ramon2020prevalencedepression}. Regarding their monitorization, context is likely to be utterly important, since it allows for a better evaluation of the data, and questionnaires can fill the gap in distinguishing both, as presented in \cite{bickman2020improvingmental}.
The symptoms of these conditions can be divided into Somatic (physical) and Psychic. For the most part, the symptoms most commonly associated with each are:
\begin{itemize}
\item Anxiety
\begin{itemize}
\item Somatic --- tremors, palpitations (increased or irregular heart rate), dizziness, nausea, shortness of breath, sweating, muscle tension, etc.;
\item Psychic --- difficulty concentrating, nervousness, Insomnia, constant worry, etc.;
\end{itemize}
\item Stress
\begin{itemize}
\item Somatic --- aches and pains, palpitations, muscle tension, digestive problems, etc.,
\item Psychic --- anxiety, irritability, depression, sadness, panic attacks, etc..
\end{itemize}
\end{itemize}
Looking at the symptoms, we can see some overlap. Given that stress can cause an anxiety response, then all the symptoms present in anxiety become targetable on stress detection. Current consumer wearables are not yet capable of distinguishing data with such precision, yet they are the most accessible way of monitoring both cases.
The production of smart devices to help individuals monitor components of their health has been on the rise during the last few years \cite{hickey2021smartdevices}. The presence of smartphones among the population is almost universal, and these two tools could be used as a way of bringing comfort and quality of life to people suffering from mental illnesses such as anxiety or chronic stress.
\subsection*{Contributions}
The Anxolotl project focuses on trying to supplement a more nuanced solution to a very nuanced problem, which is the management of mental health and follow-up of mental illnesses, namely General Anxiety Disorder (GAD) and Panic Disorder (PD). By taking advantage of consumer grade wearables, which are already very present worldwide, and using them to allow patients to better manage their mental health and well-being. The big focus is to provide a support tool, one mainly used to keep track of their mental health data, and allow them to intervene before an acute crisis settles, or chronic state in the case of stress.
\subsection*{Anxolotl - An Anxiety Companion App}
The Anxolotl --- An anxiety companion app --- presents a system that can reliably detect anxiety and stress levels, detect panic attacks (PAs) as long as a wearable is being used. Ideally, upon the detection of abnormal anxiety or stress levels, a notification would pop up, and in the case of detection of PAs, the user will be able to choose which mechanisms to use, such as automatically calling a selected person, buzzing or suggesting breathing exercises. The app is intended to run on the background and auto-start to be as frictionless as possible to use.
The main idea is to give users control over their mental health situation. This would translate into being able to check anxiety and stress levels on a smartphone, as well as being warned by a notification in the case of consistently high stress or anxiety levels. Short term solutions such as meditation, or wellness exercises could be suggested, but the main point is the detection. As long as the user leaves the app on the background, and wears the wearable, these mental health statistics can be recorded, and the user can live its life ignoring the app, until the time the app detects an abnormality.
Finally, there are some non nuclear objectives, such as the presentation of the data to a validated medical professional, and stress detection. The last one is discussed in this paper, as well as the algorithm used. It is intended to work along the anxiety detection in the way of giving users control over their situation. As stated before, anxiety and stress share symptoms, and to address this issue, questionnaires will be used such as the GAD questionnaire (GAD-7), and for stress the Perceived Stress Questionnaire (PSS). On this paper, we focus on the stress detection without context, which is tougher, given the lack of truly unique symptoms.
\subsubsection*{Environment}
The Anxolotl project starts by capturing data from the wearable. That data is sent in real time to a smartphone via Bluetooth Low Energy. The app is developed in Flutter to allow interoperability between iOS and Android, having a wider reach.
As shown in the Fig. \ref{fig:anxolotlSystem}, filtering is applied (low-pass) on the mobile app, removing any erroneous data, and some data processing is done as well. Then, the data is synced every $10$ min to the data center via HTTPS. The datacenter contains the models on an initial stage, where they are trained. On a later iteration these trained models could be loaded and applied on the smartphone to reduce latency and have a real time response. The data center is responsible of receiving the biological data and training classifier models with it, giving the mental health statistics in return. A response is then sent to the smartphone identifying stress levels and presenting them to the user. The user can then check their mental health levels on their smartphone, as well as receive notifications when the models detect unusually high levels.
\begin{figure}[t]
\centering
\framebox{\parbox{3in}{\includegraphics[width=3in]{images/Slice-Scheme1.png}}}
\caption{Anxolotl solution designed environment}
\label{fig:anxolotlSystem}
\end{figure}
\subsection*{Organization}
The remainder of this paper is structured as follows: in the section \ref{sec:relatedwork} we give an overview of the current existing research on the stress subject, as well as mention some relevant scientific projects. Section~\ref{sec:background} describes the technical context needed for an easier understanding of our solution. Section~\ref{sec:methodology} describes the methodology we use for the model design, as well as iterate on the different options. Section \ref{sec:results} presents the results and discusses some of the limitations associated with our work. Finally, Section \ref{sec:conclusion} wraps up all the work, and presents our findings as well as the next steps.
\section{Methodology}\label{sec:methodology}
The approach we use in this challenge is heavily influenced by current literature. Instead of a traditional heuristic approach based on a machine learning (ML) problem, this challenge is interpreted as a data problem. Since the provided data is not unprocessed, we opt to interpret the problem this way to try and connect the data we already have with the results we are aiming for.
While the influence of the literature is going to be relevant, another relevant feature of our work is the usage of the SKlearn framework, which brings some limitations, namely not having implementations of the most technically advanced algorithms, such as deep neural networks. With that said, we have to forfeit some of the more advanced algorithms, and focus on long established algorithms.
This section regards our analysis of the data set, features and their selection as well as an introduction to our ML algorithm choices.
\subsection*{Data}
Our solution uses the SMILE data set \cite{smileDataset}, and by extension it is designed to work well with it. A total of $45$ healthy adult Belgian participants ($39$ females and $6$ males) were recruited for SMILE. Among participants, the average age was $24.5$ years old and the standard deviation was $3.0$ years. On average, each participant contributed $8.7$ days of data.
Two types of sensors were used for the data set, one for HR, and another one for GSR and ST.
Regarding the data set itself, the data is not the original recorded data. It is anonymous and was reconstructed from a model based on the original data, and this process was made for the continuous portion of data by the data set providers. For the handcrafted portion of the data, we have $60$ min. of measurements \textit{per} stress label, which means the data has to be processed to fit into a 1-1 model --- one data point to one label.
The data also comes normalized from $0$ to $1$ and contains masking, that identifies when the captured data was unreliable, or the user was not wearing the device, and so it can be discarded. Regarding the data set organization, it is divided into deep features, in which the features are presented as close to raw as possible, while still being normalized, and handcrafted features which were calculated from not normalized features, but are presented normalized as well.
\subsubsection*{Data Filtering}
Regarding filtering the data to achieve more representative results, we opt to filter out entries presenting more than half the data as unreliable (in one minute). Not removing these points could impact our output, given that each valid point would have twice the impact on the label result. Another reason for this choice are the experimental results, given that ratio presents the best results as shown in TABLE \ref{tab:trainingDatasetComparison}.
\begin{table}[]
\centering
\caption{Accuracy and F1-score metrics in ratio of non-zero values on the training data set.
\label{tab:trainingDatasetComparison}
\begin{tabular}{|c|c|c|}
\hline
\rowcolor[HTML]{C0C0C0}
Non-zero ratio & {\cellcolor[HTML]{C0C0C0}Accuracy} & {\cellcolor[HTML]{C0C0C0}F1 Score} \\ \hline
0.3 & 0.54 & 0.58 \\
0.4 & 0.54 & 0.58 \\
\textbf{0.5} & \textbf{0.56} & \textbf{0.60} \\
0.6 & 0.53 & 0.57 \\
0.7 & 0.53 & 0.56 \\ \hline
\end{tabular}
\end{table}
\subsubsection*{Testing}
Testing is also an important part of dealing with the data set, since it is the way we validate or discard the hypothesis. For these tests we searched and found k-fold split and the k-fold stratified to be a good compromise between good output and low complexity.
We choose k-fold, since k-fold stratified removes entries to balance classes, and in the case of stress detecting, the order by which the entries are removed is important, since a timeline exists. Our solution is to balance the data set ourselves, while using k-fold. We are sticking with $3$ splits, to try and avoid overfitting while keeping the relation between train set and test set sizes near the real size relation between the train data set ($2060$ entries) and the test data set ($960$ entries).
\subsection*{Features}
The supplied data set contains features extrapolated from the original data set via ML and reassembling, as well as features based on HR, GSR and ST that were extracted from the data and presented as byproducts of the original data. The handcrafted features contain some lost granularity, but the deep features are normalized, and so it would be impossible to recalculate new features from them. With that in mind, our choice is to use only the handcrafted features.
\subsubsection*{Feature Selection}
From the provided $20$ features on the handcrafted part of the data set, we have tried and tested some, and ended up processing them to create our own features. Here we present $16$ features, some with a scientific paper support, which we cite, and some of them with an empirical evidence basis. Below we present our selected features, with citations in the case they are inspired by another paper.
\begin{itemize}
\item Heart Rate
\begin{itemize}
\item Mean HR \cite{gjoreski2017monitoringstress},
\item HR standard deviation \cite{gjoreski2017monitoringstress},
\item HR quartile deviation (percent. 75 - percent. 25) \cite{gjoreski2017monitoringstress},
\item HRV standard deviation variability \cite{can2019continuousstress},
\item HRV mean standard deviation\cite{can2019continuousstress},
\item HRV mean of root mean square of R-R differences,
\item Percentile 90 of low frequency signal,
\item Percentile 90 of low and high frequency ratio,
\item Mean of low and high frequency ratio.
\end{itemize}
\item Galvanic Skin Response
\begin{itemize}
\item Mean GSR \cite{can2019continuousstress},
\item GSR quartile deviation (percent. 75 - percent. 25) \cite{can2019continuousstress}.
\end{itemize}
\item Skin Temperature
\begin{itemize}
\item Mean ST \cite{gjoreski2017monitoringstress},
\item Mean ST Variability,
\item Max ST slope value,
\item ST Mean Slope \cite{gjoreski2017monitoringstress},
\item Percentile $90$ of ST slope.
\end{itemize}
\end{itemize}
In the case of heart activity, the unreferenced features, are added due to the fact that low and high frequencies, as well as the ratio between them are good measures of stress related activity \cite{can2019continuousstress}. The non cited data on ST, is used because increases and decreases in ST values can be indicators of stress, and the rate of increase is the biggest indicator, that is why both variability of the mean and 3 slope values are present.
All of these and more features are passed through a ridge correlation between each feature and the label, and none of them had smaller absolute correlations than $20\%$ or bigger than $200\%$, to keep balance. The data with low correlations are kept because they are often referred in literature as good indicators, and since we are still applying a feature extractor tool, not much harm can be done.
\subsubsection*{Feature Extraction}
Since the provided data set contains a minute of data per each label, features must be downsampled, but by doing that we would be losing granularity. In response, our group opted to use research used features while down sampling said data. To try and have the most important features, we tried two feature extractors, that are used with the classifier as algorithm, since we assume the same algorithm is the best feature extractor for itself.
The two tried and tested feature extractors are the recursive feature elimination with cross-validation (RFECV) \cite{iranfar2021relearn}, and the sequential feature selector, which are both provided by the SKLearn framework. They both have been positively used in the literature, and are currently regarded as trustworthy. The TABLE \ref{tab:classifierResultsTable} presents their results using Linear-SVC (C-Support Vector Classification) as the classifier, and being tested with the k-fold split. We use $5$ features minimum, as we think that is the best relation between features and the size of the data set.
\begin{figure}[thpb]
\centering
\framebox{\parbox{3in}{\includegraphics[width=3in]{images/rfecv.png}}}
\caption{Screenshot of the RFECV feature correlation results.}
\label{fig:RFECV}
\end{figure}
\begin{figure}[t]
\centering
\framebox{\parbox{3in}{\includegraphics[width=3in]{images/sequentialFeatureSelector.png}}}
\caption{Screenshot of the Sequential Feature Selector results.}
\label{fig:SFS}
\end{figure}
RFECV gives, on average, more features according to the Figs. \ref{fig:RFECV} and \ref{fig:SFS} with higher correlations between them; we believe that the data set is not big enough for so many features. On the other hand, after testing both options on the test page, the results from the RCEV are $0.59$ accuracy with $0.51$ F1-score against the results from the sequential feature selector which are around $0.62$ accuracy with $0.54$ F1-score, making the sequential feature selector a more suitable choice.
\subsection*{Machine Learning Classification}
Physiological data varies from individual to individual, and while classifying data on a per subject basis can give the ML a personal approach, our data does not have personal identifiers. With that in mind, the error rate is going to be higher, since the values that identify stress in a person are not exactly the same that identify stress on a group.
Furthermore, we approach this challenge using ML algorithms, such as K-NN with multiple neighborhoods (that is 5-NN, 7-NN and 9-NN), SVM, NB, Random Forest and Decision Trees, since those have been fairly covered in the literature, as stated before. K-NN is a method that uses k-nearest data-points and does a majority vote to predict the result, and k is used to identify the number of data-points. SVM finds hyper-planes to divide data-points into different classes \cite{Han2020objectivestress}. We use the SKlearn implementation of SVM, and mostly Linear-SVC. NB classifier predicts the result based on the probabilities of each feature's probabilistic knowledge, and Random Forest and Decision Trees work by iterating trees of questions and ending with a conclusion in the end.
The model testing evaluation is done with the training data set, as we opt to keep that variable constant. This choice was made in an effort to reduce the complexity of the system, since not having labels for the test data set proved to be a challenge, since its results do not completely correlate with our train data set results.
\section{Background}\label{sec:background}
Here we present the context we think is necessary to understand both parts of this work, both the more medical, as well as the more technological. Stress can be measured by monitoring physiological indicators such as heart activity, blood activity, skin response (GSR) or skin temperature (ST), and we address this problem with a strong theoretical background. While measuring stress on itself is tricky, we can measure indicators of such, and such events must be explained and theoretically correlated with stress itself. Regarding the machine learning (ML), we will also present the algorithms and methods we consider important. While the development is highly empirical, given a ML context, it highly relies on a basic understanding of the human body, and the relations between stress and stress related physiological data.
\subsection*{Measuring biological data}
Given that stress is a bodily response to a \textit{stimulus}, or multiple \textit{stimuli} with somatic symptoms, those same symptoms can be measured. Multiple types of symptoms allow the existence of multiple different ways of measuring, and while the most promising data seems to be heart activity and galvanic activity related \cite{gjoreski2017monitoringstress,memar2021stressclassification,Han2020objectivestress} there is also value in the monitoring of ST. All these factors play a role in the physical manifestation of stress on the human body, and these studies presented good results with accuracies of more than $90\%$ using the presented physiological data.
GSR refers to electrical changes that arise when the skin receives specific signals from the brain. These changes may be due to emotional activation, cognitive workload or physical exertion\cite{gjoreski2017monitoringstress}. While these changes can be subtle, stress can also cause sweat to happen, and as such, increase the level of GSR, which can be used for detection.
Heart activity is the most known of these biological signals, and most wearable devices can capture HR and HRV. While the HR increases upon stress, it also increases on many other ordinary phenomena, such as a scare, on the other hand HRV has a tighter relation with stress. Usually HRV is extrapolated from PPG and highly related to HR, and time-domain indices of HRV can quantify the amount of variability in measurements of the period between successive heartbeats, the Inter-Beat-Interval (IBI)\cite{gedam2021reviewmental}.
The "fight-or-flight" response restricts the blood flow from the extremities and increases the blood flow to vital organs. This peripheral vasoconstriction produces changes in ST on the extremities including hands, which can indicate stress and its intensity\cite{gjoreski2017monitoringstress}. While rises and drops in temperature are normal body functions, when correlated with other signals, ST might be a good indicator of a stress response, by using mean temperature or the slope of the temperature during a certain time frame.
It was observed that when stress occurs, HR, blood pressure, respiration rate, and GSR tend to increase while HRV and ST decrease \cite{gedam2021reviewmental}. This is not much different from a physical exercise session, and here is where context can make or break a model. But with this in mind, a good amount of features will bring better results to a model, given not a single feature can accurately detect stress.
\subsection*{Machine Learning}
In a complex problem such as stress detection, the application of some type of machine learning (ML) algorithms makes sense. The vast amount of data in a context were multiple variables, such as HR, HRV, GSR and ST, might have different outputs based on each other, makes it a prime target for the ML approach. It is no wonder it has already been applied to it, and continues to be used and researched to this day.
\subsubsection*{Feature Selection}
Machine learning algorithms are built with data that is fed to them, so it is easy to assume that the quality of the models is proportional to the amount and quality of information that is consumed. To take out any irrelevant information, it is common to apply a pre-processing step known as feature selection, in order to improve the model's performance. Following, are some of the techniques used.
Recursive feature elimination (RFE) with cross-validation is one of the algorithms used to achieve this feature selection. Recursion is the process of repeating a process multiple times. In the case of \textit{RFE}, the process consists in generating a different model, and for each iteration different features are taken away based on the generated metrics. While this process takes place, the impact of the removal of each feature is observed in the model's accuracy, to find the optimum set of features to use for the maximum results.
Sequential Feature Selector adds (forward selection) or removes (backward selection) features to form a feature subset in a greedy fashion. At each step, this estimator chooses the best feature to add or remove based on the cross-validation score of an estimator.
\subsubsection*{Classifiers}
Classification consists in predicting the class of a set of given data points; classes are sometimes called targets/labels or categories. Classification is the task of approximating a mapping function from input variables (X) to discrete output variables (Y). There are a lot of classification algorithms available, however, what dictates whether they perform accurately or not depends on the nature of the given data set and the relationships between data. Some of the most common classification algorithms are Support Vector Machines (SVM), K-Nearest Neighbours (K-NN), Random Forests (RF), Decision Trees and Naive Bayes (NB).
\subsubsection*{Classifiers in stress detection}
Many studies have applied multiple methods in feature selection and classification, but no universal algorithm has been developed for stress. With that in mind, it is a good idea to look at what came before to have a clear perspective in where to start regarding this subject.
In Gjoreski et al. \cite{gjoreski2017monitoringstress}, the best result for context stress detection, regarding F1-Score was the Decision Tree with $90\%$, followed by Random Forest with $74\%$, SVM with $69\%$ and K-NN with the same result. All these results were made in an aggregation-window with $10$ seconds. The no-context events had lower precision scores, around $7\%$ for true positives in the best model.
Memar and Mokaribolhassan \cite{memar2021stressclassification} presents a table, with a stress analysis review. Here, for data sets without context and using the data available in
our dataset (HR, GSR and ST), the best results in terms of accuracy were from a SVM with $80\%$, K-NN
with $88.6\%$ and Logistic Regression with $91.4\%$. On the other side, the measurements had a big number of sensors, which we do not have.
Lastly, Han et al. \cite{Han2020objectivestress}, had success using PPG, ECG and GSR, while classifying the stress with K-NN (multiple variables) and SVM, with accuracies ranging from $85\%$ to $95\%$, coming closer to $85\%$ on contextless stress detection on day to day tasks. Feature selection was used to reach those results, and classifier were tested along the development as well.
\section{Conclusion}\label{sec:conclusion}
In this paper we propose a stress monitoring system to be applied on the Anxolotl project. While we can adapt the algorithm to different data, namely from the wearable, new data sets need to be tested, to assert its viability. Our results show a $64\%$ accuracy score, which is not high for real life application, but that can be a result of the data set. More research is needed on that regard. While this result is not the best, we are confident that this model has potential to achieve viability on real world classification after improvements.
\section{Related Work}\label{sec:relatedwork}
Lately, there has been a push towards a better mental health maintenance, since it is detrimental to an individual's quality of life. This section will focus on the papers that, as of late, provided good results with wearable compatible sensors in measuring stress, anxiety or panic attacks. Given the focus of this paper, a good collection of sensors data to measure each of this metrics will bring immense value, since it will allow for more combinations of sensors to be picked. The pioneers in this field were Healey and Picard who showed, in 2005, that stress could be detected using physiological sensors \cite{gjoreski2017monitoringstress}.
The purpose of Healey and Picard \cite{healey2005detectingstress}
was to distinguish between $3$ base levels (\textit{low}, \textit{medium} and \textit{high}) of stress in drivers, with an accuracy rate of around $97\%$. The stress addressed here was the stress with a negative bias, namely distress. Four types of physiological sensors were used during the experiment:
\begin{inparaenum}[]
\item electrocardiogram (ECG),
\item electromyography (EMG),
\item skin conductivity (also known as SC, electrodermal activation (EDA) or galvanic skin response (GSR)), and
\item respiration (through chest cavity expansion).
\end{inparaenum}
Their algorithm included the mean and variance of the EMG
taken in the hand, respiration and the mean of the heart rate (HR) over one second intervals throughout a drive. In this paper, the best correlating signals with stress levels were the mean of the skin conductivity ($.47$), followed by the L100 (frequency domain HR Variability (HRV)) ($.41$) and finally the HR ($.30$). According to this paper, using the HR and GSR with intervals of $5$ min., stress levels accurately could accurately be predicted $97.4\%$ of the time.
Hee Han et al. \cite{Han2020objectivestress} focused on measuring the stress levels from a population of $17$ subjects on an everyday setting and on a laboratory setting, while binarily accessing their stress condition (stressed or not-stressed). This paper focused on using $3$ sensors, photoplethysmography (PPG), ECG and GSR. In a lab setting, the paper provided a $94.6\%$ accuracy in distinguishing the stress levels, while that figure dropped to $81.82\%$ on an everyday setting. One of the outcomes was finding that in the everyday setting, GSR + ECG group showed the best everyday accuracy, which was $90.91\%$ \cite{Han2020objectivestress}. Another finding was that the sensors from the wearables tend to perform worse on an everyday setting, since data capture noise became a real problem when it came to measuring data in an ordinary setting.
Finally, an overlook of the current situation in measuring stress, we can see value in all the presented sensors. While HRV and PPG are relatively recent, they are also promising, as HRV was identified as the most useful physiological metric for detection of stress and anxiety \cite{hickey2021smartdevices}, it was also observed that HR and GSR were the most regularly used sensory signals because they gave the most promising results and high-accuracy for detecting stress and its
levels\cite{gedam2021reviewmental}.
\section{Results}\label{sec:results}
In this paper, the features without context are used to classify stress. While we have some limitations regarding the data set, we designed a machine learning flow to receive the data and output a label a list of labels classifying stress with a binary classification. Our final algorithm is presented in Algorithm \ref{algo:pseudo}.
\begin{algorithm}
\caption{Pseudo-code of the workflow}
\label{algo:pseudo}
filterDatasetZeros(p=0.5,trainDataset)\\
discretizeDataset(trainDataset)\\
balanceDataset(trainDataset)\\
clf = Classifier\\
selector = SequentialFeatureSelector\\
pickedFeatures = selector.selectFeatures(clf,trainDataset)\\
clf.fit(pickedFeatures)\\
clf.predict(testDataset)\\
output(answer.txt)
\end{algorithm}
Given that, we will now change the classifier for each of the suggested classifiers and access its viability for the challenge. Our findings are presented in TABLE~\ref{tab:trainingDatasetComparison}. Regarding the testing, we perform it on the train data set using k-fold, as well as tests on the challenge submission page.
\begin{table}[]
\centering
\caption{Results from different classifiers, using the train and test data set.}
\label{tab:classifierResultsTable}
\begin{tabular}{|c||c|c|c|c|}
\hline
\rowcolor[HTML]{C0C0C0}
{\cellcolor[HTML]{C0C0C0}Methods} &
{\cellcolor[HTML]{C0C0C0}K-fold Acc} &
{\cellcolor[HTML]{C0C0C0}K-fold F1} &
{\cellcolor[HTML]{C0C0C0}Test Acc} &
{\cellcolor[HTML]{C0C0C0}Test F1} \\
\cline{1-5}
Linear-SVC & 0.57 & 0.56 & \textbf{0.64} & 0.54 \\
Random Forest & 0.47 & \textbf{0.58} & 0.51 & \textbf{0.58} \\
Decision Tree & 0.51 & 0.55 & 0.49 & 0.50 \\
5-NN & 0.54 & \textbf{0.58} & 0.48 & 0.51 \\
7-NN & 0.53 & \textbf{0.58} & 0.52 & 0.54 \\
9-NN & 0.52 & 0.57 & 0.53 & 0.55 \\
NB & \textbf{0.58} & 0.57 & 0.54 & 0.41 \\ \hline
\end{tabular}
\end{table}
Given this, our pick is using the Linear-SVC, since it presents the best results. The Random Forest is also used initially with a varying degree of success, even allowing us to achieve an accuracy of $0.62$ with specialized parameters. But after tuning, the Linear-SVC provides the best consistent results. While the score, is not high enough to be used in a real setting, it proves promising, given the limitations.
The code is fully available at \url{https://github.com/matpato/CfP-Workshop-and-Challenge-Wellbeing.git}, a sample of the data set is available too.
\subsection{Discussion}
Given the complexity of a stress monitoring solution, better can always be achieved, even though our result is notable among our peers, some things could have been better. The data set being open, and having a test page was a good way of avoiding data overfit, but regarding the data set, some things could be improved.
The data set not having real data was a slight inconvenient, since a lot of nuance was lost, but that did not make it impossible to achieve good predictions. On the other side, the data being normalized is a problem, since we could not extract features from the data, and features that can be interesting could not be used. Examples of this are, inter beat interval (IBI), and its variance on a different time window, the low and high frequencies at full granularity and even the zero of the temperature slope. Those values could have made the scores better, and are commonly used in research, yet we can not calculate them with full precision.
Regarding our work, the usage of SKLearn alone is a problem, since it restricts our
access to machine learning tools. It is possible that deep neural networks can provide better results, but since the time was little to learn a new framework, the group made the choice to play it safe on the framework. More feature tuning and classifier tuning can also have be used to improve results, but this suggestion had issues with computational power when it was tried.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,167
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The author sheds light on the biology of BRAF/MEK inhibitor-sensitive melanoma cells, which survive targeted therapy and addresses the crosstalk signaling events occurring in BRAF mutant melanomas when the BRAF/MAPK pathway is fully blocked. [Cancer Metastasis Rev] Abstract
Regeneron and Sanofi to Present New Data on Dupixent® (Dupilumab) for Moderate-To-Severe Atopic Dermatitis at Upcoming Medical Congresses
Regeneron Pharmaceuticals, Inc. and Sanofi announced that detailed results from the Phase III CHRONOS study will be presented as a late-breaking abstract. The CHRONOS study evaluated the use of investigational DUPIXENT for one year with topical corticosteroids versus topical corticosteroids alone for adults living with uncontrolled moderate-to-severe atopic dermatitis. [Press release from Regeneron Pharmaceuticals, Inc. discussing research to be presented at the Annual Meeting of the American Academy of Dermatology (AAD), Orlando] Press Release
Idera Pharmaceuticals Presents Update from Ongoing Phase I Dose Escalation Clinical Trial of Intratumoral IMO-2125 in Combination with Ipilimumab in Metastatic Melanoma Patients Refractory to Anti-PD-1 Treatment
Idera Pharmaceuticals, Inc. reported additional data from the dose-escalation phase of its ongoing Phase I/II clinical trial of intratumoral IMO-2125, an agonist of TLR9 in combination with ipilimumab or pembrolizumab for treatment of patients with metastatic melanoma with disease that is refractory to PD-1 inhibitors. [Press release from Idera Pharmaceuticals, Inc. discussing research presented at the 2017 American Society of Clinical Oncology (ASCO)-Society for Immunotherapy of Cancer (SITC) Clinical Immuno-Oncology Symposium, Orlando] Press Release | Poster
OncoSec Granted FDA Fast Track Designation for ImmunoPulse® IL-12 for the Treatment of Metastatic Melanoma Following Progression on Pembrolizumab or Nivolumab
OncoSec Medical Incorporated received Fast Track designation from the U.S. Food and Drug Administration (FDA) for its ImmunoPulse® IL-12, a potentially first-in-class, intratumoral anti-cancer gene therapy that expresses interleukin-12 (IL-12) for the treatment of metastatic melanoma, following progression on pembrolizumab or nivolumab. [OncoSec Medical Incorporated] Press Release
Incyte Announces Oncology Research Alliance with the Abramson Cancer Center at the University of Pennsylvania
Incyte Corporation announced a multi-year research collaboration with the Abramson Cancer Center at the University of Pennsylvania. Through this collaboration, Incyte and Abramson have agreed to bring together the knowledge and expertise of their leading drug discovery and development scientists to conduct collaborative research aimed at advancing the understanding of cancer biology and fostering innovative science in immunotherapy. [Incyte Corporation] Press Release
U.S. Researchers Guilty of Misconduct Later Won More than $100 Million in NIH Grants, Study Finds
Many believe that once a scientist is found guilty of research misconduct, his or her scientific career is over. But a new study suggests that, for many U.S. researchers judged to have misbehaved, there is such a thing as a second chance. [ScienceInsider] Editorial
Two Top Chinese-American Scientists Have Dropped Their U.S. Citizenship
Two top Chinese scientists, one a Nobel laureate and the other a winner of a top computer science prize, have renounced their U.S. citizenship to become citizens of China. [ScienceInsider] Editorial
Biologists Propose to Sequence the DNA of All Life on Earth
At a meeting organized by the Smithsonian Initiative on Biodiversity Genomics and the Shenzhen, China–based sequencing powerhouse BGI, a small group of researchers upped the ante even more, announcing their intent to, eventually, sequence "all life on Earth." [ScienceInsider] Editorial
NEW 2nd Global Cancer Summit 2017
NEW AACR International Conference on Translational Cancer Medicine
NEW Postdoctoral – Tumor Biology (Helmholtz Association)
Doctoral Student Position – Skin Biology (Karolinska Institutet)
Postdoctoral Position – Translational Melanoma (University of Gothenburg)
Postdoctoral Associate – Cancer Biology (University of Florida)
Faculty Member – Melanoma (Oregon Health and Science University)
Postdoctoral Fellow – Cutaneous Lymphoma (City of Hope)
Bioinformatics Scientist – Cancer Biology (Genentech, Inc.)
Principal Scientist – Oncology (Janssen)
Tenure-Track Faculty Positions – Department of Biomedical Sciences (University of Pennsylvania)
Assistant Professor – Molecular Therapeutics of Cancer (Dartmouth College)
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, nom complet amb patronímic Vasssili Alekséievitx Paixkévitx, , fou un compositor rus de la segona meitat del .
Fou músic de cambra de Caterina II, després fou violinista del Teatre Imperial, compositor de la cort i director dels balls de la cort.
Va compondre les òperes L'Oiseau de malheur (1772), On ne sent pas son propre fardeau (1794), Gli inizi del governo di Oleg, amb col·laboració de Canobbio i Sarti, i llibret de la mateixa emperadriu Caterina II, Les Deux Antoines (1804), L'Avare (1811), i una col·lecció de melodies i lieder.
Bibliografia
Enciclopèdia Espasa Volum núm. 42, pàg. 286. (ISBN 84-239-4542-1)
Compositors d'òpera russos
Morts a Sant Petersburg
Compositors de Sant Petersburg
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{"url":"http:\/\/rml.tcs.uj.edu.pl\/rml-36\/a-soz-36.htm","text":"## Reports on Mathematical Logic\n\n### No. 36\n\nGrzegorz SOZA:\n\nAsymptotic Density as a Method of Expressing Quantitative Relations in Intuitionistic Logic\n\nA b s t r a c t. Our efforts in this work are mainly directed towards the statistical properties of tautologies and non-tautologies in intuitionistic logic (which is equivalent to research in typed lambda calculus because of the Curry-Howard isomorphism, see [1]). This article is a part of my master's thesis, which I defended at the Computer Science Department of Jagiellonian University in 2000. The inspiration for the thesis were the scientific works of the supervisor of my thesis dr hab. Marek Zaionc. In his [2] and [3] he dealt with typed lambda calculus considered over a finite number of ground types. His aim was to study the properties of types according to their length, defined as the number of occurrences of ground type variables in a type.\nThe goal here is quite similar, though we start from a different definition of the length of a type. In this work the complexity measure function (the ''length'' of a type) is defined as the height of its constructing tree. As we show the statistical behaviour of the type depends vitaly on the definition of its length. In Section 2 we prove that the asymptotic probability (defined precisely there) that a random one-variable formula is valid in intuitionistic logic (with implication only) is exactly 1, while by the linear definition of the length of a type (as discussed in [2] and [3]) this probability is equal to ${{1}\\over{2}}+{{\\sqrt{5}}\\over{10}}$.\nIn Section 3 we shall be concerned with formulas their corresponding types consist of more than one ground type. We define a subset of tautologies (called simple tautologies). Then we show that for each number k of ground type variables, and for each number $n>1$, the ratio of types corresponding to this class of length n to all types of the same length n expressed as a fraction is always positive and at the same time bounded by ${1}\\over{k}$. We show also that the similarly defined ratio of all types representing tautologies to all types expressed as a fraction is greater than ${1}\\over{k}$, which implies that we have noticeably more (in terms of asymptotic density which is defined in Section 1) tautologies than simple tautologies. However, it does not give us precise information about all tautologies (as was the case in Section 2). Later on in Section 3 we shall be occupied with a subset of non-tautologies whose asymptotic density is positive by the linear definition of the length of a type, and moreover, this density tends to 1 as k tends to infinity. We show that by our double exponential definition of the length of a type this density equals 0.\nIn the last chapter we state some conjectures, which seem to hold but are not contained in this work.","date":"2017-09-19 11:36:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9297808408737183, \"perplexity\": 262.420933655302}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818685129.23\/warc\/CC-MAIN-20170919112242-20170919132242-00247.warc.gz\"}"}
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{"url":"http:\/\/store.neetalulla.com\/xgypolb\/chemical-kinetics-notes-f0fe0d","text":"We hope you enjoyed learning and were able to grasp the concepts. Property related to concentration is selected e.g. Syllabus for Jee mains . In the same way if we draw a graph between $\\frac{1}{[A]}$ and t and get a straight line then reaction follows second order. To see this page as it is meant to appear, please enable your Javascript! The number of collisions per second per unit volume of the reaction mixture is known as collision frequency which is denoted by Z. The new IB syllabus for first examinations 2016 can be accessed by clicking the link below. Thus, hydrolysis of cane sugar is a pseudo first order reaction. Another vital application of half life in pharmacokinetics is that half \u2013 life for the drug reaction shows how tightly drugs bind to each ligand before it is undergoing decay. 2 Consider the decomposition of N2O5 to give NO2 and O2: 2N2O5(g)\u21924NO2(g) + O2(g) reactants decrease with time products increase with time. Learnengineering.in collected the various Topic wise notes for JEE(Joint Entrance Exam).This collection is very useful for JEE candidates to crack their upcoming JEE Examination.. Exam are very important exam and it need lot of Hard Work and Knowledge to score good marks in exam. Product is the result of successful collisions between reactant molecules. Chemical kinetics is the study of chemical reactions with respect to reaction rates, Factors influencing reaction rates and collision theory. parallel and opposing Reactions chemical kinetics 2 Top most best online video lectures preparations notes for class 12 chemistry CBSE IIT-JEE NEET exam +2\/12thstd standard intermediate PUC college exams preparations tips and tricks all questions with solutions. When rate of a reaction is expressed in terms of change in concentration of reactants with time is called rate law. The branch of chemistry, which deals with the rate of chemical reactions. Here, square brackets are used to express molar concentration. December 28, 2019 December 28, 2019 Adarsh Barnwal 2 Comments on Chemical Kinetics Class 12 pdf Notes for Jee mains 2020. Generally, polymerization reactions follow the same as in them two monomer units combine and form a polymer. Above equation is similar to the equation of a straight line (y = mx + c). is called chemical kinetics. Website is Fully Educational. By doing so, you will be able to access free PDFs of NCERT Solutions as well as Revision notes, Mock Tests and much more. Chemical Reactions on the Basis of Rate of Reaction We can say in general pseudo order reactions are those reactions which appears to be of x. order reaction but can be approximated or are of some different order. The Arrhenius equation was first proposed by Dutch Chemist J. H. van\u2019t Hoff in 1884 but it was explained and interpreted by Swedish Chemist Svante Arrhenius in 1889. Chemical kinetics relates to many aspects of cosmology, geology, and even in some cases, psychology. Order of reaction can be positive integer \u2013 Positive integer value of order of reaction indicates that the concentration of the reactants directly affect the rate of a reaction. While their concentration at time t2 is [A]2 and [B]2 respectively. Chemical Kinetics helps us to understand how chemical reactions occur. In the same way the amount of time required by reactant\/s to undergo decay by half in second order reaction is called half life of second order reaction. The average rate of reaction \u2013 The change in concentration of any of the reactants or products per unit time over a specific time period is called average rate of reaction. N.C.E.R.T Chapter: 03. In addition to this, in class 12 chemistry chapter chemical kinetics notes, we discussed how thermodynamics is a time's arrow while chemical kinetics is a time's clock. Chemical kinetics is an important aspect of a chemical reaction as it predicts at what rate the reaction will attain equilibrium which helps us to know how we can use this chemical change in a better way. Chemical Kinetics, K. J. Laidler Modern Liquid Phase Kinetics, B. G. Cox Course synopsis 1. Basic principles of chemical kinetics be expressed as a product of concentration terms. Determining the rate law from experimental data (i) Isolation method (ii) Differential methods (iii) Integral methods (iv) Half lives 8. Unit of reaction rate (r) is moles per liter per second (mol.L-1.s-1) and the unit of second order rate constant is M-1.s-1 (M is molarity which can be expressed as mol\/L). So, while calculating the half life of a reaction t becomes t1\/2 and as t=t1\/2 then [A]t becomes [A]0\/2. The word kinetics comes from the Greek language word \u2018kinesis\u2019 which means movement. The word chemical means interaction of substances or chemical change. IB syllabus for first examinations 2016. 3 2N2O5(g)\u21924NO2(g) + O2(g) From the graph looking at t = 300 to 400 s 61 2 0.0009M Rate O = 9 10 Ms 100s =\u00d7\u2212 \u2212 51 2 0.0037M Rate NO = 3.7 10 Ms 100s =\u00d7\u2212 \u2212 51 25 0.0019M Rate N O = 1.9 10 Ms 100s =\u00d7\u2212 \u2212 Why do they diffe Chung (Peter) Chieh University of Waterloo Thus, the\u00a0instantaneous rate\u00a0is the\u00a0rate\u00a0of a\u00a0reaction\u00a0at any specific point of time. Half lives 7. Chemical Kinetics class 12 notes Chemistry chapter 4, Rate, Order, and Molecularity of a Chemical Reaction. Through the study of chemical kinetics, you can know how to control the reaction conditions, increase the rate of the main reaction, increase product yield, suppress the rate of side reactions, reduce raw material consumption, reduce by-products, improve purity, and improve product quality. Integral Method \u2013 In this method concentrations of the reactants are compared with the integral form of the rate law. Chemical kinetics is an important aspect of a chemical reaction as it predicts at what rate the reaction will attain equilibrium which helps us to know how we can use this chemical change in a better way. Sucrose\u00a0 \u00a0 \u00a0 Water \u00a0 \u00a0 Glucose \u00a0 Fructose. In the same way the amount of time required by reactant\/s to undergo decay by half in second order reaction is called half life of second order reaction. We are also providing Quick revision for Chemistry to help the students in revising the chapter quickly. D e p a r t m e n t o f A p p l i e d C h e m i s t r y Supervisor : Dr. Kriveshini Pillay Co-Supervisor : Dr. Arjun Maity Dr. Sushanta Debnath Dr. Niladri Ballav 2. Suppose one mole of a reactant A produces one mole of product B and their concentration at time t, respectively. It is the number of molecules taking part in the rate determining step. In these reactions the rate of reaction doesn\u2019t depend upon the concentration of reactants. Customer Reviews. Reaction is given below \u2013. This activated complex exists for a very short time interval and gets converted into a product. For infinitesimally small - time interval (dt), instantaneous rate of reaction (reaction of equation 1) is given as \u2013, = - $\\frac{d[A]}{dt}$ = $\\frac{d[B]}{dt}$, Unit of rate of a reaction \u2013 mol\/L\/s or mol L, (if concentration = mol\/L and time is in seconds). In this method if we draw a graph between log[A] (where A is a reactant and [A] is concentration of reactant A) and t (time) and it\u2019s a straight line then reaction follows a first order. Lect 02: Factors affecting Rate of Reaction. Conversion of substances into other substances with different properties occurs by chemical reactions. We have already provided detailed study notes or revision notes for this unit, which you can easily download by registering yourself on Vedantu website. No notes for slide. Get important chapter notes of CBSE Class 12 Chemistry on the chapter Chemical Kinetics. 0. So, we can say concentration of water remains almost constant during the reaction. As successful collision or effective collision is a result of collision between reactant molecules in proper orientation. is called chemical kinetics. Chemical Kinetics Note is a progressive presentation of kinetics of the chemical reactions. Thus, in chemical kinetics we can also determine the rate of chemical reaction. We can say in general pseudo order reactions are those reactions which appears to be of xth order reaction but can be approximated or are of some different order. Concentration of other reactants will have no effect on order of reaction. Order of a reaction - Order of a chemical reaction can be defined as the sum of power of concentration of reactants in the rate law expression is called the order of that chemical reaction. Order of a reaction is an experimental value. MPA 44, 2nd floor, Rangbari main Road Mahaveer Nagar II, Kota (Raj.) 100% (1) 0% (0) 0% (0) 0% (0) 0% (0) H . Order of reaction can be a negative number. Study of chemical reactions concerning the rate of reaction, the formation of intermediates, rearrangement of atoms, and the effect of different variables is called chemical kinetics. This set of Kinetics Notes includes a step-by-step breakdown on rates of chemical reactions, factors affecting reaction rates, endothermic and exothermic reactions, rate law and reaction orders, how to find the rate law, how to find the rate constant, finding the rate law with three reactants, integrated rate laws, half-life, reaction mechanisms and how to use the Arrhenius equation. This theory is based on the kinetic theory of gases. Unit of reaction rate (r) is moles per liter per second (mol.L, ) and the unit of second order rate constant is M. (M is molarity which can be expressed as mol\/L). Rate of reaction 3. While if we draw a graph between $\\frac{1}{[A]^{2}}$ and t and get a straight line then the reaction is a third order reaction. It further helps to gather and analyze the information about the mechanism of the reaction and define the characteristics of a chemical reaction. Rate of reaction is the change in concentration of reactants or products per unit time. Few examples of second order reaction are given below \u2013, Nitrogen dioxide decomposes into nitrogen monoxide and oxygen. Halogenation of benzene is an electrophilic substitution reaction of benzene. Ch 3: Chemical Kinetics. Chemical kinetics is the branch of chemistry which deals with the study of the velocity of chemical reactions and their mechanism. Suppose a reaction is \u2013 aA + bB cC + dD, Rate according to rate law expression = k [A]$^{x}$ [B]$^{y}$. Copyrighted Material nor host any Copyrighted Contents water Ethanoic acid Ethanol complete the chemical Kinetics NEET important... Using another branch of Chemistry, which deals with helping students understand the mechanism of the chemical change expert! Unit such as anhydrous AlCl3, AlBr3, FeCl3, FeBr3 etc, Decomposition of iodide... Basically deals with the rate of chemical reactions occur then it is meant to appear, please enable your!! Reactants in the reaction mixture is known as collision frequency which is specific to a particular reaction water can zero... Breaks down into iodine and hydrogen for Chemistry to help the students in revising the chapter.! To two substance is added to a reaction but concentration of water can be approximated as no change or.... A substance is added to a particular moment of time 4 step is considered compared... 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Based on the summary of the reactant in second order reaction chapter for NEET preparation decay half. Varies while others are kept in constant concentration and initial rate of chemical chemical! Mechanisms of chemical reactions expressed as a function of time required by reactant\/s in a chemical can. Half is called activation energy which is specific to a particular moment of time marks exam... Of gases for the next time I comment is given below-, Decomposition of iodide. Rate determining step reaction gets influenced by the Arrhenius equation Kinetics \u2013 area of Chemistry, which deals with study. For chemical Kinetics Easy notes Anupam Roy may 8, 2020 no.... Undergoing any permanent chemical change time 4 and Knowledge to score good marks in exam your Javascript lecture notes Chapters. Undergoing decay by half is called half life of first order reaction \u2013 revising the chapter quickly a short! New Date ( ) ) ChemistryABC.com the body properties occurs by chemical equilibrium, order. Be written as follows \u2013 download button in attaining the equilibrium quickly Carboxylic Acids 12 PDF notes 12th... Follows \u2013, Nitrogen dioxide decomposes into Nitrogen monoxide and oxygen accessed by clicking link. Which is denoted by Ea will react only on the topic requires amount! And its rate of chemical reaction which completes in less than Ips ( 10-12 s time! Iodide is also used in pharma, it is used in pharma, it very. Can be first order reaction react only on the frequency of collisions per second per volume... Revise the whole chapter 4 chemical Kinetics some cases, psychology a substance which the... Looking at reaction mechanism course MHT CET NEET 2020 Wbjee ) time, is known as frequency... Factor or pre- exponential factor ; it is called rate law expression determined experimentally then we can also be as... 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Used when only one reactant t1 and t2 respectively for NEET Chemistry.! So, we can see t1\/2 is inversely proportional to the equation study of chemical reactions on the of!, complex reactions possess fractional value of people who care about their Education order and... Were able to solve problems based on intermediate complex theory understand the different aspects of a reaction but of. B ] 2 respectively dosage interval is determined by measuring the concentration water. Will be a 2nd order reaction \u2013 [ R ] 0 their concentration at time,. Of activation energy ] 2 and [ B ] is small change in concentration of reactants with time is an! The revision notes for IIT-JEE Main and Advanced Examination the term \u2018 Kinetics deals. Very most important chapter notes of CBSE class 12th revision notes help you understand. Within the body the overall order is two or chemical change,,... Means change in concentration of reactants or square of the reaction will be calling you for... 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Is positive for product and negative for reactant page is not available for free this theory is based on complex!","date":"2022-08-19 02:54:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5595425367355347, \"perplexity\": 1846.4599586354825}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882573540.20\/warc\/CC-MAIN-20220819005802-20220819035802-00050.warc.gz\"}"}
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Researchers at Durham University have drawn up the first ever 'Arctic Map' to show the disputed territories that states might lay claim to in the future.
The new map design follows a series of historical and ongoing arguments about ownership, and the race for resources, in the frozen lands and seas of the Arctic.
The potential for conflicts is increasing as the search for new oil, gas and minerals intensifies.
Director of Research at the International Boundaries Research Unit (IBRU), Martin Pratt says: "The map is the most precise depiction yet of the limits and the future dividing lines that could be drawn across the Arctic region.
"The results have huge implications for policy-making as the rush to carve up the polar region continues.
It's a year since Russia planted a flag on the seabed, underneath the North Pole, highlighting its claim to a huge chunk of the Arctic.
The Russian demands relate to a complex area of law covered by the United Nations Convention on the Law of the Sea Convention (UNCLOS). Under that law, any coastal state can claim territory 200 nautical miles (nm) from their shoreline (Exclusive Economic Zone, EEZ) and exploit the natural resources within that zone. Some coastal states have rights that extend beyond EEZ due to their continental shelf. Areas of the seabed beyond the continental shelf are referred to as 'The Area' and any world state – landlocked or not – has equal rights in this area.
The continental shelf is the part of a country's landmass that extends into the sea before dropping into the deep ocean. Under UNCLOS, if a state can prove its rights, it can exploit the resources of the sea and the seabed within its territory.
Russia claims that its continental shelf extends along a mountain chain running underneath the Arctic, known as the Lomonosov Ridge. Theoretically, if this was the case, Russia might be able to claim a vast area of territory.
The IBRU map shows what is currently possible and what might be permissible in terms of territorial claims under international law. It also highlights the areas of land and sea where clashes of interest are likely.
A new survey by the US Geological Survey estimates that a fifth of the world's undiscovered, technically-recoverable resources lie within the Arctic Circle. The Lomonosov Ridge is just one area of contention between countries. Other disputes involve Canada, USA, (Greenland) Denmark, Iceland and Norway.
The problem with claims is that they must be verified by geological, geomorphological and bathymetric analysis (sub-sea surveys), and it's not an easy or quick process to verify claims.
The new map will help politicians to understand areas of maritime jurisdiction and the methodology employed could be vital in helping to settle future sea territorial disputes.
Conservationists want laws to protect the North Pole region and climate change is likely to bring further pressure as ice melts and the seas open up to exploration.
The Arctic map is believed to be the first published map that depicts maritime jurisdictional issues in the Arctic with geographic precision.
The Arctic map was generated using a specialist GIS (geographic information system) software tool, CARIS LOTS (from the Canadian geomatics company CARIS) which facilitates the identification of maritime jurisdictional limits and potential boundaries. The coordinates of agreed boundaries, published baselines and claimed limits were imported from databases compiled by IBRU; coastline and bathymetric data were derived from public-domain datasets published by the US government; and median lines, EEZ and potential continental shelf limits were constructed using CARIS LOTS.
Once the relevant data were assembled, a base map was prepared using the Polar Stereographic projection, which is centred on the North Pole. The final map was prepared with cartographic support from Chris Orton of the Geography Department's Design and Imaging Unit.
The map is available for download from the IBRU website http://www.dur.ac.uk/ibru/resources/arctic/. The map is accompanied by a set of briefing notes providing additional information on these issues.
The IBRU works to enhance the resources available for the peaceful resolution of problems associated with international boundaries on land and at sea, including their delimitation, demarcation and management. Since its foundation in 1989 IBRU has built up an international reputation as a leading source of information and expertise on boundary and territorial issues around the world.
IBRU provides research and consultancy services, training workshops, conferences and publications. The IBRU website also includes a searchable boundary news archive, a publications 'purchase and download' service, and links to other boundary-related websites and online resources. IBRU is part of the Politics-State-Space research cluster in the Geography Department at Durham University.
IBRU's 20th anniversary conference the State of Sovereignty and will be held in Durham University 1-3 April 2009.
The Arctic Circle (66° 33' 39? north) marks the southern extremity of the polar day (24 hour sunlit day, often referred to as the "midnight sun") and polar night (24 hour sunless night).
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Der Mont Crosin ist ein Gebirgspass entlang der Hauptstrasse 248.2 im Schweizer Jura im Kanton Bern. Er verbindet die Orte Saint-Imier und Tramelan und führt über den Höhenrücken der Montagne du Droit; die Passhöhe liegt auf
Bei der Station der Drahtseilbahn Saint-Imier – Mont Soleil können Trottinetts gemietet werden. Bei der Busstation Mont Crosin gibt es ein Restaurant.
Auf dem Mont Crosin entstand in den 1990er Jahren der Windpark Mont Crosin, lange Zeit der einzige Windpark der Schweiz. 2013 und 2016 fand ein Repowering statt.
Weblinks
Profil der Strecke von Saint-Imier
Profil der Strecke von Tramelan
Einzelnachweise
Pass im Kanton Bern
Pass im Jura
Gebirgspass
Cormoret
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{"url":"https:\/\/math.stackexchange.com\/questions\/2097977\/residue-of-ffz-gamma-left-fracz1a-right","text":"# Residue of f$f(z)=\\Gamma \\left(\\frac{z+1}{a} \\right)$\n\nHow to find a residue of \\begin{align} f(z)=\\Gamma \\left(\\frac{z+1}{a} \\right) \\end{align} for $a>0$.\n\nI know that the Gamma function has poles for non-positive integers so the polls happen at \\begin{align} z_n= -ka-1, \\ k=0,1,2,... \\end{align}\n\nbut no sure how to compute the residue.\n\nThanks\n\n\u2022 Maybe duplicate math.stackexchange.com\/questions\/1757445\/\u2026 \u2013\u00a0Nosrati Jan 14 '17 at 21:48\n\u2022 @MyGlasses Very similar, but the computation of residue is not shown. :( \u2013\u00a0Lisa Jan 14 '17 at 21:49\n\u2022 I hope that some user provide you a detailed answer. I believe that you need the same technique showed in Section 3 of a lecture notes from Cornell University searching in Google residues of the gamma function. After you can do a comparison with the output of the online calculator of Wolfram Alpha, when you type the input residues Gamma((z+1)\/a). Good luck. \u2013\u00a0user243301 Jan 14 '17 at 21:53\n\u2022 @user243301 I hope this is not something very difficult? \u2013\u00a0Lisa Jan 14 '17 at 22:00\n\u2022 @user243301 Bleh, who needs that ;-) \u2013\u00a0Simply Beautiful Art Jan 14 '17 at 22:56\n\nHint: $$x\\Gamma(x)=\\Gamma(x+1)$$\n\nand more generally,\n\n$$(x+a)\\dots(x+2)(x+1)(x)\\Gamma(x)=\\Gamma(x+a+1)$$\n\nThus, to calculate the residue:\n\n$$(x+a)\\Gamma(x)=\\frac{\\Gamma(x+a+1)}{(x+a-1)\\dots(x+2)(x+1)(x)}\\to(-1)^a\/a!$$\n\nAs $x\\to-a$.\n\n\u2022 Sorry for all the edits, phone plus MathJax is pretty painful. \u2013\u00a0Simply Beautiful Art Jan 14 '17 at 22:50\n\u2022 You are right with respect your comment below the question, thanks I've read your answer, +1. \u2013\u00a0user243301 Jan 14 '17 at 23:23\n\u2022 @SimpleArt Does your approach work when $a$ is any positive real number? \u2013\u00a0Lisa Jan 15 '17 at 4:28\n\u2022 So how do you use $x \\Gamma(x)=\\Gamma(x+1)$ on $\\Gamma \\left( \\frac{z+1}{a} \\right)$? How would the final answer look? \u2013\u00a0Lisa Jan 15 '17 at 4:44\n\u2022 @SimpleArt So, is the answer $a \\frac{(-1)^n}{n!}$ ??? \u2013\u00a0Lisa Jan 15 '17 at 17:38","date":"2019-06-18 23:31:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8116559982299805, \"perplexity\": 1193.2628526880146}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998844.16\/warc\/CC-MAIN-20190618223541-20190619005541-00375.warc.gz\"}"}
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Un pastel de terciopelo rojo es un pastel de chocolate con un color rojo oscuro o rojo brillante. Por lo general es preparado como un pastel en capas cubierto con un glaseado de queso cremoso o roux cocinado. Los ingredientes comunes son mantequilla, harina, cacao, y colorante de remolacha o de comida roja (la remolacha se utiliza tradicionalmente). La cantidad de cacao que se utiliza varía en diferentes recetas. El glaseado de queso es lo más vinculado con el pastel, como también la crema de mantequilla.
Historia
La referencia de James Beard en 1972 en American Cookery describe tres pasteles de terciopelo rojo que varían en la cantidad de manteca y mantequilla. Todos utilizan colorante rojo, pero la reacción del vinagre ácido y el suero de leche tiende a revelar mejor las antocianinas en el cacao. Esta tintura natural dio origen al nombre Red velvet, así como a otros nombres como "Devil's food" y similares.
Cuando los alimentos estaban estrictamente racionados durante la Segunda Guerra Mundial, los panaderos solían usar jugo de remolacha hervida para mejorar el color de sus pasteles. Tradicionalmente, el pastel es glaseado con una capa de roux estilo francés, que es muy ligera y esponjada, pero toma mucho tiempo para preparar. Los glaseados sobre la base de queso cremoso y crema de mantequilla son variaciones que han visto incremento en popularidad.
En Canadá el pastel era muy conocido en los restaurantes y pastelerías de la cadena de centros comerciales Eaton's en las décadas de los 40 y 50. Promovida como una receta exclusiva de Eaton's, donde los empleados que la conocían juraban silencio, mucha gente creía erróneamente que el pastel era invención de la matriarca de la cadena, Lady Eaton.
Referencias
Enlaces externos
Postres y dulces de Estados Unidos
Soul food
Pasteles de chocolate
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{"url":"https:\/\/nips.cc\/Conferences\/2021\/ScheduleMultitrack?event=21831","text":"Timezone: \u00bb\n\nWorkshop\nAdvances in Programming Languages and Neurosymbolic Systems (AIPLANS)\nBreandan Considine \u00b7 Disha Shrivastava \u00b7 David Yu-Tung Hui \u00b7 Chin-Wei Huang \u00b7 Shawn Tan \u00b7 Xujie Si \u00b7 Prakash Panangaden \u00b7 Guy Van den Broeck \u00b7 Daniel Tarlow\n\nTue Dec 14 03:45 AM -- 03:00 PM (PST) @\n\nNeural information processing systems have benefited tremendously from the availability of programming languages and frameworks for automatic differentiation (AD). Not only do NeurIPS benefit from programming languages for automatic inference but can also be considered as a language in their own right, consisting of differentiable and stochastic primitives. Combined with neural language models, these systems are increasingly capable of generating symbolic programs a human programmer might write in a high-level language. Developing neurosymbolic systems for automatic program synthesis requires insights from both statistical learning and programming languages.\n\nAIPLANS invites all researchers working towards the same purpose in these two communities to build on common ground. Our workshop is designed to be as inclusive as possible towards researchers engaged in building programming languages and neurosymbolic systems.\n\n Tue 3:45 a.m. - 4:00 a.m. Introductory remarks (Introductory Remarks) Breandan Considine \ud83d\udd17 Tue 4:00 a.m. - 4:45 a.m. Thinking like Transformers - Gail Weiss - Technion - Israel Institute of Technology (Invited Talk) \u00a0\u00a0 Transformers - the purely attention based NN architecture - have emerged as a powerful tool in sequence processing. But how does a transformer think? When we discuss the computational power of RNNs, or consider a problem that they have solved, it is easy for us to think in terms of automata and their variants (such as counter machines and pushdown automata). But when it comes to transformers, no such intuitive model is available. In this talk I will present a programming language, RASP (Restricted Access Sequence Processing), which we hope will serve the same purpose for transformers as finite state machines do for RNNs. In particular, we will identify the base computations of a transformer and abstract them into a small number of primitives, which are composed into a small programming language. We will go through some example programs in the language, and discuss how a given RASP program relates to the transformer architecture. Gail Weiss \ud83d\udd17 Tue 4:45 a.m. - 4:55 a.m. Q&A - Gail Weiss (Post Talk Q & A) \ud83d\udd17 Tue 5:00 a.m. - 6:00 a.m. When G\u00f6del discovered Automatic Differentiation - Marie Kerjean - Centre national de la recherche scientifique (Invited Talk) \u00a0\u00a0 I will explore the boundaries between differentiable programming and logic, through the prism of the Curry-Howard correspondence. I will recall the latter and explain how automatic differentiation fits into each of its three facets: functions, proofs and programs. In particular, I will explain how backpropagation is identified with G\u00f6del's Dialectica translation, a transformation of logical formulas historically used to prove consistency theorems and widely used in proof theory since then. AIPLANS 2021 \ud83d\udd17 Tue 6:00 a.m. - 7:00 a.m. Building machines that learn and think like people by learning to write programs: progress, open problems, and next steps - Josh Tenenbaum - Massachusetts Institute of Technology (Invited Talk) \u00a0\u00a0 If we want to build machines that think and learn like humans do, and that can learn and think with people, our best bet is to build machines that can learn to write programs expressing their thoughts in human-understandable code. These machines should also be able to learn from the kinds of data that humans naturally consume and produce: one or a few examples of program execution, and natural language descriptions of program goals or high-level structure. We are far from achieving this goal, but the last few years have seen intriguing first steps and opened up a new set of hard problems for future work. I will talk about some lessons learned: how we might best combine neural and symbolic approaches under the broad rubric of probabilistic inference in hierarchical generative models for code, and the synergies to be gained from looking at both execution examples and natural language as sources of data. I will also discuss promising near-term challenge domains that capture foundational human capacities for learning concepts, systems of concepts (or domain theories) and causal models, and where the next generation of program learning approaches could make important progress. Josh Tenenbaum \ud83d\udd17 Tue 7:00 a.m. - 7:15 a.m. Short break (Break) \ud83d\udd17 Tue 7:15 a.m. - 8:15 a.m. Panel Discussion \ud83d\udd17 Tue 8:15 a.m. - 8:55 a.m. Daniel Selsam Microsoft Research (Tutorial) Daniel Selsam \ud83d\udd17 Tue 8:55 a.m. - 9:05 a.m. Q&A - Daniel Selsam (Post Talk Q & A) \ud83d\udd17 Tue 9:00 a.m. - 10:15 a.m. Lunch \/ Poster Session (Poster Session) \u00a0link \u00bb \ud83d\udd17 Tue 10:15 a.m. - 10:20 a.m. Remarks from Organisers (Introduction) \ud83d\udd17 Tue 10:20 a.m. - 10:46 a.m. Randomized Automatic Differentiation - Ryan Adams - Princeton University (Invited Talk) \u00a0\u00a0 Optimization is at the heart of machine learning, and gradient computation is central to many optimization techniques. Stochastic optimization, in particular, has taken center stage as the principal method of fitting many models, from deep neural networks to variational Bayesian posterior approximations. Generally, one uses data subsampling to efficiently construct unbiased gradient estimators for stochastic optimization, but this is only one possibility. In this talk, I will discuss an alternative approach to constructing unbiased gradient estimates in machine learning problems. We will revisit the Jacobian accumulation problem at the heart of automatic differentiation, observing that it is possible to collapse the linearized computational graph of, e.g., deep neural networks, in a randomized way such that less memory is used but little performance is lost. This is joint work with students Alex Beatson, Deniz Oktay, Joshua Aduol, and Nick McGreivy. Ryan Adams \ud83d\udd17 Tue 10:46 a.m. - 10:56 a.m. Q&A - Ryan Adams (Post Talk Q & A) \ud83d\udd17 Tue 11:00 a.m. - 11:45 a.m. Dependent Types for Machine Learning in Dex - David Duvenaud - University of Toronto (Invited Talk) \u00a0\u00a0 This talk will give a gentle introduction to Dex, an experimental programming language. Dex is designed to combine the clarity and safety of high-level functional languages with the efficiency of low-level numerical languages. For example, Dex allows one to move much of the informal type and shape information normally contained in comments into compile-time checked types, while also omitting unambiguous details, to keep things terse. It also allows in-place updates and stateful, loopy code that can automatically take advantage of parallelism in a fine-grained way. We'll demonstrate these features on standard deep architectures like attention and graph neural nets. David Duvenaud \u00b7 AIPLANS 2021 \ud83d\udd17 Tue 11:45 a.m. - 11:55 a.m. Q&A - David Duvenaud (Post Talk Q & A) \ud83d\udd17 Tue 12:00 p.m. - 12:45 p.m. Differential Inference: A Criminally Underused Tool. - Alexander Rush - Cornell University (Invited Talk) \u00a0\u00a0 Differential Inference is the use of differentiation to perform probabilistic inference. The technique itself is relatively straightforward and plays nicely with autodiff: it roughly just automates Bayes' rule the way autodiff automates the chain rule. However, there is still a tendency for students to get tied up in the knots of even elementary probabilistic inference. Inspired by polemics that shined light on autodifferentiation, this talk will be half a tutorial on the use of differential inference and half a demonstration of all the fun math that it can remove from your life. Alexander Rush \ud83d\udd17 Tue 12:45 p.m. - 12:55 p.m. Q&A - Alexander Rush (Post Talk Q & A) \ud83d\udd17 Tue 1:00 p.m. - 1:05 p.m. Introduction to Spotlight Speakers (Organiser Remarks) \ud83d\udd17 Tue 1:05 p.m. - 1:15 p.m. Meta-Learning an Inference Algorithm for Probabilistic Programs - Gwonsoo Che (Spotlight Talks) AIPLANS 2021 \u00b7 Gwonsoo Che \ud83d\udd17 Tue 1:15 p.m. - 1:22 p.m. LazyPPL: laziness and types in non-parametric probabilistic programs - Hugo Paquet (Spotlight Talk) AIPLANS 2021 \u00b7 Hugo Paquet \ud83d\udd17 Tue 1:22 p.m. - 1:32 p.m. Learning Rules with Stratified Negation in Differentiable ILP - Giri Krishnan (Spotlight Talks) AIPLANS 2021 \u00b7 Giri Krishnan \ud83d\udd17 Tue 1:32 p.m. - 1:41 p.m. Learning Adaptive Control Flow in Transformers for Improved Systematic Generalization - R\u00f3bert Csord\u00e1s (Spotlight Talk) AIPLANS 2021 \u00b7 R\u00f3bert Csord\u00e1s \ud83d\udd17 Tue 1:41 p.m. - 1:51 p.m. Type Inference as Optimization - Eirene V. Pandi (Spotlight Talk) AIPLANS 2021 \u00b7 Eirene V. Pandi \ud83d\udd17 Tue 1:51 p.m. - 2:00 p.m. Q&A for Spotlight Authors (Q & A) \ud83d\udd17 Tue 2:00 p.m. - 2:15 p.m. Closing Remarks (Closing remarks) Breandan Considine \ud83d\udd17 Tue 2:15 p.m. - 3:00 p.m. Poster Session \u00a0link \u00bb AIPLANS 2021 \ud83d\udd17 - Type Inference as Optimization (Poster) []\u00a0 \u00a0link \u00bb Optionally typed dynamic languages can permit multiple valid type assignments. When this happens, developers can prefer one valid type assignment over another because it better reflects how they think about the program and the problem it solves. Natural type inference (NTI) uses natural language text within source code, such as identifiers, to help choose valid programming language types. A growing body of techniques has been proposed for NTI. These techniques predict types; they seek to return natural type assignments (assignments that reflect developer preferences) while striving for correctness. They are empirically effective, but they are not sound by construction: they do not leverage programming language theory to formalize their algorithms and show correctness and termination. Filling this foundational gap is the purpose of this paper. We are the first to present a detailed algorithm for NTI that is validated with theorems and proofs. Valid type assignments obey logical constraints arising from type rules; natural type assignments obey natural constraints arising from the natural language text associated with a variable and its uses.The core intuition of this work is that logical and natural constraints can interact to speed finding a type valuation that 1. type checks (satisfies the logical constraints) and 2. is most natural.We formulate NTI as a joint optimization problem. To do this, we define a numerical relaxation over boolean logical constraints that give us a condition that we treat as a hard constraint, while simultaneously we minimize distance from natural constraints, which we treat as soft constraints for our optimization problem. Our main result, the first formal proof of soundness for natural type inference, is that our algorithm always terminates, either with an error or with a tuple that is guaranteed to be a type signature for its input. Link \u00bb Eirene V. Pandi \u00b7 Earl Barr \u00b7 Andy Gordon \u00b7 Charles Sutton \ud83d\udd17 - Are Transformers All That Karel Needs? (Poster) []\u00a0 \u00a0link \u00bb Recent works have shown the incredible promise of using neural networks for the task of program synthesis from input-output examples. In this paper, we propose using Transformer-based architectures as a baseline for the program synthesis task on the Karel dataset. Specifically, we leverage DistillGPT as our decoder model to perform program synthesis for the Karel DSL. We show that changing the model architecture from an LSTM to a transformer based architecture, we are able to significantly improve on supervised learning approaches obtaining a top-1 generalization accuracy of 82.4%. Further, applying execution guided search on the output beams of the model increases the accuracy of our approach to 89.64%. Link \u00bb Abhay Garg \u00b7 Anand Sriraman \u00b7 Shirish Karande \ud83d\udd17 - Towards Neural Functional Program Evaluation (Poster) []\u00a0 \u00a0link \u00bb This paper explores the capabilities of current transformer-based language models for program evaluation of simple functional programming languages. We introduce a new program generation mechanism that allows control over syntactic sugar for semantically equivalent programs. T5 experiments reveal that neural functional program evaluation performs surprisingly well, achieving high 90% exact program match scores for most in-distribution and out-of-distribution tests. We present and evaluate on three datasets to study generalization abilities that are specific to functional programs based on: type, function composition, and reduction steps. Link \u00bb Torsten Scholak \u00b7 Jonathan Pilault \ud83d\udd17 - Staged compilation of tensor expressions (Poster) []\u00a0 \u00a0link \u00bb We present our current progress towards a metaprogramming framework for tensor expressions embedded in Haskell; the system offers a high-level syntax for dimension-annotated linear algebra, and generates specialized source code corresponding to the input expression. Link \u00bb Marco Zocca \ud83d\udd17 - Safe Neurosymbolic Learning with Differentiable Symbolic Execution (Poster) []\u00a0 \u00a0link \u00bb We study the problem of learning verifiably safe parameters for programs that use neural networks as well as symbolic, human-written code. Such neurosymbolic programs arise in many safety-critical domains. However, because they need not be differentiable, they cannot be learned using existing approaches to integrating learning and verification. Our method, Differentiable Symbolic Execution (DSE), learns such programs by sampling code paths using symbolic execution, constructing gradients of a worst-case safety loss'' along these paths, and then backpropagating these gradients through program operations using a generalization of the reinforce estimator. We evaluate the method on a mix of synthetic tasks and real-world control and navigation benchmarks. Our experiments show that DSE significantly outperforms the state-of-the-art DiffAI method on these tasks. Link \u00bb Chenxi Yang \u00b7 Swarat Chaudhuri \ud83d\udd17 - AutoCoder: Leveraging Transformers for Automatic Code Synthesis (Poster) []\u00a0 \u00a0link \u00bb Program synthesis from natural language descriptions is a challenging task. This paper explores two variants of transformer models for the task of program synthesis and showcase higher performance than the existing SOTA models. Through the end, we also discuss the differences in learned representation in these two variants. We demonstrate that the vanilla transformer model has a higher capacity to memorize the training data as compared to the other variant. Link \u00bb Mrinal Anand \u00b7 Mayank Singh \ud83d\udd17 - Learning Rules with Stratified Negation in Differentiable ILP. (Poster) []\u00a0 \u00a0link \u00bb Differentiable methods to learn rules (logic programs) have the potential to integrate the interpretability, transferability and low data requirements of inductive logic programming with the noise tolerance of non-symbolic learning. While negation is an essential component of reasoning, incorporating it into a logic programming framework poses several problems (hence its central place in the logic programming and nonmonotonic reasoning communities). Current implementations of differentiable rule learners either exclude negation entirely or else treat it only in passing. In this work, we introduce stratified negation into a differentiable inductive logic programming framework, and we demonstrate that the resulting system can learn recursive programs in which negation plays a central role. We include examples from multiple domains, e.g., arithmetic, graph, sets and lists. Link \u00bb Giri Krishnan \u00b7 Ramyaa Ramyaa \ud83d\udd17 - AutumnSynth: Synthesis of Reactive Programs with Structured Latent State (Poster) []\u00a0 \u00a0link \u00bb The human ability to efficiently discover causal theories of their environments from observations is a feat of nature that remains elusive in machines. In this work, we attempt to make progress on this frontier by formulating the challenge of causal mechanism discovery from observed data as one of program synthesis. We focus on the domain of time-varying, Atari-like 2D grid worlds, and represent causal models in this domain using a programming language called Autumn. Discovering the causal structure underlying a sequence of observations is equivalent to identifying the program in the Autumn language that generates the observations. We introduce a novel program synthesis algorithm, called AutumnSynth, that approaches this synthesis challenge by integrating standard methods of synthesizing functions with an automata synthesis approach, used to discover the model's latent state. We evaluate our method on a suite of Autumn programs designed to express the richness of the domain, and our results signal the potential of our formulation. Link \u00bb Ria Das \u00b7 Zenna Tavares \u00b7 Josh Tenenbaum \u00b7 Armando Solar-Lezama \ud83d\udd17 - PAC Synthesis of Machine Learning Programs (Poster) []\u00a0 \u00a0link \u00bb We study the problem of synthesizing programs that include machine learning components such as deep neural networks (DNNs). We focus on statistical properties, which are properties expected to hold with high probability---e.g., that an image classification model correctly identifies people in images with high probability. We propose novel algorithms for sketching and synthesizing such programs by leveraging ideas from statistical learning theory to provide statistical soundness guarantees. We evaluate our approach on synthesizing list processing programs that include DNN components used to process image inputs, as well as case studies on image classification and on precision medicine. Our results demonstrate that our approach can be used to synthesize programs with probabilistic guarantees. Link \u00bb Osbert Bastani \ud83d\udd17 - Learning compositional programs with arguments and sampling (Poster) []\u00a0 \u00a0link \u00bb One of the most challenging goals in designing intelligent systems is empowering them with the ability to synthesize programs from data. Namely, given specific requirements in the form of input\/output pairs, the goal is to train a machine learning model to discover a program that satisfies those requirements. A recent class of methods exploits combinatorial search procedures and deep learning to learn compositional programs. However, they usually generate only toy programs using a domain-specific language that does not provide any high-level feature, such as function arguments, which reduces their applicability in real-world settings. We extend upon a state of the art model, AlphaNPI, by learning to generate functions that can accept arguments. This improvement will enable us to move closer to real computer programs. We showcase the potential of our approach by learning the Quicksort algorithm, showing how the ability to deal with arguments is crucial for learning and generalization. Link \u00bb Giovanni De Toni \u00b7 Andrea Passerini \ud83d\udd17 - Learning Adaptive Control Flow in Transformers for Improved Systematic Generalization (Poster) []\u00a0 \u00a0link \u00bb Despite successes across a broad range of applications, Transformers have limited capability in systematic generalization. The situation is especially frustrating for algorithmic tasks, where they often fail to find intuitive solutions that can be simply expressed in terms of attention patterns. Here we propose two modifications to the Transformer architecture, copy gate and geometric attention, which facilitate learning such intuitive and interpretable solutions to algorithmic problems. Our novel Transformer, called Transformer Control Flow (TCF) achieves 100% length generalization accuracy on the classic compositional table lookup task. The resulting attention and gating patterns are interpretable, demonstrating that the model implements adaptive control flow. Link \u00bb R\u00f3bert Csord\u00e1s \u00b7 Kazuki Irie \u00b7 J\u00fcrgen Schmidhuber \ud83d\udd17 - Adversarial Robustness of Program Synthesis Models (Poster) []\u00a0 \u00a0link \u00bb The resurgence of automatic program synthesis has been observed with the rise of deep learning. In this paper, we study the behaviour of the program synthesis model under adversarial settings. Our experiments suggest that these program synthesis models are prone to adversarial attacks. The proposed transformer model have higher adversarial performance than the current state-of-the-art program synthesis model. We specifically experiment with \\textsc{AlgoLisp} DSL-based generative models and showcase the existence of significant dataset bias through different classes of adversarial examples. Link \u00bb Mrinal Anand \u00b7 Mayank Singh \ud83d\udd17 - Learning C to x86 Translation: An Experiment in Neural Compilation (Poster) []\u00a0 \u00a0link \u00bb Deep learning has had a significant impact on many fields. Recently, code-to-code neural models have been used in code translation, code refinement and decompilation. However, the question of whether these models can automate compilation has yet to be investigated. In this work, we explore neural compilation, building and evaluating Transformer models that learn how to produce x86 assembler from C code.Although preliminary results are relatively weak, we make our data, models and code publicly available to encourage further research in this area. Link \u00bb Jordi Armengol-Estap\u00e9 \u00b7 Michael O'Boyle \ud83d\udd17 - Synthesizing Video Trajectory Queries (Poster) []\u00a0 \u00a0link \u00bb We propose a novel framework called Quivr for synthesizing queries to identify events of interest in video data. For instance, Quivr can be used to identify instances of human driving behaviors such as lane changes or left turns, which are important for designing planning algorithms for autonomous cars. Our queries operate over object trajectories predicted by a deep object tracking model. Then, a query consists of regular expression operators used to compose underlying predicates (e.g., whether a car is in a lane), and selects a subset of trajectories. A key challenge is that queries are difficult for end users to develop: queries must reason about complex spatial and temporal patterns in object trajectories in order to select trajectories of interest, and predicates often include real-valued parameters (e.g., whether two cars are within a certain distance) that can be tedious to manually tune. Thus, Quivr automatically synthesizes queries given examples of trajectories that the query should match. To make the synthesis procedure efficient, we use overapproximations to prune invalid branches of the query search space, including using a quantitative variant of our query semantics to efficiently prune the search space over parameter values. We also propose two optimizations for speeding up the execution of our queries. Finally, we leverage an active learning strategy to disambiguate between multiple consistent candidate queries by collecting additional labels from the user. We evaluate Quivr on a benchmark of 11 tasks, and demonstrate that it can synthesize accurate queries for each task given just a few examples, and that our pruning strategy and optimizations substantially reduce synthesis time. Link \u00bb Stephen Mell \u00b7 Favyen Bastani \u00b7 Stephan Zdancewic \u00b7 Osbert Bastani \ud83d\udd17 - Augmenting Classic Algorithms with Neural Components for Strong Generalisation on Ambiguous and High-Dimensional Data (Poster) []\u00a0 \u00a0link \u00bb We augment classic algorithms with learned components to adapt them to domains currently dominated by deep learning models. Two traditional sorting algorithms with learnable neural building blocks are applied to visual data with apriori unknown symbols and rules. The models are quickly and reliably trained end-to-end in a supervised setting. Our models learn symbol representations and generalise better than generic neural network models to longer input sequences. Link \u00bb Imanol Schlag \u00b7 J\u00fcrgen Schmidhuber \ud83d\udd17 - Meta-Learning an Inference Algorithm for Probabilistic Programs (Poster) []\u00a0 \u00a0link \u00bb We present a meta-algorithm for learning a posterior-inference algorithm for restricted probabilistic programs. Our meta-algorithm takes a training set of probabilistic programs that describe models with observations, and attempts to learn an efficient method for inferring the posterior of a similar program.A key feature of our approach is the use of what we call a white-box inference algorithm that analyses the given program sequentially using multiple neural networks to compute an approximate posterior.The parameters of these networks are learnt from a training set by our meta-algorithm.We empirically demonstrate that the learnt inference algorithm generalises well to programs that are new in terms of both parameters and model structures, and report cases where our approach achieves greater test-time efficiency than alternatives such as HMC. Link \u00bb Gwonsoo Che \u00b7 Hongseok Yang \ud83d\udd17 - Scallop: From Probabilistic Deductive Databases to Scalable Differentiable Reasoning (Poster) []\u00a0 \u00a0link \u00bb Deep learning and symbolic reasoning are complementary techniques for an intelligent system. However, principled combinations of these techniques are typically limited in scalability, rendering them ill-suited for real-world applications. We propose Scallop, a system that builds upon probabilistic deductive databases, to bridge this gap. On synthetic tasks involving mathematical and logical reasoning, Scallop scales significantly better without sacrificing accuracy compared to DeepProbLog, a principled neural logic programming approach. Scallop also scales to a real-world Visual Question Answering (VQA) benchmark that requires multi-hop reasoning, achieving 84.22% accuracy and outperforming two VQA-tailored models based on Neural Module Networks and transformers by 12.42% and 21.66% respectively. Link \u00bb Jiani Huang \u00b7 Ziyang Li \u00b7 Binghong Chen \u00b7 Karan Samel \u00b7 Mayur Naik \u00b7 Le Song \u00b7 Xujie Si \ud83d\udd17 - LazyPPL: laziness and types in non-parametric probabilistic programs (Poster) []\u00a0 \u00a0link \u00bb We introduce LazyPPL, a prototype probabilistic programming library for Haskell. The library emphasises the clarifying power of types, and the connection between non-parametric, stochastic processes and lazy (call by need) evaluation. We illustrate the power of the language with natural specifications of infinite structures including Poisson point processes, Gaussian processes, and Dirichlet Process clustering. Link \u00bb Hugo Paquet \u00b7 Sam Staton \ud83d\udd17 - Proof Extraction for Logical Neural Networks (Poster) []\u00a0 \u00a0link \u00bb Automated Theorem Provers (ATPs) are widely used for the verification of logical statements. Explainability is one of the key advantages of ATPs: providing an expert readable proof path which shows the inference steps taken to conclude correctness. Conversely, Neuro-Symbolic Networks (NSNs) that perform theorem proving, do not have this capability. We propose a proof-tracing and filtering algorithm to provide explainable reasoning in the case of Logical Neural Networks (LNNs), a special type of Neural-Theorem Prover (NTP). Link \u00bb Thabang Lebese \u00b7 Ndivhuwo Makondo \u00b7 Cristina Cornelio \u00b7 Naweed A Khan \ud83d\udd17 - A Genetic Programming Approach To Zero-Shot Neural Architecture Ranking (Poster) []\u00a0 \u00a0link \u00bb Neural networks are becoming increasingly ubiquitous in a wide range of use cases. A primary hurdle in deploying neural networks in many scenarios is the tedious and difficult neural network architectural design process, which was reliant on expert knowledge and iterative design. Neural Architecture Search (NAS) reduces the human effort required for design, but still has considerable resource requirements and is extremely slow. To address the inefficiencies of conventional NAS, Zero-Shot NAS is a new paradigm, which introduces zero shot neural architecture scoring metrics (NASMs) to identify good neural network designs without training them. While applying Zero Shot NASMs is cheap and requires no training resources, we identify that there is a lack of NASMs that generalize well across neural architecture design spaces. In this paper, we present a program representation for NASMs and automate its search with genetic programming. We discover effective NASMs for Image Classification as well as Automatic Speech Recognition. We believe that our work indicates a new direction for NASM design and can greatly benefit from recent advances in program synthesis. Link \u00bb Yash Akhauri \u00b7 Juan Munoz \u00b7 Ravishankar Iyer \u00b7 Nilesh Jain \ud83d\udd17\n\n#### Author Information\n\n##### Guy Van den Broeck (UCLA)\n\nI am an Assistant Professor and Samueli Fellow at UCLA, in the Computer Science Department, where I direct the Statistical and Relational Artificial Intelligence (StarAI) lab. My research interests are in Machine Learning (Statistical Relational Learning, Tractable Learning), Knowledge Representation and Reasoning (Graphical Models, Lifted Probabilistic Inference, Knowledge Compilation), Applications of Probabilistic Reasoning and Learning (Probabilistic Programming, Probabilistic Databases), and Artificial Intelligence in general.","date":"2022-12-02 18:50:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.344417005777359, \"perplexity\": 3201.3508394376727}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710916.40\/warc\/CC-MAIN-20221202183117-20221202213117-00393.warc.gz\"}"}
| null | null |
Kentish Council är en kommun i Australien. Den ligger i delstaten Tasmanien, i den sydöstra delen av landet, omkring 180 kilometer nordväst om delstatshuvudstaden Hobart. Antalet invånare var vid folkräkningen 2016.
Följande samhällen finns i Kentish:
Sheffield
Railton
Barrington
Wilmot
Nook
Claude Road
I övrigt finns följande i Kentish:
Bare Mountain (ett berg)
Bell Mountain (ett berg)
Black Bluff (ett berg)
Black Range (ett berg)
Bond Peak (ett berg)
Bonneys Tier (ett berg)
Gog (ett berg)
Little Horn (ett berg)
Mount Claude (ett berg)
Mount Jacob (ett berg)
Mount Kate (ett berg)
Mount Misery (ett berg)
Mount Roland (ett berg)
Prospect Mountain (ett berg)
Rocky Mountain (ett berg)
Round Mountain (ett berg)
The Badgers (ett berg)
Tin Spur (ett berg)
Weindorfers Tower (ett berg)
Källor
Indelningar i Tasmanien
Kontrollbehov inkommande wikilänkar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,140
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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{
"redpajama_set_name": "RedPajamaGithub"
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package net.minecraft.util;
import net.minecraft.entity.Entity;
import net.minecraft.entity.EntityLivingBase;
import net.minecraft.item.ItemStack;
public class EntityDamageSourceIndirect extends EntityDamageSource
{
private Entity indirectEntity;
public EntityDamageSourceIndirect(String p_i1568_1_, Entity p_i1568_2_, Entity indirectEntityIn)
{
super(p_i1568_1_, p_i1568_2_);
this.indirectEntity = indirectEntityIn;
}
public Entity getSourceOfDamage()
{
return this.damageSourceEntity;
}
public Entity getEntity()
{
return this.indirectEntity;
}
/**
* Gets the death message that is displayed when the player dies
*/
public IChatComponent getDeathMessage(EntityLivingBase p_151519_1_)
{
IChatComponent ichatcomponent = this.indirectEntity == null ? this.damageSourceEntity.getDisplayName() : this.indirectEntity.getDisplayName();
ItemStack itemstack = this.indirectEntity instanceof EntityLivingBase ? ((EntityLivingBase)this.indirectEntity).getHeldItem() : null;
String s = "death.attack." + this.damageType;
String s1 = s + ".item";
return itemstack != null && itemstack.hasDisplayName() && StatCollector.canTranslate(s1) ? new ChatComponentTranslation(s1, new Object[] {p_151519_1_.getDisplayName(), ichatcomponent, itemstack.getChatComponent()}): new ChatComponentTranslation(s, new Object[] {p_151519_1_.getDisplayName(), ichatcomponent});
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,444
|
\section*{Introduction}
As already pointed out in the 18th century, exponential growth is the most prominent feature of population dynamics \cite{malthus1872essay}, and bacterial systems are probably the best-studied model system about exponential growth. However, as pointed out by J. Monod \cite{monod1949growth}, the exponential growth is only one of the growth phases of bacteria, and the stationary phase, the death phase, and the lag phase are all as important for the bacterial population dynamics.
A variety of theoretical studies on the bacterial population dynamics tend to focus on the competition for nutrients under constant environment \cite{gause1969struggle,pianka1970r}, where the competition takes place mainly in the form of exponential growth under a constant influx of substrate (combined with dilution/death to keep the environment constant). While these models have provided fundamental insights into bacterial population dynamics, in natural environments like ponds, soils, and puddles, the nutrients may be supplied by rarely happening events rather than continuous influx. Under such natural environments, bacteria experience substrate rich conditions and poor conditions alternately.
This cycle between substrate rich and poor conditions is called the feast-famine cycle \cite{hengge1993survival,simon2002microbial,datta2016microbial,kellerman2014chemodiversity,seymour2017zooming,savageau1983escherichia,merritt2018frequency}. In contrast to the continuous nutrient supply (or the constant environment) condition, under the feast-famine cycle there is no steady-state for the amount of the substrate, and accordingly for the number of the cells. While the environment becomes substrate-rich for some time after the substrate addition event, once the cells in the environment run out all the substrates, they have to tolerate until next substrate addition event which is typically highly stochastic.
The feast-famine cycle is more than just a fluctuating environment, in that the rate of the substrate consumption affects the feast and famine period. Cells starve until the substrate is supplied, and once the environment gets substrate-rich, the cells use it quickly. During the feast (substrate-rich) period the growth of the cells changes the state of the environment. If cells use the substrate slowly, the feast period lasts longer and vice versa.
In this sense, the feast-famine cycle is one example that cells are not just affected by the environmental condition, but also changing the environment. While the population dynamics without such feedback are well studied \cite{kussell2005phenotypic,kussell2005bacterial}, our understandings with the feedback between the environment and the population are still underdeveloped.
\textcolor{black}
In the seminal Lenski and coworkers' experiment of long-term bacterial evolutionary adaptation \cite{lenski1991long}, bacteria have been diluted into fresh media every 24 hours while the growth reaches the saturation after less than 10 hours, hence the cells do experience repeated feast-famine cycles. However, the famine period is rather short that there is no visible death during the period, and the major observed adaptation was to increase the growth rate and decrease lag-time during the feast period, without a significant increase of the death rate during the famine period \cite{vasi1994long}. Interestingly, however, in a separate experiment, Vasi and Lenski had isolated mutants after the $30$ to $49$ days starvation \cite{vasi1999ecological}, and most of them are found to be inferior in competing with their progenitors in fresh medium, but having better resistance to starvation. One of the five strains clearly showed an inferior fitness to its original strain for a one-day growth competition but had superior survivability in 15-day starvation. These observations suggest that evolutionary adaptation under repeated feast-famine cycles with long-enough famine periods could result in fairly non-trivial results, especially if there is a trade-off between survivability in the long famine period and the growth rate in the feast period. It is also worth mentioning that, the trade-off of growth rate and survivability has been clearly documented for {\it E. coli} mutants with genetically manipulated $rpoS$ activity \cite{yang2019temporal}; the {\it rpoS} gene encodes for the stationary/starvation sigma factor $\sigma^S$ and known to cause trade-offs in multiple stressed conditions \cite{notley2002rpoS,maharjan2013form}.
}
\textcolor{black}
In the previous works, the trade-off between the growth rate and the growth yield are well-documented \cite{jasmin2012yield,pfeiffer2001cooperation,ferenci2016trade,novak2006experimental,maclean2008tragedy,kreft2004biofilms,frick2003example}. Fast-growth but low-yield results in an extended starvation time and a reduction of the population size at the end of the feast period. It has been theoretically predicted that a fast-growth but low-yield species will wipe out a slow-growth but high-yield species in a well-mixed environment \cite{pfeiffer2001cooperation,kreft2004biofilms,frick2003example}. However, the effect of the trade-off between the growth rate and the survivability has not been theoretically analyzed yet.
}
\textcolor{black}
In this paper, we construct a population dynamics model where the population growth is driven by a discrete and stochastic substrate addition events. In order to understand the simplest case, we consider a well-mixed system with a single niche, i.e, there is only one kind of nutrient/substrate that bacteria can consume to grow. Bacteria cells divide by consuming substrates, and if there is no substrate the cells die at a constant rate. In addition to the growth/death dynamics, mutations take place to change the growth rate. We introduce a trade-off between the growth rate and the death rate or the growth yield: The fast-growing cells have a higher chance to win the competition for the nutrients in the feast period, but instead, are less tolerant in the famine period because of high death rates or a small number of the population due to the low yield.
}
\textcolor{black}
Using the model, we show that the bacterial population faces the Tragedy of The Commons (TOC)" type phenomenon \cite{hardin1968tragedy} under feast-famine cycles; the fast-grower tends to take over during the feast period, making the population less tolerant for the famine period. This evolutionary dilemma drives the population to faster growth and eventual extinction in the famine period for the growth-yield trade-off as previously predicted \cite{pfeiffer2001cooperation}, but we find that in the growth-death rate trade-off case, the long-term outcome depends on the functional form of the trade-off. By analyzing the model, we show that this is because the effective fitness under repeated feast-famine cycle is determined by the ratio of the growth rate and the death rate. We then focus on the non-trivial case of the growth-death rate trade-off where functional form drives the population to extinction in repeated feast-famine cycles, and show that the TOC effect can be prompted by increasing the amount of substrate added to the environment. Finally, we discuss the model assumptions and possible extensions in comparison with experimental results in the literature.
}
\section*{Model}
The model consists of a state vector $\vec{X}$ defined by $(M+1)$ integers that denote the population of $M$ species (either genotypes or phenotypes) and the number of substrates in the environment. It is expressed as $\vec{X}=(N_0,N_1,\cdots,N_{M-1},S)$, where $N_i$ is the number of the $i$th bacteria, and $S$ represents the number of the substrates. All elements of the state vector ${\vec X}$ are non-negative integers. Each species can have a different growth rate, growth yield, and death rate.
A single individual of the $i$th species proliferates at a constant rate $\mu_i$ being given by $\mu_i=(i+1)\Delta\mu$ if $S$ is larger than zero,
while it dies at rate $\gamma_i$ under the starving condition. Each species has a different growth yield $Y_i$, and $\lceil 1/Y_i\rceil$ substrates are consumed when a single bacterium of the $i$th species divides where $\lceil \cdot \rceil$ is the ceiling function. The number of the substrate in the environment is recovered to $S={S_m}$ when the substrate addition event takes place at a constant rate of $1/\lambda$.
We introduce the mutation among species to the model. It occurs with probability $\rho$ when an individual divides, and then, the daughter cell of the $i$th species becomes either the $(i-1)$th or $(i+1)$th species in an equal probability.
By assuming that each event occurs as the Poisson process, the master equation for the simplest one species case is given by
\begin{eqnarray}
\frac{dP(N,S)}{dt}&=&\mu(N-1)P(N-1,S+\Delta S) - (1-\theta(S))\mu N P(N,S)\nonumber\\
&+&\theta(S)\gamma\Bigl((N+1)P(N+1,0)-NP(N,0)\Bigr)\label{eq:single}\\
&-&P(N,S)/\lambda+\delta_{S,{S_m}}\sum_{i=-\infty}^{{S_m}}P(N,i)/\lambda \nonumber
\end{eqnarray}
where $P(N,S)$ is the probability of the state with $N$ bacteria and $S$ substrates. $P(N,S)=0$ holds if $N$ is smaller than zero or $S$ is larger than ${S_m}$. $\delta_{i,j}$ is Kronecker's delta, and $\hat{\delta}_{i,j}$ is given by $(1-\delta_{i,j})$.
$\theta(S)$ is unity for $S\leq 0$ else zero. $\Delta S$ is given as $\lceil 1/Y\rceil$ (the results do not change by modifying the model so that the number of the substrate to be consumed is determined stochastically with the average consumption per division as $1/Y$.). The states with negative values of $S$ are able to have non-zero values, while the right hand side of the equations for the states with negative $S$ are the same with that of $S=0$. We regard the sum of the probabilities with $S\leq 0$ as the probability of the starving state. This model construction is to avoid dividing the equation into cases that $S$ is larger and smaller than $\Delta S$ (for detailed description, see Appendix A).
Fig.\ref{fig:fig1}a shows a realization of stochastic dynamics generated by the present model Eq.(\ref{eq:single}) with the Gillespie algorithm \cite{gillespie1977exact}. The oscillation in the number of bacteria results from the feast-famine cycle.
\section*{Results}
\subsection*{The Tragedy of Commons dilemma in the bacterial evolution under the feast-famine cycle}
We study the effect of the feast-famine cycle in the multi-species system The master equation for $M\ (M>1)$ species system is obtained by just extending Eq.(\ref{eq:single}) for $M$ species and introducing mutation among the species. There is no direct interaction among the species, but the species interact via the competition for the substrate. For the exact expression, see Eq.~(\ref{eq:meq_multi}) in Appendix. We compare the dynamics of the model with the growth rate-yield ($\mu-Y$) and the growth rate-death rate ($\mu-\gamma$) trade-off separately.
We carried out stochastic simulations of the $M$-species model (Eq.(\ref{eq:meq_multi})) with a fixed $\lambda$ and ${S_m}$ value using the Gillespie algorithm. Fig.\ref{fig:fig1}b and c show evolution time courses with several trade-off relationships. There are initially $N_{\rm ini}$ cells with the lowest growth rate $\mu_0$ and the corresponding yield or the death rate, and ${S_m}$ substrates. In the $\mu-Y$ trade-off case (Fig.\ref{fig:fig1}b, with trade-off $Y=1/(\kappa+\mu)$ with a constant $\kappa$), the population-averaged growth rates keep increasing by evolution, and eventually, the whole population collapses. It is consistent with previous reports arguing that under the growth-yield trade-off, the species with a higher growth rate outcompetes others even though it leads to a reduction of the population size due to the small yield \cite{pfeiffer2001cooperation,kreft2004biofilms,frick2003example}. On the other hand, in the $\mu-\gamma$ trade-off case, the evolution of the growth rate can either leads to population collapse (Fig.\ref{fig:fig1}c top, with linear trade of $\gamma=a+b\mu$ with constant $a$ and $b$) or reach a steady state depending on the form of the trade-off (Fig.\ref{fig:fig1}c bottom, with square trade-off $\gamma=a+b\mu^2$).
To understand the differences of the outcome, we constructed a simplified version of the model (Eq.(\ref{eq:meq_multi})) which allows us analytical calculations. We approximate that the population size is a continuous quantity and consider deterministic growth and death. We denote the number of the $i$th species right before the $n$th substrate addition event by a continuous variable $N_i(n)$. Also, we regard $S$ as a continuous variable, and thus, remove the ceiling function from the yield.
After the $n$-th substrate addition, the species grow exponentially until all the substrate runs out. The length of the feast period after the $n$th addition event, $\tau(n)$, is determined by ${S_m}=\sum_{i=0}^{M-1}N_i(n)/Y_i(\exp[\mu_i\tau(n)]-1)$, because the increment of the total population divided by the growth yield should sum up with the added substrate ${S_m}$. If the interval between the $n$th and the $(n+1)$th addition events is longer than $\tau(n)$, the cells experience the famine period to die at the rate $\gamma_i$. Thus, the number of cells right before the $(n+1)$th substrate addition event is given by
\begin{equation}
N_i(n+1)=
\begin{cases}
N_i(n)e^{\mu_i \tau(n)}e^{-\gamma_i(\Delta t(n)-\tau(n))}\ (\tau(n)<\Delta t(n))\\
N_i(n)e^{\mu_i \Delta t(n)}\ ({\rm otherwise}),\\
\end{cases}
\end{equation}
where $\Delta t(n)$ is the stochastic variable representing the interval between the $n$th and the $(n+1)$th addition events, which follows the exponential distribution with average $\lambda$. By taking average of the effective growth rate, $\ln(N_i(n+1)/N_i(n))$, over the exponential distribution, we obtain a deterministic, discrete map system which describes the dynamics of the population growth as
\begin{eqnarray}
N_i(n+1)&=&N_i(n)\exp(\hat{\mu}_i\lambda)\nonumber \\
\hat{\mu}_i(n)&=&\mu_i\Bigl(1-\exp(-\tau(n)/\lambda)\Bigr)-\gamma_i\exp(-\tau(n)/\lambda).\label{eq:Mmap}
\end{eqnarray}
In Appendix, we show that the map dynamics has at least $M$ fixed points that only one species exists and the number of cells is zero for the other species, which is given by
\begin{equation}
N_i^{\rm st}(\mu_i)=\frac{{S_m} Y_i}{(1+\gamma_i/\mu_i)^{\mu_i \lambda}-1}.\label{eq:sol_map}
\end{equation}
The linear stability analysis for the fixed points showed that only one fixed point among the $M$ fixed points is stable. The condition for the fixed point to be stable is to have the largest ratio of the growth rate to the death rate $\mu_i/\gamma_i$ (see Appendix).
In the $\mu-Y$ trade-off case, the death rate $\gamma_i$ has no index dependency, and thus, the stability or the fitness is simply determined by the growth rate. There, the growth yield $Y_i$ has no effect on the competition among the species, and thus, the bacterial population evolves to increase the growth rate without caring of how the growth is efficient in terms of the substrate consumption.
In contrast, the stability criterion tells us that the functional form of $\mu-\gamma$ trade-off affects whether the evolution lasts until the whole population goes extinct or not. When the trade-off is linear ($\gamma=a+b\mu$), the ratio $\mu/\gamma$ has no upper bound, while the ratio is bounded in the square trade-off ($\gamma=a+b\mu^2$) case. Therefore, in the square trade-off case, once the growth rate reaches the optimal point (the maximum $\mu/\gamma$), the system stays at that state. On the contrary, due to the lack of the maximum, the growth rate will never stop increasing for the linear trade-off case and leads to the population collapse.
In both the growth-yield trade-off and the linear growth-death trade-off case, the growth rate increases over generation while the tolerance to the famine period gets worse due to the decrease or increase of the yield or the death rate. The evolution then leads to the reduction of total population size. Eventually, the population size becomes too small to tolerate fluctuations of the substrate addition time, and it results in the extinction of the whole population when the substrate addition times are longer than usual by chance. This can be seen as one of the typical consequences of "The Tragedy of The Commons" Dilemma.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 140 mm, angle = 0]{Fig1.eps}
\caption{(a). An example of the dynamics of one-species model (Eq.(\ref{eq:single})). The number of bacteria oscillates driven by the feast-famine cycle. (b). Evolution simulation with the growth-yield trade-off model. $Y=1/(\kappa+\mu)$ where $\kappa$ is introduced to avoid the divergence of the function. The population-averaged growth rate keeps increasing and the whole population extinct at the point indicated by the black arrow. (c). Evolution simulations with two different choices of the growth-death trade-off. (top) With the linear trade-off, $\gamma=a+b\mu$, the population-averaged growth rate keeps increasing and the whole population extinct at the point indicated by the black arrow. (bottom). On the other hand, the growth rate and the number of bacteria get stable at a certain value with the square trade-off, $\gamma=a+b\mu^2$. The stable growth rate is predicted as $\sqrt{b/a}=0.1$ by the analysis of Eq.(\ref{eq:Mmap}) which corresponds well with the numerical result. While the same parameter values of $a$ and $b$ are used for the two trade-off relationships, the outcome does not change qualitatively even if different values are used for them. The parameters are set to $\mu=1,\gamma=0.101,\lambda=10,$ and ${S_m}=10^3$ for (a), and $a=10^{-3},b=0.1,\delta\mu=10^{-2},\kappa=10^{-2},\rho=10^{-3},N_{\rm ini}=100,\lambda=1.28$, ${S_m}=128$, $\gamma=0.15$ (only for (b)) and $Y=1$ (only for (c)) for (b) and (c). }
\label{fig:fig1}
\end{center}
\end{figure}
\subsection*{Impact of the feast-famine cycle for the survival of the bacterial population}
In the previous section, we have seen that under a repeated feast-famine cycle, the TOC dilemma is evoked if the trade-off relationship is such that the death rate increase linearly or slower than linear with the growth rate, or when there is a growth rate-yield trade off. In the following sections, we study how the "degree" of the feast-famine cycle affects the survival of the bacterial population and the evolutionary dynamics when the TOC dilemma is evoked. We chose to focus on the less trivial case of linear growth-death trade-off model
$\gamma(\mu)=a+b\mu$
in the following.
Firstly, we introduce the "degree" of the feast-famine cycle. In the following, we change the value of the average famine period $\lambda$ while keeping the time-averaged substrate supply ${\bar S}={S_m}/\lambda$ constant. With this constraint, the change in $\lambda$, and accordingly in ${S_m}$, controls the severeness of the feast-famine cycle. A large $\lambda$ (and ${S_m}$) value indicates that a large amount of the substrate is supplied to the environment less often, corresponding to the severe feast-famine cycle. On the contrary, the limit of $\lambda\rightarrow0$ with keeping $\bar S$ constant would correspond to the continuous substrate-supply limit, though strictly speaking in the present model this limit cannot be taken due to the discreteness of ${S_m}$.
We compared the dynamics under the different degrees of the feast-famine cycle. Fig.\ref{fig:fig2}a shows two time courses of the population and the averaged growth rate under a moderate (top panel, $\lambda=1.28$) and a severe (bottom panel, $\lambda=81.92$) feast-famine cycle. The population-averaged growth rates commonly evolve to increase over time, and eventually, the whole populations go extinct as expected from the foregoing analysis. The difference of the degree of the feast-famine cycle appears in the length of time to extinct, which we call the survival time $T_s$, and the population average growth rate just before the extinction, which we call the critical growth rate $\langle \mu_c\rangle $.
We plotted the survival time $T_s$ and the critical growth rate $\langle \mu_c\rangle $ as a function of the degree of the feast-famine cycle ($\lambda$) in Fig.\ref{fig:fig2}b. The survival time and the critical growth rate decrease as $\lambda$ increases, reflecting the harsher environment. $\langle \mu_c\rangle$ decreases approximately proportional to $1/\lambda$, while interestingly the survival time $T_s$ shows a cross-over from $T_s\propto \lambda^{-2}$ to $T_s\propto \lambda^{-1}$ at $\lambda\approx 10$.
Qualitatively, the shorter survival time $T_s$ and the smaller critical growth rate $\langle \mu_c \rangle$ with increasing
$\lambda$ is the reflection of the asymmetry of the growth and the death in this setup. Increasing $\lambda$ increases the possible population growth per feast period linearly with $\lambda$ because of the increase of $S_m=\bar S \lambda$, but the death in the famine period affects the population exponentially as a factor $\exp(-\gamma \lambda)$. Clearly, the death effect is dominant, hence it is harder to survive with longer $\lambda$, resulting in shorter $T_s$ and smaller $\langle \mu_c \rangle$.
Quantitative analysis requires more careful consideration, which we present in the next section.\\
\begin{comment}
We consider the simplified version of the model (Eq.(\ref{eq:Mmap})) again to qualitatively understand the observed $\lambda$ dependence. When there is only one species with growth rate $\mu$ and the death rate $\gamma(\mu)$, the
non-zero steady-state solution of the population $N^{\rm st}(\mu)$ is given by
\begin{equation}
N^{\rm st}(\mu)=\frac{{S_m}}{(1+\gamma(\mu)/\mu)^{\mu \lambda}-1}=\frac{\bar{S}\lambda}{(1+\gamma(\mu)/\mu)^{\mu \lambda}-1}.\label{eq:sol_map}
\end{equation}
Clearly, this is a decreasing function of $\lambda$. When the the degree of the feast-famine cycle is increased by increasing $\lambda$, the increase of $S_m$ will increase the population size linearly with $\lambda$ (numerator of eq.~\ref{eq:sol_map}), but the prolonged famine period results in the exponential death of the population (denominator of eq.~\ref{eq:sol_map}). Therefore, in total, the number of the cells decreases for larger $\lambda$, i.e., for severer feast-famine cycles.
In addition, with the linear trade-off between the growth rate and the death rate, it is easy to see that the steady-state solution $N^{\rm st}$ is a decreasing function of $\mu$. Assuming that the extinction should occur when $N^{\rm st}(\mu)$ is less than a certain population, the critical growth rate that $N^{\rm st}(\mu)$ fall just below one should decrease with $\lambda$.
Of course, the system does not approach the critical growth rate without species competition. When there is only one species, the optimal growth rate is determined by the rate that enables the population to use all the added nutrient $S_m$ just when the famine periods end. If the growth rate is faster than that, the death in the famine period reduces the population, while if the growth rate is slower than that, there will be some nutrient left when the next feast period starts to keep accumulating as the feast-famine cycle repeats. With the competition of multispecies, however, makes the growth rate keep increasing with the linear trade-off between $\mu$ and $\gamma$ as discussed in the previous section, and make the system to finally collapse by hitting the critical growth rate, which is a decreasing function of $\lambda$ as discussed above. This also somewhat explains the decrease of the survival time with $\lambda$; since the critical growth rate decreases with $\lambda$, it will be faster to evolve to get to the critical growth rate.
This argument qualitatively explains the decrease of the survival time under the strong feast-famine cycle. However, estimating the critical growth rate by equating $N^{\rm st}(\mu)$ in eq.~(\ref{eq:sol_map}) to a small critical number does not give quantitatively correct scaling of $\langle \mu_c \rangle$ vs. $\lambda$, because of the .....
\end{comment}
\begin{figure}[htbp]
\includegraphics[width = 150 mm, angle = 0]{Fig2.eps}
\begin{center}
\caption{(a). The time courses of the total population and the average growth rate. (top) Under a weak feast-famine condition ($\lambda=1.28$), and (bottom) a strong feast-famine condition ($\lambda=81.92$). Extinction takes place in much shorter time under the strong feast-famine condition than the weak one. (b). The averaged extinction time is plotted with the standard deviation for several $\lambda$ values. The average and the standard deviation are computed from $128$ extinction events. ${S_m}$ is given as ${S_m}=\lambda\bar{S}$. The two slopes were obtained by fitting. The population-averaged growth rate achieved is also plotted in the inset. Parameter values are set to be $a=10^{-3},b=0.1,\delta\mu=10^{-2},\rho=10^{-3},$ and $\bar{S}=100$.}
\label{fig:fig2}
\end{center}
\end{figure}
\subsection*{Crossover from the directed evolution to neutral evolution}
In order to have a better understanding of the observed behavior, we now focus on the cross-over of the survival time $T_s$ shown in Fig.\ref{fig:fig2}b from being approximately proportional to $\lambda^{-2}$ to $\lambda^{-1}$.
In the moderate feast-famine cycle ($\lambda \ll 10$) depicted in the top panel of Fig.\ref{fig:fig2}a, it appears that the evolution speed of the growth rate has two regimes: The evolution speed significantly slows down after the average growth rate reaches $\approx 1$, which happens around time $1\times 10^7$ in this example. Since the mutation probability is constant, we hypothesized that the dynamics of how a new species takes over the majority of the population changes with the average growth rate.
To quantify this change of the dynamics, we studied the dynamics of taking-over among species by setting $N_{\rm ini}$ cells of the $i$th phenotype at $t=0$, and ran the population dynamics until the dominant species becomes another.
For the simulations performed in this section, we add the spontaneous migration term to the model. The spontaneous migration makes it possible to increase the number of the cells by one without consuming any nutrient even under the starved condition (for the detailed expression of the term, see Eq.(\ref{eq:meq_multi})). An introduction of the term is a mathematical treatment for gathering many trajectories efficiently for better statistics. Since the spontaneous migration rate $\epsilon$ is set to be sufficiently smaller than any other parameters in the model, it is effectively the same as introducing one cell into the system when the population extinct.
From this computation, we obtained the transition probabilities from the $i$th species to another.
With our default parameter set, it never happened that the species other than the nearest neighbors of $i$ becomes dominant before $i-1$th or $i+1$th dominates the system. Thus, the obtained transition probabilities were for increasing the growth rate by $\Delta \mu$ (probability $p(\mu)$) or decreasing it by $\Delta\mu$ (probability $1-p(\mu)$). Fig.\ref{fig:fig3}a shows the asymmetry of the probability for increasing/decreasing the growth rate,
defined by the difference between the two probabilities $(2p(\mu)-1)$. As clearly seen, the evolution of the growth rate takes place in a directed manner up to $\mu\approx 1$, whereas the dynamics of the evolution resembles the random walk when $\mu\gg 1$. This qualitative difference in the evolution dynamics above and below $\mu\approx 1$ may consistently describe the crossover of the survival time $T_s$, which is happening at around $\langle \mu_c\rangle \approx 1$. Namely, when the critical growth rate $\langle \mu_c\rangle$ is below one for long enough $\lambda$, the extinction happens relatively quickly since the growth rate systematically increase through the evolution, but when $\langle \mu_c\rangle$ is above one, the evolution takes a lot longer time due to the diffusive behavior, hence survival time $T_s$ grows faster as decreasing $\lambda$.
Where does this transition from the directed to neutral evolution come from? The simple map dynamics (Eq.(\ref{eq:sol_map})) just tells us that the species with the largest $\mu/\gamma$ dominates the population which does not explain the random walk-like behavior of the averaged growth rate. In order to gain more insights, we studied the detailed dynamics of the take over events of the population by dominant species in stochastic simulation.
Since the main purpose of this simulation was to ask how the dominant species changes one to another, we simulated the model with only two species which have a slightly different growth rate to each other.
One has a growth rate $\mu_l$ and the other has a higher growth rate, $\mu_h$ given as
$\mu_h=1.05\cdot \mu_l$. Fig.\ref{fig:fig3}b shows the time-averaged populations of the species are plotted against the growth rate of the slowly-growing species ($\mu_l$). The fast-growing species dominates the whole population and the number of individuals is much larger than that of the slow grower in the small $\mu_l$ region, whereas the difference of the population sizes of the two species shrinks as $\mu_l$ increases and it gets indistinguishably small at $\mu_l\approx 1$. Examples of time courses are plotted in Fig.\ref{fig:fig3}c. The dynamics with large $\mu_l$ shows rather stochastic changes between the fast-grower dominating and slow-grower dominating states, while with small $\mu_l$ the fast grower is stably dominates the system.
This shrinkage of the gap in the two populations explains the transition from the directed to the neutral evolution of the growth rate. Intuitively, the mechanism of this shrinkage can be understood by considering the effective fitness $\mu/\gamma$. Since we use the linear trade-off $\gamma(\mu)=a+b\mu$,
the difference of the effective fitness between the fast species with the growth rate $\mu_h=(1+\delta)\mu_l$ and
the slow species with the growth rate $\mu_l$ is given by $\mu_h/\gamma(\mu_h)-\mu_l/\gamma(\mu_l)=\delta a \mu_l/[(a+b\mu_l)(a+b(1+\delta )\mu_l]$, which approaches zero as $\mu_l$ increases. In other words, the larger the value of $\mu_l$ is, the harder it becomes for the fast species to take over the population. As a result, the growth rate performs almost a random walk through evolution for the large value of $\mu_l$.
The more quantitative understanding can be obtained by applying the Wright-Fisher (WF) model \cite{ewens2012mathematical}. The WF model is a stochastic model describing temporal changes of the population structure such as the fixation probability and the fixation time. While the set-up of the present model does not fully fit the WF framework, we can apply the framework to the model with some assumptions which are described in the Appendix. The WF framework enables us to calculate the probability of the fast grower to be fixed in the population under no-mutation no-migration condition. The fixation probability is given by
\begin{equation}
p_{\rm fix}=(1-\exp(-N_tuy))/(1-\exp(-N_tu)),\label{eq:fix_prob}
\end{equation}
where $N_t$, $y$ and $u$ represents the total number of the cells, the initial fraction of the fast growers, and the relative fitness of the fast grower defined as $u=(\mu_h/\gamma_h-\mu_l/\gamma_l)/(\mu_l/\gamma_l)$, respectively.
Fig.\ref{fig:fig3}d shows the comparison between the fixation probabilities computed by the simulation of the present model when the initial fraction of the fast-growing population $y$ set to 0.05 \footnote{To compute the fixation probability of the present model in the stochastic simulation, we set the mutation rate and the migration rate to zero, and run the dynamics from the fixed initial value of $N_l$, $N_h$, and $S$. A single run finishes if one of them extincts, and it is repeated with different random number seeds to compute the fixation probability.} and $p_{\rm fix}$ from the WF model in Eq.~(\ref{eq:fix_prob}). In the WF model, the total population size $N_t$ is replaced by the steady-state average population size of one species case for the given parameters, calculated from master equations with assuming long enough $\lambda$ so that the system typically reaches zero nutrient state in the famine period (Appendix D). The two results show good correspondence. From the analytic expression of the fixation probability obtained from the WF approach (Eq.(\ref{eq:fix_prob})), we can see that the decrease of the fixation probability is led by two effects, namely, the decrease of the relative fitness advantage and the population size-effect. One effect is the form of $u$ being a decreasing function of $\mu$, hence as discussed before the advantage of the fast growth is reduced even the population size stays constant. In addition, the population size shrinks as the growth rate increases, and it makes the population dynamics noisier, making the small fitness difference no longer be the determinant of the dynamics. The two effects similarly contribute to the change of the fixation probability as shown by the dashed lines in Fig.\ref{fig:fig3}d, where the fixation probability Eq.~(\ref{eq:fix_prob}) with a constant relative fitness $u$ or a constant total population $N_t$ are also plotted.
The WF model also in principle shows the parameter dependence of the crossover point, which should correspond to the point where fixation probability is sufficiently close to $1/2$ when starting from the equal population ($y=1/2$).
The closed-form is difficult to obtain because the complex dependence of $N_t$ on $\mu$, but the form indicates that the crossover growth rate depends on the trade-off function parameter values ($a$ and $b$).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 100 mm, angle = 0]{Fig3.eps}
\caption{(a). The asymmetry of the evolution. The evolution of the growth rate is directed up to $\mu\approx1$, while the growth rate behaves similarly to the random walker in a larger $\mu$ region. (b). The averaged population is plotted against the growth rate of the slow grower ($\mu_l$) with the error bar as the standard deviation. The growth rate of the fast grower is set to be $5\%$ higher than that of the slow grower. (c). Examples of time courses with $\mu_l=0.01$ (top) and $0.64$ (bottom). (d). The comparison of the fixation probabilities obtained from the numerical simulation and the calculation of the WF model. The comparison of the fixation time is also shown in the inset. For (d), we set $\epsilon=\rho=0$ so that one of the two species eventually be fixed. We choose the initial value of the total number of the bacteria as $N_{\rm st}(\mu_l)$ given in Eq.(\ref{eq:sol_meq}), and the initial fraction of the species with faster growth rate as $5\%$. Parameters are set to be $\lambda=1.0,{S_m}=100,a=10^{-3},b=10^{-1},\rho=10^{-3}$, and $\epsilon=10^{-8}$.}
\label{fig:fig3}
\end{center}
\end{figure}
\subsection*{Supplying more substrates leads to the quick extinction}
Finally, we show that an increase of the substrate supply ${S_m}$ with fixed $\lambda$, hence increasing the average nutrient supply $\bar S$, decreases the survival time $T_s$. This is shown in Fig.~\ref{fig:fig4}, where the survival time $T_s$ is plotted as a function of ${S_m}$ with constant $\lambda$. For the small ${S_m}$ region, the survival time of the population increases as ${S_m}$ gets larger because the amount of the substrate supply is too small at the left edge of the horizontal axis. On the other hand, the further increase of ${S_m}$ shortens the survival time even though the waiting time is kept constant meaning that the total supply of the substrate increases.
To ask what makes the survival time shorter, we estimated the survival time analytically. The survival time is approximated by the time needed to evolve the growth rate from the initial low value to the critical value with which the average population is close to one. We hypothesized that the survival time consists of the two main parts, namely, the time for gaining the new species by mutation $\tau_m$ and the time for the new species to take over the whole population.
The bacterial population needs to wait that the fast grower appears by mutation to evolve. The mean time of the emergence of the grower by mutation is given as $\tau_m=1/(1-(1-\rho/2)^N)\approx2/\rho N$, where $N$ is the population size which depends on the population structure. It is possible that the slow grower appears and takes over the population, but we ignore this small possibility for simplicity. Then, after the fast grower appears, it either takes over the population or is eliminated. The time needed for the take over ($\tau_h$) and elimination of the mutant ($\tau_l$) are estimated as the fixation time in the WF framework.
The fast grower is expected to fail the fixation $(1-p_{\rm fix})/p_{\rm fix}$ times on average, and new mutant need to appear at every fixation failure. Therefore, the time for the dominant species to become $n$ to $n+1$ is given by
\begin{equation}
T_{n,n+1}=\frac{1-p_{\rm fix}}{p_{\rm fix}}(\tau_m +\tau_l)+\tau_h.\label{eq:Ts}
\end{equation}
Note that $p_{\rm fix}$ , $\tau_h$, and $\tau_l$ are the functions of $\mu_n$ and $\mu_{n+1}$. In addition, at the every change of the dominant species that increase the average growth rate, the average population size decreases. This population size was already calculated in Appendix D from the master equation, and we assume that the extinction occurs when this population size becomes smaller than unity. Then, by summing up $T_{n,n+1}$ over $n$, until the population size reaches to unity, we obtain the estimate of the survival time.
With the present parameter values shown in Fig.\ref{fig:fig4}, the dominant term of $\sum_n T_{n,n+1}$ is the time for the fast grower appearance by mutation, i.e., $\Sigma(1-p_{\rm fix})/p_{\rm fix}\cdot \tau_m$. The comparison between the simulation and this expression is compared in Fig.~\ref{fig:fig4} shows a reasonable agreement. Also, the agreement indicates that the reduction of the survival time $T_s$ is mainly due to the increased rate of getting a fast-growing mutant with increasing ${S_m}$ because it increases the typical population size. The more detailed comparison with the rest of the terms is given in Appendix F. \\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 120 mm, angle = 0]{Fig4.eps}
\caption{The averaged survival time is plotted against ${S_m}$ with a constant $\lambda$ value ($\lambda=10$ for (a), and $\lambda=100$ for (b)). While the survival time increases with ${S_m}$, above a certain value of ${S_m}$, the further increase of ${S_m}$ causes the decrease of the survival time. The analytic estimate $\Sigma(1-p_{\rm fix})/p_{\rm fix}\cdot \tau_m$ is overlaid for each figure and captures non-monotonic behavior. Each point is obtained from $128$ independent evolution-extinction time courses and the error bars indicate the standard deviation. The parameters are set at $a=10^{-3},b=10^{-1},\rho=10^{-6}$, and $\delta \mu=10^{-2}$. $\epsilon$ is set at $0$ for the numerical simulation, while it is set at $10^{-12}$ for the analytic estimate because of the reason explained in the main text.}
\label{fig:fig4}
\end{center}
\end{figure}
\section*{Discussion}
Here, we developed a stochastic population dynamics model in which the vital substrates were supplied to the environment by discrete, stochastic events rather than continuously. This stochastic substrate addition separates the dynamics into two phases, namely, the feast and the famine phase. During the feast period with plenty of substrates, the cells with a higher growth rate increase their population more quickly than the others. On the other hand, during the famine period, the cells could not grow but die due to the lack of substrates.
With a trade-off between the growth rate and the growth yields the feast-famine cycle always led to the Tragedy of the Commons Dilemma-type result, while the outcome crucially depends on the form of the trade-off in the growth-death trade-off case.
The survival time, or the average time to extinction, was shown to have a power-law dependency to the average waiting time $\lambda$ with the constant time-averaged substrate supply $\bar{S}={S_m}/\lambda$, with a cross-over from close to $\lambda^{-2}$ to $\lambda^{-1}$. The cross over stemmed from the transition from the directed evolution to the undirected, random walk-like evolution dynamics. As the average growth rate increases, the difference in the fitness between two species with similar growth rates reduces. In addition, the population dynamics become noisier due to the decrease in the average population, and thus, the difference in fitness becomes less influential to the dynamics. Finally, it was shown that a pure increase of supplied substrate per nutrition addition event enhanced the extinction, and the reduction of the mutant appearance time due to increased population size was turned out to be the main part of the decrease of
For the cross-over from the directed to the random evolution to occur, the form of the trade-off is probably essential. With the square trade-off, one can choose the parameter values so that the evolution stops without enough shrinkage of the fitness gap and the population size, and then, only the directed evolution is expected. On the other hand, with the trade-off without the upper bound of the fitness, the population size becomes considerably small before the extinction. Also, the fitness gap always shrinks as the growth rate increases. By this argument, a cross-over being similar to what we presented above is expected to occur.
\textcolor{black}{
It is worth noting that the trade-off between the growth rate and death rate (killing rate) are well reported under a variety of stress conditions for bacteria \cite{zakrzewska2011genome,dong2009polymorphism,king2004regulatory,maharjan2013form,ferenci2016trade,porter2013trade,yang2019temporal,maharjan2013form,notley2002rpoS}, and the trade-offs between the growth and death rate are linear in some cases \cite{lee2018robust,tuomanen1986rate}. Also, the linear relationship between the growth- and the death rate is documented in the continuous culture of the fission yeast \cite{nakaoka2017aging}. With the linear trade-off, the evolution could not find an optimal point in the present model and the whole population faced extinction.
}
Undoubtedly, bacteria do not go extinct so easily as the present model has shown. Indeed, it is reported that the long-term evolution experiment of {\it E. coli} leads to the coexistence of the variety of bacterial genotypes \cite{lenski1991long,bouma1988evolution,good2017dynamics}
Since the present model is a simple toy model, there is a large gap between the model and the real experiment. For the further investigations of bacterial population dynamics under the feast-famine cycle, it might be fruitful to argue the differences between them.
First of all, the present model has no spatial degree of freedom. It is well-known fact that the introduction of the spatial structure will allow replicator models to have coexistence solutions \cite{mollison1977spatial,hufkens2009ecotones,mathiesen2011ecosystems,kerr2006local,schrag1996host,heilmann2010sustainability}. Introduction of the spacial structure was also proposed to avoid the TOC caused by the growth-yield trade-off \cite{pfeiffer2001cooperation}. Furthermore, it was experimentally shown that {\it E. coli} cells with the attenuated $rpoS$ outcompeted the wild-type cells in the well-mixed culture but coexisted with the wild-type in the spatially-structured culture \cite{hol2013spatial}. With the spatial structure introduced, it is possible that the present model shows the coexistence of multiple species that work to prevent extinction due to the TOC scenario.
Even without the effect of the space, it is possible that the physicochemical limits of the rates and the yield simply stop their values keep evolving. Another possible way to stop the extinction is to have the growth-death trade-off relationship being stronger than linear. But we would like to point out that having a stronger trade-off might be a weak strategy. Let us consider the situation where the trade-off relationship also evolves. In the population level, having the stronger trade-off relationship is preferential solution to avoid the extinction, but na\"{i}vely thinking, the bacterial population with a strong trade-off relationship is fragile to the invasion of another population with weaker trade-off relationship in the same sense as that slow growers are competed out by fast growers in the present model.
It is also worth noting that, in reality, a small fraction of the bacterial population can be non-growing persisters \cite{balaban2004bacterial,veening2008bistability,wakamoto2013dynamic}. The introduction of the persister phenotype makes the big jump between the high-growth and high-death and low-growth and low-death state possible, and it might change the evolutionary strategies. Also, the lag and stationary phases were not implemented in the model. The period of the lag phase is considered to increase with the starvation time (or the time in the stationary phase) \cite{merritt2018frequency,levin2010automated,himeoka2017theory}, and the prolonged lag time is clearly disadvantageous for the competition for substrates. The real bacteria might design their growth strategy also to cope with the length of lag time. Furthermore, it has been shown that during the death phase, the substrate provided by dead cells are utilized by alive cells to survive longer \cite{schink2019death}, and this feedback from cell death to the environment can be another factor to be considered to compare with the reality.
Another factor to consider is the stochasticity in the fixation process.
It is theoretically shown by using a Wright-Fisher-type model that the trade-off between the fitness advantage in WF scheme (e.g., the growth rate) and the carrying capacity (the maximum population of given species) makes a fixation probability of the species with smaller fitness advantage higher than that of the species with bigger fitness advantage under a certain condition \cite{houchmandzadeh2015fluctuation}. When a single cell of fast-growth and small carrying capacity is introduced into a community formed only of the other type, the chance of the former species to be selected is small due to a small frequency in the community. By contrast, in the opposite case, the population size of the community is small, and thus, the frequency and the fitness advantage are less influential to the selection. There, the randomness dominates the selection process rather than other factors.
The effect of the feast-famine cycle to the bacterial evolution has been addressed experimentally, though typically the famine period is the order of a day or shorter that the bacteria enters the stationary phase but not the death phase. In Lenski and coworkers' evolutionary adaptation experiment \cite{lenski1991long,wiser2013long,good2017dynamics,blount2012genomic,rozen2000long,blount2008historical} has been continued for more than 30 years and a variety of mutations have been confirmed. For instance, some of the new genotypes can use citrate as a sole carbon source which the ancestral strain was unable to utilize \cite{blount2008historical,blount2012genomic} and cross-feeding polymorphism to appear \cite{rozen2000long}. Those observations mean that new niches were created during the evolution time series. As only a single nutrient is considered in the present model, there is only one niche and the possibility of the cross-feeding was not taken into account. An extension of the model to have more than two types of the nutrient and secretion of the chemicals from the cells will allow the model to have more niches and hence polymorphism to appear.
In a recent experiment of bacteria under repeated feast-famine cycle with the exchange of medium \cite{merritt2018frequency}, a large amount of the bacteria cells were also flushed out when the fresh media is added. In this case, the cells that form aggregates are selected, even though the growth was in the liquid culture. Though this particular experiment imposed the selection for aggregates, in general one should have in mind that the aggregation and hence spatial heterogeneity can arise even in a liquid culture \cite{kragh2018inoculation}, which can make a deviation from the prediction of a "well-mixed" model.
Having these possible deviations from the present model in mind, it will still be interesting to perform an evolution experiment with longer famine period, to see the role of the trade-off between the growth rate and the death rate in the repeated feast-famine cycle.
As discussed, the present model has plenty of choices to be extended for emulating the strategy of the real bacterial population. But in spite of its simplicity, it provides several insights into how the feast-famine environment could affect the bacterial population dynamics, which hopefully helps the future development of our understandings of bacterial population dynamics and evolution.
\begin{acknowledgements}
The authors thank Nen Saito for fruitful discussions and Hiroshi Kori for suggesting the reference \cite{yang2019temporal}.
This work was funded by the Danish National Research Foundation (BASP: DNRF120).
\end{acknowledgements}
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ATWT/GL Cast - Mark Pinter - Welcome to Soap News.Com! - Mark Pinter was born on March 7 in Decorah, Iowa. He is 6'1" and has hazel eyes. He is married to Colleen Zenk-Pinter, who plays Barbara Ryan on As The World Turns.
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<HTML><HEAD>
<TITLE>Review for Gor (1988)</TITLE>
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<H1 ALIGN="CENTER" CLASS="title"><A HREF="/Title?0095241">Gor (1988)</A></H1><H3 ALIGN=CENTER>reviewed by<BR><A HREF="/ReviewsBy?ecl%40mtgzy.att.com+(Moderator)">ecl@mtgzy.att.com (Moderator)</A></H3><HR WIDTH="40%" SIZE="4">
<P>[The author of this review wishes to remain anonymous. Can you blame her/him?
-Moderator]</P>
<PRE> GOR
A film review by Jack Shadow
Copyright 1989 Jack Shadow</PRE>
<P> Twenty-one novels or so ago, a chap "named" John Norman starting writing a
series of Edgar-Rice-Burroughs-style adventure novels set on a counter-Earth
(opposite the Earth on the other side of the Sun). They had titles like
TARNSMAN OF GOR and RAIDERS OF GOR. Much as in ERB's classic Mars series, the
Goreans possessed an unlikely mix of high and low technology, combining
spaceships and perfect birth control with swords. A relatively minor part of
the background was slavery, as is frequently the case with hack-work adventure
novels.</P>
<P> This continued for five novels or so, and Mr. Norman achieved some
recognition as a writer of more or less enjoyable escapist adventure
literature. At some point about the fifth novel what had been a few lines
describing a slave being chained, whipped, or branded became a few pages ...
and then an entire chapter. The Gor series became an open secret - the only
source of soft-porn S&M material at Waldenbooks. Fortunately for Mr. Norman,
Falwell and Robinson have for some reason neglected his publisher, DAW Books,
and the owner of DAW is apparently well pleased with the Gor series. Rumor has
it that the Gor profits more or less pay for all the innovative, low-profit,
high-quality SF published by DAW Books.</P>
<P> The recent Gor movie must be viewed against this background. Some may have
hoped for a decent action-adventure film while others for a bevy of beauties in
chains. Both are sadly disappointed. The Gor film, is, to quote one critic,
"a travesty of a mediocrity." Gone are the SF/fantasy elements of the novels
(spaceships, alien priest-kings, tarns, etc.). Instead a magical stone
transports our hero (Tarl Cabot) to Gor where he quickly becomes an amazing
fighter in spite of a total lack of prior experience. The plot vaguely
resembles some elements of the novels, but only to a point. There are some
dancing slaves, but nothing like the preponderance in the novels. There is a
plot, but it makes even less sense than that found in the novels, and at the
end Cabot is returned to Earth via the "homestone" where he knocks down a
beach-bully-type with his new-found skills.</P>
<P> My personal nomination for the most gratuitous scene involves the main
characters sneaking though a cave. They come upon a blond woman in chains.
One character says to another, "She has the dread disease narcosis. Terrible,
isn't it?" or words to that effect. They move on and nothing further is said
of the "dread disease." Lest you get the idea that although this is a bad film
with a silly plot these flaws are compensated for by the presence of hordes of
nubile slaves, I feel compelled to point out that this is the one of a very
small number of such scenes in the entire movie.</P>
<P> Did I mention the poor acting? No? Well, there is plenty of bad acting as
well. There is also a sequel, strongly but illogically hinted at by the final
scene of the movie. This is definitely a (-2) movie, verging on (-3).</P>
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Infiniti vows to stick it out
Brand is optimistic in the wake of model cuts and job losses.
1 / 1 photos
The future may seem bleak after Infiniti confirmed it'll can its second best-selling models, yet, still, the brand remains optimistic.
Having struggled ever since it arrived in Australia in 2012, the luxury marque's sales figures dropped by 16.4 per cent in 2018 and could tumble even further after production of the Q30 and QX30 ends this year owing to the fact it is based on the previous-generation Mercedes-Benz A-Class.
The car maker has also announced it will quit markets in Western Europe in order to focus on North America, Asia and eastern Europe markets.
At this stage, the Q30 and QX30 are the only models set to be cancelled at the UK-based manufacturing plant by middle of 2019, but there is no word as to what this means for potential Aussie buyers.
Infiniti reveals QX Inspiration concept
by Alex Rae
2018 Infiniti Q60 range review
by Stephen Ottley
"There is no confirmation yet as to when sales will cease in Australia, but it'll likely to be the end of this year," says Karla Leach from Nissan Australia.
"We're sad to see Infiniti's entry model discontinued, but we're confident the current line-up will keep customers engaged in the Infiniti brand," she says.
The Q50 sedan will be offered as the entry model in the range, starting at $54,900 plus on-road costs, also the luxury marque's best selling model in 2018.
The Japanese car maker still plans to electrify its entire range by 2021 as well as launch "no less than five new models over the next five years".
And, countering the end of the Q30 and QX30, the brand says it still plans to launch the new Infiniti QX50 SUV later this year.
Alexandra Lawrence
Ali is a Motoring Reporter at Drive.com.au. She completed her Journalism degree in 2018 and has a background in the automotive industry. Ali also has a passion for racing that started in go-karts from an young age, progressing into car racing in 2016…
shoot the messenger — 15 Mar 2019 18:06
You don't have to try that hard to get a huge discount on Infiniti. Slash 30% off the Q60 in top spec and it might be worth a serious look.
(10) Comments on: Infiniti vows to stick it out
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Hinckley Journal of Politics
_Cover
_Title Page
_Board
_TableOfContents
_In Memory
_Page 1
_Page 99 Final Word
_Page 10
Title Hinckley Journal of Politics vol 7
Creator Logan P. Sisam; Bradley Curtis; Cameron B. Diehl; Sarah Green; Bryon Prince; Scott R. Rasmussen; Adam Reiser; Samuel Sutton; Dave Buhler; LaVar Christensen; Karen Hale
Subject Political Science; Political science literature; periodicals
Publisher Hinckley Institute of Politics
Digitization Specifications Originals scanned with Zeutschel OS10000 book scanner and saved as 400 ppi uncompressed tif. Display jpeg created in PhotoshopCS as 400 ppi jpeg 800 pixels in width
Contributing Institution J. Willard Marriott Library
Metadata Cataloger Kelly Taylor
ARK ark:/87278/s6z60m0q
Setname uu_hjp
Reference URL https://collections.lib.utah.edu/ark:/87278/s6z60m0q
Title _Page 89
Hingkley Journal of Politics 2006 time running out and other bills to consider, the Speaker did not recognize him. Thus, the bill died. There are several lessons I have drawn from this experience. First, as a policymaker it's important to look for opportunities to find answers that build consensus rather than extend divisiveness. This is not always possible, but when it is, it is preferable. Too frequently we can be drawn into the politics of polarization. Second, work with others to achieve a common goal. The trust and mutual respect among City Council members made it possible to coalesce around this solution. Third, in politics it is important to think ahead and even better if you can look around the corners. Our actions were constantly in the context of what others were doing or might do, whether the Mayor, the Legislature, or the courts. Most important, I am proud to have been part of taking on a controversial issue and finding a way to forge a workable solution in a way that is totally without precedent. We also took off the table a potentially divisive community issue. Working together, the City Council plowed new ground. By changing lenses, we found a way to provide fairer, equal, and inclusive health insurance coverage for all city employees. The city employee who shares a household with a sibling, parent, or long-term roommate is as deserving as one in a committed relationship to receive the same kind of health care coverage that's available to married couples. Now this is possible. References Anderson, Ross C. (2006, February 21). Statement of Objections. Anti-gay bias ordinance has a short life. (1998, January 14). Deseret Morning News. Buhler, Dave, Jill Remington Love, and Eric Jergensen. (2006, January 21). Council's plan is fairer, more inclusive. The Salt Lake Tribune. City Council crafts its own benefit plan.(2006, January 10). Deseret Morning News. Jill Remington Love's Motion. (2006, February 24). The Salt Lake Tribune. Rocky pushes for gay benefits. (2005, August 5). The Salt Lake Tribune. Rocky signs partner-benefits order. (2005, September 22). Deseret Morning News. Rocky's benefits plan lacks support. (2005, September 26). Deseret Morning News. SL Council bans gay bias-for now. (1997, December 10). Deseret Morning News Dave Buhler is currently the Chair of the Salt Lake City Council. Elected to the Council in 1999 and re-elected in 2003, he represents District 6. He served one term in the Utah State Senate, 1995-1999. A graduate of the University of Utab with a B.S. in Political Science and a B.S. in History, he also has an M.RA. from the Marriott School of Management at Brigham Young University. He has been an adjunct for the University of Utah Department of Political Science since 1990, and is an Associate Commissioner of Higher Education. 89
Reference URL https://collections.lib.utah.edu/ark:/87278/s6z60m0q/205639
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\section{Introduction}
Resonances play a central role in the control of light-matter interactions in nanophotonics. Plasmonic resonances enable such a control via large near-field enhancements \cite{maier2005,maier2007}, which allows, e.g.,
for realizing plasmonic nanoantennas to tailor the radiation from quantum emitters~\cite{novotny2011,novotny2012}. Recently, the excitation of Mie-type resonant modes in high-refractive-index
dielectric resonators has proven to be very useful for a wide range of applications, from the enhancement of nonlinear effects to a resonant control of the phase in metasurfaces~\cite{kuznetsov2016}.
One important figure of merit for measuring the effect of resonances on light-matter interactions is their quality factor ({\it $Q$-factor}), that quantifies the ability of a structure to trap light and to enhance the electromagnetic fields.
Nanoresonators act as nanoantennas for strongly localized light sources, like quantum dots or defects in crystalline lattices, which can allow for the realization of efficient single-photon sources by enhancing the emission of light~\cite{Aharonovich2011,Lodahl2015}.
Such a control of the emission via the modification of the electromagnetic environment is a concept that dates back to the pioneering work of Purcell~\cite{Purcell_1946} performed in the microwave range followed by the experiments of Drexhage \cite{drexhage1970} that demonstrated the possibility of controlling the lifetime of fluorescent molecules in the visible range. This phenomenon is ubiquitous, and it has also been used to control the resonant scattering by dielectric nanorod antennas \cite{holsteen2017}.
The figure of merit that quantifies the emission enhancement is called {\it the Purcell factor}~\cite{Purcell_1946}, and it is proportional to the $Q$-factor. Optical nanoantennas were first realized with plasmonic materials \cite{novotny2011,novotny2012}, but recently dielectric resonators have been shown to allow for large enhancements of the Purcell factor via the excitation of both electric and magnetic optically-induced Mie-type resonances~\cite{zambrana2015}. The excitation of magnetic resonances presents the advantage of enhancing light emission also via the magnetic dipole transitions. This effect was first theoretically predicted \cite{rolly2012,schmidt2012,zambrana2015} and confirmed later in experiments~\cite{sanz2018,vaskin2019,sugimoto2021}. This is a very promising application for dielectric nanoantennas as the enhancement of light emission empowered by the magnetic dipole resonances is an emerging area of research \cite{karaveli2011,baranov2017}.
The enhancement of the Purcell factor was used successfully to improve the emission of quantum dots in silicon nanoantennas~\cite{rutckaia2017} and also for metallic and hybrid nanoantennas~\cite{barreda2021}.
Control of the emission can also be achieved dynamically~\cite{casabone2021}.
Finally, nanoantennas can also be designed to enhance the performance of quantum emitters,
providing promising platforms for the realization of single-photon sources~\cite{zalogina2018}.
Bound states in the continuum (BICs) appear as a special type of nonradiating modes associated with an infinite $Q$-factor~\cite{hsu2016}. Such states can originate from different physical mechanisms~\cite{koshelev2019,tonkaev2020}. Symmetry protected BICs occur in photonic crystal slabs, and they result from the impossibility of these modes to couple to propagating fields outside the photonic crystal because of symmetry restrictions~\cite{hsu2016,lee2012}. Further, the so-called accidental BICs appear from interferences between several resonances~\cite{hsu2016,koshelev2019}. They are observed when a system parameter is varied continuously. This concept was introduced in quantum mechanics where the coupling between resonances is controlled by engineering the potential~\cite{Friedrich1985}.
In optics, one of the first attempts to study BICs was made in the physics
of photonic crystals~\cite{hsu2013}. While BICs can be realized in gratings or photonic crystals which are infinite in two directions, it is much more challenging to observe such BICs in compact structures and even more in subwavelength systems~\cite{hsu2016}. The existence of BICs, also called {\it embedded eigenstates}, was predicted theoretically in a coated nanosphere where the permittivity of the outer shell vanishes~\cite{monticone2014}.
In more realistic configurations, it is still possible to take advantage
of the coupling of resonances in nanostructures to increase the $Q$-factor of one resonance, even if it does not lead to accidental BICs with infinite $Q$-factors. In photonics, such an approach was suggested to enhance the $Q$-factors of the modes of optical micro-cavities~\cite{wiersig2006} and coupled dielectric nanopillars~\cite{song2010}. It was shown recently that high-refractive index nanodisks supporting multiple resonances are a good platform to employ this approach~\cite{rybin2017,bogdanov2019,koshelev2020c}.
Due to similarity of this approach with accidental BICs~\cite{Friedrich1985}, the large $Q$-factors achieved through the interference of several resonances are called {\it quasi-BICs}. Quasi-BICs have been observed experimentally in AlGaAs nanodisks~\cite{melik2021}, and they have been used in various applications~\cite{koshelev2019,koshelev2020b} including nonlinear optics \cite{carletti2018,carletti2019,koshelev2020} and lasing from a single nanoparticle~\cite{mylnikov2020}.
Unlike BICs, which lead to a perfect confinement of light, quasi-BICs suffer from residual radiation losses. As a consequence, for a rigorous treatment of quasi-BICs it is important to use quasinormal modes (QNMs) and associated complex eigenfrequencies which generalizes modal approaches to dissipative and non-Hermitian systems~\cite{lalanne2018,kristensen2020,wu2021}.
The influence of quasi-BICs on light-matter interactions and, in particular, their coupling to a light source can be quantified by using QNM expansions. The QNM analysis of the coupling of an electromagnetic dipole source
to an optical resonator, i.e., the modal expansion of the Purcell factor, has been carried out through several approaches~\cite{sauvan2013,ge2014,zambrana2015,muljarov2016,Zschiedrich2018}.\\
In this paper, we study quasi-BICs numerically. We consider dielectric nanoresonators either {\it with} or {\it without} a substrate and demonstrate that they can support interfering resonances with a strong coupling between a pair of modes. We carry out numerical simulations with a localized source embedded into the resonator to demonstrate different physical regimes. Modal expansions of the Purcell factor and far-field patterns reveal a complex interplay between different modal contributions interfering destructively in the spectral vicinity of quasi-BICs, yielding a strong enhancement of the Purcell factor and single modal excitation when the parameters of the source and resonator are tuned to match the quasi-BIC conditions.
\begin{center}
\begin{figure}
\includegraphics[width=0.5\textwidth]{figure_1.pdf}
\caption{Principle of the enhanced and suppressed emission with interfering resonances. (a) Schematics of GaAs nanodisks with and without a substrate. The aspect ratio $D/H$ is tuned to control the interference between the two main modes of the nanodisk. (b) Visualization of the electromagnetic field distribution resulting from a dipole emitter, represented by a white sphere, which is located below the top face of the nanodisk. Its frequency is chosen to excite the two modes of interest.
(c) and (d): 2D cross-sections through the dominant two modal fields (left) and the total field distribution (right) visualizing the real part of the y field component. Red and blue colors correspond to negative and positive fields, respectively.
The emitter position is indicated with a white circle.
(c) When the two modal fields are excited in phase they interfere constructively, leading to enhancement of dipole emission.
(d) Out-of-phase excitation of the two modal fields at a different dipole emission frequency, results in suppressed emission.
}
\label{Fig:1}
\end{figure}
\end{center}
The major steps followed in this article are illustrated in Fig.~\ref{Fig:1}. In Sec.~\ref{Sec_quasi_Bic}, we vary the aspect ratio $D/H$ of a GaAs nanodisk to control the interference between the two modes of the nanodisk with or without a substrate. In particular, the strong coupling between these modes leads to the appearance of a high-$Q$ mode: the quasi-BIC resonance. Sec.~\ref{coupling_dipole} considers the coupling of a dipole source with the nanodisk, leading to a complex electromagnetic response as seen in Fig.~\ref{Fig:1}(b). Modal expansions are employed to analyze the role of the interference between the nanodisk modes for the coupling with the dipole. These expansions enable to identify how the constructive interference between two modal contributions leads to the enhancement of the dipole emission, as illustrated in Fig.~\ref{Fig:1}(c). On the other hand, destructive interference leads to the inhibition of the dipole emission, as illustrated in Fig.~\ref{Fig:1}(d). The modal analysis of the radiation pattern is carried out in Sec.~\ref{sec:radiation}. Finally, Sec.~\ref{sec:conclusion} concludes the paper.
\section{Quasi-BICs in isolated nanodisks}\label{Sec_quasi_Bic}
To understand the appearance of quasi-BIC states, first we review the theoretical approach employed to study the mode coupling~\cite{wiersig2006,yi2019,heiss2000}. A good insight in the physics of strong coupling for interfering resonances can be gained
from a phenomenological model of mode coupling that involves the two modes with the uncoupled eigenfrequencies $\omega_{\text{un},1}$ and $\omega_{\text{un},2}$. When these two eigenfrequencies are far apart in the complex plane, there is no coupling between them. However, when the eigenfrequencies get close to each other, the coupling has to be taken into account and modifies the trajectories of these eigenfrequencies when a parameter is varied. The eigenfrequencies of the coupled modes can be found as the eigenvalues of an effective two-mode Hamiltonian, and they are equal to
\[
\omega_{\pm} = \left(\frac{\omega_{\textrm{un},1} +
\omega_{\textrm{un},2}}{2}\right)\pm\sqrt{\gamma},
\]
where
\[
\gamma = \left(\frac{\omega_{\textrm{un},1} -
\omega_{\textrm{un},2}}{2}\right)^2 + v^2 \]
with $v$ being the coupling coefficient between the modes~\cite{yi2019}.
We are interested in the regime where these two resonances are close to each other, and therefore we assume that $\Re\left(\omega_{\textrm{un},1}\right) =
\Re\left(\omega_{\textrm{un},2}\right)$ and $v$ is real as in Ref.~\cite{yi2019}.
\begin{center}
\begin{figure}
\includegraphics[width=0.49\textwidth]{figure_2.pdf}
\caption{
Real parts of two eigenfrequencies of interest (a, e) and corresponding $Q$-factors (b, f) as function of nanodisk aspect ratio $D/H$.
Avoided crossing of the real parts
and local maximum and minimum of the $Q$-factors at $D/H= 0.909$
indicate strong coupling for the nanodisk {\it without} substrate (a, b).
Crossing of the real parts of two eigenfrequencies at $D/H= 0.93$
and avoided crossing of the $Q$-factor curves at $D/H = 0.944$ leading to a peak at $D/H= 0.92$
indicate weak coupling for the nanodisk {\it with} substrate (e, f).
Field intensity maps $\lvert\mathbf{E}\rvert$ of the QNMs
in an $x-z$-cross section through the 3D field distribution.
(c), resp.~(d), corresponds to the high-$Q$ (resp.~low-$Q$) mode of the isolated nanodisk [at the aspect ratio indicated by the black, resp. green, dot in (b)].
(g), resp.~(h), corresponds to the high-$Q$ (resp.~low-$Q$) mode of the nanodisk with substrate [black, resp.~green, dot in (f)].
}
\label{Fig:2}
\end{figure}
\end{center}
As explained in \cite{wiersig2006,yi2019}, two regimes of the mode coupling may be realized depending on the relation between $v$ and $\frac{1}{2}(\omega_{\mathrm{un},1} - \omega_{\mathrm{un},2})$.
When $2v<|\Im\left(\omega_{\textrm{un},1} - \omega_{\textrm{un},2}\right)|$,
the mode eigenvalues become
\[
\omega_{\pm} = \left(\frac{\omega_{\textrm{un},1} +
\omega_{\textrm{un},2}}{2}\right)\pm i\sqrt{|\gamma|},
\]
and one observes that the coupling mostly alters the imaginary part of the eigenvalues resulting in an avoided crossing of the imaginary parts of the coupled eigenvalues and a crossing of their real parts. This behavior is a direct signature of the mode weak coupling. On the other hand, if
$2v>|\Im\left(\omega_{\textrm{un},1} - \omega_{\textrm{un},2}\right)|$,
the mode eigenvalues are presented as
\[
\omega_{\pm} = \left(\frac{\omega_{\textrm{un},1} +
\omega_{\textrm{un},2}}{2}\right)\pm \sqrt{|\gamma|},
\]
suggesting that the coupling of the eigenmodes mostly alters
the real part of the eigenfrequencies yielding, this time, an avoided crossing of the real parts of the coupled eigenvalues and a crossing of the imaginary parts. In the following, we discuss how the mode coupling may result in the appearance of a hybridized quasi-BIC mode.
We consider a Gallium Arsenide (GaAs) nanodisk resonator with a height
$H = 1260\,\mathrm{nm}$ and varying diameter $D$ in two different configurations:
The nanodisk is just surrounded by air (case 1), and, the nanodisk is placed on a glass substrate and surrounded by a super-space of air (case 2).
The optical properties of the system are investigated in the
near-infrared wavelength range; the corresponding constant relative permittivities in our model are
$\varepsilon_\textrm{GaAs} = 11.56$, $\varepsilon_\textrm{sub} = 2.25$, and
$\varepsilon_\textrm{air} = 1.0$.
The time-harmonic optical fields are modeled using Maxwell's equations,
\begin{equation}
\nabla\times\mu_0^{-1}\nabla\times\mathbf{E}(\mathbf{r},\omega)-
\epsilon(\mathbf{r})\omega^2\mathbf{E}(\mathbf{r},\omega) =
i\omega\mathbf{J}\left(\mathbf{r}\right),
\label{maxwell}
\end{equation}
where $\mu_0$ is the vacuum permeability, $\epsilon(\mathbf{r})$ is the permittivity,
and $\mathbf{J}\left(\mathbf{r}\right)$ the source current density.
For numerically solving Eq.~\eqref{maxwell}, we use an adaptive, higher-order finite element method (FEM) \cite{pomplun2007}.
For computing the eigenmodes $\mathbf{E}_n$ of the system and their associated eigenfrequencies $\omega_n$, i.e., solutions to Eq.~\eqref{maxwell} where $\mathbf{J}=0$,
the cylindrical symmetry of the system is taken into account.
Only modes with an azimuthal quantum number equal to 1 or -1
are investigated because these are the only ones excited by
a dipole located on the axis of rotation, which is the configuration
we are investigating in the second part of this study.
In order to find a quasi-BIC condition, the interference between two modes of the structure has to be tuned \cite{wiersig2006,rybin2017,huang2021}.
This is done by varying the geometry parameter, $D$, and computing eigenmodes $\mathbf{E}_n$ and their associated eigenfrequencies $\omega_n$, where $n$ is the mode index.
Note that alternatively, a perturbation approach based on QNMs may be employed for finding the quasi-BICs \cite{yan2020}.
Figure~\ref{Fig:2}(a,b,e,f) shows how
the normalized frequency, $\Re(\omega_{n} D/2c)$, and the $Q$-factor,
\[
Q= -\frac{1}{2} \frac{\Re\left(\omega_n\right)}{\Im (\omega_n)},
\]
depend on the aspect ratio $D/H$.
In Fig.~\ref{Fig:2}(a,b), the case where the GaAs nanodisk is located in air is considered. It can be observed that the real part of the eigenfrequencies is showing a repulsion behavior at $D/H= 0.909$ and an almost coinciding peak reaching $Q \approx 800$ is observed for the $Q$-factor of one of the modes while a minimum is seen for the other mode. As discussed above, this behavior is an indication of strong coupling between the two modes. The high-$Q$ mode can thus be considered to be a quasi-BIC. Figures~\ref{Fig:2}(e,f) show the results for the second case, where the nanodisk is put on a glass substrate. It can be observed that, for the investigated modes and parameter range,
the real part of the eigenfrequencies shows a crossing at $D/H = 0.933$.
We observe a peak of the $Q$-factor reaching $Q\approx 400$ at $D/H = 0.92$.
In fact, this peak is linked to the anti-crossing or level-repulsion occurring for the imaginary parts of the eigenfrequencies. This avoided crossing of the imaginary parts of the eigenfrequencies shows up in Fig.~\ref{Fig:2}(f) at about $D/H=0.944$. The qualitative analysis based on the effective Hamiltonian discussed above shows that this behavior is an indication of weak coupling between the two modes. The transition from strong to weak coupling when a substrate is added indicates that there must be an exceptional point, i.e., a condition for which the two coupled eigenvalues would become degenerated \cite{heiss2012}, when continuously varying the refractive index of the substrate from
1 to 1.5 \cite{yi2019,heiss2000,rodriguez2016,deng2021}.
To conclude the discussion on the avoided crossing of the eigenfrequencies,
we show, in Figs.~\ref{Fig:2}(c,d,g,h), the field patterns associated with both modes when the $Q$-factor is maximized. For the case without substrate, this occurs for $D/H= 0.909$ while, when the substrate is added, the maximum occurs for an aspect ratio of $D/H=0.92$. This helps to understand the level repulsion observed since, in both cases, the modes have apparently very different field patterns: The high-$Q$ mode field is concentrated in hot spots located at the top and bottom of the disk while, for the low-$Q$ mode, it is concentrated at the center of the disk. This apparent difference in the localization of the modes certainly prevents their merging.
\begin{figure*}
\includegraphics[width=0.9\textwidth]{figure_3.pdf}
\caption{Modal analysis of the wavelength ($\lambda$) dependent Purcell factor $\Gamma$ for a $y$-polarized dipole located on the symmetry axis $20\,\mathrm{nm}$ and $27\,\mathrm{nm}$ below the top face of the nanodisk in the case without substrate (a,b) and with substrate (c,d), respectively. (a)~Modal expansion for the aspect ratio $D/H= 0.909$ (maximum $Q$-factor in Fig.~\ref{Fig:2}(b). The high-$Q$ mode corresponds to the modal Purcell factor $\Gamma_2$ (a black solid curve) and is solely responsible for the peak of the total Purcell factor $\Gamma_\mathrm{tot}$ (dashed red curve) at around $1205\,\mathrm{nm}$. The contributions of the low-$Q$ mode $\Gamma_1$ (green solid curve) and of the background $\Gamma_{\text{background}}$ (dotted blue line) are negligible.
(b)~Modal expansion for $D/H= 0.933$ (crossing of $Q$-factors in Fig.~\ref{Fig:2}(b). Both modal terms $\Gamma_1$ and $\Gamma_2$ are of the same order of magnitude and destructively interfere in regions where they are of different sign. The impact of $\Gamma_{\text{background}}$ is constant and negligible in resonant regions. (c)~Modal expansion for $D/H= 0.92$ (peak of the high $Q$-factor in Fig.~\ref{Fig:2}(f). The high-$Q$ mode corresponds to $\Gamma_2$ and is responsible for the peak of $\Gamma_\mathrm{tot}$ at around $1220\,\mathrm{nm}$. (d)~Modal expansion for $D/H= 0.944$ (avoided crossing of $Q$-factors in Fig.~\ref{Fig:2}(f). The modal terms interfere, as in (b).
The markers $\Gamma_{\text{optimized}}$, $\Gamma_{\text{bd}}$, $\Gamma_{\text{resonance}}$ and $\Gamma_{\text{rd}}$ indicate the wavelengths for which far-field patterns are displayed in Fig.~\ref{Fig:4}.}
\label{Fig:3}
\end{figure*}
\section{Coupling of a point source to a nanoresonator}
\label{coupling_dipole}
Now, we turn to the study of a dipole emitter coupled to the investigated
nanoresonator considering the two cases, the nanoresonator with and without substrate. It is worth noting that the coupling of a dipole with a BIC in an array of nanoparticles have already been studied \cite{abujetas2021},
but we will here focus on the coupling of a dipole with the quasi-BIC
arising in an individual nanodisk. We consider Maxwell's equations, given by Eq.~\eqref{maxwell}, with the
current density $\mathbf{J} = \mathbf{j}\delta\left(\mathbf{r}-\mathbf{r}_\mathrm{d}\right)$ that is a point source located at
$\mathbf{r}_\mathrm{d}$.
The Purcell factor, which is used to quantify the enhancement of the emission, is defined as
$\Gamma(\omega) =
-[\Re\left(\mathbf{E}(\omega,\mathbf{r}_\mathrm{d})\cdot
\mathbf{j}^{*}(\omega,\mathbf{r}_\mathrm{d})\right)]/[2\Gamma_\mathrm{b}(\omega)]$,
where $\Gamma_\mathrm{b}(\omega)$ describes the emission of the dipole in a homogeneous medium of the permittivity $\varepsilon_s$. The interest of studying the Purcell factor and its modal analysis is twofold. On the one hand, one can see how a mode with a $Q$-factor as large as the one of the quasi-BIC can affect the dipole emission. On the other hand, looking at the modal analysis of the Purcell factor would allow to use it as a probe to study the interplay between several modes. This is particularly interesting for quasi-BICs since interferences between modes are at the origin of their formation.
To do so, we start by considering the Purcell factor for a dipole located at the maximum of the field amplitude of the high-$Q$ mode. This position is on the symmetry axis of the nanodisk, about $30\,\mathrm{nm}$ below the top face.
\begin{figure*}
\includegraphics[width=1\textwidth]{figure_4.pdf}
\caption{Modal decomposition of the $\theta$-dependent, normalized radiation patterns towards the top
for a dipole on the symmetry axis $20\,\mathrm{nm}$ and $27\,\mathrm{nm}$ below the top face of the nanodisk in the case without substrate (a-d) and with substrate (e-h), respectively.
The green (black) solid curve corresponds to the angle-resolved, far-field modal energy flux $s_1$ ($s_2$) corresponding to the low-$Q$ (high-$Q$) mode.
The red dashed curve corresponds to the total energy flux $s_{\text{tot}}$.
The upper half of each diagram shows positive contributions while the lower half in gray shows negative contributions.
The dipole emission wavelengths correspond to the different $\Gamma$ markers in Fig.~\ref{Fig:3} which are reproduced in the right bottom of each emission diagram.
(a, e) show the on-resonant far field radiation for nanodisks supporting the quasi-BIC
($D/H= 0.909$ and $\lambda = 1206\,\mathrm{nm}$, resp.~$D/H = 0.92$ and $\lambda = 1219\,\mathrm{nm}$) with clearly dominating contribution from the high-$Q$ mode.
(b-d), resp.~(f-h) show the far field radiation for nanodisks with aspect ratios of
$D/H = 0.933$, resp.~$D/H= 0.944$ (i.e., at the avoided crossing, resp.~crossing of the eigenfrequencies, cf., Figs.~\ref{Fig:2}(a, e) for on-resonant sources $(\lambda = 1217\,\mathrm{nm}$/$1237\,\mathrm{nm}$ in c/g) and off-resonant sources ($\lambda = 1204\,\mathrm{nm}/1234\,\mathrm{nm}/1219\,\mathrm{nm}/1247\,\mathrm{nm}$ in b/d/f/h).
While for on-resonant sources a single mode is predominantly contributing to the far field pattern (c, g), in off-resonant settings, the two relevant modes can interfere constructively (b) or destructively (d, f, h), as can be seen from the equal or different signs of the two dominant modal contributions in each case.
}
\label{Fig:4}
\end{figure*}
The consequences of the interplay between resonances at the origin of the quasi-BIC can be better understood by carrying out a modal analysis of the Purcell factor.
Our method for deriving modal expansions relies on the use of
Riesz projections \cite{Zschiedrich2018,Binkowski2020}.
The modal expansion of the Purcell factor reads as
\begin{equation}
\Gamma(\omega) = \sum_{n=1}^2 \Gamma_n(\omega)+\Gamma_\textrm{background}(\omega),
\end{equation}
where $\Gamma_n$ are the modal contributions to the Purcell factor that are
computed using contour integrals around the eigenfrequencies. Here, we take into account only the two interfering modes, i.e., the modes
which are also shown in Fig.~\ref{Fig:2}. The modal Purcell factors $\Gamma_{1}$ and $\Gamma_{2}$ are contributions corresponding to these two modes. The term $\Gamma_\textrm{background}$ contains the contributions of all other poles as well as the nonresonant background \cite{Zschiedrich2018,Binkowski2020}. Finally, $\Gamma_{\textrm{tot}}$ corresponds to the total expansion including both the modal and background contributions. The different black markers indicate the wavelengths at which the radiation patterns are computed in Fig.~\ref{Fig:4}. Details about the modal expansions are provided in supplemental material.
First, we look at the coupling of the dipole to the nanoresonator with the
geometry corresponding to the maximum of the $Q$-factor in Fig.~\ref{Fig:2}.
The results of the modal analysis of the Purcell factor are displayed in Fig.~\ref{Fig:3}(a), for a nanodisk in air with an aspect ratio
$D/H = 0.909$, and, in Fig.~\ref{Fig:3}(c), for a nanodisk on a substrate
with an aspect ratio $D/H = 0.92$. In both cases, the peak observed in the Purcell factor spectrum can be directly linked to the modal contribution corresponding to the high-$Q$ mode. In the region around the peak, the contributions from the low-$Q$ mode and the background are very small or even negligible. This demonstrates that, for a resonator supporting a quasi-BIC, an emitter may easily excite nearly exclusively this resonance.
We note that the quasi-BIC allows to reach a high Purcell factor of $\Gamma \approx 40$ in the case without substrate and $\Gamma \approx 20$ in the case with substrate.
It is also worth looking at configurations where one can expect that the contributions to the Purcell factor from the two main modes would be of the same order of magnitude. This would allow us to investigate the interplay between modal contributions. This is the reason for showing, in Fig.~\ref{Fig:3}(b), the Purcell factor for an aspect ratio of $D/H = 0.933$ for the disk without substrate corresponding to the crossing of the $Q$-factor of the two modes in Fig.~\ref{Fig:2}(b). For the case with substrate, we consider the aspect ratio $D/H = 0.944$ as it corresponds to the avoided crossing of the imaginary parts of the eigenvalues as can be seen from the $Q$-factor trajectories in Fig.~\ref{Fig:2}(f). This avoided crossing is caused by the interference of the interacting modes. Therefore, we expect that the interference will be seen in the contributions of the modal expansion.
The Purcell factor again shows a distinct maximum, with a value of $\Gamma \approx 8$ with substrate and $\Gamma \approx 10$ without substrate.
However, as expected, both modal contributions have the same order of magnitude. Also, the qualitative shape of both spectra of the modal Purcell factors are approximately mirror-symmetric to each other with respect to the resonance wavelength. This behavior yields the fact that, away from the resonance, the signs of the modal contributions of the two modes are opposite, leading to destructive interference in these spectral regions. This is the case in Fig.~\ref{Fig:3}(b) for wavelengths below $\sim 1210\,\mathrm{nm}$ and above $\sim 1230\,\mathrm{nm}$. For the case including a substrate in Fig.~\ref{Fig:3}(d), we observe a similar behavior for wavelengths below $\sim 1225\,\mathrm{nm}$ and above $\sim 1245\,\mathrm{nm}$. The destructive interference between modes has been used previously to qualitatively describe the appearance of quasi-BICs \cite{koshelev2020b}. In the present study, we show that the effect can be quantified by using modal expansion techniques.
\section{Modal analysis of radiation patterns}
\label{sec:radiation}
It is well known that the coupling with nanostructures can alter the radiation pattern of a quantum emitter~\cite{novotny2011}. This ability to control the emission pattern of a dipole emitter with nanostructures has a great practical interest since it can improve the collection of the emitted field with an optical system. A modal analysis allows to understand how each mode but also the interferences between modes modifies the emission pattern.
We will consider the far-field pattern of the energy flux density defined as
$s(\mathbf{r},\omega) =
\frac{1}{2}\Re\left(\mathbf{E}^{*}(\mathbf{r},\omega)\times
\frac{1}{i\omega\mu_0}\nabla\times\mathbf{E}(\mathbf{r},\omega)\right)\cdot\mathbf{n}$,
i.e., the projection of the Poynting vector on the normal vector $\mathbf{n}$
in the direction of field propagation. The modal expansion of $s(\mathbf{r},\omega)$ is computed using Riesz
projections~\cite{Binkowski2020,betz2021} leading to the expression
$s(\mathbf{r},\omega)=\sum_{n = 1}^2 s_n(\mathbf{r},\omega)+
s_{\text{background}}(\mathbf{r},\omega)$, where $\mathbf{r}$ is a point located in the far-field. We will in particular look at the dependency of the radiation pattern with $\theta$ in the x-z plane. In Fig.~\ref{Fig:4}, the field patterns radiated by the dipole upwards towards the air are plotted for different wavelengths and for different aspect ratios. In Figs.~\ref{Fig:4}(a-d), results are shown for the nanodisk without substrate for aspect ratios equal to $0.909$ and $0.933$ while Figs.~\ref{Fig:4}(g-h) display results for the nanodisk on a substrate for aspect ratios equal to $0.92$ and $0.944$. Please note that the lower region of the plot shown in gray does not correspond to the field radiated downwards but to the negative modal contributions. Negative contributions are particularly important here, since, as for the Purcell factor, they are linked to the interferences between different modal contributions. Radiation pattern towards the substrate are actually shown in the supplemental material.
In Figs.~\ref{Fig:4}(a,e), we show the radiation pattern and its modal expansion at the aspect ratio and wavelength of the quasi-BIC.
Just like for the Purcell factor, one mode has a much larger $Q$-factor than the other, it is not surprising to find that the radiation pattern can then be almost entirely understood from the contribution of the high-$Q$ mode while the contributions from the low-$Q$ mode is negligible compared to the contribution of the high-$Q$ mode.
The results of the modal expansion of the radiation pattern for the nanodisk without substrate with the aspect ratio equal to $0.933$ are plotted in Figs.~\ref{Fig:4}(b-d). These computations are made for the wavelengths on both sides of the main peak in Fig.~\ref{Fig:3}(b), with $1204\,\mathrm{nm}$, $1217\,\mathrm{nm}$, and $1234\,\mathrm{nm}$ in Figs.~\ref{Fig:4}(b, d), respectively.
For $\lambda = 1204\,\mathrm{nm}$, we obtain a positive contribution for both the modes summing up to a radiation lobe between $\sim \pm 45^\circ$. For $\lambda = 1217\,\mathrm{nm}$, the main contribution is from mode 2 leading to a quite directional emission between $\sim \pm 30^{\circ}$. Eventually, for $\lambda = 1234\,\mathrm{nm}$, an interference between the two main modal contributions is observed with a positive contribution from mode 2 between $\pm 30^{\circ}$ and a negative contribution of mode 1 in the same range.
In Figs.~\ref{Fig:4}(f-h), we show the results of the modal expansions for the nanodisk on substrate with $D/H=0.944$ for the wavelengths $1215\,\mathrm{nm}$, $1237\,\mathrm{nm}$, and $1247\,\mathrm{nm}$, respectively. In Fig.~\ref{Fig:4}(f), for $\lambda = 1215\,\mathrm{nm}$, the mode 1 has a positive contribution for angles between roughly 30 and -30 degrees while the mode 2 has a negative
contribution in the same range of angles.
Consequently, the total radiation pattern is suppressed, resulting from the interference between several modes. A very analogous behavior is observed at $\lambda = 1247\,\mathrm{nm}$ in Fig.~\ref{Fig:4}(h),
however, in this case, the mode 1 has a negative contribution while the mode 2 has a positive contribution. There is, again, a strong interference between the two modes and the far-field pattern cannot be understood without taking this interference into account. Finally, in Fig.~\ref{Fig:4}(g), for $\lambda = 1237\,\mathrm{nm}$ corresponding to the
peak of the Purcell factor in Fig.~\ref{Fig:3}(b), we observe that the contribution from both modes of interest add up leading to a larger amplitude of the radiation by the dipole and to a confined far-field pattern.
\section{Conclusion}\label{sec:conclusion}
We have numerically analyzed dielectric nanodisk resonators which support multiple resonances in overlapping frequency ranges.
Using a finite-element-method-based framework, regimes where the resonators support quasi-BIC resonances have been investigated.
The impact of the resonances on the Purcell factor describing the emission enhancement of a localized source has been shown in
the quasi-BIC regime as well as in adjacent parameter regimes where several competing resonances are excited.
The modal contributions to the Purcell factor have been computed using the Riesz projection method, and it has been shown that
a single QNM causes the strongly enhanced dipole emission in the quasi-BIC situation.
Further, we have investigated the modal, angular resolved far-field spectrum in on-resonance as well as off-resonance conditions.
This demonstrated that modal interference strongly impacts both, far-field emission strength as well as angular resolved radiation patterns.
It has been shown that micron-scale dielectric resonators supporting quasi-BICs allow for
high Purcell enhancement as well as for highly directed emission of light.
We expect that, apart from the gained insight in the complex interference behavior in multi-modal resonators, these findings
will allow for the design of efficient and robust future photonic components,
such as single-photon emitters for quantum technology applications.
\section*{Acknowledgments}
The authors acknowledge funding from the German Research Foundation (DFG, Excellence Cluster MATH+, EXC-2046/1, project 390685689), the Helmholtz Association (Helmholtz Excellence Network SOLARMATH, a strategic collaboration of MATH+ and Helmholtz-Zentrum Berlin, project ExNet-0042-Phase-2-3), and the German Federal Ministry of Education and Research (BMBF Forschungscampus MODAL, project 05M20ZBM). Also, this project has received funding from the EMPIR program co-financed by the Participating States and by the European Union's Horizon 2020 research and innovation program (project 20FUN02 "POLIGHT"). Y.K. acknowledges support from the Australian Research Council (grants DP200101168 and DP210101292) and from the Russian Science Foundation (grant 21-72-30018).
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{"url":"https:\/\/blog.acolyer.org\/2020\/02\/28\/microsoft-gandalf\/","text":"Gandalf: an intelligent, end-to-end analytics service for safe deployment in cloud-scale infrastructure\n\nModern software systems at scale are incredibly complex ever changing environments. Despite all the pre-deployment testing you might employ, this makes it really tough to change them with confidence. Thus it\u2019s common to use some form of phased rollout, monitoring progress as you go, with the idea of rolling back a change if it looks like it\u2019s causing problems. So far so good, but observing a problem and then connecting it back to a given deployment can be far from straightforward. This paper describes Gandalf, the software deployment monitor in production at Microsoft Azure for the past eighteen months plus. Gandalf analyses more than 20TB of data per day : 270K platform events on average (770K peak), 600 million API calls, with data on over 2,000 different fault types. If Gandalf doesn\u2019t like what that data is telling it, it will pause a rollout and send an alert to the development team.\n\nSince its introduction, Gandalf has significantly improved deployment times, cutting them in half across the entire production fleet. As teams gained more experience with Gandalf, and saw how it was able to detect complex failures that even human experts can miss, their initial mistrust of an automated system turned around completely:\n\nFor many teams, the deployment policy has become that the rollout will not continue to the next region unless Gandalf gives a green light decision.\n\nThe transparency that Gandalf gives to all its decisions, providing full interactive access to all of the supporting evidence, has played a big role in it gaining that trust.\n\nWhy is it so hard to spot a bad rollout?\n\nFirst off, there are a lot of them! So it\u2019s not like you\u2019re trying to assess the impact of one change against a stable base. Instead there are multiple waves of change rolling out through the fleet at any point in time.\n\nEach of those changes will go through a phase deployment typically consisting of stage, canary, light region, heavier region, half region pairs, and the other half of region pairs. More than 70% of the rollouts target multiple clusters.\n\nMore than 20% of rollouts last for over 1,000 minutes.\n\nAnd that\u2019s before we stay to layer in some of the confounding factors: there are faults going on all the time, not all of them caused by software rollouts (we don\u2019t want false alarms); there can be substantial delays between a rollout and the occurrence of problems (e.g. memory leaks that take hours to build up into an issue); and there can be problems that only exhibit themselves with certain user, hardware, or software configurations.\n\nAll of this means that Gandalf faces four key challenges in making sense of all that data:\n\n1. Gandalf needs to be able to cope with constant change in systems and signals: new components emerge, and existing components evolve, changing the failure patterns and telemetry signals.\n2. Ambient faults due to e.g. hardware faults, network timeouts, and gray failures are occurring all the time, and many of these are unrelated to deployments. Gandalf therefore operates in a very noisy environment.\n3. Detecting problems quickly, while also gathering comprehensive information over time to catch delayed issues.\n4. Once a genuine problem has been detected, figuring out which change is the likely cause!\n\nAn n-to-m mapping relationship exists between components and failures: one component may cause multiple types of failure, while a single type of failure may be caused by issues in multiple components. Figuring out which failure is likely caused by which component is not easy due to the complexity of component behaviors.\n\nGandalf system design\n\nAt a high level Gandalf looks like this:\n\nIt ingests performance data, failure signals, and component update events and passes them through both fast and slow paths (a lambda architecture). The fast path is able to detect simple, immediate issues and provide quick feedback (within 5 minutes). The batch layer can detect deferred (latent) issues, analyse more complex failure scenarios, and also provide more detailed supporting evidence.\n\nThe analysis engine in the speed layer only considers fault signals that happen 1 hour before and after each deployment in each node, and runs lightweight analysis algorithms to provide a rapid response. In Azure, most catastrophic issues happen within 1 hour after the rollout. Latent faults occurring after 1 hour will be capture by the batch layer later.\n\nThe combined outputs of the fast and slow paths are fed into Gandalf\u2019s system interface, with a front-end that developers used to view rollout status in real-time and a UI for diagnosing issues. The Azure deployment engine subscribes to Gandalf\u2019s go\/no-go decisions, and stops a rollout if a \u2018no-go\u2019 decision is published.\n\nFrom signals to decisions\n\nIn designing the algorithms for Gandalf, we considered existing options from supervised learning, anomaly detection and correlation analysis but found major limitations for each of them.\n\nSupervised learning struggles with the constant changing of signal and failure patterns in the underlying system. Anomaly detection is insufficient since there are many rollouts occurring simultaneously and it can\u2019t tell them apart. Correlation analysis based on Pearson correlation can\u2019t capture the complex causal relationships that arise.\n\nThe authors ended up with a three-phase analysis pipeline: anomaly detection, correlation analysis, and then a decision engine.\n\nGandalf does the normal trick of text clustering around log messages to generate fault signatures, and then applies anomaly detection based on the occurrences of each signature. Given the changing nature of workloads and the system itself, simple thresholding isn\u2019t sufficient and so Gandalf uses Holt-Winters forecasting to estimate a baseline using the previous 30 days worth of data, and a one hour step interval.\n\nIt\u2019s the job of the correlation analysis phase to try and figure out which component might be responsible for a detected anomaly (if any). Correlation is based on ensemble voting with vetoes. For a component change deployed on some node at time t, a fault happening after t votes for that component change as a possible cause. A fault happening before t vetoes that component change as a possible cause. Four different time windows are used for \u2018before\u2019 and \u2018after\u2019 (1 hour, 1 day, 3 days, \u2026). The votes and vetoes from each time period are weighted (exponential decay) so that smaller intervals contribute more.\n\nTo produce the final blame score the a second exponential time decay factor is incorporated that gives more weight to recent rollouts.\n\nGiven a component and a blame score, it\u2019s time to make a go\/no-go decision.\n\nWe make a go\/no-go decision for the component $c_j$ by evaluating the impacting scopes of the deployment such as the number of impacted clusters, the number of impacted nodes, number of customers impacted, etc.. Instead of setting static thresholds for each feature, the decision criteria are trained dynamically with a Gaussian discriminant classifier. The training data is generated from historical deployment cases with feedback from component teams.\n\nGandalf in action\n\nGandalf makes decisions in about 5 minutes end-to-end on the fast path, and in about 3 hours on the batch layer. In a 8 month window Gandlef captured 155 critical failures at an early stage of data plane rollout, achieving a precision of 92.4% and 100% recall (no high impact incidents got past Gandalf). For the control plane, Gandalf filed 39 incidents with 2 false alarms. Precision here was 94.9%, with 99.8% recall.\n\nThe most common issues Gandalf caught are compatibility issues and contract breaking issues. Compatibility issues arise with updates are tested in an environment with the latest hardware or software stack, but the deployed nodes may have different hardware SKUs or OS or library versions. 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I'd never considered that chickens get bored, but now I have them, I realise they do.
Our girls have a decent sized pen so they can stretch and flap, but they need more than that. They need entertaining.
Ever seen your chickens pace back and forth at the side of their pen? They're bored.
Easy ways to make their enclosure more fun is to add height, as they like to perch. We use logs of different sizes.
Straw in their pen is another good way of entertaining them. Chuck a load into their pen and they'll dig through it and throw it around for hours. Once they're bored of that, sprinkle corn through it and they'll start all over again!
This week, I also added a shallow tub of very dry topsoil. At first they were a little nervous of this new item in their pen, but after a while of 'plucking up' courage they investigated it and realised it was full of dirt. After that, they spent 4 hours in there having dust baths, digging around and squabbling over space! Safe to say most of the soil ended up outside the tub by the end, but they had great fun doing it!
The same shallow tub idea could work for bark chippings, sand or grass cuttings.
The most important thing is to make sure your chickens are entertained and happy. Have fun with them!
We bought our 3 chickens last September (2015) at the local county show. We'd just won a chicken house at a show the week before, and thought we'd better fill it.
I had considered chickens for a little while, but when we won the chicken house, we decided it was fete and time to get started.
We chose 3 different chickens, as we didn't know which breed would suit us. So we have Snowball, a white Sussex, Ginger, a Rhode Island X Barnevelder, and Barney who is a Barnevelder. You can probably guess which is which in the picture above!
What we have discovered, is that chickens have great characters – we had no idea they'd be so interesting. And now, I find myself talking about chickens a lot more than I thought I would.
They are very easy to look after and 'cheap to run' so I'd highly recommend them to anyone thinking of getting chickens. But be warned, they will destroy whatever patch of lawn you give them, and they are louder than we thought they'd be, so if you have a lot of neighbours you might want to think about if they would complain.
When we got them, I also bought the Haynes Chicken Manual. This has really taught me a lot, so I would suggest getting one.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
There is a general strategy for building bridges between combinatorics and
measure theory which we describe. Let $d\mu $ be a measure
on a interval $I \subseteq [0, \infty) \subseteq \mathbb{R}$.
We say that $d\mu$ is a combinatorial measure if for each $n \in \mathbb{N}$ the $n$-th moment
of $d\mu$
is a non-negative integer. Equivalently, let $M_{d\mu}$ be the Mellin
transform of $d\mu$ given by
$$M_{d\mu}(t)=\int_I x^{t-1}d\mu .$$ Then $d\mu$ is a combinatorial measure
if and only if $M_{d\mu}(n)\in \mathbb{N}$ for $n \in \mathbb{N}_+.$\\
Let $cmeas$ be the set of combinatorial measures, and consider the map
$m:cmeas \rightarrow \mathbb{N}^{\mathbb{N}}$ that sends a combinatorial
measure into its moment's sequence $(m_0, ..., m_n,...).$
Recall \cite{DZeilberger} that a sequence of finite sets $(s_0,...,s_n,...)$
provides a combinatorial interpretation for a sequence of integers $(m_0,...,m_n,...)$
if it is such that $|s_n|=m_n.$ By analogy we say that the sequence of
finite sets $(s_0,...,s_n,...)$ provides a combinatorial interpretation
for a measure $d\mu$ if for each $n \in \mathbb{N}$ the
following identity holds: $$|s_n|=\int_I x^n d\mu.$$
In this work we consider the reciprocal problem: given $m=(m_0,...,m_n,...) \in \mathbb{N}$
find a combinatorial interpretation for it, and furthermore find a
combinatorial measure $d\mu$ such that its sequence of
moments is $m$. Our main goal is to establish an instance
of the correspondence combinatorics/measure theory described above within the context
of $q$-calculus.
Namely, we are going to study the combinatorial and the measure theoretic
interpretations for the $k$-increasing factorial $q$-numbers
$[1]_{n,k}= [1]_q[1+k]_q[1+2k]_q. \ . \ .[1+(n-1)k]_q \in \mathbb{N}[q]^{\mathbb{N}}$
which are obtained as an instance of the $q$-analogue of the Pochhammer $k$-symbol given by
$$[t]_{n,k}= [t]_q[t+k]_q[t+2k]_q. \ . \ .[t+(n-1)k]_q =
\prod_{j=0}^{n-1}[t+jk]_q$$ where $[t]_q=\frac{1-q^t}{1-q}$ is the $q$-analogue of $t$.
The search for the combinatorial and measure theoretic interpretation for the $k$-increasing factorial $q$-numbers must be made within the context of $q$-calculus; this means that we have to broaden our techniques in order to include $q$-combinatorial interpretations, and the $q$-analogues for the Lebesgue's measure
and the Mellin's transform.
\section{The $k$-gamma measure}
Perhaps the best known example of the relation combinatorics/measure theory discussed in the introduction
comes from the factorial numbers $n!$ which count, respectively, the number of elements of $S_n$, the group of permutations of a set with $n$ elements. The Mellin transform of the measure $e^{-x}dx$ is the classical gamma function
given for $t>0$ by
$$\Gamma(t)=\int_{0}^{\infty} x^{t-1} e^{-x} dx .$$
The moments of the measure $e^{-x}dx$ are precisely the factorial numbers, indeed we have that
$$|S_n|= n!=\Gamma(n+1)=\int_{0}^{\infty} x^{n} e^{-x} dx.$$
Notice that $n!=(1)_n,$ where the Pochhammer symbol $(t)_n$ is given by
$$(t)_n =t(t+1)(t+2). \ . \ .(t+(n-1)).$$
As a second example \cite{DP} consider the combinatorial and measure theoretical interpretations for the $k$-increasing factorial numbers $(1)_{n,k}= (1+k)(1+2k). \ . \ .(1+(n-1)k),$
which arise as an instance of the Pochhammer $k$-symbol given by
$$(t)_{n,k}= t(t+k)(t+2k). \ . \ .(t+(n-1)k)=\prod_{j=0}^{n-1}(t+jk).$$
The combinatorics of the Pochhammer $k$-symbol has attracted considerable attention in the literature,
from the work of Gessel and Stanley \cite{ge} up to the quite recent works \cite{ca, ku}. Assume
$t$ is a non-negative integer and let $\mathrm{T}_{n,k}^t$ be the set of isomorphisms classes of planar rooted trees $T$ such that:
1) The set of internal vertices, i.e. vertices with one outgoing edge and at least one incoming edge, of $T$ is $\{1,2,....,n \}$;
2) $T$ has a unique vertex with no outgoing edges called the root; $T$ has a set $L(T)$ of vertices called leaves, the leaves have
no incoming edges;
3) The valence of each internal vertex of $T$ is $k+2;$ 4) The valence of the root is $t$;
5) If the internal vertex $i$ is on the path from the internal vertex
$j$ to the root, then $i < j.$ \\
Note that the set of leaves $L(T)$ comes with a natural order, and thus we can assign a number between $1$ to $|L(T)|$ to
each leave.
Figure \ref{ejemp} shows an example of a graph in $\mathrm{T}_{4,2}^2$.\\
One can show by induction that $(t)_{n,k}=|\mathrm{T}_{n,k}^t|.$
\begin{figure}[h!]
\centering
\includegraphics[width=.18\textwidth]{pp}
\caption{Example of a tree in $\mathrm{T}_{4,2}^2$.}
\label{ejemp}
\end{figure}
The Mellin transform of the measure $e^{\frac{-x^k}{k}}dx$ is the $k$-gamma function $\Gamma_k$ given for $t > 0$ by
$$\Gamma_k(t)=\int_{0}^{\infty} x^{t-1} e^{-\frac{x^k}{k}} dx .$$
The $k$-gamma function $\Gamma_k : (0, \infty ) \longrightarrow \mathbb{R}$ is univocally determined \cite{DP} by
the following properties: $\Gamma_k(t+k)=t\Gamma_k(t)$ for $t\in \mathbb{R^+};$
$\Gamma_k(k)=1;$
$\Gamma_k $ is logarithmically convex. See \cite{ko, ma} for further properties of the $k$-gamma function.\\
The $k$-increasing factorial numbers appear as moments of the $\Gamma_k$ function as follows:
$$|\mathrm{T}_{n,k}^1|=(1)_{n,k}=\Gamma_k(1+ nk)= \frac{1}{\Gamma_k(1)} \int_{0}^{\infty} x^{nk} e^{\frac{-x^k}{k}} dx=
\frac{k^{\frac{k-1}{k}}}{\Gamma(\frac{1}{k})} \int_{0}^{\infty} x^{nk} e^{\frac{-x^k}{k}} dx.$$
Indeed the following more general identity holds:
$$|\mathrm{T}_{n,k}^t| = (t)_{n,k} =\frac{\Gamma_k(t+nk)}{\Gamma_k(t)}=
\frac{1}{\Gamma_k(t)}\int_{0}^{\infty} x^{t+nk-1} e^{\frac{-x^k}{k}} dx.$$
\section{Review of $q$-calculus}\label{Basic}
In this section we introduce some useful basic definitions \cite{RA, Ch, HY}. We
begin introducing the $q$-derivative and the Jackson $q$-integral.
Let $\mathrm{Map}(\mathbb{R},\mathbb{R})$ be the real vector
space of functions from $\mathbb{R}$ to $\mathbb{R}$. Fix a real number
$0 \leq q<1,$ the $q$-derivative is the linear operator
$$\partial_q:\mathrm{Map} (\mathbb{R},\mathbb{R})\rightarrow
\mathrm{Map}(\mathbb{R}\setminus \{0\},\mathbb{R}) \ \ \ \ \ \mbox{given by}$$ $$\partial_q f(x)=\frac{f(qx)-
f(x)}{(q-1)x}. \ \ \ \ \mbox{For example we have that}\ \ \partial_0 f(x)=\frac{f(x)-
f(0)}{x}.$$ Notice that $\partial_q f$ is not a priori well-defined at $x=0$. Nevertheless, it is often the
case that $\partial_q f$ can be extended by continuity over the whole real line, e.g.
when $f$ is a polynomial function.\\
For $0\leq a < b \leq +\infty$ the Jackson $q$-integral from $a$ to $b$ of $f \in
\mathrm{Map}(\mathbb{R},\mathbb{R})$ is given by
$$\int_{a}^{b}f(x)d_qx=(1-q)b\sum_{n=0}^{\infty}q^nf(q^nb)-(1-q)a\sum_{n=0}^{\infty}q^nf(q^na).$$
For example we have that
$$\int_a^{b} f(x) d_0x =bf(b)-af(a).$$
Set $I_q f(x)=f(qx).$ The following properties hold for $f,g \in \mathrm{Map}(\mathbb{R},\mathbb{R}):$
\begin{eqnarray*}
\partial_q(fg)&=& \partial_qf g + I_q f \partial_q g \\
\partial_q(f(a x^{b}))&=& a[b]_qx^{b-1}\partial_{q^{b}}f(ax^{b}) \\
f(b)g(b)-f(a)g(a)&=& \int_a^b \partial_qf gd_qx + \int_a^b I_q g \partial_q fd_qx,
\end{eqnarray*}
For $0 <q<1$, $x,y \in\mathbb{R}$, $n \in \mathbb{N}_+$, and
$t \in \mathbb{R}$ we set
$$(x+y)_{q^k}^n = \prod_{j=0}^{n-1} (x+ q^{jk}y), \ \ \ (x+y)_{q^k}^{\infty} = \prod_{j=0}^{\infty} (x+ q^{jk}y) \mbox{\ \ \ and \ \ \ }
(1+x)_{q^k}^{t}=\frac{(1+x)_{q^k}^{\infty}}{(1+q^{kt}x)_{q^k}^{\infty}}.$$
\section{$q$-Analogue of the $k$-gamma function}
We proceed to study the $q$-analogue of the $k$-increasing factorial numbers
$$[1]_{n,k}= [1]_q[1+k]_q[1+2k]_q. \ . \ .[1+(n-1)k]_q $$
which are an instance of the $q$-analogue of the Pochhammer $k$-symbol $[t]_{n,k}$ given
for $t \in \mathbb{R}$ by
$$[t]_{n,k}= [t]_q[t+k]_q[t+2k]_q. \ . \ .[t+(n-1)k]_q =
\prod_{j=0}^{n-1}[t+jk]_q.$$
The motivation behind our definition of the $q$-analogue of the $k$-gamma function comes from the
work of De Sole and Kac \cite{So}, where they introduced a
$q$-deformation of the gamma function given by the $q$-integral:
$$\Gamma_{q}(t)=\int_{0}^{\frac{1}{1-q}}x^{t-1}E_{q}^{-q
x}d_qx,$$
where the $q$-analogue $E_{q}^{x}$ of the exponential function is given by
$$E_{q}^{x}= \sum_{n=0}^{\infty}q^{\frac{n(n-1)}{2}}\frac{x^n}{[n]_{q}!}.$$ For example we have that $ E_0^x=1 +x,$ $E_0^{-0x}=1$, and therefore $\Gamma_0(t)=1.$\\
We define the $q$-analogue of the $k$-gamma function $\Gamma_{q,k}$ by
demanding that it satisfies the $q$-analogues of the properties
of the $\Gamma_k$ function. Thus $\Gamma_{q,k}$ is such that
$\Gamma_{q,k}(t+k)=[t]_q \Gamma_{q,k}(t)$ and $\Gamma_{q,k}(k)=1.$
Several applications of the former property show that $$\Gamma_{q,k}(nk)=\prod_{j=1}^{n-1}[jk]_q=
\prod_{j=1}^{n-1} \frac{(1-q^{jk})}{{(1-q)}}
=\frac{(1-q^k)_{q^k}^{n-1}}{(1-q)^{n-1}}.$$
After a change of variables the function $\Gamma_{q,k}$ may be written as follows:
$$\Gamma_{q,k}(t)=\frac{{(1-q^k)_{q^k}^{{\frac{t}{k}}
-1}}}{{(1-q)^{\frac{t}{k}-1}}}. \ \ \ \ \mbox{For example we have that} \ \ \Gamma_{0,k}(t)=1.$$ The previous formula
implies an infinite product expression for $\Gamma_{q,k}$ given by
$$\Gamma_{q,k}(t)=\frac{(1-q)^{1-\frac{t}{k}}(1-q^k)_{q^k}^{\infty}}{(1-q^t)_{q^k}^{\infty}},$$ and also the following result.
\begin{lema}
{\em The $q,k$-gamma function $\Gamma_{q,k}$ and the $q^k$-gamma function $\Gamma_{q^k}$ are related by the identity
$\ \ \Gamma_{q,k}(t) = [k]_q^{\frac{t}{k}-1}\Gamma_{q^k}(\frac{t}{k}) .$
}
\end{lema}
The following result \cite{CTT} provides an integral representation for $\Gamma_{q,k}$.
\begin{prop}
$$\Gamma_{q,k}(t)=\int_{0}^{\frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}}
x^{t-1}E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx.$$
\end{prop}
This integral representation for $\Gamma_{q,k}$ may be regarded as a $q$-analogue of the
Mellin transform, therefore one is entitled to consider the $q$-measure
$$E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx \ \ \ \mbox{defined on the interval} \ \ \
\big[ 0, \frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}} \big]$$
as the inverse Mellin $q$-transform of the $\Gamma_{q,k}$ function. Figure \ref{eje} shows
the graph of $E_{q^k}^{-\frac{q^k x^k}{[k]_q}}$ for $q=0.6$ and $1\leq k \leq 5$.
\begin{figure}[h!]\label{eje}
\centering
\includegraphics[width=10cm,height=5cm]{campconkvariandokyq=06}
\caption{Display of $E_{q^k}^{-\frac{q^k x^k}{[k]_q}}$ for $q=0.6$ and $1\leq k \leq 5$.}
\end{figure}
One can check that for $0 \leq x \leq \frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}$
the function $E_{q^k}^{-\frac{q^k x^k}{[k]_q}}$ is given by
$$E_{q^k}^{-\frac{q^k x^k}{[k]_q}}=
\sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{kn(n+1)}{2}}x^{kn}}{[k]_q^n [n]_{q^k}! }.$$
\begin{thm}\label{ya}
{\em The function $\Gamma_{q,k}$ is given by
$$ \Gamma_{q,k}(t)=(1-q)^{1-\frac{t}{k}}
\sum_{n=0}^{\infty} \frac{ q^{ \frac{kn(n+1)}{2} } }{ (1-q^{kn+t})(q^k -1)^n[n]_{q^k} ! } .$$}
\end{thm}
\begin{proof}
From Theorem \ref{cool} below we know that
$$\int_{0}^{x}
s^{t-1}E_{q^k}^{-\frac{q^k s^k}{[k]_q}}d_qs =
(1-q)x^t\sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{kn(n+1)}{2}} x^{kn} }
{(1-q^{kn+t}) [k]_q^n [n]_{q^k}! }.$$
The desired result follows taking $x=\frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}$.
\end{proof}
\begin{cor}
{\em
$$(1-q^k)_{q^k}^{\frac{t}{k}} = \sum_{n=0}^{\infty} \frac{ q^{ \frac{kn(n+1)}{2} } }{ (1-q^{kn+t})(q^k -1)^n[n]_{q^k} ! }.$$}
\end{cor}
\begin{proof}
Follows from Theorem \ref{ya} and the identity $\Gamma_{q,k}(t)=\frac{{(1-q^k)_{q^k}^{{\frac{t}{k}}
-1}}}{{(1-q)^{\frac{t}{k}-1}}}.$
\end{proof}
By definition the cumulative distribution function associated with the measure $$E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx \ \ \ \ \mbox{is given for} \ \ \ 0 \leq x \leq \frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}\ \ \ \ \mbox{by} \ \ \ \int_{0}^{x}
E_{q^k}^{-\frac{q^k s^k}{[k]_q}}d_qs.$$
\begin{prop}
{\em $$\int_{0}^{x}
E_{q^k}^{-\frac{q^k s^k}{[k]_q}}d_qs=(1-q)x
\sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{kn(n+1)}{2}}x^{kn+1}}{(1-q^{kn+1}) [k]_q^n [n]_{q^k}! }.$$}
\end{prop}
\begin{proof}
The result follows from Theorem \ref{cool} below taking $t=1.$
\end{proof}
\section{Combinatorial interpretation of the Pochhammer $q,k$-symbol}
Just as in combinatorics one studies the cardinality of finite sets, in
$q$-combinatorics one studies the cardinality of $q$-weighted finite sets, i.e.
pairs $(x,\omega)$ where $x$ is a finite set and the $q$-weight is an arbitrary map
$\omega:x \rightarrow \mathbb{N}[q]$ from $x$ to $\mathbb{N}[q]$ the algebra of polynomials in
$q$ with non-negative integer coefficients. The
cardinality of the pair $(x, \omega)$ is by definition given by
$$|x, \omega|=\sum_{i\in x}\omega(i) \in \mathbb{N}[q].$$
To provide a $q$-combinatorial interpretation for the Pochhammer $k$-symbol $[t]_{n,k}$ we
let again $t$ be a positive integer and consider the set $\mathrm{T}_{n,k}^t$ of planar rooted trees introduced above.
Next we define a $q$-weight $\omega$ on $\mathrm{T}_{n,k}^t$. The construction of $\omega$
is based on the following elementary facts:\\
\begin{figure}[h!]
\centering
\includegraphics[width=.30\textwidth]{pt}
\caption{Display of the tree $r_3$ and a tree $c_i$ with $5$ leaves.}
\label{nuevo}
\end{figure}
\noindent 1) Let $r_t$ be the rooted tree with $t$ leaves and no internal vertices. See Figure 3. \\
\noindent 2) For $1 \leq i \leq n$, let $c_i$ be the rooted tree with $i$ as its unique internal vertex and $k+1$ leaves.
See Figure 3.\\
\noindent 3) If $T$ is a planar rooted tree and $l$ is a number between $1$ and $|L(T)|$, then
there is a well-defined rooted planar tree $T \circ_l c_i$ obtained by gluing the root of $c_i$ with the leave $l$
of $T$ to form a new edge.\\
\noindent 4) Clearly each tree $T \in \mathrm{T}_{n,k}^t$ can be written in a unique way as
$$T=(...((r_t \circ_{l_1} c_1)\circ_{l_2} c_2)...)\circ_{l_{n}} c_n .$$
\noindent 5) The weight $\omega(T)$ of a tree $T$ written in the form above is given by
$$\omega(T)=\prod_{i=1}^{n-1} q^{l_i -1} \in \mathbb{N}[q].$$
For the tree $T$ from Figure 1 we have that
$$T = ((( r_2 \circ_1 c_1) \circ_3 c_2 ) \circ_6 c_3) \circ_7 c_4 \ \ \mbox{ and }\ \ \omega(T) = q^0 q^2 q^5 q^6 = q^{13}.$$
\begin{thm}{\it
$$[t]_{n,k} = |\mathrm{T}_{n,k}^t, \omega | .$$
}
\end{thm}
\begin{proof}
The proof goes by induction on $n$. We have the following chain of identities
$$|\mathrm{T}_{n+1,k}^t, \omega | = \sum_{T \in \mathrm{T}_{n+1, k}^t}\omega(T)=
\sum_{S\in \mathrm{T}_{n,k}^t }\sum_{l \in L(S)}\omega(S \circ_l c_{n+1})=
\sum_{S\in \mathrm{T}_{n,k}^t }\sum_{l=1}^{t+kn}\omega(S)q^{l-1}=$$
$$=\left(\sum_{S\in \mathrm{T}_{n,k}^t}\omega(S) \right) \left(\sum_{l=1}^{t+nk}q^{l-1} \right)
=|\mathrm{T}_{n+1,k}^t, \omega |[t+nk]_q = [t]_{n,k}[t+nk]_q=[t]_{n+1,k}.$$
In the computation above we used two main facts: 1) Each tree $\mathrm{T}_{n,k}^t$ has exactly
$t +nk$ leaves; 2) Each tree $T \in \mathrm{T}_{n+1, k}^t$ can be written in a unique way
as $T= S \circ_l c_{n+1}$ where $S \in \mathrm{T}_{n, k}^t$, $l$ is a leaf of $S$, and
$c_{n+1}$ is the rooted tree with $k+1$ leaves and $n+1$ as its unique internal vertex.
\end{proof}
\section{$k$-Gamma $q$-distribution}
We are ready to define the $k$-gamma $q$-distribution. From the identity
$$\Gamma_{q,k}(t)=\int_{0}^{\frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}}
x^{t-1}E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx$$
we see that the function
$$ x^{t-1}\frac{E_{q^k}^{-\frac{q^k x^k}{[k]_q}}}{\Gamma_{q,k}(t)}$$ defines a $q$-density
on the interval $[0, \frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}} ]$, in the sense that it is a non-negative function whose $q$-integral is equal to one. Consider the case $t=1$ an $k=3$.
Figure \ref{dng} shows the graph of $ \frac{E_{q^3}^{-\frac{q^3 x^3}{[3]_q}}}{\Gamma_{q,3}(1)}$ for $q\in [0,1)$.
\begin{figure}[h!]\label{dng}
\centering
\includegraphics[width=10cm,height=5cm]{densidadgamanormalizadak3t1}
\caption{Display of $ \frac{E_{q^3}^{-\frac{q^3 x^3}{[3]_q}}}{\Gamma_{q,3}(1)}$ for $q \in [0, 1)$.}
\end{figure}
\begin{thm}\label{cool}
{\em The cumulative distribution of the $k$-gamma $q$-density is given by
$$\frac{1}{\Gamma_{q,k}(t)}\int_{0}^{x}
s^{t-1}E_{q^k}^{-\frac{q^k s^k}{[k]_q}}d_qs =
\frac{(1-q)x^t}{\Gamma_{q,k}(t)}\sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{kn(n+1)}{2}} x^{kn} }
{[k]_q^n [n]_{q^k}! (1-q^{kn+t}) }.$$}
\end{thm}
\begin{proof}
\begin{eqnarray*}
\frac{1}{\Gamma_{q,k}(t)} \int_{0}^{x}
s^{t-1}E_{q^k}^{-\frac{q^k s^k}{[k]_q}}d_qs
&=& \frac{(1-q)x}{\Gamma_{q,k}(t)} \sum_{m=0}^{\infty} q^m \sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{kn(n+1)}{2}}(q^m x)^{kn +t-1}}{[k]_q^n [n]_{q^k}!} \\
&=& \frac{1-q}{\Gamma_{q,k}(t)}
\sum_{n=0}^{\infty} \frac{(-1)^n q^{\frac{kn(n+1)}{2}}x^{kn + t}}{[k]_q^n [n]_{q^k}! }
\sum_{m=0}^{\infty} q^{m(kn+t)}\\
&=& \frac{1-q}{\Gamma_{q,k}(t)} \sum_{n=0}^{\infty} \frac{ (-1)^n q^{ \frac{kn(n+1)}{2} } x^{kn+t}}{ [k]_q^n [n]_{q^k}!(1-q^{kn+t}) }
\end{eqnarray*}
\end{proof}
Consider the case $t=1$ an $k=3$.Figure \ref{dng2} shows the cumulative distribution associated to the $q$-density $ \frac{E_{q^3}^{-\frac{q^3 x^3}{[3]_q}}}{\Gamma_{q,3}(1)}$ for $q\in [0,1)$.
\begin{figure}[h!]\label{dng2}
\centering
\includegraphics[width=10cm,height=5cm]{distgammak3t1corregida}
\caption{Cumulative distribution of the $q$-density $ \frac{E_{q^3}^{-\frac{q^3 x^3}{[3]_q}}}{\Gamma_{q,3}(1)}$ for $q \in [0, 1)$.}
\end{figure}
The previous considerations imply our next result which establishes an example of the link between $q$-combinatorics and $q$-measure theory
promised in the introduction.
\begin{thm} {\em The $k$-increasing factorial $q$-numbers appear as moments of the $\Gamma_{q,k}$ function as follows: $$|\mathrm{T}_{n,k}^1, \omega| = [1]_{n,k}=\Gamma_{q,k}(1+ nk)=\frac{1}{\Gamma_{q,k}(1)}\int_{0}^{\frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}}
x^{nk}E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx.$$
Indeed the following more general identity also holds:
$$|\mathrm{T}_{n,k}^t, \omega | = [t]_{n,k} =
\frac{\Gamma_{q,k}(t+nk)}{\Gamma_{q,k}(t)}= \frac{1}{\Gamma_{q,k}(t)}\int_{0}^{\frac{[k]_q^{\frac{1}{k}}}{(1-q^k)^{\frac{1}{k}}}}
x^{t+nk-1}E_{q^k}^{-\frac{q^k x^k}{[k]_q}}d_qx.$$}
\end{thm}
\section{ $k$-Beta $q$-distribution}
Recall that the classical beta function is given for $s,t >0$ by
$$B(t,s)= \frac{\Gamma(t)\Gamma(s)}{\Gamma(t+s)}=\int_{0}^{1} x^{t-1}(1-x)^{s-1}dx .$$
The $q$-analogue of the $k$-beta function is correspondingly defined by
$$B_{q,k}(t,s)=\frac{\Gamma_{q,k}(t)\Gamma_{q,k}(s)}{\Gamma_{q,k}(t+s)}=
\frac{(1-q)(1-q^k)_{q^k}^{\frac{s}{k}-1}}{(1-q^t)_{q^k}^{\frac{s}{k}}}.$$
Notice that $B_{0,k}(t,s)= 1$. One can show \cite{CTT} that the function $B_{q,k}$ has the following integral representation
$$B_{q,k}(t,s)=[k]_q^{-\frac{t}{k}}\int_{0}^{[k]_{q}^{\frac{1}{k}}}x^{t-1}\left(
1-q^k\frac{x^k}{[k]_q}\right)_{q^k}^{\frac{s}{k}-1}d_qx.$$
Because of the factor $[k]_q^{-\frac{t}{k}}$ this integral representation is not quite a Mellin
transform. However we see that the $q$-measure $$\left(
1-q^k\frac{x^k}{[k]_q}\right)_{q^k}^{\frac{s}{k}-1}d_qx$$ is a Mellin $q$-transformation inverse
of the function $$B_{q,k}(t,s)[k]_q^{\frac{t}{k}} .$$
On the other hand we see that the function
$$\frac{x^{t-1}\left(
1-q^k\frac{x^k}{[k]_q}\right)_{q^k}^{\frac{s}{k}-1}}{B_{q,k}(t,s)[k]_q^{\frac{t}{k}}} $$
defines a $q$-density on the interval $[\ 0, \ [k]_q^{\frac{1}{k}} \ ],$ indeed it defines
a $q$-analogue for the $k$-beta density. Figure \ref{dng3} below shows the graph of the $q$-density
$$\frac{x^{-0.5}\left(
1-q^3\frac{x^3}{[3]_q}\right)_{q^3}^{\frac{0.5}{3}-1}}{B_{q,3}(0.5,0.5)[3]_q^{\frac{0.5}{3}}} $$
\begin{figure}[h!]\label{dng3}
\centering
\includegraphics[width=10cm,height=5cm]{densidadnormalizadak3ts05}
\caption{Display of the function $\frac{x^{-0.5}\left(
1-q^3\frac{x^3}{[3]_q}\right)_{q^3}^{\frac{0.5}{3}-1}}{B_{q,3}(0.5,0.5)[3]_q^{\frac{0.5}{3}}} $ for $0 \leq q < 1$ and $0 \leq x \leq 1$.}
\end{figure}
Our final result provides an explicit formula for the cumulative beta $q$-distribution. If follows as an easy consequence of the
definition of the Jackson integral and the definition of $B_{q,k}(t,s)$.
\begin{thm}{\em
The cumulative beta $q$-distribution is given by
$$\frac{1}{B_{q,k}(t,s)[k]_q^{\frac{t}{k}}}\int_0^x s^{t-1}\left(
1-q^k\frac{s^k}{[k]_q}\right)_{q^k}^{\frac{s}{k}-1} d_qx =
\frac{(1-q)x^t}{B_{q,k}(t,s)[k]_q^{\frac{t}{k}}}\sum_{n=0}^{\infty}q^{nt}\left(
1-q^{k(n+1)}\frac{x^k}{[k]_q}\right)_{q^{k}}^{\frac{s}{k}-1} .$$}
\end{thm}
\begin{figure}[h!]\label{dng4}
\centering
\includegraphics[width=10cm,height=5cm]{distbetak3ts05}
\caption{Cumulative distribution of the $q$-density $\frac{x^{-0.5}\left(
1-q^3\frac{x^3}{[3]_q}\right)_{q^3}^{\frac{0.5}{3}-1}}{B_{q,3}(0.5,0.5)[3]_q^{\frac{0.5}{3}}} $ for $0 \leq q < 1$ and $0 \leq x \leq 1$.}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,065
|
\section{INTRODUCTION}\label{sec:introduction}
Learning-based control is seeing growing interest due to the abundance of data being collected in today's control systems. Especially reinforcement learning has demonstrated that controllers can be learned for complex or even uncertain cost functions and system models, see e.g. \cite{ng2006autonomous}, \cite{vanhasselt2016deep}. However, these methods often lack safety guarantees, i.e., the proposed control actions of the learning-based algorithm could lead the system into unsafe regions of the state space, e.g., a quadrotor approaching the ground with high speed, especially during exploration. This limits their application to safety-critical systems, e.g., autonomous transportation systems or medical applications, where certain state and input constraints are required to be satisfied for safety.
In order to leverage the advantages from learning-based control while ensuring constraint satisfaction, modular, invariance-based safety frameworks have been developed using control barrier functions, see e.g. \cite{wieland2007constructive} and \cite{ames2019control}, or Hamilton-Jacobi reachability, as discussed, e.g., in \cite{gillulay2011guaranteed} and \cite{chen2018hamilton}.
As these approaches can be computationally challenging or difficult to design in the case of larger scale systems, they have been extended using Model Predictive Control (MPC) techniques, see e.g. \cite{rawlings2017model}, providing a scalable safety framework for linear dynamics in \cite{wabersich2018linear}, with extensions for probabilistic, nonlinear or distributed systems in \cite{wabersich2021probabilistic}, \cite{wabersich2021safe} and \cite{muntwiler2020distributed}, respectively. Here, a predictive control problem is solved at every time step to find the closest input to a proposed learning-based input together with a trajectory satisfying all state and input constraints and leading to a terminal safe set. This safe set is a set in the state space, which ensures that constraint satisfaction can be guaranteed at all future time steps through the use of a safety controller. The approach itself implicitly defines a safe set through the feasible set of the predictive control problem, ensuring the existence of a safe backup trajectory for the system.
While existing formulations \cite{wabersich2021probabilistic,wabersich2021safe,muntwiler2020distributed} are tailored to specific model classes, they do not provide a principled mechanism to adaptively refine the underlying system model using incoming state measurements, while maintaining recursive feasibility guarantees.
In this paper, we propose an adaptive Model Predictive Safety Certification (MPSC) scheme, which considers linear models with parametric uncertainties and unknown but bounded additive disturbances. The proposed scheme allows to augment any learning-based controller such that state and input constraint satisfaction properties are ensured for all future time steps.
Instead of performing episodic model learning updates, we leverage recent results from adaptive MPC literature, see \cite{lorenzen2019robust}, \cite{lu2019robust}, \cite{kohler2019linear} and \cite{kohler2020nonlinear}, to estimate uncertain parameters in the system dynamics online. By using set-membership estimation, implausible model parametrisations are recursively eliminated, see \cite{milanese1991optimal}.
This results in a rigorous adaptive refinement of the MPSC scheme, which ensures safety with respect to the uncertain parameters as well as exogenous disturbances through recursive feasibility and guarantees a non-deteriorating performance.
Through less restrictive assumptions on the terminal set used in the predictive control problem, the design procedure for the proposed adaptive MPSC scheme is simplified compared to previous robust adaptive MPC schemes.
Additionally, we propose a terminal safe set enlargement similar to \cite{bujarbaruah2018adaptive}, which reduces the effect of potentially short planning horizons on the performance due to real-time computation requirements. The terminal safe set enlargement can be performed online using solved instances of the MPSC optimisation problem by using the convex hull of the corresponding tubes.
In this paper we focus on a linear system model as specified in Section~\ref{sec:preliminaries}, for which we derive the proposed method in Section~\ref{sec:ampsc}.
A discussion of an efficient design procedure for the adaptive MPSC using polytopic disturbance and parameter sets and homothetic tubes is provided in Section~\ref{sec:effampsc}. Finally, a numerical example for a chain of mass-spring-damper systems is provided in Section~\ref{sec:numericalexample} to illustrate the increase in size of the resulting safe sets and a comparison to the MPSC in \cite{wabersich2018linear} is provided.
\section{PRELIMINARIES}\label{sec:preliminaries}
\textit{Notation:}
The set of integers ranging from $a$ to $b$ is denoted by $\mathcal{I}_{[a,b]}$, the set of all positive integers is $\mathcal{I}_{\geq 0}$ and $2^{\mathbb{A}}$ denotes the power set of the set $\mathbb{A}$.
We define the unit hypercube as $\mathbb{B}_n=\{x\in\mathbb{R}^n|\;\Vert x \Vert_\infty\leq 0.5\}$. The Minkowski sum of two sets $\mathbb{A}\subseteq\mathbb{R}^n$ and $\mathbb{B}\subseteq\mathbb{R}^n$ is given by $\mathbb{A}\oplus\mathbb{B}=\{a+b|\;a\in\mathbb{A}, b\in\mathbb{B}\}$ with $a,b\in\mathbb{R}^n$.
The convex hull of a set $\mathbb{A}$ is denoted as $\textup{co}(\mathbb{A})$ and the $i$-th entry of the vector $a$ is denoted $[a]_i$. The projection of a set $\mathbb{A}\subset\mathbb{R}^m$ onto the first $n$ dimensions, where $m\geq n$, is given by $\textup{Proj}^n(\mathbb{A})$ and onto the last $n$ dimensions by $\textup{Proj}_n(\mathbb{A})$.
\subsection{Problem Description}\label{sec:problemdescription}
We consider uncertain discrete-time linear dynamics of the form
\begin{equation} \label{eq:lindynamics}
x_{k+1}=A(\theta)x_k+B(\theta)u_k+w_k,
\end{equation}
with states $x_k\in\mathbb{R}^n$, inputs $u_k\in\mathbb{R}^m$, disturbances $w_k\in\mathbb{W}\subseteq\mathbb{R}^n$ and uncertain parameters $\theta\in\mathbb{R}^p$.
We assume that the true system dynamics are captured by \eqref{eq:lindynamics} with parameters equal to their true value $\theta=\theta^*$.
The considered disturbance is bounded in a compact set $\mathbb{W}$ and the parameters $\theta$ lie within an a priori known, compact set of parameters $\Theta_0$, which includes the true value $\theta^*$.
\begin{remark}
The considered problem description also captures nonlinear systems, where a range of parameters $\theta$ can explain the system evolution if the disturbance set $\mathbb{W}$ is enlarged to encompass the error between the considered linear model and the true nonlinear dynamics. An extension to fully support nonlinear dynamics models is given in Appendix~\ref{sec:nonlinearextension}.
\end{remark}
This is a common problem setup used in robust adaptive model predictive control frameworks, see e.g. \cite{lorenzen2019robust}, \cite{lu2019robust} and \cite{kohler2019linear}. The uncertain parameters $\theta$ are assumed to enter the dynamics \eqref{eq:lindynamics} affinely as follows:
\begin{assumpt}\label{ass:bounded}
The system matrices $A(\theta)$ and $B(\theta)$ depend affinely on the parameter vector $\theta\in\mathbb{R}^p$ such that
\begin{equation}
(A(\theta),B(\theta))=(A_0,B_0)+\sum_{i=1}^p(A_i,B_i)[\theta]_i,
\end{equation}
where $A_0, A_i\in\mathbb{R}^{n\times n}$ and $B_0,B_i\in\mathbb{R}^{n\times m}$
\end{assumpt}
Note that such a model description can be derived from a linear system model \eqref{eq:lindynamics} by reformulating parameters which affect the system matrices nonlinearly as new parameters $\theta$ if their influence can be bounded, as is done, e.g., in \cite{didier2021robust}.
The system \eqref{eq:lindynamics} is subject to polytopic safety-critical state and physical input constraints given by
\begin{equation}\label{eq:stateinputconstr}
(x_k,u_k)\in\mathbb{Z}=\{(x,u)\in\mathbb{R}^n\times\mathbb{R}^m|\;Fx+Gu\leq z\},
\end{equation}
where $F\in\mathbb{R}^{n_z\times n}$, $G\in\mathbb{R}^{n_z\times m}$ and $z\in\mathbb{R}^{n_z}$.
The projection of the constraint set $\mathbb{Z}$ onto the state space $\mathbb{R}^n$ and input space $\mathbb{R}^m$ is defined as
$\mathbb{X}=\textup{Proj}^n(\mathbb{Z})$ and
$\mathbb{U}=\textup{Proj}_m(\mathbb{Z})$, respectively.
\subsection{Parameter Identification}\label{sec:parameteridentification}
Instead of inferring parameter estimates a priori from data as done in, e.g., \cite{wabersich2021safe}, we begin with a set of possible parameters, which will iteratively be refined online using incoming state measurements. More precisely, starting from an initial uncertainty set $\Theta_0$, which could arise in practice from, e.g., production tolerances or tasks with uncertain parameters like lifting an object with uncertain mass as in \cite{didier2021robust}, new sets $\Theta_k$ are inferred with the properties given in the following assumption.
\begin{assumpt}\label{ass:parameter}
The parameter identification method fulfils for all $k\geq0$
\begin{enumerate}
\item Consistency of the identification method, i.e., if the true parameter $\theta^*\in\Theta_0\Rightarrow\theta^*\in\Theta_k$
\item Recursive set estimate inclusion, i.e., $\Theta_{k+1}\subseteq\Theta_{k}\subseteq\Theta_0$
\end{enumerate}
\end{assumpt}
Note that this assumption encompasses any parameter identification method for which a set of parameters is guaranteed to contain the true parameter. If consecutive sets are not recursively contained within each other, e.g., due to restrictions on $\Theta_k$ for computational reasons, the sets can be updated only when they are a subset of the previously used set. For example, if confidence sets obtained via Bayesian Linear Refression are used for the parameters such that $\textup{Pr}(\theta^*\in\Theta_k)\geq p_\theta$ for some desired probability level $p_\theta$, then recursive inclusion of the set estimates is not guaranteed given new data and needs to be verified online.
Different set-membership estimation methods exist that fulfil the properties in Assumption~\ref{ass:parameter} by construction, such as a polytopic formulation in \cite{milanese1991optimal} and a spherical formulation in \cite{dhaliwal2012set}. By using such an adaptive model refinement, we derive the adaptive MPSC scheme in the following section. The computation of polytopic parameter sets using set-membership estimation is detailed in Section~\ref{sec:efficientAMPSC}, which allows for a computationally efficient adaptive MPSC scheme.
\section{ADAPTIVE MODEL PREDICTIVE SAFETY CERTIFICATION}\label{sec:ampsc}
The proposed adaptive MPSC scheme is a modular framework, which takes as an input a learning-based control action $u_k^\mathscr{L}$ and the current state in order to verify the safety of the proposed action based on computing a safe forward plan using a sequentially improved data-driven model.
A schematic of this framework can be seen in Figure \ref{fig:schematic}, where the applied control input corresponds to the MPSC policy, i.e., $u_k=\pi_{\textup{MPSC}}(u_k^\mathscr{L},x_k,\Theta_k,k)$.
\begin{figure}[h]
\vspace{0.2cm}
\centering
\begin{tikzpicture}[node distance=2cm, every text node part/.style={align=center}]
\node (System) [rectangle, text centered, draw=black, minimum height=1.3cm, minimum width=4cm] {System \\ $x_{k+1}=A(\theta)x_k+B(\theta)u_k+w_k$};
\node (Learning) [rectangle, text centered, draw=black, minimum height=1.3cm, minimum width= 3.5cm,below of=System, xshift=2.2cm, yshift=0.2cm] {Learning-based \\ Controller};
\node (Parameter) [rectangle, text centered, draw=black, minimum height=1.3cm, minimum width= 3.5cm,below of=Learning, yshift=0.2cm] {Parameter Estimation};
\node (Adaptive) [rectangle, text centered, draw=black, minimum height=1.3cm, minimum width= 3.5cm,below of=System, xshift=-2.2cm, yshift=-0.7cm] {Adaptive MPSC};
\draw [->, to path={-| (\tikztotarget)}] (System) -- node[anchor=south] {$x_k$} (4.2cm,0) |- (Learning.east);
\draw [->, to path={-| (\tikztotarget)}] (System) -- (4.2cm,0) |- (Parameter.east);
\draw [->, to path={-| (\tikztotarget)}] (System) -- (4.2cm,0) |- (Adaptive.east);
\draw [->, to path={-| (\tikztotarget)}] (Learning.west) -- node[anchor=south] {$u_k^\mathscr{L}$} (-0.2cm, -1.8cm) |- ([yshift=0.3cm]Adaptive.east);
\draw [->, to path={-| (\tikztotarget)}] (Parameter.west) -- node[anchor=north] {$\Theta_k$} (-0.2cm, -3.6cm) |- ([yshift=-0.3cm]Adaptive.east);
\draw [->, to path={-| (\tikztotarget)}] (Adaptive.west) -| (-4.2cm, 0cm) -- node[anchor=south] {$u_k$} (System.west);
\end{tikzpicture}
\caption{Schematic of the adaptive Model Predictive Safety Certification framework. Given the current state of the system $x_k$, a learning-based control input $u_k^\mathscr{L}$ and the set of possible parameters $\Theta_k$, resulting from the parameter estimation,
the input $u_k$ to be applied to the system is provided by the adaptive Model Predictive Safety Certification scheme.}
\label{fig:schematic}
\end{figure}
The proposed method is based on computing a state and input backup trajectory from the current state to a terminal safe set, with the goal of matching the first element of the input backup sequence with the desired learning input at each time step.
If the final predicted backup states are contained in the terminal safe set, constraint satisfaction can be guaranteed at all further time steps.
In the following, we begin by formalising the terminal safe set, which is used to define the MPSC algorithm in Section~\ref{sec:ampscalg}. The optimisation problem, which is solved at every time step in order to compute the backup trajectory is then discussed and the algorithm of the adaptive MPSC is provided. Finally, extensions of the scheme are presented by updating the terminal safe set.
\subsection{Adaptive Model Predictive Safety Certification Algorithm} \label{sec:ampscalg}
In order to guarantee that the constraints \eqref{eq:stateinputconstr} can be satisfied for all times, the concept of a safe set is used, as defined in \cite{wabersich2018linear}, \cite{wabersich2021probabilistic}, \cite{wabersich2021safe} and \cite{wabersich2018scalable}. A safe set is a set in the state space, which ensures constraint satisfaction through the use of a safe control policy $\pi_\mathbb{S}$.
\begin{defn}\label{def:safeset}
A set $\mathbb{S}\subseteq\mathbb{X}$ is called a safe set for system \eqref{eq:lindynamics} if a safe backup control law $\pi_\textup{B}:\mathbb{R}^m\times\mathbb{R}^n\times 2^{\mathbb{R}^p}\times\mathcal{I}_{\geq0}\rightarrow\mathbb{U}$ is available such that for an arbitrary (learning-based) action $u_k^\mathscr{L}\in\mathbb{R}^m$, the application of the safe control policy
\begin{gather}
\begin{align}
&\pi_{\mathbb{S}}(u_k^\mathscr{L}, x_k, \Theta_k, k)=\nonumber \\
&\begin{cases}
u_k^\mathscr{L}, \textup{ if }(x_k, u_k^\mathscr{L}){\in}\mathbb{Z} {\wedge}\{A(\theta)x_k{+}B(\theta)u_k^\mathscr{L}\}{\oplus}\mathbb{W}{\subseteq}\mathbb{S}\;\forall\theta{\in}\Theta_k \\
\pi_\textup{B}B(u_k^\mathscr{L}, x_k, \Theta_k, k), \textup{ otherwise}
\end{cases}\nonumber
\end{align}
\end{gather}
guarantees constraint satisfaction of the system state and inputs, i.e., $(x_k, \pi_{\mathbb{S}}(u_k^\mathscr{L}, x_k, \Theta_k, k))\in\mathbb{Z}$ for all $k\geq\bar{k}$ if $x_{\bar{k}}\in\mathbb{S}$.
\end{defn}
A safe set thus provides a guarantee that the state and input constraints are satisfied for all times $k\geq\bar{k}$ by using the safe control policy $\pi_{\mathbb{S}}(u_k^\mathscr{L}, x_k, \Theta_k, k)$ if the state $x_{\bar{k}}$ is in the safe set $\mathbb{S}$ at time step $\bar{k}$.
Note that for the convex polytopes resulting from set-membership estimation, it suffices to check the condition $\{A(\theta^j_k)x_k+B(\theta^j_k)u_k^\mathscr{L}\}\oplus\mathbb{W}\subseteq\mathbb{S}$ at every vertex $\theta_k^j$ of the polytope $\Theta_k$.
\begin{remark}
While a Robust Positively Invariant (RPI) set used in robust MPC, see e.g. \cite[Chapter 2.6]{rawlings2017model}, requires that any possible state evolution starting inside the set will be contained in the set, Definition~\ref{def:safeset} of a safe set only requires that starting from a certain subset implies safety for all future times. This allows, e.g., to define a safe set using expert knowledge without the need for expensive offline computations.
However, principled robust invariant set computations can be employed and are available for parametric uncertainties, e.g., according to the algorithm provided in \cite{pluymers2005efficient}, even though they suffer from limited scalability. Ellipsoidal RPI sets for linear feedback controllers can be computed through semi-definite programming, see e.g. \cite{wabersich2018scalable} and \cite[Appendix]{kohler2019linear}.
\end{remark}
Due to the uncertain model, the computation of the required backup trajectory can be conservative, motivating the use of recent advances in robust adaptive MPC schemes \cite{lorenzen2019robust}, \cite{lu2019robust}, \cite{kohler2019linear} and \cite{kohler2020nonlinear}. At every time step $k$, we compute a tube in the state space starting from the current state measurement $x_k$, which is guaranteed to contain the future states for any disturbances in $\mathbb{W}$ and uncertain parameters in $\Theta_k$ through the use of a tube control law $\kappa:\mathbb{R}^n\times\mathbb{R}^{n_v}\rightarrow\mathbb{R}^m$ with $n_v$ parameters. The tube then consists of sets $\mathbb{X}_{l|k}$, which are predicted at time $k$, given the tube control law $\kappa(x,v_{l|k})$, for $l\in\mathcal{I}_{[0,N]}$ future time steps given a horizon $N$, with the last polytope being constrained to lie in a terminal safe set $\mathbb{S}_f$. The last predicted set $\mathbb{X}_{N|k}$ is required to be a subset of a terminal safe set $\mathbb{S}_f$, which fulfils Definition~\ref{def:safeset} with the safe control policy $\pi_{\mathbb{S}_f}(u_k^\mathscr{L}, x_k, \Theta_k, k)$. This allows to ensure constraint satisfaction for further time steps if
for all $x\in\mathbb{X}_{l|k}$, it holds that $(x,\kappa(x,v_{l|k}))\in\mathbb{Z}$.
The adaptive MPSC algorithm is a modular framework which uses a learning-based controller for performance, i.e., the goal is to apply the learning-based input if constraint satisfaction can be ensured. This objective is realised by minimising the norm of the difference of the first control input of the planned tube and the proposed learning-based input $u_k^\mathscr{L}$. The optimisation problem we solve at every time step is thus given as
\begin{subequations} \label{eq:optimisation}
\begin{alignat}{1}
\min_{v_{\cdot |k}, \mathbb{X}_{\cdot|k}} & \Vert u_k^\mathscr{L}-\kappa(x_k, v_{0|k})\Vert \\
\textup{s.t. } & \forall l\in\mathcal{I}_{[0,N-1]} \nonumber \\
& x_k \in\mathbb{X}_{0|k}, \label{eq:initialconstr} \\
& A(\theta)x+B(\theta)\kappa(x,v_{l|k})+w \in \mathbb{X}_{l+1|k}, \nonumber\\
&\qquad\qquad \forall x\in\mathbb{X}_{l|k}, w\in\mathbb{W}, \theta\in\Theta_k,\label{eq:tubeincconstr} \\
&(x,\kappa(x,v_{l|k}))\in\mathbb{Z}, \quad \forall x \in\mathbb{X}_{l|k}, \label{eq:stateconstr} \\
&\mathbb{X}_{N|k}\subseteq \mathbb{S}_f. \label{eq:terminalconstr}
\end{alignat}
\end{subequations}
As \eqref{eq:optimisation} is not guaranteed to be recursively feasible due to the weak terminal safe set assumption compared to a robust invariant set, a switching mechanism is introduced similar to \cite{wabersich2018linear} in case the optimisation problem becomes infeasible. The mechanism then switches to the last computed optimal solution of \eqref{eq:optimisation} at time step $\bar{k}$, i.e., $\kappa(x_{\bar{k}+l}, v^*_{l|\bar{k}})$ for $l\in\mathcal{I}_{[1,N]}$. This input sequence guarantees that the state reaches the terminal safe set according to \eqref{eq:terminalconstr}. At this point, if \eqref{eq:optimisation} remains infeasible, the backup controller according to Definition~\ref{def:safeset} is used, such that safety is ensured for all time steps. The described procedure is formalised in Algorithm~1.
\begin{remark}
For a less intrusive safety filter algorithm, Line~10 in Algorithm~1 can be replaced with \\
\centerline{\textup{10: Solve \eqref{eq:optimisation} with horizon $N-k_{\textup{inf}}$,}}\\
which preserves the safety guarantees, similar to \cite{wabersich2021safe}.
\end{remark}
If the initial state $x_0$ of the system lies within the feasible set of \eqref{eq:optimisation} for the initial unknown parameter set $\Theta_0$, denoted as $\mathbb{X}_{\textup{feas}}(\Theta_0)$, or within the terminal safe set $\mathbb{S}_f$, Algorithm~1 guarantees constraint satisfaction for all time steps $k\geq0$ by construction. This follows from the set update of $\Theta_k$ in Line 3 of Algorithm~1, which ensures that $\theta^*\in\Theta_k$ for all $k$ if $\theta^*\in\Theta_0$ under Assumption~\ref{ass:parameter}. It is thus possible to show that the feasible set of \eqref{eq:optimisation} implicitly describes a safe set. Additionally, through the update of the parameter set $\Theta_k$ under Assumption~\ref{ass:parameter}, the size of the feasible set increases as the parameter estimate improves, i.e., $\mathbb{X}_{\textup{feas}}(\Theta_{k-1})\subseteq\mathbb{X}_{\textup{feas}}(\Theta_k)$ as $\Theta_{k-1}\supseteq\Theta_k$.
\vspace{-0.2cm}
\begin{algorithm}[H]\label{alg:1}
\caption{Adaptive Model Predictive Safety Certification Scheme.}
\begin{algorithmic}[1]
\State $k_{\textup{inf}}\leftarrow N-1$
\For{$k=0,1,\dots$}
\State Update $\Theta_k$ using the state measurement $x_k$
\If{\eqref{eq:optimisation} is feasible}
\State Apply $u_k\leftarrow \kappa(x_k,v^*_{0|k})$ to \eqref{eq:lindynamics}
\State $k_{\textup{inf}}\leftarrow0$
\Else
\State $k_{\textup{inf}}\leftarrow k_{\textup{inf}}+1$
\If{$k_{\textup{inf}}\leq N-1$}
\State Apply $u_k\leftarrow \kappa(x_k,v^*_{k_{\textup{inf}}|k-k_{\textup{inf}}})$ to \eqref{eq:lindynamics}
\Else
\State Apply $u_k\leftarrow \pi_{\mathbb{S}_f}(u_k^\mathscr{L}, x_k, \Theta_k, k)$ to \eqref{eq:lindynamics}
\EndIf
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\vspace{-0.5cm}
\begin{thrm}\label{thrm:AMPSC}
If Assumptions \ref{ass:bounded} and \ref{ass:parameter} hold, the control law $\pi_{\textup{MPSC}}(u_k^\mathscr{L},x_k,\Theta_k,k)$ resulting from Algorithm~1 is a safe backup controller and the set $\mathbb{X}_{\textup{feas}}(\Theta_k)\cup\mathbb{S}_f$ is the corresponding safe set at time step $k$ according to Definition~\ref{def:safeset}. Additionally, it holds that $\mathbb{X}_{\textup{feas}}(\Theta_{0})\subseteq\mathbb{X}_{\textup{feas}}(\Theta_{1})\subseteq\dots\subseteq\mathbb{X}_{\textup{feas}}(\Theta_{k})$ for all time steps $k>0$.
\end{thrm}
\begin{proof}
The first part of this proof is analogous to the proof of \cite[Theorem~III.5]{wabersich2018linear}. Consider $x_0\in\mathbb{S}_f\setminus\mathbb{X}_{\textup{feas}}(\Theta_0)$, through the initialisation of $k_{\textup{inf}}$, $\pi_{\mathbb{S}_f}(u_k^\mathscr{L}, x_k, \Theta_k, k)$ is applied to the system, which according to Definition~\ref{def:safeset} ensures constraint satisfaction for all future time steps. If $x_0\in\mathbb{X}_{\textup{feas}}(\Theta_0)$ and \eqref{eq:optimisation} is feasible for all $k\geq0$, it follows that safety is ensured through the constraints (\ref{eq:initialconstr}-d) as $\theta^*\in\Theta_k$ under Assumption~\ref{ass:parameter}, see e.g. \cite[Theorem~14]{lorenzen2019robust}. If at any given time step $\bar{k}$, \eqref{eq:optimisation} becomes infeasible, the optimal control input $ \kappa(x_{\bar{k}+k_{\textup{inf}}},v^*_{k_{\textup{inf}}|\bar{k}-1})$ from time step $\bar{k}-1$ is used until $x_{\bar{k}-1+N}\in\mathbb{X}_{\bar{k}-1+N|\bar{k}-1}\subseteq\mathbb{S}_f$ according to \eqref{eq:tubeincconstr} and \eqref{eq:terminalconstr}. At this point, Algorithm~1 switches to using the safe control input $\pi_{\mathbb{S}_f}(u_k^\mathscr{L}, x_k, \Theta_k, k)$. Thus constraint satisfaction is guaranteed by \eqref{eq:stateconstr} and the definition of the terminal safe set.
Through the parameter set update it holds that $\Theta_{k-1}\supseteq\Theta_k$, as follows from Assumption~\ref{ass:parameter}. It therefore holds that any state $x\in\mathbb{X}_{\textup{feas}}(\Theta_{k-1})$ must fulfil $x\in\mathbb{X}_{\textup{feas}}(\Theta_k)$ as constraint \eqref{eq:tubeincconstr} is fulfilled for all $\theta\in\Theta_{k}\subseteq\Theta_{k-1}$.
\end{proof}
\subsection{Iterative Enlargement of the Terminal Safe Set}\label{sec:ittermset}
While the terminal safe set can be enlarged using previously solved instances for adaptive MPC with unknown constant offset as is done in \cite{bujarbaruah2018adaptive}, it has not been discussed for adaptive MPC with parametrised system matrices to the best of the authors' knowledge.
Using a convex formulation \eqref{eq:optimisation}, it is possible to show that the convex hull of all initial polytopes $\mathbb{X}_{0|k}$ can be added to the terminal safe set. The convex hull of the set of time steps, where \eqref{eq:optimisation} was successfully solved, is denoted as $\mathcal{M}(k)=\{i\in\mathcal{I}_{[0,k]}|\; x_i\in\mathbb{X}_{\textup{feas}}(\Theta_i)\}$ and we use
\begin{equation}
\mathbb{X}^*_{0|\mathcal{M}(k)}=\textup{co}\left(\{\mathbb{X}^*_{0|i}\}_{i\in\mathcal{M}(k)}\right).
\end{equation}
The terminal safe set $\mathbb{S}_f$ can then be enlarged as follows.
\begin{thrm}\label{thrm:safesetenlargement}
If Assumptions \ref{ass:bounded} and \ref{ass:parameter} hold and \eqref{eq:optimisation} is convex, then the set
\begin{equation}
\mathbb{S}_f^{\mathcal{M}(k)}=\mathbb{X}^*_{0|\mathcal{M}(k)}\cup\mathbb{S}_f
\end{equation}
is again a safe set according to Definition~\ref{def:safeset} with a safe backup controller given by Algorithm~1 with terminal safe set $\mathbb{S}_f$.
\end{thrm}
\begin{proof}
As \eqref{eq:optimisation} is assumed to be convex, it follows that for a fixed parameter set $\Theta_k$, the feasible set $\mathbb{X}_{\textup{feas}}(\Theta_k)$ of \eqref{eq:optimisation} is also convex at every time step $k>0$, see \cite{boyd2004convex}. It then follows that $\mathbb{X}^*_{0|\mathcal{M}(k)}\subseteq\mathbb{X}_{\textup{feas}}(\Theta_k)$ as any $x\in\mathbb{X}^*_{0|i}$ admits a feasible solution to \eqref{eq:optimisation} for all $i\in\mathcal{M}(k)$. As it holds that $\mathbb{X}_{\textup{feas}}(\Theta_{k})\subseteq\mathbb{X}_{\textup{feas}}(\Theta_{k+1})$ if the parameter set is updated and that the union of two safe sets is a safe set, the result follows.
\end{proof}
\subsection{A Recursively Feasible MPSC Scheme}\label{sec:recursiveMPSC}
While the safe set according to Definition~\ref{def:safeset} supports an easier design, the resulting implementation becomes more complex due to the required switching mechanism. As an alternative, we additionally consider the case of requiring an RPI terminal set, for which we additionally provide a data-driven design using past data in Section~\ref{sec:ittermset}. In order to provide a recursively feasible optimisation problem \eqref{eq:optimisation}, we require that under the control law $\kappa(x,v)$, a $v$ exists such that all possible uncertain state evolutions from the last predicted state polytope $\mathbb{X}^*_{N|k}$ will be robustly contained in the terminal safe set.
\begin{assumpt} \label{ass:termset}
Consider a non-empty terminal set $\mathbb{X}_f$ and a tube control law $\kappa(x,v)$ in \eqref{eq:optimisation}. For every set $\mathcal{X}\subseteq\mathbb{X}_f$, there exists a $v$, such that $(x,\kappa(x,v))\in\mathbb{Z}$ for all $x\in\mathcal{X}$ and such that for all $\theta\in\Theta_0$ it holds that
\begin{equation}A(\theta)\mathcal{X}\oplus B(\theta)\kappa(\mathcal{X},v)\oplus\mathbb{W}\subseteq\mathbb{X}_f. \nonumber
\end{equation}
\end{assumpt}
Under Assumption~\ref{ass:termset}, recursive feasibility of \eqref{eq:optimisation} can be shown. Note that this generalised assumption contains specific robust adaptive MPC formulation such as \cite{lorenzen2019robust}, \cite{lu2019robust}, \cite{kohler2019linear} and \cite{kohler2020nonlinear} as special cases.
\begin{thrm}
Let $\mathbb{S}_f=\mathbb{X}_f$. If Assumptions \ref{ass:bounded}, \ref{ass:parameter} and \ref{ass:termset} hold, then $\kappa(x_k,v^*_{0|k})$
is a safe backup control law and $\mathbb{X}_{\textup{feas}}(\Theta_k)$ a corresponding safe set according to Definition~\ref{def:safeset}. In addition, the set $\mathbb{X}_{\textup{feas}}(\Theta_k)$ is a robust positively invariant set for a fixed $\Theta_k$.
\end{thrm}
\begin{proof}
The proof follows standard recursive feasibility arguments similar to, e.g., \cite{lorenzen2019robust}. Consider \eqref{eq:optimisation} feasible at time step $\bar{k}$. The optimal input sequence $\kappa(x_{\bar{k}},v^*_{l|\bar{k}})$ for $l\in\mathcal{I}_{[1,N]}$ ensures that $x_{l-1|\bar{k}+1}\in\mathbb{X}^*_{l|\bar{k}}$ since $x_{0|\bar{k}+1}\in\mathbb{X}^*_{1|\bar{k}}$ and according to Assumption~\ref{ass:parameter}, $\Theta_{\bar{k}+1}\subseteq\Theta_{\bar{k}}$. As $\mathbb{X}^*_{N|\bar{k}}\subseteq\mathbb{X}_f$, we can set $\mathbb{X}^*_{N|\bar{k}+1}=\mathbb{X}_f$ according to Assumption~\ref{ass:termset}, which fulfills the terminal constraint \eqref{eq:terminalconstr} with $\mathbb{S}_f=\mathbb{X}_f$, such that state and input constraints are satisfied.
Robust positive invariance follows directly from recursive feasibility, as $x_k\in\mathbb{X}_{\textup{feas}}(\Theta_k)\Rightarrow x_{k+1}\in\mathbb{X}_{\textup{feas}}(\Theta_{k})$.
\end{proof}
The design of a terminal set $\mathbb{X}_f$ fulfilling Assumption~\ref{ass:termset} for homothetic tube sets $\mathbb{X}_{l|k}$ is discussed in \cite{lorenzen2019robust}, \cite{kohler2019linear} and a low-complexity terminal set for a 12-dimensional quadrotor example is presented in \cite{didier2021robust}, whereas the condition is implemented as a constraint in the optimisation problem in \cite{lu2019robust}. Note that if a terminal set $\mathbb{X}_f$ fulfills Assumption~\ref{ass:termset}, a terminal safe set enlargement similar to Section~\ref{sec:ittermset} can be performed using the convex hull of all computed solutions $\mathbb{X}_{l|k}^*$ and the terminal set $\mathbb{X}_f$, as feasibility of \eqref{eq:optimisation} is guaranteed. The resulting set is then a safe set according to Definition~\ref{def:safeset}, but does not verify Assumption~\ref{ass:termset}, for which we need a different approach tailored to a specific tube structure as presented in Section~\ref{sec:ittermrecfeas}.
\section{EFFICIENT DESIGN USING POLYTOPIC SETS}\label{sec:effampsc}
In this section, we provide details on how a computationally efficient adaptive MPSC problem can be designed for the linear case by leveraging the formulations in \cite{lorenzen2019robust}, \cite{lu2019robust} and \cite{kohler2019linear}. We then show how the specific structure can be exploited to obtain a data-driven terminal set enlargement, resulting in a recursively feasible optimisation problem \eqref{eq:optimisation}.
\subsection{Homothetic Tube Formulation}\label{sec:efficientAMPSC}
The considered formulation makes use of recent reformulations of the constraints in \eqref{eq:optimisation} into linear constraints with respect to the optimisation variables in \cite{lorenzen2019robust}, \cite{lu2019robust} and \cite{kohler2019linear}. The considered sets $\mathbb{W}=\{w\in\mathbb{R}^n |\; H_ww\leq h_w\}$ and $\Theta_0=\{\theta\in\mathbb{R}^p |\; H_{\theta_0}\theta\leq h_{\theta_{0}}\}$ are assumed to be polytopic, with $H_w\in\mathbb{R}^{n_w\times n}$, $h_w\in\mathbb{R}^{n_w}$, $H_{\theta_0}\in\mathbb{R}^{n_\theta \times p}$ and $h_{\theta_0}\in\mathbb{R}^{n_\theta}$. In order to ensure polytopic sets $\Theta_k$, polytopic set-membership estimation is used, which consists of computing the set of all possible parameters that explain the system evolution given a set of possible disturbances $\mathbb{W}$.
For the considered dynamics \eqref{eq:lindynamics}, given state measurements $x_{k-1}$ and $x_k$, this non-falsified set of parameters is given by
\begin{equation}
\Delta_k =\{\theta\in\mathbb{R}^p |\; x_k-(A(\theta)x_{k-1}+B(\theta)u_{k-1})\in\mathbb{W}\},
\end{equation}
which is polytopic and whose explicit formulation is given in \cite{lorenzen2019robust}.
The parameter set $\Theta_k$ is updated by taking the intersection of the previous set $\Theta_{k-1}$ and the non-falsified parameter set
$\Theta_k=\Theta_{k-1}\cap\Delta_k.$
A major drawback of the proposed identification scheme is the potential increase in complexity of the resulting parameter sets through the addition of new half-spaces at every set update, which increases the computational complexity of the proposed adaptive MPSC scheme.
This issue can be addressed by fixing the shape of the parameter polytopes, e.g., by fixing the half-spaces, i.e., $\Theta_k=\{\theta\in\mathbb{R}^p|\;H_{\theta}\theta\leq h_{\theta_k}\}$, and only recomputing the right-hand side of the polytope inequality $h_{\theta_k}$ through the solution of a linear program (LP), as is shown in \cite{lorenzen2019robust}.
To further increase the computational update efficiency of the parameter identification as well as the efficiency of the proposed adaptive MPSC scheme, the set of parameters can be restricted to hypercubes with centre $\bar{\theta}_k\in\mathbb{R}^p$ and size $\eta_k\geq0$, i.e., $\Theta_k=\{\bar{\theta}_k\}\oplus\eta_k\mathbb{B}_p$ as described in \cite{kohler2019linear} and \cite{kohler2020nonlinear}.
This parametrisation results in $2p$ LPs to find the minimal and maximal values of $\theta$ in $\Theta_{k-1}\cap\Delta_k$ in every parameter dimension, thereby computing the smallest bounding hypercube of the intersection.
By using a tube controller $\kappa(x,v_{l|k})=Kx+v_{l|k}$ and a homothetic tube formulation for the sets
$\mathbb{X}_{l|k}=\{z_{l|k}\}\oplus\alpha_{l|k}\mathbb{X}_0$, with $\mathbb{X}_0=\{x\in\mathbb{R}^n|H_xx\leq\mathbf{1}\}$, $H_x\in\mathbb{R}^{n_x\times n}$ and $\alpha_{l|k}\geq0$, the optimisation problem \eqref{eq:optimisation} can be formulated as a quadratic program if $\mathbb{S}_f$ is also a polytope, with optimisation variables $v_{l|k}, z_{l|k}$ and $\alpha_{l|k}$. In \cite{kohler2019linear}, $z_{l|k}$ are computed according to dynamics \eqref{eq:lindynamics} with the center of the hypercube $\Theta_k$ as parameters, allowing for a more computationally efficient reformulation.
Note that in the homothetic tube formulations, the terminal constraint \eqref{eq:terminalconstr} is given by $(z_{N|k},\alpha_{N|k})\in\mathbb{X}_f$, where the terminal set $\mathbb{X}_f$ is a set of translations and dilations $(z,\alpha)$. which can be iteratively enlarged through previously solved instances of \eqref{eq:optimisation} as shown in the next section.
\subsection{Iterative Terminal Set Enlargement for Recursive Feasibility}\label{sec:ittermrecfeas}
A recursively feasible MPSC problem facilitates the implementation of Algorithm~1, however it introduces the design task of finding a possibly large terminal set in order to reduce conservative safety interventions. We thus propose a mechanism to iteratively enlarge a terminal set for the homothetic tube formulation as described in Section~\ref{sec:efficientAMPSC} such that recursive feasibility is guaranteed when employing this new terminal set.
As discussed in Section~\ref{sec:recursiveMPSC}, we select the terminal set $\mathbb{X}_f$ such that for every translation and dilation $(z,\alpha)\in\mathbb{X}_f$, a translation and dilation in the terminal set exists at the next time step, ensuring recursive feasibility of \eqref{eq:optimisation}. This assumption on the terminal set is common in the robust adaptive MPC literature and is stated explicitly in \cite{lorenzen2019robust}, \cite{kohler2019linear} and \cite{kohler2020nonlinear} and used implicitly in \cite{lu2019robust} in the online optimisation problem.
\begin{assumpt} \label{ass:homothetictubeterm}
Let $\mathbb{W}$, $\bar{\Theta}$ be polytopic and let $\mathbb{X}_{l|k}$ be of the form $\{z_{l|k}\}\oplus\alpha_{l|k}\mathbb{X}_0$ with polytopic $\mathbb{X}_0=\{H_xx\leq\mathbf{1}\}$ and $H_x\in\mathbb{R}^{n_x\times n}$.
Consider a non-empty terminal set $\mathbb{X}_f=\{(z,\alpha)| H_Tz+h_T\alpha\leq\mathbf{1}\}$, with $H_T\in\mathbb{R}^{n_T\times n}$ and $h_T\in\mathbb{R}^{n_T}$, and a tube control law $\kappa(x,v)=Kx+v$ in \eqref{eq:optimisation}. For every $(z,\alpha)\in\mathbb{X}_f$ , there exists a $v$ and $(z^+,\alpha^+)\in\mathbb{X}_f$, such that for all $\theta\in\bar{\Theta}$ and $x\in\{z\}\oplus\alpha\mathbb{X}_0$, it holds that $(x,\kappa(x,v))\in\mathbb{Z}$ and $A(\theta)(\{z\}\oplus\alpha\mathbb{X}_0)\oplus B(\theta)K(\{z\}\oplus\alpha\mathbb{X}_0)\oplus\{ B(\theta)v\}\oplus\mathbb{W}\subseteq\{z^+\}\oplus\alpha^+\mathbb{X}_0.$
\end{assumpt}
By using solved instances of \eqref{eq:optimisation} with optimal $(z_{l|k}^*,\alpha_{l|k}^*)$, the terminal set can then be enlarged, such that a recursively feasible optimisation problem is recovered.
\begin{thrm}
Let Assumptions~\ref{ass:bounded},\ref{ass:parameter} and \ref{ass:homothetictubeterm} hold and \eqref{eq:optimisation} be convex, then the set
\begin{equation}\label{eq:homotheticsetenlargement}
\mathbb{X}_f^{\mathcal{M}(\bar{k})}=\textup{co}\left(\{(z^*_{l|k},\alpha^*_{l|k})\}_{l\in\mathcal{I}_{[0,N]},k\in\mathcal{M}(\bar{k})},\mathbb{X}_f\right)
\end{equation}
satisfies Assumption~\ref{ass:homothetictubeterm} with respect to $\bar{\Theta}=\Theta_{\bar{k}}$
\end{thrm}
\begin{proof}
We denote the $n_X$ vertices of $\mathbb{X}_f$ as $\{(z^*_{0|k},\alpha^*_{0|k})\}_{k\in\mathcal{I}_ {[-n_X,-1]}}$ and construct corresponding tuples $(z^*_{l|k},\alpha^*_{l|k}){\in}\mathbb{X}_f$ for $l{\in}\mathcal{I}_{[1,N+1]}$ that satisfy Assumption~\ref{ass:homothetictubeterm} for consecutive pairs $l$ and $l{+}1$. We then denote as $\mathcal{N}{=}\mathcal{M}(\bar{k})\!\cup\mathcal{I}_{[-n_X,-1]}$ the set of solved time steps $\mathcal{M}(\bar{k})$, together with the $n_X$ constructed solutions for each vertex of $\mathbb{X}_f$. Through the solutions of \eqref{eq:optimisation} and the constructed solutions we have for all $l\in\mathcal{I}_{[0,N]}$, $k\in\mathcal{N}$ and $\theta\in\Theta_{\bar{k}}\subseteq\Theta_k$ , that it holds that
\vspace{-0.1cm}
\begin{equation*}
\vspace{-0.1cm}
A_{\!cl}(\theta)(\!\{z_{l|k}^*\}{\oplus}\alpha^*_{l|k}\mathbb{X}_0\!){\oplus}\{\!B(\theta)v_{l|k}^*\!\}{\oplus}\mathbb{W}{\subseteq}\{\!z_{l+1|k}^*\!\}{\oplus}\alpha_{l+1|k}^*\mathbb{X}_0,
\end{equation*}
where we define $A_{cl}(\theta){=}A(\theta){+}B(\theta)K$ and use the fact that $(z^*_{N+1|k},\alpha^*_{N+1|k}){\in}\mathbb{X}_f$ exists according to Assumption~\ref{ass:homothetictubeterm} as $(z^*_{N|k},\alpha^*_{N|k}){\in}\mathbb{X}_f$.
For any $(z,\alpha)\in\mathbb{X}_f^{\mathcal{M}(\bar{k})}$, we can write $(z,\alpha)=\sum_{l\in\mathcal{I}_{[0,N]}}\sum_{k\in\mathcal{N}}\lambda_{l|k}(z_{l|k}^*,\alpha_{l|k}^*)$ due to the convex hull, where it holds that $\sum_{l\in\mathcal{I}_{[0,N]}}\sum_{k\in\mathcal{N}}\lambda_{l|k}{=}1$, $\lambda_{l|k}{\geq}0$.
We then choose $v{=}\sum_{l\in\mathcal{I}_{[0,N]}}\sum_{k\in\mathcal{N}}\lambda_{l|k}v^*_{l|k}$ where $v^*_{l|k}$ corresponds to the input solution of \eqref{eq:optimisation} at time step $l|k$, and the corresponding $(z^+,\alpha^+){=}\sum_{l\in\mathcal{I}_{[0,N]}}\sum_{k\in\mathcal{N}}\lambda_{l|k}(z^*_{l+1|k},\alpha^*_{l+1|k})$.
It then follows that for all $\theta\in\Theta_{\bar{k}}$,
\begin{gather}
\begin{align}
&A_{cl}(\theta)\left(\{z\}\oplus\alpha\mathbb{X}_0\right)\oplus\{B(\theta)v\} \oplus\mathbb{W}\nonumber\\
=&A_{cl}(\theta)\Big(\{{\sum_{l\in\mathcal{I}_{[0,N]}}}{\sum_{k\in\mathcal{N}}}\lambda_{l|k}z_{l|k}^*\}
{\oplus}\big({\sum_{l\in\mathcal{I}_{[0,N]}}}{\sum_{k\in\mathcal{N}}}\lambda_{l|k}\alpha_{l|k}^*\big)\mathbb{X}_0\Big) \nonumber\\
&{\oplus}\{B(\theta){\sum_{l\in\mathcal{I}_{[0,N]}}}{\sum_{k\in\mathcal{N}}}\lambda_{l|k}v_{l|k}^*\}
{\oplus}\big({\sum_{l\in\mathcal{I}_{[0,N]}}}{\sum_{k\in\mathcal{N}}}\lambda_{l|k}\big)\mathbb{W}\nonumber\\
=&{\bigoplus_{l\in\mathcal{I}_{[0,N]}}}{\bigoplus_{k\in\mathcal{N}}}\lambda_{l|k}\big(A_{cl}(\theta)(\{z_{l|k}^*\}{\oplus}\alpha_{l|k}^*\mathbb{X}_0){\oplus}\{B(\theta)v_{l|k}^*\}{\oplus}\mathbb{W}\big) \nonumber \\
\subseteq&{\bigoplus_{l\in\mathcal{I}_{[0,N]}}}{\bigoplus_{k\in\mathcal{N}}}{\lambda_{l|k}}(\{z_{l+1|k}^*\}\oplus\alpha_{l+1|k}^*\mathbb{X}_0)\nonumber\\
=&\{z^+\}\oplus\alpha^+\mathbb{X}_0,\nonumber
\end{align}
\end{gather}
where step 2 is shown in detail for convex sets $\mathbb{A}$ and $\mathbb{B}$: \\
$(\sum_i\lambda_i)\mathbb{A}\oplus(\sum_i\lambda_i)\mathbb{B}=\{\sum_i\lambda_ia+\sum_i\lambda_ib|a\in\mathbb{A}, b\in\mathbb{B}\} \\
=\{\sum_i\tilde{a}_i+\sum_i\tilde{b}_i|\tilde{a}_i\in\lambda_i\mathbb{A},\tilde{b}_i\in\lambda_i\mathbb{B}\}=\bigoplus_i(\lambda_i\mathbb{A}\oplus\lambda_i\mathbb{B})\\
=\bigoplus_i\lambda_i(\mathbb{A}\oplus\mathbb{B})$.
As $(z,a)\in\mathbb{X}_f^{\mathcal{M}(\bar{k})}$, it follows that the tuple $(z^+,\alpha^+)\in\mathbb{X}_f^{\mathcal{M}(\bar{k})}$ from a convex combination of the tuples $(z^*_{l+1|k},\alpha^*_{l+1|k}){\in}\mathbb{X}_f^{\mathcal{M}(\bar{k})}$. Similarly through the convex combination of $v$, the combined state and input constraints are guaranteed to hold
\end{proof}
\begin{remark}
Given a representation of the set $\mathbb{X}_0{=}\textup{co}(x^1, x^2, \dots, x^{n_{X_0}})$ with $n_{X_0}$ vertices, the terminal set enlargement in \eqref{eq:homotheticsetenlargement} can be further improved with the vertices of the previously computed $\mathbb{X}^*_{l|k}$ by using $\textup{co}\left(\mathbb{X}_f^{\mathcal{M}(\bar{k})}, \{(z^*_{l|k}+\alpha^*_{l|k}x^j, 0)\}_{j\in\mathcal{I}_{[0,n_{X_0}]}, l\in\mathcal{I}_{[0,N]},k\in\mathcal{M}(\bar{k})}\right)$ as for all $j{\in}\mathcal{I}_{[0,n_{X_0}]}$, $z^*_{l|k}{+}\alpha^*_{l|k}x^j{\in}\mathbb{X}^*_{l|k}$, which implies that $\forall\theta\in\Theta_{\bar{k}}$, $A_{cl}(\theta)(z^*_{l|k}{+}\alpha^*_{l|k}x^j){\oplus}\{B(\theta)v^*_{l|k}\}{\oplus}\mathbb{W}{\subseteq}\mathbb{X}^*_{l+1|k}$.
\end{remark}
\section{NUMERICAL EXAMPLE}\label{sec:numericalexample}
We consider a chain of $n_{\textup{MSD}}$ mass elements connected by $n_{\textup{MSD}}-1$ springs and dampers. The discrete-time dynamics of the mass element $i$ are given by
\begin{gather}
\begin{align}
p_{k+1,i}&{=}p_{k,i}+T_sv_{k,i} \nonumber\\
v_{k+1,i}&{=}v_{k,i}{-}T_sc_{i-1,i}(p_{k,i}{-}p_{k,i-1}){-}T_sd_{i-1,i}(v_{k,i}{-}v_{k,i-1}) \nonumber \\
&{+}T_sc_{i,i+1}(p_{k,i+1}{-}p_{k,i}){-}T_sd_{i,i+1}(v_{k,i+1}{-}v_{k,i}){+}u_{k,i} \nonumber
\end{align}
\end{gather}
with the position of element $i$ at time step $k$ denoted by $p_{k,i}$ and the element velocity $v_{k,i}$, sampling time $T_s=0.2s$, spring and damping constants $c_{i,i+1}$ and $d_{i,i+1}$ of the springs and dampers connecting elements $i$ and $i+1$.
All damping coefficients $d_{i,i+1}=0.1$, with $d_{0,1}$ and $d_{n_{\textup{MSD}},n_{\textup{MSD}}+1}$ and the corresponding spring constants being 0. The remaining spring constants are randomly drawn between $[0.05, 0.25]$ and are considered as uncertain parameters $\theta$ with the initial set of parameters $\Theta_0=[0.05, 0.25]^{n_{\textup{MSD}}-1}$, such that $\theta^*\in\Theta_0$ and an additive disturbance on the positions and velocities with $\abs{w}\leq1\textup{e}-3$ is used. The dynamics can thus be defined as $x_{k+1}=A(\theta)x_k+Bu_k+w_k$ and Assumption~\ref{ass:bounded} is fulfilled.
\begin{figure*}[thpb]
\vspace{-0.2cm}
\hspace{-2cm}\includegraphics[width=1.2\textwidth]{3MSD_Sim_3.pdf}
\vspace{-0.7cm}
\caption{Simulation of 3 mass-spring-damper elements using the adaptive MPSC scheme. The approach is compared against \cite{wabersich2018linear} (left) demonstrating less frequent safety filter interventions while ensuring constraint satisfaction and successfully identifying the unknown parameters (bottom right). Additionally, the initial and final feasible set as well as the feasible set after terminal set enlargement $\mathbb{X}_{\textup{feas}}(\Theta_0)$, $\mathbb{X}_{\textup{feas}}(\Theta_{150})$ and $\mathbb{X}_{\textup{feas,f}}(\Theta_{150})$, respectively, are shown for the element positions for a fixed velocity (top right).}
\vspace{-0.2cm}
\label{fig:sim}
\end{figure*}
The system is simulated for $30$s with $3$ and $8$ elements from the origin using the adaptive MPSC scheme with the constraint reformulation in \cite{kohler2019linear} with decoupled state and input constraints $\mathbb{X}=[-2.3,2.3]^{2n_{\textup{MSD}}}$ and $\mathbb{U}=[-3.5,3.5]^{n_{\textup{MSD}}}$. The problem is solved using YALMIP \cite{lofberg2004yalmip} and MOSEK \cite{anderson2000mosek} with an average computation time of $6$ms and $340$ms for 3 and 8 elements respectively. A PGSD controller \cite{kolter2009PGSD} is used with random initial control parameters which are trained during the simulation.
The terminal constraints used are $z_{N|k}=0$ and $0\leq\alpha_{N|k}\leq1$. The simulation results with 3 elements and a comparison to the method in \cite{wabersich2018linear} where the parametric uncertainty is included by enlarging the disturbance set can be seen in Figure \ref{fig:sim}. The adaptive MPSC scheme successfully prevents constraint violations of the system and interferes less conservatively than \cite{wabersich2018linear}. The enlargement of the implicitly defined safe set of the MPSC through parameter adaptation and through an additional terminal set enlargement using all available data after the simulation is finished is also shown in Figure \ref{fig:sim}. A section of the feasible sets $\mathbb{X}_{\textup{feas}}(\Theta_0)$, $\mathbb{X}_{\textup{feas}}(\Theta_{150})$ and with the enlarged terminal set, $\mathbb{X}_{\textup{feas,f}}(\Theta_{150})$, for a fixed velocity is shown, which is computed through gridding of the state space. In order to compute the volume of the implicitly defined safe sets in $2n_{\textup{MSD}}$ dimensions, Monte Carlo Integration is used with $10^5$ randomly drawn samples. The results and a comparison with the feasible set $\mathbb{X}_{\textup{feas,\cite{wabersich2018linear}}}$ of \cite{wabersich2018linear} and the total volume within the constraints are shown in Table 1, where a $21\%$ increase in volume is observed for the case of $3$ mass-spring-damper elements after the parameter estimation and a total increase of $28\%$ with the terminal set enlargement. For $8$ elements, an increase of $100\%$ and $120\%$, respectively, is observed.
\section{CONCLUSION}
An adaptive Model Predictive Safety Certification scheme was proposed, which ensures safety of dynamical systems controlled by any learning-based controller. This modular framework uses set-membership estimation in order to sequentially improve the set in which uncertain parameters can possibly lie. The parameter estimation allows to enlarge the feasible set of the MPSC, and thereby the safe set of operation, in an online manner with recursive feasibility guarantees. We provide a possible enlargement of the terminal safe set used in the MPSC optimisation problem using previously solved instances, in order to further increase the feasible set of the MPSC and present a design method allowing for a computationally efficient optimisation problem. The adaptive MPSC scheme was applied to a chain of mass-spring-damper elements, which showed a significant increase in the implicit safe set volume through the parameter estimation and the terminal safe set enlargement and interfered less often than the nominal method in \cite{wabersich2018linear}.
\begin{table}
\vspace{0.22cm}
\setlength{\tabcolsep}{4.7pt}
\caption{Volume of the feasible set of the adaptive MPSC Optimisation Problem through Monte Carlo Integration}
\label{table:ex}
\vspace{-0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
&\!$\mathbb{X}_{\textup{feas}}(\Theta_0)$\!&\!$\mathbb{X}_{\textup{feas}}(\Theta_{150})$\!&\!$\mathbb{X}_{\textup{feas,f}}(\Theta_{150})$\!&\!$\mathbb{X}_{\textup{feas,\cite{wabersich2018linear}}}$\!&\!Constraint\!\\
\!\#\! MSD\!& Volume & Volume & Volume & Volume & Volume\\
\hline
$3$ & $5.86\textup{e}3$ & $7.10\textup{e}3$ & $7.46\textup{e}3$ & $3.62\textup{e}3$ & $9.47\textup{e}3$\\
\hline
$8$ & $9.64\textup{e}9$ & $1.93\textup{e}10$ & 2.12\textup{e}10 & $\diagup$ \footnotemark & $4.02\textup{e}10$\\
\hline
\end{tabular}
\end{center}
\end{table}
\footnotetext{The RPI set for \cite{wabersich2018linear} could not be computed due to the complexity of the disturbance set resulting from the use of nominal linear dynamics matrices.}
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<?xml version="1.0" encoding="utf-8"?>
<LinearLayout
xmlns:android="http://schemas.android.com/apk/res/android"
android:orientation="vertical" android:layout_width="wrap_content"
android:layout_height="wrap_content">
<!--Game Name-->
<AutoCompleteTextView
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:id="@+id/dialog_abbr_add_game"
android:dropDownHeight="wrap_content"
android:dropDownVerticalOffset="0dp"
android:singleLine="true"
android:hint="@string/dialog_abbr_add_hint_game"
/>
<!--Game Abbreviation-->
<EditText
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:id="@+id/dialog_abbr_add_abbr"
android:singleLine="true"
android:hint="@string/dialog_abbr_add_hint_abbr"
/>
</LinearLayout>
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Q: is there any simpler way to refresh a page in rails Guys i have tried to refresh my page with a button..
<input type="reset" value="Refresh" />
but it not refreshing, not to refresh the textboxes jst to refresh the entire page.
A: Try this:
<INPUT TYPE="button" VALUE="Reload Page" onClick="history.go(0)">
A: I know this post is four years old, but I wanted to put out the snippet I used using Rails formatting.
<%= button_to "Refresh", "", { :onclick => "history.go(0)" } %>
A: in my case this works very well:
= link_to "close", "", :onclick => '$("#ajax_form").hide(); history.go(0); return false;', :style => "color:white"
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Hanging ten and riding on cloud nine, describes Puerto de la Cruz Tenerife to a tee. Great water temperature year around, a plentiful variety of fresh caught fish, great beaches, a large relaxing water park, and a quaint old town. Words just aren't good enough. Is there a negative? With so many positives, who cares?
|
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| 6,978
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Esdeveniments
Benet V és escollit papa però deposat el mateix any
S'escriu el primer text en castellà
Naixements
Garcia III Sanxes II de Navarra, rei de Navarra i comte d'Aragó
Necrològiques
Joan XII, Papa
Anys del segle X
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131 Playing
4.9K Backlogs
108 Replays
3.9% Retired
Join the battle for Bionis to repel the invasion of a terrifying mechanical army and uncover the secrets of a mystical sword called the Monado in Xenoblade. Xenoblade Chronicles is an epic role-playing game which takes place on an immense game world on the remains of two giant titans, and features fast-paced real-time battling. The game heavily foc ...Read Moreuses on exploring vast landscapes and immerses you into the game with cinematic cutscenes. The story focuses on Shulk and the Monado, a mystical sword that gives his bearer great powers, and a war between humans and robots. The game's real-time battles have an action oriented approach, giving you the chance to unleash special attacks and strategies by selecting them from a command gauge; the battling also is interlinked with the affinity between characters.
When focusing on the main objectives, Xenoblade Chronicles is about 58½ Hours in length. If you're a gamer that strives to see all aspects of the game, you are likely to spend around 143 Hours to obtain 100% completion.
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Any% 9 19h 44m 18s 4h 46m 56s 2h 50m 56s 60h
100% 4 46h 39m 42s 23h 22m 7s 19h 54m 36s 120h
Emulated 66 57h 13m 80h 15m 165h 21m 30h 210h
Nintendo 3DS 302 64h 48m 96h 3m 152h 24h 38m 346h 35m
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Wii 735 70h 30m 103h 45m 160h 47m 21h 42m 350h 27m
Wii U 89 72h 6m 89h 28m 164h 36m 38h 251h
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Janusz Słowikowski ur. 19 marca 1937 roku w Łodzi, zginął tragicznie 16 maja 1971 roku. Poeta, autor znakomitych tekstów piosenek, tłumacz (pierwszy tłumacz tekstów Bułata Okudżawy) dziennikarz.
Życiorys
Janusz Słowikowski ukończył III Liceum Ogólnokształcące im. Tadeusza Kościuszki w Łodzi, był absolwentem Wydziału Filologicznego Uniwersytetu Łódzkiego. Jego rodzicami byli: matka – Czesława ( z domu Woźniak), wieloletnia nauczycielka, ojciec – Bolesław Słowikowski, ekonomista i działacz społeczny. Brat – Piotr Słowikowski (1949), dziennikarz, autor filmów dokumentalnych. Żona – Maria Machan. Córka – Dorota.
W czasie studiów Janusz Słowikowski (na przełomie 1955/56 roku) współpracować zaczął ze Studenckim Teatrem Satyry "Pstrąg", założonym w 1954 roku. Początkowo w brygadzie sceny. Wkrótce wykazał się kunsztem pisania tekstów piosenek, które przyniosły mu sławę, jak np. "Parasolki, parasolki" z muzyką Piotra Hertla, wykonywane w "Pstrągu" przez Ewę Nagurską, zdobyły rozgłos dzięki Marii Koterbskiej. Na Festiwalu Sopot 1963 "Parasolki, parasolki" wykonywała Lulu Porter z USA, gdzie tę piosenkę nagrała firma DECCA.
W końcu lat pięćdziesiątych i w następnym dziesięcioleciu powstały jego najlepsze teksty, pisane do "Pstrąga" i na sceny zawodowe – Teatru Nowego, Teatru 7.15 w Łodzi, do kabaretów "Poddasze" i "Figa".
Janusz Słowikowski pracował przez wiele lat w Łódzkiej Rozgłośni Polskiego Radia, w redakcjach programów estradowych: "Wesoły Autobus" i "Program z dywanikiem"
Był autorem szeregu satyrycznych piosenek pisanych do audycji radiowych, autorem – wespół z Januszem Kłosińskim – sztuki teatralnej dla dzieci "Ali Baba i 40 rozbójników" – premiera w Teatrze Nowym w Łodzi w 1965 roku. Napisał wszystkie piosenki do musicalu "Kariera Nikodema Dyzmy" według Tadeusza Dołęgi-Mostowicza, z muzyką Andrzeja Hundziaka, librettem – Stanisława Powołockiego. Niestety, tej premiery w reżyserii Danuty Baduszkowej w Teatrze Muzycznym w Gdyni – nie doczekał.
Zginął 16 maja 1971 roku na jeziorze Śniardwy po kilkunastu godzinach oczekiwania na pomoc uczestnikom rejsu na przewróconej żaglówce. Janusz Słowikowski miał 34 lata.
Pierwszy program telewizyjny z piosenek Janusza Słowikowskiego powstał już w 1963 roku pt. "Kredą na płocie" w reżyserii Romana Sykały.
Następny pt. "Ballady Janusza Słowikowskiego" w reżyserii Jerzego Woźniaka i Piotra Hertla, nadany został w 1974 roku. W roku 2002 opublikowany zastał najpełniejszy przegląd twórczości pt. "Parasolki... parasolki... wiersze, piosenki, ballady" – Wydawnictwo "Papier-service". Spektakl teatralny pt. "Małe miasteczka" w reżyserii Zbigniewa Szczapińskiego w Teatrze Powszechnym w Łodzi w 2004 roku był najpełniejszą prezentacją piosenek Janusza Słowikowskiego z muzyką Piotra Hertla.
Twórczość – najważniejsze piosenki
Piosenki z muzyką kompozytorów
Z muzyką Piotra Hertla: "Parasolki, parasolki", "Moja Bezsenność", "Lorenzo", "Żołnierze w każdej potrzebie", "Powroty", "A ja jestem zadowolony", "My trzej ( Trzej z Bałut)", "Poezjusz", "Kredą na płocie", "Ballada o armacie", "Pułkownik", "Echa corridy", "Colt" "Za ścianą sąsiad", "Pieśń hiszpańskiego rzeźnika o corridzie w Fiqueras", "Arabskie lamentacje na temat piszczącego kokosa", "Strzelnica", "Gdy gra orkiestra", "Tango o dwoistości pojęcia władzy", "W otwartym oknie", "Ach jak bym coś przeskrobać chciał", "Ślepy Tommy", "Serenada kominiarza", "Trzy słowa", "Nie chodź do domu", "W ostatniej trójce"," Sami sobie literaci", "Jubileusz", "Pan Marcin", "Opowieść o fortepianie", "Napoleon nie był wielki", "Dwaj znikąd".
Z muzyką Czesława Majewskiego: "Tam, gdzie byłem", "Trzydziestego".
Z muzyką Włodzimierza Korcza: "Zieleń z lasu odleciała", "Koloryści jesieni", "Dopóki cisza nie pęknie".
Z muzyką Janusza Kaźmierczaka: "Piosenka myśliwych", "Stara muzyka", "Wędkarski walczyk".
Z muzyką Piotra Marczewskiego: "Za lasami dębowymi", "Pozłacany budzik".
Z muzyką Jerzego Abratowskiego: "Przedmieściowa ballada o kocie i oprychu".
Z muzyką Krzysztofa Cwynara: "Cienie".
Z muzyką Włodzimierza Wandera: "Przeciera się", "Nie wierzę".
Z muzyką Tadeusza Woźniakowskiego: "Zawołaj", "Pan Antoni", "Automobilem".
Piosenki z filmów
Z muzyką Jerzego Krzemińskiego "Gramofon", "Opowiadał wróbel strachom" (z Piotrem Janczerskim) z filmu "Milion za Laurę" w reżyserii Hieronima Przybyła, 1971.
Piosenki dla zawodowych scen teatralnych
"Ballada o kulawym sercu" i "Piosenka Rozalki" z muz. Piotra Hertla do spektaklu "Nie igra się z miłością" Alfreda de Musset w reżyserii Jerzego Antczaka w Teatrze Nowym w Łodzi 1959 r.
"Rozchoruj się" z muz. Piotra Hertla, ze spektaklu "Lekarz mimo woli" Moliera, w reżyserii Jerzego Waldena w Teatrze 7.15 w Łodzi 1964.
Z przekładów z Bułata Okudżawy
"Błękitny balonik", "Czarny kot", "Nie wierzcie piechocie", "Piosenka o durniach", "Papierowy żołnierzyk", "Trzy razy", "Piosenka amerykańskiego żołnierza", "Żołnierskie buty".
Bibliografia
Janusz Słowikowski "Parasolki... Parasolki... wiersze, piosenki, ballady", wybór i opracowanie Leszek Skrzydło, Piotr Słowikowski "Papier-service", Łódź 2002.
"Teatry studenckie w Polsce" opracowanie zbiorowe, (opracowanie redakcyjne Jerzy Koenig) Wydawnictwa Artystyczne i Filmowe, Warszawa 1968.
Wiesław Machejko, "Pstrąg. Studencki Teatr Satyry", Dom Wydawniczy Elipsa, Warszawa 2005.
Adrjański Zbigniew, "Kalejdoskop estradowy. Leksykon polskiej rozrywki 1944-1989. Artyści, Twórcy, Osobistości". Dom Wydawniczy Bellona, Warszawa 2002, hasło: Słowikowski Janusz s. 420-421.
"Pamiętnik Teatralny" Rok XXV Zeszyt 2(82), Instytut Sztuki Polskiej Akademii Nauk, Warszawa 1972 (Zofia Śliwińska, Studencki Teatr Satyry "Pstrąg" s. 218–223).
"Kronika Miasta Łodzi nr 1/2002. Wyd. Urząd Miasta Łodzi 2002 r.(Wiesław Machejko, Wspomnienie o Januszu Słowikowskim, s. 119-124)
Katarzyna Janicka, "Cechy językowe poezji i przekładów Janusza Słowikowskiego", praca magisterska napisana pod kierunkiem prof. dr hab. Kazimierza Michalewskiego w Katedrze Współczesnego Języka Polskiego Uniwersytetu Łódzkiego, Nr albumu: 104248/S, Łódź 2005.
"Rewia piosenek 30-lecia" PWM 1975. Redakcja: Wanda Doleżal, Danuta Idzik, Jerzy W.Martin ("Parasolki, parasolki" s. 79)
"Estrada. Materiały repertuarowe dla estrad" WAiF, Warszawa 1975 (piosenka "Nas trzech" s. 68)
Agnieszka Barczyk-Sitkowska, "Telewizyjna twórczość Piotra Słowikowskiego w kontekstach kulturotwórczej roli TVP Łódź", Wydawnictwo Primum Verbum, Łódź 2017.
Filmpolski.pl – Janusz Słowikowski
Przypisy
Ofiary katastrof w ruchu wodnym
Polscy tłumacze literatury rosyjskojęzycznej
Poezja śpiewana i piosenka literacka
Urodzeni w 1937
Zmarli w 1971
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Fire safety rules and regulations are designed to save lives and protect both people and property. As the owner of a business, organisation or facility, you are responsible for fire safety and the health and safety of all people on the premises.
A crucial part of fire safety is the fire risk assessment, which is designed to flag up potential hazards and help you to eliminate them through implementing fire safety measures. TutorCare offer a BSC award in Fire Risk Assessment training course online.
One of the first things you are required to do when you (or an appointed consultant) conduct a fire risk assessment is to assess who is most at risk from fire. In a normal business, i.e. an office, the people most at risk would be children, elderly or disabled people, those who work in isolated areas or close to fire hazards.
In a care environment, however, the number of people most at risk drastically increases. This is because most of the residents in care homes are likely to be disabled, elderly, in poor health or vulnerable in some other way. In case of a fire, these people may find it harder to escape the danger, and they may also be more at risk from the effects of fire (i.e. smoke inhalation).
This makes fire safety in care homes doubly important, meaning that all rules and regulations must be met, fire awareness training must be carried out and overall, standards must be kept exceptionally high.
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\section{Introduction}
Heavy-ion collision experiments at RHIC and LHC lead to thermalized dense
matter at small but non-zero baryon density, or equivalently chemical
potential. Therefore it is necessary to study the bulk thermodynamics of QCD at
finite chemical potentials. In this work, we use the Taylor expansion method
\cite{method} to study the equation of state, the number density and
fluctuations of various quantum numbers on the lattice. We study \(2+1\) flavor
QCD with tree level Symanzik-improved gauge action and p4fat3-improved
staggered fermion action \cite{Karsch:2000kv}. The simulations are carried out
on \(16^{3}\times4\) and \(24^{3}\times6\) lattices on a line of constant
physics with almost physical quark masses; the pion mass is about \(220\) MeV
and the strange quark mass is adjusted to its physical value. We have scanned
a temperature range approximately from \(170\) MeV to \(500\) MeV. We are using
the exact RHMC algorithm \cite{RHMC} to update configurations. Details on our
simulation parameters can be found in \cite{Jan}.
\section{Taylor expansions of thermodynamic quantities}
For a large homogeneous system, the pressure of QCD with \(u\), \(d\) and
\(s\) quarks can be expressed
as\begin{equation}
\frac{p}{T^{4}}=\frac{1}{VT^{3}}\ln Z\left(V,T,\mu_{u},\mu_{d},\mu_{s}\right),
\label{eq:Pdef}
\end{equation}
where the partition function $Z$ is a function of the volume $V$, temperature
$T$ and chemical potentials of $u$, $d$, and $s$ quarks. We have not considered
other species of quarks whose masses are much heavier. Due to the sign problem,
the difficulty of a direct lattice calculation at non zero chemical potentials
arises. We perform a Taylor expansion in terms of the chemical potentials
\begin{equation}
\frac{p}{T^{4}}
=\sum_{i,j,k}c_{ijk}(T)\left(\frac{\mu_{u}}{T}\right)^{i}
\left(\frac{\mu_{d}}{T}\right)^{j}\left(\frac{\mu_{s}}{T}\right)^{k},
\label{eq:PTaylor}
\end{equation}
and compute the coefficients $c_{ijk}$ at zero chemical potentials. When the
sum $i+j+k$ is odd, the coefficient $c_{ijk}$ is given as expectation value of
purely imaginary operators and therefore vanishes exactly. This reflects the
invariance of the QCD partition function under change of particle and
anti-particle. The leading term $c_{000}$ gives the pressure at vanishing
baryon density and can be calculated via the integral method. Results for the
parameter values considered here have been presented in \cite{Jan}. In
this work, we will concentrate on the part of the pressure
\begin{equation}
\Delta p=p(\vec\mu)-p(\vec\mu=0),
\end{equation}
that arises due to non-zero chemical potentials, where
\(\vec\mu=(\mu_{u},\mu_{d},\mu_{s})\). For \(i+j+k>0\), the coefficients
\begin{equation}
c_{ijk}=\left.\frac{1}{i!j!k!}
\frac{\partial^{i}}{\partial\hat\mu_{u}^{i}}
\frac{\partial^{j}}{\partial\hat\mu_{d}^{j}}
\frac{\partial^{k}}{\partial\hat\mu_{s}^{k}}
\left( p/T^{4}\right)\right|_{\mu=0},
\end{equation}
are derivatives of the partition function, and can be calculated on the
lattice, where \(\hat\mu=\mu/T\). These coefficients provide information about
other thermal quantities as well. For example, the strange quark number density
expands in chemical potentials as
\begin{equation}
\frac{n_{s}}{T^{3}}=\sum_{i,j,k} \left( k+1 \right)
c_{ij(k+1)}\hat\mu_{u}^{i}\hat\mu_{d}^{j}\hat\mu_{s}^{k},
\label{eq:ns}
\end{equation}
and similarly the light up and down quark numbers. We can further consider
fluctuations in these quantities.
Alternatively, one can introduce chemical potentials for the conserved quantities
baryon number \(B\), electric charge \(Q\) and strangeness \(S\), which are
related to \(\mu_u, \mu_d ,\mu_s \) via
\begin{equation}
\mu_u=\frac{1}{3}\mu_{B} +\frac{2}{3}
\mu_{Q},\qquad
\mu_{d}=\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q},\qquad
\mu_{s}=\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q}-\mu_{S},
\label{eq:chempot}
\end{equation}
and compute e.g. the baryon density as
\begin{equation}
n_{B}=\frac{1}{3}\left( n_{u}+n_{d}+n_{s}\right).
\end{equation}
Then we can study densities and fluctuations in \(B \), \(Q\) and \(S\).
In the following, we will regard $u$ and $d$ quarks as degenerate and consider
$2+1$ flavor QCD. With the definition $\mu_{q}\equiv\mu_{u}=\mu_{d}$ for the
light quarks, the coefficients are
\begin{equation}
c_{ij}^{qs}=\left.\frac{1}{i!j!}
\frac{\partial^{i}}{\partial\hat\mu_{q}^{i}}
\frac{\partial^{j}}{\partial\hat\mu_{s}^{j}}
\left(p/T^{4}\right)\right|_{\vec \mu=\mathbf{0}},
\label{eq:cqs}
\end{equation}
where the subscripts denote the order of the derivative and the superscripts
indicate the corresponding flavors. If not specified, the default superscripts
will be \(qs\) and will often be left out. It is evident from Eqs.
(\ref{eq:chempot}) that choosing $\mu_u\equiv\mu_d$ is equivalent to a
vanishing electric charge potential $\mu_Q\equiv 0$.
Now we discuss how to evaluate these coefficients on the lattice. Inserting
Eq. (\ref{eq:Pdef}) into Eq. (\ref{eq:cqs}) and integrating out the fermion
fields in the partition function yields the coefficients as expectation
values of operators that contain derivatives of the determinant of the fermion
matrix \(M\). For example the formula for \(c_{20}\) reads \begin{equation}
c_{20}
=\frac{N_{\tau}}{2N_{\sigma}^{3}}\left({\frac{n_{f}}{4}}
\left\langle {\frac{\partial^{2}\ln\det M}{\partial\vec \mu_{q}^{2}}}\right\rangle
+{\left(\frac{n_{f}}{4}\right)^{2}}
\left\langle \left({\frac{\partial\ln\det M}{\partial\vec \mu_{q}}}\right)^{2}\right\rangle \right),
\end{equation}
where \(N_{\tau }\) and \(N_{\sigma}\) are temporal and spacial extent of the
lattice, \(n_{f} \) is the number of quark flavors in question (here
\(n_{f}=2\)), and \(\left\langle \cdots\right \rangle\) indicates taking the
thermal average over the ensemble. On each configuration, derivatives of
$\ln\det M$ need to be evaluated up to the same order as the order of the
expansion coefficients. These derivatives lead to the appearances of the
inverse fermion matrix \(M^{-1}\) inside traces
\begin{eqnarray}
\frac{\partial\ln\det M}{\partial\mu}
& = & \mbox{Tr}\left({M^{-1}}\frac{\partial M}{\partial\mu}\right),\\
\frac{\partial^{2}\ln\det M}{\partial\mu^{2}}
& = & \mbox{Tr}\left({M^{-1}}\frac{\partial^{2}M}{\partial\mu^{2}}\right)
-\mbox{Tr}\left({M^{-1}}\frac{\partial M}{\partial\mu}{M^{-1}}\frac{\partial M}{\partial\mu}\right).
\end{eqnarray}
To avoid full matrix inversions, we use the random noise method in estimating
such traces. Suppose we have generated a set of \(N\) random noise vectors \(R^{(a)},
a=1,\ldots,N\), then the trace can be estimated as
\begin{equation}
\text{Tr}\left( \mathcal{O}\ M^{^{-1}}\right)
\approx\frac{1}{N}\sum^{N}_{a=1}R^{(a)}\mathcal{O}M^{-1}R^{(a)},
\end{equation}
where \(\mathcal{O}\) is some arbitrary matrix. For each vector \(R^{(a)}\)
only the linear system \(MX=R^{(a)}\) needs to be solved. It is still quite
expensive to compute all necessary operators, since a large number of
random vectors is needed in order to get a satisfactory accuracy. Also, higher order
coefficients are more expensive, because more operators are needed. For the 4th
order coefficients one has
\begin{equation}
\begin{split}
c_{40} \ = \ &\frac{1}{4!N_{\sigma}^{3}N_{\tau}}\left\{ {\frac{n_{f}}{4}}\left\langle {\frac{\partial^{4}\ln\det M}{\partial\mu_{q}^{4}}}\right\rangle\right.\\
& +4{\left(\frac{n_{f}}{4}\right)^{2}}\left\langle {\frac{\partial^{3}\ln\det M}{\partial\mu^{3}_{q}}}\frac{\partial\ln\det M}{\partial\mu_{q}}\right\rangle+3{\left(\frac{n_{f}}{4}\right)^{2}}\left\langle \left(\frac{\partial^{2}\ln\det M}{\partial\mu^{2}_{q}}\right)^{2}\right\rangle\\ &+6{\left(\frac{n_{f}}{4}\right)^{3}}\left\langle \frac{\partial^{2}\ln\det M}{\partial\mu^{2}_{q}}\left(\frac{\partial\ln\det M}{\partial\mu_{q}}\right)^{2}\right\rangle +{\left(\frac{n_{f}}{4}\right)^{4}}\left\langle \left(\frac{\partial\ln\det M}{\partial\mu_{q}}\right)^{4}\right\rangle \\
&\left. -3\left({\frac{n_{f}}{4}}\left\langle \frac{\partial^{2}\ln\det M}{\partial\mu^{2}_{q}}\right\rangle +{\left(\frac{n_{f}}{4}\right)^{2}}\left\langle \left(\frac{\partial\ln\det M}{\partial\mu_{q}}\right)^{2}\right\rangle \right)^{2}\right\},
\end{split}
\end{equation}
where
\begin{eqnarray}
\frac{\partial^{3}\ln\det M}{\partial\mu^{3}} & = & \mbox{Tr}\left(M^{-1}\frac{\partial^{3}M}{\partial\mu^{3}}\right)-3\mbox{Tr}\left(M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial^{2}M}{\partial\mu^{2}}\right)\notag\\
&&+2\mbox{Tr}\left(M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}\right),\\
\frac{\partial^{4}\ln\det M}{\partial\mu^{4}} & = & \mbox{Tr}\left(M^{-1}\frac{\partial^{4}M}{\partial\mu^{4}}\right)-4\mbox{Tr}\left(M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial^{3}M}{\partial\mu^{3}}\right)\notag\\
& & \mspace{-36.0mu}-3\mbox{Tr}\left(M^{-1}\frac{\partial^{2}M}{\partial\mu^{2}}M^{-1}\frac{\partial^{2}M}{\partial\mu^{2}}\right)+12\mbox{Tr}\left(M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial^{2}M}{\partial\mu^{2}}\right)\notag\\
& & \mspace{-36.0mu}-6\mbox{Tr}\left(M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}M^{-1}\frac{\partial M}{\partial\mu}\right).
\end{eqnarray}
Five matrix inversions per random vector are necessary here, while for the 6th
order, 12 matrix inversions are needed. Depending on quark mass, temperature
and particular operator, different numbers of random vectors are needed to
obtain that the errors arising from the stochastic estimator are smaller than
or of the same magnitude as the statistical fluctuations within the ensemble.
\section{Pressure and densities}
In this section, we will first show results for the coefficients, then
use them in computing pressure and quark number densities.
In Fig. \ref{fig1},
\begin{figure}
\centering
\includegraphics[width=0.48\linewidth]{figs/c2l.eps}
\includegraphics[width=0.48\linewidth]{figs/c4s.eps}
\caption{\(c_{200}\) on the left and \(c_{004}\)on the right for \(N_\tau=4\)
and \(6\). The second order coefficients increase rapidly from confined
phase to deconfined phase at around \(200\) MeV, while the fourth order
ones develop a peak there. Stephen-Boltzmann limits of the free case for
the action that we use are marked for both quantities and matched very well
in the high temperature region. }
\label{fig1}
\end{figure}
we show the coefficients \(c_{200}\) and \(c_{004}\) on both \(N_{\tau}=4\) and
\(6\) lattices. \(c_{200}\), also known as the fluctuation in \(u\) (\(d\))
quark number density, increase rapidly through the phase transition region. As
one can see, the lattice cut-off effect is small and seems to be under control.
Results for $c_{002}$ from $N_\tau=8$ lattices \cite{HotQCD} further support
this statement. The fourth order coefficient \(c_{4}\) shows a pronounced peak
around $T_c$. To compare the quark mass dependence, \(c_{200}\) and \(c_{002}\)
for \(u\) and \(s\) quarks respectively are shown in Fig. \ref{fig2}. The slope
is steeper for light than for the strange quarks, which indicates a stronger
sensitivity to the chiral transition for lighter quark masses.
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=0.99\linewidth]{figs/c2ls.eps}
\caption{Second order coefficients \(c_{200}\) and \(c_{002}\) for
\(u\) and \(s\) quark respectively on \(N_{\tau}=6\) lattice.}
\label{fig2}
\end{minipage}
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=0.99\linewidth]{figs/c11qs.eps}
\caption{\(c^{qs}_{11}\) on \(N_{\tau}=4\) and \(6\) lattices. }
\label{fig3}
\end{minipage}
\end{figure}
We also show \(c^{qs}_{11}\) in Fig. \ref{fig3}, which approaches zero from
below in the high temperature limit.
Combining all the measured coefficients, we obtain pressure and number density
according to formula (\ref{eq:PTaylor}) and (\ref{eq:ns}). In Fig. \ref{fig4},
we show the pressure difference \(\Delta p/T^{4}\) and light quark number
density \(n_{q}/T^{3}\) at finite light quark chemical potential but zero
strange quark chemical potential \(\mu_{s}=0\), up to the 4th order.
\begin{figure}
\includegraphics[width=0.49\textwidth]{figs/pmus0.eps}
\includegraphics[width=0.49\textwidth]{figs/nqmus0.eps}
\caption{Pressure \(\Delta p/T^{4}\) and light quark number density
\(n_{q}/T^{3}\) at \(\mu_{s}=0\) and \(\mu_{q}/T=0.2\) , \(0.4\), \(0.6\) and
\(1.0\). Small differences are observed between \(N_{\tau}=4\) and \(6\),
especially when \(\mu_{q}/T\) is small. Light quark number density seems to
develop a peak around \(200 \)MeV when \(\mu_{q}/T\) increases.}
\label{fig4}
\end{figure}
This should be compared to the pressure at vanishing chemical potential
\cite{Jan}, which rises rapidly to a value of about \(p/T^4\approx 14\) above
the transition. The finite density contribution to the pressure adds to this
less than 10\% for \(\mu_q/T<1\).
\section{Hadronic fluctuations at zero and non-zero chemical potential}
Fluctuations of charge densities $n_{B,S,Q}$ are related by the fluctuation
dissipation theorem to the second derivatives of the partition function with
respect to the corresponding chemical potentials $\mu_{B,S,Q}$. Here $B,S,Q$
denote baryon number, strangeness and electrical charge, respectively. Using
Eqs. (\ref{eq:chempot}) we can rearrange the expansion coefficients
$c^{uds}_{ijk}$ of the pressure to get the coefficients of an expansion in
$\mu_{B,S,Q}$, defined as
\begin{equation}
\frac{p}{T^{4}} =
\sum_{i,j,k}c^{BSQ}_{ijk}(T)
\left(\frac{\mu_{B}}{T}\right)^{i}
\left(\frac{\mu_{S}}{T}\right)^{j}
\left(\frac{\mu_{Q}}{T}\right)^{k}.
\label{eq:PTaylorBSQ}
\end{equation}
E.g., the following two relations hold for $c^{BSQ}_{200}\equiv c^{B}_2$ and
$c^{BSQ}_{400}\equiv c^B_4$
\begin{equation}
c^{B}_2
=\frac{1}{9}\left(c^{qs}_{20}
+c^{qs}_{11}+c^{qs}_{02}\right), \qquad
c^{B}_4
=\frac{1}{81}\left(c^{qs}_{40}
+c^{qs}_{31}+c^{qs}_{22}+c^{qs}_{13}+c^{qs}_{04}\right).
\end{equation}
\begin{figure}
\includegraphics[width=0.49\textwidth]{figs/c2_B_MEV.eps}
\includegraphics[width=0.49\textwidth]{figs/c4_B_MEV.eps}
\caption{Quadratic and quartic baryon number fluctuations at vanishing net
density as function of temperature. Preliminary data from (2+1)-flavor
simulations with almost realistic quark masses are compared with previous
2-flavor simulations [2]. Both results have been obtained on $16^3\times 4$
lattices. }
\label{fig:cB}
\end{figure}
\begin{figure}
\includegraphics[width=0.49\textwidth]{figs/c2_Q_MEV.eps}
\includegraphics[width=0.49\textwidth]{figs/c4_Q_MEV.eps}
\caption{Quadratic and quartic electric fluctuations at vanishing net density
as function of temperature. Preliminary data from (2+1)-flavor simulations
with almost realistic quark masses are compared with previous 2-flavor
simulations [2]. Both results have been obtained on $16^3\times 4$ lattices.}
\label{fig:cQ}
\end{figure}
In Fig.~\ref{fig:cB} we show the first two diagonal expansion coefficients in
$\mu_B/T$ as function of temperature, which can also be interpreted as the quadratic
and quartic baryon number fluctuations. We compare our preliminary results for
(2+1)-flavor and almost realistic quark masses to earlier results with 2-flavor
and a pion mass $m_\pi\approx 700 MeV$ \cite{Allton:2005gk}. The normalization
is such that in both cases the same Stefan-Boltzmann value for large temperatures
is reached, i.e. we have divided by the number of flavors. An obvious shift in
the curves reflects the shift in the transition temperature from about 220~$MeV$
to 200~$MeV$. Moreover the sudden change in the quadratic fluctuations is
more pronounced for the smaller masses and the Stefan-Boltzmann value is
reached faster. Correspondingly, the peak in the quartic fluctuations is higher
for smaller masses.
The expansion coefficients in $\mu_S/T$ are identical to that in $\mu_s/T$ --
although the strangeness chemical potential differs from the strange quark
chemical potential by a different sign -- and are shown in Fig.~\ref{fig1} and
\ref{fig2}. In Fig.~\ref{fig:cQ} we show the first two diagonal expansion
coefficients in $\mu_Q/T$. The qualitative picture is very similar to $\mu_B/T$
although the quark mass dependence of the peak height is significantly weaker.
Using the expansion coefficients in $\mu_{B,S,Q}/T$, one can construct hadronic
fluctuations at non-zero baryon number density. Up to fourth order correction
in $\mu_B/T$ we have the following relations for baryon number, strangeness and
electric charge fluctuations $\chi_{BSQ}$,
\begin{eqnarray}
\frac{\chi_B(\mu_B/T)}{T^2} &=& 2c^B_2
+ 12c^B_4\left(\frac{\mu_B}{T}\right)^2
+ \mathcal{O}\left[\left(\frac{\mu_B}{T}\right)^4\right]\\
\frac{\chi_S(\mu_B/T)}{T^2} &=& 2c^S_2
+ 2c^{BS}_{22}\left(\frac{\mu_B}{T}\right)^2
+ \mathcal{O}\left[\left(\frac{\mu_B}{T}\right)^4\right]\\
\frac{\chi_Q(\mu_B/T)}{T^2} &=& 2c^Q_2
+ 2c^{BQ}_{22}\left(\frac{\mu_B}{T}\right)^2
+ \mathcal{O}\left[\left(\frac{\mu_B}{T}\right)^4\right] .
\end{eqnarray}
\begin{figure}
\includegraphics[width=0.49\textwidth]{figs/chi_B.eps}
\includegraphics[width=0.49\textwidth]{figs/chi_s.eps}
\caption{Baryon number and strangeness fluctuations at finite baryon number
density, controlled by a finite baryon chemical potential. Results are
correct up to fourth order corrections in chemical potential and have been
obtained on $16^3\times 4$ lattices.}
\label{fig:chi}
\end{figure}
In Fig.~\ref{fig:chi} we show baryon number and strangeness fluctuations at
finite baryon number density. It is obvious that both quantities are developing
a peak for increasing $\mu_B/T$. However, the peak in $\chi_B$ is much more
pronounced since this quantity eventually diverges at the critical point in the
$(T-\mu_B)$-plane. As we anticipated from Fig.~\ref{fig:cB}, the peak height in
$\chi_B$ is about twice as large as in earlier calculations with larger quark
masses \cite{Allton:2005gk}. Note that higher order corrections are still
important, especially the position of the peak will be $\mu_B$-dependent only
by including the next higher order. This has to be analyzed in more detail and
eventually will allow to limit the range of values for $\mu_B/T$ where the
leading order result is reliable.
The off-diagonal coefficients in Eq.~\ref{eq:PTaylorBSQ} are usually connected to
correlations between baryon number, strangeness and electrical charge.
The correlation of baryon number and strangeness can be expressed in terms of
expansion coefficients as
\begin{equation}
\frac{1}{T^2}\left(\left< n_B n_S \right>-\left< n_B \right>\left< n_S \right>\right)
= c^{BS}_{11}+3c^{BS}_{31}\left(\frac{\mu_B}{T}\right)^2
+\mathcal{O}\left[\left(\frac{\mu_B}{T}\right)^4\right]
\end{equation}
and is shown in Fig.~\ref{fig:corr}.
\begin{figure}
\includegraphics[width=0.49\textwidth]{figs/corr_BS.eps}
\includegraphics[width=0.49\textwidth]{figs/strangeness_fluc.eps}
\caption{Correlation between baryon number and strangeness for several values
of the baryon chemical potential from $16^3\times 4$ lattices (left) and the
linkage between baryon number and electric charge with strangeness respectively
(right). On the right panel we compare or preliminary data (full symbols) to
previously obtained results from partially quenched calculations (open
symbols) [7], both obtained on $N_\tau=4$ lattices.}
\label{fig:corr}
\end{figure}
We find that also this quantity is developing a peak for increasing chemical
potential, thus the enhanced correlations suggest the vicinity of a critical
point. Another interesting quantity is the ``linkage'' of strangeness and
baryon number or electric charge \cite{Gavai:2005yk}, which is defined as
$C_{SX}=c^{SX}_{11}/c^{S}_2$, where $X=B,Q$. It is known to be a robust
quantity, i.e. the cut-off effects are small. In Fig.~\ref{fig:corr} (right) we
compare our preliminary results with almost realistic quark masses with
previously obtained partially quenched results and slightly larger light quark
masses \cite{Gavai:2005yk}. The two calculations show good agreement, thus also
the quenching and quark mass effects seem to be small in this quantity. Both
results on correlation and linkage between the different quantum numbers
suggest that the basic charges are carried by quasi-free quark directly above
the transition. This seems to rule out the existence of bound states as
dominant degrees of freedom in this regime \cite{Shuryak:2004tx}.
\section{Conditions at heavy ion colliders and constrained densities}
In general, the pressure, or higher derivatives of the partition functions
with respect to chemical potentials, are dependent on at least 3 variables
$\mu_{u,d,s}$ or equivalently $\mu_{B,S,Q}$. So far we chose $\mu_B>0$, while
holding $\mu_S=\mu_Q=0$. To compare with experiment, for instance heavy ion
collisions, the chemical potentials might need to be adjusted to meet the
conditions of particular event-by-event fluctuation analyzes
\cite{Begun:2006uu}. A very natural choice of the chemical potentials is
to constrain the strange quark density to zero. Due to the existence of non
zero off-diagonal coefficients in Eq.~\ref{eq:PTaylorBSQ} we find an increasing
strangeness with increasing $\mu_B$, even for $\mu_S=0$. In heavy ion experiments
the total strangeness is zero. Below we outline a procedure to constrain
the net strange quark number density $n_s$ to zero, subsequently order by order
in our $\mu_B$ expansion. The procedure can be easily generalized to constrain
other charge densities to arbitrary values. This might be of importance, since
experiments are often restricted to certain rapidity windows, which may alter
expectation values of charge densities.
We can express the strange quark number density ($n_s$) in terms of the expansion
coefficients of the pressure. Up to the 4th order, it reads
\begin{equation}
n_{s}=-n_{S}\left(\hat{\mu}_{B},\hat{\mu}_{S}\right)
= -c^{BS}_{11}\hat{\mu}_{B}
-2c^{BS}_{02}\hat{\mu}_{S}
- c^{BS}_{31}\hat{\mu}_{B}^{3}
-2c^{BS}_{22}\hat{\mu}_{B}^{2}\hat{\mu}_{S}
-3c^{BS}_{13}\hat{\mu}_{B} \hat{\mu}_{S}^{2}
-4c^{BS}_{04}\hat{\mu}_{S}^{3}\equiv0 ,
\end{equation}
where \(\hat \mu=\mu/T\), which means that the strangeness chemical potential
\(\mu_{S}\) is no longer a free parameter but depends on
\(\mu_{B}\),
\begin{equation}
\hat{\mu}_{S}\left(\hat{\mu}_{B}\right)
=\left(-\frac{c^{BS}_{11}}{2c^{BS}_{02}}\right)\hat{\mu}_{B}
+\left(\frac{2c^{BS}_{04}{c^{BS}_{11}}^{3}
-3c^{BS}_{02}{c^{BS}_{11}}^{2}c^{BS}_{13}
+4{c^{BS}_{02}}^{2}c^{BS}_{11}c^{BS}_{22}
-4{c^{BS}_{02}}^{3}c^{BS}_{31}}{8{c^{BS}_{02}}^{4}}\right)\hat{\mu}_{B}^{3}
+\mathcal{O}\left(\hat{\mu}_{B}^{5}\right)
.\end{equation} Therefore, the formula for the pressure is modified to
\begin{equation}
\frac{\Delta p}{T^{4}}
=\left(c^{BS}_{20}-{\frac{{c^{BS}_{11}}^{2}}{4c^{BS}_{02}}}\right)\hat{\mu}_{B}^{2}
+\left(c^{BS}_{40}+{\frac{c^{BS}_{04}{c^{BS}_{11}}^{4 }}{16{c^{BS}_{02}}^{4}}
-\frac{{c^{BS}_{11}}^{3}c^{BS}_{13}}{8{c^{BS}_{02}}^{3}}+\frac{{c^{BS}_{11}}^{2}c^{BS}_{22}}{4{c^{BS}_{02}}^{2}}
-\frac{c^{BS}_{11}c^{BS}_{31}}{2c^{BS}_{02}}}\right)\hat{\mu}_{B}^{4}
+\mathcal{O}\left(\hat{\mu}_{B}^{6}\right),
\end{equation}
which contains off-diagonal coefficients \(c_{11}\),\(c_{13}\), etc. On the
quark level those coefficients are generally small numbers since they are not
present in the free theory. However, on the hadronic level they contain the
diagonal strange quark coefficients which have -- at least in leading order --
a non-zero Stefan-Boltzmann limit. Hence the constraints \(n_{S}=0 \) and
\(\mu_{S}=0\) lead to a quite different dependence of the pressure on $\mu_B/T$,
as can be seen in Fig.\ref{fig:constrain} (left).
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{figs/pv2.eps}
\includegraphics[width=0.49\textwidth]{figs/rchi_B.eps}
\caption{The pressure \(\Delta p/T^{4} \) up to the second order for both constraints as labeled
(left) and the ratio $\mathcal{N} \hat\chi_B/\chi_B$ as explained in the text
(right) for various values of $\mu_B/T$. The differences between the two
constraints are of the order of 30\% for both quantities. Results have been
obtained on $24^3\times 6$ lattices (left) and $16^3\times 4$ lattices (right).}
\label{fig:constrain}
\end{figure}
The difference is almost negligible, when performing an expansion in the light
quark chemical potential $\mu_q/T$ instead. It is interesting to mention that
with the constraint $n_s=0$, the pressure expansion in $\mu_q/T$ and $\mu_B/T$
are identical up to a trivial factor between the two chemical potentials, i.e.
the relation $\mu_B=3\mu_q$ holds in this case and we have
\begin{equation}
\left. \Delta p/T^4(\mu_q/T)\right|_{n_s=0}
\equiv
\left. \Delta p/T^4(\mu_B/T,\mu_Q=0)\right|_{n_S=0}.
\end{equation}
We have also computed the constrained baryon number fluctuations at finite
baryon chemical potential $\hat\chi_B$. Qualitatively, the two cases of
$\mu_S=0$ and $n_S=0$ are very similar. However, it is interesting to remark
that the two cases reach different Stefan-Boltzmann limits for high
temperatures ($T\to\infty$). Taking this into account we show in
Fig.~\ref{fig:constrain} (right) the ratio $\mathcal{N}\hat\chi_B /\chi_B$,
where $\mathcal{N}$ is the ratio of the corresponding Stefan-Boltzmann values.
As one can see, the difference below $T_c$ is as high as 30\%.
\section{Summary and conclusions}
We have presented a method to rigorously compute corrections to bulk
thermodynamic quantities at non vanishing chemical potential, by performing a
Taylor expansion in $\mu/T$. Our new preliminary results improved previous
calculations in many ways: we went to smaller quark masses, finer lattice
spacings and 2+1 dynamical quark flavor. We also showed how to calculate
various hadronic fluctuations, starting from a theory which naturally is
formulated in terms of quark fields, as QCD is. The Taylor expansion method
provides a variety of input to heavy ion phenomenology.
Our findings are that the finite chemical potential contribution to the pressure
is blow 10\%, up to a chemical potential of $\mu_B/T<3$ and that various
hadronic fluctuations develop a peak with increasing baryon chemical potential.
This seems to hold true also for strangeness fluctuations, although the
peak is much less pronounced in this case. Correlations between strangeness and
other charges increase as well when approaching the critical point.
\section*{Acknowledgments}
We would like to thank all members of the RBC-Bielefeld Collaboration for
helpful discussions and comments. The work has been supported in parts by the
U.S. Department of Energy under Contract No. DE-AC02-98CH10886 and by the
Deutsche Forschungsgemeinschaft under grant GRK 881. Numerical simulations have
been performed on the QCDOC computer of the RIKEN-BNL research center the DOE
funded QCDOC at BNL and the APEnext at Bielefeld University.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
\noindent
The quantization of gravity in $2+1D$ is notoriously a solvable problem \cite{WittenQuantumGravity2+1D}, as it has been emphasised over the last four decades in the literature on topological quantum field theory. There are two fundamental directions to achieve this goal in reduced space-time dimensions. Focusing on the Euclidean $SU(2)$ symmetric version of $2+1D$ gravity, it is possible either to accomplish the quantization of the reduced phase-space of the classical theory, imposing classical constraints before quantization \cite{WittenQuantumGravity2+1D}, or to achieve directly the quantization of the kinematical Hilbert space of the theory, on which only the gauge constraint has been imposed.
Having the cosmological constant involved in the analysis allows for a novel symmetric structure to arise, expressed in terms of the axioms of Hopf algebras. These axioms in turn can be cast resorting to the versatility of the Reidemeister moves \cite{Kauffman:1990am, Kauffman1994}. Jones polynomials are crucial in recovering the link to quantum Hopf algebras, and to show how these latter ones provide link invariants via solutions to the Yang-Baxter equations.
Relevant topological invariants, including the Jones polynomials \cite{WittenJonesPolinomials}, are defined in terms of their properties under Reidemeister moves. This in general characterises a wide field of studies, encoding topological quantum field theory.
In particular, the novel symmetric structures that emerge
are called quantum groups. These are non-trivial Hopf algebras, characterised by the deformation of the product rules in the algebraic sector and the deformation of the Leibnitz rule at the level of the co-algebra, which in turn enters the bi-algebra structure of Hopf algebras.
On the side of the quantization of the reduced phase-space of gravity in $2+1D$ with cosmological constant, it was proven by Witten \cite{WittenJonesPolinomials} that the path-integral quantization of the theory, equivalent to the quantization of two uncoupled Chern-Simons theories, provides the Turaev-Viro topological invariant \cite{TV}. Nonetheless, it has remained unclear hitherto whether a different procedure of quantization, accounting for the imposition at the quantum level of the curvature constraint on the states of the kinematical Hilbert space, would entail the same theory.
\section{Canonical 3D gravity with cosmological constant}
\noindent
The (first order formalism) three-dimensional Riemannian theory of gravity with cosmological
constant $\Lambda$ that we are considering is defined on a space-time $\mathcal{M}$, which we assume to be a three-dimensional oriented smooth manifold, through the expression for the action
\nopagebreak[3]\begin{equation} \label{ac}
S[e,\omega]=\int_{\mathcal{M}} {\rm Tr}[{e} \wedge F({\omega})] + \frac{\Lambda}{3} \,{\rm Tr}[e \wedge e \wedge e]\,,
\end{equation}
where $e$ stands for the triad, which is an $\mathfrak{su}(2)$-valued $1$-form, $\omega$ is an $SU(2)$ three dimensional connection, $F(\omega)$ is the curvature of $\omega$ and the trace ``Tr'' denotes the Killing form on $\mathfrak{su}(2)$. With no loss of generality, we can adopt the usual decomposition and assume the space-time topology to be $\mathcal{M} = \Sigma \times \mathbb{R}$, where $\Sigma$ is a Riemann surface of arbitrary genus.
Suppose now to pull back to $\Sigma$ the spin-connection $\omega$ and the triad $e$, then we can express the new variables in the local coordinates to be the two-dimensional connection $A^i_a$ and the triad field $e_b^j$, in which $a=1,2$ are space coordinate indices and $i,\,j=1,\,2,\,3$ are $\mathfrak{su}(2)$ indices. The Poisson brackets among these variables now provide the symplectic structure
\nopagebreak[3]\begin{equation}
\{ A^i_a(x),\,e_b^j(y)\}=\epsilon_{ab}\,\delta^{ij} \, \delta^{(2)}(x,y)\,.
\end{equation}
The phase space of the theory can also be parametrized in terms of the desitized triad $E^b_j=\epsilon^{bc}\,e^k_c\,\eta_{jk}$, {\it i.e.}
\nopagebreak[3]\begin{equation}
\{ A^i_a(x),\,E^b_j(y)\}=\delta^b_a\,\delta^i_j \delta^{(2)}(x,y)\,.
\end{equation}
Varying the action in terms of the pull back of the fields, {\it i.e.} with respect to the independent fields $A^i_a(x)$ and $e_b^j(x)$, we get the first class local constraints $D_A\,e\simeq 0$ --- here $\simeq$ denotes validity on the constraints surface, namely weak equality --- and $F(A)+ \Lambda e \wedge e \simeq 0$. In terms of the components, we find
\nopagebreak[3]\begin{equation}
D_A^b\,e_b^j=0\,, \qquad
F^i_{ab}(A)+ \Lambda \epsilon^{i}_{\;jk} e_a^j\,e_b^k = 0\,.
\end{equation}
These constraints generate local symmetries. In particular, smearing out $D_A^b\,e_b^j$ with the test field $\alpha_j$ we get the Gau\ss\, constraint
\nopagebreak[3]\begin{equation}
G[\alpha,\, A,\, e]=\int_{\Sigma} \alpha_j\, D_A^b\,e_b^j =0\,,
\end{equation}
which generate infinitesimal $SU(2)$ gauge transformations
\nopagebreak[3]\begin{eqnarray}
\delta_\alpha A^i_a&=&\{ A^a_i,\, G[\alpha,\, A,\, e]\} = (D_a\alpha)^i\,, \nonumber\\
\delta_\alpha e_a^i&=&\{ e_a^i,\, G[\alpha,\, A,\, e]\} = \alpha_k e_{aj} \epsilon^{ijk}\,.
\end{eqnarray}
Smearing out $F^i_{ab}(A)+ \Lambda \epsilon^{i}_{\;jk} e_a^j\,e_b^k$ with the test function $\beta_j$, we get the curvature constraint $C_\Lambda[\beta, A]$, which reads
\nopagebreak[3]\begin{equation}
C_\Lambda[\beta, A,e]=\int_{\Sigma} \beta_i\,(F^i_{ab}(A)+ \Lambda \epsilon^{i}_{\;jk} e_a^j\,e_b^k)=0
\end{equation}
and generates transformations that contain diffeomorphisms, namely
\nopagebreak[3]\begin{eqnarray}
\delta_\beta A^i_a&=&\{ A^i_a,\,C_\Lambda[\beta, A,e] \}= \Lambda \epsilon^{ijk} e^j_a \beta^k\,, \nonumber\\
\delta_\beta e^i_a&=&\{ e^i_a,\,C_\Lambda[\beta, A,e] \}=D_c \beta^i\,,
\end{eqnarray}
provided that the triad fields $e^i_a$ are assumed to be non degenerate.
Indeed, if we consider the vector field $v=v^a\partial_a$ on the surface $\Sigma$ and hence define the parameters $\alpha^i=v^a\,A^i_a$ and $\beta^i=e_a^i \,v^b$, the previous transformations become\footnote{We recall that spatial diffeomorphism along a vector field $v^a$ are defined by $\delta_v A^i_a=\{ A^i_a, V(v^a) \}=\mathcal{L}_v A^i_a$ and $\delta_v e^i_a=\{ e^i_a, V(v^a) \}=\mathcal{L}_v e^i_a$, where $V$ is the canonical vector constraint of general relativity.}
\nopagebreak[3]\begin{equation}
(\mathcal{L}_vA)^i_a \simeq \delta_{\alpha(v)} A_a^i\,, \quad (\mathcal{L}_v e)^i_a \simeq \delta_{\alpha(v)} e^i_a + \delta_{\beta(v)} e^i_a\,,
\end{equation}
where $\mathcal{L}_v$ is the Lie derivative along the vector field $v$.
\section{$SU(2)$ kinematical Hilbert space. }
\noindent
The theory above can be quantized {\it \`a la loop} by a way \cite{Rov, Thi} that follows the Dirac's procedure. Indeed, we can first construct an auxiliary Hilbert space on which we provide a representation of the basic variables we are going to deal with and on which constraints will be represented. In our scheme, connections are represented in terms of holonomies $h_\gamma[A]$ along path $\gamma\in \Sigma$, that are in turn defined by $h_\gamma[A]=P \exp \int_\gamma A$, where $P$ denotes here path ordering. Thus functional of connections will be represented in terms of functionals of holonomies. Triad fields $e^i_a$, associated to the densitized electric field $E_i^a$, will be smeared, as usual, along co-dimension one surfaces. These canonical variables are then promoted to operators acting on the auxiliary Hilbert space of functionals of holonomies. The physical Hilbert space corresponds to those states that are annihilated by the constraints. These states are distributional, as they are not normalizable with respect to the auxiliary Hilbert space and hence no more in it.
The auxiliary Hilbert space $\mathcal{H}_{aux}$ is the Cauchy completion of the space of cylindrical functions $Cyl$. These latter ones are defined on the space of generalized connections $\mathcal{A}$, which provide in turn a map from the set of paths $\gamma\in \Sigma$ to $SU(2)$, and hence represent an extension of the notion of holonomy $h_\gamma[A]$. Elements $\Psi_{\Gamma,f}[A]$ of the space $Cyl$ are defined as follows
\nopagebreak[3]\begin{equation}
\Psi_{\Gamma,f}[A]= f(h_{\gamma_1}[A],\cdot \cdot\cdot h_{\gamma_l}[A],\cdot \cdot\cdot h_{\gamma_L}[A] )\,.
\end{equation}
Consequently, these states are functionals of $A$ labeled by a finite graph $\Gamma\in \Sigma$ and a continuous function $f: \, SU(2)^L \rightarrow{} \mathbb{C}$, where $L$ denotes the number of links $\gamma_l$ of $\Gamma$.
Therefore, the inner product adopted in order to define the completion of the auxiliary Hilbert space is that one used for any two cylindrical functions $\Psi_{\Gamma_1,f_1}[A]$ and $\Psi_{\Gamma_2,f_2}[A]$, namely the Ashtekar-Lewandowski measure
\nopagebreak[3]\begin{eqnarray} \label{Ash-Lew}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\phantom{a}\!\!&\mu(\overline{\Psi_{\Gamma_1,f_1}[A]}\Psi_{\Gamma_2,f_2}[A])\equiv <\Psi_{\Gamma_1,f_1}[A],\, \Psi_{\Gamma_2,f_2}[A] >=\nonumber\\
&=&\int \prod \limits_{\tilde{l}=1}^{\tilde{L}} dh_{\tilde{l}} \, \overline{\tilde{f}_1(\cdot \cdot\cdot h_{\gamma_{\tilde{l}}}[A],\cdot \cdot\cdot h_{\gamma_{\tilde{L}}}[A] )}\cdot \nonumber\\ &\,\,\phantom{a}\,\,&\cdot \tilde{f}_2(\cdot \cdot\cdot h_{\gamma_{\tilde{l}}}[A],\cdot \cdot\cdot h_{\gamma_{\tilde{L}}}[A] )\,, \nonumber
\end{eqnarray}
in which $\tilde{l}$ labels links of $\tilde{\Gamma}=\Gamma_1 \cup \Gamma_2 $ (whose total number of links is $\tilde{L}$) and $\tilde{f}_1$ and $\tilde{f}_2$ denote the extension of the functions $f_1$ and $f_2$, defined respectively on $\Gamma_1$ and $\Gamma_2$, on $\tilde{\Gamma}$, and $dh_{\tilde{l}}$ stands for the invariant $SU(2)$-Haar measure.
Quantization on $\mathcal{H}_{aux}$ of generalized connections is achieved by promoting holonomies to act as operators on $\mathcal{H}_{aux}$, namely
\nopagebreak[3]\begin{equation}
\widehat{h_\gamma[A]}\,\Psi[A]=h_\gamma[A]\, \Psi[A]\,,
\end{equation}
whose procedure defines a self adjoint operator in $\mathcal{H}_{aux}$. In a similar way, the triad $e^i_a$ is promoted to a self adjoint operator valued distribution acting as a derivative with respect to $A$, {\it i.e.}
\nopagebreak[3]\begin{equation}
\widehat{e}_i^a= - i\,L_P \, \frac{\partial}{\partial A^i_a}\,,
\end{equation}
and equivalently the densitized Ashtekar electric field becomes
\nopagebreak[3]\begin{equation}
\widehat{E}^i_a= - i\,L_P \epsilon_{ab} \eta^{ij} \frac{\partial}{\partial A^j_b}\,,
\end{equation}
in which $L_P=\hbar G$ ($G$ being the Newton constant) is the Planck length in three dimensions.
Imposition of the Gau\ss\, constraint corresponds to the selection of elements of $Cyl$ invariant under $SU(2)$ gauge transformations. Concretely, gauge transformations act
on the cylindrical functions by acting on the holonomies as
\nopagebreak[3]\begin{equation}
h_l[A] \rightarrow{} g_{s(l)} \,h_l[A]\, g_{t(l)}^{-1}\,,
\end{equation}
in which $g_{s(l)},\,g_{t(l)}\in SU(2)$ are group elements associated, respectively, to the source and target nodes of the link $l$. The kernel of the Gau\ss\, constraint, namely the projection into the $SU(2)$ gauge invariant subspace of the auxiliary Hilbert space, defines the kinematical Hilbert space $\mathcal{H}_{kin} \subset \mathcal{H}_{aux}$.
Harmonic analysis on $SU(2)$, and specifically the Peter-Weyl theorem, enables us to expand any square integrable function $f: SU(2) \rightarrow \mathbb{C}$ in terms of unitary irreducible representations of $SU(2)$
\nopagebreak[3]\begin{equation}
f(h)=\sum_j f_j \stackrel{j}{\Pi}(h)\,, \quad {\rm with} \quad f_j=\int dh \overline{ \stackrel{j}{\Pi}(h)}\, f(h)\,,
\end{equation}
in which $f_j$ can be seen as an element of the tensor product vector space $ \mathcal{H}_j^* \otimes \mathcal{H}_j $ (where $ \mathcal{H}_j$ denotes the vector space in the $j$ representation and $ \mathcal{H}_j^*$ represents its complex conjugated copy), and magnetic indices contraction is understood.
This procedure clearly enables to introduce an orthonormal basis of states in $\mathcal{H}_{aux}$. Any element of $Cyl$ can be now expressed as a linear combination of tensor product of $L$ $SU(2)$-irreducible representations. Orthogonality of such elements of $Cyl$ is checked by using the physical inner-product (\ref{Ash-Lew}). The action of the $SU(2)$ gauge transformations generator, {\it i.e.} the Gau\ss\, constraint, on Fourier modes is given by
\nopagebreak[3]\begin{equation}
\stackrel{j}{\Pi}(h) \rightarrow{} \stackrel{j}{\Pi}(g_{s(l)}) \, \stackrel{j}{\Pi}(h_l)\, \stackrel{j}{\Pi}(g_{t(l)}^{-1})\,,
\end{equation}
and allows to construct a basis of gauge invariant functions by contraction of Wigner representation matrices with $\mathfrak{su}(2)$-invariant tensor or $\mathfrak{su}(2)$-intertwiners. Intertwiners that are $\mathfrak{su}(2)$-invariant admit an orthomormal basis $\iota_n\in {\rm Inv}[\mathcal{H}_{j_1}\otimes\mathcal{H}_{j_2}\otimes \cdot \cdot \cdot \otimes \mathcal{H}_{j_L}]$ and are labeled by $n$. Once we have introduced such a notation, we are able to define a basis of gauge-invariant elements of $Cyl$ that corresponds to the spin-network basis, whose element are labeled by a graph $\Gamma$, a set of spin $\{j_l\}$ for each link $l\in \Gamma$ and a set of intertwiners $\iota_n$ labeling nodes $n\in \Gamma$
\nopagebreak[3]\begin{equation}
\psi_{\Gamma,\{j_l\}, \{\iota_n\} }[A]= \bigotimes_{n \in \Gamma} \iota_n\, \bigotimes_{l \in \Gamma}\, \stackrel{j_{\gamma}}{\Pi}(h_l [A]) \,.
\end{equation}
\section{The quantum symmetrizer in the loop basis}
\noindent
Historically, the loop basis was introduced by Rovelli and Smolin in \cite{RS1,RS2}, in order to implement the Wilsonian quantization of the Einstein-Hilbert theory of gravity, recast in term of the gauge ``Ashtekar'' variables \cite{Ash}. The elements of this basis are the Wilson loops, which are traces of closed holonomies of the gravitational $SU(2)$ gauge connection. These are automatically gauge invariant due to the properties of traces. It was then shown in \cite{Rovelli:1995ac} that the same basis is equivalent to the spin-network basis. Roughly speaking, the equivalence of the two bases is obtained by undoing the symmetrizer at the edges of given spin-network basis elements, therefore projecting each bundle of strands onto the Temperley-Lieb algebra with the same number of strands. This same principle will be used below as well, as we shall see.
Loops introduced in this way are kinematical objects --- they do not account for the imposition of the space diffeomorphism and time re-parametrization constraints that implement the dynamics of the Einstein-Hilbert action --- and apply to the quantization of the Hilbert space of the $2+1D$ reduction of the same theory, which reduces to a topological quantum field theory.
The building block for our considerations is the loop in the fundamental representation (rep) of $SU(2)$. This is expressed by the Wigner matrix ${\Pi}(g)$, which provides the fundamental representation of an element $g\in SU(2)$. When the dependence on the group element is frozen by setting it equal to the unit element $g=e$, the trace of the Wigner matrix, namely the Wilson loop of the fundamental rep, provides the dimension of this latter one.
We may introduce for convenience graphical notations for the irreducible representations (irreps) of $SU(2)$. Spinor indices are denoted as $A,B\in\{0,1\}$. The trivial (identity) intertwiner $\delta^{\ A}_{B}$ is graphically denoted as a straight line, while the Levi-Civita tensors $\epsilon_{AB}$ and $\epsilon^{AB}$ are denoted respectively as bottom-up and bottom-down arches. Within this notation, a Wilson loop reads
\nopagebreak[3]\begin{equation} \label{s2}
\begin{tikzpicture}[baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0) circle (15pt);
\end{tikzpicture}
=
\delta^A_B \stackrel{\frac{1}{2}}\Pi\!\!\!\!\!\! \phantom{\Pi}^B_A \, (g) |_{g=e}\equiv {\chi}_{\frac{1}{2}}(g)\,,
\end{equation}
where $\chi(g)$ denotes the character of the irrep ${\Pi}(g)$ of a group element $g\in SU(2)$. Furthermore, taking into account this diagrammatics, we can construct the Jones-Wenzl projector moving from the realization of the symmetrizer for two fundamental reps, so as to realize an irrep of spin $1$. The coefficients at the right hand-side of Eq.~\eqref{s2} are determined by the properties of the Wigner matrices (instantiating the irreps of $SU(2)$) with $g=e$, once the symmetrizer of the two fundamental reps is required to be a projector --- by iteration for $n>2$ fundamental reps, the Jones-Wenzl projector for a generic irrep of spin $j=n/2$ is recovered.
We observe that setting $g\!=\!e$ is equivalent to imposing the curvature flatness condition on the elements of the kinematical Hilbert space of the theory that is considered. In particular, in $2+1D$ this corresponds to imposing the curvature constraint of the Einstein-Hilbert action. Technically, this is realized by considering the action of the Dirac delta function $\delta(g)$ on loops, or equivalently on holonomies. The dimension of the fundamental representation of $SU(2)$ reads in this case
\nopagebreak[3]\begin{equation} \label{abc}
d= \delta(g ) \triangleright \chi_{\frac{1}{2}}(g) \,,
\end{equation}
where the action $\triangleright$ of $\delta(g )$ is normalized by integration with respect to the Haar measure $dg$. From now on, we focus on the $2+1D$ Einstein-Hilbert action with cosmological constant $\Lambda$. The imposition of the curvature constraint to the loops, in the fundamental representation, entails to calculate the trace of the holonomy of the Ashtekar connection in a $SU(2)$ group element $H_{\Lambda}$ that represents the space-time curvature induced by $\Lambda$.
As we will show in section \ref{reg}, the action of the curvature constraint when a non-vanishing $\Lambda$ is added to the Einstein-Hilbert action amounts to the multiplication of the Hilbert space elements by the Dirac delta function $\delta(g\, H_{\Lambda}^{-1})$, namely
\begin{eqnarray}
\begin{tikzpicture}[baseline={([yshift=-0.3cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0) circle (15pt);
\draw[fill=black] (-0.52,0) circle (2pt);
\node (a) at (0.5,0.7) {$\frac{1}{2}$};
\end{tikzpicture}
&=& \delta(gH_\Lambda^{-1})\chi^{\frac{1}{2}}(g) = \stackrel{\frac{1}{2}}\Pi\!\!\!\!\!\! \phantom{\Pi}^{\alpha}_{\alpha} (gH_\Lambda^{-1}) \,,\label{eq:diag_loop} \\
\chi^{\frac{1}{2}}(H_\Lambda) &=& {\rm Tr}_{\frac{1}{2}}(e^{i\tau_3\sqrt{\Lambda}n^3}) = \cos(\sqrt{\Lambda}n) \,,\label{eq:trace_loop}
\end{eqnarray}
with $\tau_3$ a basis element of $\mathfrak{su}(2)$ in the fundamental representation, and the appearance of $\sqrt{\Lambda}$ in Eq.~\eqref{eq:trace_loop} clarified in section \ref{reg}. This immediately provides the fundamental representation
\nopagebreak[3]\begin{equation}
d_q= {\chi}_{\frac{1}{2}} (H_{\Lambda})
\end{equation}
of $SU_q(2)$, i.e. the Chebyschev polynomial of degree one in which the parameter $q$ is a function of $\sqrt{\Lambda}$. Projecting two strands on the loop basis, i.e. projecting two open strands on the Temperley-Lieb algebra, along with the value of the trace just computed provides the Jones-Wenzl projector at $q\neq -1$. In fact, in the Temperley-Lieb algebra we have
\begin{eqnarray}\label{eq:qJW_2_strings}
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.3cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (-1,-2) -- (-1,0);
\draw (1,-2) -- (1,0);
\draw (-1.5,0) rectangle (1.5,1);
\draw (-1,1) -- (-1,3);
\draw (1,1) -- (1,3);
\draw[fill=black] (1,-1) circle (4pt);
\draw[fill=black] (-1,-1) circle (4pt);
\end{tikzpicture}
&=&
a\ \
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.3cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (-1,-2) -- (-1,3);
\draw (1,-2) -- (1,3);
\draw[fill=black] (-1,-1) circle (4pt);
\draw[fill=black] (1,-1) circle (4pt);
\end{tikzpicture}
+
b \ \
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.3cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (-1,2) arc (180:360:1.5cm and 1.5cm);
\draw (2,-2) arc (0:180:1.5cm and 1.5cm);
\draw[fill=black] (-0.75,-1.2) circle (4pt);
\draw[fill=black] (1.75,1.2) circle (4pt);
\end{tikzpicture} \,.
\end{eqnarray}
Imposing that the symmetrizer is a projector ({\it i.e.} it is idempotent) fixes the coefficients to be $a=1$ and $b=-d_q^{-1}$, which shows the claim.
We consider now the symmetrizer on a number of strands larger than two, showing an iterative version of the reasoning given in Eq.~\eqref{eq:qJW_2_strings}. In fact, using the symmetrizer to project on Temperley-Lieb algebra elements, along with the value of the ``fundamental loop'' computed in Eq.~\eqref{eq:diag_loop} and Eq.~\eqref{eq:trace_loop}, automatically forces the Kauffman smoothing relations where the smoothing factor is provided by the quantum dimension. This is essentially a consequence of the fact that the Kauffman bracket is unique, and the value as computed in Eq.~\eqref{eq:trace_loop} fixes the value of the coefficients to be the quantum dimension. We want to prove that using the symmetrizer with group element corresponding to the cosmological constant necessarily satisfies the inductive equations at the end of Section~3.2 of \cite{KL}, which shows that this is the Jones-Wenzl projector with arbitrary $q$.
We proceed inductively, using as base for induction the result already displayed for two strands. Suppose that the statement holds for some $k>2$. Applying the symmetrizer on $k+1$ strands we find two types of elements of the Temperley-Lieb algebra. Those where the $(k+1)^{\rm th}$-strand is straight, and those where it is not. We depict this situation diagrammatically as
\begin{eqnarray}
\begin{tikzpicture}[baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1);
\draw (0.2,2) -- (0.2,1);
\draw[dashed] (0.25,1.5) -- (0.95,1.5);
\draw (1,2) -- (1,1);
%
\draw (-0.2,1) rectangle (1.2,0.5);
\draw (0,0.5) -- (0,-0.5);
\draw (0.2,0.5) -- (0.2,-0.5);
\draw[dashed] (0.25,0) -- (0.95,0);
\draw (1,0.5) -- (1,-0.5);
%
\draw[fill=black] (0,0) circle (2pt);
\draw[fill=black] (0.2,0) circle (2pt);
\draw[fill=black] (1,0) circle (2pt);
\end{tikzpicture}
&=& \label{eq:higher_smoothing}
A\ \ \begin{tikzpicture}[baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1);
\draw (0.2,2) -- (0.2,1);
\draw[dashed] (0.25,1.5) -- (0.9,1.5);
\draw (0.95,2) -- (0.95,1);
\draw (1.3,2) -- (1.3,-0.5);
%
\draw (-0.2,1) rectangle (1.15,0.5);
\draw (0,0.5) -- (0,-0.5);
\draw (0.2,0.5) -- (0.2,-0.5);
\draw (0.95,0.5) -- (0.95,-0.5);
\draw[dashed] (0.25,0) -- (0.9,0);
%
\draw[fill=black] (0,0) circle (2pt);
\draw[fill=black] (0.2,0) circle (2pt);
\draw[fill=black] (1.3,0) circle (2pt);
\draw[fill=black] (0.95,0) circle (2pt);
\end{tikzpicture}
+
B\ \
\begin{tikzpicture}[baseline={([yshift=-0.2cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1.7);
\draw (0.2,2) -- (0.2,1.7);
\draw (0.95,2) -- (0.95,1.7);
\draw[dashed] (0.3,1.9) -- (0.9,1.9);
\draw (-0.2,1.7) rectangle (1.15,1.2);
\draw (0,1.2) -- (0,0.3);
\draw (0.2,1.2) -- (0.2,0.3);
\draw[dashed] (0.3,0.75) -- (0.9,0.75);
\draw (-0.2,0.3) rectangle (1.15,-0.2);
\draw (0,-0.2) -- (0,-0.7);
\draw (0.2,-0.2) -- (0.2,-0.7);
\draw[dashed] (0.3,-0.45) -- (0.95,-0.45);
\draw (0.95,-0.2) -- (0.95,-0.7);
\draw[rounded corners] (1.2,0.8) ..controls(1.4,1).. (1.5,2);
\draw[rounded corners] (0.95,1.2) .. controls(0.95,0.85).. (1.2,0.8);
\draw[rounded corners] (1.2,0.6) ..controls(1.4,0.8).. (1.5,-0.7);
\draw[rounded corners] (0.95,0.3) .. controls(1.05,0.5).. (1.2,0.6);
\draw[fill=black] (0,-0.5) circle (2pt);
\draw[fill=black] (0.2,-0.5) circle (2pt);
\draw[fill=black] (0.95,-0.5) circle (2pt);
\draw[fill=black] (1.48,-0.5) circle (2pt);
\end{tikzpicture}\,,
\end{eqnarray}
where the rectangles are still to be determined, as in the step with $k=2$. Also, the rectangles that symmetrize the $k$ strands are in principle not necessarily equal to the symmetrizer obtained from the inductive step at $k$. Let us now apply the symmetrizer twice, to obtain the diagrammatic equation
\begin{eqnarray}
\begin{tikzpicture}
[scale=0.7,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1);
\draw (0.2,2) -- (0.2,1);
\draw[dashed] (0.25,1.5) -- (0.95,1.5);
\draw (1,2) -- (1,1);
%
\draw (-0.2,1) rectangle (1.2,0.5);
\draw (0,0.5) -- (0,-0.5);
\draw (0.2,0.5) -- (0.2,-0.5);
\draw[dashed] (0.25,0) -- (0.95,0);
\draw (1,0.5) -- (1,-0.5);
%
\draw[fill=black] (0,0) circle (2pt);
\draw[fill=black] (0.2,0) circle (2pt);
\draw[fill=black] (1,0) circle (2pt);
\draw (0,3) -- (0,4);
\draw (0.2,3) -- (0.2,4);
\draw[dashed] (0.25,2.5) -- (0.95,2.5);
\draw (1,3) -- (1,4);
%
\draw (-0.2,3) rectangle (1.2,2.5);
\draw (0,2.5) -- (0,1.5);
\draw (0.2,2.5) -- (0.2,1.5);
\draw[dashed] (0.25,2) -- (0.95,2);
\draw (1,2.5) -- (1,1.5);
%
\draw[fill=black] (0,2) circle (2pt);
\draw[fill=black] (0.2,2) circle (2pt);
\draw[fill=black] (1,2) circle (2pt);
\end{tikzpicture}
&=&\label{eq:concatenate}
A^2\ \
\begin{tikzpicture}[scale=0.7,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1);
\draw (0.2,2) -- (0.2,1);
\draw[dashed] (0.25,1.5) -- (0.9,1.5);
\draw (0.95,2) -- (0.95,1);
\draw (1.3,2) -- (1.3,-0.5);
%
\draw (-0.2,1) rectangle (1.15,0.5);
\draw (0,0.5) -- (0,-0.5);
\draw (0.2,0.5) -- (0.2,-0.5);
\draw (0.95,0.5) -- (0.95,-0.5);
\draw[dashed] (0.25,0) -- (0.9,0);
%
\draw[fill=black] (0,0) circle (2pt);
\draw[fill=black] (0.2,0) circle (2pt);
\draw[fill=black] (1.3,0) circle (2pt);
\draw[fill=black] (0.95,0) circle (2pt);
\draw (0,4.5) -- (0,3.5);
\draw (0.2,4.5) -- (0.2,3.5);
\draw[dashed] (0.25,4) -- (0.9,4);
\draw (0.95,4.5) -- (0.95,3.5);
\draw (1.3,4.5) -- (1.3,2);
%
\draw (-0.2,3.5) rectangle (1.15,3);
\draw (0,3) -- (0,2);
\draw (0.2,3) -- (0.2,2);
\draw (0.95,3) -- (0.95,2);
\draw[dashed] (0.25,2.5) -- (0.9,2.5);
%
\draw[fill=black] (0,2.5) circle (2pt);
\draw[fill=black] (0.2,2.5) circle (2pt);
\draw[fill=black] (0.95,2.5) circle (2pt);
\end{tikzpicture}
+
AB\ \
\begin{tikzpicture}[scale=0.7,baseline={([yshift=-0.2cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]]
\draw (0,2) -- (0,1.7);
\draw (0.2,2) -- (0.2,1.7);
\draw (0.95,2) -- (0.95,1.7);
\draw[dashed] (0.3,1.9) -- (0.9,1.9);
\draw (-0.2,1.7) rectangle (1.15,1.2);
\draw (0,1.2) -- (0,0.3);
\draw (0.2,1.2) -- (0.2,0.3);
\draw[dashed] (0.3,0.75) -- (0.9,0.75);
\draw (-0.2,0.3) rectangle (1.15,-0.2);
\draw (0,-0.2) -- (0,-0.7);
\draw (0.2,-0.2) -- (0.2,-0.7);
\draw[dashed] (0.3,-0.45) -- (0.95,-0.45);
\draw (0.95,-0.2) -- (0.95,-0.7);
\draw[rounded corners] (1.2,0.8) ..controls(1.4,1).. (1.5,2);
\draw[rounded corners] (0.95,1.2) .. controls(0.95,0.85).. (1.2,0.8);
\draw[rounded corners] (1.2,0.6) ..controls(1.4,0.8).. (1.5,-0.7);
\draw[rounded corners] (0.95,0.3) .. controls(1.05,0.5).. (1.2,0.6);
\draw[fill=black] (0,-0.5) circle (2pt);
\draw[fill=black] (0.2,-0.5) circle (2pt);
\draw[fill=black] (0.95,-0.5) circle (2pt);
\draw[fill=black] (1.48,-0.5) circle (2pt);
\draw (0,4.5) -- (0,3.5);
\draw (0.2,4.5) -- (0.2,3.5);
\draw[dashed] (0.25,4) -- (0.9,4);
\draw (0.95,4.5) -- (0.95,3.5);
\draw (1.5,4.5) -- (1.5,2);
%
\draw (-0.2,3.5) rectangle (1.15,3);
\draw (0,3) -- (0,2);
\draw (0.2,3) -- (0.2,2);
\draw (0.95,3) -- (0.95,2);
\draw[dashed] (0.25,2.5) -- (0.9,2.5);
%
\draw[fill=black] (0,2.5) circle (2pt);
\draw[fill=black] (0.2,2.5) circle (2pt);
\draw[fill=black] (0.95,2.5) circle (2pt);
\draw[fill=black] (1.5,2.5) circle (2pt);
\end{tikzpicture}
\\ \notag
\\ \notag
&+&
AB
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+
B^2\ \
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\end{eqnarray}
Imposing the symmetrizer to be idempotent, it follows that the term with coefficient $A$ has idempotent boxes of degree $k$, which automatically forces this to be the symmetrizer obtained for $k$, since this is unique by induction hypothesis. It also follows that $A=1$. Then, let us indicate by $\Phi$ and $\Psi$ the diagrams whose coefficients are $A$ and $B$ in Eq.~\eqref{eq:higher_smoothing}, respectively. Then, by equating the terms where the last strand is not straight, in the idempotence condition, we have the equation $B\Psi = AB (\Phi\Psi + \Psi\Phi) + B^2 \Psi^2$, {\it i.e.} the second term of Eq.~\eqref{eq:higher_smoothing} is equal to the last three terms of Eq.~\eqref{eq:concatenate}. Using the idempotence in the known results for the Jones-Wenzl projector for degree $k$, which hold true by inductive hypothesis, it now follows that the symmetrizer at degree $k+1$ coincides with the quantum Jones-Wenzl projector, where the value of $d$ is given by the quantum dimension computed in Eq.~\eqref{eq:trace_loop}. This shows that the value of the fundamental loop with group element being the first Chebyschev polynomial, along with projecting onto the loop basis, substantially determines our effective recoupling theory to be the quantum recoupling theory of \cite{KL} at $A\neq -1$. Moreover, we observe that we have not assumed $A$ be a root of unity at any step.
We therefore obtain a diagrammatics that implies an effective recoupling, which is equivalent to the Kauffman-Lins recoupling with $A\neq -1$, namely to the recoupling theory of $SU_q(2)$ that is implemented in the Turaev-Viro model. Here strands with bullets indicate representations with the insertion of the group element, and the symmetrizer on such strands is the quantum Wenzl-Jones symmetrizer, following the procedure given above.
We emphasize that this is entirely due to the dynamical implementation of the curvature constraint at the quantum level, and the assumption that projecting onto the loop basis satisfies the idempotence condition. Thus the effective quantum representations that are found within this scheme are the by-product of the quantization of the Einstein-Hilbert action with cosmological constant in $2+1D$. \\
\section{Spin-foam dynamics}
\noindent
Quantum gravity in $2+1D$ could have been completely solved in \cite{NoPe} by regularizing the generalized projector $P$. The projection operator $P$ encodes the quantum evolution due to the presence of the Hamiltonian constraint and provides a physical scalar product in the spin-foam representation, which produces the Ponzano-Regge model. In this section we consider the procedure of extending the regularization of the projector $P$ in the presence of nontrivial cosmological constant. We start from the setting of \cite{NoPe}, and we consider the dynamics from the spin-foam perspective, as a covariant way to implement the quantization of loop quantum gravity.
The extension of these results to the $3+1D$ case presents several difficulties that have
not yet been overcome \cite{Haggard:2015nat,Haggard:2014xoa,Han:2011aa,Han:2010pz,Noui:2002ag,Fairbairn:2010cp}. \\
The generalized projector $P$ defining the generic physical scalar product $<s, P\,s'>$ (between spin-network states $s$ and $s'$ of the physical Hilbert space) implements the curvature constraint in the spin-foam formalism representation of $<s, P\,s'>$, and is expressed by
\nopagebreak[3]\begin{eqnarray} \label{prolam}
&P=``\prod \limits_{x\in \Sigma} \delta\left( \hat{F}(\omega) + \Lambda \, \hat{e} \wedge \hat{e} \right)" =\nonumber\\
& \int \mathcal{D} [N] \exp i \int_\Sigma {\rm Tr} [N \left( F(\omega) + \Lambda \, \hat{e} \wedge \hat{e} \right)]\,,
\end{eqnarray}
where $N=N^i \tau_i\in \mathfrak{su}(2)$ and $\tau_i$ are basis elements of $\mathfrak{su}(2)$ in the fundamental representation.\\
Following \cite{NoPe}, we pick a cellular decomposition $\Sigma_\delta(\Gamma, \Gamma')$ of $\Sigma$ depending on an infinitesimal parameter $\delta \in \mathbb{R}$. The cellular decomposition $\Sigma_\delta(\Gamma, \Gamma')$ consists of $0$-cells called vertices, $1$-cells that consist of edges connecting the $0$-cells, and $2$-cells called plaquettes, and denoted by $p$. The latter are squares delimited by $1$-cells between $0$-cells.
The union of $0$-cells and $1$-cells contains the graphs $\Gamma$ and $\Gamma'$ on which the spin-networks $s$ and $s'$ are supported, respectively. Furthermore, we assume that there is a covering by open balls $\mathcal{B}_\delta$ of radii $\delta$ such that each plaquette $p$ is contained in some $\mathcal{B}_\delta$. Therefore, as $\delta \rightarrow 0$, the plaquettes shrink to points.
This allows us to define a regularization for the physical inner product as in \cite{NoPe}. Indeed, once an ordering for the set of plaquettes $p^i\in \Sigma_\delta(\Gamma, \Gamma')$ with $i=1,...N_p^\delta$ has been introduced --- $N_p^\delta$ being the total number of plaquettes for an assigned value of the regulator $\delta$ of the cellular decomposition --- the physical inner product between two spin-network states $s$ and $s'$, respectively supported on $\Gamma$ and $\Gamma'$, is
\nopagebreak[3]\begin{eqnarray} \label{sv}
<s\!\!\!\!&,&\!\!\! \!s'>_{\rm Phys}= <s, P s'> =\\
&=& \!\!{\rm lim}_{\delta \rightarrow 0} \sum \limits_{j_{p^i}} {\rm dim}\,j_{p^i} <\prod_{p^i} \chi_{j_{p^i}}(U_{p^i}\, H^{-1}_\Lambda) s, s' >\,, \nonumber
\end{eqnarray}
where ${\rm dim}\,j_{p^i}$ stands for the dimension of the irrep of spin $j_{p^i}$, $U_{p^i}$ for the holonomy around the plaquette $p^i$, $H_\Lambda$ is a $SU(2)$ group element encoding space-time curvature and $ \chi_{j_{p^i}}(U_{p^i}\, H^{-1}_\Lambda)$ denotes the trace of the irrep $\Pi^j$ of spin $j$ of the $SU(2)$ group element ``$U_{p^i}\, H^{-1}_\Lambda$''.\\
\section{Regularization of the Hamiltonian constraint in loop basis} \label{reg}
\noindent
We can regularize the curvature constraint operator, extending the $2+1D$ analysis for the Einstein-Hilbert action developed in \cite{NoPe} to the case encoding the cosmological constant.
The main difference with respect to the previous literature is that we first smear at the classical level triads on the edge of the lattice dual to the the square tessellation of the space surfaces, and then obtain an element of the $\mathfrak{su}(2)$ algebra. We then regularize the curvature constraint as the product of two $SU(2)$ group elements, and implement the quantization in terms of holonomy operators in the fundamental representation. The effective (``quantum group'' like) recoupling theory that is induced by the quantum dynamics, as previously commented, will then extend the result to loops in any irrep of $SU(2)$. \\
In more detail, we can motivate Eq.~(\ref{sv}) by considering that, for a local patch $X\in \Sigma$ in which the cellular decomposition is made out of square cells of coordinate length $\delta$, the curvature constraint reads
\nopagebreak[3]\begin{eqnarray}
\!\!\!\!F_{\Lambda}[N]&\!=\!&\int_X {\rm Tr} [N \,( F(\omega)+ \Lambda \, e \wedge e)]=\nonumber\\
&\!=\!&\!\lim \limits_{\delta \rightarrow 0}\, \sum \limits_{p^i} \delta^2\, {\rm Tr}[N_{p^i}( F_{p^i} + \Lambda\, n_{p^i} )]\,,
\end{eqnarray}
in which the $\mathfrak{su}(2)$ algebra elements $N_{p^i}$ and $F_{p^i}$ and $n_{p^i}$ stand respectively for the value of $N=N^j\, \tau_j$ and $\tau_j \epsilon^{ab} F_{ab}^j(\omega)$ and $\epsilon^{ab}e_a^i e_b^k \epsilon_{ik}^{\,\,\,\,\,j} \tau_j$ at an interior point $p^i$ of the $i$-th plaquette. It was already noticed in \cite{NoPe} that the holonomy $U_{p^i}$ undergoes in the $\delta\rightarrow 0$ limit the approximation
\nopagebreak[3]\begin{equation}
U_{p^i}(\omega)= 1\!\!1+ \delta^2 F_{p^i}(\omega)+ O(\delta^2)\,,
\end{equation}
and that as a consequence
\nopagebreak[3]\begin{equation}
F[N]=\int_X {\rm Tr} [N F(\omega)]=\lim \limits_{\delta \rightarrow 0} \sum \limits_{p^i} {\rm Tr}[N_{p^i} U_{p^i}(\omega)]\,.
\end{equation}
We notice that a similar argument can be deployed to recast the term $\Lambda \epsilon_{jk}^{i} e^j_a \,e^k_b$, which amounts to the action of the triads on the dual lattice, given that we perform the expansion of triads around the base point $p^i$ positioned at the center of the $i$-th plaquette as it follows:
\nopagebreak[3]\begin{equation}
e^i_a(x)\big|_{p^i}\simeq \delta^i_a + O(\delta). \label{exe}
\end{equation}
This observation allows us to define a $SU(2)$ group element $H^{\Lambda}_{p^i}$, which expanding in the infinitesimal $\delta$ parameter reads
\nopagebreak[3]\begin{equation} \label{expH}
H^{\Lambda}_{p^i}= 1\!\!1+ \delta^2 \Lambda n_{p^i} + O(\delta^2)\,.
\end{equation}
Within this expression the smeared (on the $\mathfrak{su}(2)$ algebra element $N$) flux of $\epsilon^{ab}e_a^i e_b^k \epsilon_{ik}^{\,\,\,\,\,j} \tau_j$ appears. Indeed
\nopagebreak[3]\begin{eqnarray}
\Lambda \int_X {\rm Tr} [N e\wedge e]&=& \Lambda \int_X \epsilon_{ijk} N^i \epsilon^{ab}\, e_a^j e_b^k =\nonumber\\
&=&\lim \limits_{\delta \rightarrow 0} \sum \limits_{p^i} {\rm Tr}[N_{p^i} H^\Lambda_{p^i}(\omega)]\,,
\end{eqnarray}
provided that on the loop states
\nopagebreak[3]\begin{equation} \label{smd}
\delta^2 n^j_{p^i} \sim
\lim \limits_{\delta \rightarrow 0} \Phi_{X} (\tilde{E}^j)=\lim \limits_{\delta \rightarrow 0} \int_{X} \epsilon^{ab}e_a^i e_b^k \epsilon_{ik}^{\,\,\,\,\,j} \,.
\end{equation}
In Eq.~\eqref{smd} we have used the definition $E^b_i=\epsilon^{ab} e^s_a \eta_{is}$ of the Ashtekar electric field on the $2$-dimensional surface of pull-back $\Sigma$, we have performed the triadic projection with respect to $e^k_b$ and finally contracted the internal indices with the Levi-Civita tensor $\epsilon_{k}^{\;\;ij}$, namely $\tilde{E}^j=E^b_i e_b^k \epsilon_{k}^{\;\;ij}$.
Because of the gauge invariance of $\chi_{j_{p^i}}(U_{p^i}\, H^{-1}_\Lambda)$, we can rewrite the $SU(2)$ group element encoding the space-time curvature as
\nopagebreak[3]\begin{equation}
H^\Lambda_{p^i}=\exp (\Lambda n^3_{p^i} \tau_3)\,.
\end{equation}
At each plaquette we can renormalize $\Lambda$, and rewrite $H^{\Lambda'}_{p^i}=\exp (\Lambda_{p^i}' \tau_3)$. For the regularization of the projector on the loop basis we can write $H^{\Lambda'}_{p^i}=\exp (\Lambda' \tau_3)$, where the discretization is independent on the $i$-th plaquette that has been chosen. We then notice that a rescaling on the connection $\omega$ by $1/G$, so to make it dimensionless, as well as a rescaling of the coordinates by $1/\sqrt{\Lambda}$, allows to recast the action in terms of only dimensionless quantities, as
\nopagebreak[3]\begin{equation}\label{res_act}
S'=\frac{1}{G\, \sqrt{\Lambda}} \int_{\mathcal{M}} \frac{1}{G\, \sqrt{\Lambda}} \, {\rm Tr}[{e} \wedge F({\omega})] + \frac{1}{3} \,{\rm Tr}[e \wedge e \wedge e]\,.
\end{equation}
The peculiarity of Eq.~\eqref{res_act} now traces back at the level of the spin-foam dynamics to the definition of a $SU(2)$ group element encoding space-time curvature of the form $H^{\Lambda}\!=\!\exp (G \sqrt{\Lambda}\, \tau_3 n^3)$. For a generic ``quantum-group effective'' irrep $j$, using the effectively induced Jones-Wenzl projector, the evaluation of the trace of $H_{\Lambda}$ provides the Chebyschev $\Delta_{2j}^{\Lambda}$ polynomial of degree-$2j$, evaluated in $\sqrt{\Lambda}$. \\
We finally comment that the curvature group element $H_\Lambda$ converges to the unity of the group $U=e$ in the vanishing cosmological constant limit $\Lambda \rightarrow 0$. Hence the standard flat curvature constraint is recovered \cite{NoPe}, which induces the convergence of the recoupling theory of $SU_q(2)$ to the standard recoupling theory of $SU(2)$.\\
\section{Two loops calculation}
\noindent
As an instructive study case, we inspect the physical scalar product of two loops, for the case without cosmological constant, as in the framework of \cite{NoPe}, and subsequently repeat the calculation for the case with non-vanishing cosmological constant.
For two cylindrical functions $\Psi_{\Gamma_1, f[A]}$ and $\Psi_{\Gamma_2, g[A]}$, the inner product is defined by the AL measure
\begin{eqnarray}
\mu_{AL}(\overline{\Psi_{\Gamma_1, f[A]}}\Psi_{\Gamma_2, g[A]})=\left\langle \Psi_{\Gamma_1, f[A]},\Psi_{\Gamma_2, g[A]}\right\rangle\\
=\int \Pi_{i=1}dh_i\overline{f(h_{\gamma_1},....,h_{\gamma_{N_{\ell}}})}g(h_{\gamma_1},....,h_{\gamma_{N_{\ell}}}),
\end{eqnarray}
where $dh_i$ corresponds to the invariant $SU(2)$ Haar measure.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{Two_loops.png}
\caption{\footnotesize{Diagrammatic definition of inner product on two loops (left) and result of integration (right). }} \label{fig:Two_loops}
\end{figure}
We calculate the inner product between two loops with spin $j$ and $j'$, inserting an extra loop corresponding to the projector of \cite{NoPe}. Specifically, we compute
\begin{eqnarray}
\left\langle O_j, O{j'}\right\rangle_{\rm Ph} &=& \int dU\Sigma_k \Delta_k\chi^*_j(U)\chi_{j'}(U)\chi_k(U)\\
\nonumber &=& \Sigma_k\Delta_k \int dU \stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\alpha}_{\alpha}(U)\stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\beta}_{\beta}(U)\stackrel{j'}\Pi\!\!\!\!\!\!\phantom{\Pi}^{\gamma}_{\gamma}(U) \,.\\
\end{eqnarray}
Notice that such integration can be directly solved without expanding the Dirac delta on the group, but in stead imposing the curvature constraint. In absence of cosmological constant, this latter reads $U=e$, with $U$ group element around the loop and $e$ unit element of $SU(2)$. In this case, using the composition rule $\chi^*_j(U)\chi_{j'}(U)=\sum_k \chi_k(U)$, where the sum is over the compatible spin $k$ such that $|j-j'|<k<j+j'$, we can immediately find
\begin{eqnarray}
\left\langle O_j, O{j'}\right\rangle_{\rm Ph} &=& \int dU \chi^*_j(U)\chi_{j'}(U)\delta(U)\\
\nonumber &=& \Sigma_k \int dU \chi_k(U) \delta(U)= \Sigma_k \Delta_k \,.
\end{eqnarray}
We can now show that the same result can be recovered by expanding the Dirac delta function, as in \cite{NoPe}. In this case, integration over the three representations of the group elements along each link of the squared loop provide four pairs of trivalent intertwiners. This is shown on the right hand side of Figure~\ref{fig:Two_loops}. We represent these intertwiners in terms of Jones-Wenzl projectors and renormalize the trivalent vertices by the $\theta$-net evaluations. Specifically, the internal gauge indices of these tensors are contracted with one another according to a specific combinatorial path of contractions that respect the symmetries of the Jones-Wenzl projector, namely
\begin{eqnarray} \label{C6}
\int &dU& \stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\alpha}_{\alpha}(U)\stackrel{j'}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\beta}_{\beta}(U)\stackrel{j''}\Pi\!\!\!\!\!\!\phantom{\Pi}^{\gamma}_{\gamma}(U)
= \upsilon^{\alpha \beta \gamma}\ \upsilon_{\alpha \beta \gamma} \nonumber \\
&=& \frac{\overline{\upsilon}^{\alpha \beta \gamma}}{\sqrt{\theta(a,b,c)}} \, \ \frac{\overline{\upsilon}_{\alpha \beta \gamma}}{\sqrt{\theta(a,b,c)}} \,
\,,
\end{eqnarray}
where $\upsilon$ is the trivalent intertwiner among the $j$, $j'$ and $j''$ representations --- assumed to be compatible to provide a non-trivial non-vanishing result, namely $j+j'+j''=\mathbb{N}$ --- of the spin-network basis, and $\overline{\upsilon}$ denotes the trivalent intertwiner in the Kauffman-Lins formalism among $a$, $b$ and $c$ fundamental representations, having introduced $a=2j$, $b=2j'$ and $c=2j''$.
Finally, since contraction over the indices provides
\begin{eqnarray}
\overline{\upsilon}^{\alpha \beta \gamma} \overline{\upsilon}_{\alpha \beta \gamma} = \theta(a,b,c)\,,
\end{eqnarray}
we obtain the result
\begin{eqnarray}
\left\langle O_j, O_{j'}\right\rangle_{\rm Ph} = \Sigma_k \Delta_k \,,
\end{eqnarray}
where the sum is over all the $k$ under the restrictions imposed by the compatibility conditions. We observe that these are finitely many, thus the result is finite.
We now repeat the same steps, accounting for a non-vanishing cosmological constant.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{Lambda_two_loops.png}
\caption{\footnotesize{Diagrammatic definition of inner product on two loops (left) and result of integration (right), where the black dot indicates the presence of the group element.}}
\label{fig:Lambda_two_loops}
\end{figure}
We calculate the inner product between two loops with spin $j$ and $j'$
\begin{eqnarray}
\left\langle O_j, O_{j'}\right\rangle_{\rm Ph} &=& \int dU\Sigma_k \Delta_k\chi^*_j(U)\chi_{j'}(U)\chi_k(UH_{\Lambda}^{-1})\\
\nonumber &=& \Sigma_k\Delta_k \int dU \stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\alpha_1}_{\beta_1}(U)\stackrel{j'}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\beta_1}_{\alpha_1}(U)\stackrel{j'}\Pi\!\!\!\!\!\!\phantom{\Pi}^{\alpha_2}_{\beta_2}(U)\\
&\phantom{a}& \qquad \qquad \ \ \times \stackrel{j'}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\beta_2}_{\alpha_2}(U) \stackrel{k}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\alpha_3}_{\beta_3}(U)\stackrel{k}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\beta_3}_{\alpha_3}(H^{-1}_{\Lambda}), \nonumber \\
\notag &=& \Sigma_k\Delta_k \int dU \stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\alpha_1}_{\beta_1}(U)\stackrel{j'}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\beta_2}_{\alpha_2}(U)\stackrel{k}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\alpha_3}_{\beta_3}(U) \nonumber \\
&\phantom{a}& \qquad \qquad \ \ \times
\stackrel{k}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\beta_3}_{\alpha_3}(H^{-1}_{\Lambda})\frac{\delta_{jj'}}{\Delta_j}\delta_{\beta_1\beta_2}\delta_{\alpha_1\alpha_2}\,, \nonumber
\end{eqnarray}
where we have used
\begin{eqnarray}
\stackrel{j}{\Pi^*}\!\!\!\!\!\!\phantom{\Pi}^{\alpha}_{\beta}(U)\stackrel{j'}{\Pi}\!\!\!\!\!\!\phantom{\Pi}^{\gamma}_{\delta}(U)=\frac{\delta_{jj'}}{\Delta_j}\delta_{\alpha\delta}\delta_{\beta\gamma}=\delta_{jj'}=1,\label{eqn:produreps}
\end{eqnarray}
and
\begin{eqnarray}
\overset{j_p}{\chi}(U_{p}H_{\Lambda}^{-1})=\overset{j_p}\Pi\!\!\!\!\phantom{\Pi}^{\alpha}_{\beta}(U_p)\overset{j_p}{\Pi}\!\!\!\!\phantom{\Pi}^{\beta}_{\alpha}(H_{\Lambda}^{-1}).
\end{eqnarray}
The definition is represented diagrammatically in the right hand side of Figure~\ref{fig:Lambda_two_loops}.
Among the four pairs of trivalent intertwiners, only one of them will encapsulate the representations of the group element implementing the curvature constraint, namely
\begin{eqnarray}
&&\upsilon^{\alpha_1 \alpha_2 \alpha_3}\ \overset{j''}{\Pi}\!\!\!\!\phantom{\Pi}^{\beta_3}_{\alpha_3}(H^{-1}_{\Lambda}) \upsilon_{\alpha_1 \alpha_2 \beta_3} \\
&&=\overline{\upsilon}^{\alpha_1 \alpha_2 \alpha_3}\ \overset{\frac{c}{2}}{\Pi}\!\!\!\!\phantom{\Pi}^{\beta_3}_{\alpha_3}(H^{-1}_{\Lambda}) \ \overline{\upsilon}_{\alpha_1 \alpha_2 \beta_3} \frac{1}{ \theta(a,b,c)}
= \frac{\theta_{\Lambda}(a,b,c)}{ \theta(a,b,c)}\,. \nonumber
\end{eqnarray}
Making use of Lemma~7 in \cite{KL}, by inserting the group element corresponding to the cosmological constant, we obtain the expression in Figure~5, from which it is immediate to compute the value of the $\theta$-nets with one insertion of group element with cosmological constant, $H_\Lambda$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{Lemma_7.png}
\caption{\footnotesize{Diagrammatic reformulation of Lemma~7 in \cite{KL}, with quantum group representation here induced by the cosmological constant.}} \label{fig:Lemma7}
\end{figure}
Therefore it follows that
\begin{eqnarray} \label{C8}
\frac{\theta_{\Lambda}(a,b,c)}{ \theta(a,b,c)}
=\frac{\overset{k}{\chi}(H_{\Lambda}^{-1})}{\Delta_k}\,.
\end{eqnarray}
Notice that contractions inside and outside an integral follows respectively from
\begin{eqnarray}
\int dg\overset{j}{\Pi^{\alpha_1} _{\alpha_2}}(g^{-1})\overset{j'}{\Pi^{\beta_1} _{\beta_2}}(g)=\frac{\delta_{jj'}}{\Delta j}\delta^{\alpha_1\beta_1}\delta_{\alpha_2\beta_2}\,,
\end{eqnarray}
which diagrammatically recasts into
\begin{eqnarray}
\begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,0.3)--(4,0.3);
\draw (1,0)-- (4,0);
\filldraw [fill = black] (2.3,-0.3) rectangle (2.7,0.6);
\end{tikzpicture} = \frac{\delta_{jj'}}{\Delta_j} \ \ \begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (2,3) arc (90:270:0.3cm and 0.25cm);
\draw (1,2.5) -- (0.5,2.5);
\draw (1,3) -- (0.5,3);
\draw (2,3) -- (2.5,3);
\draw (2,2.5) -- (2.5,2.5);
\end{tikzpicture}
\,,
\end{eqnarray}
and follows from Eq.~(\ref{eqn:produreps}).
\\
Finally, taking into account Eqs. \eqref{C6} and \eqref{C8}, we are able to recover the evaluation of the inner product, as in
\begin{eqnarray}
\left\langle O_j, O_{j'}\right\rangle &=& \int dU\Sigma_k \Delta_k\chi^*_j(U)\chi_{j'}(U)\chi_k(UH_{\Lambda}^{-1})\nonumber \\
&=& \sum_k \Delta_k \frac{\overline{\upsilon} \cdot \overline{\upsilon}}{\theta(a,\,b,\,c)} \, \frac{{\chi_k}(H_{\Lambda})}{\Delta_k} \nonumber \\
&=& \sum_k {\chi_k}(H_{\Lambda}) \,.
\end{eqnarray}
We observe that this is formally the same as in the case without cosmological constant, where the classical dimension has been replaced by the quantum dimension, as calculated in Eq.~(\ref{eq:trace_loop}). Furthermore, we observe that this procedure, by expanding the Dirac delta function in representations of $SU(2)$, retains a spurious dependence on the $SU(2)$ group elements that might eventually render more difficult the interpretation of the results in term of the recoupling theory of $SU_q(2)$.\\
A more intuitive path to recognize the emergence of the recoupling theory of $SU_q(2)$ amounts to directly integrating out the $SU(2)$ elements. This corresponds, from a physical perspective, to imposing the constraints at the quantum level on the loop elements.
\begin{center}
\begin{figure}[h!]
\begin{tikzpicture}[scale=1.3]
\draw (0,0) circle (20pt);
\draw (0,0) circle (25pt);
\draw[fill=black] (-0.87,0) circle (2pt);
\draw[fill=black] (-0.70,0) circle (2pt);
\end{tikzpicture}
\caption{Two loops with only group element $H_\Lambda$ inserted, as resulting from applying the Dirac delta-function imposing the curvature constraint.}
\label{fig:loops_with_lambda}
\end{figure}
\end{center}
As in the standard case, we may opt for imposing the curvature constraint without expanding the Dirac delta function. In this case:
\begin{eqnarray}
\left\langle O_j, O_{j'}\right\rangle &=& \int dU \chi^*_j(U)\chi_{j'}(U)\delta(UH_{\Lambda})\nonumber \\
&=& \sum_k \int dU \chi_k(U)
\delta(U H_{\Lambda}^{-1})
\nonumber \\
&=& \sum_k \chi_k(H_{\Lambda}) \,,
\end{eqnarray}
see Figure~\ref{fig:loops_with_lambda}.
\section{Diffeomorphism invariance}
\noindent
We can now check how the projector extended so to include the cosmological constant, namely Eq.~\eqref{prolam}, naturally incorporates diffeomorphism invariance, as for the physical projector introduced in \cite{NoPe}.
\subsection{Case without vertices involved}
\noindent
We shall first inspect the case in which spin-network states, or their sub-states, enclose no intertwiners. In this case, considering two holonomies with different shapes, and using the spurious notation that arises from expanding the Dirac delta-functions imposing the curvature constraint, we can easily convince ourselves that
\begin{eqnarray}
&\sum_k \Delta_k&
\begin{tikzpicture}[scale=0.50,baseline={([yshift=0cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,3) -- (7,3);
\draw (0,2.5) -- (2,2.5);
\draw (2,2.5) arc (180:360:1.5cm and 1.5cm);
\draw (5,2.5) -- (7,2.5);
\draw (2.5,2.5) arc (180:360:1cm and 1.2cm);
\draw (2.5,2.5) -- (4.5,2.5);
\filldraw [fill = black] (1,2.2) rectangle (1.3,3.2);
\filldraw [fill = black] (3.4,0.7) rectangle (3.7,1.5);
\filldraw [fill = black] (3.4,2.2) rectangle (3.7,3.2);
\filldraw [fill = black] (5.8,2.2) rectangle (6.1,3.2);
\filldraw[fill=black] (4.5,2.5) circle (3pt) ;
\node (a) at (0.5,3.6) {$j$};
\node (a) at (6.5,3.6) {$j$};
\node (a) at (3,2) {$k$};
\end{tikzpicture}\\
&=& \sum_k \frac{\Delta_k}{\Delta_j^3}\ \begin{tikzpicture}[scale=0.50,baseline={([yshift=0cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (6,3) arc (90:270:0.3cm and 0.25cm);
\draw (1,2.5) -- (0.5,2.5);
\draw (1,3) -- (0.5,3);
\draw (6,3) -- (6.5,3);
\draw (6,2.5) -- (6.5,2.5);
\draw (2,2.5) arc (180:360:1.5cm and 1.5cm);
\draw (2.5,2.5) arc (180:360:1cm and 1.2cm);
\draw (2,3) arc (90:270:0.3cm and 0.25cm);
\draw (5,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (2,3) -- (3,3);
\draw (2.5,2.5) --(3,2.5);
\draw (3,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (4.5,2.5) -- (4,2.5);
\draw (5,3) -- (4,3);
\draw (4,3) arc (90:270:0.3cm and 0.25cm);
\filldraw[fill=black] (4.5,2.5) circle (3pt);
\filldraw [fill = black] (3.4,0.7) rectangle (3.7,1.5);
\node (a) at (0.5,3.6) {$j$};
\node (a) at (6.5,3.6) {$j$};
\node (a) at (3,2) {$k$};
\end{tikzpicture} \\
&=& \sum_k \delta_{jk}\frac{\Delta_k}{\Delta^4_j}\ \begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (6,3) arc (90:270:0.3cm and 0.25cm);
\draw (1,2.5) -- (0.5,2.5);
\draw (1,3) -- (0.5,3);
\draw (6,3) -- (6.5,3);
\draw (6,2.5) -- (6.5,2.5);
\draw (3,2.7) circle (10pt);
\draw (4.5,2.7) circle (10pt);
\filldraw[fill=black] (4.2,2.5) circle (3pt);
\node (a) at (0.5,3.6) {$j$};
\node (a) at (6.5,3.6) {$j$};
\node (a) at (4.5,2) {$k$};
\node (a) at (3.5,2) {$j$};
\end{tikzpicture}\\
&=& \frac{\Delta^{\Lambda}_j}{\Delta^2_j}\
\begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,2.5) arc (270:450:0.3cm and 0.25cm);
\draw (2,3) arc (90:270:0.3cm and 0.25cm);
\draw (1,2.5) -- (0.5,2.5);
\draw (1,3) -- (0.5,3);
\draw (2,3) -- (2.5,3);
\draw (2,2.5) -- (2.5,2.5);
\end{tikzpicture}\\
&=& \frac{\Delta^{\Lambda}_j}{\Delta_j}\ \ \begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,0.3)--(4,0.3);
\draw (1,0)-- (4,0);
\filldraw [fill = black] (2.3,-0.3) rectangle (2.7,0.6);
\end{tikzpicture}\\
&=& \begin{tikzpicture}[scale=0.50,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (1,0.3)--(6,0.3);
\draw (1,0)-- (6,0);
\filldraw[fill = black] (2.3,-0.3) rectangle (2.7,0.6);
\filldraw[fill=black] (3.5,0) circle (2pt);
\filldraw[fill = black] (4.3,-0.3) rectangle (4.7,0.6);
\end{tikzpicture}
\,.
\end{eqnarray}
\subsection{Case with vertices involved}
\noindent
We now consider the case of diffeomorphism invariance when spin-network states contain vertices. First, we observe that due to the translation invariance of the Haar measure, it follows that the following (diagrammatic) equations hold
\begin{eqnarray}\label{eqn:box_slide}
\begin{tikzpicture}[scale=0.3,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,1.5) -- (5,1.5);
\draw (0,0) -- (5,0);
\draw[fill=black] (2.2,-0.5) rectangle (2.8,2);
\filldraw[fill=black] (1,0) circle (5pt);
\end{tikzpicture}\ \ &=&
\begin{tikzpicture}[scale=0.3,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,1.5) -- (5,1.5);
\draw (0,0) -- (5,0);
\draw[fill=black] (2.2,-0.5) rectangle (2.8,2);
\filldraw[fill=black] (4,1.5) circle (5pt);
\end{tikzpicture}
\end{eqnarray}
\begin{eqnarray}\label{eqn:vertex_slide}
\begin{tikzpicture}[scale=0.3,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (-3,1.5) -- (0,0) -- (3,1.5);
\draw (0,0) -- (0,-3);
\filldraw[fill=black] (0,-2) circle (5pt);
\end{tikzpicture}\ \ &=&
\begin{tikzpicture}[scale=0.3,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (-3,1.5) -- (0,0) -- (3,1.5);
\draw (0,0) -- (0,-3);
\filldraw[fill=black] (-1,0.5) circle (5pt);
\filldraw[fill=black] (1,0.5) circle (5pt);
\end{tikzpicture}
\end{eqnarray}
where the group element on the left hand side of Eq.~\eqref{eqn:vertex_slide} gives two copies of its inverse on the right hand side. Consider a transition of type
\begin{center}
\begin{tikzpicture}[scale=0.4]
\draw (0,0) -- (7,0);
\draw (0,0.5) -- (4,0.5);
\draw (4,0.5) -- (4.2,0.3);
\draw (4.5,-0.2) -- (8,-4);
\draw (4,0.5) -- (8,4.5);
\draw (7,-2) -- (7,2.5);
\draw (7,2.5) -- (8.5,4);
\draw (7,-2) -- (8.5,-3.5);
\draw[dashed] (-1.5,0.25) -- (9,0.25);
\draw[fill=black] (1,-0.3) rectangle (1.5,0.8);
\draw[fill=black,rotate=45] (7.2,-3.4) rectangle (7.7,-2.2);
\draw[fill=black,rotate=-45] (7,3.8) rectangle (7.5,2.6);
\node (a) at (2.5,1.2) {$j$};
\node (a) at (2.5,-0.7) {$j$};
\node (a) at (5,2.5) {$k$};
\node (a) at (5,-2) {$k$};
\node (a) at (7.5,1.2) {$m$};
\node (a) at (7.5,-0.7) {$n$};
\end{tikzpicture}
\end{center}
Now, by inserting the projector with group element, we compute
\begin{eqnarray}
\lefteqn{\sum_{p,q} \Delta_p\Delta_q\
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
\draw (0,0) -- (1,0);
\draw[fill=black] (1,-0.2) rectangle (1.2,0.7);
\draw (1,0.5) -- (3,0.5);
\draw (1,0) -- (7.5,0);
\draw (3,0.5) -- (3.8,0.5);
\draw (4.2,-0.1) -- (8.3,-4);
\draw (3.8,0.5) -- (4.05,0.22);
\draw (5,-0.3) -- (7,-0.3) -- (7,-2.2) -- (5,-0.3);
\draw (4.8, 0.5) -- (7,0.5) -- (7,2.3) -- (4.8,0.5);
\draw (3.8,0.5) -- (8.8,4.5);
\draw (7.5,-2.6) -- (7.5,2.75);
\draw (7.5,2.75) -- (9.2,4.1);
\draw (7.5,-2.6) -- (8.5, -3.55);
\draw (8.5,-3.55) -- (9,-4);
\draw (8.3,-4) -- (8.7,-4.4);
\draw[fill=black] (6,-0.5) rectangle (6.3,0.7);
\draw[fill=black] (6.8,1.2) rectangle (7.7,1.5);
\draw[fill=black] (6.8,-1.2) rectangle (7.7,-1.5);
\draw[rotate=45,fill=black] (5,-2.45) rectangle (5.35,-3.35);
\draw[rotate=135,fill=black] (-5.2,-2.7) rectangle (-5.5,-3.6);
\draw[rotate=45,fill=black] (8,-2.8) rectangle (8.4,-3.65);
\draw[rotate=135,fill=black] (-8,-2.8) rectangle (-8.4,-3.65);
\node (a) at (0.5,1) {$j$};
\node (a) at (0.5,-0.5) {$j$};
\node (a) at (6,2.7) {$k$};
\node (a) at (6.5,-2.9) {$\ell$};
\node (a) at (8,2) {$m$};
\node (a) at (8,-2) {$n$};
\node (a) at (6.5,1.5) {$p$};
\node (a) at (6.5,-1) {$q$};
\draw[fill=black] (7,2.3) circle (2.5pt);
\draw[fill=black] (7,-0.3) circle (2.5pt);
\end{tikzpicture}}\label{eqn:initial_vertex} \\
&=&
\sum_{p,q} \Delta_p\Delta_q\ \begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
\draw (0,0) -- (1,0);
\draw[fill=black] (1,-0.2) rectangle (1.2,0.7);
\draw (1,0.5) -- (3,0.5);
\draw (1,0) -- (7.5,0);
\draw (3,0.5) -- (3.8,0.5);
\draw (4.2,-0.1) -- (8.3,-4);
\draw (3.8,0.5) -- (4.05,0.22);
\draw (5,-0.3) -- (7,-0.3) -- (7,-2.2) -- (5,-0.3);
\draw (4.8, 0.5) -- (7,0.5) -- (7,2.3) -- (4.8,0.5);
\draw (3.8,0.5) -- (8.8,4.5);
\draw (7.5,-2.6) -- (7.5,2.75);
\draw (7.5,2.75) -- (9.2,4.1);
\draw (7.5,-2.6) -- (8.5, -3.55);
\draw (8.5,-3.55) -- (9,-4);
\draw (8.3,-4) -- (8.7,-4.4);
\draw[rotate=45,fill=black] (5,-2.45) rectangle (5.35,-3.35);
\draw[rotate=135,fill=black] (-5.2,-2.7) rectangle (-5.5,-3.6);
\draw[rotate=45,fill=black] (8,-2.8) rectangle (8.4,-3.65);
\draw[rotate=135,fill=black] (-8,-2.8) rectangle (-8.4,-3.65);
\node (a) at (0.5,1) {$j$};
\node (a) at (0.5,-0.5) {$j$};
\node (a) at (6,2.7) {$k$};
\node (a) at (6.5,-2.9) {$\ell$};
\node (a) at (8,2) {$m$};
\node (a) at (8,-2) {$n$};
\node (a) at (6.5,1.5) {$p$};
\node (a) at (6.5,-1) {$q$};
\draw[fill=black] (7,2.3) circle (2.5pt);
\draw[fill=black] (7,-0.3) circle (2.5pt);
\end{tikzpicture}\label{eqn:vertex_gauge_fixing} \\
&=&
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
\draw (0,0) -- (1,0);
\draw[fill=black] (1,-0.2) rectangle (1.2,0.7);
\draw (1,0.5) -- (3,0.5);
\draw (1,0) -- (7.5,0);
\draw (3,0.5) -- (3.8,0.5);
\draw (4.2,-0.1) -- (5.5,-1.5);
\draw (6,-2) -- (8.3,-4);
\draw (3.8,0.5) -- (4.05,0.22);
\draw (5,-0.3) -- (7,-0.3)--(7,-2.2);
\draw (5,-0.3)-- (5.8,-1.2);
\draw (7,-2.2) -- (6.5,-1.8);
\draw (6.5,-1.8) arc (45:210:0.3cm and 0.2cm);
\draw (5.5,-1.5) arc (225:415:0.24cm and 0.2cm);
\draw (4.8, 0.5) -- (7,0.5) -- (7,2.3);
\draw (4.8,0.5)-- (5.7,1.2);
\draw (7,2.3) -- (6.3,1.75);
\draw (3.8,0.5) -- (5.3,1.5);
\draw (6,2.1)--(9,4.5);
%
\draw (6,2.1) arc (135:297:0.28cm and 0.21cm);
\draw (5.7,1.2) arc (-45:150:0.25cm and 0.25cm);
\draw (7.5,-2.6) -- (7.5,2.75);
\draw (7.5,2.75) -- (9.2,4.1);
\draw (7.5,-2.6) -- (8.5, -3.55);
\draw (8.5,-3.55) -- (9,-4);
\draw (8.3,-4) -- (8.7,-4.4);
\draw[rotate=45,fill=black] (8,-2.8) rectangle (8.4,-3.65);
\draw[rotate=135,fill=black] (-8,-2.8) rectangle (-8.4,-3.65);
\node (a) at (0.5,1) {$j$};
\node (a) at (0.5,-0.5) {$j$};
\node (a) at (6,2.7) {$k$};
\node (a) at (6.5,-2.9) {$\ell$};
\node (a) at (8,2) {$m$};
\node (a) at (8,-2) {$n$};
\node (a) at (6.5,1.5) {$p$};
\node (a) at (6.5,-1) {$q$};
\draw[fill=black] (7,2.3) circle (2.5pt);
\draw[fill=black] (7,-0.3) circle (2.5pt);
\end{tikzpicture}
\end{eqnarray}
\begin{eqnarray}
&=&
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
\draw (0,0) -- (1,0);
\draw[fill=black] (1,-0.2) rectangle (1.2,0.7);
\draw (1,0.5) -- (3,0.5);
\draw (1,0) -- (7.5,0);
\draw (3,0.5) -- (3.8,0.5);
\draw (4.2,-0.1) -- (8.3,-4);
\draw (3.8,0.5) -- (8.8,4.5);
\draw (3.8,0.5) -- (4.05,0.22);
\draw (7.5,-2.6) -- (7.5,2.75);
\draw (7.5,2.75) -- (9.2,4.1);
\draw (7.5,-2.6) -- (8.5, -3.55);
\draw (8.5,-3.55) -- (9,-4);
\draw (8.3,-4) -- (8.7,-4.4);
\draw[rotate=45,fill=black] (8,-2.8) rectangle (8.4,-3.65);
\draw[rotate=135,fill=black] (-8,-2.8) rectangle (-8.4,-3.65);
\node (a) at (0.5,1) {$j$};
\node (a) at (0.5,-0.5) {$j$};
\node (a) at (6,2.9) {$k$};
\node (a) at (6.5,-2.9) {$\ell$};
\node (a) at (8,2) {$m$};
\node (a) at (8,-2) {$n$};
\draw[fill=black] (5,1.45) circle (2.5pt);
\draw[fill=black] (5,-0.85) circle (2.5pt);
\end{tikzpicture}\label{eqn:vertex_integration}
\\
&=&\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
\draw (0,0) -- (1,0);
\draw[fill=black] (1,-0.2) rectangle (1.2,0.7);
\draw (1,0.5) -- (3,0.5);
\draw (1,0) -- (7.5,0);
\draw (3,0.5) -- (3.8,0.5);
\draw (4.2,-0.1) -- (8.3,-4);
\draw (3.8,0.5) -- (8.8,4.5);
\draw (3.8,0.5) -- (4.05,0.22);
\draw (7.5,-2.6) -- (7.5,2.75);
\draw (7.5,2.75) -- (9.2,4.1);
\draw (7.5,-2.6) -- (8.5, -3.55);
\draw (8.5,-3.55) -- (9,-4);
\draw (8.3,-4) -- (8.7,-4.4);
\draw[rotate=45,fill=black] (8,-2.8) rectangle (8.4,-3.65);
\draw[rotate=135,fill=black] (-8,-2.8) rectangle (-8.4,-3.65);
\node (a) at (0.5,1) {$j$};
\node (a) at (0.5,-0.5) {$j$};
\node (a) at (6,2.9) {$k$};
\node (a) at (6.5,-2.9) {$\ell$};
\node (a) at (8,2) {$m$};
\node (a) at (8,-2) {$n$};
\draw[fill=black] (2.5,0.5) circle (2.5pt);
\end{tikzpicture}
\end{eqnarray}
\begin{eqnarray}
&=&
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-0.1cm]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw (0,0.5) -- (1,0.5);
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\draw (3.8,0.5) -- (8.8,4.5);
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\draw (7.5,-2.6) -- (7.5,2.75);
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%
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\end{tikzpicture}
\end{eqnarray}
where we have used the gauge fixing identity (see Appendix of \cite{NoPe}) in the first equality, and then we have integrated to obtain the second equality. We observe that, upon using Eq.~\eqref{eqn:box_slide}, {\it i.e.} sliding the group elements appropriately, we can indeed obtain a configuration where we can draw loops intersecting the spin-network only through Haar integration boxes, where the group elements do not appear inside the loops; this ensures that we can apply the gauge fixing identity even though the group elements appear in the spin-network. Finally, the double integration on the left part of the last diagrammatic equality provides a factor $\Delta_j^\Lambda/\Delta_j$ that multiplies the result without cosmological constant derived in \cite{NoPe}. \\
\section{Relation to the Turaev-Viro model}
\noindent
We now consider the relation between the present theory and the Turaev-Viro model. In particular, we show that the quantum group recoupling theory of Kauffman and Lins \cite{KL} arises by introducing the cosmological constant in the original computation of Noui and Perez in \cite{NoPe}. Explicitly, we find that inserting the group element that arises from the cosmological constant, the tetra-net corresponding to certain transition amplitudes becomes the quantum $6j$ symbol.
If we proceed by imposing the curvature constraint through the Dirac delta function $\delta(UH_\Lambda^{-1})$, the result is immediate:
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{Hom_tetra.png}
\caption{\footnotesize{Tetra-net where non-vanishing curvature has been imposed homogeneously around the circles.}} \label{fig:Hom_tetra}
\end{figure}
where the intertwiners are now compatible with the recoupling theory of $SU_q(2)$, by assumption that the symmetrizer of irreps is a projector.
In order to evaluate this tetra-net, we employ the chromatic evaluation of \cite{KL}, Theorem~4. Therefore, the value of the tetrahedron coincides with the quantum $6j$ symbol, as expected. \\
\section{Conclusions}
\noindent
We have analyzed the Riemannian Einstein-Hilbert theory of gravity in $2+1D$, entailing $SU(2)$ internal symmetry, and shown that, when an additional cosmological constant term is considered, imposing constraints at the quantum level induces an effective recoupling theory that is the one proper of the $SU_q(2)$ quantum group. This amounts, at the quantum level, to replace standard expressions for the amplitudes encoding elements of the recoupling theory of $SU(2)$ with elements of the recoupling theory of $SU_q(2)$. This has brought to verify the dynamical emergence of the Turaev-Viro model, as expected by comparison with the different perspective of quantization provided in \cite{WittenJonesPolinomials}. \\
Implementing the physical projector with cosmological constant, we have provided explicit computations of the physical inner product of two loop states, showing in details the emergence of the deformed $SU_q(2)$ recoupling theory. We have further discussed how the physical projector implements the diffeomorphism symmetry, and finally described the emergence of the Turaev-Viro model in the theory. \\
Instead of quantizing the reduced phase-space of the theory, we have shown here that the action of the curvature constraint at the quantum level induces the emergence of the effective recoupling theory, both at the level of the representation of the fundamental loop and at the level of the higher spin loops representations. Switching from the loop to the spin-network basis, adopting the very same symmetrization of representations, finally entails the effective equivalent expression for the intertwiners of the theory. This latter observation sheds light on the possible way to deform the internal $SL(2, \mathbb{C})$ symmetry of Lorentzian theories of gravity in $3+1D$, providing a constructive argument based on a physical insight.
\acknowledgements
\noindent
The authors acknowledge interesting discussions with Alejandro Perez and Carlo Rovelli that have brought to develop this analysis. NG was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China Programme grant No. 19KJB140018 and Xi'an Jiaotong-Liverpool University through grant No. REF-18-02-03. A.M.~wishes to acknowledge support by the Natural Science Foundation of China, through the grant No. 11875113, the Shanghai Municipality, through the grant No.~KBH1512299, and by Fudan University, through the grant No.~JJH1512105.
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Gobitrichinotus är ett släkte av fiskar. Gobitrichinotus ingår i familjen Kraemeriidae.
Kladogram enligt Catalogue of Life:
Källor
Externa länkar
Abborrartade fiskar
Gobitrichinotus
|
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{"url":"http:\/\/www.talkstats.com\/threads\/in-statistical-learning-is-the-learning-function-a-random-variable-or-a-constant.72834\/","text":"# In statistical learning, is the learning function a random variable or a constant?\n\n#### arkm25\n\n##### New Member\nHi\n\nConsider a predictor x and a response Y, where the true relationship between them is given by Y = f(x) + e. e is a random error term.\n\nA training data set (x_1, Y_1), ..., (x_n, Y_n) is collected and from this an estimated learning function f_hat is fitted. Then Y_hat = f_hat(x) becomes an estimate for the true response Y.\n\nMy question is about the derivation of the error of this estimate. This derivation shows that the total error can be divided into a reducible and irreducible component and can be summarized as ...\n\nE(Y-Y_hat)^2 = E(f(x) - f_hat(x))^2 + var(e).\n\nFor the reducible component, in some derivations I've seen, they simply write E(f(x) - f_hat(x))^2 = (f(x) - f_hat(x))^2.\n\nThis treats the the fitted\/estimated learning function f_hat(x) as constant and not a random variable. My question is why is this the case?\n\nThe estimated learning function f_hat is constructed from a training data set which is a random sample since each response Y_i is a random variable. Therefore if you collected a different sample, you should get a different estimate for f_hat.\n\nShouldn't f_hat be a random variable then?\n\nAppreciate any insight.\n\nLast edited:\n\n#### Buckeye\n\n##### Active Member\nWe take a random sample and gather data on it (height, weight, occupation, residence, etc.). These are the x's or inputs to the model. They are fixed because we know what they are. The parameters of the model (the beta coefficients) are also fixed, but unknown. We use statistics to study the parameters of a population \/ model (and characterize variability). There is random unexplained variation from sample to sample which is captured in the error term.\n\nLast edited:","date":"2022-05-27 13:27:48","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8665555715560913, \"perplexity\": 1061.4891771054345}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662647086.91\/warc\/CC-MAIN-20220527112418-20220527142418-00497.warc.gz\"}"}
| null | null |
namespace RISE
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namespace Implementation
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class Sampling2D : public virtual ISampling2D
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protected:
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Scalar dSpaceHeight;
unsigned int numSamples;
Sampling2D( ){numSamples=0;}
virtual ~Sampling2D( ){};
public:
void SetNumSamples( const unsigned int num_samples ){ numSamples = num_samples; }
unsigned int GetNumSamples(){ return numSamples; }
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\section{Introduction}\label{sec:introh2h}
We study the dynamics of the \emph{collinear hydrogen exchange reaction} $\text{H}_2+\text{H}\rightarrow \text{H}+\text{H}_2$, which is an invariant subsystem of the spatial hydrogen exchange reaction, using the potential provided by Porter and Karplus in \cite{PorterKarplus64}. In literature it is considered a paradigm system for understanding chemical reactions due to its simplicity and variety of exhibited dynamics. Because the system consists of three identical atoms confined to a line, it is the simplest imaginable system with $2$ degrees of freedom modeling a chemical reaction.
The hydrogen atoms themselves are the simplest atoms in the universe. Because each consist of one proton and one electron only, an accurate potential energy surface for this reaction can be obtained via the Born-Oppenheimer approximation. Intriguingly enough, this system exhibits behaviour that is still not well understood.
The phenomenon we examine here is the counterintuitive observation that the reaction rate decreases as energy increases beyond a critical value. After all, one would expect to break bonds more easily using more energy. So far a satisfactory explanation of this phenomenon is missing and only an upper bound and a lower bound to the rate have been found. The upper bound is obtained by means of \emph{transition state theory} (TST), due to \cite{Wigner37}. TST is a standard tool for studying reaction rates due to its simplicity and accuracy for low energies, but it does not capture the decline of the reaction rate. The improvement brought by \emph{variational transition state theory} (VTST) \cite{Horiuti38}, does not capture this behaviour either.
\emph{Unified statistical theory}, due to \cite{Miller76}, which is in a certain sense an extension of TST to more complicated system, does capture the culmination of the reaction rate, but does not yield higher accuracy. The lower bound on the other hand does come quite close. It is obtained using the so-called \emph{simple-minded unified statistical theory} \cite{PollakPechukas79UST}.
A review of reaction rate results including TST can be found in \cite{Keck67}. \cite{Pechukas81} and \cite{Truhlar84} review various extensions of TST.
Using lobe dynamics (introduced in \cite{Rom-Kedar90}) we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive bounds of the reaction rate, which are desirable for practical reasons.
In the remainder of this Section we introduce the system, give an overview of TST and explain the current state of affairs with regards to the collinear hydrogen exchange reaction. Section \ref{sec:orbits} focuses on relevant periodic orbits and definition of regions of phase space. In Section \ref{sec:define sos} we introduce new coordinates using which we define a surface of section. In Section \ref{sec:transport barriers} we explain how we study invariant manifolds on the surface of section. In Section \ref{sec:tangles reaction} we give a detailed insight into the structures formed by invariant manifolds and their role in the reaction. Section \ref{sec:intricate interval} is devoted to a novel way of breaking down heteroclinic tangles to provide a better understanding of the interplay of invariant manifolds of three TSs. In Section \ref{sec:bounds} we calculate various upper and lower bounds of the reaction rate.
\begin{figure}
\centering
\includegraphics{H3.pdf}
\caption{Collinear hydrogen atoms and distances.}
\label{fig:H3}
\end{figure}
\subsection{Porter-Karplus potential}\label{subsec:potential}
The collinear hydrogen exchange system consists of three hydrogen atoms confined to a line, as shown in Fig. \ref{fig:H3}, where $r_1$ and $r_2$ denote the distances in atomic units between neighbouring atoms. Forces between the atoms are given by the Porter and Karplus potential \cite{PorterKarplus64} is the standard potential for the hydrogen exchange reaction (collinear and spatial) used for example in \cite{MorokumaKarplus71,ChapmanHornsteinMiller75,PollakPechukas78,PollakPechukas79UST,PollakChildPechukas80,Davis87,Inarrea11}. The system is considered to react, if it passes from the region of reactants ($r_1>r_2$) to the region of products ($r_1<r_2$) and remains there.
We point out two key properties of the Porter-Karplus potential:
\begin{itemize}
\item the discrete reflection symmetry with respect to the line $r_1=r_2$,
\item saddle point at $r_1=r_2=R_s:=1.70083$.
\end{itemize}
The symmetry expresses the fact that we cannot distinguish between three identical hydrogen atoms, we can only measure distances between them. Hence, any statement referring to $r_1<r_2$ automatically also holds for $r_1>r_2$.
Potential saddle points represent the activation energy needed for a reaction to be possible. In the all of this work we give energies as values in atomic units above the minimum of the system. In this convention the energy of the saddle point is $0.01456$.
From a configuration space perspective, such a potential barrier is the sole structure separating reactants from products and the sole obstacle the system needs to overcome in order to react. This perspective implicitly assumes that the system does not recross the potential barrier back into reactants. Dynamical structures that cause recrossings are only visible from a phase space perspective.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{PotEnSurf3DCont.png}
\includegraphics[width=0.49\textwidth]{PotFix.png}
\caption{The Porter-Karplus potential energy surface with contours and its cross sections for fixed values of $r_2=1.70083$ (cyan), $2$ (blue), $2.5$ (red), $3$ (green), $4$ (black), $50$ (yellow).}
\label{fig:potential}
\end{figure}
Figure \ref{fig:potential} shows the potential energy surface near the potential saddle and cross sections of the potential at various values of $r_2$. Due to diminishing forces between the atom and the molecule over large distances the differences between the cross sections fade after $r_2=4$ and are indistinguishable in double precision beyond $r_2=40$.
\subsection{Definitions}\label{subsec:definitions}
The collinear hydrogen exchange reaction is described by the Hamiltonian
\begin{equation}\label{eq:Ham}
H(r_1,p_{r_1},r_2, p_{r_2})=\frac{p_{r_1}^2+p_{r_2}^2-p_{r_1}p_{r_2}}{m_H} + U(r_1,r_2),
\end{equation}
where $p_{r_1}$, $p_{r_2}$ are the momenta conjugate to interatomic distances $r_1$, $r_2$, $m_H$ is the mass of a hydrogen atom and $U$ is the Porter-Karplus potential described above.
The equations of motion associated to $H$ are as follows:
\begin{equation}
\begin{split}
\dot{r}_1 &= \frac{2p_{r_1}-p_{r_2}}{m_H},\\
\dot{p}_{r_1} &= -\frac{\partial U(r_1,r_2)}{\partial r_1},\\
\dot{r}_2 &= \frac{2p_{r_2}-p_{r_1}}{m_H},\\
\dot{p}_{r_2} &= -\frac{\partial U(r_1,r_2)}{\partial r_2}.
\end{split}
\label{eq:eqHam}
\end{equation}
The discrete symmetry of the potential translates into the invariance of $H$ and the equations of motion under the map $(r_1,p_{r_1},r_2, p_{r_2}) \mapsto (r_2,p_{r_2},r_1, p_{r_1})$.
The Hamiltonian flow generated by equations \eqref{eq:eqHam} preserves the energy of the system $E=H(r_1,p_{r_1},r_2, p_{r_2})$ and the phase space of this system is therefore foliated by energy surfaces $H=E$.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{traj_config.png}
\caption{Examples of reactive (black) and nonreactive (red, blue) trajectories in configuration space at energy $0.02400$.}
\label{fig:traj}
\end{figure}
\begin{definition}\label{def:trajectories}
A trajectory passing through the point $\bigl(r_1^0,p_{r_1}^0,r_2^0, p_{r_{2}}^0\bigr)$ is said to be a \emph{reactive trajectory} if the solution
$(r_1(t),p_{r_1}(t),r_2(t), p_{r_2}(t))$
of the system with the initial condition
$$(r_1(0),p_{r_1}(0),r_2(0), p_{r_2}(0))=\bigl(r_1^0,p_{r_1}^0,r_2^0, p_{r_{2}}^0\bigr),$$
satisfies
$r_1(t)<\infty$ and $r_2(t)\rightarrow\infty$ as $t\rightarrow\infty$ and $r_1(t)\rightarrow\infty$ and $r_2(t)<\infty$ as $t\rightarrow-\infty$ or vice versa.
A \emph{nonreactive trajectory} is one for which the solution satisfies $r_1(t)\rightarrow\infty$ and $r_2(t)<\infty$ as $t\rightarrow\pm\infty$ or $r_1(t)<\infty$ and $r_2(t)\rightarrow\infty$ as $t\rightarrow\pm\infty$.
\end{definition}
Examples of reactive and nonreactive trajectories are shown in Figure \ref{fig:traj}. Note that nonreactive trajectories may cross the potential barrier in the sense that they cross the line $r_1=r_2$.
From the above it follows that the reaction rate at a fixed energy $E$ can be calculated using a brute force Monte Carlo method as the proportion of initial conditions of reactive trajectories at infinity. Since the system decouples in a numerical sense around $r_2=40$, it is enough to sample a sufficiently remote surface in the reactants ($r_1>r_2$) that is transversal to the flow, for example
\begin{equation}
r_1+\frac{r_2}{2}=50,\quad p_{r_2}<0.
\label{eq:MCsurface}
\end{equation}
Since $r_1$, $r_2$ is not a centre of mass frame, $r_2=const$ is not transversal to the flow.
We remark that $(r_2, p_{r_2}-\frac{p_{r_1}}{2})$ are canonical coordinates on $r_1+\frac{r_2}{2}=50$ that yield a uniform random distribution of initial conditions.
\subsection{Transition state theory}\label{subsec:TST}
Since its formulation in \cite{Wigner37}, TST became the standard tool for estimating rates of various processes not only in chemical reactions \cite{Keck67}. It has found use in many fields of physics and chemistry, such as celestial mechanics \cite{Henrard82}, \cite{Jaffe02}, plasma confinement \cite{Meissetal85} and fluid mechanics \cite{Ottino89}.
Key element of TST is the \emph{transition state} (TS), a structure that is between reactants and products. There is no single generally accepted definition unfortunately, because in some publications concerning systems with $2$ degrees of freedom TS refers to an unstable periodic orbit while in others TS is a dividing surface (DS) associated with the unstable periodic orbit.
We adopt the following definition of a TS from \cite{MacKay2014}:
\begin{definition}[TS]
A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-$2$ submanifold of the energy surface that can be spanned by two surfaces (the TS is the surfaces' boundary) of unidirectional flux, whose union divides the energy surface into two components and has no local recrossings.
\end{definition}
For a system with $2$ degrees of freedom as considered in this work, a closed, invariant, oriented, codimension-$2$ submanifold of the energy surface is a periodic orbit and it can be shown that the periodic orbit must be unstable \cite{Pechukas76}, \cite{PollakPechukas78}, \cite{Sverdlik78}.
In general, the TS has to be a normally hyperbolic invariant manifolds (NHIM), an invariant manifolds with linearised transversal instabilities that dominate the linearised tangential instabilities (\cite{Fenichel71}, \cite{Hirsch77}).
\begin{theorem}[TST]
\label{th:TST}
In a system that admits a TS and all trajectories that pass from reactants to products the DS precisely once, the flux across a DS is precisely the reaction rate.
\end{theorem}
We remark that in general the flux through a DS associated with a TS is an upper bound to the reaction rate \cite{Wigner37}, \cite{Pechukas81}.
Since its early applications, developments in the field led to a shift in the understanding of the TS to be an object in phase space rather than configuration space \cite{Wigginsetal01}, \cite{Uzeretal02}, \cite{Waalkensetal04a}, \cite{Waalkens04}, \cite{Waalkensetal04b}, \cite{Waalkensetal05a}, \cite{Waalkensetal05c}, \cite{Waalkens08}.
All relevant periodic orbits in this system are self-retracing orbits whose configuration space projections oscillate between equipotential lines, so called brake orbits (\cite{Ruiz75}). As suggested by \cite{PollakPechukas78}, let $(r_1^{po},r_2^{po})$ be the configuration space projection of a brake orbit at energy $E$, then the associated DS is the set of all phase space points $(r_1^{po},p_{r_1},r_2^{po}, p_{r_2})$ that satisfy $H(r_1^{po},p_{r_1},r_2^{po}, p_{r_2})=E$. For constructions of a DS near a saddle type equilibrium point in systems with more than $2$ degrees of freedom see \cite{Wigginsetal01}, \cite{Uzeretal02}, \cite{Waalkens04}.
Hydrogen exchange results and evolution of understanding of TST follow.
\subsection{Known results}\label{subsec:known results}
In $1971$, Morokuma and Karplus \cite{MorokumaKarplus71} evaluated three representatives of different classes of reactions. They found the collinear hydrogen exchange reaction to be the best suited for a study of the accuracy of TST due to smoothness, symmetry and simplicity. They found that TST agreed with Monte Carlo calculations up to a certain energy, but became inaccurate rather quickly after that.
In $1973$ \cite{PechukasMcLafferty73} Pechukas and McLafferty stated that for TST to be exact, every trajectory passing through the DS does so only once. In other words, TST fails in the presence of trajectories that oscillate between reactants and products.
In $1975$ Chapman, Hornstein and Miller \cite{ChapmanHornsteinMiller75} present numerical results showing that transition state theory ``fails substantially'' for the hydrogen exchange reaction (collinear and spatial) above a certain threshold.
Pollak and Pechukas \cite{PollakPechukas78} proved in $1978$ that flux through a DS constructed using an unstable brake orbit gives the best approximation of the reaction rate. In the presence of multiple TSs the authors introduce \emph{Variational TST} (VTST) - using the DS with the lowest flux to approximate the reaction rate. These results detach TST from potential saddle points. The authors find for the collinear hydrogen exchange reaction that when TST breaks down, VTST can be significantly more accurate, even though both fail to capture the reaction rate decrease.
In $1979$ Pollak and Pechukas \cite{PechukasPollak79TST} proved that TST is exact provided there is only one periodic orbit. Simultaneously, they derived the best estimate of the reaction rate so far for the collinear hydrogen exchange reaction in \cite{PollakPechukas79UST} using what they called \emph{Simple-minded unified statistical theory} (SMUST).
\emph{Unified statistical theory} (UST), due to Miller \cite{Miller76}, attempts to take advantage of the difference of fluxes through all DSs and essentially treat regions of simple and complicated dynamics separately. The authors of \cite{PollakPechukas79UST} found that UST captures the drop in the reaction rate and elaborate on the deviation of UST from the actual rate. The derivation of a lower bound (subject to assumptions) of the rate using the difference between TST and VTST is presented in the appendix of \cite{PollakPechukas79UST}.
A rigorous lower bound is presented in \cite{PollakChildPechukas80}. It uses a DS constructed using a stable periodic orbit between two TS to estimate the error of TST. The accuracy of this lower bound for the hydrogen exchange reaction is remarkable.
In $1987$ M. Davis \cite{Davis87} studied the hydrogen exchange reaction in phase space and considered the role of invariant structures. For low energies he showed that TST can be exact even if several TSs are present, provided that their invariant manifolds do not intersect. At higher energies he made some numerical observations of heteroclinic tangles of invariant manifolds and nearby dynamics. At high energies Davis found that a particular heteroclinic tangle grows in size and by assuming that it contains exclusively nonreactive trajectories he found a very accurate lower bound. The idea of this lower bound is very similar to \cite{PollakChildPechukas80}, but Davis endures a computational cost to quantify trajectories instead of fluxes through DSs.
Davis also formulated an estimate of the reaction rate based on the observation that not many trajectories undergo a complicated evolution, as found by \cite{PollakPechukas79UST}. The estimate assumes that beyond a certain time dynamics in the heteroclinic tangle is randomised and $50\%$ of the remaining trajectories are reactive.
Davis' observations hint at the crucial role played by invariant manifolds, but the precise manner in which this happens is not understood. Our aim is to explain the role of invariant manifolds in the reaction mechanism and extending it to the energy interval that Davis did not study, the interval with three TSs. We provide new understanding of the interactions between invariant manifolds of two and three TSs and consequently explain the counterintuitive reaction rate decrease.
\section{Periodic orbits and geometry}\label{sec:orbits}
\subsection{Local geometry}\label{subsec:local geometry}
Before we introduce periodic orbits that are relevant to the reaction mechanism, we describe the local energy surface geometry near a potential saddle point. We show that the neighbourhood necessarily contains an unstable periodic orbit and we highlight the importance of invariant manifolds to the local dynamics. The description remains true near unstable periodic orbits that do not lie near saddle points.
Consider the Williamson normal form \cite{Williamson36}, \cite{Uzeretal02} of a system near a saddle point.
In the neighbourhood $V$ of a potential saddle point, the system is accurately described in some suitable canonical coordinates $(q_1,p_1,q_2,p_2)$ by
\begin{equation*}
H_2(q_1,p_1,q_2,p_2)=\frac{1}{2}\lambda (p_1^2-q_1^2)+\frac{1}{2}\omega (p_2^2+q_2^2),
\end{equation*}
where $\lambda,\omega>0$. For a fixed energy $H_2=h_2$, this is equivalent to
\begin{equation}
h_2+\frac{1}{2}\lambda q_1^2=\frac{1}{2}\lambda p_1^2+\frac{1}{2}\omega (p_2^2+q_2^2).
\label{eq:sphere}
\end{equation}
\begin{figure}
\centering
\includegraphics{bottleneck.png}
\caption{Illustration of local energy surface geometry in the neighbourhood of a saddle point. Sections for fixed values of $q_1$ define spheres (with $\pm p_1$ given implicitly by $H_2(q_1,p_1,q_2,p_2)=h_2$), shown are $q_1=1.5,-.25,-2$.}
\label{fig:bottleneck}
\end{figure}
For a each fixed $q_1$ such that $h_2+\frac{1}{2}\lambda q_1^2>0$ this defines a sphere, as shown in Figure \ref{fig:bottleneck}. Depending on $h_2$, the energy surface has the following characteristics:
\begin{itemize}
\item If $h_2<0$, the energy surface consists of two regions locally disconnected near $q_1=0$, reactants ($q_1>0$) and products ($q_1<0$).
\item Reactants and products are connected by the saddle point for $h_2=0$.
\item For $h_2>0$, the energy surface is foliated by spheres. The radius of the spheres increases with $|q_1|$. Locally the energy surface has a wide-narrow-wide geometry usually referred to as a \emph{bottleneck}.
\end{itemize}
We remark that $q_1$ can be referred to as a \emph{reaction coordinate}.
To understand transport through a bottleneck, fix an energy $h_2$ slightly above $0$ and consider the Hamiltonian equations for $H_2$:
\begin{align*}
\dot{q}_1 &= \lambda p_1,\qquad\qquad\dot{q}_2 = \omega p_2,\\
\dot{p}_1 &= \lambda q_1,\qquad\qquad\dot{p}_2 = -\omega q_2.
\end{align*}
The degrees of freedom are decoupled with hyperbolic dynamics in $(q_1,p_1)$ and elliptic in $(q_2,p_2)$. Moreover $q_1=p_1=0$ defines an unstable periodic orbit and $q_1=0$ defines a DS separating reactants from products. This DS, similarly to the one defined in Sec. \ref{subsec:TST}, is a sphere that is due to the instability of $q_1=p_1=0$ transversal to the flow and does not admit local recrossings. The sphere itself is divided by its equator $q_1=p_1=0$ into two hemispheres with unidirectional flux - trajectories passing from reactants to products cross the hemisphere $p_1>0$, while trajectories from products to reactants cross $p_1<0$. Therefore $q_1=p_1=0$ satisfies the definition of a TS. We remark that the DS can be perturbed and as long as its boundary remains fixed and transversality is not violated, the flux through the perturbed and unperturbed DS remains the same.
This description breaks down at high energies, when the periodic orbit may become stable, an event commonly referred to as loss of normal hyperbolicity. Then TST is inaccurate due to local recrossings of the DS. Loss of normal hyperbolicity occurs in the hydrogen exchange reaction, yet TST breaks down at lower energies due the presence of multiple transition states.
Having the same energy distribution between the degrees of freedom as the periodic orbit $q_1=p_1=0$, its invariant manifolds are given by $$p_1^2-q_1^2=0,$$ the stable being $q_1=-p_1$ and the unstable $q_1=p_1$. They consist of two branches each - one on the reactant side with $q_1>0$, one on the product side with $q_1<0$. These manifolds are cylinders with the periodic orbit as its base. They are codimension-$1$ in the energy surface and separate reactive and nonreactive trajectories - reactive ones inside the cylinders
$$\frac{1}{2}\lambda (p_1^2-q_1^2)>0,$$
and nonreactive outside
$$\frac{1}{2}\lambda (p_1^2-q_1^2)<0.$$
Only reactive trajectories reach the DS.
Note that in a configuration space projection, the separation between reactive and nonreactive trajectories is not as natural/obvious as in a phase space perspective. Therefore we study the structures made up of invariant manifolds that cause the reaction rate decrease in phase space.
We remark that bottlenecks are related to TSs rather than potential saddle points. Sec. \ref{sec:tangles reaction} contains examples of bottlenecks unrelated to potential saddle points and a saddle point without a bottleneck.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{orbits.png}
\caption{The projections of the periodic orbits of $F_0$ (black), $F_1$ (blue) and $F_2$ (green) onto configuration space at energies $0.02210$, $0.02300$, $0.02400$, $0.02500$ and $0.02600$ and the corresponding equipotential lines (grey).}
\label{fig:orbits}
\end{figure}
\subsection{Periodic orbits}\label{subsec:po}
For energies $E$ above $0.01456$, the energy of the saddle point, the system \eqref{eq:Ham} admits periodic orbits that come in one-parameter families parametrised by energy. Initially we focus on each family separately and subsequently we investigate the interplay that governs the complicated dynamics exhibited by this system. We adopt the notation of \cite{Inarrea11} for different families of periodic orbits $F_n$, where $n\in\mathbb{N}$, and briefly describe their evolution with increasing energy. We remark that many families come in pairs related by symmetry and for simplicity we restrict ourselves to the $r_1\geq r_2$ half plane. We will refer to orbits of the family $F_n$ on the other half plane by $\widehat{F}_n$.
By $F_0$ we denote the family of Lyapunov orbits associated with the potential saddle, which as explained in Sec. \ref{subsec:local geometry} must be unstable for energies slightly above the saddle. The orbits lie on the axis of symmetry of the system $r_1=r_2$, see Fig. \ref{fig:orbits}. Orbits of this family were used in TST calculations in many of the previous works.
A saddle-centre bifurcation at approximately $0.02204$ results in the creation of two families - the unstable $F_1$ and the initially stable $F_2$. The configuration space projections of these orbits are shown in Fig. \ref{fig:orbits}.
The unstable family $F_1$ is the furthest away from $F_0$ and does not undergo any further bifurcations. The $F_2$ family is initially stable, but undergoes a period doubling bifurcation at $0.02208$ creating the double period families $F_{21}$ and $F_{22}$. Unlike reported by \cite{Inarrea11}, we do not find these families disappear in an inverse period doubling bifurcation of $F_2$ at $0.02651$. Instead $F_{21}$ and $F_{22}$ persist with double period until $0.02654$, when they collide together with $F_2$ and $F_0$, see Fig. \ref{fig:bif2}. Consequently $F_0$ becomes stable. We would like to enhance the findings of \cite{Inarrea11} by remarking that $F_{21}$ and $F_{22}$ are briefly stable between switching from hyperbolic to inverse hyperbolic and vice versa, see Fig. \ref{fig:bif1}.
At $0.02661$, $F_0$ is involved in a bifurcation with a double period family $F_4$ that originates in a saddle-centre bifurcation at $0.02254$. $F_4$ is a family symmetric with respect to $r_1=r_2$. For dynamical purposes we point out that above $0.02661$ $F_0$ is inverse hyperbolic.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{ER.png}%
\includegraphics[width=0.48\textwidth]{ES.png}
\caption{Bifurcation diagrams showing the evolution of $F_0$ (black), $F_1$ (blue), $F_2$ (light green), $F_{21}$ (dark green), $F_{22}$ (red) and $F_4$ (orange) on the energy-residue ($E,R$) and the energy-action ($E,S$) plane. The residues of other families and the action of orbits of period higher than $1$ are omitted for the sake of clarity.
}\label{fig:bif1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{ER_min.png}%
\includegraphics[width=0.48\textwidth]{ER_zoom.png}
\caption{Details of the evolution of $F_0$ (black), $F_1$ (blue), $F_2$ (light green), $F_{21}$ (dark green), $F_{22}$ (red, identical with $F_{21}$) and $F_4$ (orange) on the energy-residue ($E,R$) plane.
}\label{fig:bif2}
\end{figure}
Fig. \ref{fig:bif1} and \ref{fig:bif2} show bifurcation diagrams of most of the families on the energy-residue and the energy-action plane. By residue $R$ we mean the Greene residue as introduced by J. M. Greene in \cite{Greene68}, where $R<0$ means that the periodic orbit is hyperbolic, $0<R<1$ means it is elliptic and $R>1$ means it is inverse hyperbolic.
The residue is derived from a matrix that describes the local dynamics near a periodic orbit - the monodromy matrix. Let $\Gamma$ be a periodic orbit with the parametrisation $\gamma(t)$ and period $T$, and $M(t)$ be the matrix satisfying the variational equation
\begin{equation}
\dot{M}(t)=JD^2 H(\gamma(t))M(t),
\label{eq:Monod}
\end{equation}
where $J=\begin{pmatrix} 0 & Id\\ -Id & 0\end{pmatrix}$,
with the initial condition $M(0)=Id.$
The monodromy matrix is defined by $M=M(T)$ and it describes how a sufficiently small initial deviation $\delta$ from $\gamma(0)$ changes after a full period $T$:
\begin{equation*}
\Phi_H^T(\gamma(0)+\delta)=\gamma(T)+M\delta+O(\delta^2),
\end{equation*}
where $\Phi_H^t$ is the Hamiltonian flow.
According to \cite{Eckhardt91}, if $\delta$ is an initial displacement along the periodic orbit $\delta\parallel J\nabla H$, then $\delta$ is preserved after a full period $T$, i.e. $M\delta=\delta$. Similarly an initial displacement perpendicular to the energy surface $\delta\parallel \nabla H$ is preserved. Consequently, two of the eigenvalues of $M$ are
\begin{equation}
\lambda_1=\lambda_2=1.
\label{eq:lambda1}
\end{equation}
As \eqref{eq:Monod} is Hamiltonian, the preservation of phase space volume following Liouville's theorem implies $\det M(t)=\det M(0)=1$ for all $t$. Therefore the two remaining eigenvalues must satisfy $\lambda_3\lambda_4=1$ and we can write them as $\lambda$ and $\frac{1}{\lambda}$. $\Gamma$ is hyperbolic if $\lambda>1$, it is elliptic if $|\lambda|=1$ and it is inverse hyperbolic if $\lambda<-1$.
\begin{definition}
The Greene residue of $\Gamma$ is defined as $R=\frac{1}{4}(4-Tr M),$
where $M$ is the monodromy matrix corresponding to the periodic orbit $\Gamma$.
\end{definition}
Using \eqref{eq:lambda1} we can write $R$ as
$$R=\frac{1}{4}\left(2-\lambda-\frac{1}{\lambda}\right).$$
By definition $R<0$ if $\Gamma$ is hyperbolic, $0<R<1$ if it is elliptic and $R>1$ if it is inverse hyperbolic.
Davis \cite{Davis87} mostly focused on the energy interval below $0.02214$ and above $0.02655$, the interval where TST is exact and the interval where two TSs exist, respectively.
In the light of normal form approximation described in Sec. \ref{subsec:local geometry}, we remark that the approximation breaks down completely when $F_0$ loses normal hyperbolicity at $0.02655$ at the latest. The loss of normal hyperbolicity is not the cause for the overestimation of the reaction rate by TST as it starts to deviate from the Monte Carlo rate well before $0.02300$.
\subsection{Phase space regions}\label{subsec:regions}
We would like to give up the binary partitioning of an energy surface into reactants and products in favour of defining an interaction region inbetween into which trajectories can only enter once.
As explained in Sec. \ref{subsec:local geometry}, TSs give rise to bottlenecks in phase space. Because $F_1$ gives rise to the bottleneck the furthest away from the potential barrier, we use it to delimit regions as follows. Denote DS$_1$ and DS$_{\widehat{1}}$ the DSs constructed using $F_1$ and $\widehat{F}_1$ according to Sec. \ref{subsec:TST}. The interaction region is the region of the energy surface between the two DSs and it contains all other periodic orbits. Reactants and products are the regions on the $r_1>r_2$-side and the $r_1<r_2$-side of the interaction region respectively, see Figure \ref{fig:reg_config}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{regions.png}
\caption{Regions in configuration space at energy $0.02400$. The interaction region (red) bounded by two orbit from the family $F_1$ (blue), the region of reactants (blue) and the region of products (green). The orbit $F_0$ (black) is also included.
}\label{fig:reg_config}
\end{figure}
The advantages of this partition of space are immediate.
\begin{itemize}
\item All TSs and bottlenecks are in the interaction region or on its boundary. The dynamics in reactants and products has no influence on reactivity and to fully understand the hydrogen exchange reaction, it is enough to restrict the study to the interaction region.
\item Trajectories that leave the interaction region never return. This is true in forward and backward time.
\item It is impossible for a trajectory to enter reactants and products in the same time direction, unlike in the binary partitioning, where trajectories may oscillate between reactants and products.
\end{itemize}
\section{Definition of a Poincar\'e surface of section}\label{sec:define sos}
Invariant manifolds are $2$ dimensional objects on the $3$ dimensional energy surface embedded in $4$ dimensional phase space. To facilitate the study of intersections of invariant manifolds, we define a $2$ dimensional surface of section on the energy surface that is transversal to the flow and intersects invariant manifolds in $1$ dimensional curves.
\subsection{Reaction coordinate and minimum energy path}\label{subsec:mep}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{MEP.png}
\caption{Comparison of the MEP (red), the coordinate line $q_1=0$ (black) and the coordinate line $\tilde{q}_1=0$ (cyan). Equipotential lines of the potential energy surface correspond to energies $0.01200$, $0.01456$, $0.02000$, $0.02800$ and $0.03500$
}\label{fig:minE}
\end{figure}
Here we define a reaction coordinate, using which we can monitor the progress along a reaction pathway. Frequently a reaction coordinate is closely related to a \emph{minimum energy path} (MEP) connecting the potential wells of reactants and products via the potential saddle. The coordinate as such is not a solution of the Hamiltonian system and, as remarked in \cite{Pechukas1976mep}, is of no dynamical significance to the system.
A MEP can be defined as the union of two paths of steepest descend, the unique solutions of the gradient system
\begin{equation*}
\dot{r}_1 = -\frac{\partial U}{\partial r_1},\qquad \dot{r}_2 = -\frac{\partial U}{\partial r_2},
\end{equation*}
one connecting the saddle $(R_s,R_s)$ to the potential well $(\infty,R_{min})$, the other connecting $(R_s,R_s)$ to $(R_{min},\infty)$. Fig. \ref{fig:minE} shows the MEP on a contour plot of $U$.
\subsection{Surface of section}\label{subsec:sos}
The MEP as defined above does not have an analytic expressing, but can be approximated using $q_1=0$, where
\begin{equation*}
q_1=(r_1-R_{min})(r_2-R_{min})-(R_s-R_{min})^2,
\end{equation*}
as used by \cite{Davis87} and shown in Figure \ref{fig:minE}. Invariant manifolds are always transversal to the MEP and transversal to $q_1=0$ for the energy interval considered in this work. At higher energies Davis used $q_1=-0.04$, $q_1=-0.07$ and $q_1=-0.084$ to avoid tangencies.
We found that
\begin{equation}
\tilde{q}_1=(r_1-R_{min})(r_2-R_{min})-(R_s-R_{min})^2e^{-2((r_1-R_s)^2+(r_2-R_s)^2)},
\end{equation}
approximates the MEP significantly better, but a coordinate system involving $\tilde{q}_1$ is rather challenging to work with.
Throughout this work we use the surface of section $\Sigma_0$ defined by $q_1=0$, $\dot{q}_1>0$. The condition $\dot{q}_1>0$ determines the sign of the momenta and guarantees that each point on $\Sigma_0$ corresponds to a unique trajectory. We remark that the boundary of $\Sigma_0$ does not consist of invariant manifolds and therefore it is not a surface of section in the sense of Birkhoff \cite[Chapter 5]{Birkhoff27}.
For the sake of utility, we define the other coordinate $q_2$ such that $(q_1,q_2)$ is an orthogonal coordinate system on $\mathbb{R}^2$ and the coordinate lines of $q_2$ are symmetric with respect to $r_1=r_2$. These conditions are satisfied by
\begin{equation}
q_2=\frac{1}{2}(r_1-R_{min})^2-\frac{1}{2}(r_2-R_{min})^2.
\end{equation}
Note that $q_2=0$ is equivalent to $r_1=r_2$ and $q_2$ is a reaction coordinate - it captures progress along $q_1=0$ and $q_2>0$ contains reactants, while $q_2<0$ contains products. We remark that $q_1$ can locally considered a \emph{bath coordinate} capturing oscillatory motion near the potential barrier. For a fixed energy, the energy surface is bounded in $q_1$ and unbounded in $q_2$.
\subsection{Symplectic coordinate transformation}\label{subsec:momenta}
Here we define a coordinate system in phase space, such that the coordinate transformation is symplectic.
This requires finding the conjugate momenta $p_1$, $p_2$ corresponding to $q_1$, $q_2$. For this purpose we use the following generating function (type 2 in \cite{Arnold76}):
\begin{multline*}
G(r_1,r_2,p_1,p_2)= \big((r_1-R_{min})(r_2-R_{min})-(R_s-R_{min})^2\big)p_1\\ +\frac{1}{2}\big((r_1-R_{min})^2-(r_2-R_{min})^2\big)p_2.
\end{multline*}
Then
\begin{equation*}
\frac{\partial G}{\partial r_i}=p_{r_i},\qquad \frac{\partial G}{\partial p_i}=q_i.
\end{equation*}
One finds that
\begin{equation*}
\begin{split}
p_{r_1}&=\frac{\partial G}{\partial r_1}=(r_2-R_{min})p_1+(r_1-R_{min})p_2,\\
p_{r_2}&=\frac{\partial G}{\partial r_2}=(r_1-R_{min})p_1-(r_2-R_{min})p_2.
\end{split}
\end{equation*}
From this we obtain
\begin{equation*}
\begin{split}
p_1&=\frac{(r_2-R_{min})p_{r_1}+(r_1-R_{min})p_{r_2}}{(r_1-R_{min})^2+(r_2-R_{min})^2},\\
p_2&=\frac{(r_1-R_{min})p_{r_1}-(r_2-R_{min})p_{r_2}}{(r_1-R_{min})^2+(r_2-R_{min})^2}.
\end{split}
\end{equation*}
This transformation has a singularity at $r_1=r_2=R_{min}$, but $U(R_{min},R_{min})=0.03845$ is inaccessible at energies we consider. By straightforward calculation one finds that the symplectic $2$-form $\omega_2$ is indeed preserved:
$$\omega_2=\mathrm{d} p_{r_1}\wedge \mathrm{d} r_1 + \mathrm{d} p_{r_2}\wedge \mathrm{d} r_2 = \mathrm{d} p_1\wedge \mathrm{d} q_1 + \mathrm{d} p_2 \wedge \mathrm{d} q_2.$$
We remark that $(q_2,p_2)$ as defined above are the canonical coordinates on $\Sigma_0$.
\section{Transport and barriers}\label{sec:transport barriers}
In this section we discuss the dynamics on the surface of section $q_1=0$ under the return map. This involves investigating structures formed by invariant manifolds via lobe dynamics due to \cite{Rom-Kedar90}.
\subsection{Structures on the surface of section}
The return map $P$ associated with $\Sigma_0$ is defined as follows. Every point $(q^0,p^0)$ on $\Sigma_0$ is mapped to
$$P(q^0,p^0)=(q_2(T),p_2(T)),$$
where $T>0$ is the smallest for which $q_1(T)=0$ along the solution $$(q_1(t),p_1(t),q_2(t),p_2(t)),$$ with the initial condition
$$(q_1(0),p_1(0),q_2(0),p_2(0))=(0,p_1,q^0,p^0),$$
where $p_1$ is given implicitly by the fixed energy $E$.
$P$ is symplectic because it preserves the canonical $2$-form restricted to $\Sigma_0$,
\begin{equation}
\omega_2\bigr\vert_{\Sigma_0}=\mathrm{d} p_2\wedge\mathrm{d} q_2,
\label{eq:omega2sigma}
\end{equation}
see \cite{Binney85}. Because the Hamiltonian flow is reversible, $P^{-1}$ is well defined.
Each periodic orbit intersects $\Sigma_0$ in a single point that is a fixed point of $P$. Its stability follows from the eigenvalues of the monodromy matrix, as explained in Sec. \ref{subsec:po}. Due to conservation laws, the eigenvalues can be written as $\lambda$, $\frac{1}{\lambda}$, $1$, $1$, see \cite{Eckhardt91}. For TSs, the eigenvectors corresponding to $\lambda$, $\frac{1}{\lambda}$ define stable and unstable invariant manifolds under the linearisation of $P$ near a fixed point.
\subsection{Barriers formed by invariant manifolds}\label{subsec:barrier}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{1900_traj.png}
\includegraphics[width=0.49\textwidth]{t_1900.png}
\caption{Disjoint invariant manifolds of $F_0$ forming a barrier on $\Sigma_0$ at $0.01900$ and examples of a nonreactive (black) and a reactive (blue) trajectory on $\Sigma_0$ and in configuration space.
}\label{fig:1900}
\end{figure}
In the following we discuss invariant manifolds of TSs and their impact on dynamics with increasing energy. Let $F_i$ be a TS, we denote $W_{F_i}$ its invariant manifolds as a whole, stable and unstable invariant manifolds are denoted $W^s_{F_i}$ and $W^u_{F_i}$ respectively. An additional $+/-$ subscript indicates the branch of the invariant manifold with larger/smaller $q_2$ coordinate in the neighbourhood of $F_i$, for example $W^s_{F_i+}$ and $W^s_{F_i-}$. Recall from Sec. \ref{subsec:local geometry} that invariant manifolds of unstable brake orbits are cylinders of codimension-$1$ on the energy surface and they intersect $\Sigma_0$ in curves that divide $\Sigma_0$ into two disjoint parts each.
As mentioned in Sec. \ref{subsec:po}, the system has a single periodic orbit $F_0$ between $0.01456$ and $0.02204$. Its invariant manifolds do not intersect and act as separatrices or \emph{barriers} between reactive and nonreactive trajectories, as shown at $0.01900$ in Fig. \ref{fig:1900}. Reactive trajectories are characterised by a large $|p_2|$ momentum and are located above and below $W_{F_0}$. Nonreactive ones have a smaller $|p_2|$ momentum and are located between $W^s_{F_0}$ and $W^u_{F_0}$. Consequently DS$_0$, the DS associated with $F_0$, has the no-return property and TST is exact (\cite{Davis87}).
$F_1$ and $F_2$ come into existence at $0.02204$, but the reaction mechanism is governed entirely by $W_{F_0}$. $W_{F_1}$ form a homoclinic tangle, but it only contains nonreactive trajectories. TST remains exact until $0.02215$, when a heteroclinic intersection of $W_{F_0}$ and $W_{F_1}$ first appears. In the following we introduce the notation for homoclinic and heteroclinic tangles and subsequently introduce lobe dynamics due to \cite{Rom-Kedar90} on the example of the homoclinic tangle formed by $W_{F_1}$, the $F_1$ tangle.
\subsection{Definitions and notations}\label{subsec:def manif}
Let $F_i$ and $F_j$ be fixed points and assume $W^s_{F_i}$ and $W^u_{F_j}$ intersect transversally, as is the case in this system. The \emph{heteroclinic point} $Q\in W^s_{F_i}\cap W^u_{F_j}$ converges to $F_i$ as $t\rightarrow\infty$ and to $F_j$ as $t\rightarrow-\infty$. The images and preimages of $Q$ under $P$ are also heteroclinic points and therefore $W^s_{F_i}$ and $W^u_{F_j}$ intersect infinitely many times creating a \emph{heteroclinic tangle}. If $i=j$, we speak of homoclinic points and homoclinic tangles.
Homoclinic and heteroclinic tangles are chaotic, since dynamics near its fixed points is locally conjugate to Smale's horseshoe dynamics (see \cite{Hirsch04}).
Denote the segment of $W^s_{F_i}$ between $F_i$ and $Q$ by $S[F_i,Q]$ and the segment of $W^u_{F_j}$ between $F_j$ and $Q$ by $U[F_j,Q]$.
\begin{definition}
If $S[F_i,Q]$ and $U[F_j,Q]$ only intersect at $Q$ (and $F_i$ if $i=j$), then $Q$ is a \emph{primary intersection point} (pip).
\end{definition}
It should be clear that every tangle necessarily has pips. If $Q$ is a pip, then $PQ_0$ is a pip too, because if $S[F_i,Q]\cap U[F_j,Q]=\{Q\}$, then $S[F_i,PQ]\cap U[F_j,PQ]=\{PQ\}$. Similarly $P^{-1}Q$ is a pip. We remark that by definition all pips lie on $S[F_i,Q]\cup U[F_j,Q]$.
\begin{definition}
Let $Q_0$ and $Q_1$ be pips such that $S[Q_1,Q_0]$ and $U[Q_0,Q_1]$ do not intersect in pips except for their end points. The set bounded by $S[Q_1,Q_0]$ and $U[Q_0,Q_1]$ is called a \emph{lobe}.
\end{definition}
Note that the end points of the segments are ordered, the first being closer to the fixed point along corresponding the manifold in terms of arclength on $\Sigma_0$. Clearly $P$ preserves this ordering. It follows that if $S[Q_1,Q_0]$ and $U[Q_0,Q_1]$ do not intersect in pips except for the endpoints, $S[PQ_1,PQ_0]$ and $U[PQ_0,PQ_1]$ cannot intersect in pips other than the end points. Therefore $P$ always maps lobes to lobes.
\subsection{A partial barrier}\label{subsec:partial barrier}
Without knowing about invariant manifolds, the influence of a tangle on transport between regions of a Hamiltonian system may seem unpredictable and random. The role of invariant manifolds is well known and the transport mechanism may be intricate, yet understandable.
We explain this mechanism on the example of the $F_1$ tangle. The analogue in heteroclinic tangles will be apparent. The choice of the $F_1$ tangle at $0.02206$ is due to the logical order in terms of increasing energy and its relative simplicity. Of the invariant manifolds, $W^s_{F_1+}$ and $W^u_{F_1+}$ form barriers similar to those discussed in Sec. \ref{subsec:barrier} at all energies, while $W^s_{F_1-}$ and $W^u_{F_1-}$ form a homoclinic tangle. All branches of $W_{F_1}$ lie in the region of nonreactive trajectories on the reactant side of $F_0$, see Figure \ref{fig:2206_all}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{2206_all.png}
\caption{Invariant manifolds of $F_0$, $F_1$ and $\widehat{F}_1$ at $0.02206$.}\label{fig:2206_all}
\end{figure}
Choose a pip $Q_0\in W^s_{F_1-}\cap W^u_{F_1-}$, we will comment on the negligible consequences of choice later. The segments $S[F_1,Q_0]$ and $U[F_1,Q_0]$ delimit a region that we denote in reference to $F_1$ by $R_1$. The complement to $R_1$ in the region bounded by $W^s_{F_0+}$ and $W^u_{F_0+}$ is denoted $R_0$, see Figure \ref{fig:2206}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{2206_def.png}
\includegraphics[width=0.49\textwidth]{2206_detail.png}
\caption{Definition of a region and highlighted lobes in the $F_1$ tangle at $0.02206$.}\label{fig:2206}
\end{figure}
There is only one pip between $Q_0$ and $PQ_0$, denote it $Q_1$. In general the number of pips between $Q_0$ and $PQ_0$ is always odd (see \cite{Rom-Kedar90}.
We define lobes using $Q_0$, $Q_1$ and all of their (pre-)images. The way lobes guide trajectories in and out of regions can be seen on the lobe bounded by $S[Q_1,Q_0]$ and $U[Q_0,Q_1]$. The lobe is located in $R_0$, but its preimage bounded by $S[P^{-1}Q_1,P^{-1}Q_0]$ and $U[P^{-1}Q_0,P^{-1}Q_1]$ lies in $R_1$. This area escapes from $R_1$ to $R_0$ after $0$ iterations of the map $P$, we denote the lobe by $L_{1,0}(0)$. Analogously, by $L_{0,1}(0)$ we denote the lobe that is captured in $R_1$ from $R_0$ after $0$ iterations and is bounded by $S[PQ_0,Q_1]$ and $U[Q_1,PQ_0]$. We refer to images and preimages of $L_{1,0}(0)$ and $L_{0,1}(0)$ as \emph{escape lobes} and \emph{capture lobes} respectively.
Note that due to the no-return property of the interaction region, escape and capture lobes cannot intersect beyond DS$_1$.
Denote the lobe that leaves $R_i$ for $R_j$, $i\neq j$, immediately after $n$ iterations of the map $P$ by
$$L_{i,j}(n).$$
In this notation we have for all $k,n\in\mathbb{Z}$ the relation
\begin{equation}\label{eq:map lobes}
P^kL_{i,j}(n)=L_{i,j}(n-k).
\end{equation}
Transition between $R_0$ and $R_1$ is closely connected to $Q_0$ and the transition from $L_{i,j}(1)$ to $L_{i,j}(0)$. All other lobes are confined by the barrier consisting of invariant manifolds to their respective regions. Near $Q_0$, however, the barrier has a gap through which trajectories can pass. MacKay, Meiss and Percival \cite{MacKay84} described this mechanism by saying that it ``acts like a revolving door or turnstile.'' The term \emph{turnstile} was born and lives on, see \cite{Meiss15}.
While $W^s_{F_1-}$ contracts exponentially near the $F_1$, $W^u_{F_1-}$ stretches out. It is easy to see that $S[F_1,Q_0]$ is a rigid barrier - nearly linear and guiding all trajectories in its vicinity. $W^u_{F_1-}$ is a more flexible barrier in forward time - the manifold itself twists and stretches, alternately lying in $R_0$ and $R_1$. The fluid shape of $W^u_{F_1-}$ is the result of complicated dynamics and the influence of $S[F_1,Q_0]$. Stable manifolds behave similarly in backward time and the transition from rigid to flexible results in the turnstile mechanism.
The same is true for heteroclinic tangles. These imperfect barriers are responsible for nonreactive trajectories with high translational energy and reactive trajectories with surprisingly low translational energy. Due to this strangely selective mechanism we speak of a \emph{partial barrier}.
Choosing any other pip than $Q_0$ for the definition of the regions merely affects the time in which lobes escape. Compared to definitions based on $Q_0$, if we chose $PQ_0$ instead, escape/capture of lobes would be delayed by $P$, if we chose $Q_1$, only escape lobes would be affected. This has implications for notation, not for dynamics or its understanding.
\subsection{Properties of lobes}
Here we state some of the basic properties of lobes that will be relevant in the following sections. The following statements assume that we study transport between two regions that are separated by a homoclinic tangle or a heteroclinic tangle and involves no other invariant manifolds. This provides useful insight into the complex dynamics of homoclinic and heteroclinic tangles.
If the intersection $L_{i,j}(0)\cap L_{j,i}(0)$ is non-empty, it does not leave the respective region and is not subject to transport. In this case we may redefine lobes to be $$\tilde{L}_{i,j}(k):=L_{i,j}(k)\setminus \left(L_{i,j}(k)\cap L_{j,i}(k)\right),$$
where $\tilde{L}_{i,j}(k)\cap\tilde{L}_{j,i}(k)=\emptyset$. This justifies the following assumption.
\begin{assumption}\label{assum:lobe}
We assume that the lobes $L_{i,j}(0)$ and $L_{j,i}(0)$ are disjoint.
\end{assumption}
Equivalently we could assume $L_{i,j}(1)\subset R_i$ and $L_{i,j}(0)\subset R_j$. In case of transport between several regions, we can only make statements based on the two regions that are separated by manifolds of the given tangle.
Each homoclinic and heteroclinic tangle involves a region bounded by segments of invariant manifolds, such as $R_1$ in Sec. \ref{subsec:partial barrier}. Since $P$ is symplectic, almost all trajectories that enter the bounded region must eventually leave it. This can be formulated as
\begin{lemma}\label{lemma:partition}
Let at least one of $R_i$ and $R_j$ be bounded. Then $L_{i,j}(0)$ can be partitioned, except for a set of measure zero $O$, as
$$L_{i,j}(0)\setminus O=\bigcup_{n\in\mathbb{Z}} L_{i,j}(0)\cap L_{j,i}(n).$$
\end{lemma}
\begin{remark}
The region $R_j$ has the no-return property iff escape lobes ($L_{j,i}$) are disjoint, or equivalently iff capture lobes are disjoint. Automatically then for all $n>0$
$$L_{i,j}(0)\cap L_{j,i}(-n) = \emptyset.$$
\end{remark}
Some of the intersections in Lemma \ref{lemma:partition} $L_{i,j}(0)\cap L_{j,i}(n)$ for $n>0$ are empty sets. We are going to show that finitely many are empty at most.
\begin{lemma}\label{lemma:lobes intersect}
For all $n_0>0$
$$L_{i,j}(0)\cap L_{j,i}(n_0)\neq \emptyset \Rightarrow L_{i,j}(0)\cap L_{j,i}(n_0+1)\neq \emptyset.$$
\end{lemma}
Using Fig. \ref{fig:2206} as an example,
$$L_{0,1}(-1)\cap L_{1,0}(2)\neq \emptyset \Rightarrow L_{0,1}(-1)\cap L_{1,0}(3)\neq \emptyset,$$
because $L_{0,1}(0)\cap L_{1,0}(3)\neq \emptyset$ and $W^s_{F_1-}$ can only reach $L_{0,1}(0)$ by passing through $L_{0,1}(-1)$.
\begin{proof}
Without loss of generality assume $R_j$ is bounded and fix $n_0>0$. If
$$L_{i,j}(0)\cap L_{j,i}(n_0)\neq \emptyset,$$
then its image under $P$
$$L_{i,j}(-1)\cap L_{j,i}(n_0-1)\neq \emptyset.$$
We are going to argue that the only way for $L_{i,j}(-1)$ to reach $L_{j,i}(n_0-1)$ is by intersecting $L_{j,i}(n_0)$.
Denote $Q_1$ and $Q_2$ the pips that define $L_{i,j}(0)$ and $P^{-n_0}Q_0$ and $P^{-n_0}Q_1$ the pips that define $L_{j,i}(n_0)$.
Let $\widetilde{Q}\in U[Q_1,Q_2]\cap S[P^{-n_0}Q_1,P^{-n_0}Q_0]$.
$L_{i,j}(-1)$ lies inside $R_j$ (possibly partially in $R_i$ via another escape lobe) and so does $U[PQ_1,PQ_2]$, the part of $\partial L_{i,j}(-1)$ that does not coincide with $\partial R_j$. Note that as all pips, $PQ_1,PQ_2\in\partial R_j$. The intersection point $\widetilde{Q}$ lies in the interior of the region bounded by $U[P^{-n_0}Q_1,\widetilde{Q}]$ and $S[P^{-n_0}Q_1,\widetilde{Q}]$, while $PQ_1$ is located outside. Because a invariant manifold cannot reintersect itself, $U[PQ_1,P\widetilde{Q}]$ has to cross $S[P^{-n_0}Q_1,\widetilde{Q}]$, which is part of $\partial L_{j,i}(-n_0)$. Therefore
$$L_{i,j}(-1)\cap L_{j,i}(n_0)\neq \emptyset,$$
and when mapped backward,
$$L_{i,j}(0)\cap L_{j,i}(n_0+1)\neq \emptyset.$$
\end{proof}
Note for $n_0<0$, time reversal yields using a similar argument
$$L_{i,j}(0)\cap L_{j,i}(n_0)\neq \emptyset \Rightarrow L_{i,j}(0)\cap L_{j,i}(n_0-1)\neq \emptyset.$$
Following Lemma \ref{lemma:partition} and Lemma \ref{lemma:lobes intersect}, for $k$ large enough $L_{i,j}(k)$ lies simultaneously in both regions forming a complicated structure. Since pips are mapped exclusively on $\partial R_j$, they aid identification of parts of lobes.
Due to (\ref{eq:map lobes}), for $n$ small we may study lobe intersections of the form
$$L_{0,1}(k)\cap L_{1,0}(k+n),$$
that tend to be heavily distorted by the flow simply by mapping them forward or backward to less distorted intersections. However this does not work for
$$L_{0,1}(-k)\cap L_{1,0}(k),$$
for large $k$. On the other hand, we can expect the area of this intersection to shrink considerably with $k$, so their quantitative impact is limited.
We remark that while almost the entire area of a capture lobe must escape at some point, this does not apply to entire regions. Regions may contain stable fixed points surrounded by KAM curves (sections of KAM tori) that never escape.
The picture of a heteroclinic tangle as a structure consisting of only two manifolds is oversimplified. In general heteroclinic tangles in a Hamiltonian system with $2$ degrees of freedom can be expected to involve four branches of invariant manifolds. It takes four segments and two pips to define a region and consequently there will always be two turnstiles. The oversimplification is justified for tangles where the two turnstiles are made up of mutually disjoint lobes. Tangles with two intersecting turnstiles admit transport between non-neighbouring regions and we approach them differently.
\subsection{Content of a lobe}
In this section we use show how lobes guide trajectories in their interior.
Denote by $\mu$ the measure on $\Sigma_0$, that is proportional to $\omega_2\bigr\vert_{\Sigma_0}$ \eqref{eq:omega2sigma}. Under area preservation we understand that for any set $A$ and for all $k\in\mathbb{Z}$ $$\mu\left(A\right)=\mu\left(P^kA\right).$$
As a direct consequence of area preservation of a region we have for all $k,n\in\mathbb{Z}$
$$\mu\left(L_{i,j}(n)\right)=\mu\left(L_{j,i}(k)\right).$$
\begin{assumption}\label{assum:nonzero}
Throughout this work we assume that $\mu(L_{i,j}(0))\neq0$.
\end{assumption}
Combining Assumptions \ref{assum:lobe} and \ref{assum:nonzero} implies that $L_{i,j}(0),L_{j,i}(1)\subset R_j$ and if
$$2\mu\left(L_{i,j}(0)\right)>\mu\left(R_j\right),$$
then necessarily $L_{i,j}(0)\cap L_{j,i}(1)\neq \emptyset$.
All other lobes may partially lie in both $R_i$ and $R_j$, depending on the intersections of escape and capture lobes.
\begin{definition}
Assume $R_j$ is bounded. The \emph{shortest residence time} in a tangle is a number $k_{srt}\in\mathbb{N}$, such that
$$L_{i,j}(0)\cap L_{j,i}(k) = \emptyset,$$
for $0<k<k_{srt}$ and
$$L_{i,j}(0)\cap L_{j,i}(k_{srt})\neq \emptyset.$$
\end{definition}
\begin{remark}\label{rem:intersections}
The first lobe to lie partially outside $R_j$ is $L_{i,j}(-k_{srt})$, because it intersects $L_{j,i}(0)\subset R_i$. The lobes $L_{i,j}(-k)$ and $L_{j,i}(k)$ are entirely contained in $R_j$ for $0\leq k<k_{srt}$.
\end{remark}
Note that in a homoclinic tangle, since $L_{i,j}(-k)$ for $0\leq k<k_{srt}$ must be mutually disjoint and all contained in $R_j$, necessarily
$$\mu\left(R_j\right)>k_{srt}\mu\left(L_{i,j}(0)\right).$$
Once $L_{i,j}(-k_{srt})$ where $k_{srt}>0$ lies partially in $R_i$ by Lemma \ref{lemma:lobes intersect}
$$L_{i,j}(-k_{srt})\cap L_{j,i}(n)\neq \emptyset,$$
for all $n>0$ and therefore $L_{i,j}(-k)$ intersects $L_{j,i}(0)\subset R_i$ for all $k>k_{srt}$. Due to reentries and Assumption \ref{assum:lobe}, the statement is not true for $L_{i,j}(k)$ with $k>0$, but an analogue holds in reverse time.
Reentries are possible in tangles where escape (and capture) lobes are not mutually disjoint, hence the following Lemma.
\begin{lemma}
Let $k_1<k_3$ be such that $L_{i,j}(k_1)\cap L_{i,j}(k_3) \neq\emptyset$ with $i=0,1$ and $j=1-i$. Then
\begin{equation*}
L_{i,j}(k_1)\cap L_{i,j}(k_3) = \bigcup\limits_{k_2=k_1+1}^{k_3-1} L_{i,j}(k_1)\cap L_{j,i}(k_2)\cap L_{i,j}(k_3).
\end{equation*}
\end{lemma}
\begin{proof}
Let $p\in L_{i,j}(k_1)\cap L_{i,j}(k_3)$, $P^{k_1}p\in R_j$ and $P^{k_3-1}p\in R_i$ follow from Assumption \ref{assum:lobe}. Necessarily there exists $k_2$, such that $k_1<k_2<k_3$ and $p\in L_{j,i}(k_2)$. Since $k_2$ may be different for every $p$, the union over $k_2$ follows.
\end{proof}
The argument can be easily generalised for tangles that govern transport between multiple regions. One only needs to observe that $p$ can return to $R_i$ from any region.
In the $F_1$ tangle at $0.02215$, reentries can be deduced from the intersection $L_{0,1}(1)\cap L_{1,0}(0)$ that lies completely in $R_0$. See Figure \ref{fig:tangle intersection} for comparison of a tangle at $0.02215$ with reentries and at $0.02210$ without. Note that both tangles have $k_{srt}=1$.
\begin{figure}
\centering
\includegraphics{2210_detail2.png}\\
\includegraphics{2215_detail2.png}
\caption{The $F_1$ tangle at $0.02210$ (above) and at $0.02215$ (below). Both homoclinic tangles have $k_{srt}=1$, that can be seen by $L_{0,1}(0)\cap L_{1,0}(1)\neq\emptyset$ shown in cyan. At $0.02215$ the tangle admits reentries.}\label{fig:tangle intersection}
\end{figure}
Instantaneous transport between regions is described by the turnstile mechanism. Transport on a larger time scale can be studied using a measureless and weightless entity (species, passive scalars or contaminants \cite{Sreenivasan91}, \cite{Sreenivasan97}) that is initially contained and uniformly distributed in a region, as done in \cite{Rom-Kedar90}. Its role is to retain information about the initial state without influencing dynamics indicate escapes and reentries via lobes.
The challenge of studying lobes over large timescales is to determine which regions a lobe lies in and correctly identifying the interior of a lobe. For this we propose a partitioning of heteroclinic tangles into regions of no return outside of which the evolution of lobes is of no interest.
\section{Influence of tangles on the reaction rate}\label{sec:tangles reaction}
In this section we discuss the evolution of homoclinic and heteroclinic tangles in the entire energy interval $0<E\leq0.03000$ and their influence on dynamics in the interaction region. The dynamics for higher energies is due to the lack of bifurcations analogous. The study of invariant manifolds employs lobe dynamics and a new partitioning based on dynamical properties. An in-depth review of invariant manifolds in a chemical system and structural changes in tangles caused by bifurcations has to our knowledge not been done before.
\subsection{Energy interval where TST is exact}
TST is exact in the presence of a single TS (due to \cite{PechukasPollak79TST}) and remains exact in case of multiple TSs provided their invariant manifolds do not intersect (due to \cite{Davis87}). Therefore results of TST and Monte Carlo agree on the interval from $0$ to $0.02215$. $W_{F_0}$ separate reactive and nonreactive trajectories, see Sec. \ref{subsec:barrier}, while the $F_1$ tangle captures nonreactive trajectories only.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2206_all.png}\\
\includegraphics[width=\textwidth]{2214.png}
\caption{Invariant manifolds at $0.02206$ and $0.02214$.}\label{fig:2206 and 2214}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2206_detail.png}\\
\includegraphics[width=\textwidth]{2214_detail.png}
\caption{Lobe structure of the $F_1$ tangle at $0.02206$ and $0.02214$.}\label{fig:homoclinic 2206 and 2214}
\end{figure}
Some properties of the $F_1$ tangle are carried over to higher energies, such as shape of lobes or $k_{srt}$. Fig. \ref{fig:2206 and 2214} shows $W_{F_0}$ and $W_{F_1}$ approaching prior to the intersection at $0.02215$ and the failure of TST.
Each change of structure seems to coincide with a bifurcation of a periodic orbit. The decrease $k_{srt}$ from $3$ to $1$ over the energy interval, shown in Fig. \ref{fig:homoclinic 2206 and 2214}, coincides with the period doubling of $F_2$ at $0.02208$ and the period doubling of $F_{21}$ before $0.02209$. From a quantitative perspective, the tangle and its lobes grow larger in area.
\subsection{Point where TST fails}
At $0.02215$, $W_{F_0}$ and $W_{F_1}$ interact through heteroclinic intersections. Instead of minor changes in the overall topology of the invariant manifolds, we come across something that is better described as a chain reaction.
Firstly, we observe that $W_{F_0+}$ and $W_{F_1-}$ intersect forming a heteroclinic tangle, see Fig. \ref{fig:2230 R0}. Consequently, TST starts to fail (see \cite{Davis87}) and the Monte Carlo reaction rate is lower than TST. $W^s_{F_0}$ and $W^u_{F_0}$ form a partial barrier and this enables the $F_1$ tangle to capture reactive trajectories. We also find heteroclinic intersections of $W_{F_1-}$ and $W_{\widehat{F}_1+}$ as shown in Fig. \ref{fig:2230 F1F1}, as well as $W_{F_1-}$ and $W_{F_0-}$. Recall that statements for $F_1$ also hold for $\widehat{F}_1$.
\begin{figure}
\centering
\includegraphics{2230_R0.png}
\caption{The regions $R_0$ and $R_1$ at $0.02230$.}
\label{fig:2230 R0}
\end{figure}
Choose two pips in the $F_0$-$F_1$ tangle, so that the region bounded by $W_{F_0+}$ and $W_{F_1-}$ denoted $R_0$ satisfies $R_1\subset R_0$ (Fig. \ref{fig:2230 R0}) and define $\widehat{R}_0$ using symmetry.
As $L_{0,1}(0)$ and $L_{1,0}(1)$ in the $F_1$ tangle contain heteroclinic points that converge towards $F_0$ (forward or backward time), they necessarily intersect in $R_0$ (see Fig. \ref{fig:tangle intersection}). By definition, $L_{0,1}(0)\cap L_{1,0}(1)$ contains trajectories that reenter $R_1$ after they have escaped and consequently $R_1$ (and $\widehat{R}_1$) loses its no-return property.
In particular, trajectories that periodically reenter $R_1$ may exist and if they do, they will be located in $L_{0,1}(0)\cap L_{1,0}(\tilde{k}) \cap \dots$ for some $\tilde{k}$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2230_F11.png}
\caption{The $F_1$-$\widehat{F}_1$ tangle at $0.02230$, $W_{F_1}$ are shown as solid lines, $W_{\widehat{F}_1}$ as dashed.}
\label{fig:2230 F1F1}
\end{figure}
By symmetry $L_{\hat{0},\hat{1}}(0)$ and $L_{\hat{0},\hat{1}}(1)$ also contain heteroclinic points that converge towards $F_0$ and they cannot avoid intersecting $L_{1,0}(1)$ and $L_{0,1}(0)$ respectively. Figure \ref{fig:2230 F1F1} portraits the intersecting invariant manifolds. These intersections guide trajectories that may cross DS$_0$ multiple times and result in an overestimation of the reaction rate by TST. Due to the size of the lobe intersections, the overestimation is small but increases with energy. VTST suffers from recrossings too as it estimates the rate using the DS with lowest flux, but none of the DSs is recrossing-free.
Due to a high $k_{srt}$ and small area of lobes, we avoid details of the $F_1$-$\widehat{F}_1$ tangle until higher energies. We remark that lobes in the $F_1$-$\widehat{F}_1$ tangle do not intersect outside of the bounded region.
\subsection{Definitions of important regions}
We have established that TST fails at $0.02215$ due to recrossings. In this section we give a detailed description of homoclinic and heteroclinic tangles at $0.02230$ and explain the transport mechanism in these tangles using lobes. The energy $0.02230$ is representative for the interval between TST failure at $0.02215$ and one of several period doubling bifurcations of $F_{21}$ at $0.02232$. Moreover, lobes at $0.02230$ are sufficiently large to study.
For the sake of simple notation, in what follows $Q_0$, $Q_1$, $Q_2$ and $Q_3$ denote pips that differ from tangle to tangle. To avoid confusion, we always clearly state which tangle is discussed.
First we discuss the homoclinic tangles of $F_0$, $F_1$ and $\widehat{F}_1$ at $0.02230$. We define regions relevant to these homoclinic tangles shown in Figure \ref{fig:2230 regions} as follows.
Denote $R_0$, the region bounded by $W_{F_0+}$ and $W_{F_1-}$. The $F_0$-$F_1$ tangle is responsible for most of the complicated evolution of reactive trajectories at $0.02230$. The regions above and below the $F_0$-$F_1$ tangle are $R_2$ and $R_3$ respectively.
The region inside the $F_1$ tangle bounded by $W_{F_1-}$ is denoted $R_1$. Further we denote $R_4$ the region bounded by $W_{F_0+}$ that is relevant for the $F_0$ tangle. A near-intersection of $W_{F_0+}$ in $R_1$ suggests that $R_4$ is smaller after the period doubling bifurcation of $F_{21}$ at $0.02232$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2230.png}
\caption{Various region at $0.02230$.}\label{fig:2230 regions}
\end{figure}
\subsection{Homoclinic tangles}\label{subsec:homoclinic}
First we concentrate on the $F_0$ tangle at $0.02230$, followed by the $F_1$ tangle, both depicted in Fig. \ref{fig:homoclinic labeled}. In both it is possible to identify a number of lobes that explain the dynamics within.
The $F_0$ tangle govern transport from $R_3$ to $R_4$ and from $R_4$ to $R_2$. The lobes in this tangle consist of two disjoint parts. $L_{3,4}(0)$, for example, is bounded by $S[Q_1,Q_0]\cup U[Q_0,Q_1]$ and $S[Q_3,Q_2]\cup U[Q_2,Q_3]$. Note that $L_{4,2}(1)$ and $L_{3,4}(1)$ intersect near $Q_0$ and recall that $L_{4,2}(1) \cap L_{3,4}(1)$ does not leave $R_4$. $L_{3,4}(0) \cap L_{4,2}(1)$ near $Q_3$ implies $k_{srt}=1$.
By far the largest intersection in the $F_0$ tangle is $L_{3,4}(-1) \cap L_{4,2}(2)$. It comprises most of the white area in $R_4$ occupied by nonreactive trajectories and we can deduce the structure of the intersection from $L_{3,4}(0)$ and $L_{4,2}(1)$ as follows. As an image of $L_{3,4}(0)$, the larger part of $L_{3,4}(-1)$ is bounded by $S[PQ_1,PQ_0]\cup U[PQ_0,PQ_1]$ with pips indicated in Fig. \ref{fig:homoclinic labeled}. This is nearly a third of the entire region $R_4$.
Similarly the larger part of $L_{4,2}(1)$ is bounded by $S[Q_0,P^{-1}Q_3]\cup U[P^{-1}Q_3,Q_0]$. Its preimage, the larger part of $L_{4,2}(2)$, is bounded by $S[P^{-1}Q_0,P^{-2}Q_3]\cup U[P^{-2}Q_3,P^{-1}Q_0]$. Thanks to pips we are able to deduce that the majority of trajectories in the $F_0$ tangle is due to the intersection of these two lobes.
Note that part of an escape lobe extends to the product side of $F_0$ and contains reactive trajectories. This part of the lobe enters $R_4$ via $L_{3,4}(1)$, most of which is mapped to $L_{3,4}(0)\cap L_{4,2}(2)$ and escapes into $R_2$ via $L_{4,2}(1)$. Using an analogous argument we find that the part of a capture lobe lies on the product side of $F_0$ and carries reactive trajectories that escaped from $R_4$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2230_F0_detail2.png}\\
\includegraphics[width=\textwidth]{2230_F1_detail.png}
\caption{Homoclinic tangles associated with $F_0$ and $F_1$ respectively at $0.02230$.}\label{fig:homoclinic labeled}
\end{figure}
The $F_1$ tangle has only one pip between $Q_0$ and $PQ_0$ and therefore a simpler structure. $L_{0,1}(0) \cap L_{1,0}(1)$ implies $k_{srt}=1$, therefore trajectories pass through this tangle quickly. Most nonreactive trajectories of the $F_0$ tangle pass inbetween $L_{1,0}(0)$ and $L_{0,1}(1)$ and avoid the $F_1$ tangle. This follows from its adjacency to $Q_0$, which is only mapped along the boundary of $R_1$ always on the reactant side of $F_0$. Similarly we can follow the area between $L_{1,0}(0)$ and $L_{0,1}(2)$ on the product side of $F_0$ using the $\widehat{F}_1$ tangle and symmetry.
The considerable size of lobes on the product side of $F_0$ carries information about nonreactive trajectories. The part of $L_{0,1}(1)$ on the product side of $F_0$ enters $R_1$ via the upper part of $L_{0,1}(0)$, just above the indicated intersection with $L_{1,0}(-1)$. Since this area does not lie in $L_{1,0}(1)$, it is has to be mapped to $L_{0,1}(-1)\setminus L_{1,0}(0)$ that remains in $R_1$ and is defined by the pips $PQ_1$ and $P^2Q_0$ located on $S[F_1,PQ_0]$. Further this area will be mapped in $L_{1,0}(1)\setminus L_{0,1}(0)$ and, unlike the part of $L_{1,0}(1)$ bordering $S[P^{-1}Q_1,P^{-1}Q_0]$, back into products.
In contrast, we can follow the part of $L_{0,1}(2)$ near its boundary $U[P^{-1}Q_0,P^{-2}Q_1]$ in reactants being mapped to $L_{0,1}(1)$ near its boundary $U[Q_0,P^{-1}Q_1]$ and via $L_{0,1}(0)$ near its boundary $U[PQ_0,Q_1]$ into products.
As energy increases, we observe that the nonreactive mechanism of the $F_0$ tangle grows slower than the nonreactive mechanism in the $F_1$ tangle or even shrinks. The later involves crossing the axis $q_2=0$, which on $\Sigma_0$ coincides DS$_0$. Due to symmetry the same happens in the $\widehat{F}_1$ tangle. Therefore the flux across DS$_0$ grows twice as quickly as across DS$_1$. Therefore eventually DS$_1$ becomes the surface of minimal flux.
\subsection{Heteroclinic tangles}\label{subsec:heteroclinic}
Heteroclinic tangles partially share shapes, lobes and boundaries with homoclinic tangles and their description of transport must agree. Recall heteroclinic tangles have two turnstiles and two sets of escape and capture lobes.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2230_R0_detail.png}\\
\includegraphics[width=\textwidth]{2230_F1F1.png}
\caption{The $F_0$-$F_1$ tangle and the outline of $F_1$-$\widehat{F}_1$ tangle at $0.02230$.}\label{fig:heteroclinic labeled}
\end{figure}
For the sake of simplicity, we rely on pips and prior knowledge from Sec. \ref{subsec:homoclinic} to interpret Fig. \ref{fig:heteroclinic labeled}. Define $R_0$ in the $F_0$-$F_1$ tangle using $W_{F_0+}$ and $W_{F_1-}$ and the pips $Q_0$ and $Q_2$. A single pip is located on $\partial R_0$ between $Q_0$ and its image, the same is true for $Q_2$.
$L_{3,0}(0)$ bounded by $S[Q_1,Q_0]\cup U[Q_0,Q_1]$ is significantly larger than $L_{0,3}(1)$ bounded by $S[Q_0,P^{-1}Q_1]\cup U[P^{-1}Q_1,Q_0]$. Similarly $L_{0,2}(1)$ is larger than $L_{2,0}(0)$. Also note that $L_{3,0}(0)\cap L_{0,2}(1)$ takes up most of $R_0$. Hence most of $R_0$ originates in $R_3$ and escapes into $R_2$ after $1$ iteration. The trajectories contained therein are nonreactive.
It is worth mentioning that the lobes governing transport from $R_2$ to $R_3$, $L_{0,3}(1)$ and $L_{2,0}(0)$, are disjoint. Nonreactive trajectories originating in $R_2$ spend some time in $R_0$. This agrees with our conclusions on the nonreactive mechanism in the $F_1$ tangle.
The reactive mechanism in the $F_0$-$F_1$ tangle involves the capture lobe $L_{3,0}(1)$ part of which is mapped to $L_{3,0}(0)\setminus L_{0,2}(1)$ and on to $L_{3,0}(1)\cap R_0$, part of which lies in $L_{0,3}(1)$. The area of this intersection is small in $R_0$.
Understanding the $F_1$-$\widehat{F}_1$ tangle is very involved, as the boundary of the tangle requires several segments of $W_{F_1-}$ and $W_{\widehat{F}_1+}$. We propose a different point of view. In all tangles above, we have found that escape from the bounded region in a tangle, all area above the uppermost and below the lowermost stable invariant manifold escapes without further delay. For example in the $F_0$-$F_1$ tangle, $L_{0,2}(1)$ located above $W^s_{F_1-}$ and $L_{0,3}(1)$ located below $W^s_{F_0+}$ escape to reactants and products respectively, because as the stable manifold bounding the lobe contracts, the unstable manifold is unobstructed to leave the interaction region. In this sense that we propose only stable invariant manifolds to be considered a barrier in forward time.
Using this reasoning, concentrate on the area between $S[\widehat{F}_1,\widehat{Q}_0]$ and $S[F_1,Q_0]$ in the $F_1$-$\widehat{F}_1$ tangle. Everything above $S[Q_3,Q_2]$ and below $S[\widehat{Q}_3,\widehat{Q}_2]$ may pass through the tangle, but evolves in a regular and predictable manner from $R_3$ to $R_2$ or vice versa. We remark that this area is the intersection of two turnstiles. The same argument applies to the areas above $S[Q_1,Q_0]$ and below $S[\widehat{Q}_1,\widehat{Q}_0]$. Complicated dynamics is restricted to $R_1$, as defined in the $F_1$ tangle, $\widehat{R}_1$ and an island near $F_0$ and should be treated separately from predictable areas.
Using this line of thought enables us to formulate bounds and estimates of the reaction rate. Before we proceed to quantitative results, we conclude this section by describing the evolution of tangles with increasing energy.
\subsection{Higher energies}\label{subsec:higher energies}
The based on the analysis in Sections \ref{subsec:homoclinic} and \ref{subsec:heteroclinic} for tangles at $0.02230$, here we discuss on the evolution of tangles at higher energies and their impact on dynamics in the interaction region. As the mechanisms have been described, most of our comments concern sizes of lobes and duration of escape from a tangle.
An interesting question arises from the connection between bifurcations and changes in geometry of invariant structures. The causal relationship is not evident. Also bifurcations are mostly thought of as local events. However as they seem to affect invariant manifolds, a change in tangles propagates instantaneously throughout the whole space. This phenomenon reminds of the infinite propagation speed in the heat equation.
The next bifurcation above $0.02230$ according to Sec. \ref{subsec:po} is a period doubling of $F_{21}$ at $0.02232$, followed by a saddle-centre bifurcation that creates $F_3$ and $F_4$ at $0.02254$ and a bifurcation of $F_3$ where $F_{31}$ and $F_{32}$ are created at $0.02257$. At around $0.02523$ follows another period doubling of $F_{21}$, $F_{21}$ collides with $F_2$ at $0.02651$ and subsequently $F_2$ collides with $F_0$ at $0.02654$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2253_F0.png}\\
\includegraphics[width=\textwidth]{2253_R0.png}
\caption{The $F_0$ tangle and the $F_1$ tangle at $0.02253$.}\label{fig:2253}
\end{figure}
The major consequence of the bifurcation of $F_{21}$ at $0.02232$ is a new intersection of $W^u_{F_0+}$ and $W^s_{F_0+}$ labeled $Q_0$ in Fig. \ref{fig:2253}. This reduces the number of pips between $Q_0$ and $PQ_0$ to one and therefore lobes are no longer made up of two disjoint sets. The $F_0$ tangle resembles the $F_0$-$F_1$ tangle at $0.02230$. Also the size of $R_4$ is reduced.
In the $F_1$ tangle we see $L_{0,1}(2)$ cross DS$_0$ twice as shown in Fig. \ref{fig:2253}. All $L_{0,1}(k)$ for $k>2$ and also $L_{1,0}(k)$ with $k<-2$ therefore pass through $\widehat{R}_1$. Moreover, the tip of $L_{0,1}(2)$ approaching $R_1$ can be expected to pass cross $R_1$ after the bifurcations at $0.02254$ and $0.02257$.
A small remark regarding notation. At this energy $L_{0,1}(2)$ lies in $R_0$, $\widehat{R}_0$, $R_1$, $\widehat{R}_1$, $R_2$ and $R_3$, but we maintain the notation for consistency.
At $0.02400$, $L_{0,1}(2)$ in the $F_1$ tangle passes through $R_1$ twice and the number increases at higher energies. Almost all lobes lie in almost all regions, but the mechanism for fast entry and exit of the tangles remain the same.
Fig. \ref{fig:2400 regions} shows $R_0$ and $\widehat{R}_0$. While $R_4$ is considerably larger than $R_1$ at $0.02253$, the opposite is true at $0.02400$. Recall that $R_4$ contains predominantly nonreactive trajectories that do not cross DS$_0$, whereas $R_1$ mostly contains ones that do. The overestimation of the reaction rate follows.
The capture lobes in the $F_1$ tangle guide predominantly trajectories from products into $R_1$, as shown in Fig. \ref{fig:2400 regions}. A significant portion of $R_1$ is taken up by $L_{0,1}(0)\cap L_{1,0}(1)$ and it is prevented by $W^s_{F_1-}$ from escaping into reactants. Moreover, the a large part of the intersection lies below $W^s_{\widehat{F}_1+}$, see Fig. \ref{fig:2400 regions}, that guides it into back products as $W^s_{\widehat{F}_1+}$ contracts.
Heteroclinic tangles mirror the changes of the homoclinc tangles (Fig. \ref{fig:2400 tangles}).
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2400_short.png}\\
\includegraphics[width=\textwidth]{2400_F1.png}
\caption{Indication of boundaries of $R_0$ and $R_4$ (above) and the $F_1$ tangle at $0.02400$ (below). The area in the $F_1$ tangle highlighted in cyan is the part of $L_{0,1}(0)\cap L_{1,0}(1)$ that originates in products and is guided by $W^s_{\widehat{F}_1+}$ (dashed) into products.}\label{fig:2400 regions}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{2400_F01.png}
\includegraphics[width=0.49\textwidth]{2400_F11.png}
\caption{Structure of the heteroclinic tangles at $0.02400$. $W_{F_0+}$ and $W_{F_1-}$ making up the $F_0$-$F_1$ tangle (left) and the $F_1$-$\widehat{F}_1$ tangle (right). Unstable invariant manifolds are as indicated red and green, stable are blue and orange.}\label{fig:2400 tangles}
\end{figure}
\subsection{Loss of normal hyperbolicity}
\label{subsec:lossNH}
$F_0$ loses normal hyperbolicity and becomes stable at $0.02654$, in a bifurcation involving $F_2$, $\widehat{F}_2$, $F_{21}$, $\widehat{F}_{21}$. TST cannot be based on $F_0$ and $W_{F_0}$ cease to exist. The sudden disappearance of invariant manifolds has no dramatic consequences. As can be deduced from Fig. \ref{fig:2400 tangles}, $W_{F_0}$ are at energies below $0.02654$, very close to $W_{F_1-}$ and $W_{\widehat{F}_1+}$ and naturally take over the role of $W_{F_0}$.
Throughout the energy interval from $0.02206$ when $F_1$ appears to the loss of normal hyperbolicity at $0.02654$, we see a transition of dominance from $F_0$ to $F_1$-$\widehat{F}_1$.
The loss of normal hyperbolicity of $F_0$ simplifies dynamics due to the presence of fewer TSs, for example compare Figures \ref{fig:2400 tangles} and \ref{fig:2700}.
At $0.02661$, $F_0$ collides with $F_4$ and becomes inverse hyperbolic. Due to the inverse hyperbolicity, $W_{F_0}$ exist, but they must contain a twist that is manifested as a reflection across the $F_0$ (see \cite{OzoriodeAlmeida90}), i.e. have the geometry of a M\"{o}bius strip. At the same time $W_{F_0}$ are enclosed between $W_{F_1-}$ along with $W_{\widehat{F}_1+}$, but with cylindrical structure. Consequences of the geometry of $W_{F_0}$ are unknown.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2700.png}\\
\includegraphics[width=\textwidth]{2700_detail.png}
\caption{The $F_1$-$\widehat{F}_1$ tangle at $0.02700$ and an indication how certain parts of lobes are mapped in this tangle.}\label{fig:2700}
\end{figure}
There are no more significant bifurcations above $0.02661$ and therefore apart from growing tangles and lobes, the tangles remains structurally the same.
Together with $W_{F_0}$ we observe the disappearance of $R_4$ and of the mechanism that carries nonreactive trajectories through the $F_0$ tangle without crossing DS$_0$. Consequently, all trajectories that pass through the $F_1$-$\widehat{F}_1$ tangle cross DS$_0$ at least twice. Each hemisphere of DS$_1$ still possesses the no-return property, which means trajectories cross DS$_1$ at most twice. Trajectories that avoid the tangle cross both DSs once or not at all.
Similarly to lower energies, $R_1$ is predominantly made up of $L_{0,1}(0)\cap L_{1,0}(1)$ in the $F_1$ tangle or $L_{2,0}(0)\cap L_{0,3}(1)$ in the $F_1$-$\widehat{F}_1$ tangle, as shown in Figure \ref{fig:2700}. The argument that trajectories in the $F_1$-$\widehat{F}_1$ tangle below and above all stable manifolds leave the interaction region is still valid. Capture lobes are disjoint, therefore it is not possible to reenter the bounded region. Although $R_1$ and $\widehat{R}_1$ admit return, $R_1\cup\widehat{R}_1$ possesses the no-return property.
\subsection{Known estimate}\label{subsec:known estimate}
Davis \cite{Davis87} formulated bounds and an estimate of the reaction rate based on numerical observation of dynamics. He observed that a significant portion of trajectories leave the heteroclinic tangle above $0.02654$ after one iteration and imposed the assumption of fast randomization on the remaining trajectories.
As described above, Davis' observation is due a property of the $F_1$-$\widehat{F}_1$ tangle - $R_1$ is mostly occupied by $L_{0,1}(0)\cap L_{1,0}(1)$. We quantify this proportion below.
The assumption of fast randomization of the other trajectories and a $50\%$ probability of them reacting is more difficult to support. From the analysis of lobes we know that however intricate the dynamics is, there is no reason for precisely half of the remaining trajectories to leave to reactants and half to products. Instead we find that for small energies, trajectories that spend $2$ and more iterations $R_0$ and $\widehat{R}_0$ make up a significant part of the tangles (up to half at $0.02350$), but their total proportion is very small and only grows slowly with increasing energy. In the interval up to $0.03000$, these trajectories make up at most $3\%$ of the total, $2\% $ below $0.02650$, see Table \ref{tab:areas}. Consequently, any estimate of the reaction rate that takes trajectories escaping after $1$ iteration into account is accurate to within $3\%$ below $0.03000$ and when we include trajectories escaping after $2$ iterations, this number drops to less than $1\%$.
The difficulty lies in accurately calculating the amount of trajectories. At the cost of accuracy, Davis used VTST as a measure of trajectories entering the interaction region, $\mu(L_{3,0}(0))$ to estimate the size of the tangle and $\mu(L_{3,0}(0)\cap L_{0,2}(1))$ to subtract trajectories escaping after $1$ iteration. The upper and lower estimates assume all, respectively none, of the trajectories that escape after $2$ or more iterations are reactive.
\section{The intricate energy interval}\label{sec:intricate interval}
The energy interval $0.02215<E<0.02654$, when TST is not exact and $F_0$ is a TS, has been largely avoided in the past. The interaction of invariant manifolds of two TSs posed enough difficulties. Dividing tangles using pieces of invariant manifolds and following pips to understand dynamics within make this task possible. We divide tangles differently to the lobe dynamics approach, because we aim to describe and measure parts of heteroclinic tangles that do not necessarily fall into a single lobe.
\subsection{Division of a tangle}\label{subsec:division}
Davis \cite{Davis87} calculated pieces of invariant manifolds in this interval at an energy of $0.7\text{eV}\approx 0.02572$, but complexity of their intersections did not admit deeper insight. With current understanding it is not possible to consider all the invariant manifolds at once, because even identifying lobes is challenging, not to speak of their intersections.
We use the approach outlined in Sec. \ref{subsec:heteroclinic} and concentrate on $W_{F_1}$ and $W_{\widehat{F}_1}$, while keeping $W_{F_0}$ in mind near $F_0$. A similar approach may be used for homoclinic tangles. We separate predictably evolving trajectories from chaotic ones, for example trajectories escaping after $1$, $2$ or $3$ iterations from the rest of the tangle. To our knowledge, tools for identifying particular lobe intersections and determining the area, a heteroclinic tangle surgery toolbox, have not been previously presented or reported.
There is one more important property of the manifolds that stands out from all previous figures. Inside the $F_1$-$\widehat{F}_1$ tangle, $W^u_{F_1-}$ and $W^u_{\widehat{F}_1+}$ are restricted to the stripe between two pieces of unstable manifold, e.g. $U[\widehat{F}_1,Q_3]$ and $U[F_1,Q_1]$ at $0.02700$ in Fig. \ref{fig:2700} or $U[\widehat{F}_1,PQ_1]$ and $U[F_1,P\widehat{Q}_1]$ at $0.02400$ in Fig. \ref{fig:2400}. Similarly $W^s_{F_1-}$ and $W^s_{\widehat{F}_1+}$ are confined to a single stripe. We remark that $W_{F_0}$ are located between $W_{F_1-}$ and $W_{\widehat{F}_1+}$ and thereby confined as well. It therefore makes sense to study this stripe in detail.
Consider the $F_1$-$\widehat{F}_1$ tangle at $0.02400$, where $R_1$ and $R_4$ are reasonably sized and nonreactive trajectories that do not cross DS$_0$ exist. Following the motto \emph{divide et impera}, we take the following steps:
\begin{itemize}
\item We identify new regions that have the no-return property.
\item We use as few pieces of invariant manifolds as possible.
\item We define subsets of regions containing reactive/nonreactive trajectories.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{2400_F1F1.png}
\includegraphics[width=0.49\textwidth]{2400_he.png}
\caption{The $F_1$-$\widehat{F}_1$ tangle at $0.02400$ (left) and its simplification (right). $W_{F_1-}$ are drawn with solid lines, $W_{\widehat{F}_1+}$ are dashed.}\label{fig:2400}
\end{figure}
Define $R_5$ as the bounded region inside the tangle, the upper part of the boundary is made up of $U[\widehat{F}_1, Q_2]$, $S[Q_2, Q_3]$, $U[Q_3,PQ_0] $ and $S[PQ_0,F_1]$, see Figure \ref{fig:2400}, and the lower part is symmetric to it. Each lobe consists of two disjoint sets, for example, $L_{2,5}(0)$ is bounded by $S[PQ_1,PQ_0]$, $U[PQ_0,PQ_1]$ and $S[Q_3,Q_2]$, $U[Q_2,Q_3]$. We remark that lobes do not intersect outside $R_5$ and leave the interaction region. Disjoint capture lobes imply:
\begin{remark}
$R_5$ has the no-return property.
\end{remark}
As found in Sec. \ref{subsec:higher energies}, a large part of $R_5$ behaves regularly and leaves the tangle within $1$ iteration. As argued in Sec. \ref{subsec:heteroclinic}, stable manifolds contract in forward time and thereby act as a barrier. Everything above $W^s_{F_1-}$ leaves at the next iteration to reactants, everything below $W^s_{\widehat{F}_1+}$ leaves to products. This agrees with the lobes $L_{5,3}(1)$ and $L_{5,2}(1)$ that leave $R_5$ by definition.
The remainder of $R_5$ is the stripe between $W^s_{F_1-}$ and $W^s_{\widehat{F}_1+}$, the only part of $R_5$ where stable manifolds can lie. We refer to it as the \emph{capture stripe} and denote it $R_6$, see Fig. \ref{fig:2400}. Its boundary consists of $S[\widehat{F}_1,\widehat{Q}_1]$, $U[F_1,\widehat{Q}_1]$, $S[F_1,Q_1]$ and $U[\widehat{F}_1,Q_1]$.
In backward time, the roles of stable and unstable manifolds switch - everything below $W^u_{F_1-}$ and above $W^u_{\widehat{F}_1+}$ escapes $R_5$. Define $R_7$, the \emph{escape stripe} bounded by $S[\widehat{F}_1,P\widehat{Q}_1]$, $U[F_1,P\widehat{Q}_1]$, $S[F_1,PQ_1]$ and $U[\widehat{F}_1,PQ_1]$. $R_5\setminus R_7$ escapes $R_5$ after $1$ iteration in backward time.
We conclude that all complicated and chaotic dynamics is confined to $R_6\cap R_7$ and due to the no-return property of $R_5$:
\begin{remark}
$R_6$ and $R_7$ have the no-return property.
\end{remark}
Note that the boundary of $R_7$ is the image of the boundary of $R_6$. Necessarily
$$PR_6=R_7,$$
and due to preservation of area $\mu(R_6)=\mu(R_7)$.
There are more regions with the no-return property in the $F_1$-$\widehat{F}_1$ tangle. Obviously, $R_5\setminus (R_6\cup R_7)$ must be a no-return region as it escapes the $R_5$ immediately after entering. Also $R_6\setminus R_5$ as the entry point to $R_7$ must have the no-return property as well as capture and escape lobes.
\subsection{Dynamical properties}
\label{subsec:dynamical properties}
To shorten and facilitate the description of reactive and dynamical properties of $R_5$, $R_6$ and $R_7$, we introduce the following classification of trajectories.
\begin{definition}
We call the set of trajectories:\\
\emph{directly reactive} ($DR$) if they remain in $R_2$ or $R_3$,\\
\emph{directly nonreactive} ($DN$) if they do not enter the interaction region,\\
\emph{captured reactive} after $n$ iterations ($CR_n$) if they react after $n$ iterations in $R_5$,\\
\emph{captured nonreactive} after $n$ iterations ($CN_n$) if they return to the region of origin after $n$ iterations in $R_5$.
\end{definition}
Clearly $DR$ and $DN$ never enter $R_5$. Following Sec. \ref{subsec:higher energies} and Sec. \ref{subsec:division}, $R_5\setminus (R_6\cup R_7)$ is the region of $CN_1$ and $CR_1$ is always empty. $CR_2$ and $CN_2$ are pass through $R_6\setminus R_7$ and $R_7\setminus R_6$ and therefore never enter $R_6\cap R_7$.
This leaves the complicated evolution and chaotic behaviour restricted to $R_6\cap R_7$. Below $0.02500$, $R_6\cap R_7$ consists of $5$ squares near $F_0$, $F_1$, $\widehat{F}_1$, $F_2$ and $\widehat{F}_2$. As $F_2$ and $\widehat{F}_2$ approach the bifurcation with $F_0$, the three squares near them merge into one around $0.02523$ when $F_{21}$ bifurcates, see Figures \ref{fig:2500} and \ref{fig:2550}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{2500_short.png}
\includegraphics[width=0.49\textwidth]{2500_detail.png}
\caption{The $F_1-\widehat{F}_1$ tangle at $0.02500$ and a detail of the diminishing part of $R_5\setminus(R_6\cup R_7)$ highlighted in cyan.}\label{fig:2500}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{2550_short.png}
\includegraphics[width=0.49\textwidth]{2550_detail.png}
\caption{The $F_1-\widehat{F}_1$ tangle at $0.02550$ and a detail of the diminished part of $R_5\setminus(R_6\cup R_7)$.}\label{fig:2550}
\end{figure}
Trajectories enter $R_5$ via $R_6\setminus R_7$ and escape via $R_7\setminus R_6$, hence every trajectory crosses $R_6\setminus R_7$ and $R_7\setminus R_6$ at most once. The same is true for $R_5\setminus (R_6\cup R_7)$ consisting of $CN_1$. Therefore of $R_5$ only the size $R_6\cap R_7$ does not reflect the number of trajectories it contains. It follows that the area of $R_6\setminus R_7$, $R_7\setminus R_6$ and $R_5\setminus (R_6\cup R_7)$ on the surface of section $\Sigma_0$ is the same of their images on DS$_1$ and DS$_{\widehat{1}}$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2400_dyn.png}\\
\includegraphics[width=\textwidth]{2400_detail.png}
\caption{Coloured sets in $R_6$ showing how part of the capture stripe is mapped at $0.02400$. $W_{F_1-}$ are drawn with solid lines, $W_{\widehat{F}_1+}$ are dashed. $CN_1$ are shown in orange, $CR_2$ green and yellow, $CN_2$ red. Part of blue also belongs to $CN_2$. Blue, yellow, red and green are separated by white stripes that are mapped to $R_6\cap R_7$.}\label{fig:2400 dyn}
\end{figure}
Fig. \ref{fig:2400 dyn} shows a more detailed partitioning of $R_6$ and $R_7$. Essentially, $R_6$ is divided into finer stripes by pieces of $W^s_{F_1-}$ and $W^s_{\widehat{F}_1+}$ that are nearly parallel to the boundary. The boundary of $R_6$ illustrates how the content of the stripe is deformed when mapped into $R_7$. It is compressed along the stable manifolds towards the fixed points, e.g.
$$P(S[F_1,Q_1])=S[F_1,PQ_1],$$
and stretched along the unstable manifolds away from the fixed points. We remark that the whole highlighted set in $R_5\setminus R_7$ of Fig. \ref{fig:2400 dyn} is connected, only separated by stable invariant manifolds. When mapped forward it is stretched, but remains connected. The sets labeled by yellow, red and green are alternately mapped above and below the capture stripe.
There is a connection between these coloured stripes and lobe intersections, but lobes do not distinguish how often trajectories cross DS$_0$, which is necessary to understand overestimation of the reaction rate by TST.
The connected components of $R_6\cap R_7$ contain dynamics similar to Smale's horseshoe dynamics, \cite{Hirsch04}. As a consequence we observe a fractal structure, as can be seen in Fig. \ref{fig:2500 isl}. $R_6\cap R_7$ accounts for less than $12\%$ of the all trajectories that pass through $R_5$ below $0.02400$ when the dynamics is relatively slow. The proportion drops to roughly $7\%$ of $R_5$ at $0.03000$ and remains below $1\%$ of the total amount of trajectories.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{2500_isl.png}
\caption{Detail of the island near $\widehat{F}_1$ and its content at $0.02500$. $CN_3$ are purple, $CR_3$ are magenta and the rest of the island is plain. $CN_2$ (green) and $CR_2$ (cyan) contained in the adjacent regions $R_6$ and $R_7$ are shown for completeness.}\label{fig:2500 isl}
\end{figure}
\subsection{Areas}
In this system, determining the area of $R_5$, $R_6$, $R_7$, $R_5\setminus (R_6\cup R_7)$, $R_6\setminus R_7$ and $R_6\cap R_7$, is significantly easier than calculating lobe intersections. We employ a Monte Carlo based method that is expensive, yet simple. Ultimately the cost and accuracy depend on the level of detail in $R_6\cap R_7$, i.e. it can be determined a priori. We also tune initial and terminal conditions to obtain a high accuracy at a reasonable cost.
Previous works seem to consider initial conditions on $r_1+\frac{r_2}{2}=50$, $p_{r_1}<0$, which is a surface near $q_2=1181$. We prefer to sample the hemisphere of DS$_1$, through which trajectories enter the interaction region. Directly we have that the difference between the inward hemisphere of DS$_1$ and $r_1+\frac{r_2}{2}=50$, $p_{r_1}<0$ corresponds to DN trajectories.
The slowest of $DR$ are located near the boundary of $R_5$, and those near pips evolve similarly to pips. Using pips on $W^u_{F_1}$ we define \emph{checkpoints}, that mark distance these pips are mapped, i.e. the least distance $DR$ cover in the interaction region in $1$ iteration. Then all trajectories that pass the second checkpoint $1$ iteration after they pass the first checkpoint, are $DR$ and are not captured in $R_5$. Recall that all captured reactive trajectories spend at least $2$ iterations in $R_5$. Trajectories that have a delay of $n$ iteration between crossing of the checkpoints are $CR_n$. Since $R_5$ is symmetric, we use the symmetric counterpart of the second checkpoint to identify $CN_n$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2400_checkpoints.png}
\caption{Checkpoints defined in the $F_1$-$\widehat{F}_1$ tangle at $0.02400$.}\label{fig:2400 chk}
\end{figure}
At $0.02400$, $Q_1$ and $PQ_1$ are the natural choice for checkpoints, because mark the endpoints of $R_6\setminus R_7$ via which trajectories enter $R_5$, see Figure \ref{fig:2400 chk}. We define checkpoint $Ch_{Q_1}$ as a vertical line passing through $Q_1$.
It is necessary that $Ch_{Q_1}$ avoids capture lobes, therefore at energies above $0.02900$ the computationally most efficient solution is to use another vertical line between $Q_1$ and $F_0$.
The role of the second checkpoint, $Ch_{PQ_1}$, is to distinguish trajectories in $R_5$ from those outside $R_5$. We use a linear approximation of $S[\widehat{F}_1,PQ_1]$, the boundary between the escape lobe and $R_5$, in conjunction with a vertical line passing through $PQ_1$. The checkpoint symmetric to $Ch_{PQ_1}$ is defined analogously and denoted $Ch_{P\widehat{Q}_1}$. If desired, we can track the number crossings of DS$_0$ using the sign of $q_2$.
We can measure individual components of $R_5$: $\mu(R_5\setminus R_7)$ corresponds to the number of captured trajectories, $\mu(R_5\setminus(R_6\cup R_7))$ is given by $CN_1$. Then
$$\mu(R_6\setminus R_7)=\mu(R_5\setminus R_7)-\mu(R_5\setminus(R_6\cup R_7)),$$
and $R_6\cap R_7$ can be deduced from $CR_n$ and $CN_n$ where $n\geq3$. The latter follows from the fact that $CR_2$ and $CN_2$ do not pass through $R_6\cap R_7$.
This method is not computationally cheap, but the computational difficulty can be easily estimated a priori. Determining the distribution up to $CR_n$ and $CN_n$ with $N$ initial conditions requires approximately $nN$ iterations of the map $P$, but considering the prevalence of $DR$ and $DN$, this number will be considerably lower.
Alternative approaches to calculating lobe areas face the obstacle in distinguishing the inside from the outside of a lobe, not to mention their intersections. Recent developments \cite{krajnak2018phase,krajnak2018influence} suggest that a reactive island approach can be used to calculate areas of intersections in a cheaper and simpler manner.
\section{Bounds of the reaction rate}\label{sec:bounds}
\subsection{Quantification}\label{subsec:quantification}
In Tab. \ref{tab:areas} we present proportions of areas of classes of trajectories on the plane $r_1+\frac{r_2}{2}=50$, $p_{r_1}<0$. Between $10^7$ and $2.10^8$ initial conditions were used to obtain these values.
\begin{table}
\centering
\begin{tabular}{r|*{6}{c}}
Energy & $DR$ & $DN$ & $CR_2$ & $CN_1$ & $CN_2$ & $Other$ \\
\hline
0.02205 & 0.590 & 0.410 & 0 & 0 & 0 & 0\\
0.02214 & 0.595 & 0.405 & 0 & 0 & 0 & 0\\
\hline
0.02215 & 0.687 & 0.296 & 0 & 0.016 & 0.001 & 0.000\\
0.02230 & 0.693 & 0.290 & 0.001 & 0.013 & 0.001 & 0.001\\
0.02253 & 0.703 & 0.282 & 0.001 & 0.011 & 0.001 & 0.002\\
0.02300 & 0.717 & 0.266 & 0.003 & 0.008 & 0.003 & 0.002\\
0.02350 & 0.725 & 0.255 & 0.004 & 0.010 & 0.005 & 0.003\\
0.02400 & 0.733 & 0.239 & 0.004 & 0.015 & 0.006 & 0.003\\
0.02450 & 0.737 & 0.227 & 0.005 & 0.021 & 0.006 & 0.004\\
0.02500 & 0.739 & 0.216 & 0.006 & 0.028 & 0.007 & 0.005\\
0.02550 & 0.739 & 0.206 & 0.006 & 0.037 & 0.008 & 0.006\\
0.02600 & 0.737 & 0.197 & 0.007 & 0.046 & 0.008 & 0.006\\
0.02650 & 0.734 & 0.188 & 0.007 & 0.055 & 0.009 & 0.007\\
\hline
0.02662 & 0.734 & 0.186 & 0.008 & 0.057 & 0.009 & 0.007\\
0.02700 & 0.731 & 0.180 & 0.008 & 0.064 & 0.010 & 0.007\\
0.02800 & 0.723 & 0.166 & 0.009 & 0.083 & 0.011 & 0.008\\
0.02900 & 0.714 & 0.153 & 0.010 & 0.101 & 0.012 & 0.009\\
0.03000 & 0.705 & 0.145 & 0.011 & 0.116 & 0.013 & 0.010
\end{tabular}
\caption{Proportions of areas of classes of trajectories on the plane $r_1+\frac{r_2}{2}=50$, $p_{r_1}<0$. Directly reactive (DR) and directly nonreactive ($DN$) trajectories do not enter $R_5$. Captured reactive ($CR_2$) and captured nonreactive ($CN_1$, $CN_2$) enter and leave $R_5$ after $1$ or $2$ iterations. $Other$ trajectories do not leave $R_5$ within $2$ iterations after their entry and are inside $R_6\cap R_7$. Horizontal lines represent the creation of the homoclinic tangles and loss of normal hyperbolicity of $F_0$.}
\label{tab:areas}
\end{table}
The proportion of $DN$ decreases steadily over the whole interval presented in Tab. \ref{tab:areas} and beyond. This is not surprising given that widening bottlenecks allow more trajectories enter the interaction region. Thereby nonreactive heavily oscillating trajectories enter the interaction region and consequently $R_5\setminus(R_6\cup R_7)$ grows faster than the rest of $R_5$.
Note that the proportion of $DR$ culminates between $0.02500$ and $0.02550$. At this energies the geometry of the $F_1$-$\widehat{F}_1$ tangle simplifies with the consequence that all captured trajectories cross DS$_1$.
The proportion of $DR$ above $0.02550$ decreases predominantly in favour of $CN_1$. We observe the growth of capture lobes mainly in the area of large $|p_2|$ momentum, containing predominantly $CN_1$ trajectories (Tab. \ref{tab:areas}), approaching the maximal values of $|p_2|$ at the given energy. This implies that for a given (small) $|p_1|$ momentum, trajectories are more likely to react at a lower energy due to smaller capture lobes.
In the physical world large values of $|p_2|$ correspond by definition (Sec. \ref{subsec:momenta}) to large $|p_{r_1}|$ on the reactant side and large $|p_{r_2}|$ on the product side. Since trajectories are less likely to react at higher energies, the mechanism for transfer of kinetic energy between the degrees of freedom in the interaction region must be failing at high energies. Consequently, the energy passed from the incoming $H$ to the $H_2$ may be so high, that it repels the whole molecule instead of breaking its bond. This may be true for a whole class of collinear atom-diatom reactions, provided it is possible to define an interaction region multiple TSs.
\begin{table}[ht]
\centering
\begin{tabular}{r|*{6}{c}}
Energy & $P_{\text{TST}}$ & $P_{\text{VTST}}$ & $P_{\text{MC}}$ & $L_2$ & $U_2$ & $U_1$ \\
\hline
0.01600 & 0.181 & 0.181 & 0.181 & & & \\
0.01800 & 0.383 & 0.383 & 0.383 & & & \\
0.01900 & 0.469 & 0.469 & 0.469 & & & \\
0.02000 & 0.545 & 0.545 & 0.545 & & & \\
0.02100 & 0.615 & 0.615 & 0.615 & & & \\
0.02205 & 0.681 & 0.681 & 0.681 & & & \\
\hline
0.02215 & 0.687 & 0.687 & 0.687 & 0.687 & 0.687 & 0.689\\
0.02230 & 0.696 & 0.696 & 0.695 & 0.694 & 0.696 & 0.697\\
0.02253 & 0.709 & 0.709 & 0.705 & 0.704 & 0.706 & 0.707\\
0.02300 & 0.736 & 0.734 & 0.721 & 0.720 & 0.722 & 0.725\\
0.02350 & 0.763 & 0.748 & 0.732 & 0.728 & 0.731 & 0.736\\
0.02400 & 0.789 & 0.761 & 0.739 & 0.737 & 0.741 & 0.746\\
0.02450 & 0.814 & 0.773 & 0.744 & 0.742 & 0.746 & 0.753\\
0.02500 & 0.838 & 0.784 & 0.746 & 0.744 & 0.749 & 0.756\\
0.02550 & 0.860 & 0.794 & 0.747 & 0.744 & 0.750 & 0.758\\
0.02600 & 0.883 & 0.804 & 0.747 & 0.743 & 0.750 & 0.758\\
0.02650 & 0.904 & 0.812 & 0.745 & 0.742 & 0.748 & 0.757\\
\hline
0.02662 & 0.909 & 0.814 & 0.744 & 0.741 & 0.748 & 0.757\\
0.02700 & 0.924 & 0.820 & 0.743 & 0.739 & 0.746 & 0.756\\
0.02800 & 0.963 & 0.835 & 0.736 & 0.732 & 0.740 & 0.751\\
0.02900 & 0.999 & 0.847 & 0.729 & 0.725 & 0.734 & 0.746\\
0.03000 & 1.033 & 0.858 & 0.720 & 0.716 & 0.726 & 0.739\\
\hline
0.04000 & 1.278 & 0.960 & 0.626 & & & \\
0.05000 & 1.428 & 1.002 & 0.542 & & &
\end{tabular}
\caption{Comparison of results of TST and VTST with the actual reaction rate computed via Monte Carlo and our upper and lower estimates.}
\label{tab:myP}
\end{table}
\subsection{MC based bounds}
Using Tab. \ref{tab:areas} we are able to formulate estimates of the reaction rate up to arbitrary precision. The idea is similar to \cite{Davis87}. An upper/lower bound on the reaction rate is obtained by assuming that all/none of the trajectories that remain in $R_5$ after $n$ iterations react. Tab. \ref{tab:myP} contains the resulting bounds.
Denote $U_1$ the rate estimate obtained by assuming all trajectories in $R_5\setminus(R_6\cup R_7)$ react, or equivalently only $CN_1$ do not react. Since $CN_2$ and some of $Other$ do not react, the true reaction rate is lower.
\begin{lemma}
$U_1=\mu(DR)+\mu(CR_2)+\mu(CN_2)+\mu(Other)$ is an upper bound of the reaction rate.
\end{lemma}
$U_1$ can be easily improved by acknowledging that $CN_2$ are nonreactive. Denote this bound by $U_2$. Because the $R_6\cap R_7$ contain reactive as well as nonreactive trajectories, the true reaction rate is lower.
\begin{lemma}
$U_2=\mu(DR)+\mu(CR_2)+\mu(Other)$ is an upper bound of the reaction rate.
\end{lemma}
A lower bound $L_2$ is obtained assuming all of $R_6\cap R_7$ are nonreactive trajectories.
\begin{lemma}
$L_2=\mu(DR)+\mu(CR2)$ is a lower bound of the reaction rate.
\end{lemma}
The difference between $L_2$ and $U_2$ is precisely $\mu(Other)$. This gives us an upper bound on the error of both estimates. An estimate of the reaction rate can be obtained using $L_2$ and $U_2$.
\section{Conclusion}
We have studied invariant manifolds of TSs to find an explanation for the decrease of the reaction rate. In the process of understanding how energy surface volume passes through homoclinic and heteroclinic tangles formed by these invariant manifolds we found the need for tools that would allow us to work with the tangles and not get lost in details of its chaotic structure.
We introduced a suitable division of homoclinic and heteroclinic tangles that is simple and understandable based on reactive properties of trajectories.
Once divided, the heteroclinic tangles decompose into areas of simple and more complicated dynamics. We were able to identify a large class of trajectories that are merely diverted by the tangles and areas of fractal horseshoe-like structure near hyperbolic or inverse-hyperbolic periodic orbits. In addition to a better understanding, the division provides an easy way calculating the corresponding areas.
Contrary to expectations, the decline of the reaction rate is not a result of loss of normal hyperbolicity. We may consider the decrease of the reaction rate and loss of normal hyperbolicity to be consequences of insufficient transfer of kinetic energy between the degrees of freedom. In physical terms, the single atom has so much kinetic energy, that it repels the whole molecule instead of becoming part of it.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 932
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interface class ICallbackHandler
{
public:
virtual void OnCallback();
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,505
|
Q: Saxon-HE Java Extension - How to I access the value of a xsl-variable which is passed as a parameter? I have created a function using the Saxon documentation which has 3 parameters. The function takes an input string and pads it out to a specific size using an integer and string values.
padStringLeft(inputStr,size,padChar)
If I put this in my XSLT and hard wire the parameters the function works.
<debug1><xsl:value-of select="c4j_XSLT_Ext_padStringLeft:padStringLeft('1',4,'0')" /></debug1>
The output from the above would be '0001'
When I pass the contents of a XSLT variable however and set a debug / break point in my java function I can see that I'm getting param0 as a lazysequence.
<debug2><xsl:value-of select="c4j_XSLT_Ext_padStringLeft:padStringLeft($myvar,4,'0')" /></debug2>
Java function
As my code is attempting to treat it as a string it does not work.
How should I be handling this scenario, how do I access the value or the xsl-variable/param and what if sometimes I want to use a literal string instead of a variable?
public class XSLT_Ext_padStringLeft extends ExtensionFunctionDefinition
{
@Override
public SequenceType[] getArgumentTypes()
{
return new SequenceType[]{SequenceType.SINGLE_STRING,SequenceType.SINGLE_INTEGER, SequenceType.SINGLE_STRING};
}
@Override
public StructuredQName getFunctionQName()
{
return new StructuredQName("c4j_XSLT_Ext_padStringLeft", "http://com.commander4j.Transformation.XSLT_Ext_padStringLeft", "padStringLeft");
}
@Override
public SequenceType getResultType(SequenceType[] arg0)
{
return SequenceType.SINGLE_STRING;
}
@Override
public ExtensionFunctionCall makeCallExpression()
{
return new ExtensionFunctionCall() {
@Override
public Sequence call(XPathContext context, Sequence[] arguments) throws XPathException {
String inputStr;
try
{
inputStr = ((StringValue)arguments[0]).getStringValue();
} catch (ClassCastException ex)
{
inputStr = "";
}
long size;
try
{
String temp =arguments[1].toString();
size = Integer.valueOf(temp);
} catch (ClassCastException ex)
{
size = 1;
}
String padStr;
try
{
padStr = ((StringValue)arguments[2]).getStringValue();
} catch (ClassCastException ex)
{
padStr = "";
}
String result = inputStr;
while (result.length() < size)
{
result = padStr + result;
}
return StringValue.makeStringValue(result);
}
};
}
}
Thanks
Dave
A: In general the parameters are passed as instance of the class net.sf.saxon.om.Sequence, and you should only use the methods on the interface Sequence, rather than examining what particular kind of Sequence it is, because that could change in the future.
If you're expecting a singleton sequence (that is, a single item), call head() to get the first item in the sequence (this will return null if the sequence is empty). You will then have an instance of net.sf.saxon.om.Item. (The Sequence might already be an Item, because an item is a sequence, but you can't rely on that, and calling head() is safer than casting.) If you're expecting a string, you can safely call getStringValue() on this item to get the value as a string.
Also note, Saxon uses lazy evaluation wherever possible, which means that the string might not actually be computed until someone asks for its value. This means that innocent-looking calls like head() and getStringValue() can actually throw exceptions, and you need to be prepared for this.
So in short, you should replace
inputStr = ((StringValue)arguments[0]).getStringValue();
with
inputStr = arguments[0].head().getStringValue();
A:
Also note, Saxon uses lazy evaluation wherever possible, which means that the string might not actually be computed until someone asks for its value. This means that innocent-looking calls like head() and getStringValue() can actually throw exceptions, and you need to be prepared for this.
So if I understand you correctly - when I call Transform to process the XSLT transformation it will call each of my custom java external functions as needed but the reference to
inputStr = arguments[0].head().getStringValue();
could generate an exception?
I would then need to do something within the java function to force it to get the value - or would I let the exception propogate back to the calling Transformation and catch it there ?
Dave
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,833
|
Morgan Sparks (6 de julio de 1916 - 3 de mayo de 2008) fue un científico e ingeniero estadounidense que ayudó a desarrollar el transistor de unión bipolar de microwatt en 1951, lo que fue un paso crítico en la fabricación de transistores usables para la electrónica diaria. Sparks dirigió Sandia National Laboratories.
Vida y educación tempranas
Sparks nació en Pagosa Springs, Colorado y fue un estudiante no licenciado en la Universidad Rice y después hizo su trabajo de PhD en química física en la Universidad de Illinois en Urbana-Champaign.
Carrera
Sparks fue a trabajar en los Laboratorios Bell en donde John Bardeen, Walter Brattain y William Shockley estaban desarrollando el primer transistor. Sparks permaneció en los Laboratorios Bell y trabajó allí para desarrollar el transistor de unión bipolar de microvatio que ayudó a hacer los transistores lo suficientemente prácticos para el uso común. Más adelante, Sparks dejó los Laboratorios Bell para convertirse en el director de Sandia National Laboratories.
Véase también
Transistor
Transistor de unión bipolar
Historia del transistor
Referencias
Fallecidos en Fullerton (California)
Científicos de Estados Unidos
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,001
|
Resultados do Grande Prêmio da Itália de Fórmula 1 realizado em Monza em 11 de setembro de 1994. Décima segunda etapa da temporada, teve como vencedor o britânico Damon Hill, da Williams-Renault.
Classificação da prova
Tabela do campeonato após a corrida
Classificação do mundial de pilotos
Classificação do mundial de construtores
Nota: Somente as primeiras cinco posições estão listadas.
Itália
1994
Desporto na Itália em 1994
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,640
|
FTPEditor FTPEditor is an FTP client combined with a multi-document editor that allows you to edit remote files as if they were on your local hard drive.FTPEditor opens text files directly from your FTP server, lets you edit them and saves the changed files again. You may open multiple files from different directories and FTPEditor will always save them into their correct place. ..
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,418
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{
"redpajama_set_name": "RedPajamaC4"
}
| 6,213
|
We can help students of all levels improve their skills in English Language Arts. Whether your student needs help improving their reading skills, or is looking to improve their AP English exam score, our experienced tutors are here to help.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,677
|
Documentation
=============
[Getting started](getting-started.md)
-------------------------------------
Get started running your project.
[Configuration](configuration.md)
---------------------------------
The configuration file is a single YAML file that dictates Vagrant box settings
as well as chef recipe attributes. For additional information about available
settings and an indepth description of each see [configuration.md](configuration.md).
[Cookbooks](cookbooks.md)
-------------------------
For a rundown of all dStack specific cookbooks see the [cookbooks.md](cookbooks.md)
file. This will give an overview of what is included in dStack and links to
specific documentation for each of them.
[Makefile](makefile.md)
-------------------------
The make file used to accomplish development and installation tasks.
[Packer](../packer/README.md)
-------------------------
Creating new boxes with packer.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,970
|
Q: How do I access Apache error logs via the Terminal in Mac OS X 10.6.8? I am having trouble with serving up my rails app on a remote computer after upgrading from Rails 3.0 to 3.1. So I want to take a look at what exactly is going wrong when Apache attempts to start up my app and it fails. How can I find the errors that Apache (or maybe Passenger?) is throwing via the Terminal?
A: in terminal, have you tried
cat /var/log/apache2/error_log
A: only to complemente the others answers, I'm using the version 10.9.3 and to me, the address is:
/private/var/log/apache2
A: Try:
cat /private/var/log/apache2/error_log
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 315
|
Rajasthan Forest Department invites online application for the post of Drivers and Surveyors in various forests departments offices in the state. Department of Forest under Government of Rajasthan released the notification for the recruitment of Drivers and Surveyors.
Pay Scale: Rs 5200-20200/- (P.B-1) Grade pay 2400/- . Monthly Rs 7900/- will be paid during 2 years probation period.
Candidates must pass 8th class with Light & Heavy Motor Vehicle Driving License for Driver Post, Secondary with ITI Certificate in Civil Survey or Diploma in Civil Engineering from a recognised institution for Surveyor Post.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,020
|
Q: Modifying java array within a JSP input form I am trying to modify a java array within an <input> tag in JSP
String[] userProperties = {"one", "two", "three"};
String[] propValues = new String[userProperties.length];
.
.
.
<% for(int i = 0; i < userProperties.length; i++) { %>
<tr>
<td>
<input type="text" size="30" maxlength="150" name="<%=propValues[i]%>" value="somevalue">
</td>
</tr>
<% } %>
.
.
.
I would like for there to be 3 input forms in this example, and for each form, the value entered by the user would be bound to the appropriate position in the propValues array once the submit button is clicked. I am modifying this based on older code that had name set to a local Java variable, and it was successful in being able to modify that variable. Is this not possible with arrays in JSP? I am aware that JSTL has a <c:forEach> tag that makes this simpler, but since I am working with a single file in a fairly aged codebase I want to keep the libraries it has access to consistent. Is this possible using <%> Java code blocks?
A: //length will 3 but initialize with default value null.
String[] propValues = new String[userProperties.length];
what you are trying to do is you are initializing variable userProperties and try to access array Variable propValues which actually contains null.
name="<%=propValues[i]%>" // set your name as a null.
So, all your three input have same name as a null. better to try this code
<input type="text" size="30" maxlength="150" name="<%=userProperties[i]%>" value="somevalue">
instead of
<input type="text" size="30" maxlength="150" name="<%=propValues[i]%>" value="somevalue">
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,843
|
Q: Не выполняется update-запрос Laravel На страничку выгружаются данные из базы.У каждой строки есть своя кнопка, которая отвечает за отображение данных на главной странице (1 - отобразить, 0 - скрыть).
@if($product->visible == 1)
<form action="{{ route('visibleFalse', $product->id) }}" method="post">
{{ csrf_field() }}
{{ method_field('patch') }}
<button type="submit" class="btn btn-warning btn-xs" title="Скрыть"><i class="fa fa-eye" aria-hidden="true"></i></button>
</form>
@elseif($product->visible == 0)
<form action="{{ route('visibleTrue', $product->id) }}" method="post">
{{ csrf_field() }}
{{ method_field('patch') }}
<button type="submit" class="btn btn-default btn-xs" title="Показать"><i class="fa fa-eye-slash" aria-hidden="true"></i></button>
</form>
@endif
Соответственно, работает только кнопка с title="Скрыть". Почему не работает вторая - не могу понять..Вот код контроллера и роутов.
Контроллер:
public function visibleTrue(Product $product)
{
$product->update(request(['visible', 1]));
return redirect('/adminpanel/products');
}
public function visibleFalse(Product $product)
{
$product->update(request(['visible', 0]));
return redirect('/adminpanel/products');
}
Роуты:
Route::patch('/adminpanel/products/false/{product}', 'Admin\ProductController@visibleFalse')->name('visibleFalse');
Route::patch('/adminpanel/products/true/{product}', 'Admin\ProductController@visibleTrue')->name('visibleTrue');
Помогите пожалуйста разобраться в данной проблеме. Всем заранее большое спасибо!
A: Ответ найден через познание)
public function visibleTrue(Product $product)
{
$product->update(['visible' => 1]);
return redirect('/adminpanel/products');
}
public function visibleFalse(Product $product)
{
$product->update(['visible' => 0]);
return redirect('/adminpanel/products');
}
Моя ошибка была в том, что я пытался установить разные значения по дефолту в request(), а надо было в массив параметров для update().
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,146
|
Passionels Tagebuch è un cortometraggio muto del 1914 diretto da Louis Ralph. Fu l'esordio cinematografico di Emil Jannings e il secondo film della carriera di Adele Sandrock, una famosa attrice teatrale.
Trama
Note
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,262
|
Schinhorn – szczyt w Alpach Berneńskich, części Alp Zachodnich. Leży w Szwajcarii w kantonie Valais. Należy do głównego łańcucha Alp Berneńskich. Można go zdobyć ze schroniska Anenhütte (1734 m) lub Hollandiahütte (3238 m).
Bibliografia
Schinhorn
Szczyty Alp Berneńskich
Szczyty Szwajcarii
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,518
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\section{Materials and methods}
\subsection{Preparation of DNA molecules}
A 105-bp-long DNA molecule was extracted from yeast genomic DNA by polymerase chain reaction (PCR) to serve as a control DNA template without any structural defect. To probe the effect of a permanent defect, we planned to introduce a DNA mismatch to the control molecule by mixing it with its mutant followed by a strand-exchange reaction. To do so, we additionally prepared a set of mutated DNA molecules that differ from the control only in a certain location in which we put a mutation of size equal to 1bp, 3bp, or 5bp. To make such molecules, first, the mutated templates of the control DNA were synthesized from Eurofins Genomics (EXTREMer oligos) and duplexed via PCR. Each of the duplexed products was then incorporated into a pJET1.2$\backslash$blunt vector (ThermoFisher) and cloned into DH5\textrm{$\alpha$} \textit{Escherichia coli} cells. Finally, the cloned fragments of DNA were extracted via colony PCR from the cells and were sequenced to ensure the correct mutation was made at the desired location.
To modify these molecules to carry a FRET pair (i.e. Cy3 and Cy5), biotin, and single-stranded sticky ends, we followed our standard preparation protocol \cite{Le2014jove}, which involves a series of PCR and strand exchange reactions that can be found in elsewhere. For introducing a DNA mismatch in the final construct, we mixed the Cy3-labled control molecule with one of the Cy5-labeled mutated molecules with a ratio of 4:1 in the strand-exchange reaction.
The final DNA construct generated by this protocol carries a 5$^\prime$ protruding sticky end on each end and makes a hairband loop upon end-annealing as shown in Figure 2(a) of the main text. We also made hairpin loops by having sticky ends on the same DNA strand (Figure 3(a) of the main text). A complete list of all DNA sequences can be found in Tables \ref{supp-tab1} and \ref{supp-tab2} below.
\subsection{single-molecule FRET looing and unlooping assay}
We followed our previous single-molecule FRET assay that employs the sudden salt-exchange protocol \cite{Le2014,Jeo}. For cyclization, DNA molecules were deposited on a passivated surface of a flow-cell and were incubated at a low salt (10 mM [NaCl) imaging buffer containing the PCD-PCA oxygen scavenging system \cite{Aitken2008} for 10 minutes. We then injected a high salt (1 M [NaCl]) imaging buffer into the flow-cell to promote sticky ends to capture the loop configuration. Decyclization measurements were done similarly, except that the NaCl concentration was changed from 2 M to 75 mM. The immobilized molecules were excited by a 532-nm laser continuously through an objective-type TIR microscope from the beginning of the buffer exchange. The time trajectories of FRET signals (Figure 1(b) of the main text) from the molecules were recorded by an EMCCD camera (DU-897ECS0-\# BV, Andor) at a rate of 100 ms per frame for the mismatch-free molecules and 50 ms per frame for the molecules with a mismatch.
\subsection{Minicircle simulations}
The Monte Carlo simulation of a minicircle was implemented as previously described \cite{zheng2009theoretical,Le2014}. A set of 105 connected nodes was used to create a coarse-grained representation of a DNA minicircle of 105 bp. The bending energy at each node was described by the kinkable worm-like chain model \cite{zheng2009theoretical} with the parameters of b = 0.3 and h = 12 following the same notation used in Ref. \cite{Vologodskii2013a}. We performed the simulation with and without a flexible defect of zero bending energy placed at a fixed location. For the case of no flexible spot, we first initialized the simulation without allowing the kink formation. Once the kink-free simulation was equilibrated, we allowed spontaneous kinks to appear. To construct the probability density of kink positions, we ran the simulation and stop at the first appearance of a kink. We then recorded the position of this kink and equilibrated back to the kink-free state. This procedure was repeated until we collected a distribution of 1000 kink positions. The same procedure was repeated in the presence of the hyperflexible spot to predict the effect of a flexible spot on the probability distribution of kink.
\clearpage
\onecolumngrid
\begin{longtable*}{| P{0.14\textwidth}| p{0.75\textwidth}| }
\caption{DNA sequences of hairband molecules.}\label{supp-tab1} \\
\hline \hline
\endfirsthead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\endlastfoot
No mismatch & 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCAACGAGGTCGCACACGCCCCACACCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGTGTGGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
1bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCAACGAGGTCGCACACGCCCCAGACCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGTCTGGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
3bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCAACGAGGTCGCACACGCCCCGGGCCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGCCCGGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
5bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCAACGAGGTCGCACACGCCCGCGCGCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGTTGCTCCAGCGTGTGCGGGCGCGCGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
3bp-mismatch (10 bp off-center)& 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCAACGAGGTCGTGGACGCCCCACACCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGTTGCTCCAGCACCTGCGGGGTGTGGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
3bp-mismatch (20 bp off-center) & 5$^\prime-$\underline{TGAATTTAC}\seqsplit{G}\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCGCGATCGCCATGGCGGTGAGGTCGCACACGCCCCACACCCAGACCTCCCTGCGAGCGGGCATGGGTACAATCATTCGAGCTCGTTGTAG}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGCGCTAGCGGTACCGCCACTCCAGCGTGTGCGGGGTGTGGGTCTGGAGGGACGCTCGCCCGTACCCATGTTAGTAAGCTCGAGCAACA}\textcolor{green}{\textbf{T}}C\underline{ACTTAAATG}-5$^\prime$
\\ \hline
\end{longtable*}
\begin{longtable*}{| P{0.14\textwidth}| p{0.75\textwidth}| }
\caption{DNA sequences of hairpin molecules.}\label{supp-tab2} \\
\hline \hline
\endfirsthead
\hline \multicolumn{2}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\endlastfoot
No mismatch & 5$^\prime-$\underline{TGAATTTACG}(CT)G\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCACATCGCCATGGCAACGAGGTCGCACACGCCCCACACCCAGACCTCCCTGCGAGCGGGCATGGGTTGCATGTCAGCTATGGATCCATTCGTAAATTCA}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGTGTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGTGTGGGTCTGGAGGGACGCTCGCCCGTACCCAACGTACAGT}(CG)\underline{ATACCTAGGT}-5$^\prime$[\textcolor{green}{Cy3}]
\\ \hline
1bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTACG}(CT)G\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCACATCGCCATGGCAACGAGGTCGCACACGCCCCAGACCCAGACCTCCCTGCGAGCGGGCATGGGTTGCATGTCAGCTATGGATCCATTCGTAAATTCA}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGTGTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGTCTGGGTCTGGAGGGACGCTCGCCCGTACCCAACGTACAGT}(CG)\underline{ATACCTAGGT}-5$^\prime$[\textcolor{green}{Cy3}]
\\ \hline
3bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTACG}(CT)G\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCACATCGCCATGGCAACGAGGTCGCACACGCCCCGGGCCCAGACCTCCCTGCGAGCGGGCATGGGTTGCATGTCAGCTATGGATCCATTCGTAAATTCA}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGTGTAGCGGTACCGTTGCTCCAGCGTGTGCGGGGCCCGGGTCTGGAGGGACGCTCGCCCGTACCCAACGTACAGT}(CG)\underline{ATACCTAGGT}-5$^\prime$[\textcolor{green}{Cy3}]
\\ \hline
5bp-mismatch (central) & 5$^\prime-$\underline{TGAATTTACG}(CT)G\textcolor{red}{\textbf{T}}\seqsplit{GCCAGCAACAGA[T]AGCCACATCGCCATGGCAACGAGGTCGCACACGCCCGCGCGCCAGACCTCCCTGCGAGCGGGCATGGGTTGCATGTCAGCTATGGATCCATTCGTAAATTCA}-3$^\prime$ \newline
3$^\prime-$\seqsplit{CACGGTCGTTGTCTATCGGTGTAGCGGTACCGTTGCTCCAGCGTGTGCGGGCGCGCGGTCTGGAGGGACGCTCGCCCGTACCCAACGTACAGT}(CG)\underline{ATACCTAGGT}-5$^\prime$[\textcolor{green}{Cy3}]
\\ \hline
\end{longtable*}
\begin{minipage}{0.95\textwidth}
Both top (5$^\prime$ to 3$^\prime$) and bottom (3$^\prime$ to 5$^\prime$) sequences are shown. The underlined sequences represent sticky ends. A Cy5 fluorophore is internally attached at the thymine base colored in red. A Cy3 fluorophore is either at the green thymine base or the 5$^\prime$ end of the bottom strand. A biotin molecule is linked to the thymine base shown as [T]. Hairpin molecules includes a 2-nt gap (indicated by sequences in parentheses) near each end of the top strand before sticky ends.
\end{minipage}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,489
|
TO THE MEMORY
OF
JENNY MURRAY, 1987–2007
* * *
## CONTENTS
* * *
1. Cover
2. Title Page
3. Dedication
4. INTRODUCTION
5. BREAKFAST & BRUNCH
1. Poached Eggs Three Ways
1. Prosciutto, Preserved Lemon Cream and Parsley
2. Black Pudding and Mushroom à la Crème
3. Seared Spring Onions and Spring Onion Vinaigrette
2. Kippers with Kombu Butter and Toast
3. Macroom Oatmeal with Milk and Salt
4. Rye Spring Onion Pancakes with Lime and Chilli Butter
5. Buckwheat Pancakes
6. Blueberry Buttermilk Drop Scones
6. BREAD
1. Irish Stout and Treacle Loaf
2. Seeded Dillisk Loaf
3. Plain Malted Milk Loaf
4. Thyme and Kombu Focaccia
5. Macroom Brown Soda Bread
6. Wholemeal Pitta Bread
7. Irish Cider and Honey Loaf
8. Rosemary, Apple and Buttermilk Loaf
7. SOUP
1. Parsnip and Fennel Soup
2. Very Green Asparagus Soup
3. Vegetable Broth with Chard and Orzo
4. Chicken Broth with Potato, Spring Onion and Courgette
5. Coconut and Lime Soup with Hake and Coriander
6. Cleansing Nettle Broth
8. SALAD
1. Baked Feta, Roasted Lemon and Trofie Salad with Sesame Toasts
2. Crushed New Potatoes with Pink Peppercorns, Capers and Parsley
3. Smoked Mackerel with Tabasco and Lime Potato Salad
4. Balsamic Cucumber with Kalamata Olives
5. Yellow and Green Courgetti with Toasted Almonds and Feta
6. Chicory Salad with Pomegranate, Grilled Halloumi and Lime Dressing
7. Orzo with Preserved Lemon and Thyme Cream
8. Citrus Salad with Honeyed Buttermilk
9. Hedgerow Berry Salad with Lime Syrup
10. Fresh Figs with Black Pepper and Honey
9. VEGETABLE & PASTA MAINS
1. Cavolo Nero, Feta and Butternut Squash Filo Pie
2. Rustic Rye Galette with Leeks, Fennel, Goat's Cheese and Toasted Pine Nuts
3. Homemade Gnocchi with Buffalo Mozzarella, Pickled Walnuts and Green Herb and Lemon Dressing
4. Baked Potatoes with Ricotta and Green Herb and Lemon Dressing
5. Ruby Chard Korma
6. Chard and Ricotta Lasagne
7. Breda's Cauliflower Cheese
10. FISH
1. Pan-fried Hake with Candy Beetroot and Orange Salad
2. Slow-roasted Salmon with Blood Orange, Lemon, Fennel and Parsley
3. Breda's Fish Pie
4. Barbecued Mackerel
5. Salmon, Samphire, Broccoli and Cream Pappardelle
6. Salmon with Baby Potatoes, Capers and Garlic, Lemon and Parsley Butter
7. Smoked Rainbow Trout with Fennel, Goat's Cheese, Pink Peppercorns and Dill
8. Asparagus and Smoked Rainbow Trout with a Herb Sauce
11. MEAT
1. Roast Chicken with Harissa Butter
2. Penne Pasta with Ham, Cabbage, Wild Garlic Pesto and Pickled Walnut
3. Ham in Juniper and Apple Juice
4. Steak Sandwich with Fried Onions, Cashel Blue Cheese and Mushroom Ketchup
5. Lamb Steak Sandwich with Garlic, Lemon and Parsley Butter and Wicklow Bán Brie
6. Turkey Burgers with Chanterelles and Gruyère
7. Ferryhouse Cottage Pie
8. Fried Cabbage and Ham Sandwich
9. Pappardelle with Speck, Ricotta, Beetroot Tops and Walnuts
10. Sausage and Thyme Stuffing
12. SIDE DISHES
1. Gags's Potato Gratin
2. Colcannon with Curly Kale
3. Ginger-braised Leeks
4. Mushroom à la Crème
5. Buttery Purple Carrots
6. Mustard Parsnip Mash
7. Potato and Chive Cake
8. Buttered Cabbage with Caraway or Fennel Seeds
13. PRESERVES, DIPS & CONDIMENTS
1. Kale, Cashew and Wakame Pesto
2. Beetroot and Seaweed Hummus
3. Harissa
4. Spicy Pickled Carrot
5. Spicy Cucumber Pickle
6. Tahini Dressing
7. Flavoured Butters
1. Burnt Onion Butter
2. Orange and Cinnamon Butter
3. Kombu Butter
4. Honey and Redcurrant Butter
5. Garlic, Lemon and Parsley Butter
8. Mushroom Ketchup
9. Walnut and Feta Dip
10. Green Herb and Lemon Dressing
11. Spring Onion Vinaigrette
12. Lime and Coriander Mayo
13. Bretagne Sauce
14. Fried Apple and Sage
14. SWEET THINGS
1. Pear and Frangipane Tart
2. Fresh Blueberry Pie with a Lemon Curd Cream
3. Flourless Dark Chocolate and Sea Salt Cake
4. Lemon and Lavender Cake
5. Soaked Orange Cake
6. Wholemeal Spelt Carrot Loaf with Orange Mascarpone Icing
7. Pecan and White Chocolate Banana Loaf
8. Glamnilla Shortbread Biscuits
9. Rock Sugar Biscuits
10. Molasses Biscuits
11. Lemon and Rosemary Biscuits
12. Orange Shortbread with Salted Dark Chocolate
13. Honey Biscuits
14. Currabinny Brown Apple Tray Bake
15. Patrick O'Hara's Elderflower Cordial
15. SETTING THE SCENE
16. ACKNOWLEDGEMENTS
17. FOLLOW PENGUIN
18. COPYRIGHT PAGE
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1. Cover
2. Table of Contents
3. Begin Reading
* * *
## INTRODUCTION
* * *
'All sorrows are less with bread'
_Miguel de Cervantes_
Welcome to _The Currabinny Cookbook_!
For us, food is life's greatest pleasure, so it has been a labour of love to bring together recipes to answer that simplest of questions, _What do I want to eat?_
This question will arise in all kinds of environments and moods, whether you're living in the city or the countryside, getting home late from a long day's work and using mainly store-cupboard ingredients to make something simple, or – with the luxury of time – shopping for fresh produce and preparing ingredients for a more complicated recipe. Whatever your circumstances, we believe your food should taste great and be a source of enjoyment and comfort.
But first, who are we?
Well, we're not chefs or any kind of experts. We're just devoted foodies and home cooks. Both of us grew up in homes – William in Currabinny on Cork Harbour, James in Rathfarnham in South Dublin – where delicious, freshly prepared meals and baking were par for the course (thanks, Breda and Gags!).
After completing his Fine Art degree, William decided to explore his passion for cooking by doing the three-month course at the legendary Ballymaloe Cookery School. At the same time, James was working in public relations and two of the clients he worked with were iconic Irish brands, Barry's Tea and Kerrygold Butter. Having to think about what made these brands great – quality and tradition – sowed a seed in James that came to fruition when we got together.
Our shared obsession with great food and cooking made us realize that it was more than just an interest – it was something we wanted to make our life's work.
After spending a summer in Currabinny, talking and dreaming about selling our food at markets, catering, doing pop-up food events, one day owning a café, writing a cookbook, we started brainstorming names for our baby. And of course the perfect name was staring us in the face...
Currabinny is a peninsula that projects into Cork Harbour between the port of Ringaskiddy and the fishing and sailing village of Crosshaven. Currabinny Woods rise steeply behind houses that are strung along the water's edge, looking across to Crosshaven. It's a quiet and picturesque place, dominated by trees, rocks and water.
In Currabinny there is a huge interest in food. While there is a respect for tradition, and families have their own recipes that have been handed down from one generation to the next, there is also an appetite for experimentation and innovation. People are in the habit of foraging for ingredients from both land and sea. Recipes, techniques and new discoveries are generously shared between neighbours. (Indeed, here local botanical artist Patrick O'Hara shares his wonderful elderflower cordial recipe.)
Fishing and tending a small vegetable patch are both second nature to William. And inspired by his mother, he grew up experimenting with and inventing ways to use the produce that came from outside the kitchen window.
So Currabinny is not just a place we love, but the food culture there embodies everything we admire and value as an approach to cooking and eating – drawing on the best of local produce and old traditions while being willing to try new things.
For us, cooking is about curiosity, community and, above all else, taste. We tend to toggle between countryside and city, taking things we've seen and learned in Dublin to Cork to tease out and develop – and vice versa.
* * *
CURRABINNY IS NOT JUST A PLACE WE LOVE, BUT THE FOOD CULTURE THERE EMBODIES EVERYTHING WE ADMIRE
* * *
When we're in the city, nothing pleases us more than exploring cafés and restaurants, seeing classics done brilliantly or reinvented, and also getting a feel for emerging flavours and innovative methods of cooking.
And when we're in Currabinny, we will while away a day sailing around Cork Harbour, pulling into new places and trying out whatever is on offer. You'll find us picking samphire along the shore, buying vegetables at a farmers' market and some fish or meat from a fishmonger or butcher, and then discussing what we're going to make for dinner all the way home.
We love the fact that when we're in Cork we have uninterrupted time to develop our food ideas and come up with recipes that we can refine further in Dublin, where things are a little faster paced for us. This call-and-response pattern is something that has arisen naturally out of our lifestyle, but it has been a crucial influence on the evolution of Currabinny as a company and has shaped our food vision.
* * *
WE LIKE TO BE SPONTANEOUS AND FRESH AND WE ALSO LIKE TO MAKE OLD FAVOURITES AGAIN AND AGAIN WITH CONTEMPORARY, BOLD TWISTS
* * *
In our cooking we like to be spontaneous and fresh and we also like to make old favourites again and again with contemporary, bold twists. We have made all kinds of food; some dishes have been resounding successes and others not so much.
By taking the time to experiment, to be curious and to take risks, to do it the wrong way and then correct course, adding a little more of this or less of that, we have developed a repertoire of recipes that we love and that we know that other people love too. For us, this is the essence of cooking and enjoying food: trying out things, refining recipes until they are truly delicious, sharing.
As we said, we're not experts. We don't so much have rules as guiding principles. We believe in making an effort to source ingredients locally and to support small producers. And we like to eat seasonably and sustainably.
We have no interest in advising people on their diets and what to eat. Of course, we strongly believe in a varied diet and a healthy attitude to eating, but we don't demonize any ingredients, especially when they might be simply misunderstood.
We stand for balance, but not sacrifice. Food is about pleasure – eat well and not just to survive. There are no villains in these pages, only heroes.
* * *
WE DON'T DEMONIZE ANY INGREDIENTS
* * *
Some ingredients you will see popping up repeatedly because they are the bedrock of our cooking. For instance, butter. Butter got quite a hammering in the past due to the fierce marketing of manufactured 'spreads'. But butter is back – and we're delighted. We strongly believe in butter, and why wouldn't we when Ireland produces the best and richest butter in the world?
Use butter sensibly, of course, but remind yourself that it is completely natural and full of good fats and nutrients. (Where we include butter we mean regular salted butter; we sometimes reduce the amount of salt in a recipe to compensate. And of course all our dairy ingredients are full fat!)
Something that might seem a world away from butter and more conventional ingredients is seaweed, something we also love to cook with. But seaweed could be as ordinary an ingredient in the Irish kitchen as it is in the Japanese. Not only that, but Irish seaweed is softer, more palatable and less mass produced than other versions. It is also incredibly good for you and a natural sustainable product. You won't be surprised to hear that we have a recipe for a seaweed butter!
We cover all the bases – breakfast and brunch, breads, soups, salads, meat, fish, vegetable and pasta-based mains, side dishes, sweet things.fn1 Something we think is a little unique is our chapter on preserves, condiments and dips. We have tons of recipes for these and have shared a selection of them here. They are all delicious in their own right for simple snacking, but we also use them lavishly in our cooking to enhance other dishes, so you'll see them cross-referenced through the book.
* * *
WE HOPE THE RECIPES IN THESE PAGES WILL NOURISH, COMFORT AND DELIGHT YOU AS MUCH AS THEY HAVE US!
* * *
Finally, anyone who knows us – or is aware of us via social media – knows how much we like to create a sense of occasion. So with that in mind we have shared our thoughts on how to set the scene for a truly special dinner.
Please leaf through, get hungry, get excited, experiment. We hope the recipes in these pages will nourish, comfort and delight you as much as they have us!
## Recipe List
1. Poached Eggs Three Ways
* Prosciutto, Preserved Lemon Cream and Parsley
* Black Pudding and Mushroom à la Crème
* Seared Spring Onions and Spring Onion Vinaigrette
2. Kippers with Kombu Butter and Toast
3. Macroom Oatmeal with Milk and Salt
4. Rye Spring Onion Pancakes with Lime and Chilli Butter
5. Buckwheat Pancakes
6. Blueberry Buttermilk Drop Scones
## Poached Eggs Three Ways
Waking up on a cold morning to find two glass bottles of milk and half a dozen fresh eggs delivered to the door is something that we think should be revived – for the good of the nation. Have one of these poached egg recipes with lots of buttery toast for breakfast and you'll be set up for the day!
##### SERVES 1
* 2 very fresh organic eggs
* white vinegar
* sea salt and freshly ground black pepper
##### METHOD
Make sure you use really good-quality eggs. Bring a medium-sized saucepan of water to a simmer and add a few dashes of white vinegar.
We find it best to crack the eggs first into a small cup or bowl. Create a whirlpool in the water with a spoon, then slowly tip in the eggs one at a time. Cook for around 3 minutes.
Remove the eggs on to kitchen paper using a slotted spoon, then transfer to a warmed plate. Cut away any scraggly bits of egg, and season lightly with salt and pepper.
They are perfect just like this, with some sourdough toast, but here are three of our favourite ways to enhance them.
## Prosciutto, Preserved Lemon Cream and Parsley
* 4 thin slices of prosciutto
* around 150ml preserved lemon and thyme cream (see here)
* 1 tablespoon chopped fresh flat-leaf parsley leaves
Get a warm plate ready. Place the prosciutto in a hot frying pan and cook for a few seconds until it crisps up.
In a small saucepan, gently heat the preserved lemon and thyme cream and drizzle it over the warm plate using a large spoon. Place the crispy prosciutto in the middle, your poached eggs on top, and sprinkle everything with parsley.
Or, again, you could serve the lot on buttery toast – eggs, prosciutto, and then the lemon cream as a dressing.
## Black Pudding and Mushroom à la Crème
* 2 large slices of good-quality black pudding, such as O'Herlihy's or Rosscarbery
* mushroom à la crème (see here – use the full quantity from that recipe)
Get a warm plate ready. Place the black pudding slices under a hot grill for around 10 minutes, turning once. Place them on the warm plate, put your poached eggs on top and pour the mushroom à la crème over everything.
This is great in the winter when it is cold outside and you need something to stick to your bones!
## Seared Spring Onions and Spring Onion Vinaigrette
* 4–6 spring onions
* olive oil
* 1 slice of toasted sourdough, buttered
* spring onion vinaigrette (see here)
* a few fresh flat-leaf parsley leaves, chopped
Trim and cut the spring onions lengthways. Put a griddle or grill pan on a high heat, place the spring onions into the pan and drizzle with a little olive oil. Cook until wilted and lightly charred.
Drape over buttered sourdough, place your poached eggs on top and drizzle generously with the zingy spring onion vinaigrette, finishing with some parsley. By far our favourite way to have poached eggs.
Poached Eggs Three Ways
## Kippers with Kombu Butter and Toast
Most mornings William's grandmother made smoked kippers, fried with a little butter, for his grandad. So this recipe is in memory of them. We love them with a nice bit of cider and honey loaf, toasted and spread generously with some salty, smoky kombu butter.
##### SERVES 1
* 1 tablespoon olive or rapeseed oil
* 2 smoked kippers
* 2 thick slices of tasty bread, such as Irish cider and honey loaf (see here)
* 100g kombu butter (see here)
* ½ a lemon, for squeezing
* sea salt and freshly ground black pepper
* 1 tablespoon chopped fresh flat-leaf parsley leaves
##### METHOD
In a large frying pan, heat the oil over a medium flame. Add the kippers and cook for 2 to 5 minutes per side.
Toast the bread until golden. Smother the slices with the kombu butter and arrange on a plate, placing the kippers on top. Squeeze some lemon juice over the kippers and season with salt, pepper and the chopped parsley.
Kippers with Kombu Butter and Toast
## Macroom Oatmeal with Milk and Salt
You will see Macroom Oatmeal mentioned more than once in this book – and not just because it's from Co. Cork! It's because it's the king of oatmeal. It's wonderful on its own with a pinch of salt or with some fruit and honey. Use whatever fruit is good, ripe and in season. In winter, homemade jam is a perfect addition to porridge.
##### SERVES 1–2
* 1 teacup of water
* 1½ teacups of milk
* a good pinch of sea salt
* ½ teacup of Macroom Oatmeal
* soft brown sugar (optional)
##### METHOD
Put all the ingredients except the sugar into a saucepan on a medium heat and stir continuously. The oats will absorb the liquid very fast, so be careful not to let the mixture stick to the bottom.
After about 5 or 6 minutes, when the oatmeal has softened and is at the consistency of porridge, transfer to a bowl and sprinkle with a little bit of brown sugar if desired.
##### WITH SATURN PEACHES, POMEGRANATE AND RAW HONEY
Follow the same recipe as above, but use just a tiny pinch of sea salt. When you have transferred the oatmeal to your bowl, arrange some slices of good ripe Saturn peaches (also called doughnut peaches) in the middle, sprinkle a few pomegranate seeds over and drizzle generously with good-quality raw honey. Orange blossom or wildflower varieties work really well.
Macroom Oatmeal with Milk and Salt
## Rye Spring Onion Pancakes with Lime and Chilli Butter
These pancakes are extremely healthy but still packed with flavour. The rye flour really complements the spinach and spring onion, and is full of fibre and lower in gluten than regular flour. Especially good when paired with the zingy lime and chilli butter!
##### SERVES 2
##### _FOR THE LIME AND CHILLI BUTTER:_
* 1 small clove of garlic
* a handful of fresh coriander leaves
* 2½cm piece of fresh red chilli
* juice of 1 lime
* a pinch of sea salt
* 100g butter, softened
##### _FOR THE PANCAKES:_
* 8 spring onions
* a handful of fresh chives
* 200g fresh baby spinach
* 100g rye flour
* 2 tablespoons baking powder
* 2 medium organic eggs
* 50g butter, melted
* 1 teaspoon ground coriander
* 75ml milk
* sea salt and freshly ground black pepper
* 1 tablespoon rapeseed oil, for frying
##### METHOD
First prepare the lime and chilli butter. Peel and crush the garlic, chop the coriander and finely dice the chilli. Put these into a bowl with the lime juice, salt and butter, and mash together with a fork or wooden spoon. When well combined, roll the butter into a log shape, wrap in baking parchment and cool in the fridge.
Finely slice the spring onions and chop the chives. Cook the baby spinach with a splash of water in a small pan until completely wilted. Drain and squeeze excess water out by pressing into wads of kitchen paper. Chop the spinach and put into a large bowl with the rest of the pancake ingredients, apart from the rapeseed oil. Mix well with a wooden spoon.
Put the oil into a frying pan on a medium-high heat. When hot, drop ladlefuls of the mixture into the pan – you should be able to cook 3 pancakes at a time. Cook each side for around 3 minutes until golden. Serve each pancake with a round of lime and chilli butter (cut from the roll and parchment removed) melting on top.
Rye Spring Onion Pancakes with Lime and Chilli Butter
## Buckwheat Pancakes
Our neighbour Naomi gave us this recipe, which comes all the way from New Zealand. The combination of nutty buckwheat – full of vitamins and protein – along with the probiotics in the yoghurt makes for an amazing start to the day.
##### SERVES 2
* 100g buckwheat flour
* 1 tablespoon caster sugar
* a pinch of sea salt
* 250ml milk
* 1 tablespoon natural yoghurt
* 1 medium organic egg
* 125ml water
* butter and rapeseed oil, for frying
##### METHOD
Whisk the flour, sugar, salt, milk, yoghurt and egg together in a large mixing jug. Slowly whisk in all the water, until the batter is smooth and runny.
Put a saucepan of water on to simmer and get a plate that will fit comfortably over the top.
Place a large frying pan over a high heat, melt a tablespoon of butter and a little rapeseed oil in it and lower the heat to medium. Pour in the batter to the size you want your pancake to be and cook until the underside is golden, then flip it over and cook the other side.
The whole process should take only around 2 minutes, a minute on each side. Repeat with the rest of the batter.
As you cook them, stack your pancakes on the plate placed over the simmering water. Serve with lemon juice, sugar, golden or maple syrup, Nutella, jam or butter.
## Blueberry Buttermilk Drop Scones
'Go on, invite them down.' Ten minutes later the sliding door in the kitchen in Currabinny would be pulled back and four or five kids would march in, chattering away about what time the tide would be in for swimming. And soon the warm smell of the drop scones would fill the kitchen. It's a ritual that continues to this day when we're visiting Currabinny!
##### SERVES 4
* 75g cream flour
* 500ml buttermilk
* 2 teaspoons caster sugar
* 1 teaspoon bicarbonate of soda
* 1 medium organic egg
* 1 tablespoon butter, melted
* 80g fresh blueberries
* milk, for thinning the batter (optional)
* butter and rapeseed oil, for frying
##### METHOD
In a large mixing bowl, whisk the flour and buttermilk together with the sugar, bicarbonate of soda, egg and melted butter. Stir in the blueberries and leave the mixture to stand for at least 30 minutes.
The mixture should be the consistency of thick syrup when you come to cook the scones. If the mixture is too thick, add a little milk to thin it out again. Put a saucepan of water on to simmer and get a plate that will fit comfortably over the top.
Place a frying pan over a high heat, melt some butter and a little rapeseed oil in it and lower the heat to medium. Using a tablespoon as your measure, drop a spoonful of the batter on to the pan – there might be room to cook 2 or 3 at a time, depending on the size of your pan. Let the drop scones cook for 1 to 2 minutes on each side until golden brown all over.
Pile batches of your scones on to the plate placed over the simmering water until all of them are done. Drop scones should be served when everyone is sitting around the table, with freshly squeezed orange juice and a choice of black coffee or Barry's Tea.
Maple or golden syrup, lemon juice and even a little butter are all acceptable options with which to adorn your drop scones.
## Recipe List
1. Irish Stout and Treacle Loaf
2. Seeded Dillisk Loaf
3. Plain Malted Milk Loaf
4. Thyme and Kombu Focaccia
5. Macroom Brown Soda Bread
6. Wholemeal Pitta Bread
7. Irish Cider and Honey Loaf
8. Rosemary, Apple and Buttermilk Loaf
## Irish Stout and Treacle Loaf
Is there anything better than rich, salty Irish butter spread generously on a slice of nutty homemade bread? We think it's an irresistible combination, and this brown bread – dense and slightly sweet and malty – is the perfect companion to butter, though it's delicious on its own too.
##### MAKES 10 SLICES
* Butter for greasing
* 200g strong white flour
* 375g Macroom Stoneground Wholewheat Flour (extra coarse)
* 1 teaspoon bicarbonate of soda
* 1½ teaspoons sea salt
* 50g rolled oats, plus extra to sprinkle on top
* 125ml Guinness or other Irish stout
* 300ml buttermilk
* 100ml treacle
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Butter and line a 450g loaf tin with baking parchment.
In a large bowl, mix the flours, bicarbonate of soda, salt and oats. Mix the Guinness, buttermilk and treacle in a large jug and pour into the dry ingredients, making sure to scrape the remaining treacle off the jug with a spatula.
Mix well, then scoop the mixture into the prepared loaf tin and top with a sprinkle of oats.
Bake in the oven for 25 minutes, then lower the temperature slightly to 150ºC fan/gas 3 and bake for a further 30 minutes.
Make sure the loaf is cooked through by removing from the tin and tapping the bottom – if it sounds hollow it should be cooked. You could always turn it upside down and put it in the oven for a further 5 minutes just to make sure. When finished, take it out of the tin and cool on a wire rack.
_The stout and treacle loaf is shown on the left in the picture; the picture also shows kombu butter (seehere)_
Irish Stout and Treacle Loaf
## Seeded Dillisk Loaf
The combination of seeds, buttermilk and salty dillisk seaweed makes for an unusual bread, both in flavour and appearance. The veins of purple that run through the bread look quite striking, and it's really delicious and earthy. One of our favourite breads to make and eat.
##### MAKES 8–10 SLICES
* butter, for greasing
* 30g dried dillisk
* 340g wholemeal flour
* 60g sunflower seeds
* 60g pumpkin seeds
* 2 teaspoons bicarbonate of soda
* a pinch of sea salt
* 300ml buttermilk
* 2 medium organic eggs
* 2 tablespoons rapeseed oil
##### METHOD
Preheat the oven to 210ºC fan/gas 8 and butter a 20cm round baking tin.
Finely chop the dillisk. Mix this with the flour, seeds, bicarbonate of soda and salt in a large bowl.
In a large jug, whisk the buttermilk, eggs and rapeseed oil together and pour into the dry ingredients. Using your hand as a claw, stir the mixture in a circular motion until it is well combined.
Pour into the prepared tin and bake in the oven for 40 to 45 minutes. Remove from the tin and pat the bottom – it should sound hollow when it's ready. Cool fully on a wire rack.
_The Seeded Dillisk Loaf is shown at the top in the picturehere._
## Plain Malted Milk Loaf
This is an everyday staple, an easy, versatile bread that's great for toasting and slathering with whatever you fancy. Perfect with soup for a quick lunch or supper. We always keep a loaf or two in the bread bin.
##### MAKES 2 LOAVES
* 250ml milk
* 25g butter, plus extra for greasing
* 2 tablespoons honey
* 4 tablespoons malt extract
* 300g cream flour, plus extra for dusting
* 150g wholemeal flour
* 1 teaspoon sea salt
* 14g dried yeast
##### METHOD
In a saucepan, gently heat the milk, butter, honey and malt extract until warm, whisking lightly to incorporate all the ingredients.
In a large bowl, combine both flours, the salt and the yeast. Pour the wet ingredients from the saucepan into the dry ingredients and use your hands to knead it gently in the bowl into a well-combined dough.
Tip on to a lightly floured surface and sprinkle it with flour. Knead gently for 2 to 3 minutes, adding a little more flour if needed.
Butter two 450g loaf tins and divide the dough between them. Cover with a clean tea towel and leave to rise for 2 hours until the dough reaches the top of the tins.
Preheat the oven to 160ºC fan/gas 4.
Put the loaves into the oven and bake for 40 to 50 minutes. Keep an eye on them – if they start to brown too much on the top, cover them with tinfoil. Test with a skewer – it should come out clean. Remove from the tins and cool on wire racks.
## Thyme and Kombu Focaccia
A classic Italian bread, great for tearing and having with dips or oils. The dried kombu adds a real flavour of the Irish coast, and with the woody aroma of the thyme this is the essence of Currabinny in a bread!
##### SERVES 4
* 1 tablespoon dried yeast
* 180ml warm water
* 2 tablespoons olive oil, plus extra for greasing and drizzling
* 1 teaspoon honey
* 320g Italian Tipo 00 flour, plus extra for dusting
* 2 teaspoons sea salt, plus extra for the top
* 15g dried kombu, finely chopped
* 5–6 sprigs of fresh thyme
##### METHOD
In a large jug dissolve the yeast in the warm water and leave for 5 minutes until it starts to get foamy. Add the olive oil and the honey and whisk gently to combine.
Sift the flour into a large mixing bowl and add the 2 teaspoons of salt, the finely chopped kombu and the leaves from the thyme sprigs. Pour the yeast and honey mix into the flour and mix to form a dough.
Tip the dough on to a lightly floured surface and knead for 5 to 10 minutes until smooth, soft and elastic.
Clean the mixing bowl and rub oil all over the inside. Place the dough back in the bowl and cover with a clean cloth. After 30 minutes, tip the dough on to a lightly floured surface again and stretch it into a rectangle shape, folding it and reshaping it three or four times. The dough should now be a very thick small rectangle shape.
Oil a baking sheet (about 25cm x 35cm) and place the folded dough in the middle, cover with a clean tea towel and leave for another hour – it should double in size in this time.
Once risen, stretch the dough out to cover the baking sheet and sprinkle with extra salt. Cover with a cloth and leave to rest for another 20 minutes before using your fingertips to imprint dents all over the dough.
Drizzle with a little olive oil and leave it to rest for a further 20 minutes. While it's resting, preheat the oven to 180ºC fan/gas 6.
Bake in the oven for 25 minutes until golden and crisp-looking. Serve warm, cut into squares.
Thyme and Kombu Focaccia
## Macroom Brown Soda Bread
Could there be anything more Irish and down-to-earth than a classic soda bread made with wholewheat flour from the legendary Walton's Mill in Macroom, Co. Cork, Ireland's only surviving stone mill? We don't think so!
##### MAKES 8–10 SLICES
* butter, for greasing
* 180g cream flour
* 340g Macroom Stoneground Wholewheat Flour (extra coarse)
* 2 teaspoons bicarbonate of soda
* 1 teaspoon sea salt
* 70g Macroom Oatmeal
* 1 medium organic egg
* 575ml buttermilk
##### METHOD
Preheat the oven to 180ºC fan/gas 6. Butter a 450g loaf tin.
In a large mixing bowl, mix the flours, bicarbonate of soda, salt and oatmeal to combine, then make a well in the centre.
Whisk together the egg and buttermilk in a jug, and pour into the dry mix. Using your hand as a claw, mix the ingredients together in a circular motion until well combined.
Pour the mixture into the loaf tin and bake in the oven for 40 to 50 minutes, until a skewer comes out clean. When you remove the loaf from the tin, make sure to tap the bottom too, listening for that hollow sound just to be sure. Cool on a wire rack.
Macroom Brown Soda Bread
## Wholemeal Pitta Bread
Pitta bread is both delicious and incredibly versatile, and it's also surprisingly easy to make at home. The key is to use finely milled wholewheat flour along with the very fine Tipo 00 flour. This combination prevents the dough from becoming too wet and hard to work with.
##### MAKES 10 PITTAS
* 240ml warm water
* 1 teaspoon honey
* 2 teaspoons dried yeast
* 1 tablespoon olive oil, plus extra for greasing
* 170g wholewheat flour
* 170g Italian Tipo 00 flour, plus extra for dusting
* 1 teaspoon sea salt
##### METHOD
In a large jug, mix the warm water, honey and yeast and leave for 5 minutes until it starts to get foamy.
Add the olive oil and gently whisk to combine. Mix the two flours with the salt in a large bowl. Pour the yeast and honey mixture into the flour and bring together with your hands to form a dough.
Turn out your dough on to a lightly floured surface and knead for 5 to 10 minutes until the dough is smooth.
Clean out your mixing bowl and rub oil lightly all over the inside. Form the dough into a ball and put it into the bowl, then cover with a clean tea towel and leave to rise for about an hour.
Knock back the dough and divide into 10 individual balls. Roll out each ball on a lightly floured surface until you have 5mm-thick rounds. Leave on a floured surface and cover with a damp cloth for 20 minutes while you preheat the oven to 210ºC fan/gas 8.
Cook 3 or 4 at a time in the oven on an oiled baking sheet for around 5 minutes (or you could put in two baking sheets at the same time), until the pitta breads start to balloon and turn light golden in colour. Cool and serve.
Wholemeal Pitta Bread
## Irish Cider and Honey Loaf
The combination of yeast, really dry cider and honey gives this bread a distinctive tartness with an undercurrent of mellow sweetness. Lovely at any time of the year, but particularly comforting when the leaves start changing colour and the days are getting shorter.
##### MAKES 12–15 SLICES
* 150ml milk
* 2 teaspoons honey
* 30g fresh yeast
* 250g strong white flour, plus extra for dusting
* 250g Macroom Stoneground Wholewheat Flour (extra coarse)
* 1 teaspoon sea salt
* 250ml dry Irish cider, such as Longueville or Llewellyn's
* olive or rapeseed oil, for greasing
##### METHOD
Gently warm the milk and honey in a small saucepan – you should be able to place a finger in the milk and comfortably leave it there without burning yourself.
Take off the heat and crumble the yeast in, stirring with a fork. Leave for 5 to 7 minutes until the milk is biscuit-coloured and foamy.
In a large bowl, mix the two flours and the salt. Pour the milk mixture into the flour mixture and then pour the cider in, using a wooden spoon to mix the ingredients as you go. A sticky dough should form.
Tip on to a lightly floured surface and knead for at least 5 minutes or until the dough is relatively smooth. You will need to sprinkle more flour on to the dough, but try not to add too much. The dough will be sticky and hard to handle, but stick with it!
Clean out your mixing bowl and rub oil lightly all over the inside. Place the dough in the bowl, cover with a clean tea towel and leave to rise in a warm place for around 1 hour. After this time the dough should have doubled in size.
Tip on to a lightly floured surface once again and knead for 1 to 2 minutes. Return the dough to the bowl and leave covered for a further 30 minutes until it has risen again.
Preheat the oven to 180ºC fan/gas 6. Line a baking sheet with baking parchment and lightly flour it.
Flip the dough once more on to a lightly floured surface and knock back, then knead lightly and form into a smooth ball. Place on the baking sheet, lightly flour the top and cover with the tea towel to rise for another 20 minutes while the oven heats up.
Bake in the oven for 25 to 30 minutes until brown and the crust is starting to tear and split at the top. Using a tea towel, flip the bread over and tap the bottom – if there's a good hollow sound then your bread is done.
Cool on a wire rack before slicing.
## Rosemary, Apple and Buttermilk Loaf
Rosemary is both pungent and woody, giving this loaf a lovely earthy flavour. It's an intriguing combination of savoury and sweet, so it's perfect for guests who don't have a very sweet tooth. It combines lovely sharp apples and the tang of buttermilk along with the rosemary. Go gently with the rosemary the first time you bake this and then you can adjust the level to suit to your taste.
##### MAKES 8–10 SLICES
* butter, for greasing
* 65g wholemeal spelt flour
* 65g cream flour
* 2 teaspoons baking powder
* a pinch of sea salt
* 80g golden caster sugar, plus extra to sprinkle on top
* 2 teaspoons finely chopped fresh rosemary, or to taste
* zest of 1 lemon
* 150ml buttermilk
* 60ml rapeseed oil
* 1 medium organic egg
* 1 large cooking apple
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Butter a 450g loaf tin.
In a large bowl, mix the flours, baking powder, salt, sugar, rosemary and lemon zest. Whisk the buttermilk, rapeseed oil and egg in a large jug. Peel and core the apple, and chop into small pieces.
Make a well in the centre of the flour mixture and pour in the wet ingredients from the jug. Mix gently until smooth, then fold in the apple pieces.
Pour the mixture into the loaf tin, sprinkle a little golden caster sugar on top and place in the oven for around 40 minutes. When a skewer comes out clean, the loaf should be done – you want a nice pale golden colour on top.
Leave to cool in the tin for 10 minutes before gently flipping the loaf out to cool further on a wire rack. You can serve it warm if you like, with a little butter.
_The rosemary, apple and buttermilk loaf is shown to the right in the picturehere_
Rosemary, Apple and Buttermilk Loaf
## Recipe List
1. Parsnip and Fennel Soup
2. Very Green Asparagus Soup
3. Vegetable Broth with Chard and Orzo
4. Chicken Broth with Potato, Spring Onion and Courgette
5. Coconut and Lime Soup with Hake and Coriander
6. Cleansing Nettle Broth
## Parsnip and Fennel Soup
In this soup the natural sweetness of parsnip combines beautifully with the delicate aniseed flavour of fennel. The result is smooth, velvety and very elegant.
##### SERVES 4–6
* 1 medium-sized onion
* 4 medium-sized parsnips
* 2 large fennel bulbs, stalks removed
* 1 stick of celery
* 15g fresh flat-leaf parsley
* 70g butter
* sea salt and freshly ground black pepper
* 1½ litres vegetable stock
* 200ml milk
##### _TO SERVE:_
* fresh cream
* fresh fennel fronds
##### METHOD
Peel the onion and parsnips. Chop finely, together with the fennel bulbs and celery, to roughly the same size dice. Roughly chop the parsley leaves.
Melt the butter in a large pot or casserole dish. Add the onion, parsnips, fennel and celery, and season well with salt and pepper. Stir so that everything in the pot is well coated in the butter.
Construct a cartouche by cutting a circle of greaseproof paper which perfectly covers the inside of your pot. Press this down on the vegetables, sealing them in to cook. Put the lid on the pot and cook for around 10 minutes on a gentle heat. Check and stir at least once to make sure nothing is catching on the bottom.
Meanwhile, in another pot, heat up your vegetable stock until it comes to the boil. This will shorten the cooking time considerably.
When it's boiling, remove the cartouche from the other pot and pour your hot stock over the vegetables, stirring the contents to make sure nothing is stuck to the bottom.
Simmer on a medium heat for around 20 minutes until the vegetables are completely soft and tender.
Add the milk and parsley, and blend with a stick blender until completely smooth and creamy.
Check the seasoning and serve with a swirl of cream and some fennel fronds sprinkled on top of each bowl.
Parsnip and Fennel Soup
## Very Green Asparagus Soup
Though we share a love of food and experimentation, William is the chef in the house and James is more the sous-chef. However, James has his star recipes too, and this asparagus soup is a favourite – just the ticket when you are craving a big bowl of green goodness.
##### SERVES 2–3
* 350g asparagus
* 3 shallots
* 2 cloves of garlic
* 25g butter
* a dash of rapeseed oil
* 2 large handfuls of spinach
* 700ml vegetable stock
* sea salt and freshly ground black pepper
##### _TO SERVE:_
* crème fraîche
* olive oil
##### METHOD
Remove the woody ends from the asparagus spears, then chop the stalks into 2cm pieces and reserve the tips. Peel and finely slice the shallots, and peel and crush the garlic.
Put the butter and rapeseed oil into a large saucepan on a medium-high heat. When foaming, add the asparagus tips and fry for a few minutes to soften. Remove the asparagus tips and set aside.
Add the shallots, asparagus stalks and garlic to the pan, and cook for 5 to 10 minutes until softened but still bright in colour.
Next, stir through the spinach, pour over the stock and bring to the boil. Remove from the heat and blitz with a hand blender. Season generously with salt and pepper, and add some hot water to loosen if needed.
Ladle into bowls and swirl through some crème fraîche or drizzle some olive oil. Scatter the asparagus tips over each serving and serve with chunks of bread.
Very Green Asparagus Soup
## Vegetable Broth with Chard and Orzo
This is a gorgeous, hearty, healthy soup, featuring one of William's favourite vegetables, chard, and also orzo – a tiny pasta shape that looks like rice – to give it extra body. It's totally comforting and fills the belly nicely on a chilly night!
##### SERVES 2
* 2 leeks
* 3 cloves of garlic
* 3 medium-sized carrots
* 2 sticks of celery
* 4 medium-sized potatoes
* 30g butter
* olive oil
* sea salt and freshly ground black pepper
* 500ml vegetable stock, or more if you prefer
* 2 bay leaves
* 4 sprigs of fresh thyme
* 100g orzo pasta
* 3 tablespoons chopped fresh flat-leaf parsley
* 250g chard, stalks removed, leaves washed
* juice of ½ a lemon
##### METHOD
Remove the green part of the leeks (keep aside for stocks and soups), then wash the white part and chop into 2cm rounds. Peel and slice the garlic. Cut the carrots and celery into large chunks, keeping them separate, and peel and cut the potatoes into medium-sized chunks.
Put the butter and a little drizzle of olive oil into a heavy-bottomed saucepan or casserole dish on a medium heat. Add the leeks, garlic and celery and season well with salt and pepper.
After 5 minutes, add the potatoes and carrots and cook for another 5 minutes, stirring everything occasionally. Allow the vegetables to brown slightly and stick to the bottom every now and then.
Bring the stock to the boil in a separate pot, then pour this over the browned vegetables. Add the bay leaves and thyme, and simmer for about 20 minutes until everything is tender. If you feel you'd like more liquid in your broth, you can always add a little more stock to taste.
Add the orzo, 2 tablespoons of the chopped parsley and the chard leaves (tear the larger ones), and cook for another 10 minutes or until the orzo is al dente. Add the lemon juice and the rest of the parsley.
Check the seasoning, and serve hot with some hunks of good buttered bread.
Vegetable Broth with Chard and Orzo
## Chicken Broth with Potato, Spring Onion and Courgette
Everyone knows that chicken broth is the ultimate cure-all. This version has a lighter, more spring–summer feel than other recipes – but it's just as comforting and restorative.
##### SERVES 2–4
##### _FOR THE CHICKEN STOCK:_
* 1 free-range chicken carcass, leftover meat removed
* 1 stick of celery, chopped
* 1 carrot, chopped into thick rounds
* 1 large onion, peeled and quartered
* 3–4 black peppercorns
* 1 bay leaf
* 1 sprig of fresh thyme
* sea salt and freshly ground black pepper
##### _FOR THE FINISHED BROTH:_
* 2 medium-sized floury potatoes, such as Red Rooster
* 2 courgettes
* 6 spring onions
* 1 tablespoon olive oil
* 25g butter
* as much leftover chicken as you have
* fresh chives, to garnish
##### METHOD
First prepare the stock. Place the chicken carcass in a large, heavy-based saucepan and cover with 1½ to 2 litres of cold water. Bring to the boil and add the rest of the stock ingredients. Bring to the boil again, then reduce to a rolling simmer and leave to bubble away with a lid askew for about an hour. You can add more water if necessary.
Now reduce the heat to low and leave to simmer gently for another 40 minutes to an hour. Strain the broth into a bowl, so you are left with a delicious concentrated stock. You'll get about 1 to 1½ litres.
If you're not eating straight away, you can leave it to cool, put it in the fridge and skim off any excess fat that rises to the top when you get the chance – though we don't mind a little fat personally!
To make the finished broth, peel the potatoes and chop into chunks, then chop the courgettes into similar-sized chunks and thinly slice the spring onions.
Put the olive oil and the butter into a heavy casserole dish over a medium heat. Add the potatoes and courgettes, and soften for around 5 minutes.
Throw in the leftover chicken, add 500ml of your chicken stock and bring to the boil. Reduce the heat to a simmer and leave for 10 to 15 minutes until the potatoes are cooked. When the potatoes are nearly ready, add the spring onions and cook for a minute or two.
Divide between your bowls, garnishing with some roughly chopped chives.
## Coconut and Lime Soup with Hake and Coriander
Hake tastes so delicate that it can easily be overpowered if it's put with strong flavours. The South East Asian ingredients in this soup – coconut milk, lime, coriander, fish sauce – make it light and tangy and are the perfect accompaniment to the fresh clean flavour of the fish. It's a beautiful aromatic soup that you'll return to again and again.
##### SERVES 2–3
* 1 x 400ml tin of organic coconut milk
* 150ml vegetable stock
* 1 tablespoon fish sauce
* juice of 1 lime
* 2 x 150g hake fillets, cut into chunks
* sea salt and freshly ground black pepper
* a handful of fresh coriander leaves
* 2 spring onions
##### METHOD
In a large saucepan, heat the coconut milk, vegetable stock, fish sauce and lime juice until they come to the boil. Season the hake chunks with salt and pepper and add to the pan, reducing the heat to a simmer.
Cook for 5 to 10 minutes.
Roughly chop the coriander, finely slice the spring onions and sprinkle them over the soup. Serve piping hot in large bowls, with chunks of rustic bread.
## Cleansing Nettle Broth
As this is light and delicate, it's best to use only young nettle leaves and buds, which will give a gentle texture and flavour.
##### SERVES 4–6
* 4 onions
* 3 leeks
* 1 stalk of celery
* 100g young spring nettle leaves
* a small bunch of fresh chives
* 3 tablespoons olive oil
* 1½ litres water
* sea salt and freshly ground black pepper
##### METHOD
Peel and slice the onions. Remove the green part of the leeks (keep aside for stocks and soups), then wash the white part and slice into 1–2cm pieces.
Slice the celery into 1–2cm pieces. Wash the nettle
leaves and chop the chives.
Pour the olive oil into a large saucepan and place on a medium heat. Add the onions, leeks and celery, and sauté gently until softened. Add the water, season with salt and pepper, bring to the boil and simmer for around 35 minutes.
Add the nettle leaves and chives (reserving some of the chives for later) and cook for a further 5 minutes. Serve hot, sprinkled generously with the remaining chives.
Cleansing Nettle Broth
## Recipe List
1. Baked Feta, Roasted Lemon and Trofie Salad with Sesame Toasts
2. Crushed New Potatoes with Pink Peppercorns, Capers and Parsley
3. Smoked Mackerel with Tabasco and Lime Potato Salad
4. Balsamic Cucumber with Kalamata Olives
5. Yellow and Green Courgetti with Toasted Almonds and Feta
6. Chicory Salad with Pomegranate, Grilled Halloumi and Lime Dressing
7. Orzo with Preserved Lemon and Thyme Cream
8. Citrus Salad with Honeyed Buttermilk
9. Hedgerow Berry Salad with Lime Syrup
10. Fresh Figs with Black Pepper and Honey
## Baked Feta, Roasted Lemon and Trofie Salad with Sesame Toasts
Imagine a large bowl filled with the freshest, brightest and most colourful ingredients, each one offering a different taste and texture – that's this salad, bursting with the flavour of summertime.
##### SERVES 4 AS A LIGHT LUNCH, 2 AS A MAIN MEAL
* 300g trofie pasta
* 2 lemons, sliced into thin rounds
* olive oil
* sea salt
* 2 sprigs of fresh marjoram
* 1 small raw beetroot
* juice of 1 lemon
* 2 tablespoons white wine vinegar
* 1 tablespoon honey
* 150g radishes, quartered
* 1 small red onion, sliced thinly
* 50g rocket
* a handful of fresh basil leaves
* a handful of fresh flat-leaf parsley leaves
* a handful of fresh dill, chopped
* a few cornflowers, to garnish (optional)
##### _FOR THE DRESSING:_
* zest of 1 orange
* 100ml olive oil
* 4 tablespoons lemon juice
##### _FOR THE BAKED FETA:_
* 200g feta cheese
* a few fresh marjoram leaves
##### _FOR THE SESAME TOASTS:_
* 2 cloves of garlic
* a small handful of fresh flat-leaf parsley leaves
* 50g butter, softened
* juice of ½ a small lemon
* 2 tablespoons sesame seeds
* 4 slices of sourdough bread
##### METHOD
Preheat the oven to 200ºC fan/gas 7.
Cook the trofie pasta in salted boiling water until al dente. Drain and rinse with cold water to stop the pasta from cooking further. Set aside.
Lay out the lemon slices on a roasting tray, drizzle with olive oil, season with salt and place the sprigs of marjoram on top. Put into the oven for 20 minutes or until caramelized. Keep an eye on them so they don't burn!
While the lemon slices are in the oven, scrub, top and tail and slice the beetroot as thinly as possible. Lay the slices on a plate, mix together the lemon juice, white wine vinegar and honey with a little salt, and drizzle over. Leave to marinate for 30 minutes. When the lemon slices are ready, remove them from the oven and leave to cool, but don't turn the oven off yet.
Make the dressing by whisking all the ingredients together with a pinch or two of salt. In a large bowl, mix together the cooked trofie, radishes, onion, rocket, most of the basil, parsley and dill, and the cooled lemon slices. Stir the dressing through gently, along with the beetroot slices.
Cut the feta into large chunks and place in an ovenproof dish. Drizzle with olive oil and sprinkle with the marjoram leaves. Bake for about 10 minutes, until soft.
While the feta is cooking, make the sesame toasts. Peel and crush the garlic, chop the parsley and mash these into the butter with some salt and the lemon juice. Melt in a shallow pan and add the sesame seeds.
When the seeds turn golden, turn the heat to moderate and add the bread to the pan, coating it in the sesame seeds. Cook both sides until crisp and golden. Drain on paper towels.
Assemble the salad on a large plate. Arrange the warm baked feta in the middle, garnish with cornflowers and the leftover herb leaves, and serve with the sesame toasts.
Baked Feta, Roasted Lemon and Trofie Salad with Sesame Toasts
## Crushed New Potatoes with Pink Peppercorns, Capers and Parsley
New potatoes have thinner skins, a waxier texture and a sweeter flavour than older types and are therefore perfectly suited to salads. This recipe keeps things simple and shows them off to best effect.
##### SERVES 6 AS A GENEROUS SIDE
* 1kg new potatoes, unpeeled
* 2 cloves of garlic
* 4 spring onions, white parts only
* 300g Greek yoghurt
* 100ml olive oil
* 30g fresh flat-leaf parsley, leaves picked and chopped
* 1 tablespoon chopped fresh chives
* 1 tablespoon pink peppercorns
* 2 tablespoons capers
* sea salt and freshly ground black pepper
##### METHOD
Wash the new potatoes in cold water and put them into a pan. Cover with water and add plenty of salt. Bring to the boil and simmer for 20 to 25 minutes until a knife can go through them with just a little resistance.
Drain the cooked potatoes, place in a large mixing bowl and crush roughly with a fork.
Peel and crush the garlic, then slice the spring onions. Mix the yoghurt, olive oil and crushed garlic together in a small bowl. Add this mixture to the potatoes and stir, making sure to coat all the potatoes.
Stir in the parsley, chives, spring onions, pink peppercorns and capers, and season with salt and pepper before serving.
## Smoked Mackerel with Tabasco and Lime Potato Salad
Robust flavours and textures make this salad punchy and distinctive. Use as much or as little Tabasco as you can handle!
##### SERVES 4
* 340g new potatoes
* zest and juice of 1 lime
* a pinch of pink peppercorns
##### _FOR THE DRESSING:_
* a handful of fresh flat-leaf parsley leaves
* a handful of fresh coriander leaves
* juice of 1 lemon
* 3 anchovies
* 1 teaspoon Dijon mustard
* a few drops of Tabasco
* 1 clove of garlic, peeled
* 1 tablespoon white wine vinegar
* sea salt and freshly ground black pepper
* 2 tablespoons olive oil
##### _TO SERVE:_
* 2 smoked mackerel fillets
* 1 lime
* a handful of fresh coriander leaves, chopped
##### METHOD
Cut the potatoes in half, place in a large saucepan and cover with cold salted water. Bring to the boil and simmer for 15 minutes or until cooked through.
Drain and toss with the lime juice and zest.
Blitz all the dressing ingredients in a food processor, adding more olive oil if needed to get a good pouring consistency. Put the potatoes into a large bowl, drizzle generously with the dressing and sprinkle over the pink peppercorns.
Pull the smoked mackerel fillets gently apart into large pieces and arrange on top with wedges of lime and some chopped coriander.
Smoked Mackerel with Tabasco and Lime Potato Salad
## Balsamic Cucumber with Kalamata Olives
Bright, salty, complex flavours make this salad an addictive and healthy lunch. It's great on its own, but also delicious with focaccia or other good bread and some beetroot and seaweed hummus (see here). It also works very well with barbecued mackerel (see here). To get so much taste packed into something that's so simple to create makes it extra rewarding!
##### SERVES 4–6
* 2 cucumbers
* 2 spring onions
* 2 tablespoons balsamic vinegar
* 1 tablespoon olive oil
* sea salt and freshly ground black pepper
* juice of ½ a lemon
* 8–10 Kalamata olives, unstoned*
* 1 tablespoon chopped fresh dill
##### METHOD
Cut the cucumbers in half lengthways and use a teaspoon to gouge out the watery seeds. Chop the cucumber halves into good-sized chunks. Finely slice the spring onions.
In a medium-sized bowl, whisk together the balsamic vinegar, olive oil, some salt and pepper and the lemon juice until well combined. Put the cucumber chunks into this bowl with the olives, sliced spring onions and dill. Cover with cling film and leave to marinate in the fridge for at least 30 minutes before serving.
* _Unless making something like a tapenade, we leave olives unstoned to preserve all the flavours and juices until the moment of eating._
## Yellow and Green Courgetti with Toasted Almonds and Feta
In Currabinny we always grew too many courgettes, so we have a lot of courgette recipes! In this salad you are effectively 'cooking' the flesh in the acid of the lemon juice. It's a perfect way to enjoy courgette, as it retains its bright summery flavour and crunchy texture.
##### SERVES 4
* 2 green courgettes
* 1 yellow courgette
* sea salt and freshly ground black pepper
* 1 tablespoon rapeseed oil
* juice of 1 lemon
* a handful of fresh mint leaves
* a handful of fresh basil leaves
* 1 tablespoon pink peppercorns
* 200g feta cheese
* 50g flaked almonds
##### METHOD
Trim the courgettes and spiralize into flat, long ribbons. If you don't own a spiralizer, you can use a vegetable peeler instead.
In a large bowl, mix the courgette ribbons with a good pinch of salt, some pepper, the rapeseed oil and the lemon juice. Leave for 10 minutes to let the lemon juice 'cook' the courgettes.
Chop the mint and basil leaves and add these to the courgettes, along with the pink peppercorns. Crumble in the feta.
Put a frying pan on a medium-high heat and add the flaked almonds. Keep the almonds moving in the pan for around 2 minutes, until they start to turn golden around the edges and you can smell them. Remove from the pan and add to the salad bowl. This salad is best served while the almonds are still warm.
_The yellow and green courgetti salad is shownhere_
## Chicory Salad with Pomegranate, Grilled Halloumi and Lime Dressing
The bitterness of chicory can make it difficult to work with. Using a zingy dressing – like this lime and balsamic mix – balances the bitterness beautifully. The halloumi turns this salad into a proper meal – its saltiness can hold its own in this symphony of intense flavours!
##### SERVES 4 AS A SIDE DISH OR 2 AS A MAIN
* 3 small radishes
* 2 heads of white chicory
* 1 head of red chicory
* 100g rocket
* 50g walnuts, chopped roughly
* 1 pomegranate
##### _FOR THE DRESSING:_
* juice of 1 lime
* 2 tablespoons rapeseed oil
* a pinch of sea salt and freshly ground black pepper
* 1 tablespoon chopped fresh coriander leaves
* 1 tablespoon balsamic vinegar
##### _FOR THE HALLOUMI:_
* 225g halloumi cheese
* rapeseed oil
* 1 lime
##### METHOD
Trim and thinly slice the radishes. Cut the ends off the heads of chicory, cut in half lengthways and separate the leaves. Put the radishes and chicory into a large salad bowl with the rocket and chopped walnuts.
Slice the pomegranate in half and hold one of the halves, cut side down, over the bowl. With your hand covering the cut half, beat the back of the pomegranate with a wooden spoon so the seeds fall in between your fingers into the bowl. Make sure to remove any bits of white pith that fall into the salad. Repeat with the other half.
Whisk together the dressing ingredients until well combined and pour into a jar or dressing bottle.
Slice the halloumi into chunky 2cm-sized pieces. Heat a little rapeseed oil in a frying pan or griddle until the pan is very hot, then turn the heat down to medium. Fry the halloumi in batches for around 2 minutes on each side, being careful not to let it burn. Squeeze lime juice over both sides of the halloumi pieces as you cook them.
Arrange the salad on plates, top with the hot halloumi and drizzle with the dressing.
Chicory Salad with Pomegranate, Grilled Halloumi and Lime Dressing
## Orzo with Preserved Lemon and Thyme Cream
This is a really elegant pasta dish that can be served hot, or cold as a salad. The sumac is colourful, slightly bitter and wonderfully fragrant, and it, along with the salty preserved lemon, gives this dish an exotic edge.
##### SERVES 4
* 350g orzo pasta
##### _FOR THE PRESERVED LEMON AND THYME CREAM:_
* 1 shallot
* skin of ½ a preserved lemon
* olive oil
* 250ml double cream or crème fraîche
* sea salt and freshly ground black pepper
* 1 tablespoon fresh thyme leaves
* 1 teaspoon sumac
##### METHOD
To make the cream, peel and finely chop the shallot, then finely chop the lemon skin. Put a little olive oil into a frying pan, add the shallot and cook for 2 minutes on a medium heat to soften. Add the preserved lemon and stir for another 2 minutes until fragrant.
Pour in the double cream and leave to cook for 5 minutes until bubbling, then season the cream with salt and pepper and add the thyme leaves. Cook for a further 5 minutes, at which point the cream will start to thicken.
Take off the heat and stir in the sumac.
Cook the orzo as per the packet instructions, then drain, and pour the sauce over, mixing thoroughly.
Orzo with Preserved Lemon and Thyme Cream
## Citrus Salad with Honeyed Buttermilk
When William was growing up there was always some buttermilk in the fridge, so it's very much part of the Currabinny story. Finding uses for it apart from in baking can be hard, but here it goes wonderfully with fruit. Think of this as a breakfast salad – combining juicy citrus fruit with the sweet but tangy buttermilk.
##### SERVES 2
* 1 pink grapefruit
* 1 yellow grapefruit
* 2 blood oranges
* 2 large Seville oranges
* a few chopped fresh mint or basil leaves, to serve
##### _FOR THE HONEYED BUTTERMILK DRESSING:_
* 120ml buttermilk
* ½ teaspoon vanilla extract
* 2 tablespoons honey
* 3 tablespoons natural yoghurt
* juice of ½ a lemon
* 1 tablespoon peanut oil
* a pinch of sea salt
##### METHOD
Using a sharp knife, slice the tops and bottoms off each of the citrus fruits, then stand them up on your chopping board. Slice or peel away their peel and pith until just the flesh remains.
Turn each fruit on its side and slice into ½cm rounds. Arrange in a single layer on a large platter or two plates.
Whisk all the ingredients for the honeyed buttermilk dressing together in a bowl and pour into a serving jug. Pour a little of the dressing over the fruit and sprinkle with the chopped mint or basil leaves.
## Hedgerow Berry Salad with Lime Syrup
Blackberry-picking is part of every Irish country childhood –hedgerows heavy with fruit, purple-stained fingers, faces and clothes, the odd prick of a thumb! Combined with the other fruit and drizzled with a tangy syrup made from a sweet herb, this is about as simple as it gets, a real celebration of berries. As it's so fresh and easy, we've included this in our salad section, but you'll probably serve it as a refreshing dessert. Still, there's nothing to stop you having it for breakfast or lunch!
##### SERVES 4
* 300g fresh hedgerow blackberries
* 100g redcurrants
* 50g strawberries
* 50g raspberries
* 5–6 fresh mint leaves
##### _FOR THE SYRUP:_
* 150ml water
* 150g caster sugar
* 4–5 sweet geranium leaves (mint, lemon balm or sweet cicely leaves also work well)
* juice and peel of 1 lime
##### METHOD
Wash all the berries, then remove the stems from the strawberries and cut them into quarters. Assemble the berries on a large platter or in a bowl. Chop the mint finely and sprinkle over the berries.
Make the syrup by heating the water and sugar up to boiling point. Turn the heat down, add the geranium leaves and lime peel and simmer for 5 minutes. Take off the heat and add the lime juice, stirring it through to combine. Leave to cool.
Serve the berries with a jug of the syrup on the side, so that people can pour it over the fruit themselves.
Hedgerow Berry Salad with Lime Syrup
## Fresh Figs with Black Pepper and Honey
How beautiful and perfect are figs? When ripe they are sticky, sweet and heavy with flavour. Here they're sliced thinly – skin and all – and dressed with warm peppery honey to create the perfect amuse-bouche or starter for a summer dinner.
##### SERVES 4–6
* 8–10 fresh figs
* 100ml honey
* 4–5 sprigs of fresh thyme
* 1 teaspoon freshly ground black pepper
##### METHOD
Twist the stems off the figs and cut each one into 3 or 4 slices. Or you can cut a cross from the top down to the base, without cutting through, and open out into quarters. Arrange on a platter.
In a small saucepan, gently heat the honey, sprigs of thyme and black pepper until just about simmering.
Drizzle directly over the fig slices and serve.
Fresh Figs with Black Pepper and Honey
## Recipe List
1. Cavolo Nero, Feta and Butternut Squash Filo Pie
2. Rustic Rye Galette with Leeks, Fennel, Goat's Cheese and Toasted Pine Nuts
3. Homemade Gnocchi with Buffalo Mozzarella, Pickled Walnuts and Green Herb and Lemon Dressing
4. Baked Potatoes with Ricotta and Green Herb and Lemon Dressing
5. Ruby Chard Korma
6. Chard and Ricotta Lasagne
7. Breda's Cauliflower Cheese
## Cavolo Nero, Feta and Butternut Squash Filo Pie
This is a hearty, comforting pie that makes the most of winter vegetables and has a gentle kick of chilli heat. It's particularly great for dinner when the days are closing in and it's cold and wet.
##### MAKES 8 GOOD-SIZED SLICES
* 1 large butternut squash (about 900g unpeeled weight)
* 2 medium-sized red onions
* 4 cloves of garlic
* 1 small fresh red chilli
* olive oil
* sea salt and freshly ground black pepper
* 15g fresh rosemary spears
* 400g cavolo nero
* 200g feta cheese
* 4 sheets of filo pastry
* melted butter, for brushing the pastry
##### METHOD
Preheat the oven to 200ºC fan/gas 7.
Peel the squash, scoop out any seeds and cut the flesh into bite-sized pieces – you should get around 500g of flesh. Peel and slice the onions and garlic. Finely chop the chilli, removing the seeds if you don't like too much heat.
Place the butternut squash, red onions and chilli in an even layer on a baking tray, drizzle with olive oil and season with salt and pepper. Pick and chop the rosemary leaves, scatter them over the top and bake in the oven for 20 minutes until the butternut squash is tender. Leave the oven on for baking the pie.
While the squash is roasting, wash and roughly chop the cavolo nero, removing the heavy stalks. Cook with a good splash of water and a sprinkle of salt in a heavy-based saucepan until thoroughly wilted but still dark green. Squeeze out the moisture using kitchen paper.
Place the roasted veg in the bottom of a casserole dish, scatter with the cavolo nero and crumble the feta on top.
Cover with the filo pastry, cutting away any excess pieces you don't need. Brush the top with a little melted butter and place in the oven for 15 minutes until the pastry is golden. Serve immediately.
Cavolo Nero, Feta and Butternut Squash Filo Pie
## Rustic Rye Galette with Leeks, Fennel, Goat's Cheese and Toasted Pine Nuts
You construct this traditional free-form savoury tart by rolling out the pastry into a large round, dolloping the buttery filling in the middle, and gathering up the pastry around the filling to form a border, not caring if things are uneven or if a piece breaks off. This is a celebration of simple, unfussy, hands-on cooking at its best.
##### SERVES 4–6
##### _FOR THE RYE PASTRY:_
* 80g cream flour
* 90g rye flour
* 1 teaspoon caster sugar
* ½ teaspoon sea salt
* 1 medium organic egg
* 40ml double cream
* 120g cold butter
* 2 teaspoons lemon juice
* ½ teaspoon lemon zest
##### _FOR THE FILLING:_
* 4 medium-sized leeks
* 2 fennel bulbs
* 15g butter
* olive oil
* 2 teaspoons fresh thyme leaves, plus extra for sprinkling
* sea salt and freshly ground black pepper
* 60ml white wine
* 60ml double cream
* 2 tablespoons chopped fresh flat-leaf parsley leaves
* 150g soft goat's cheese, such as St Tola or Ardsallagh
* toasted pine nuts, to garnish
##### METHOD
First, make the pastry dough. Combine the two flours, sugar and salt in a large mixing bowl. Whisk the egg with the cream in a large jug. Cut the butter into small chunks and rub into the flour mix until you have a breadcrumb-like consistency.
Drizzle the egg and cream mixture into the crumbs slowly, mixing it with your hands as you go. You may not need all the egg and cream – you are looking for a smooth dough that comes together nicely without sticking to everything.
Reserve a small amount of the cream mixture for finishing the galette later. Add the lemon juice and zest to the dough and knead in, sprinkling more flour on if it becomes too wet. Shape the dough into a disc, cover with cling film and refrigerate for at least 2 hours.
Preheat the oven to 200ºC fan/gas 7.
Remove the green part of the leeks (keep aside for stocks and soups), then wash the white part and slice into 1cm rounds. Finely slice the fennel. In a heavy-bottomed saucepan, heat the butter with a little olive oil and add the leeks, fennel and thyme leaves. Season with salt and pepper and cook on a medium heat for around 10 minutes until everything is nicely softened.
Add the white wine and continue cooking until it has reduced, then add the cream and parsley. Cook until the sauce is nicely coating the leeks and fennel and isn't too runny. Take off the heat and leave to cool.
Roll the pastry out into a big round about ½cm thick on a large baking tray (about 40cm x 35cm) lined with baking parchment. Cut off any excess dough – the pastry should be roughly 30cm in diameter.
Spread the leek and fennel mixture in the middle of the dough, leaving a good 5cm gap along the edges. Spoon dollops of goat's cheese all over the top, then sprinkle with salt and pepper and some more thyme leaves.
Fold the uncovered dough in on itself until you have a rustic, rough-and-ready open pie. Use the remaining egg and cream to brush the dough, then place in the oven for around 40 minutes until golden and bubbling. Sprinkle the toasted pine nuts over before serving.
Rustic Rye Galette with Leeks, Fennel, Goat's Cheese and Toasted Pine Nuts
## Homemade Gnocchi with Buffalo Mozzarella, Pickled Walnuts and Green Herb and Lemon Dressing
Sometimes you come across pickled walnuts on a dusty shelf, as a kind of oddity. However, there's nothing odd about the flavour – it is a wonderful combination of sweet and tangy nuttiness. In this recipe it pairs beautifully with mozzarella and a herby dressing. Making gnocchi is so simple, it is a shame anyone buys them pre-made.
##### SERVES 4
* 500g floury potatoes, such as Maris Piper
* 125g Italian Tipo 00 flour, plus extra for dusting
* 4 buffalo mozzarella balls
* 2 pickled walnuts, sliced
* 2 tablespoons grated pecorino
* green herb and lemon dressing (see here – use the full quantity from that recipe)
##### METHOD
Peel the potatoes and boil them in salted water for around 30 minutes until cooked through. Drain, then use a wooden spoon to push the potatoes through a fine-mesh sieve into a bowl.
Gently sprinkle the flour in batches into the mashed potato, folding through to combine each time until you are left with an elastic, dough-like mash.
Lightly flour a board and roll the dough out into long sausages, roughly the circumference of a butcher's sausage. Cut the sausages into 3cm pieces and place on a floured plate or tray until ready to use. Cover with cling film so they don't dry out.
Bring a large, heavy-bottomed saucepan of salted water to the boil and cook the gnocchi in batches, 10 or 12 at a time, so you don't overcrowd the pan.
The gnocchi cook quickly – you will know they are ready when they float and bob on the surface of the water. Drain on paper towels when done.
Divide the gnocchi into bowls, then tear over the mozzarella into large pieces. Sprinkle with the sliced pickled walnuts and grated pecorino, and drizzle generously with the green herb and lemon dressing.
Homemade Gnocchi with Buffalo Mozzarella, Pickled Walnuts and Green Herb and Lemon Dressing
## Baked Potatoes with Ricotta and Green Herb and Lemon Dressing
This recipe takes the lonely image of the baked potato dinner and makes it the treat it truly is. Between creamy ricotta and a lovely sharp dressing packed with herby freshness, in this version the floury flesh of the potato gets the kind of celebratory treatment it deserves.
##### SERVES 4
* 4 good-sized floury potatoes, such as Rooster or Kerr's Pink
* olive oil
* sea salt and freshly ground pepper
* 1 clove of garlic
* 5–6 fresh chives
* 250g buffalo ricotta
* green herb and lemon dressing (see here – use the full quantity from that recipe)
##### METHOD
Preheat the oven to 200ºC fan/gas 7.
Prick the potatoes several times with a fork and rub all over with olive oil. Place on a baking tray, sprinkle with salt and bake in the oven for 1 hour.
When the potatoes are done, remove from the oven, cut in half and scoop out a little of the flesh of each potato half. Return the potato halves to the oven for another 8 minutes.
Peel and finely chop the garlic, chop the chives and combine in a bowl with the scooped-out potato and ricotta. Season with salt and pepper.
Remove the potato halves from the oven and fill each half with the ricotta mix, then return to the oven for a final 10 minutes. Serve hot with the dressing spooned over.
## Ruby Chard Korma
William's mother, Breda, has always kept a patch of ground at Currabinny for the largely unsuccessful cultivation of vegetables, berries and fruit trees. The combination of salty gale-force winds from Cork Harbour and the relentless encroachment of tree roots from Currabinny Woods just behind the house present her with some serious challenges. However, among the fruitless plum trees, the gooseberry bushes with their bitter, unripe berries, and the small, flattened carrot tops, one thing grows in abundance: chard. Thus, in Currabinny, chard gets put into every dish imaginable.
Chard is a highly nutritious leaf, commonly used in the same way as spinach, although it's more closely related to beetroot. William didn't see chard much after moving to Dublin (though it's more common now), so, in a fit of nostalgic enthusiasm when he came across it, and with an unshakable belief that chard works in anything, he decided to add it to a korma. And the result was amazing – as he knew it would be!
##### SERVES 4–6
* 3 onions
* 3 cloves of garlic
* a thumb-sized piece of fresh ginger
* 700g chestnut mushrooms
* a large knob of butter
* sea salt and freshly ground black pepper
* seeds from 10 cardamom pods, crushed
* 1 teaspoon ground cumin
* 1 teaspoon ground turmeric
* a few pinches of ground cinnamon
* a few pinches of chilli powder
* 3 bay leaves
* 200ml water
* 350g ruby chard
* 200g natural yoghurt
* 150g crème fraîche
##### _TO SERVE:_
* toasted flaked almonds
* pomegranate seeds
* basmati rice
##### METHOD
Peel the onions, garlic and ginger. Slice the onions and mushrooms, grate the ginger and crush the garlic with some salt. Melt the butter in a large pan and add the onions, garlic and ginger with some salt and pepper.
When the onions have softened a bit, add the cardamom, cumin, turmeric, cinnamon, chilli powder and bay leaves. Now add the sliced mushrooms to the pan and cook for a couple of minutes, stirring regularly. Pour in the water, stir, and simmer for 15 minutes, then check the seasoning.
Meanwhile, remove the stalks from the chard* and add the leaves in batches to the pot until it is all wilted. Turn the heat to low and gently stir in the yoghurt and crème fraîche.
Serve with rice and top with the almonds and pomegranate seeds.
* _Don't throw away the stalks! You can use them to make stock (they'll keep in the freezer), but they are also really nice chopped roughly, sautéed with butter and served as a side dish with a main course._
Ruby Chard Korma
## Chard and Ricotta Lasagne
This is a break from the traditional lasagne, which has a lot of elements and involves a lot of cooking. Of course, a classic lasagne is lovely, but this veggie version is quick, easy and just as tasty.
##### SERVES 6–8
* 400g rainbow chard
* 400g chestnut mushrooms
* 3 cloves of garlic
* sea salt and freshly ground black pepper
* 50g butter
* 1 tablespoon fresh thyme leaves
* a small handful of fresh flat-leaf parsley leaves, chopped
* 200g hard goat's cheese, such as aged Ardsallagh
* 2 medium organic eggs
* 300g buffalo ricotta
* 300ml single cream
* ¼ teaspoon freshly grated nutmeg
* zest of ½ a small lemon
* 180g dried lasagne sheets
##### METHOD
Preheat the oven to 180ºC fan/gas 6.
Wash the chard, remove the heavy stalks and shred the leaves. Wipe and thinly slice the mushrooms, then peel and slice the cloves of garlic.
Place the chard leaves in a large saucepan with a little cold water, salt and pepper. Cook on a high heat for 2 to 3 minutes until the water stars to boil. Drain and place the chard in wads of kitchen paper or clean tea towels, pressing down hard to drain the liquid out.
Heat the butter in a large frying pan and add the mushroom slices, seasoning with salt and pepper. Add the garlic, thyme and parsley. When the mushrooms start to caramelize in the hot butter, check the seasoning, then remove from the heat and set aside.
Grate the goat's cheese and combine three quarters of it with the eggs, ricotta, cream, nutmeg and lemon zest in a large bowl. Reserve the remaining cheese for the topping.
Spread a third of the ricotta mixture over the bottom of a large 28cm x 22cm ovenproof dish. Place half the chard on top and then a layer of the lasagne sheets. Put another third of the ricotta mixture on top, then layer with half the herby mushrooms.
Add the rest of the chard and then the remaining lasagne sheets. Spread the rest of the mushrooms on top, along with a final layer of ricotta sauce. Sprinkle the top with the remaining goat's cheese.
Bake in the oven for 30 to 40 minutes until golden brown on top, bubbling and ready to serve.
Chard and Ricotta Lasagne
## Breda's Cauliflower Cheese
Every week for as long as William can remember, his mother, Breda, has made this recipe. It is about as classic a cauliflower cheese as it gets, and all the more wonderful for that very fact – when something is this good it would be a crime to interfere with it!
##### SERVES 4 AS A SIDE DISH
* 1 large cauliflower
* sea salt and freshly ground black pepper
* 150g mature Irish Cheddar cheese
* 50g Parmesan cheese
* 1 small carrot
* 1 small onion
* 50g butter
* 50g cream flour
* 700ml milk
* 3 peppercorns
* 2 sprigs of fresh thyme
* a small bunch of fresh flat-leaf parsley
* 2 teaspoons Dijon mustard
* freshly grated nutmeg (optional)
* ground mace, to finish
##### METHOD
Preheat the oven to 200ºC fan/gas 7.
Remove the thick base stalk from the cauliflower and discard, then break the cauliflower into florets. Steam the florets until tender. Season lightly with salt and place in an ovenproof dish. Grate the Cheddar and the Parmesan, keeping them separate.
Next make the béchamel sauce. Peel and chop the carrot, then peel and slice the onion. Make a roux by melting the butter in a small saucepan and adding the flour, stirring with a wooden spoon until thickened but not burned. Set aside.
Put the milk into a medium-sized saucepan with the carrot, onion, peppercorns, thyme and parsley. Bring to the boil, then simmer for 5 minutes.
Strain the milk into a jug, removing all the vegetables, herbs and peppercorns. Pour the milk slowly back into the saucepan with the roux and bring to the boil again, thoroughly whisking in the roux until thickened. Take off the heat and stir in the mustard, some salt and pepper, and a little nutmeg if desired.
Add the grated Cheddar to the béchamel while still hot, stirring gently until it is melted and combined. Check the seasoning and pour the sauce over the cauliflower florets.
Sprinkle the Parmesan on top and bake in the oven for 40 minutes until bubbling and golden. Sprinkle a little ground mace on top before serving.
Breda's Cauliflower Cheese
## Recipe List
1. Pan-fried Hake with Candy Beetroot and Orange Salad
2. Slow-roasted Salmon with Blood Orange, Lemon, Fennel and Parsley
3. Breda's Fish Pie
4. Barbecued Mackerel
5. Salmon, Samphire, Broccoli and Cream Pappardelle
6. Salmon with Baby Potatoes, Capers and Garlic, Lemon and Parsley Butter
7. Smoked Rainbow Trout with Fennel, Goat's Cheese, Pink Peppercorns and Dill
8. Asparagus and Smoked Rainbow Trout with a Herb Sauce
## Pan-fried Hake with Candy Beetroot and Orange Salad
Not only is this dish simple and delicious, but the pinks, oranges and yellows of the salad make it very beautiful. Because of its subtle fresh flavour and versatility, hake is William's favourite white fish to cook with.
##### SERVES 4
##### _FOR THE SALAD:_
* 4 medium-sized candy beetroots
* juice of 1 lemon and zest of ½ a lemon
* juice of ½ an orange
* sea salt and freshly ground black pepper
* 4 medium-sized oranges
##### _FOR THE HAKE:_
* 4 hake fillets, skin on (roughly 175g each)
* 1 tablespoon olive oil
* juice of ½ a lemon
##### _TO GARNISH:_
* 15g fresh flat-leaf parsley
* pink peppercorns, whole or lightly crushed, to taste
##### METHOD
First, make the salad. Slice the beetroots as thinly as possible into almost transparent rounds. In a shallow bowl, combine the slices with the lemon juice and zest, the orange juice and a pinch of salt and pepper.
Leave in the fridge for at least 30 minutes to marinate thoroughly and absorb the flavours. The beetroot should soften in this time but should retain a little crunch.
Peel the oranges and slice thinly into rounds, assembling them on a large platter with the marinated slices of candy beetroot. Spoon over a little of the marinating juices and season again with some salt and pepper.
Season your hake fillets with salt and pepper. Heat the olive oil in a frying pan and place the hake in skin side down, cooking them for around 3 minutes until the skin starts to crisp.
Turn them over carefully and cook for a further 3 or 4 minutes until they are cooked through but not overdone. Arrange the hake fillets on top of the beetroot and orange slices, squeeze the lemon juice over and sprinkle with parsley and pink peppercorns.
Pan-fried Hake with Candy Beetroot and Orange Salad
## Slow-roasted Salmon with Blood Orange, Lemon, Fennel and Parsley
Blood orange may seem a surprising ingredient to pair with salmon, but it really enhances the subtle flavours of the dish and makes everything more zingy, fresh and vibrant.
##### SERVES 2
* 1 fennel bulb
* 2 lemons
* sea salt and freshly ground black pepper
* 1 blood orange
* 30g fresh flat-leaf parsley
* 2 salmon fillets, skin on (about 300g)
* olive oil
* a couple of sprigs of fresh dill to taste
##### METHOD
Preheat the oven to 150ºC fan/gas 3.
Slice the fennel as thinly as possible and arrange on a plate. Squeeze the juice of half a lemon over the slices and sprinkle with salt. Carefully peel the blood orange and slice the flesh into thin rounds, arranging them on the plate with the fennel.
Pick and chop the parsley leaves and sprinkle half on top of the fennel and blood orange. Cover with cling film and leave in the fridge to marinate while you cook the salmon.
Season the salmon fillets with salt and pepper. Place skin side down in a small baking dish and squeeze the juice of half a lemon over the salmon. Slice the remaining lemon into thin rounds and arrange around and on top of the fish. Sprinkle with the remaining parsley and drizzle with olive oil.
Bake in the oven for 25 to 30 minutes until the salmon is slightly opaque. Place the salmon on top of the plated fennel and blood orange, scatter with the dill and serve.
Slow-roasted Salmon with Blood Orange, Lemon, Fennel and Parsley
## Breda's Fish Pie
Fish pie on a Friday evokes strong memories of growing up in Currabinny. William's mum, Breda, would make it, and all the family looked forward to the smells from the stove, the fogged-up windows and taking turns to mash the buttery potato until it was smooth and velvety. Delicious smoky fish with a crusty potato topping – it was like a warm hug at the end of a long week.
##### SERVES 4–6
* 1 large onion
* 10 cloves
* 450ml milk
* 300ml double cream
* 3 sprigs of fresh thyme
* a small bunch of fresh flat-leaf parsley, plus extra to garnish
* 1 bay leaf
* 6 black peppercorns
* 750g good white fish fillets, such as cod or hake, skin on
* 250g smoked haddock, skin on
* 1 medium-sized leek
* 250g butter, plus extra for the leek
* rapeseed oil
* 4 medium organic eggs, plus 1 egg yolk
* 50g cream flour
* 1¼kg floury potatoes, such as Maris Piper
* sea salt and freshly ground black pepper
* 100g grated Irish Cheddar cheese
##### METHOD
Peel the onion, cut it in half and stud with the cloves. Put it in a large, heavy-based saucepan with the milk, cream, herbs and peppercorns. Add the fish fillets, bring to the boil gently and simmer uncovered for around 6 minutes.
Remove from the heat, pop the lid on and leave for a further 2 or 3 minutes, until cooked through.
Remove the fish when cool enough to handle, keeping the poaching liquid in the pan. Slip the skins off gently while removing any bones you might see, then flake the fish into large pieces.
Meanwhile, remove the green part of the leek (keep aside for stocks and soups), then wash the white part and thickly slice. Put a little butter and rapeseed oil into a saucepan on a medium heat to melt, then add the leek. Cook for around 6 minutes, until softened.
Boil 4 eggs in another saucepan for 8 minutes, take off the heat, run under cold water and remove the shells.
Strain the fish poaching liquid and return to the pan on a low heat. Make a roux in a small saucepan with 50g of the butter and the flour, whisking them together to form a thick brown paste. Whisk this roux into the poaching liquid until it starts to thicken.
Put the flaked fish back in, along with the leeks. Slice the eggs into quarters or eighths and add to the liquid too. Place everything in a large casserole dish or ovenproof dish. Preheat the oven to 180ºC fan/gas 6.
Peel the potatoes and boil in salted water until a knife or skewer goes easily through them. Drain and mash together with the rest of the butter and some salt and pepper. Leave to cool slightly before stirring in the egg yolk.
Spoon the mash over the fish, run the tines of a fork all over the surface and sprinkle with the grated Cheddar.
Bake in the oven for around 30 minutes, until golden brown and bubbling. Leave to settle after removing it from the oven, sprinkle with some chopped parsley and serve.
Breda's Fish Pie
## Barbecued Mackerel
Catching mackerel is a powerful Currabinny memory. William recalls long summer days, high tides, and the schools of mackerel that would herd millions of tiny sprat into shallower waters. Amateur fishermen would descend upon the piers, slipways and low cliff edges around Cork Harbour. Those with boats could get behind the schools of mackerel and would reap the easiest catches. By the end of the summer, the freezer in Currabinny would be filled to bursting point.
Although completely native to Irish waters, there is something exotic about the marbled and tiger-striped, blue, green and black bodies of mackerel. If you can get them fresh during the summer, their rich oily flesh can take centre stage as part of a delicious and easily prepared lunch or dinner.
##### SERVES 2–4
* 2 whole fresh mackerel or 4 fresh fillets, skin on
* olive oil
* sea salt and freshly ground black pepper
* 1 lemon
* 8–10 sprigs of fresh thyme
##### METHOD
You can of course buy mackerel fillets in your local fishmonger, prepared and ready to cook. If you do have whole fish though, it is quite straightforward to fillet them yourself, as you'll see below. Make sure you light your barbecue before you prepare the fillets, so that it has enough time to heat up.
Using a sharp filleting knife, make an incision behind both fins behind the head of the fish. Flip the fish on to its belly and cut the head off, straight through the backbone. Throw the head away or keep for a fish stock – you could even use them as bait to catch other fish. Slice down the backbone of the fish's body, dragging the knife as close to it as possible. Repeat this on the other side until both fillets are removed. Use tweezers to remove the pin-bones from the middle of each fillet. Rinse the fillets with cold water and pat dry with kitchen paper.
Rub some olive oil into the skin of the fish and season well with salt and pepper on both sides. Cut the lemon in half and slice one half into a few rounds. Squeeze the other half over the mackerel fillets.
Brush some olive oil over the hot grill (or a hot griddle pan) and arrange the sprigs of thyme as a sort of bed on the grill. Place the mackerel fillets skin side down on the thyme sprigs. The sprigs will likely burn and even catch fire, but this will all add to the smoky aroma you want the fish to absorb. After 3 to 4 minutes, turn the fillets over and cook for a further 3 to 4 minutes, until the flesh has turned grey-white. Garnish with the lemon rounds.
We like to serve this with a spicy cucumber pickle (see here) or, for a more substantial meal, a balsamic cucumber and Kalamata olive salad (see here) with some lemony Bretagne sauce (see here).
Barbecued Mackerel
## Salmon, Samphire, Broccoli and Cream Pappardelle
When William was growing up in Currabinny, one of his neighbours used to harvest samphire from the little cliffs that jut down from the forest and crumble into the sea. While he didn't think much of it then, he has grown to love this salty little weed, which is a perfect companion to fish.
##### SERVES 2
* 400g tenderstem broccoli
* 100g samphire
* 400g dried pappardelle pasta
* 2 tablespoons olive oil
* 350g salmon fillets, skin removed
* sea salt and freshly ground black pepper
* 150ml cream
* zest of 1 lemon
* 15g fresh flat-leaf parsley, leaves picked and chopped
##### METHOD
Bring some salted water to the boil in a small saucepan, add the broccoli and samphire and simmer for 2 minutes. Drain well, rinse with cold water and set aside.
In another, larger pan, bring some well-salted water to the boil and add the pasta. Simmer for 8 to 10 minutes until al dente. Drain and return to the pan, stir through 1 tablespoon of olive oil and set aside.
Cut the salmon into large pieces and season with salt and pepper. Heat the other tablespoon of oil in a large frying pan and add the salmon, together with the broccoli and samphire. Cook for around 1 minute before adding the cream.
Bring the cream to a simmer, then add the lemon zest, chopped parsley, and salt and pepper to taste. Make sure the salmon has been cooked all the way through before removing from the heat. Stir the pasta through the sauce and serve.
## Salmon with Baby Potatoes, Capers and Garlic, Lemon and Parsley Butter
This is such a classic dish! It celebrates the purity of simple ingredients and is full of flavours that work together beautifully. In the summer use new-season baby potatoes, as they will be in abundance and at their very best.
##### SERVES 2
* 2 salmon fillets, skin on (about 150g each)
* sea salt and freshly ground black pepper
* 2 tablespoons olive oil
* 600g new potatoes
* 50g garlic, lemon and parsley butter (see here)
* 30g fresh flat-leaf parsley
* 2 tablespoons capers
##### METHOD
Preheat your oven to 150ºC fan/gas 3.
Season the salmon fillets with salt and pepper. Drizzle the olive oil into a small baking dish and add the fillets skin side down. Bake in the oven for 25 to 30 minutes until the salmon is opaque and tender in the middle.
Meanwhile, put the new potatoes into a large pot of salted water and bring to the boil. Simmer for 20 to 25 minutes until just tender. Drain the potatoes well and add the garlic, lemon and parsley butter, coating thoroughly.
Pick and chop the parsley leaves and sprinkle, along with the capers, over the buttery new potatoes. Divide between two plates and top with the salmon fillets.
## Smoked Rainbow Trout with Fennel, Goat's Cheese, Pink Peppercorns and Dill
Delicious flakes of pink trout and the luxury of soft goat's cheese make for surprisingly hearty eating. When making this, we like to use the very best ingredients: smoked trout from Goatsbridge Trout Farm in Kilkenny and goat's cheese from Ardsallagh Goat Farm in Cork – both delicious prize-winning Irish products and widely available.
##### SERVES 2–4
* 15g fresh dill
* 2 medium-sized fennel bulbs
* juice of ½ a lemon
* sea salt and freshly ground black pepper
* 1 tablespoon rapeseed oil
* 300g smoked rainbow trout
* 165g soft goat's cheese, such as Ardsallagh
* 2 teaspoons pink peppercorns
##### METHOD
Chop the dill. Slice the fennel as thinly as possible and arrange on a large plate. Squeeze the lemon juice over, sprinkle with salt, pepper and the chopped dill, and drizzle with the rapeseed oil.
Cover the plate with cling film and leave to marinate in the fridge for 20 to 30 minutes – the fennel should soften slightly as the acidic lemon juice 'cooks' it.
When the fennel has marinated, flake the smoked rainbow trout over it, add the goat's cheese in dollops all around the plate and sprinkle with the pink peppercorns. Simple, fresh and delicious, this is best served with hunks of decent bread to mop up the flavours.
Smoked Rainbow Trout with Fennel, Goat's Cheese, Pink Peppercorns and Dill
## Asparagus and Smoked Rainbow Trout with a Herb Sauce
This recipe is extremely simple, tasty and light. If you make the garlic, lemon and parsley butter for the ciabattas very heavy on the garlic, it plays well with the greenness of the asparagus and the herb sauce. This is perfect for a hot day when all you want is to be outside with a plate of something delicious and light on your lap!
##### SERVES 4
* 500g super-fine asparagus, untrimmed if young
* 1 large ciabatta loaf
* 75g garlic, lemon and parsley butter (see here)
* 4 smoked rainbow trout fillets
* 1 tablespoon of capers
##### _FOR THE SAUCE:_
* a good handful of fresh basil leaves
* a small bunch of fresh mint, leaves picked
* a good handful of fresh flat-leaf parsley leaves
* 4–5 anchovy fillets, drained
* 2 teaspoons Dijon mustard
* 4 tablespoons olive oil
* juice of ½ a lemon
##### METHOD
Blitz all the ingredients for the sauce in a food processor until smooth.
Cook the asparagus in a small pan with a splash of water until tender but still with a crunch – this should take around 2 minutes.
Slice the ciabatta thickly and spread generously with the garlic, lemon and parsley butter. Toast under a grill until it turns golden and the butter has melted and soaked through the bread.
Divide the slices of toast between four plates, top with some of the asparagus spears and flake a trout fillet over each then drizzle with the sauce and scatter the capers on top.
Asparagus and Smoked Rainbow Trout with a Herb Sauce
## Recipe List
1. Roast Chicken with Harissa Butter
2. Penne Pasta with Ham, Cabbage, Wild Garlic Pesto and Pickled Walnut
3. Ham in Juniper and Apple Juice
4. Steak Sandwich with Fried Onions, Cashel Blue Cheese and Mushroom Ketchup
5. Lamb Steak Sandwich with Garlic, Lemon and Parsley Butter and Wicklow Bán Brie
6. Turkey Burgers with Chanterelles and Gruyère
7. Ferryhouse Cottage Pie
8. Fried Cabbage and Ham Sandwich
9. Pappardelle with Speck, Ricotta, Beetroot Tops and Walnuts
10. Sausage and Thyme Stuffing
## Roast Chicken with Harissa Butter
A roast chicken dinner should be special, like it was in our grannies' time. When people you love are gathered around the table, a plump roast chicken is the most comforting of dishes. This is one of our favourite ways to prepare it. Anointing the bird generously with harissa butter keeps it moist and gives the flesh a subtle heat and fragrance.
##### SERVES 4–6
* 100g butter, at room temperature
* 2 teaspoons harissa (see here)
* 1 large organic chicken (about 1.5kg)
* sea salt and freshly ground black pepper
* 1 lemon
* 6 cloves of garlic, unpeeled
* 5 or 6 sprigs of fresh thyme
##### METHOD
Preheat the oven to 200ºC fan/gas 7.
Make the harissa butter by mashing the butter in a bowl with the harissa until well combined.
Rub the chicken all over with the harissa butter, making sure you coat all of the skin, including the legs and wings. Use a knife to separate the skin from the flesh and rub some harissa butter under there as well. Season the chicken inside and out with salt and pepper.
Put the lemon, pierced several times with a knife, inside the chicken, together with 2 cloves of garlic and 2 sprigs of thyme. Place the chicken in a large roasting tin with the remaining thyme and garlic cloves scattered around it.
Put into the oven and cook for around 1 hour 20 minutes or until the juices run clear – the cooking time will depend on the size of the chicken. The skin might start to burn, so keep an eye on it and cover with tinfoil if necessary.
Remove from the oven and place on a large wooden board for carving. Here we've served it on a bed of herby couscous, but it would be equally good with roast potatoes and your favourite vegetables.
Roast Chicken with Harissa Butter
## Penne Pasta with Ham, Cabbage, Wild Garlic Pesto and Pickled Walnut
This is another variation – an elevation, we go so far as to say! – of bacon and cabbage. Leftover ham is one of the most useful things you can have in your fridge. It's unbeatable here paired with its traditional companion – cabbage – and a garlicky pesto. Wild garlic pops up everywhere in early spring; make a big batch of this pesto when it's in season and you'll use it year round.
##### SERVES 2
* 200g penne pasta
* 1 tablespoon butter
* olive oil
* 1 savoy cabbage, shredded
* 100g cooked ham, or speck, prosciutto or pancetta
* 1 pickled walnut, sliced
* grated Parmesan cheese, to serve
##### _FOR THE PESTO:_
* 75g wild garlic, stalks removed
* 30g fresh flat-leaf parsley, stalks removed
* juice of ½ a lemon
* 80ml rapeseed oil, plus extra to seal the jar
* 30g Parmesan cheese, grated
* 30g walnuts
* sea salt and freshly ground black pepper, to taste
##### METHOD
Put the ingredients for the pesto into a food processor and blitz to a smooth paste. Decant into a sterilized jar and pour a little rapeseed oil over the top to seal. Refrigerate until needed (it will keep this way for 2 weeks).
Bring a large pot of salted water to the boil, add the penne and cook according to the packet instructions.
Meanwhile, put the butter and a drop of olive oil into a large, heavy-based frying pan on a medium heat and add the shredded cabbage. Season with a little salt and pepper. Cook the cabbage for 3 to 4 minutes until wilted,* then tear in the ham and cook for a further 2 minutes. Add a little of the pasta water if it looks dry.
Add 2 tablespoons of the wild garlic pesto and stir to combine, then cook for a further 2 to 3 minutes.
When the pasta is cooked, drain in a colander and add to the pan, stirring the sauce into the pasta until it is all well coated. Serve with a few slices of pickled walnut on top, and offer some extra grated Parmesan.
* _We would usually cook the cabbage so that it's softer and less crisp than the cabbage in the photograph. You can cook it to taste._
Penne Pasta with Ham, Cabbage, Wild Garlic Pesto and Pickled Walnut
## Ham in Juniper and Apple Juice
While there is nothing quite like a great boiled him, sometimes it can be a bit salty. A delicious solution is to cook it in a good cloudy apple juice. The mellow sweetness of the juice balances the saltiness of the meat beautifully.
##### SERVES 4–6
* 2kg boneless ham or gammon
* 2 litres cloudy apple juice
* 1 leek
* 1 stick of celery
* 1 carrot
* 12 juniper berries
* 12 black peppercorns
* a large bunch of fresh flat-leaf parsley
* 2 bay leaves
##### METHOD
Place the ham in a large pot and fill with cold water.
Bring the water to boil, then take off the heat, drain the water and rinse the ham. Return the ham to the pot and pour in the apple juice, bringing it slowly to the boil.
Meanwhile, wash and trim the leek and cut it in half. Cut the celery stick in half too, and cut the carrot into large chunks. Add it all to the pot. Crush the juniper berries and peppercorns lightly with a knife and add these, along with the parsley (stems and all) and bay leaves.
When the apple juice has come to the boil, turn the heat down to a gentle simmer and scoop off any scum that has formed on the top.
Put a lid askew on the pot and leave to simmer for 2 hours, then check the ham to see if it is cooked through – to do this, remove it from the cooking liquid, pierce it with a knife and check if the juices are running clear.
Once it's ready, leave the ham to cool slightly in the cooking liquor for up to 20 minutes before carving. Lift out the ham, cut into thick slices and drizzle a little of the cooking liquor over the top (discarding the vegetables). This will keep the ham lovely and moist.
We like to serve this with some mustard parsnip mash (see here) and some fried apple and sage (see here).
## Steak Sandwich with Fried Onions, Cashel Blue Cheese and Mushroom Ketchup
Living in the city, maybe juggling lots of daily commitments or working shifts, trying to grab dinner when running out the door again for work or meetings... this is the reality of life for many of us nowadays. This is William's favourite 'grab and go' meal. You can put it together super-quickly to eat in the back of a taxi on the way to your destination – but ask nicely before you tuck in!
##### SERVES 2
* 1 clove of garlic
* 1 medium-sized onion
* a handful of fresh flat-leaf parsley leaves
* 2 medium-sized striploin steaks
* sea salt and freshly ground black pepper
* olive oil
* 1 baguette, cut into two pieces
* 2 teaspoons Dijon mustard
* mushroom ketchup (see here)
* 50g Cashel Blue cheese
##### METHOD
Peel the garlic and onion and slice both thinly. Roughly chop the parsley. Place a griddle pan or frying pan on a medium heat and season the steaks with salt and pepper. When the pan is hot, cook the steaks in a little olive oil for 2 to 3 minutes on each side. Transfer to a board to rest for 5 minutes.
Meanwhile, put the sliced onion and a little more oil into the pan and cook for around 8 to 10 minutes until the onion is brown and softened. Add the sliced garlic and cook for another minute or so. Stir the parsley into the onion and garlic and take off the heat.
Open up the baguettes and spread a little mustard on one side of each and a little mushroom ketchup on the other. Cut the steaks into thick pieces and place inside the baguette. Crumble the cheese on top, then add the onions and sandwich together. Serve immediately.
## Lamb Steak Sandwich with Garlic, Lemon and Parsley Butter and Wicklow Bán Brie
Imagine: molten cheese, juicy lamb, the crunch of sourdough bread, garlicky butter running down your fingers... this sandwich is pure indulgence!
##### SERVES 2
* 2 boneless lamb steaks
* sea salt and freshly ground black pepper
* 4 medium-sized chestnut mushrooms
* 2 tablespoons garlic, lemon and parsley butter (see here)
* olive oil
* 1 large sourdough baguette
* 2 tablespoons mayonnaise
* 4 thick slices of Wicklow Bán Brie or any good young Brie
##### METHOD
Preheat your grill. Season the lamb steaks liberally on both sides with salt and pepper. Wipe the mushrooms and slice thinly. Put the garlic, lemon and parsley butter and a little olive oil into a large frying pan on a high heat.
When the butter is bubbling, add the steaks and reduce the heat to medium. Cook for 2 minutes, then add the mushrooms and flip the lamb steaks over.
Cook for a further 4 minutes while spooning the butter over the top of the steaks and stirring the mushrooms occasionally.
Cut the baguette into two pieces widthways, then slice each piece in half lengthways. Put them under the hot grill for a couple of minutes until they turn golden.
Remove from the grill, then spread the mayonnaise on the top pieces of bread and drizzle some of the buttery juices from the frying pan over the bottom pieces, letting them soak into the bread.
Roughly chop the lamb steaks and place them on the bottom pieces of bread. Top with the mushrooms, then the slices of Brie. Put the top pieces of bread on and eat immediately, letting the juices run down your chin!
Lamb Steak Sandwich with Garlic, Lemon and Parsley Butter and Wicklow Bán Brie
## Turkey Burgers with Chanterelles and Gruyère
It's hard to make a case for turkey when you can just as easily have chicken – except in the case of this burger. Here, turkey – which can be dry and bland – is made juicy with some added bacon and gains depth of flavour from the Worcester sauce, saltiness from the Parmesan and a great aroma from the thyme and lemon zest. Topped with melted Gruyère, woody chanterelles and a little mayonnaise, and sandwiched between fluffy brioche buns, it is the ultimate burger.
##### SERVES 4
* 2 shallots
* 25g butter, plus extra to fry the shallots
* 450g lean turkey mince
* 150g streaky bacon, finely chopped
* 1 medium organic egg
* 30g breadcrumbs (made from slightly stale bread)
* 1 tablespoon fresh thyme leaves
* 1 tablespoon Worcester sauce
* 2 teaspoons lemon zest
* 2 tablespoons grated Parmesan cheese
* a good pinch of sea salt and freshly ground black pepper
* 200g chanterelle mushrooms
* 1 tablespoon chopped fresh flat-leaf parsley leaves
* olive oil
* 4 thin slices of Gruyère cheese
* 4 brioche buns
* mayonnaise
* a handful of mixed leaves, such as rocket, watercress, baby chard
##### METHOD
Peel and dice the shallots, then sauté in a frying pan with a little butter until softened. Put into a large mixing bowl with the turkey mince, bacon, egg, breadcrumbs, thyme, Worcester sauce, lemon zest and Parmesan. Season well with salt and pepper and mix thoroughly.
Form 4 equal-sized burgers with your hands and place on a plate in the fridge for 30 minutes.
Preheat your grill.
Put 25g of butter into a large pan on a medium-high heat, add the chanterelles and cook for around 5 minutes until soft. Add the parsley and season with salt and pepper. Transfer to a plate and drizzle over a little olive oil.
Fry the burgers in the pan over a medium heat, cooking for around 5 minutes on each side or until cooked through. Place a slice of Gruyère on top of each burger for the final 2 or 3 minutes of cooking.
Split the brioche buns and toast under the hot grill, then spread mayonnaise on both sides and add some salad leaves. Place a cheesy burger on the bottom half of each, load with the chanterelles and put the top piece of brioche on. Serve immediately.
Turkey Burgers with Chanterelles and Gruyère
## Ferryhouse Cottage Pie
When William was growing up, cottage pie was a favourite way to use up the leftover beef from the Sunday roast. On the Monday the cold meat went through a mincer, was added to gravy, topped with potato mash and baked in the oven. Delicious! This recipe is our more up-to-date version made from raw minced beef, and is sure to become a comforting favourite.
The recipe is named in honour of William's home in Currabinny (here) – the house is called Ferryhouse because in olden times it was the home of the man who ran the ferry across the harbour from Currabinny to Crosshaven.
##### SERVES 4–6
* 1 large onion
* 2 medium-sized carrots
* 1 large stick of celery
* 500g lean minced beef
* sea salt and freshly ground black pepper
* 45g cream flour
* 200g butter
* 40ml Worcester sauce
* 750ml chicken stock
* 4 sprigs of fresh thyme
* a small bunch of fresh flat-leaf parsley, leaves picked and chopped
* cooked chard or spinach (optional)
* 1.2kg floury potatoes, such as Maris Piper
* 150ml milk
* 50g Irish Cheddar cheese, grated
* 50g Parmesan cheese, grated
##### METHOD
Peel and chop the onion and carrots. Finely chop the celery. Fry the mince in a large frying pan until brown and add the trio of onion, carrots and celery. Cook until the vegetables have softened, and season everything well with salt and pepper.
Add the flour, 50g of the butter, and the Worcester sauce, and cook for a further minute.
Heat the chicken stock in a saucepan and add to the frying pan, then bring to the boil and reduce to a simmer for 5 minutes. Add the thyme and parsley, then check the seasoning for salt and pepper.
Everything should have thickened nicely at this point. Transfer to a large casserole or ovenproof dish. For an extra dimension, you could layer some cooked chard or spinach on top.
Preheat the oven to 180ºC fan/gas 6. Peel and chop the potatoes. Boil them in salted water until cooked through, then drain and mash together with the remaining butter and the milk.
Season the mash and scoop on top of the meat sauce, spreading to cover all the meat. Sprinkle with the grated Cheddar and Parmesan, then bake in the oven for 25 to 30 minutes until golden and bubbling.
Ferryhouse
## Fried Cabbage and Ham Sandwich
This recipe reinvents the traditional bacon and cabbage dinner. Forget about over-boiled, greasy meat and veg. In this recipe, cabbage and ham are fried in garlicky, lemony butter. Then a crusty baguette soaks up the pan juices before being stuffed with the fried ingredients. You may want to eat straight from the pan, rather than bothering with a plate – we totally understand!
##### SERVES 1
* 2 tablespoons garlic, lemon and parsley butter (see here)
* olive oil
* a good handful of green or York cabbage
* 6 rough slices of cooked ham, or charcuterie ham such as speck, prosciutto or capicola
* a hunk of good-quality French baguette, cut in half
##### METHOD
In a large frying pan on a medium heat, melt the garlic, lemon and parsley butter with a drop of olive oil.
Shred the cabbage thinly and add to the pan, softening it for 2 or 3 minutes (you can wilt it more than in the photograph). Add the ham and cook for a further minute. Push the contents to one side and place the pieces of bread on the hot pan, allowing them to soak up the hot garlicky butter.
Spoon the ham and cabbage on to the bottom piece of baguette, put the top on and eat immediately.
Fried Cabbage and Ham Sandwich
## Pappardelle with Speck, Ricotta, Beetroot Tops and Walnuts
This recipe came about when William brought home a big bunch of mucky beetroots. The inky purple had spread into the veins of the beautiful leaves and he couldn't bear to throw them away. Beetroot tops can be harder work than other leaves (even kale), but spending a little time – cooking them down until the purple stalks become tender and sweet – will reward you with a feast of peppery greenness. You can of course use chard instead, which is a relative of beetroot.
##### SERVES 4
* 20 or so beetroot tops, or 20 or so chard leaves (stalks removed), roughly chopped
* 1 clove of garlic
* 15g fresh flat-leaf parsley
* 80g shelled walnuts
* 400g dried pappardelle pasta
* 2 tablespoons butter
* 100g speck, sliced thinly
* juice of ½ a lemon
* 1 tablespoon grated Parmesan cheese
* sea salt and freshly ground black pepper
* 250g ricotta
##### METHOD
Bring a large pot of salted water to boil for the pasta. Wash the beetroot tops well in cold water and leave to drain in a colander. Peel and thinly slice the garlic clove, then roughly chop the parsley leaves and walnuts.
Add the pasta to the pot and cook for 7 to 10 minutes until al dente. While the pasta is cooking, heat the butter in a saucepan and add the sliced garlic. When the garlic starts to turn golden and fragrant, add the beetroot tops, wilting them in the butter.
Next add the speck and stir through the pan for around 1 minute before adding the lemon juice and Parmesan.
Drain the pasta and add it to the pan, then take the pan off the heat and add some salt and pepper to taste. Stir through the chopped walnuts and parsley. Divide the pasta between four plates and garnish each one with several blobs of ricotta.
## Sausage and Thyme Stuffing
Like many people in their twenties, we return to our parents for Christmas – you can't beat the traditional family Christmas! But in the weeks beforehand we like to gather with friends to have our own celebration and we all prepare something for the table. William came up with this sausage and thyme stuffing. It was meant to be just a small side dish; now it's a main in its own right. Such is its popularity that every year he has to bring a bigger dish, so he's very proud of it. A stuffing that outshines the turkey, the ham and the spiced beef – now that is a Christmas miracle!
##### SERVES 4 AS A SIDE DISH
* 2 medium-sized onions
* 150g stale white bread
* sea salt and freshly ground black pepper
* 50g butter, melted, plus extra to fry the onions
* 50g cooked chestnuts, roughly chopped
* 800g good-quality sausage meat
* 2 tablespoons fresh thyme leaves
##### METHOD
Preheat the oven to 180ºC fan/gas 6.
Peel and finely chop the onions. Blitz the bread with some salt and pepper in a food processor to make breadcrumbs. In a large bowl, mix the breadcrumbs, melted butter, chestnuts and sausage meat together.
Sauté the chopped onions until soft in a frying pan with a little butter. Leave to cool slightly and add to the sausage meat mixture. Stir through the thyme leaves and scoop the mixture into a ceramic baking dish.
Bake in the oven for 30 to 40 minutes, until golden brown on top.
Sausage and Thyme Stuffing
## Recipe List
1. Gags's Potato Gratin
2. Colcannon with Curly Kale
3. Ginger-braised Leeks
4. Mushroom à la Crème
5. Buttery Purple Carrots
6. Mustard Parsnip Mash
7. Potato and Chive Cake
8. Buttered Cabbage with Caraway or Fennel Seeds
## Gags's Potato Gratin
James's mum, Gags, used to call into his old workplace once a week with a big dish of her potato gratin, still hot from the oven. The whole office would go wild for it, and there would be a dash to the kitchen to get plates and a spatula to divide and conquer. It's rich and irresistible, and a guaranteed way to people's hearts!
##### SERVES 4–6
* Butter for greasing
* 110g strong Cheddar cheese
* 55g Parmesan cheese
* 55g cold butter
* 2 cloves of garlic
* 4 large or 7 medium potatoes
* 500ml fresh cream
* sea salt and freshly ground black pepper
##### METHOD
Preheat the oven to 190ºC fan/gas 6 and lightly butter a gratin dish.
Grate the two cheeses together, chop the cold butter into small pieces, and peel and finely chop the garlic. Peel and thinly slice the potatoes.
Put a layer of potatoes in the prepared dish and sprinkle on some cheese and pieces of butter, continuing to layer in this way until all the potato is used. Reserve some cheese for the topping.
Mix the cream, garlic and some salt and pepper in a jug and pour over the potatoes in the dish. Sprinkle the remaining cheese on top, cover with tinfoil and bake in the oven for 1 hour.
Remove the tinfoil and bake for about 15 minutes until bubbling and nicely browned. Leave to set in the dish for about 15 minutes before serving.
Gags's Potato Gratin
## Colcannon with Curly Kale
Curly kale is perhaps the least attractive, toughest and most misunderstood of the kale or brassica varieties. It deserves more love, and here is a recipe that gives it the treatment it deserves: wilted down until tender and mixed with buttery mashed potatoes in this wonderful traditional dish.
##### SERVES 4–6
* 4–5 medium-sized floury potatoes, such as Red Rooster
* 2 leeks
* 100g butter, plus an extra knob to serve
* 200g curly kale
* a handful of chopped wild garlic (optional)
* 250ml milk
* 175ml double cream
* sea salt and freshly ground black pepper
* 2 spring onions, sliced thinly
##### METHOD
Peel the potatoes, wash and trim the leeks and chop both roughly, keeping them separate. Boil the potatoes in a large pot of salted water for 15 to 20 minutes until cooked through. Remove and drain.
In a large saucepan, heat 100g of butter and add the leeks. Sauté gently for around 10 minutes until softened. Meanwhile, remove the stalks from the curly kale and discard, then roughly chop the leaves. Add the wild garlic to the pan if using, then add the chopped kale and cook until wilted.
Pour in the milk and cream and bring to a simmer. Tip in the cooked potatoes and season with salt and pepper. Mash the mixture with a potato masher until smooth. Stir through an extra knob of butter for good measure, together with the spring onions, and transfer to a serving dish.
_Colcannon with curly kale is shownhere_
## Ginger-braised Leeks
This is perfect at any time, but particularly as part of a winter dinner. Ginger is not only warm and aromatic, it's also good for you. Using it like this makes for a really comforting side dish.
##### SERVES 2–4
* 2 medium-sized leeks
* 1 thumb-sized piece of fresh ginger
* 1 tablespoon rapeseed oil
* 25g butter
* 4–5 sprigs of fresh thyme
* juice of ½ a lemon
* 80ml white wine
* sea salt and freshly ground black pepper
##### METHOD
Preheat the oven to 180ºC fan/gas 6.
Remove the green part of the leeks (keep aside for stocks and soups), then wash the white part and chop into thickish rounds. Peel and grate the ginger and set aside.
Heat the rapeseed oil and butter in a heavy-based frying pan until sizzling. Add the leeks, stirring to coat in the oil and butter. Gently soften over a medium-low heat, being careful to keep the rounds intact.
Transfer the leek rounds to a roasting pan, filling the bottom in a single layer. Cover with the ginger, thyme and lemon juice, and pour the wine over everything. Drizzle with oil if desired. Season with salt and pepper.
Roast in the oven for 25 to 30 minutes until the edges of the leeks start to brown and almost all of the liquid has evaporated.
## Mushroom à la Crème
Much as William loves fresh local ingredients, he has a love–hate relationship with mushrooms. On the positive side, there are many varieties to choose from and the fact you can forage for them However, they can be pungent and overpowering when left to their own devices. This recipe – using butter, herbs, cream and lemon juice – treats mushrooms with respect and brings out their natural earthy nuttiness. It's a truly delicious side dish.
##### SERVES 2–4
* 225g chestnut mushrooms (or a mix of chestnut and chanterelles)
* 1 shallot
* 15g fresh flat-leaf parsley
* 5 fresh chives
* 50g butter
* sea salt and freshly ground black pepper
* olive oil
* 125ml cream
* juice of ½ a small lemon
##### METHOD
Wipe and thinly slice the mushrooms. If using chanterelles, you can leave them whole. Peel and finely dice the shallot, and chop the parsley leaves and chives. Place a large, heavy-based saucepan on a medium heat and put the butter into the pan. When melted, add the shallot and cook for 5 to 10 minutes until softened.
Season the mushrooms well with salt and pepper. Add them to the pan and stir through the shallot and butter. Add a little drizzle of olive oil and increase the heat. Cook for around 10 minutes until the mushrooms are soft, brown and smelling nutty.
Pour in the cream and allow to bubble, then reduce the heat and add the lemon juice, parsley and chives, stirring them through. Serve immediately.
Mushroom à la Crème
## Buttery Purple Carrots
It's worth looking out for purple carrots. Their peppery sweetness is a wonderful match for roast pork or game birds like pheasant or duck. But don't worry if you can't find the purple variety – the regular orange ones are also delicious when prepared this way!
##### SERVES 4
* 6 purple carrots (or a mix of purple and orange)
* 1 tablespoon butter
* olive oil
* 1 tablespoon caster sugar
* 50ml water
* sea salt and freshly ground black pepper
* 1 tablespoon roughly chopped fresh flat-leaf parsley leaves
##### METHOD
Wash the carrots and thickly slice them at an angle. Heat the butter and a drizzle of olive oil in a large, heavy-based saucepan until bubbling. Add the sugar and water and stir for a moment. Tip in the slices of carrot and stir to coat, seasoning with salt and pepper.
Put a lid slightly askew on the saucepan and cook on a medium-high heat for 10 minutes. Remove the lid, throw in the parsley and cook for another 5 minutes or so, stirring occasionally. It is ready when the liquid has mostly evaporated, leaving the carrots coated in a syrupy, buttery sauce.
Buttery Purple Carrots
## Mustard Parsnip Mash
A satisfying combination of sweet parsnip and warming Dijon mustard, this is a perfect accompaniment to ham, pork, game and other wintry, hearty meats.
##### SERVES 2–4
* 800g parsnips
* 80g butter
* a drizzle of rapeseed oil
* a pinch of sea salt and freshly ground black pepper
* 1 teaspoon lemon juice
* 1 tablespoon Dijon mustard
##### METHOD
Peel the parsnips and cut into small cubes. Melt the butter and rapeseed oil in a large pan on a medium heat. Add the parsnips and stir to coat all of the pieces with butter. Cook for 8 to 10 minutes until soft and sticking to the pan.
Reduce the heat to low and season with the salt, pepper and lemon juice. With a potato masher, crush the parsnips into a mash, adding a little more butter if needed. Stir the mustard through and serve.
## Potato and Chive Cake
This potato cake is gentle and comforting and has subtle chive flavours. It works really well with most things and it's also very satisfying on its own. Throw in a bit of salad on the side and you have a perfect light lunch.
##### SERVES 4
* 800g large potatoes
* 3 onions
* a bunch of fresh chives
* 4 medium organic eggs
* sea salt and freshly ground black pepper
* 1 tablespoon olive oil
##### METHOD
Peel and grate the potatoes, then peel and slice the onions thinly. Chop the chives finely. In a medium-sized bowl, lightly beat the eggs and stir in the grated potato, onions and chives. Season with salt and pepper.
Heat the olive oil in a large frying pan and add the potato mixture, flatten with a spatula and cook over a gentle heat for around 8 minutes. Turn over carefully and cook for another 8 minutes.
## Buttered Cabbage with Caraway or Fennel Seeds
Caraway and fennel seeds are both earthy and sweet. They add subtly different anise flavours to the cabbage – fennel is a simple flavour and well liked; caraway is more complex, earthy and aromatic and tends not to be quite as popular – so you can experiment to see which you prefer. We never blanch the cabbage beforehand as the butter, oil and heat of the pan combine to tenderize the thin shreds perfectly. This dish would go perfectly with the ham in juniper and apple juice here.
##### SERVES 2–4
* ½ a savoy cabbage
* 75g butter
* rapeseed oil
* ½ teaspoon caraway seeds or 1 teaspoon fennel seeds
* sea salt and freshly ground black pepper
##### METHOD
Shred the cabbage into long thin ribbons. Melt the butter with a small drizzle of rapeseed oil in a medium, heavy-based pan. Add the cabbage and cook on a medium-high heat for 5 minutes until it is all wilted and well coated in the butter.
Add the caraway or fennel seeds and some salt and pepper, reduce the heat to medium and cook for a further 8 to 10 minutes, stirring continuously. You want the cabbage to brown slightly and become almost sticky and caramelized.
## Recipe List
1. Kale, Cashew and Wakame Pesto
2. Beetroot and Seaweed Hummus
3. Harissa
4. Spicy Pickled Carrot
5. Spicy Cucumber Pickle
6. Tahini Dressing
7. Flavoured Butters
* Burnt Onion Butter
* Orange and Cinnamon Butter
* Kombu Butter
* Honey and Redcurrant Butter
* Garlic, Lemon and Parsley Butter
8. Mushroom Ketchup
9. Walnut and Feta Dip
10. Green Herb and Lemon Dressing
11. Spring Onion Vinaigrette
12. Lime and Coriander Mayo
13. Bretagne Sauce
14. Fried Apple and Sage
## Kale, Cashew and Wakame Pesto
Curly kale sometimes seems like the poor relation of leafy green vegetables. For years, people avoided it. Then, because it's so fabulously good for you, it became trendy and everyone pretended to enjoy it raw, covered in lemon juice – as if that was going to soften the tough leaves. In our opinion, kale is best eaten cooked, not raw – except in this recipe, where it's blitzed with oil, cheese, lemon juice and nuts to create a delicious peppery pesto that brims with goodness and flavour. The addition of wakame seaweed gives a lovely hint of the sea.
##### MAKES ABOUT 250G
* 2 cloves of garlic, peeled
* 60g curly kale, stalks removed
* 30g dried wakame, soaked in water and drained
* juice of 1 lemon
* 60g cashew nuts
* 80ml rapeseed oil
* 60g hard cheese such as Gruyère or Parmesan, or a hard mature sheep's cheese such as Cáis na Tíre or Cratloe Hills, grated
##### METHOD
Put all the ingredients into a food processor and whiz until well combined but still textured and chunky. Add more oil to loosen up the mixture if needed.
It will keep in the fridge for up to 2 weeks in a sterilized jar and is delicious stirred through pasta, dolloped on top of soups or spread on crackers.
_The pesto is shown middle right in the picturehere_
## Beetroot and Seaweed Hummus
Hummus wasn't something we ate traditionally in Currabinny. But we grew beetroot in the garden. It was scrubbed hard, boiled, allowed to cool and then either pickled in spicy vinegar or sliced into a salad. Adding beetroot to hummus, with flakes of dillisk seaweed and some horseradish for a little background heat, gives an earthy, sweet, salty and decidedly Irish twist to this Middle Eastern delicacy.
##### MAKES ABOUT 800G
* 2 tablespoons finely chopped dillisk, plus extra flakes to garnish
* 200g beetroot, cooked and peeled
* 1 x 400g tin of chickpeas, drained
* juice of ½ a lemon
* 3 tablespoons tahini
* 2 teaspoons ground coriander
* 100ml rapeseed oil
* a handful of fresh flat-leaf parsley leaves, plus extra to garnish
* 1 clove of garlic, peeled
* sea salt and freshly ground black pepper, to taste
* 1 tablespoon horseradish (or more to taste)
##### METHOD
Put all of the ingredients into a food processor apart from the horseradish and the garnishes. Whiz until smooth and thick.
Scoop the mixture into a serving bowl and stir the horseradish through. Garnish with a few chopped parsley leaves and seaweed flakes.
_The hummus is shown on top right in the picturehere_
## Harissa
Spices toasting in a dry pan, the scent of roses, the intense heat of red chillies... though harissa will transport you to foreign lands, it will also become a staple in your kitchen. Harissa is a North African chilli paste, but we use it for all kinds of things. Whether whipped into good Irish butter that's spread over a chicken for roasting (see here), or combined with oil and vinegar to make a lively salad dressing, it can make any number of dishes more luxurious and special.
##### MAKES 1 X 190ML JAR
* ½ teaspoon cumin seeds
* ½ teaspoon fennel or caraway seeds
* ½ teaspoon coriander seeds
* 5 small fresh red chillies
* 5 small dried chillies
* 1 clove of garlic, peeled
* 2 teaspoons Pedro Ximénez sherry vinegar
* 2 tablespoons rapeseed oil
* 2 teaspoons tomato purée
* a pinch of sea salt
* ½ teaspoon preserved lemon skin
* 1 teaspoon rose water
* ¼ teaspoon dried rose petals
##### METHOD
Toast the cumin, fennel/caraway and coriander seeds in a dry pan for a few minutes until a fragrant aroma rises up. Remove the stalks from the fresh chillies and if you don't like it too hot, slice them all in half and remove the seeds.
Blitz all the ingredients, including the toasted seeds, in a food processor until smooth and well combined.
Transfer into a small sterilized jar and refrigerate – it will keep for at least 2 weeks.
_The harissa is shown on bottom right in the picturehere_
## Spicy Pickled Carrot
The mix of spices and textures in this pickle makes it rich and aromatic. It's best eaten a few days after pickling when the peppercorns have softened and the flavours have really intensified. Perfect on a flatbread with hummus and lamb or in a sandwich or salad.
##### MAKES ABOUT 650G
* 500g carrots
* 1 tablespoon rapeseed oil
* 1½ teaspoons cumin seeds
* 2 teaspoons mustard seeds
* ½ teaspoon fennel seeds
* 5 curry leaves
* 1 teaspoon black peppercorns
* 150ml apple cider vinegar
##### _FOR THE PASTE:_
* 2 teaspoons cumin seeds
* 2 fresh red chillies, finely chopped
* a pinch of sea salt
* 4 cloves of garlic, crushed
* 50g fresh ginger, peeled and chopped
* 100g soft brown sugar
* 3 tablespoons rapeseed oil
##### METHOD
First make the paste. Using a large pestle and mortar, grind all the ingredients for the paste together until smooth and pungent.
Peel and grate the carrots into a large bowl. Heat the oil in a large, heavy-based saucepan and add the cumin, mustard and fennel seeds, cooking for a few seconds until you hear the seeds start to pop. Add the curry leaves and peppercorns and cook for 2 minutes.
Add the spice paste next and cook for a further 2 minutes before adding the vinegar and bringing to a simmer. Add the carrots and stir well to coat. Cook for a further 15 minutes on a medium-low heat. Leave to cool before placing in sterilized jars. It will keep for several months in the fridge.
_The pickled carrot is shown on the left in the picturehere_
Spicy Pickled Carrot
## Spicy Cucumber Pickle
Adding a little chilli heat to cucumber seems to intensify its subtle and refreshing flavour. This is a gorgeous pickle that goes well with a slice of stout and treacle loaf (see here) or makes as a great accompaniment to a sandwich or salad.
##### MAKES 1 X 200ML JAR
* ½ a small fresh red chilli
* 2 cucumbers
* 60ml white wine vinegar
* 2 teaspoons sea salt
* 1 teaspoon caster sugar
##### METHOD
Scrape out the seeds from the chilli and chop as finely as possible. Slice the cucumbers as thinly as possible into rounds. Put these into a bowl together with the vinegar, salt and sugar. Cover with cling film and leave to marinate for at least 2 hours.
This keeps for several months at least if stored in a sterilized jar in a cool, dark place.
_The cucumber pickle is shown at the top in the picturehere_
Kale, Cashew and Wakame Pesto, Beetroot and Seaweed Hummus, Harissa, Spicy Pickled Carrot and Spicy Cucumber Pickle
## Tahini Dressing
Combining the nuttiness of tahini with the bright zing of lemon, this creamy dressing is perfect drizzled over any combination of leaves, salad vegetables or roasted root vegetables.
##### MAKES 120ML
* juice of ½ a lemon
* 1 tablespoon honey
* 5 tablespoons tahini
* 2 tablespoons rapeseed oil
* 1 teaspoon sea salt
* 2 cloves of garlic, crushed
* freshly ground black pepper, to taste
##### METHOD
Put all the ingredients into a large bowl and whisk thoroughly to combine. It will keep for at least a month in the fridge in a sterilized jar.
Tahini Dressing
## Flavoured Butters
Butter makes everything taste better. In Ireland we are lucky to have the best butter in the world. This is something to be proud of and to celebrate – it's part of what we are, and our ancestors probably had it running through their veins. We now know that butter is a good fat, full of nutrients and vitamins, so we needn't be afraid of it. While butter on its own is great, adding flavours – such as garlic, onion, seaweed, herbs, honey, fruits, spices – can make it special. At our markets we always have a good selection of flavoured butters...
## Burnt Onion Butter
* 2 small onions
* olive oil
* 125g butter, softened
* a pinch of sea salt
Peel the onions. Cut one into dice and slice the other one finely. In a frying pan on a medium-low heat, cook the diced onion in a little olive oil for around 20 minutes, until golden brown and starting to stick to the pan. Add more oil or a splash of water to the pan if the onion starts to become too dry or burn. Leave to cool, and preheat your grill.
Lay out the sliced onion on some baking parchment on a baking tray, put under the hot grill and cook until completely dry, black and burnt beyond recognition. Remove from the grill, tip the charred onion into a pestle and mortar, and grind to a fine black powder.
In a bowl, mix the butter, cooled caramelized onion and salt until well combined. Mould into a ramekin and, using a brush, gently dust the top with the burnt onion dust until completely covered. Place in the fridge to harden. Take out and spread on everything.
Burnt Onion Butter
## Orange and Cinnamon Butter
* ½ teaspoon ground cinnamon
* zest of 1 orange
* 125g butter, softened
Beat the cinnamon and most of the orange zest into the butter until well combined. Mould into a ramekin, cover with the remaining orange zest and refrigerate. Delicious slathered on toast or scones.
## Kombu Butter
* 10g dried kombu
* 125g butter, softened
* sea salt and freshly ground black pepper
* juice of ½ a lemon
* 1 clove of garlic, crushed
Break the kombu into small pieces and place in a cup of water for a few moments to soften. Remove from the water and dry on paper towels. Put the butter into a bowl and mix together with salt, pepper, the lemon juice, garlic and kombu until well combined. Transfer to a ramekin and chill in the fridge to set. Kombu butter is amazing on the seeded dillisk loaf here. It is also perfect tossed into steamed samphire or green beans and served with fish.
## Honey and Redcurrant Butter
* 125g butter, softened
* 1 tablespoon honey
* 2 tablespoons fresh redcurrants
In a bowl, mix the butter with the honey, combining it well until smooth. Using a fork, incorporate the redcurrants into the honeyed butter trying to keep them whole. If they release their juice into the butter they won't mix well. Transfer the butter into a ramekin and refrigerate until set. Spread on pastries, bread or anything you desire.
## Garlic, Lemon and Parsley Butter
* 10g fresh flat-leaf parsley
* 125g butter, softened
* sea salt and freshly ground black pepper
* juice of ½ a lemon
* 1 clove of garlic, crushed
Chop the parsley leaves finely. Place the butter in a bowl and mix together with some salt, pepper, the lemon juice, garlic and parsley until well combined. Transfer to a ramekin and chill in the fridge to set. This is probably the most versatile butter we make, as it works well on anything from pasta to steak and is also delicious spread generously on any type of bread.
Flavoured Butters
## Mushroom Ketchup
This is a deep, rich, woody alternative to traditional tomato ketchup. In fact, ketchup was first made with mushrooms instead of tomatoes, so technically this is a more traditional version! It is perfect for steak or anything that you would usually put tomato ketchup on.
##### MAKES ABOUT 350ML
* 600g chestnut mushrooms
* 2 tablespoons sea salt
* 25g dried porcini mushrooms
* 100ml sherry vinegar
* ¼ teaspoon freshly grated nutmeg
* 2 shallots, finely diced
* 1 thumb-sized piece of fresh ginger, peeled and sliced
* 1 bay leaf
* 1 teaspoon black peppercorns
* ½ teaspoon allspice berries
##### METHOD
You will need to start making this a week before you want to use it.
Wipe and slice the chestnut mushrooms and put into a large bowl with the salt. Cover with cling film and leave for 24 hours. Every so often, press down the mushrooms and stir with a wooden spoon until they start to release their juices and break up slightly.
Put the porcini mushrooms into 125ml of boiling water and leave to soak for an hour or so. Remove the porcini mushrooms and pour the water into a jug through a fine sieve to remove any grit.
Place the chestnut mushrooms along with any liquid they have released in a large pan with the vinegar, then add the porcini mushrooms along with their soaking water, the nutmeg and the shallots.
Put the ginger, bay leaf, peppercorns and allspice berries into a piece of muslin tied with string and place in the pan. Bring to the boil and then simmer gently, uncovered, for around 1½ hours. Stir the mixture regularly until it has started to thicken.
Discard the bag of spices, then transfer the mixture from the pot into a food processor and blitz until very smooth. You can also use a hand blender for this, but you may need to blitz the ingredients for longer to reach the right consistency.
Return the smooth sauce to the pan and bring to the boil, then simmer for 5 minutes. Pour into sterilized jars and leave to develop its flavour for around one week before using. It should keep unopened in a cool place for 3 months.
## Walnut and Feta Dip
This is an impressive-tasting dip. Pairing walnuts with feta deepens and mellows the mix and balances the tanginess and sharpness of the cheese. Though it tastes complex, it couldn't be easier to make.
##### MAKES ABOUT 300G
* 120g walnuts, roughly chopped
* 2 tablespoons chopped fresh flat-leaf parsley leaves
* 150g feta cheese
* 1 clove of garlic
* 2 tablespoons rapeseed oil
* juice of ½ a lemon
* a little freshly ground black pepper
* a little smoked paprika and a drizzle of olive oil, to garnish
##### METHOD
Put 100g of the walnuts and 1 tablespoon of the parsley into a food processor, then add the rest of the ingredients except for the paprika and olive oil.
Blitz until smooth, then stir in the reserved walnuts and parsley. Place in a nice bowl and drizzle with a little olive oil and a small sprinkling of paprika. Great on toasted flatbreads.
## Green Herb and Lemon Dressing
This is a super-bright, zingy dressing that is perfect for fresh garden leaves. The recipe makes a small quantity, but it is packed with flavour so it goes a long way.
##### MAKES 100ML
* 2 tablespoons fresh flat-leaf parsley leaves
* 2 tablespoons fresh chervil leaves
* 1 tablespoon fresh French tarragon leaves
* 1 teaspoon fresh marjoram leaves
* juice of 1 lemon
* 1 tablespoon capers
* 3 tablespoons rapeseed oil, plus extra if needed
* sea salt and freshly ground black pepper
##### METHOD
Combine all the ingredients in a food processor or blender and blitz until you have a bright green sauce. Add more rapeseed oil if needed – the consistency should be quite loose and runny.
Green Herb and Lemon Dressing
## Spring Onion Vinaigrette
William likes this bright, fresh vinaigrette on everything, especially his poached eggs in the morning!
##### SERVES 1
* 4 spring onions
* a pinch of sea salt and freshly ground black pepper
* juice of 1 lemon
* 1 tablespoon rapeseed oil
##### METHOD
Slice 3 of the spring onions thinly and place in a pestle and mortar with the salt and pepper. Grind to a rough paste and transfer to a small bowl with the lemon juice and oil. Whisk gently with a fork. Cut the remaining spring onion into small rounds and add to the bowl, stirring to combine. This is particularly good drizzled over poached eggs (see here).
## Lime and Coriander Mayo
We love this version of mayonnaise – the lime juice gives it such a refreshing zing! Sadly, coriander isn't for everyone, so if it's not your thing just leave it out. This would be amazing in a halloumi burger or with some freshly grilled mackerel.
##### MAKES 320ML
* 15g fresh coriander
* 2 medium egg yolks
* 1 tablespoon white wine vinegar
* ¼ teaspoon Dijon mustard
* a pinch of sea salt
* 250ml sunflower oil
* juice of 1 lime
##### METHOD
Roughly chop the coriander leaves. In a large bowl, whisk the egg yolks together with the vinegar, mustard and salt. Slowly pour the oil into the bowl in a thin but steady stream while whisking vigorously. Be careful not to add the oil too fast or the mixture will curdle.
The mixture will quickly thicken until you are left with a nice pale yellow homemade mayonnaise. Stir in the lime juice and coriander, mixing until well combined. This keeps for 2 weeks in a sterilized jar in the fridge.
## Bretagne Sauce
This is a Breton variation of a classic hollandaise sauce that's great with seafood and particularly good for oily fish like mackerel or kippers. It's also a bit easier to make than hollandaise.
##### MAKES ABOUT 200ML
* 2 medium egg yolks
* juice of ½ a lemon
* 1 teaspoon Dijon mustard
* 2 tablespoons white wine vinegar
* 1 tablespoon chopped fresh chervil or flat-leaf parsley leaves (or both)
* 55g butter
##### METHOD
Put the egg yolks, lemon juice, mustard, vinegar and herbs into a large bowl. Melt the butter in a small pan and slowly drizzle it into the egg yolk mixture, whisking vigorously to combine.
Continue to whisk until the sauce thickens and becomes silky smooth in texture. Be careful not to allow the mixture to split by adding the butter too quickly.
This sauce keeps for 3 to 4 days in the fridge.
## Fried Apple and Sage
The sweetness of apples goes wonderfully with savoury dishes like pork, cold meats, and cheese and pickles. But these fried apples also work well when given a starring role – for instance, on top of a rye cracker or inside a wholemeal pitta.
##### SERVES 2–4 AS SIDE DISH, MORE ON CRACKERS
* 4 apples (preferably Cox's)
* 25g butter
* olive oil
* 12–20 fresh sage leaves
* 1 tablespoon fresh thyme leaves
* sea salt and freshly ground black pepper
##### METHOD
Core the apples, unpeeled, and cut into quarters. In a frying pan over a medium heat, melt the butter with a little olive oil. Add the apples, sage and thyme leaves, stirring to coat with the butter. Season with salt and pepper.
Keep turning in the frying pan for around 4 minutes, until the apples start to caramelize around the edges and the flesh is soft but not falling apart.
Leave to cool and decant to sterilized jars, then leave in a cool place until ready to serve. It will keep for up to a week.
_This recipe is picturedhere_
Fried Apple and Sage
## Recipe List
1. Pear and Frangipane Tart
2. Fresh Blueberry Pie with a Lemon Curd Cream
3. Flourless Dark Chocolate and Sea Salt Cake
4. Lemon and Lavender Cake
5. Soaked Orange Cake
6. Wholemeal Spelt Carrot Loaf with Orange Mascarpone Icing
7. Pecan and White Chocolate Banana Loaf
8. Glamnilla Shortbread Biscuits
9. Rock Sugar Biscuits
10. Molasses Biscuits
11. Lemon and Rosemary Biscuits
12. Orange Shortbread with Salted Dark Chocolate
13. Honey Biscuits
14. Currabinny Brown Apple Tray Bake
15. Patrick O'Hara's Elderflower Cordial
## Pear and Frangipane Tart
This is a classic recipe that everyone who goes to the Ballymaloe Cookery School will know and love. William's version makes the pastry extra short, because that's how we like it! And lacing your frangipane filling with a good slosh of kirsch brings out the subtle flavours of the pears and gives the tart more kick!
##### MAKES 8 SLICES
##### _FOR THE SIMPLE SHORTCRUST PASTRY:_
* 240g cream flour, plus extra for dusting
* a pinch of sea salt
* 2 teaspoons caster sugar
* 180g cold butter
* 1 medium organic egg yolk
* 1–2 tablespoons chilled water
##### _FOR THE POACHED PEARS:_
* 1 litre water
* 4 Conference pears
* 1 vanilla pod
* 3 x 2cm pieces of lemon peel
* juice of 1 lemon
* 200g caster sugar
##### _FOR THE FRANGIPANE:_
* 80g butter, softened
* 80g caster sugar
* 1 medium organic egg
* 100g ground almonds
* 1 tablespoon cream flour
* 1–3 tablespoons kirsch, to taste
##### _FOR THE APRICOT GLAZE:_
* 2 tablespoons apricot jam
* 2 teaspoons water
##### METHOD
Sift the flour for the pastry into a large bowl and add the salt and sugar. Cut the cold butter into cubes and rub into the flour with your fingers until the mixture resembles breadcrumbs.
Add the egg yolk and just enough of the water to bring the mixture together into a ball. We like it as short as possible, so don't worry if it's a little crumbly and hard to handle. Shape into a thick disc, cover in cling film and put in the fridge for 30 to 40 minutes.
Put the water for the poached pears on to boil in a large, heavy-based saucepan. Peel the pears and cut them in half, then remove the core with a melon baller or teaspoon. We like to keep the stalks intact if the pears still have them.
Split the vanilla pod, leaving it attached at the top, and add it to the pot of boiling water, together with the pears, lemon peel, lemon juice and sugar. Bring back to the boil, then reduce to a simmer.
Construct a cartouche by cutting a circle of baking parchment which perfectly covers the inside of your pan. Place it inside the pan, put a lid slightly askew over it and leave to simmer gently for 20 minutes.
Test the pears with a sharp knife – if it slides through them easily, they are done. Remove the pan from the heat and leave the pears to cool in their cooking liquid. When everything has cooled, remove the pears with a slotted spoon and drain on kitchen paper.
To make the frangipane, cream the butter and sugar together in a large bowl until smooth and pale.
Add the egg, beating vigorously to avoid the mixture splitting. Stir in the ground almonds and flour until well incorporated. Add the kirsch at the end – we use 3 tablespoons, which is quite strong, so try adding a tablespoon at a time and tasting the mixture as you go to get the flavour you like.
Preheat the oven to 180ºC fan/gas 6.
Roll out your chilled pastry on a lightly floured surface to around 1cm thick. The pastry will be a little difficult to handle, but stick with it and you will be rewarded with the most mouth-watering crumbly crust. Carefully flip the pastry on to a 20cm fluted loose-bottomed tart tin – there will be breakages, cracks and gaps, but don't panic; just use the excess pastry to patch everything up. Press gently into the sides of the tin and leave a rim of about 1cm hanging over the edges.
Line the pastry case with baking parchment, fill with baking beans and place in the oven for 10 to 15 minutes, until golden. Remove from the oven, lift out the paper and beans, and leave to cool.
When cool, fill the case with the frangipane. Slice the poached pears from stalk to base in 1cm pieces, retaining their pear shape. Arrange the pieces in a tight circle on top of the frangipane, stalks facing inwards.
Bake in the oven for 20 minutes, then reduce the temperature to 160ºC fan/gas 4 and bake for another 10 minutes until golden brown on top. Leave to cool slightly before removing from the tart tin.
Prepare the apricot glaze by warming the jam with the water over a low heat. Pass the mixture through a sieve to remove any lumps and brush over the top of the tart. Allow the tart to cool completely before serving.
Pear and Frangipane Tart
## Fresh Blueberry Pie with a Lemon Curd Cream
Blueberries tend to burst and fall apart when cooked, so cooks often add a load of sugar to compensate, which makes most blueberry pies tooth-achingly sweet. In our version you cook just a quarter of the blueberries in a syrupy sauce and then fold in the remaining fresh blueberries. This way the gorgeous sharp blueberry flavour remains the star of the show!
##### SERVES 8–10
##### _FOR THE SHORTCRUST PASTRY:_
* 240g cream flour
* a pinch of sea salt
* 2 teaspoons caster sugar
* 180g cold butter
* 1 medium organic egg
* 1–2 tablespoons chilled water
##### _FOR THE FRESH BLUEBERRY FILLING:_
* 600g blueberries
* 120ml water, plus an extra 2 tablespoons
* 2 tablespoons corn starch
* 100g caster sugar
* 1 tablespoon lemon juice
* a pinch of sea salt
##### _FOR THE LEMON CURD CREAM:_
* 50g butter
* 110g caster sugar
* grated zest and juice of 2 lemons
* 2 medium eggs and 1 egg yolk, beaten
* 110g mascarpone cheese
* 2 tablespoons icing sugar
##### METHOD
Sift the flour for the pastry into a large bowl and add the salt and sugar. Cut the cold butter into cubes and rub into the flour with your fingers until the mixture resembles breadcrumbs. Add the egg yolk (reserving the white) and just enough of the water to bring the mixture together into a ball. We like it as short as possible, so don't worry if it's a little crumbly and hard to handle. Shape into a thick disc, cover in cling film and put in the fridge for 30 to 40 minutes.
Preheat the oven to 180ºC fan/gas 6.
Roll out your chilled pastry on a lightly floured surface into a 5mm-thick round. Flip this on to a 25cm loose-bottomed tart tin, patching any pieces that have crumbled or torn. You want to make the walls of the pastry come slightly over the side of the tart tin.
Line the pastry case with baking parchment, fill with baking beans and place in the oven for 20 minutes. Remove from the oven, lift out the paper and beans, prick the base with a fork and return to the oven for 10 minutes, until golden. Let the pie case cool in its tin for 3 minutes on a rack, then brush with the reserved egg white. Remove the pie case from the tin and place on a plate.
To make the filling, put 150g of the blueberries in a saucepan with 120ml of water, cover and bring to the boil. Meanwhile, in a small bowl, whisk together the corn starch and about 2 tablespoons of water.
When the blueberries and water have come to the boil, lower the heat to a simmer, stirring constantly until the blueberries start to burst and the juices begin to thicken. Add the corn starch mixture, caster sugar, lemon juice and salt. Simmer for 1 minute until the mixture becomes translucent. Remove from the heat and stir in the remaining blueberries.
Spoon the mixture into the pie case and allow to sit for 2 hours.
To make the lemon curd cream, melt the butter in a small saucepan over a very low heat. Add the caster sugar, lemon zest and juice, then add the beaten eggs. Stir carefully with a straight-ended wooden spatula until the mixture coats the back of it. Remove from the heat and allow to cool.
Put the mascarpone into a bowl, add the lemon curd, sift over the icing sugar and gently mix together. You can spoon this mixture on top of the pie or serve it on the side.
Fresh Blueberry Pie with a Lemon Curd Cream
## Flourless Dark Chocolate and Sea Salt Cake
This is an incredibly rich, salty-sweet chocolate cake that's perfect for dessert. The egg whites provide just enough rise and lightness. The darkness of the chocolate is up to you. We've gone for 70 per cent here, as any higher will make your cake a little bitter. The combination of rich dark chocolate and salt manages to be both sophisticated and sweet – delicious and chic!
##### MAKES 8 SLICES
* 170g butter, plus extra for greasing
* 350g dark chocolate
* 150g caster sugar
* 5 medium organic eggs
* 50g ground almonds
* 2 teaspoons sea salt, plus extra for decoration
* edible gold dust (optional)
* whipped cream, to serve
##### METHOD
Preheat the oven to 160°C fan/gas 4. Grease a 23cm springform cake tin and line with baking parchment.
Put the butter, the chocolate and the sugar into a large glass bowl over a pan of barely simmering water until the chocolate and butter melt. Be careful not to overheat the mixture. Leave this to cool while you separate the eggs.
Add the egg yolks to the bowl one at a time, beating into the mixture as you go. Whisk the egg whites into stiff peaks in a separate bowl.
Fold the ground almonds into the chocolate along with the sea salt and half of the whisked egg whites. Carefully incorporate the rest of the egg whites, folding them through the mixture.
Transfer the mix into the prepared cake tin and bake in the oven for 30 to 40 minutes. The top should be well set, with cracks around the circumference so the middle seems to be breaking away from the sides. Leave to cool for at least 15 minutes, then transfer to a plate and decorate with large flakes of sea salt (or edible gold dust, as we like to). Serve with big dollops of softly whipped cream.
Flourless Dark Chocolate and Sea Salt Cake
## Lemon and Lavender Cake
Lavender in cooking poses a bit of a problem: how do you capture that background flavour and aroma – the one that transports you to a summer's day in Provence – without it tasting like soap? Some people recommend staying away from dried lavender and sticking to products like lavender extract paste. However, this is hard to find. Dried lavender works wonderfully, as long as you grind it up so that people aren't biting into whole buds! Combining lavender with lemon and yoghurt makes this cake sticky, subtle and utterly delicious.
##### MAKES 8–10 SLICES
* butter, for greasing
* 1 tablespoon dried lavender flowers
* 250g caster sugar
* 175g cream flour
* ½ teaspoon baking powder
* ½ teaspoon bicarbonate of soda
* a pinch of sea salt
* 2 medium organic eggs
* 250g Greek yoghurt
* 125ml rapeseed oil
* finely grated zest and juice of 1 lemon
* dried lavender sprigs, to decorate
##### _FOR THE ICING:_
* 200g icing sugar
* juice of 1 lemon
* 1 medium egg white
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Butter a 20cm springform cake tin and line with baking parchment.
Crush the lavender in a pestle and mortar. Put the caster sugar into a large bowl and mix the lavender through. Add the flour, baking powder, bicarbonate of soda and salt, and stir to combine.
In another bowl, mix the eggs with the yoghurt and rapeseed oil and pour this into the dry ingredients, stirring well. Add the lemon zest and juice.
Pour the mixture into the cake tin and bake in the oven for around 50 minutes until golden brown and firm to the touch. Leave to cool in the tin for a minute, then turn the cake out to cool fully on a wire rack.
Sieve the icing sugar into a bowl and add the lemon juice, whisking until smooth. Add the egg white gradually to loosen the mixture until it is quite runny and pourable. The icing should be extremely sharp and lemony. Spoon this icing over the top of the cake until it covers the top and starts to drip down the sides.
Arrange some dried lavender sprigs on the top as decoration.
Lemon and Lavender Cake
## Soaked Orange Cake
Come hail, rain or shine, this cake will always be sitting on the kitchen table of William's mother, Breda, in Currabinny. It is perpetually being baked, eaten and replaced. This cake just says 'home'. It is always moist, and the flavour is even more orangey the next day. William likes to add Campari for a deeper orange flavour, but the original recipe leaves it out.
##### MAKES 8–10 SLICES
* butter, for greasing
* 50g breadcrumbs (made from slightly stale bread)
* 200g caster sugar
* 110g ground almonds
* 2 teaspoons baking powder
* zest of 1 orange
* zest of 1 lemon
* 4 medium organic eggs
* 200ml rapeseed oil
* candied orange peel (optional), to decorate
##### _FOR THE SYRUP:_
* juice of 1 large orange
* juice of 1 lemon
* 50ml Campari or Aperol
* 70g caster sugar
* 1 cinnamon stick
##### METHOD
Butter and line a 20cm springform cake tin. In a large bowl, combine the breadcrumbs, caster sugar, ground almonds, baking powder and orange and lemon zests.
In a jug, whisk the eggs with the oil and pour into the bowl of dry ingredients, then stir well to combine.
Pour into your prepared tin and place in a cold oven. Turn the oven to 170ºC fan/gas 5 and bake for 45 to 60 minutes. You want the cake to be firm, golden brown and definitely not soggy in the middle. If the cake is browning too quickly, cover it with some tinfoil or even turn the oven down a little. You'll know the cake is done when a skewer comes out clean.
While the cake is baking, make the syrup. Heat the ingredients in a pan over a medium heat until gently simmering, then turn the heat down to low and simmer for 5 minutes.
When the cake is ready, leave in the tin, pierce all over with a fork and pour the syrup over the cake until it has been soaked up, reserving the cinnamon stick.
When the cake has cooled completely, release it from the tin and decorate with the cinnamon stick or some candied orange peel.
Soaked Orange Cake
## Wholemeal Spelt Carrot Loaf with Orange Mascarpone Icing
This manages to be both indulgent and really healthy (well, as healthy as any cake gets!). The recipe includes wholemeal spelt flour, which gives the loaf a moreish nuttiness.
##### MAKES 8 SLICES
* 75ml rapeseed oil, plus extra for greasing
* 175g carrots
* 2 medium organic eggs
* 110g soft brown sugar
* 100g wholemeal spelt flour
* 1 teaspoon baking powder
* 1½ teaspoons bicarbonate of soda
* 50g desiccated coconut
* 1 teaspoon ground cinnamon
* ½ teaspoon ground nutmeg
##### _FOR THE ICING:_
* 250g mascarpone cheese
* 2 tablespoons icing sugar
* juice of 1 orange
* a few chopped walnuts (for decoration)
* some orange zest (for decoration – use as much as you like)
##### METHOD
Preheat the oven to 190ºC fan/gas 6. Grease and line a 450g loaf tin with baking parchment.
Peel the carrots and grate them finely. In a large bowl, whisk the eggs and brown sugar until thick and creamy. Continuing to whisk, slowly pour the oil into the egg mixture until well combined.
In another bowl, gently mix the flour, baking powder, bicarbonate of soda, coconut, cinnamon and nutmeg together. Add to the batter in three batches, folding in well each time, then add the grated carrots, stirring gently to combine. Pour into the lined loaf tin and bake in the oven for 25 minutes.
Meanwhile, put the ingredients for the icing into a bowl and mix well. Use a skewer to test if the loaf is done. When it comes out clean, remove the loaf from the oven and cool on a wire rack. Once cooled, use a spatula to cover the loaf with icing and sprinkle over some chopped walnuts and some orange zest if you like.
Wholemeal Spelt Carrot Loaf with Orange Mascarpone Icing
## Pecan and White Chocolate Banana Loaf
If someone is coming over to the house for catch-ups, this is our go-to recipe. We can whip it up in no time, and a few slices go down a treat with a pot of tea. Also, it's full of bananas, so it's one of your five-a-day! If you can get your hands on some dried flower petals, sprinkle them over the top once it's done to get extra marks for effort.
##### MAKES 8 SLICES
* 125g butter, plus 1 tablespoon for the tin
* 175g cream flour, plus extra for dusting
* 4 small, ripe bananas
* 100g white chocolate
* 60g pecans
* 2 teaspoons baking powder
* ½ teaspoon bicarbonate of soda
* ½ teaspoon sea salt
* 150g caster sugar
* 2 medium organic eggs
* 1 teaspoon vanilla extract
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Melt all the butter in a small saucepan on a low heat. Brush the inside of a 900g loaf tin with 1 tablespoon of the butter, then dust with flour.
Mash the bananas, chop the white chocolate into chunks and roughly chop the pecans. Mix the flour with the baking powder, bicarbonate of soda and salt in a bowl.
In a separate, large bowl, whisk the rest of the melted butter and sugar together. Beat in the eggs, one at a time, then stir in the mashed bananas, white chocolate chunks, pecans and vanilla extract.
Add the dry ingredients to the wet ingredients in three batches, stirring after each addition. Pour into the loaf tin and bake in the oven for 1 to 1¼ hours, or until a skewer comes out clean. Slide a spatula around the edge of the loaf and leave in the tin to cool before turning out on to a wire rack.
Now, devour.
Pecan and White Chocolate Banana Loaf
## Glamnilla Shortbread Biscuits
We love sitting at home in the evening, watching documentaries with a pot of tea and a big plate of vanilla shortbread biscuits. To add a bit of glamour when we sold them at our first market stall, we coated them with some edible gold dust, and they looked so gorgeous that James Snapchatted them. One of his followers said they should be called 'Glamnillas' – and that was that! If you've friends over for tea, arrange some on a white plate: you'll be totally channelling Marie Antoinette.
##### MAKES 20
* 170g cream flour, plus extra for dusting
* 1 vanilla pod
* 110g cold butter
* 55g caster sugar
* edible gold dust, to finish (optional)
##### METHOD
Put the flour into a chilled bowl. Slice open the vanilla pod lengthways, and with the back of a knife scrape the seeds out of the pod and add the seeds to the flour. Cut the butter into small chunks and add to the flour, along with the sugar. Rub together with your fingers until the mixture starts to come together. The idea is to create a very short, crumbly dough with very little moisture.
Pat the dough into a flat disc, cover in cling film and chill for 15 minutes in the fridge. Preheat the oven to 160ºC fan/gas 4.
Roll out the dough on a lightly floured surface to about 3mm thick (i.e. quite thin). Cut out shapes with pastry cutter/s of your choice; we like to keep things as simple as possible with a small round one.
Arrange on a baking tray lined with baking parchment, leaving around 1cm between each biscuit to allow for expansion. Gather up the trimmings, roll out and cut your shapes out again until you've used up all the dough.
Bake in the oven for 10 to 15 minutes, until the biscuits are a very light golden colour. Keep a watchful eye on them, as they will brown very quickly. When ready, immediately remove from the baking tray and arrange on a cooling rack. Dust lightly with edible gold dust for an extra-special touch. They will keep for a couple of days in an airtight jar or tin in a cool place.
Glamnilla Shortbread Biscuits
## Rock Sugar Biscuits
Making these lovely-looking textured biscuits is simple and a real treat – there's nothing like doing a bit of bashing with a rolling pin to get rid of stress!
##### MAKES 10–15
* 180g butter, softened
* 140g caster sugar
* 2 medium egg whites
* 1 tablespoon mixed spice
* 1 teaspoon sea salt
* 300g cream flour
* 150g white sugar cubes
##### METHOD
Cream the butter, caster sugar and half the egg whites with the mixed spice and salt until smooth and pale. Sift the flour into the mixture in three batches, beating it in each time with a wooden spoon until well incorporated and a dough has formed. Shape the dough into a disc, wrap in cling film and refrigerate for half an hour.
Wrap the sugar cubes in a clean tea towel and beat with a rolling pin until they're all broken up into individual grains.
Preheat the oven to 150ºC fan/gas 3.
Roll the dough out on to a large piece of baking parchment into a 3mm-thick, roughly square shape. Brush the dough with the remaining egg white and sprinkle the sugar granules over it. Score the surface of the dough into biscuit-sized squares, making sure you don't cut all the way through.
Lift the parchment carefully on to a baking tray and bake in the oven for 30 to 40 minutes. Remove from the oven when golden and leave to cool on the parchment. When cool you can break along the pre-cuts into their individual squares. They will keep for a couple of days in an airtight jar or tin in a cool place.
## Molasses Biscuits
The rich, full-bodied flavour of molasses makes these biscuits the perfect accompaniment to strong coffee.
##### MAKES AROUND 40
* 180g butter
* 100g molasses
* 180g soft brown sugar
* 280g cream flour
* 1 teaspoon ground cinnamon
* ½ teaspoon ground ginger
* 1 teaspoon bicarbonate of soda
* a pinch of sea salt
* 1 medium organic egg
* 50g rolled oats
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Line two baking trays with baking parchment.
Melt the butter in a large, heavy-based saucepan over a medium heat, then add the molasses and sugar and stir until the sugar has dissolved. Remove from the heat and leave to cool.
In a large bowl, mix the flour, cinnamon, ginger, bicarbonate of soda and salt together.
Whisk the egg into the cooled molasses mixture until smooth, add the oats and stir through. Add the dry mixture to the molasses mixture in batches, stirring to combine into a firm but sticky dough.
Get tablespoons of the dough and roll into balls with your hands. Place on the baking trays about 2½cm apart. Bake in the oven for 10 to 15 minutes until hard and medium brown in colour. After cooling for a few minutes on the trays, transfer to wire racks to cool fully. They will keep for a couple of days in an airtight jar or tin in a cool place.
## Lemon and Rosemary Biscuits
Moving from Cork to Dublin, William naturally brought some staples to remind him of home – including Barry's Tea and a big bag of stoneground Macroom Oatmeal. Luckily, James was already a Barry's convert, which made moving in together a lot easier. The oatmeal was a harder sell. James is not one for porridge, and this hardy oatmeal is apparently considered an acquired taste in Dublin. He is, however, a huge fan of biscuits. In this recipe the natural nuttiness and roughness of the oatmeal goes beautifully with the earthy fragrance of rosemary and the sharpness of lemon.
##### MAKES 20
* 225g butter, softened
* 100g caster sugar
* 1 medium organic egg
* 1 tablespoon lemon zest
* 1 teaspoon vanilla bean paste
* 200g cream flour
* 85g Macroom Oatmeal
* a pinch of sea salt
* 1 tablespoon finely chopped fresh rosemary
##### METHOD
Cream the butter and sugar together in a large bowl until light and fluffy, then beat the egg in slowly with the lemon zest and vanilla, being careful not to split the mixture. In another bowl, mix the flour, oatmeal, salt and rosemary.
Gently add the dry ingredients to the butter mixture, stirring slowly until well combined into a smooth dough. Shape the dough into two logs and cover in cling film. Freeze the logs for 1 hour.
Preheat the oven to 180ºC fan/gas 6.
Remove the logs from the freezer, take off the cling film and slice into rounds 6mm thick. Place on a baking tray lined with baking parchment, leaving 2cm between each biscuit.
Bake in the oven for 15 to 20 minutes until the biscuits are golden around the edges. Cool on the baking tray for a few minutes, then transfer to a wire rack to cool fully. They will keep for a couple of days in an airtight jar or tin in a cool place.
Lemon and Rosemary Biscuits
## Orange Shortbread with Salted Dark Chocolate
This pairing of orange and dark chocolate is a classic. You can think of these as inspired by Jaffa Cakes – they are like a fancier version!
##### MAKES 25–30 BISCUITS
* 115g butter, softened
* 65g caster sugar
* 2 teaspoons fresh orange juice
* 130g cream flour
* 2 teaspoons orange zest, plus extra for sprinkling
##### _FOR THE SALTED DARK CHOCOLATE:_
* 350g dark chocolate (70%)
* 2 teaspoons sea salt
##### METHOD
Put the butter and sugar into a large bowl and beat vigorously with a wooden spoon until creamy (you could use a hand-held mixer if you prefer). Add the orange juice slowly, being careful that the mixture does not split.
Sift in the flour, then add the orange zest, working the two ingredients into the butter and sugar mixture until a smooth dough forms. If the dough is too sticky and wet, add a little more flour.
Roll the dough into a sausage shape around 4cm in diameter and wrap in baking parchment. Chill the dough in the freezer for 1 hour.
Preheat the oven to 160ºC fan/gas 4. Line two baking sheets with baking parchment.
After the dough has hardened, remove the baking parchment and slice into rounds 3mm thick. Place these rounds on your baking sheets and bake in the oven for 10 to 15 minutes until lightly golden. Transfer to a wire rack, being careful they don't break – they will be quite soft and delicate – and leave to cool until crisp and hard.
While your shortbread is cooling, melt the dark chocolate in a heatproof bowl over a pan of gently simmering water. Keep stirring the chocolate as it melts. Turn the heat off but leave the bowl over the hot water to keep warm. Stir the salt through the melted chocolate.
When the biscuits have cooled, dip them halfway into the chocolate, leaving one half bare. Leave to set on sheets of baking parchment.
Before the chocolate on the biscuits has completely hardened, sprinkle some orange zest over them. You can store them for a couple of days in an airtight jar or tin in a cool place.
## Honey Biscuits
These biscuits could not be easier to make. But the inclusion of beautiful wildflower honey raises their flavour to another level and makes them really special.
##### MAKES 50–60
* 300g cream flour
* 1 teaspoon bicarbonate of soda
* a pinch of sea salt
* 240g butter
* 150g caster sugar
* 2 generous tablespoons good honey, such as wildflower or orange blossom
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Line two baking trays with baking parchment.
Sift the flour and bicarbonate of soda into a large bowl and add the salt. In a small saucepan, heat the butter, sugar and honey until the sugar has dissolved. Remove from the heat and pour into the flour mixture. Mix together with a fork to form a smooth dough.
Using a teaspoon, place small balls of dough on the prepared baking trays and press down with the back of the spoon into thick discs.
Bake in the oven for 10 to 12 minutes until golden. Allow to cool on the trays for a few minutes, then transfer to a wire rack to cool completely. They will keep for a couple of days in an airtight jar or tin in a cool place.
## Currabinny Brown Apple Tray Bake
Four gnarled and hardy apple trees grow behind the house at Currabinny. On windy nights the sounds of apples getting blown on to the roof can give people sleeping underneath quite a fright! This simple, soothing, warmly spiced tray bake is a lovely home for the apples that make it through the stormy nights.
##### SERVES 6–8
* butter, for greasing
* 225g wholemeal self-raising flour
* 25g ground almonds
* 1 teaspoon mixed spice
* 1 teaspoon ground cinnamon
* ½ teaspoon freshly grated nutmeg
* a pinch of sea salt
* 100g brown sugar, plus extra to sprinkle on top
* 100g cold butter
* 400g Bramley apples
* 2 medium organic eggs
* 2 tablespoons milk
* 25g flaked almonds
##### METHOD
Preheat the oven to 160ºC fan/gas 4. Butter a square (25cm x 25cm) deep-sided baking tray or rectangular (24cm x 28cm) casserole dish.
In a large bowl, mix together the flour, ground almonds, mixed spice, cinnamon, nutmeg, salt and sugar. Cube the cold butter and add to the bowl, rubbing it into the mixture with your fingertips until it's a fine breadcrumb-like consistency.
Peel and core the apples and chop them into chunks. Beat the eggs in a small bowl with the milk. Add the apple, then the egg mixture to the bowl of dry ingredients, stirring with a wooden spoon as you do it. This will create a soft batter.
Pour the batter into your prepared tray or dish and scatter the flaked almonds and some brown sugar over the top. Bake in the oven for 40 to 45 minutes until risen and firm to the touch – the colour should be dark brown but watch out for burning at the edges. Remove from the oven and leave to cool before cutting into squares.
## Patrick O'Hara's Elderflower Cordial
* * *
WILLIAM'S NEIGHBOUR IN CURRABINNY, ARTIST PATRICK O'HARA, MAKES A GORGEOUS ELDERFLOWER CORDIAL EVERY SUMMER. HERE PATRICK KINDLY SHARES HIS TOP TIPS AND RECIPE SO THAT YOU CAN MAKE THIS REFRESHING DRINK YOURSELF.
* * *
At the tip of the Currabinny peninsula there is a 90-acre forested sugarloaf hill which is topped by an ancient stone circle and cairn–reputed to be a focal point of magical and strange happenings. Around its edges are some of our more exotic shrubs and bushes, thriving in the balmy microclimate created by the Gulf Stream, which washes around the shoreline. Among them are some of the finest examples of our native Irish elder shrubs.
Elder reaches its peak flowering time around 21 June – Midsummer's Day – and its flowers can produce one of the most delicious and refreshing soft drinks. Elder branches are brittle and easily broken, so you must be careful when you pick the 40 or so umbrella-shaped heads of creamy white flowers needed for this recipe. They are best gathered on a hot, dry, sunny day, as close as possible to midday, when the tiny flowers are newly opened and giving off maximum fragrance.
Put them into a clean plastic bucket with a lid and add about 3 litres of boiling water, a couple of roughly chopped oranges and lemons, about 1½ kilos of granulated or caster sugar and an 85g sachet of citric acid.
Cover and leave for 48 hours. Then stir daily for 10–14 days with a potato masher, with which you can also gently squash the citrus fruit. After this time, leave it unstirred for a day before pouring it all through a large, coarse sieve into a sterilized bowl.
Next, pour that cloudy liquid through a jelly-bag or very fine sieve into a sterilized jug, before decanting it into clean, sterilized bottles (preferably the kind with stoppers). You can sterilize glass bottles by washing them in warm, soapy water, rinsing in boiling water, then placing to dry in a preheated oven at 120°C fan/gas 1 for 15 minutes. When you've filled your bottles with cordial, seal immediately.
The cordial is best served with ice and diluted in the ratio of about 1 part cordial and 5 parts sparkling spring water.
Keep in a cool, dark place and it could last for up to 9 months, but it won't – your friends and family will have drunk it all long before that!
I have heard it said that it makes quite a stimulating aftershave as well!
## Because you – and your guests – are worth it...
* * *
WHETHER SETTING THE TABLE FOR ONE, TWO OR EIGHT PEOPLE, IT ALWAYS LIFTS THE MOOD AND SOMEHOW MAKES YOUR FOOD TASTE EVEN BETTER IF YOU MAKE AN EFFORT WITH YOUR SURROUNDINGS.
* * *
We are big fans of vintage shops for sourcing unusual and attractive cups, plates, bowls, sauce boats, platters and all kinds of intriguing tableware to showcase our food.
When it comes to cutlery, we like the warmth and softness of bronze, coppers and muted silvers (we like to support Irish companies and are delighted we can get great cutlery from Newbridge Silverware). We love to use tablecloths and cloth napkins, and we find that natural colours look best – greys, browns and beige tend to work well in any situation (again, we try to go local for dressing our table and we get lovely table linen from STABLE of Ireland).
Candlelight is the best light. Candelabras and candlesticks look gorgeous– often far more impressive than they cost. Again, you'll get great ones with a lovely weathered look in vintage shops. Add them to your dining armoury to pull out for special occasions. Place them along the centre of the table and everyone will be bathed in a glow of soft golden light.
No table setting is complete without flowers. We love purples, whites and greens, things like flowering cabbage, baby's breath, thistle and hydrangea. We like our flowers to look as if they have been picked from an overgrown country garden. Uniformity when it comes to flowers isn't a good look!
Now, this may sound a bit over the top, but never underestimate the value of a place card when you have people over. Even if you're just having a few friends around for a casual dinner, taking a bit of extra time to write their names on cards will create a lovely talking point (and some nice social media content for guests who are sharing your dinner with their followers!).
Depending on the occasion you can customize your place cards, or come up with simple but ingenious card-holders; for example, when the colder months come around, pine cones add a beautiful seasonal touch to the table.
* * *
NEVER UNDERESTIMATE THE VALUE OF A PLACE CARD WHEN YOU HAVE PEOPLE OVER
* * *
Another great benefit of place cards is social engineering! You may be entertaining people who don't know each other well, or who you know will chat more easily with one guest than another; using place cards allows you to orchestrate the atmosphere around the table.
Those are our tips for setting the scene for a simple and memorable meal. But the main thing to do is relax. Plan to serve something that you're happy making and something you can prepare in advance. And if something goes wrong, don't panic; just pass around some nice bread and toppings and fill up everyone's glasses. Sláinte!
## Thanks...
To everyone below – we could not have written our first cookbook without you!
First, to our parents – Gags and Alan, Breda and Peter – for all the lifts and the general support, both moral and practical, you've given us. Special thanks to Gags and Breda for the recipes you shared with us, for your advice, and for cooking your signature dishes for the book. And thanks to Breda and Peter for letting us take over the house in Currabinny for the shoot.
To our neighbours in Phibsboro – Bang Bang Café and the Vintage Shop – for allowing us to take pictures in the café and shop for the book. Thanks to Daniel and Grace in Bang Bang for feeding us your delicious Brunch Burgers and to the Vintage for having the coolest dinnerware.
Thanks also to all the stallholders at the Mahon Point Farmers' Market for your wonderful produce and being so game for being photographed for the book. A special thanks to Patrick O'Hara of Currabinny for sharing your elderflower cordial recipe and agreeing to have your picture included.
To everyone who worked on the shoot: Bríd O'Donovan for taking the most delicious photographs; Jette Virdi for your flawless food-styling skills; Ciara Nolan – no one chops vegetables quite like you; and Aoife Datta for your great help in the kitchen. And thanks to Andrew McLaughlin, who took some great shots and footage during the making of this book and is a whizz on social media.
Also, to those who supplied wonderful props and produce for the shoot: Helen James for the beautiful plates and crockery; STABLE of Ireland for the lovely fabrics that helped set the scene in many of our photographs; Lynn Hunter for the plates (that we still need to return to you – sorry!); and Nudie Foods for all the lovely fruit and vegetables.
Thanks to Susan McKeever and Caroline Pretty for getting the text into such good shape. And to Danielle O'Connell for your gorgeous work designing the book and for helping bring our vision to life.
Thanks to Penguin Ireland for believing in us – to our editor, Patricia Deevy; to Michael McLoughlin, MD; to Cliona Lewis, Aimée Johnston and Aislin Reddie on the publicity team; and to Carrie Anderson and Brian Walker in sales. And in London thanks to Sara Granger, Natalie Wall and Emma Brown, who made sure the book stayed on track in the production process.
Thanks to our housemate Edel, who heard a lot about this project over the last couple of years – you've been very patient! William wants to specially thank Lorra Kent, his manager at l'Gueuleton, and also his colleagues there for all the support – for time off to work on the book and for putting up with him when he was getting stressed about it!
Thanks to Darina, Rory, Rachel and the teachers at Ballymaloe, who helped to instil in us a great food ethos. Ballymaloe is the spiritual home of Irish food and we are honoured to have been able to spend time with the great team there, and to return frequently – you constantly inspire us. Thanks also to Blathnaid Bergin for your superb advice on the business of food.
To all the people who have contributed to our food journey from childhood onwards. To William's 'second parents', Pat 'the potter' Cunningham and Ann O'Regan, for your passion for food, learning and sharing. To all the people who have helped us at the markets, events and suppers that have brought us to this point: Anna Moloney, Ciaran Murphy, Igor Brodecki, Joy Freeman, Silvio Barletti, Shaylyn Gilheaney, Anna Burke, Kyle Cheldon Barnett, Lynda Burke, Kate Mcelroy, Mark Geraghty, Jill and Gill and so many more. You have encouraged us, been generous with your knowledge and advice, and shaped the approach to food that we share in this book. We look forward to continuing the journey alongside all of you.
Finally, a massive thanks to our Currabinny followers – sure, we wouldn't be able to do this if you didn't like our food!
## THE BEGINNING
Let the conversation begin...
Follow the Penguin Twitter.com@penguinUKbooks
Keep up-to-date with all our stories YouTube.com/penguinbooks
Pin 'Penguin Books' to your Pinterest
Like 'Penguin Books' on Facebook.com/penguinbooks
Listen to Penguin at SoundCloud.com/penguin-books
Find out more about the author and
discover more stories like this at Penguin.co.uk
##### PENGUIN IRELAND
UK | USA | Canada | Ireland | Australia
India | New Zealand | South Africa
Penguin Ireland is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com.
First published 2018
Copyright © James Kavanagh and William Murray, 2018
Photography copyright © Bríd O'Donovan, 2018
The moral right of the authors has been asserted
Colour reproduction by Altaimage Ltd
ISBN: 978-1-844-88415-5
##### INTRODUCTION
fn1 _Temperatures are for fan ovens. If you_ ' _re using a conventional oven, please increase the temperature given by 20 degrees._
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Objective Proficiency Second edition provides official preparation for the revised 2013 Cambridge English: Proficiency exam, also known as Certificate of Proficiency in English (CPE). A variety of challenging, lively topics provide thorough training in exam skills and high-level language development. Each unit contains three double-page lessons ensuring flexibility, even pacing and progress. This motivating material is also suitable for high-level students keen to improve their general English. The Workbook with answers provides opportunities for further practice of new language and exam skills either at home or in the classroom. The CD contains the audio material for the Workbook listening tasks.
Opiniones "OBJECTIVE PROFICIENCY (2ND ED.): WORKBOOK WITH ANSWERS WITH AUDIO CD"
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The materials transported include coal, limestone, kaolin clay, China clay, iron concentrate, copper and nickel concentrates, phosphate concentrates, gold ore, fly ash, and mineral tailings. Slurry pipelines are often exposed to high internal abrasion of the pipe wall caused by the movement of mineral slurry through the line.
Ferrexpo produces iron ore pellets mostly with 65% iron content. A small proportion of pellets have 62% Fe content. The company also has its own capability to beneficiate iron ore and to process iron ore concentrate into iron ore pellets.
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Les Jeux mondiaux des sports de l'esprit (en anglais : World Mind Sport Games) est une compétition organisée par l'Association internationale des sports de l'esprit (AISE) dont la première édition s'est déroulée à Pékin du 3 au , dans la foulée des Jeux olympiques et des Jeux paralympiques. La dernière édition a eu lieu à Lille du 9 au .
Sports pratiqués
En 2019, huit sports de l'esprit seront présents. Sept fédérations sont membres de la Fédération mondiale des sports de l'esprit :
Sur les 500 millions d'amateurs de sports de l'esprit dans le monde, plus de se sont affrontés à Pékin.
Les jeux mathématiques ont été présentés pour la première fois, hors compétition, aux Jeux de Lille en 2012.
Pays participants
Notes et références
Liens externes
Site officiel 2008wmsg.org
The first international mind sports games "IMSA Cup" FIDE
China to host 1st World Mind Sports Games latestchess.com
un article décrivant la compétition
Compétition d'échecs en Asie
Compétition de bridge
Compétition de go
Compétition fondée en 2008
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{"url":"https:\/\/tlk-energy.de\/en\/tools\/air-density","text":"# Air Density\n\nbar\n\n\u00b0C\n\n%\n\nkg\/m\u00b3\n\n## How is the air density calculated?\n\nThe density of air depends on pressure, temperature and humidity. Dry air at 20\u00b0C and standard atmospheric pressure (1.013 bar) has a density of 1.204 kg\/m\u00b3.\n\nAir behaves like an ideal gas and the density $$\\rho$$ (rho) can be calculated accordingly with the following formula:\n$$\\rho = \\frac{p \\cdot M}{R \\cdot T}$$\nWhere $$p$$ is the absolute pressure in Pascal (Pa), $$M$$ is the molar mass of air in (kg\/mol), $$R$$ is the universal gas constant, and $$T$$ is the temperature in Kelvin (K).\n\nThe density of air therefore increases with increasing pressure and decreases with increasing temperature. The lower density of warm air is also the reason why a hot air balloon rises.\n\n## Density of humid air\n\nThe density of air also depends on the molar mass and thus on the composition of the air. The dry air density at sea level is calculated with a molar mass of about $$M_{\\rm A}$$ = 28.96 g\/mol.\n\nAs water has a lower molar mass with $$M_{\\rm W}$$ = 18.02 g\/mol, the density of air decreases with increasing humidity.\n\nThe molar mass of moist air is calculated from the mass fractions and molar masses of dry air and water:\n$$M = x_{\\rm A} \\, M_{\\rm A} + x_{\\rm W} \\, M_{\\rm W}$$\nThe mass fraction $$x_{\\rm W}$$ of water can be calculated from the relative humidity rH and the saturated vapor pressure $$p_{\\rm sat}$$:\n$$x_W = {\\rm rH} \\cdot \\frac{p_{\\rm sat}}{p}$$\nThe saturated steam pressure $$p_{\\rm sat}(T)$$ in turn is a function of temperature. Here we use the Antoine equation for calculation.\n\nFor humid air we also have a dew point calculator and a detailed hx diagram.","date":"2022-09-25 07:26:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8087103366851807, \"perplexity\": 336.0627710526866}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334515.14\/warc\/CC-MAIN-20220925070216-20220925100216-00221.warc.gz\"}"}
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UEM SUNRISE BERHAD ("UEM Sunrise" or the "Company") (Company No. 830144-W) is a public listed company and one of Malaysia's leading property developers. It is the flagship company for township and property development businesses of UEM Group Berhad ("UEM Group") and Khazanah Nasional Berhad ("Khazanah"). UEM Group is wholly-owned by Khazanah, an investment fund of the Government of Malaysia.
The Company has core competencies in macro township development; high rise residential, commercial, retail and integrated developments; as well as property management and project & construction services.
UEM Sunrise is currently undertaking various residential, commercial and mixed-use developments in Iskandar Puteri (formerly known as Nusajaya), one of the five flagship zones of Iskandar Malaysia. Upon completion, Iskandar Puteri will become the largest fully integrated urban development in Southeast Asia that will provide significant investment, financial and business opportunities.
Embracing innovation and technology, Iskandar Puteri will be the role model for an economically, socially and environmentally sustainable city. With modern infrastructure and cutting edge architectural masterplan, the expected social and foreign investment inflows into Iskandar Puteri will propel economic growth and transform the Southern Peninsular of Malaysia into an exciting centre of economic development in the region.
The thrust of Iskandar Puteri lies in the array of signature and catalytic developments including Kota Iskandar, the Johor State administrative centre which houses the State and Federal Government offices; Puteri Harbour, an integrated waterfront development; Southern Industrial and Logistics Clusters ("SiLC"), a managed, clean and green industrial park; and Afiat Healthpark, a comprehensive medical park offering modern, traditional, and complementary medicine and wellness. Together with a mix of residential, commercial and industrial properties; hotels, resorts and many other amenities, Iskandar Puteri will emerge as a vibrant and dynamic destination offering holistic and integrated lifestyle, with immense potential growth for investors.
UEM Sunrise is also cementing its commitment in delivering affordable homes with the launch of Bayu Nusantara and Denai Nusantara, which are integral to the Company's pledge of developing 10,000 units of affordable homes in Iskandar Puteri.
Gerbang Nusajaya, the second phase development of Iskandar Puteri is a 4,551-acre project which will also feature various catalytic developments and will be developed over a period of 25 years with components such as lifestyle & retail parks including FASTrack Iskandar; campus offices & industrial parks including Nusajaya Tech Park; as well as residential precincts including Melia Residences, and Gerbang Nusantara.
In the Central Region of Peninsular Malaysia, UEM Sunrise is renowned for its award winning and up-market high rise residential and commercial developments largely in the Mont'Kiara, Kuala Lumpur enclave; featuring projects such as Residensi Sefina Mont'Kiara, Residensi22 Mont'Kiara, Arcoris Mont'Kiara and 28 Mont'Kiara and many more. It is also responsible for introducing the concept of creative retail in Solaris Dutamas, known as Publika.
UEM Sunrise is also developing the 448-acre integrated township of Serene Heights Bangi that offers life's simple pleasures with nature-inspired environment apart from Symphony Hills, an exclusive residential development equipped with the Country's first smart-home features and community connectivity via the Connected Intelligent Community ("CIC") system. The Company also has other numerous award-winning residential, commercial and mixed-use developments within the Kuala Lumpur City Centre, Shah Alam and Seremban.
Internationally, UEM Sunrise's presence extends into Richmond in Vancouver, Canada via its 4.8-acre mixed-use development, Quintet and the newly acquired site at Alderbridge.
Australia has become an increasingly strong market for the Company, having launched the 92-storey Aurora Melbourne Central, the tallest development in the CBD of Melbourne and the inspiring 42-storey Conservatory located on Mackenzie Street in Melbourne with panoramic views over the historic UNESCO World Heritagelisted Royal Exhibition Building and Carlton Gardens. UEM Sunrise has also acquired its first city-fringe property at 412 St. Kilda Road in Melbourne.
The Company retains a land bank in Durban, South Africa that is poised to be developed into a luxurious mixeduse project known as Durban Point Waterfront.
UEM Sunrise is also the appointed Project Manager (Marketing) for developer, M+S Pte. Ltd., a company owned by Khazanah and Temasek Pte. Ltd. for its Marina One and DUO mixed-use developments in Singapore.
At UEM Sunrise, sustainability lies at the heart of our business. We are conscious of the impact we have on the lives we touch through our residential, commercial, retail, hospitality and industrial developments. Accordingly, we are committed to incorporating designs and technological elements that enhance a sense of well-being, be it for work, living or leisure. Going above and beyond our commitment to quality products, we seek to balance our economic ambitions with sustainable initiatives that have positive impacts on our stakeholders. Our sustainability principles focus on the Economic, Environmental and Social initiatives in our continuous effort to create inherent value to our customers, investors, employees and suppliers as well as the environment and communities which not only help us deliver excellence in our services, products and maximising value, but more importantly, ensure that our business meets our sustainability objective of Building Beyond Buildings.
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South Africa. 2017. 120 mins. Sesotho, and English with English subtitles.
This intriguing South African Western-Thriller observes Tau who returns to his hometown of Marseilles after 20 years, only to find it – and some of his former friends – in the grip of a violent oppressive power, one he must reluctantly fight to liberate.
South by South is a series of quarterly film screenings showcasing the work of African artists alongside a programme of masterclasses for young filmmakers.
Michael Matthews' love for cinema gives him a strong focus on emotive, visual storytelling and performance. He really enjoys finding the core of a concept and making it feel more unique and powerful. He has been nominated and won awards both locally and internationally for his work in commercials, short films and music videos, and worked with brands such as MTV, Axe/Lynx, Nike, Kulula (airlines), TAB, Wrigleys and Smirnoff. Directing high and low budget, technically complex productions, has earned Michael a reputation for executing ambitious projects. Five Fingers for Marseilles is Michael's first feature film.
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Transport Times Events | News/Blog | Time for a Capital Region Transport Body?
This is a time of change for transport planning, with a move to breaking down the artificial barriers created by local authority boundaries. The creation of bodies such as Transport for the North has generated impetus to improve transport links and enable economic growth.
Amongst all this change, the elephant in the room is London. The 2011 Census shows that 18 percent of London workers (nearly 800,000 people) live outside of the Greater London Authority (GLA) area, and a very high percentage of these people are in highly-skilled jobs. The London Plan acknowledges that the capital is "inextricably linked" with a wider south-east England, and commuting is predicted to grow sharply as London's economy outstrips its housing supply. However, while the new Sub Regional Transport Bodies have boundaries related to journey-to-work and other travel patterns, Transport for London's (TfL) remit is firmly linked to the Greater London Authority boundary.
The result for the commuter (and theatre-goer, shopper, and business-tripper) is marked disparity in the ease of journey depending on whether your starting point is within or without the GLA boundary. The 'withouters' are deprived of the benefit of zonal fare systems and have patchy access to Oyster. My own county town has the absurdity of Oyster to Hertford East but not to Hertford North, creating confusion and chaos for those using these alternative routes, especially in times of disruption.
The current transport planning arrangements fail to acknowledge the wider picture. Clearly, there is a requirement to meet London's need to draw in workers and other visitors as befits its World City status. But this must not be at the expense of meeting the local transport needs of those authorities in the 'doughnut' around London. At present authorities such as Hertfordshire have no say in the rail franchise process and little influence over strategic transport schemes.
TfL has little incentive to prioritise cross-boundary schemes where the benefits are in favour of its neighbours. A case in point is the Metropolitan Line Extension which will bring significant economic development and transport opportunities to Hertfordshire, but is not a game-changer for the capital. The result is a scheme that is caught in a funding ping-pong between TfL and DfT. Any coordinated transport planning system would recognise the wider benefits to the country.
The irony is that the current system does not fully meet London's needs. DfT's rejection of further rail devolution to London, despite clear support from many of its surrounding authorities, demonstrates that sensible transport planning decisions are in short supply.
So what is the answer? A geographical extension to London as an administrative body would be seen in many parts of the country as fuelling the capital's dominance. So the solution would seem to a new transport body which works with existing authorities but provides the desperately needed coordination. My plan is for a Capital Region Transport Body which would cover all public transport in the capital and its hinterland, coordinating day to day services and fares, and providing the framework for long-term transport planning. This body would ensure that the capital has an efficient commuting network, and also crucially that all parts of the region (including London) have public transport that meets its local needs.
I can't claim that this is an original idea. The London Passenger Transport Board set up in 1933 had a remit for buses, trams and underground far greater than the then London County Council boundary, stretching to Baldock in the north, east to Shenfield, south to Horsham and to High Wycombe in the west. A modern version may well need to be larger to reflect modern commuting patterns, but the LPTB boundaries would be a good starting point.
There are plenty of details still to be debated as to how the new Capital Region Transport Body would work with existing local authorities and emerging transport bodies, but the time has come to re-think how public transport is planned in the London and the south-east.
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{"url":"http:\/\/tex.stackexchange.com\/questions\/74478\/latex-command-incantation-for-r","text":"# LaTeX \u201ccommand\u201d \/incantation for R\n\nI am using the tufte-handout class in LaTeX and I want to refer to R using the \"correct\" sans-serif font. I am currently using\n\n\\newcommand{\\R}{ {\\bf \\sffamily R } }\n\n\nused as ...\n\n... function in the \\verb|spatstat| \\R package ...\n\n\nWhat LaTeX \"command\" \/ incantation would you use?\n\n-\n\n## migrated from stackoverflow.comSep 28 '12 at 9:59\n\nThis question came from our site for professional and enthusiast programmers.\n\nRather than flagging to close just because it's easy to do, why not flag for moderator attention to migrate, even though it takes a little more effort?? \u2013\u00a0 Brent.Longborough Sep 28 '12 at 9:32\nThis is surely a duplicate; anyway, the answer is \\newcommand{\\R}{\\textbf{\\textsf{R}}}. Never use the two letter commands \\bf, \\sf or \\it, they are obsolete. \u2013\u00a0 egreg Sep 28 '12 at 10:02\n@egreg: I agree, I just thought the attitude of the folks @ SO a bit less than constructive \u2013\u00a0 Brent.Longborough Sep 28 '12 at 11:31\nRelated (but here it appears that the \"R\" is not desired in sans serif font): tex.stackexchange.com\/questions\/73089\/\u2026 Indeed in the R manual the \"R\" is just in the roman font (possibly bold in a bold context). So \\newcommand{\\R}{\\textup{R}} might be better. \u2013\u00a0 egreg Sep 28 '12 at 11:37\n@egreg Thanks for the link. Opinion seems mixed, the Springer books (including MASS4) use sans. \u2013\u00a0 Sean Sep 28 '12 at 11:47\n\nIf you want to ensure a sans serif upright bold font, just define\n\n\\newcommand{\\R}{\\textnormal{\\sffamily\\bfseries R}}}\n\n\nIf you want to avoid typing \\R{} is a nice program, then\n\n\\usepackage{xspace}\n\\newcommand{\\R}{\\textnormal{\\sffamily\\bfseries R}}\\xspace}\n\n\nwill allow\n\n\\R is a nice program\n\n-","date":"2015-08-31 18:21:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9634010791778564, \"perplexity\": 6706.276343649383}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644066275.44\/warc\/CC-MAIN-20150827025426-00220-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
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Ордина́рні ви́на — вина, що випускаються без витримки, але не раніше ніж за три місяці з дня переробки винограду. Це звичайні, дешеві вина, що не відрізняються якими-небудь особливо високими якостями.
Столові білі сухі ординарні вина частіше мають назву винограду, з якого вироблені (Рислінг, Ркацителі, Фетяска, Совіньйон, Аліготе та ін.). Якщо використовується суміш сортів винограду, то вино називається «Столове біле».
Столові червоні сухі ординарні вина, як і білі ординарні, мають назви сортів винограду (Сапераві, Каберне, Матраса). Якщо виготовлене з сумішей сортів винограду, називається «Столове червоне».
Вина
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{"url":"https:\/\/gmatclub.com\/forum\/why-the-various-generals-of-the-army-of-the-potomac-before-145397.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 20 Jan 2019, 07:37\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\nYour Progress\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n## Events & Promotions\n\n###### Events & Promotions in January\nPrevNext\nSuMoTuWeThFrSa\n303112345\n6789101112\n13141516171819\n20212223242526\n272829303112\nOpen Detailed Calendar\n\u2022 ### FREE Quant Workshop by e-GMAT!\n\nJanuary 20, 2019\n\nJanuary 20, 2019\n\n07:00 AM PST\n\n07:00 AM PST\n\nGet personalized insights on how to achieve your Target Quant Score.\n\u2022 ### GMAT Club Tests are Free & Open for Martin Luther King Jr.'s Birthday!\n\nJanuary 21, 2019\n\nJanuary 21, 2019\n\n10:00 PM PST\n\n11:00 PM PST\n\nMark your calendars - All GMAT Club Tests are free and open January 21st for celebrate Martin Luther King Jr.'s Birthday.\n\n# Why the various Generals of the Army of the Potomac before\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMagoosh GMAT Instructor\nJoined: 28 Dec 2011\nPosts: 4485\nWhy the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n03 Dec 2012, 11:20\n1\n6\n00:00\n\nDifficulty:\n\n25% (medium)\n\nQuestion Stats:\n\n66% (00:58) correct 34% (01:07) wrong based on 876 sessions\n\n### HideShow timer Statistics\n\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than\n(B) are debated in no less than\n(C) is debated about in no fewer than\n(D) is debated in no fewer than\n(E) is debated in no less than\n\nSee a full discussion of the grammar involved at:\nhttp:\/\/magoosh.com\/gmat\/2012\/substantiv ... -the-gmat\/\n\n_________________\n\nMike McGarry\nMagoosh Test Prep\n\nEducation is not the filling of a pail, but the lighting of a fire. \u2014 William Butler Yeats (1865 \u2013 1939)\n\n##### Most Helpful Expert Reply\nMagoosh GMAT Instructor\nJoined: 28 Dec 2011\nPosts: 4485\nWhy the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n07 Jan 2013, 17:34\n5\n7\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than\n(B) are debated in no less than\n(C) is debated about in no fewer than\n(D) is debated in no fewer than\n(E) is debated in no less than\n\nFor a complete discussion of substantive clauses (a.k.a nominal clauses, a.k.a noun clauses) ---including the very tricky issue of noun-clauses and subject-verb agreement --- as well as a full explanation of this particular question, see this blog:\nhttp:\/\/magoosh.com\/gmat\/2012\/substantiv ... -the-gmat\/\n\nMike\n_________________\n\nMike McGarry\nMagoosh Test Prep\n\nEducation is not the filling of a pail, but the lighting of a fire. \u2014 William Butler Yeats (1865 \u2013 1939)\n\n##### Most Helpful Community Reply\nManager\nJoined: 16 Aug 2013\nPosts: 50\nConcentration: Finance, Real Estate\nGPA: 3.73\nWE: Analyst (Consulting)\nWhy the various Generals of the Army\u00a0 [#permalink]\n\n### Show Tags\n\n17 May 2015, 08:05\n2\n4\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n\na)are debated about in no less than\nb)are debated in no less than\nc)is debated about in no fewer than\nd)is debated in no fewer than\ne)is debated in no less than\n##### General Discussion\nBoard of Directors\nJoined: 01 Sep 2010\nPosts: 3280\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\nUpdated on: 05 Dec 2012, 08:57\nClearly fewer refers to journals that are countable\n\nC is ackward and not fluent\n\nD it is\n\nCiao Mike\n_________________\n\nOriginally posted by carcass on 03 Dec 2012, 11:59.\nLast edited by carcass on 05 Dec 2012, 08:57, edited 1 time in total.\nManager\nJoined: 05 May 2011\nPosts: 65\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\n03 Dec 2012, 13:25\nmikemcgarry wrote:\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than\n(B) are debated in no less than\n(C) is debated about in no fewer than\n(D) is debated in no fewer than\n(E) is debated in no less than\n\nSee a full discussion of the grammar involved at:\nhttp:\/\/magoosh.com\/gmat\/2012\/substantiv ... -the-gmat\/\n\nWhy X were so singularly successful against Y IS blah blah. Therefore \"are\" is incorrect and hence A & B are out.\n\nC - \"Is debated about\" is unidiomatic, wordy and just plain awkward\n\nE - \"is debated in no less than\" is incorrect since we are using numbers (five hundred) so \"fewer\" is preferred.\n\nD is the only remaining choice that fixes all the errors.\nManager\nJoined: 20 Aug 2012\nPosts: 51\nSchools: Jones '15\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\n03 Dec 2012, 13:27\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than should be singular is instead of are\n(B) are debated in no less than should be singular is instead of are\n(C) is debated about in no fewer than debated about is unidiomatic. should be debated\n(D) is debated in no fewer than Correct!!!\n(E) is debated in no less thanless should not be used with countable quantity\n\nIMO, answer is D.\nManager\nJoined: 20 Apr 2010\nPosts: 206\nLocation: Hyderabad\nWE 1: 4.6 years Exp IT prof\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\n04 Dec 2012, 07:53\nSubject Verb error :\nis\/are -- \"is\" is correct as the subject is Singular. This cancels option A and B\nless\/ fewer -- \"fewer\" is to be used in case of countable quantity. This cancels option E\nWe are now left with C and D.\n\nDebated about could be unidiomatic or awkward construction. Hence the incorrect choice.\nThe choice D conveys the correct meaning.\nHence D is correct\n_________________\n\nI will give a Fight till the End\n\n\"To dream anything that you want to dream, that is the beauty of the human mind. To do anything that you want to do, that is the strength of the human will. To trust yourself, to test your limits, that is the courage to succeed.\"\n- Bernard Edmonds\n\nA person who is afraid of Failure can never succeed -- Amneet Padda\n\nDon't Forget to give the KUDOS\n\nIntern\nJoined: 18 Dec 2012\nPosts: 21\nLocation: Singapore\nConcentration: Strategy, Entrepreneurship\nGMAT 1: 620 Q48 V27\nWE: Supply Chain Management (Energy and Utilities)\nRe: Why the various Generals of the Army before Grant\u00a0 [#permalink]\n\n### Show Tags\n\n07 Jan 2013, 19:41\n1\nI go with (C) , though I still wonder the need for \"about\"\nMagoosh GMAT Instructor\nJoined: 28 Dec 2011\nPosts: 4485\nRe: Why the various Generals of the Army before Grant\u00a0 [#permalink]\n\n### Show Tags\n\n07 Jan 2013, 22:15\nvickymancer07 wrote:\nI go with (C) , though I still wonder the need for \"about\"\n\nWhat is your logic for picking (C) over (D)?\n_________________\n\nMike McGarry\nMagoosh Test Prep\n\nEducation is not the filling of a pail, but the lighting of a fire. \u2014 William Butler Yeats (1865 \u2013 1939)\n\nIntern\nJoined: 18 Dec 2012\nPosts: 21\nLocation: Singapore\nConcentration: Strategy, Entrepreneurship\nGMAT 1: 620 Q48 V27\nWE: Supply Chain Management (Energy and Utilities)\nRe: Why the various Generals of the Army before Grant\u00a0 [#permalink]\n\n### Show Tags\n\n07 Jan 2013, 23:11\nHi Mike,\n\nWith my initial round of POE , it was either (C) or (D) but I was not sure whether to go for \"Debate\"or \"Debate about\".\n\nFrom your blog I just read that \"debate about\"is awkward and not idiomatic , so then its (D)\n\nCheers!\nBoard of Directors\nJoined: 01 Sep 2010\nPosts: 3280\nRe: Why the various Generals of the Army before Grant\u00a0 [#permalink]\n\n### Show Tags\n\n08 Jan 2013, 06:33\n1\nI like this sentence with multiple modifiers.\n\nBasically the key words is generals so plural and this can lead us to think \"are\" but is misleading.\n\nhre we are talking about a question so the right verb is $$\"is\"$$\n\nThen we have to take in account generals so fewer (countable) and C is ackward\n\nD is the best\n\n_________________\nManager\nJoined: 31 May 2012\nPosts: 112\nRe: Why the various Generals of the Army before Grant\u00a0 [#permalink]\n\n### Show Tags\n\n08 Jan 2013, 20:08\n1\nmikemcgarry wrote:\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than\n(B) are debated in no less than\n(C) is debated about in no fewer than\n(D) is debated in no fewer than\n(E) is debated in no less than\n\n\"Why the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee\" is a single fact and should proceed with singular verb is. Eliminating A,B\n\nAs five hundred is countable number, So, we can't use less than. Eliminate E.\n\nBetween C & D, Usage debated in is correct as we are only referring the time. So, Option D looks correct.\nSenior Manager\nJoined: 21 Jan 2010\nPosts: 253\nRe: Why the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n27 Aug 2013, 09:34\n1\nmikemcgarry wrote:\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n(A) are debated about in no less than\n(B) are debated in no less than\n(C) is debated about in no fewer than\n(D) is debated in no fewer than\n(E) is debated in no less than\n\nFor a complete discussion of substantive clauses (a.k.a nominal clauses, a.k.a noun clauses) ---including the very tricky issue of noun-clauses and subject-verb agreement --- as well as a full explanation of this particular question, see this blog:\nhttp:\/\/magoosh.com\/gmat\/2012\/substantiv ... -the-gmat\/\n\nMike\n\nsubject is singular + historical journals can be counted. In C about and in both are prepositions. That's incorrect, we want a noun phrase after a preposition. This eliminates C. D wins.\nSenior Manager\nJoined: 12 Mar 2010\nPosts: 277\nConcentration: Marketing, Entrepreneurship\nGMAT 1: 680 Q49 V34\nRe: Why the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n27 Aug 2013, 21:05\nPicked C over D. I knew \"debate about\" is the idiom.\n\nWith this I got a silly question - Do I say \"I knew \"debate about\" was\/is the idiom\". Because the truth of the idiom still holds in present.\nManager\nJoined: 25 Mar 2013\nPosts: 56\nWE: Project Management (Telecommunications)\nRe: Why the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n27 Aug 2013, 21:54\n1\nChose C and it's wrong\n\neliminated A and B because of \"are\"\neliminated E because of \"less\"\n\nhad to choose between C and D, but did not know which one is right.\nIntern\nStatus: I'm trying to GMAT?\nJoined: 12 Feb 2013\nPosts: 23\nLocation: United States\nConcentration: Finance, General Management\nGMAT Date: 06-22-2013\nWE: Engineering (Consulting)\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\n28 Nov 2013, 08:53\naim730 wrote:\nSubject Verb error :\nis\/are -- \"is\" is correct as the subject is Singular. This cancels option A and B\nless\/ fewer -- \"fewer\" is to be used in case of countable quantity. This cancels option E\nWe are now left with C and D.\n\nDebated about could be unidiomatic or awkward construction. Hence the incorrect choice.\nThe choice D conveys the correct meaning.\nHence D is correct\n\nI had the right answer by eliminating the less\/ fewer option, but what is the subject of this sentence? Can't spot it.\nMagoosh GMAT Instructor\nJoined: 28 Dec 2011\nPosts: 4485\nRe: Why the various Generals of the Army of the Potomac\u00a0 [#permalink]\n\n### Show Tags\n\n29 Nov 2013, 11:31\nsadovskiya wrote:\nI had the right answer by eliminating the less\/ fewer option, but what is the subject of this sentence? Can't spot it.\n\nDear sadovskiya,\nI'm happy to respond.\n\nThe subject is hard to spot because the main subject of the sentence is not an individual noun, but an entire clause. This is called a \"substantive clause\" --- sometimes also called a \"noun clause\" or a \"relative nominal clause\". This is a kind of clause that can act in the role of a noun and hence be the subject of the sentence.\n\nSee this blog for a full explanation:\nhttp:\/\/magoosh.com\/gmat\/2012\/substantiv ... -the-gmat\/\n\nMike\n_________________\n\nMike McGarry\nMagoosh Test Prep\n\nEducation is not the filling of a pail, but the lighting of a fire. \u2014 William Butler Yeats (1865 \u2013 1939)\n\nManager\nJoined: 28 Nov 2013\nPosts: 73\nConcentration: General Management\nGMAT 1: 760 Q49 V46\nRe: Why the various Generals of the Army of the Potomac before\u00a0 [#permalink]\n\n### Show Tags\n\n29 Nov 2013, 21:09\n'Is' brings me down to C, D, and E. 'Fewer' brings me down to C and D. Never heard of 'debated about', so D it is!\n_________________\nSenior Manager\nStatus: Verbal Forum Moderator\nJoined: 17 Apr 2013\nPosts: 480\nLocation: India\nGMAT 1: 710 Q50 V36\nGMAT 2: 750 Q51 V41\nGMAT 3: 790 Q51 V49\nGPA: 3.3\nWhy the various Generals of the Army of the Potomac before Ulysses S.\u00a0 [#permalink]\n\n### Show Tags\n\n15 Sep 2014, 15:40\n2\nWhy the various Generals of the Army of the Potomac before Ulysses S. Grant were so singularly unsuccessful against Robert E Lee are debated about in no less than five hundred historically oriented journals.\n\nare debated about in no less than\nare debated in no less than\nis debated about in no fewer than\nis debated in no fewer than\nis debated in no less than\n_________________\n\nLike my post Send me a Kudos It is a Good manner.\nMy Debrief: http:\/\/gmatclub.com\/forum\/how-to-score-750-and-750-i-moved-from-710-to-189016.html\n\nSenior Manager\nStatus: Verbal Forum Moderator\nJoined: 17 Apr 2013\nPosts: 480\nLocation: India\nGMAT 1: 710 Q50 V36\nGMAT 2: 750 Q51 V41\nGMAT 3: 790 Q51 V49\nGPA: 3.3\nRe: Why the various Generals of the Army of the Potomac before Ulysses S.\u00a0 [#permalink]\n\n### Show Tags\n\n15 Sep 2014, 15:41\nNoun Clause is The Subject here!\n_________________\n\nLike my post Send me a Kudos It is a Good manner.\nMy Debrief: http:\/\/gmatclub.com\/forum\/how-to-score-750-and-750-i-moved-from-710-to-189016.html\n\nRe: Why the various Generals of the Army of the Potomac before Ulysses S. &nbs [#permalink] 15 Sep 2014, 15:41\n\nGo to page \u00a0 \u00a01\u00a0\u00a0\u00a02\u00a0 \u00a0 Next \u00a0[ 33 posts ]\n\nDisplay posts from previous: Sort by\n\n# Why the various Generals of the Army of the Potomac before\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\n\n Powered by phpBB \u00a9 phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2019-01-20 15:37:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7402277588844299, \"perplexity\": 14596.754343122766}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583722261.60\/warc\/CC-MAIN-20190120143527-20190120165527-00196.warc.gz\"}"}
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Q: Correctly accounting for multiple backslashes when tokenizing custom mini-format I am writing a small tokenizer in Python for a custom mini-format which looks like this (it can be nested too):
<tag:some_text>
tag is a combination of a finite set of values and some_text is just text. The delimiters <, : and > can be escaped by a single \ if they appear in, and as, text.
I used the regex r"((\\)?[<:>])" along with re.finditer to find delimiters and then remove the backslash if necessary by checking with token.startswith('\\'). The problem is that if more backslashes come before the delimiter, the regex is wrong, e.g. "<tag:Some \\\\< text>" -> ['<', 'tag', ':', 'Some \\\\', '<', ' text', '>'].
I cannot find a sensible solution using regexes, and I am considering just writing the tokenization in pure Python, i.e. no regex magic etc (but that may be slow?) or am I overcomplicating this? Any suggestions?
A: Your regular expression will only match the last backslash and delimiter \< in \\\\<.
Just add the + quantifier which means (1 or more times)
((\\+)?[<:>])
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The days of the steamer trunk are over! And remember these? We've all had them!
Suitcases are now much lighter and made of sturdier fabrics made for wear and tear! There are some great new packing ideas, especially for the Cruisers! Let's see the new bags and how they can be used for the cruising crowd! First, do you want to see my carry-on bag? Here it is!!! Oh my, what do we have here? It is very small! Welcome to the FOLD!
Folding Shopping Bag by Biaggi: Will be My Carry-On Bag!
And a sleeve to hook the bag on your larger baggage handle too………….
The bag is very wide and deep to carry lots of STUFF!
Now what else can we fold? Why our luggage of course! Here is the back of the 31″ Foldable Spinner Zipsak, unpacked. Notice all the outer pockets? It's also on very nice spinner wheels!
Here is the side of the empty Zipsak so you can see the side handle too. Good for lifting if you need to!
Now watch the video to see how convenient this bag is, especially for the cruising set. Unpack that suitcase and Fold it UP and get it out of your way!
Now for the ZipCubes to put in your Zipsak……….. The 31″ Zipsak will hold 7 of the 31″ ZipCubes and more!
And here are six pair of Tieks shoes packed in 1/3 of a ZipCube!!!! Awesome! And I have six more ZibCubes to pack!
Or, I could pack my Tieks in the Shoebag that comes with every three Zipcubes you buy! More that 9 pair of Tieks will fit in that shoebag!!!! Who needs nine pair of shoes on a cruise? What else can I put in that bag? Hmmmm…. my must haves on vaca….. plastic bags in assorted sizes, you never know when you'll need a plastic ziploc bag. Good for treats in your purse when you are off on an excursion…….. or wet clothes that did not dry in time……..and souvenirs won't be all jumbled together. Oh, and I take small pieces of bubble wrap too, in case I buy something fragile that needs to be protected. Yep, all that could go in a shoebag!
Here is my Foldable Spinner Zipsak suitcase and the ZipCubes ready to be filled. The suitcase opens from the side for easy packing or from the top to get something out you have packed on top!
My Biaggi ZipSak with 7 ZipCubes to Fill!
And when I get to the cruise ship I will take out the ZipCubes and place them in the drawers and fold up my Zipsak and Carry-on bag and store them……..wherever, out of my way!
I think the Biaggi Bags are awesome! When I get home my Biaggi bags are tucked away in the closet, not taking up tons of room!
Now, about the cute Smart Recovery Tag as seen on my Zipsak. It is embedded with information that I put in it and have access to! Only me! However, I can track that bag wherever it goes. So, if I landed in Spain and my luggage is still at Heathrow, or wherever, I will know it! The tag can be scanned and read, with info going to the Smart Tag Recovery site which I have access to. The person who finds the bag does not have direct access to me. They can message me through the site and let me know what they plan to do with my luggage! Ha! Now if a thief gets my luggage he probably won't scan the tag, but I'll know where he cut the tag off! Get out those strong wire clippers though! Just sayin'! All of the bags and cubes can be found on the Biaggi website, HERE. They also come in many colors, not just the gray, that I have selected! The Smart Recovery Tags can be found on Amazon! I am getting ready to go! I hope you have enjoyed my packing tips!
I am going on a trip to Canada in September, and I will save this post for future reference. Thanks.
I think you will like these bags! I do!
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{"url":"https:\/\/math.stackexchange.com\/questions\/2885348\/tight-upper-tail-bound-for-normal-distribution","text":"# Tight upper tail bound for Normal distribution\n\nThe following is a well-known chain of inequalities for the tail of the normal distribution when $a = 1:$ $$\\Big(\\frac{1}{x} - \\frac{a}{x^3}\\Big) \\phi(x) \\leq \\Big(\\frac{x}{a + x^2}\\Big) \\phi(x) \\leq \\Phi(-x) \\leq \\frac{1}{x}\\phi(x), \\qquad x > 0.$$ where $\\phi(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp(-x^2\/2)$ and $\\Phi(-x) = \\int_x^\\infty \\phi(t) \\,\\mathrm{d}t$ are the normal pdf and cdf respectively. Moreover, observe that setting $a = 0$ recovers the upper bound for the normal cdf on the right-hand side of the above chain. My question, therefore, is whether a tighter upper bound for the normal tail exists for some $a \\in (0,1)$ when we restrict to $x \\geq 1$. Specifically, this yields the following two questions\n\nProve or disprove: There exists $a \\in (0, 1)$ such that for all $x \\geq 1$ $$\\Phi(-x) \\leq \\Big(\\frac{1}{x} - \\frac{a}{x^3}\\Big) \\phi(x)$$\n\nProve or disprove: There exists $a \\in (0,1)$ such that for all $x \\geq 1$, $$\\Phi(-x) \\leq \\Big(\\frac{x}{a + x^2}\\Big) \\phi(x)$$\n\nSo far, I've exhausted most of my \"painless\" tricks, and thought I'd post a fun problem that looks to me like it might have a positive answer.\n\n\u2022 It might help to look at the proofs of the inequalities in the case $a=1$ and inspect the looseness of each inequality that appears in the proof. (Although I have not done this myself.) \u2013\u00a0angryavian Aug 17 '18 at 3:56\n\nLet $$G(x) = \\left(\\frac {1} {x} - \\frac {a} {x^3}\\right)\\phi(x) - \\Phi(-x)$$ for $a \\in (0, 1)$. Then \\begin{align} G'(x) &= \\left(-\\frac {1} {x^2} + \\frac {3a} {x^4}\\right)\\phi(x) + \\left(\\frac {1} {x} - \\frac {a} {x^3}\\right)(-x)\\phi(x) + \\phi(-x) \\\\ &= \\left(-\\frac {1} {x^2} + \\frac {3a} {x^4} - 1 + \\frac {a} {x^2} + 1\\right)\\phi(x) \\\\ &= \\frac {\\phi(x)} {x^4}\\left[-(1-a)x^2 + 3a\\right] \\\\ &\\begin{cases} <0 & \\text{when} & \\displaystyle x < -\\sqrt{\\frac {3a} {1-a}} \\\\ =0 & \\text{when} & \\displaystyle x = -\\sqrt{\\frac {3a} {1-a}} \\\\ >0 & \\text{when} & \\displaystyle -\\sqrt{\\frac {3a} {1-a}} < x < \\sqrt{\\frac {3a} {1-a}}\\\\ =0 & \\text{when} & \\displaystyle x = \\sqrt{\\frac {3a} {1-a}} \\\\ <0 & \\text{when} & \\displaystyle x > \\sqrt{\\frac {3a} {1-a}} \\\\ \\end{cases} \\end{align}\n\nSince we are interested in $x > 1$ only, we can safely ignore the first two negative cases. Note that\n\n$$\\sqrt{\\frac {3a} {1-a}} > 1 \\iff a > \\frac {1} {4}$$\n\nSo we can conclude that when $\\displaystyle 0 < a \\leq \\frac {1} {4}$, $G$ is strictly decreasing on $[1, +\\infty)$. And it is trivial to check that $\\displaystyle \\lim_{x \\to +\\infty} G(x)= 0$ which implies $G(x) \\geq 0$ for all $x \\in (1, +\\infty)$. And this proves the first claim - it holds when $a$ is small enough.\n\nWe can also check the case when $\\displaystyle \\frac {1} {4} < a < 1$, $G$ first strictly increasing on $\\displaystyle \\left[1, \\sqrt{\\frac {3a} {1-a}}\\right)$, attains the maximum at $\\displaystyle x = \\sqrt{\\frac {3a} {1-a}}$, and then strictly decreasing on $\\displaystyle \\left(\\sqrt{\\frac {3a} {1-a}}, +\\infty\\right)$. So we also need to check the value of $G(1)$ as the boundary point. Note that\n\n$$G(1) = (1 - a)\\phi(1) - \\Phi(-1) \\geq 0 \\iff a \\leq 1 - \\frac {\\Phi(-1)} {\\phi(1)} \\approx 0.3443205$$\n\nSo the inequality also holds when $\\displaystyle \\frac {1} {4} < a \\leq 1 - \\frac {\\Phi(-1)} {\\phi(1)}$\n\nThe another claim can also be checked similarly I think.\n\nThis is also worth a read: https:\/\/mathoverflow.net\/questions\/19404\/approximation-of-a-normal-distribution-function\n\n\u2022 Phenomenal! I bungled a sign when I tried this myself the first time. I'm happy to see it works out. \u2013\u00a0bashfuloctopus Aug 18 '18 at 0:18","date":"2019-11-19 03:21:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9999136924743652, \"perplexity\": 286.35903023470115}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496669967.80\/warc\/CC-MAIN-20191119015704-20191119043704-00440.warc.gz\"}"}
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